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config/Typora/backups/1

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+---
+Author: Antoine Foucault-Castelli
+Date: 23/03/2026
+Update: 31/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 5.0
+---
+
+> [!BUG]
+> Some equations has been put in latex block code as they didnt show properly on gitlab
+
+# Gearbox simulation
+## Structure
+```
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+-- main
+main.m
+>> used to launch all run_*.m at once
+main_parameters.m
+>> charge model's parameters 
+
+-- runners
+run_main.m
+>> PURPOSE: launch every class_func functions in the right order to model the system
+run_display.m
+>> PURPOSE: launch every class_displayer functions
+run_print.m
+>> PURPOSE: print the figures made by run_display
+>> OUTPUT: .svg
+
+-- classes
+class_display.m
+>> displays the inputs and outputs of the model
+    [col1,col2,col3,col4,col5,col6,col7,col8,lw1,lw2] = charge_aff()
+    [figname,namex,namey,namet,ylabelx,ylabelt,ylabelfft,ylabelfftT] = type_choice(type)
+    listOfFig = disp_x(t,x,type,listOfFig)
+    listOfFig = excitations_ext(t,T_aero,k_var,k_mean,listOfFig)
+    listOfFig = spectral_analysis(x,tf,dt,maxf,type,listOfFig)
+    % TODO   
+    plot_fep(t,Fep,i)
+    listOfFig = plotThetaDiff(t,x,listOfFig)
+
+class_func.m
+>> model functions (explicit cf. comments in code)
+    [t, dt, tf, t_mesh, f_mesh, fs] = 
+        time_sampling(fs_div,n_periods,Z1,N1)
+    [Jin, J1, J2, Jout, J_all] =
+        inertia(m_all,r_blade,di1,di2)
+    [c12, db1, db2] = 
+        gearbox_geometry(alpha0, Z1, Z2, m0, a)
+    [k_var, k_mean] = 
+        mesh_stiffness(epsilon, t, t_mesh, k_mesh_mean)
+    [T_aero,T_out] = 
+        wind_excitation(t,rho_air,r_blade,v_wind,T_fluct_in,f_ext,omega_in,cp,ratio)
+    [M] = 
+        mass_matrix(m_all, J_all)
+    [K_var, K_const, K_mean] = 
+        stiffness_matrix(db1, db2, alpha0, kx, ky, ktheta, k_mean)
+    [K] = 
+        stiffness_update(K_const,K_var,k)
+    [C] = 
+        damping_matrix(M,K_mean,damp_eta,damp_beta)
+    [F] = 
+        forces_matrix(T_in,T_out)
+    [x,v,a] =
+        initialization(N1,t,ratio,cp)
+    [x,v,a] = 
+        newmark(M,C,K_const,K_var,K_mean,k_var,dt,t,T_in,T_out,x,v,a,omega_in,gamma,beta,F_ep)
+    [F_ep] = 
+        error_profile_excitation(e12,f_mesh,t,alpha0,db1,db2,k_var)
+    [damp_eta, damp_beta] = 
+        damping_coeff(M,K,xi1,xiN)
+
+-- standalone
+_convergence.m
+>> purpose: test the numerical convergence of the model
+>> output: 3 graphs + print/_convergence/*.svg
+_gear-mesh-var.m
+>> purpose: shows the variation of gear mesh with central distance changes
+>> output: 1 graph + print/_gear-mesh-var/*.svg
+_rms-analysis.m
+>> purpose: analysis of RMS, STD and kurtosis for many setups
+>> output: data/rms.svg
+```
+
+## How does it work ?
+**For Matlab under windows :** 
+classic matlab
+
+> [!WARNING]
+> NOT TESTED UNDER MATLAB
+> remove all occurences of `graphics_toolkit("gnuplot")` in the code which is only useful for Octave visualization
+
+**For Octave/Linux :**
+*May need a few package installation (follow octave recommandations after first(s) run*
+
+```baswind_turbine_gearbox/octave/h
+cd wind_turbine_gearbox/octave/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class objects)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [x] Errors/defects implementation
+    - [x] profile error
+    - [x] assembly defect (change on gear stiffness)
+- [x] README
+    - [x] Maths
+    - [~] Images
+    - [x] Source(s)
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+    - [x] Printing pdf and svg
+- [x] Convergence verification
+- [x] Display effect of defects
+- [ ] Verify convergence for other input
+- [ ] Preparation Monte Carlo Parallel calculation
+- [x] Decide a number of samples per mesh period
+- [ ] Random models implementation
+- [x] Implemented easy disp and print
+- [ ] Add a "peak finder" for FFT
+
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+| Newmark parameters                               | Value     | **Unit**        |
+| ------------------------------------------------ | --------- | --------------- |
+| $\gamma$                                         | 0.7       |                 |
+| $\beta$                                          | 0.3       |                 |
+| **Gears**                                        | **Value** |                 |
+| Pinon 1 tooth number $Z1$                        | 72        |                 |
+| Pinion 2 tooth number $Z2$                       | 18        |                 |
+| Pinion modulus $m_0$                             | 0.016     | m               |
+| Pinion width $b$                                 | 0.1       | m               |
+| Internal diameter pinion 1 $d_{i1}$              | 0.5       | m               |
+| Internal diameter pinion 2 $d_{i_2}$             | 0.3       | m               |
+| Rotor radius $r_{rot}$ | 0.5 | m |
+| Input theoretical speed $N_1$           | 17        | m/s             |
+| Contact angle $\alpha$                           | 20        | deg          |
+| Steel density $\rho_{steel}$                     | 7860      | kg/m3           |
+| **Masses**                                       | **Value** | **Unit**        |
+| Input mass $m_{in}$ (hub + blades)               | 54000     | kg              |
+| Output mass $m_{out}$ (rotor)                    | 10000     | kg              |
+| **Stiffness**                                    |           |                 |
+| Flexion x : $k_x=k_{x_1}=k_{x_2}$                | 1e8       | N/m             |
+| Flexion y : $k_y=k_{y_1}=k_{y_2}$                | 1e8       | N/m             |
+| Torsion z : $k_\theta=k_{\theta_1}=k_{\theta_2}$ | 1e6      | N/m             |
+| Mean mesh stiffness $k_{mesh_{mean}}$            | 2.8e8    | N/rad           |
+| **Damping**                                      | **Value** | **Unit**        |
+| Mass damping ratio                              | 0.05      |    |
+| Stiffness damping ratio                        | 0.01      | |
+| **Wind excitation**                              | **Value** | **Unit**        |
+| Blade radius $r_{blade}$                         | 6         | m               |
+| Air density $\rho_{air}$                         | 1.225     | kg/m3           |
+| Wind speed (mean) $v_{wind_{mean}}$              | 37.5        | m/s             |
+| Fluctuating external torque $T_{fluct}$          | 50        | N               |
+| Fluctuating external frequency $f_{fluct}$       | 6         | Hz              |
+| Performance efficiency $cp$                      | 16/27     |                 |
+| **Errors**                                       | **Value** | **Unit**        |
+| Profile error (approx)                                   | 1e-5     | m               |
+| Center distance error (approx)                           | 1e-2     | m               |
+
+
+
+### Energy equations
+
+$$
+        \left[\frac{\partial E_p}{\partial q_i}\right] 
+        + \left[\frac{d}{dt}\left(\frac{\partial E_k}{\partial\dot{q}_i}\right)\right] 
+        + \frac{\partial C}{\partial\dot{q}_i} 
+        = \frac{\partial W}{\partial q_i}
+        \label{eq:reformlagrange}
+\\
+        E_k = 	\frac{1}{2}
+                \left(
+                        m_1{\dot{x}_1}^2+
+                        m_1{\dot{y}_1}^2+
+                        I_{11}{\dot{\theta}_{11}}^2+
+                        I_{12}{\dot{\theta}_{12}}^2+
+                        m_2{\dot{x}_2}^2+
+                        m_2{\dot{y}_2}^2+
+                        I_{22}{\dot{\theta}_{22}}^2+
+                        I_{21}{\dot{\theta}_{21}}^2
+                \right)
+\\
+        E_p = \frac{1}{2}
+                \left( 
+                        k_{x_1}{x_{1}}^2+
+                        k_{y_1}{y_{1}}^2+
+                        k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+                        k_{x_2}{x_{2}}^2+
+                        k_{y_2}{y_{2}}^2+
+                        k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+                        K_e(t){\delta}^2
+                \right)
+$$
+
+### Equations of movement
+
+The equations of movement are given by putting $E_k$ and $E_p$ into the Lagrangian formula. 
+
+$$
+\begin{cases}
+\frac{\partial E_p}{\partial x_1} + \frac{d}{dt}\left( \frac{\partial E_k}{\partial \dot{x}_1} \right) + \frac{\partial C}{\partial \dot{x}_1}
+                = m_1\ddot{x}_1
+                + k_{x_1}x_1 
+                + \sin(\alpha)K_e(t)\delta 
+                + C_{x_1}\dot{x}_1
+                = 0 \\
+        \frac{\partial E_p}{\partial y_1} + \frac{d}{dt}\left( \frac{\partial E_k}{\partial \dot{y}_1} \right) + \frac{\partial C}{\partial \dot{y}_1}
+                = m_1\ddot{y}_1
+                + k_{y_1}y_1 
+                + \cos(\alpha)K_e(t)\delta 
+                + C_{y_1}\dot{y}_1
+                = 0 \\
+                \frac{\partial E_p}{\partial \theta_{11}} + \frac{d}{dt}\left( \frac{\partial E_k}{\partial \dot{\theta}_{11}} \right) + \frac{\partial C}{\partial \dot{\theta}_{11}}
+                = I_{11}\ddot{\theta}_{11}
+                + k_{\theta_{11}}(\theta_{11}-\theta_{12}) 
+                + C_{\theta_{11}}\dot{\theta}_{11}
+                = T_{aero} \\
+                \frac{\partial E_p}{\partial \theta_{12}} + \frac{d}{dt}\left( \frac{\partial E_k}{\partial \dot{\theta}_{12}} \right) + \frac{\partial C}{\partial \dot{\theta}_{12}}
+                = I_{12}\ddot{\theta}_{12}
+                - k_{\theta_{11}}(\theta_{11}-\theta_{12}) 
+                + {r_b}_1K_e(t)\delta 
+                + C_{\theta_{12}}\dot{\theta}_{12}
+                = 0 \\
+
+        \frac{\partial E_p}{\partial x_2} + \frac{d}{dt}\left( \frac{\partial E_k}{\partial \dot{x_2}} \right) + \frac{\partial C}{\partial \dot{x_2}}
+                = m_2\ddot{x}_2
+                + k_{x_2}x_2 
+                - \sin(\alpha)K_e(t)\delta 
+                + C_{x_2}\dot{x}_2
+                = 0 \\
+        \frac{\partial E_p}{\partial y_2} + \frac{d}{dt}\left( \frac{\partial E_k}{\partial \dot{y_2}} \right) + \frac{\partial C}{\partial \dot{y_2}}
+                = m_2\ddot{y}_2
+                + k_{y_2}y_2 
+                - \cos(\alpha)K_e(t)\delta 
+                + C_{y_2}\dot{y}_2
+                = 0 \\
+        \frac{\partial E_p}{\partial \theta_{22}} + \frac{d}{dt}\left( \frac{\partial E_k}{\partial \dot{\theta}_{22}} \right) + \frac{\partial C}{\partial \dot{\theta}_{22}}
+                = I_{22}\ddot{\theta}_{22}
+                - k_{\theta_{22}}(\theta_{21}-\theta_{22})
+                + C_{x_1}\dot{\theta}_{22}
+                = T_{gen} \\
+        \frac{\partial E_p}{\partial \theta_{21}} + \frac{d}{dt}\left( \frac{\partial E_k}{\partial \dot{\theta}_{21}} \right) + \frac{\partial C}{\partial \dot{\theta}_{21}}
+                = I_{21}\ddot{\theta}_{21}
+                + k_{\theta_{21}}(\theta_{21}-\theta_{22}) 
+                - {r_b}_2K_e(t)\delta 
+                + C_{\theta_{21}}\dot{\theta}_{21}
+                = 0
+\end{cases}
+$$
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}-\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+
+$$
+\left[M\right]\{\ddot{q}\} + \left[C\right] \{\dot{q}\}+ \left(\left[K_{var}\right]+\left[K_{cst}\right]\right) \{q\} = \left\{F\right\}
+$$
+
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+
+$$
+K_{var} = K_e(t)
+        \begin{bmatrix}
+                s^2     &       sc      &       0       &       sr_1    &       -s^2    &       -sc     &       0       &       -sr_2   \\
+                cs      &       c^2     &       0       &       cr_1    &       -cs     &       -c^2    &       0       &       -cr_2   \\
+                0       &       0       &       0       &       0       &       0       &       0       &       0       &       0       \\
+                r_1s    &       r_1c    &       0       &       r_1^2   &       -r_1s   &       -r_1c   &       0       &       -r_1r_2 \\
+                -s^2    &       -sc     &       0       &       -sr_1   &       s^2     &       sc      &       0       &       sr_2    \\
+                -cs     &       -c^2    &       0       &       -cr_1   &       cs      &       c^2     &       0       &       cr_2    \\
+                0       &       0       &       0       &       0       &       0       &       0       &       0       &       0       \\
+                -r_2s   &       -r_2c   &       0       &       -r_2r_1 &       r_2s    &       r_2c    &       0       &       r_2r_1
+        \end{bmatrix}
+$$
+
+With
+
+-  $s=sin(\alpha)$
+- $c=cos(\alpha) $
+- $r1=r_{b_1} $
+- $r2=r_{b_2}$
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+
+Damping matrix is : $C = \eta M+\beta K_{mean}$
+
+**Forces**
+
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{aero}	\\	0	\\	0\\	0\\ T_{gen}\\	0
+\end{bmatrix}
+$$
+
+**Mesh stiffness model**
+
+The mesh stiffness is modelized by a trapezoïdal signal between a low value ($=k_{mesh_{mean}}$ for one tooth in contact) and its double value as a high value. Length of both phase is given by $T_{high}=(\varepsilon-1)T_{mesh}$ and $T_{low}=(2-\varepsilon)T_{mesh}$. 10% of the start and of the end of the “high” phase is a linear slope from low value to high value.
+
+> [!NOTE]
+> $T_{mesh}=1/f_{mesh}=1/(Z1*N1/60)$
+> $\varepsilon$ is an approximation using the following formula :
+
+$$
+c_1=\frac{1+\frac{1}{Z1}}{\left(\pi cos(\alpha)\left(\frac{sin(\alpha)}{2}+\sqrt{\frac{sin(\alpha)^2}{4}+\frac{1}{Z1^2}+\frac{1}{Z1}}\right)\right)} \\
+c_1=\frac{1+\frac{1}{Z2}}{\left(\pi cos(\alpha)\left(\frac{sin(\alpha)}{2}+\sqrt{\frac{sin(\alpha)^2}{4}+\frac{1}{Z2^2}+\frac{1}{Z2}}\right)\right)} \\
+c_{12}=c_1+c_2 \approx \varepsilon
+$$
+
+
+## Numerical model, wind model and errors
+
+**Time :** time is sampled on the $T_{mesh}$, with $fs$ the number of points for each period, and $n_{periods}$ the number of periods calculated. The number of points is $fs\times n_{periods}$.
+
+**Speed :** rotational speed is initialized at its “stable” value.
+
+> if no automatic implementation, it has to be done “by hand” by executing the program once, get the speeds after the transient zone, and put them by hand in “initialization” source code (`func.m`)
+
+**Wind :** cosine variation around mean torque value
+$$
+ T_{in} = \rho_{air}\pi(r_{blade}^2)(v_{wind}^3)*Cp)/(2\omega) \\
+ T_{aero}(t) = T_{in} + T_{fluct}\cos(2\pi f_{ext}t)
+$$
+
+
+**Initialization :** everything is set at 0 for initialization (except speed)
+
+**Model :** a basic Newmark scheme is used to compute the model over time
+
+**Error implementation :** 
+
+- error profile is implemented as ($+\infty$ is approximated as $1000$) :
+
+$$
+ep(t) = e_{12}+\sum ^{+\infty}_{n=1}e_{12}sin(2n\pi f_m t) \\
+\left\{F_{ep}\right\} = \frac{\partial \delta (t)}{\partial q_i}k(t)ep(t) \\
+\frac{\partial \delta(t)}{\delta q_i} =
+\begin{bmatrix}
+sin(\alpha) \\ cos(\alpha) \\ 0 \\ rb_1 \\ -sin(\alpha) \\ -cos(\alpha) \\ 0 \\ rb_2
+\end{bmatrix}
+$$
+
+- center distance error is implemented as a transformation of $\alpha$ depending on a variation of $a$, in `gearbox_geometry`. It does only change the values of contact ratio $\varepsilon$ and then the profile of the mesh stiffness (length of phases and max/min values).
+  $$
+  \alpha'=acos\left(\frac{rb_1+rb_2}{E+a}\right)
+  $$
+  
+  
+  
+
+## TO FIX
+
+> [!CAUTION]
+> Speed not right
+
+## Machine specifications
+For the computational speed, here’s my machine’s spec. :
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.10-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

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+  "searchService": [
+    ["Search with Google", "https://google.com/search?q=%s"]
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+  // Custom key binding, which will override the default ones.
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+    // All other options are the menu items 'text label' displayed from each typora menu
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+  "monocolorEmoji": false, //default false. Only work for Windows
+  "maxFetchCountOnFileList": 500,
+  "flags": [] // default [], append Chrome launch flags, e.g: [["disable-gpu"], ["host-rules", "MAP * 127.0.0.1"]]
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config/Typora/draftsRecover/2026-3-19 .typora_readme 122640.md

@@ -0,0 +1,309 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [ ] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Images
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Auto speed initialization
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+| Newmark parameters                               | Value     | **Unit**        |
+| ------------------------------------------------ | --------- | --------------- |
+| $\gamma$                                         | 0.7       |                 |
+| $\beta$                                          | 0.3       |                 |
+| **Gears**                                        | **Value** |                 |
+| Pinon 1 tooth number $Z1$                        | 72        |                 |
+| Pinion 2 tooth number $Z2$                       | 18        |                 |
+| Pinion modulus $m_0$                             | 0.016     | m               |
+| Pinion width $b$                                 | 0.1       | m               |
+| Internal diameter pinion 1 $d_{i1}$              | 0.5       | m               |
+| Internal diameter pinion 2 $d_{i_2}$             | 0.3       | m               |
+| Input speed $N_1$                                | 17        | m/s             |
+| Contact angle $\alpha$                           | 20        | degree          |
+| Steel density $\rho_{steel}$                     | 7860      | kg/m3           |
+| **Masses**                                       | **Value** | **Unit**        |
+| Input mass $m_{in}$                              | 54000     | kg              |
+| Output mass $m_{out}$                            | 10        | kg              |
+| **Stiffness**                                    |           |                 |
+| Flexion x : $k_x=k_{x_1}=k_{x_2}$                | 1e8       | N/m             |
+| Flexion y : $k_y=k_{y_1}=k_{y_2}$                | 1e8       | N/m             |
+| Torsion z : $k_\theta=k_{\theta_1}=k_{\theta_2}$ | 1e5       | N/m             |
+| Mean mesh stiffness $k_{mesh_{mean}}$            | 1e8       | N/rad           |
+| **Damping**                                      | **Value** | **Unit**        |
+| Mass related $\eta$                              | 5e-2      | N/(m/s)/(kg)    |
+| Stiffness related $\beta$                        | 1e-6      | N/(m/s)/(N/rad) |
+| **Wind excitation**                              | **Value** | **Unit**        |
+| Blade radius $r_{blade}$                         | 6         | m               |
+| Air density $\rho_{air}$                         | 1.225     | kg/m3           |
+| Wind speed (mean) $v_{wind_{mean}}$              | 10        | m/s             |
+| Fluctuating external torque $T_{fluct}$          | 50        | N               |
+| Fluctuating external frequency $f_{fluct}$       | 6         | Hz              |
+| Performance efficiency $cp$                      | 16/27     |                 |
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1\dot{x}_1^2+
+		\frac{1}{2}m_1\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2\dot{x_2}^2+
+		\frac{1}{2}m_2\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+
+**Mass matrix :**
+
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+
+$$
+\frac{K(t)}{K_e(t)}= 
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+
+Damping matrix is : $C = \eta M+\beta K_{mean}$
+
+**Forces**
+
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+**Mesh stiffness model**
+
+The mesh stiffness is modelized by a trapezoïdal signal between a low value ($=k_{mesh_{mean}}$ for one tooth in contact) and its double value as a high value. Length of both phase is given by $T_{high}=(\varepsilon-1)T_{mesh}$ and $T_{low}=(2-\varepsilon)T_{mesh}$. 10% of the start and of the end of the “high” phase is a linear slope from low value to high value.
+
+> [!NOTE]
+> $T_{mesh}=1/f_{mesh}=1/(Z1*N1/60)$
+> $\varepsilon$ is an approximation using the following formula :
+> $$
+> c_1=\frac{1+\frac{1}{Z1}}{\left(\pi cos(\alpha)\left(\frac{sin(\alpha)}{2}+\sqrt{\frac{sin(\alpha)^2}{4}+\frac{1}{Z1^2}+\frac{1}{Z1}}\right)\right)} \\
+> c_1=\frac{1+\frac{1}{Z2}}{\left(\pi cos(\alpha)\left(\frac{sin(\alpha)}{2}+\sqrt{\frac{sin(\alpha)^2}{4}+\frac{1}{Z2^2}+\frac{1}{Z2}}\right)\right)} \\
+> c_{12}=c_1+c_2 \approx \varepsilon
+> $$
+
+
+## Model
+
+**Time :** time is sampled on the $T_{mesh}$, with $fs$ the number of points for each period, and $n_{periods}$ the number of periods calculated. The number of points is $fs\times n_{periods}$.
+
+**Speed :** rotational speed is initialized at its “stable” value.
+
+> if no automatic implementation, it has to be done “by hand” by executing the program once, get the speeds after the transient zone, and put them by hand in “initialization” source code (`func.m`)
+
+**Initialization :** everything is set at 0 for initialization (except speed)
+
+**Model :** a basic Newmark scheme is used to compute the model over time
+
+## TO FIX
+
+> [!CAUTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 104658.md

@@ -0,0 +1,246 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!CAUTION]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [ ] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+
+
+### Energy
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F = \bma{0}
+$$
+
+
+
+
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 104724.md

@@ -0,0 +1,249 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!CAUTION]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [ ] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+
+
+### Energy
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+za
+\end{bmatrix}
+$$
+
+
+
+
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 104753.md

@@ -0,0 +1,249 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!CAUTION]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [ ] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+
+
+### Energy
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	&	0	&	T_{in}	&	0	&	0&	0&	0&	0
+\end{bmatrix}
+$$
+
+
+
+
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 104825.md

@@ -0,0 +1,249 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!CAUTION]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [ ] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+
+
+### Energy
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	&	0&	0&	T_{out}&	0
+\end{bmatrix}
+$$
+
+
+
+
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 104843.md

@@ -0,0 +1,249 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!CAUTION]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [ ] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+
+
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105011.md

@@ -0,0 +1,250 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!ATT]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [ ] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+
+
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105400.md

@@ -0,0 +1,246 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105540.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Parameter        | Value |
+| ---------------- | ----- |
+| Newmark $\gamma$ |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105607.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Parameter        | Value |
+| ---------------- | ----- |
+| Newmark $\gamma$ |       |
+| Newmark $\beta$  |       |
+| $Z1$             |       |
+| $Z2$             |       |
+| $m_0$            |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105638.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Parameter        | Value |
+| ---------------- | ----- |
+| Newmark $\gamma$ |       |
+| Newmark $\beta$  |       |
+| **Gear**         |       |
+| $Z1$             |       |
+| $Z2$             |       |
+| $m_0$            |       |
+| $b$              |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105713.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark          | Value |
+| ---------------- | ----- |
+| Newmark $\gamma$ |       |
+| Newmark $\beta$  |       |
+| **Gears**        |       |
+| $Z1$             |       |
+| $Z2$             |       |
+| $m_0$            |       |
+| $b$              |       |
+| $d_{i1}$         |       |
+| $d_{i_2}$        |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105746.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark          | Value |
+| ---------------- | ----- |
+| Newmark $\gamma$ |       |
+| Newmark $\beta$  |       |
+| **Gears**        |       |
+| $Z1$             |       |
+| $Z2$             |       |
+| $m_0$            |       |
+| $b$              |       |
+| $d_{i1}$         |       |
+| $d_{i_2}$        |       |
+| $N_1$            |       |
+| $\alpha$         |       |
+| $\rho_{steel}$   |       |
+| **Masses**       |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+|                  |       |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105811.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark          | Value     |
+| ---------------- | --------- |
+| Newmark $\gamma$ |           |
+| Newmark $\beta$  |           |
+| **Gears**        | **Value** |
+| $Z1$             |           |
+| $Z2$             |           |
+| $m_0$            |           |
+| $b$              |           |
+| $d_{i1}$         |           |
+| $d_{i_2}$        |           |
+| $N_1$            |           |
+| $\alpha$         |           |
+| $\rho_{steel}$   |           |
+| **Masses**       | **Value** |
+| $$               |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105824.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark          | Value     |
+| ---------------- | --------- |
+| Newmark $\gamma$ |           |
+| Newmark $\beta$  |           |
+| **Gears**        | **Value** |
+| $Z1$             |           |
+| $Z2$             |           |
+| $m_0$            |           |
+| $b$              |           |
+| $d_{i1}$         |           |
+| $d_{i_2}$        |           |
+| $N_1$            |           |
+| $\alpha$         |           |
+| $\rho_{steel}$   |           |
+| **Masses**       | **Value** |
+| $m_{input}$      |           |
+| $m_{pinion}$     |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105845.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark          | Value     |
+| ---------------- | --------- |
+| Newmark $\gamma$ |           |
+| Newmark $\beta$  |           |
+| **Gears**        | **Value** |
+| $Z1$             |           |
+| $Z2$             |           |
+| $m_0$            |           |
+| $b$              |           |
+| $d_{i1}$         |           |
+| $d_{i_2}$        |           |
+| $N_1$            |           |
+| $\alpha$         |           |
+| $\rho_{steel}$   |           |
+| **Masses**       | **Value** |
+| $m_{in}$         |           |
+| $m_{12}$         |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105912.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark          | Value     |
+| ---------------- | --------- |
+| Newmark $\gamma$ |           |
+| Newmark $\beta$  |           |
+| **Gears**        | **Value** |
+| $Z1$             |           |
+| $Z2$             |           |
+| $m_0$            |           |
+| $b$              |           |
+| $d_{i1}$         |           |
+| $d_{i_2}$        |           |
+| $N_1$            |           |
+| $\alpha$         |           |
+| $\rho_{steel}$   |           |
+| **Masses**       | **Value** |
+| $m_{in}$         |           |
+| $m_{pinion12}$   |           |
+| $m_{pinion21}$   |           |
+| $m_{pinio}$      |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 105943.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark          | Value     |
+| ---------------- | --------- |
+| Newmark $\gamma$ |           |
+| Newmark $\beta$  |           |
+| **Gears**        | **Value** |
+| $Z1$             |           |
+| $Z2$             |           |
+| $m_0$            |           |
+| $b$              |           |
+| $d_{i1}$         |           |
+| $d_{i_2}$        |           |
+| $N_1$            |           |
+| $\alpha$         |           |
+| $\rho_{steel}$   |           |
+| **Masses**       | **Value** |
+| $m_{in}$         |           |
+| $m_{pinion12}$   |           |
+| $m_{pinion21}$   |           |
+| $m_{out}$        |           |
+| **Stiffness**    |           |
+| $k_x=k_{x}$      |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+|                  |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110000.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark               | Value     |
+| --------------------- | --------- |
+| Newmark $\gamma$      |           |
+| Newmark $\beta$       |           |
+| **Gears**             | **Value** |
+| $Z1$                  |           |
+| $Z2$                  |           |
+| $m_0$                 |           |
+| $b$                   |           |
+| $d_{i1}$              |           |
+| $d_{i_2}$             |           |
+| $N_1$                 |           |
+| $\alpha$              |           |
+| $\rho_{steel}$        |           |
+| **Masses**            | **Value** |
+| $m_{in}$              |           |
+| $m_{pinion12}$        |           |
+| $m_{pinion21}$        |           |
+| $m_{out}$             |           |
+| **Stiffness**         |           |
+| $k_x=k_{x_1}=k_{x_2}$ |           |
+| $k_y=k_{y_1}=k_{y_2}$ |           |
+| $k_x=k_{x_1}=k_{x_2}$ |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+|                       |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110027.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark                              | Value     |
+| ------------------------------------ | --------- |
+| Newmark $\gamma$                     |           |
+| Newmark $\beta$                      |           |
+| **Gears**                            | **Value** |
+| $Z1$                                 |           |
+| $Z2$                                 |           |
+| $m_0$                                |           |
+| $b$                                  |           |
+| $d_{i1}$                             |           |
+| $d_{i_2}$                            |           |
+| $N_1$                                |           |
+| $\alpha$                             |           |
+| $\rho_{steel}$                       |           |
+| **Masses**                           | **Value** |
+| $m_{in}$                             |           |
+| $m_{pinion12}$                       |           |
+| $m_{pinion21}$                       |           |
+| $m_{out}$                            |           |
+| **Stiffness**                        |           |
+| $k_x=k_{x_1}=k_{x_2}$                |           |
+| $k_y=k_{y_1}=k_{y_2}$                |           |
+| $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| $k_{mesh__{mean}}$$                  |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110102.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark                              | Value     |
+| ------------------------------------ | --------- |
+| Newmark $\gamma$                     |           |
+| Newmark $\beta$                      |           |
+| **Gears**                            | **Value** |
+| $Z1$                                 |           |
+| $Z2$                                 |           |
+| $m_0$                                |           |
+| $b$                                  |           |
+| $d_{i1}$                             |           |
+| $d_{i_2}$                            |           |
+| $N_1$                                |           |
+| $\alpha$                             |           |
+| $\rho_{steel}$                       |           |
+| **Masses**                           | **Value** |
+| $m_{in}$                             |           |
+| $m_{pinion12}$                       |           |
+| $m_{pinion21}$                       |           |
+| $m_{out}$                            |           |
+| **Stiffness**                        |           |
+| $k_x=k_{x_1}=k_{x_2}$                |           |
+| $k_y=k_{y_1}=k_{y_2}$                |           |
+| $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| $k_{mesh_{mean}}$                    |           |
+| **Damping**                          |           |
+| $\eta$                               |           |
+| $\beta$                              |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110121.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                   | Value     |
+| ------------------------------------ | --------- |
+| $\gamma$                             |           |
+| $\beta$                              |           |
+| **Gears**                            | **Value** |
+| $Z1$                                 |           |
+| $Z2$                                 |           |
+| $m_0$                                |           |
+| $b$                                  |           |
+| $d_{i1}$                             |           |
+| $d_{i_2}$                            |           |
+| $N_1$                                |           |
+| $\alpha$                             |           |
+| $\rho_{steel}$                       |           |
+| **Masses**                           | **Value** |
+| $m_{in}$                             |           |
+| $m_{pinion12}$                       |           |
+| $m_{pinion21}$                       |           |
+| $m_{out}$                            |           |
+| **Stiffness**                        |           |
+| $k_x=k_{x_1}=k_{x_2}$                |           |
+| $k_y=k_{y_1}=k_{y_2}$                |           |
+| $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| $k_{mesh_{mean}}$                    |           |
+| **Damping**                          |           |
+| $\eta$                               |           |
+| $\beta$                              |           |
+| **Wind excitat**                     |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110144.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                   | Value     |
+| ------------------------------------ | --------- |
+| $\gamma$                             |           |
+| $\beta$                              |           |
+| **Gears**                            | **Value** |
+| $Z1$                                 |           |
+| $Z2$                                 |           |
+| $m_0$                                |           |
+| $b$                                  |           |
+| $d_{i1}$                             |           |
+| $d_{i_2}$                            |           |
+| $N_1$                                |           |
+| $\alpha$                             |           |
+| $\rho_{steel}$                       |           |
+| **Masses**                           | **Value** |
+| $m_{in}$                             |           |
+| $m_{pinion12}$                       |           |
+| $m_{pinion21}$                       |           |
+| $m_{out}$                            |           |
+| **Stiffness**                        |           |
+| $k_x=k_{x_1}=k_{x_2}$                |           |
+| $k_y=k_{y_1}=k_{y_2}$                |           |
+| $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| $k_{mesh_{mean}}$                    |           |
+| **Damping**                          |           |
+| Mass related $\eta$                  |           |
+| Stiffness relat$\beta$               |           |
+| **Wind excitation**                  |           |
+| Blade radius $r_{blade}$             |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+|                                      |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110212.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                    | Value     |
+| ------------------------------------- | --------- |
+| $\gamma$                              |           |
+| $\beta$                               |           |
+| **Gears**                             | **Value** |
+| $Z1$                                  |           |
+| $Z2$                                  |           |
+| $m_0$                                 |           |
+| $b$                                   |           |
+| $d_{i1}$                              |           |
+| $d_{i_2}$                             |           |
+| $N_1$                                 |           |
+| $\alpha$                              |           |
+| $\rho_{steel}$                        |           |
+| **Masses**                            | **Value** |
+| $m_{in}$                              |           |
+| $m_{pinion12}$                        |           |
+| $m_{pinion21}$                        |           |
+| $m_{out}$                             |           |
+| **Stiffness**                         |           |
+| Flexion k_x=k_{x_1}=k_{x_2}$          |           |
+| $k_y=k_{y_1}=k_{y_2}$                 |           |
+| $k_\theta=k_{\theta_1}=k_{\theta_2}$  |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$ |           |
+| **Damping**                           |           |
+| Mass related $\eta$                   |           |
+| Stiffness related $\beta$             |           |
+| **Wind excitation**                   |           |
+| Blade radius $r_{blade}$              |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+|                                       |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110228.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                          | Value     |
+| ------------------------------------------- | --------- |
+| $\gamma$                                    |           |
+| $\beta$                                     |           |
+| **Gears**                                   | **Value** |
+| $Z1$                                        |           |
+| $Z2$                                        |           |
+| $m_0$                                       |           |
+| $b$                                         |           |
+| $d_{i1}$                                    |           |
+| $d_{i_2}$                                   |           |
+| $N_1$                                       |           |
+| $\alpha$                                    |           |
+| $\rho_{steel}$                              |           |
+| **Masses**                                  | **Value** |
+| $m_{in}$                                    |           |
+| $m_{pinion12}$                              |           |
+| $m_{pinion21}$                              |           |
+| $m_{out}$                                   |           |
+| **Stiffness**                               |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$          |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$          |           |
+| Torsion$k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$       |           |
+| **Damping**                                 |           |
+| Mass related $\eta$                         |           |
+| Stiffness related $\beta$                   |           |
+| **Wind excitation**                         |           |
+| Blade radius $r_{blade}$                    |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+|                                             |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110301.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                                 | Value     |
+| -------------------------------------------------- | --------- |
+| $\gamma$                                           |           |
+| $\beta$                                            |           |
+| **Gears**                                          | **Value** |
+| $Z1$                                               |           |
+| $Z2$                                               |           |
+| $m_0$                                              |           |
+| $b$                                                |           |
+| $d_{i1}$                                           |           |
+| $d_{i_2}$                                          |           |
+| $N_1$                                              |           |
+| $\alpha$                                           |           |
+| $\rho_{steel}$                                     |           |
+| **Masses**                                         | **Value** |
+| $m_{in}$                                           |           |
+| $m_{pinion12}$                                     |           |
+|                                                    |           |
+| Output mass $m_{out}$                              |           |
+| **Stiffness**                                      |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$                 |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$                 |           |
+| Torsion $z$ : $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$              |           |
+| **Damping**                                        |           |
+| Mass related $\eta$                                |           |
+| Stiffness related $\beta$                          |           |
+| **Wind excitation**                                |           |
+| Blade radius $r_{blade}$                           |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110335.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                                 | Value     |
+| -------------------------------------------------- | --------- |
+| $\gamma$                                           |           |
+| $\beta$                                            |           |
+| **Gears**                                          | **Value** |
+| $Z1$                                               |           |
+| $Z2$                                               |           |
+| $m_0$                                              |           |
+| $b$                                                |           |
+| $d_{i1}$                                           |           |
+| $d_{i_2}$                                          |           |
+| Inpu$N_1$                                          |           |
+| Contact angle $\alpha$                             |           |
+| Steel density $\rho_{steel}$                       |           |
+| **Masses**                                         | **Value** |
+| Input mass $m_{in}$                                |           |
+| Output mass $m_{out}$                              |           |
+| **Stiffness**                                      |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$                 |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$                 |           |
+| Torsion $z$ : $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$              |           |
+| **Damping**                                        |           |
+| Mass related $\eta$                                |           |
+| Stiffness related $\beta$                          |           |
+| **Wind excitation**                                |           |
+| Blade radius $r_{blade}$                           |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110349.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                                 | Value     |
+| -------------------------------------------------- | --------- |
+| $\gamma$                                           |           |
+| $\beta$                                            |           |
+| **Gears**                                          | **Value** |
+| $Z1$                                               |           |
+| $Z2$                                               |           |
+| $m_0$                                              |           |
+| $b$                                                |           |
+| $d_{i1}$                                           |           |
+| Internal diameter pinion 2 $d_{i_2}$               |           |
+| Input speed $N_1$                                  |           |
+| Contact angle $\alpha$                             |           |
+| Steel density $\rho_{steel}$                       |           |
+| **Masses**                                         | **Value** |
+| Input mass $m_{in}$                                |           |
+| Output mass $m_{out}$                              |           |
+| **Stiffness**                                      |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$                 |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$                 |           |
+| Torsion $z$ : $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$              |           |
+| **Damping**                                        |           |
+| Mass related $\eta$                                |           |
+| Stiffness related $\beta$                          |           |
+| **Wind excitation**                                |           |
+| Blade radius $r_{blade}$                           |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110407.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                                 | Value     |
+| -------------------------------------------------- | --------- |
+| $\gamma$                                           |           |
+| $\beta$                                            |           |
+| **Gears**                                          | **Value** |
+| $Z1$                                               |           |
+| $Z2$                                               |           |
+| $m_0$                                              |           |
+| Pini$b$                                            |           |
+| Internal diameter pinion 1 $d_{i1}$                |           |
+| Internal diameter pinion 2 $d_{i_2}$               |           |
+| Input speed $N_1$                                  |           |
+| Contact angle $\alpha$                             |           |
+| Steel density $\rho_{steel}$                       |           |
+| **Masses**                                         | **Value** |
+| Input mass $m_{in}$                                |           |
+| Output mass $m_{out}$                              |           |
+| **Stiffness**                                      |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$                 |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$                 |           |
+| Torsion $z$ : $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$              |           |
+| **Damping**                                        |           |
+| Mass related $\eta$                                |           |
+| Stiffness related $\beta$                          |           |
+| **Wind excitation**                                |           |
+| Blade radius $r_{blade}$                           |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110430.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                                 | Value     |
+| -------------------------------------------------- | --------- |
+| $\gamma$                                           |           |
+| $\beta$                                            |           |
+| **Gears**                                          | **Value** |
+| $Z1$                                               |           |
+| Pinion 2 tooth number $Z2$                         |           |
+| Pinion modulus $m_0$                               |           |
+| Pinion width $b$                                   |           |
+| Internal diameter pinion 1 $d_{i1}$                |           |
+| Internal diameter pinion 2 $d_{i_2}$               |           |
+| Input speed $N_1$                                  |           |
+| Contact angle $\alpha$                             |           |
+| Steel density $\rho_{steel}$                       |           |
+| **Masses**                                         | **Value** |
+| Input mass $m_{in}$                                |           |
+| Output mass $m_{out}$                              |           |
+| **Stiffness**                                      |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$                 |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$                 |           |
+| Torsion $z$ : $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$              |           |
+| **Damping**                                        |           |
+| Mass related $\eta$                                |           |
+| Stiffness related $\beta$                          |           |
+| **Wind excitation**                                |           |
+| Blade radius $r_{blade}$                           |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110519.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                                 | Value     |
+| -------------------------------------------------- | --------- |
+| $\gamma$                                           |           |
+| $\beta$                                            |           |
+| **Gears**                                          | **Value** |
+| Pinon 1 tooth number $Z1$                          |           |
+| Pinion 2 tooth number $Z2$                         |           |
+| Pinion modulus $m_0$                               |           |
+| Pinion width $b$                                   |           |
+| Internal diameter pinion 1 $d_{i1}$                |           |
+| Internal diameter pinion 2 $d_{i_2}$               |           |
+| Input speed $N_1$                                  |           |
+| Contact angle $\alpha$                             |           |
+| Steel density $\rho_{steel}$                       |           |
+| **Masses**                                         | **Value** |
+| Input mass $m_{in}$                                |           |
+| Output mass $m_{out}$                              |           |
+| **Stiffness**                                      |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$                 |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$                 |           |
+| Torsion $z$ : $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$              |           |
+| **Damping**                                        |           |
+| Mass related $\eta$                                |           |
+| Stiffness related $\beta$                          |           |
+| **Wind excitation**                                |           |
+| Blade radius $r_{blade}$                           |           |
+| Air density $rho_{air}$                            |           |
+| Wind                                               |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110549.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                                 | Value     |
+| -------------------------------------------------- | --------- |
+| $\gamma$                                           |           |
+| $\beta$                                            |           |
+| **Gears**                                          | **Value** |
+| Pinon 1 tooth number $Z1$                          |           |
+| Pinion 2 tooth number $Z2$                         |           |
+| Pinion modulus $m_0$                               |           |
+| Pinion width $b$                                   |           |
+| Internal diameter pinion 1 $d_{i1}$                |           |
+| Internal diameter pinion 2 $d_{i_2}$               |           |
+| Input speed $N_1$                                  |           |
+| Contact angle $\alpha$                             |           |
+| Steel density $\rho_{steel}$                       |           |
+| **Masses**                                         | **Value** |
+| Input mass $m_{in}$                                |           |
+| Output mass $m_{out}$                              |           |
+| **Stiffness**                                      |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$                 |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$                 |           |
+| Torsion $z$ : $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$              |           |
+| **Damping**                                        |           |
+| Mass related $\eta$                                |           |
+| Stiffness related $\beta$                          |           |
+| **Wind excitation**                                |           |
+| Blade radius $r_{blade}$                           |           |
+| Air density $\rho_{air}$                           |           |
+| Wind speed (mean) $v_{wind_{mean}}$                |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110612.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                                 | Value     |
+| -------------------------------------------------- | --------- |
+| $\gamma$                                           |           |
+| $\beta$                                            |           |
+| **Gears**                                          | **Value** |
+| Pinon 1 tooth number $Z1$                          |           |
+| Pinion 2 tooth number $Z2$                         |           |
+| Pinion modulus $m_0$                               |           |
+| Pinion width $b$                                   |           |
+| Internal diameter pinion 1 $d_{i1}$                |           |
+| Internal diameter pinion 2 $d_{i_2}$               |           |
+| Input speed $N_1$                                  |           |
+| Contact angle $\alpha$                             |           |
+| Steel density $\rho_{steel}$                       |           |
+| **Masses**                                         | **Value** |
+| Input mass $m_{in}$                                |           |
+| Output mass $m_{out}$                              |           |
+| **Stiffness**                                      |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$                 |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$                 |           |
+| Torsion $z$ : $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$              |           |
+| **Damping**                                        |           |
+| Mass related $\eta$                                |           |
+| Stiffness related $\beta$                          |           |
+| **Wind excitation**                                |           |
+| Blade radius $r_{blade}$                           |           |
+| Air density $\rho_{air}$                           |           |
+| Wind speed (mean) $v_{wind_{mean}}$                |           |
+| Fluctuating external torque $T_{fluct}$            |           |
+| Fluctuating external frequency $$                  |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```

config/Typora/draftsRecover/2026-3-19 README 110643.md

@@ -0,0 +1,286 @@
+---
+Author: Antoine Foucault-Castelli
+Date: 17/03/2026
+Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
+Version: 4.0 
+---
+| Newmark parameters                                 | Value     |
+| -------------------------------------------------- | --------- |
+| $\gamma$                                           |           |
+| $\beta$                                            |           |
+| **Gears**                                          | **Value** |
+| Pinon 1 tooth number $Z1$                          |           |
+| Pinion 2 tooth number $Z2$                         |           |
+| Pinion modulus $m_0$                               |           |
+| Pinion width $b$                                   |           |
+| Internal diameter pinion 1 $d_{i1}$                |           |
+| Internal diameter pinion 2 $d_{i_2}$               |           |
+| Input speed $N_1$                                  |           |
+| Contact angle $\alpha$                             |           |
+| Steel density $\rho_{steel}$                       |           |
+| **Masses**                                         | **Value** |
+| Input mass $m_{in}$                                |           |
+| Output mass $m_{out}$                              |           |
+| **Stiffness**                                      |           |
+| Flexion $x$ : k_x=k_{x_1}=k_{x_2}$                 |           |
+| Flexion $y$ : k_y=k_{y_1}=k_{y_2}$                 |           |
+| Torsion $z$ : $k_\theta=k_{\theta_1}=k_{\theta_2}$ |           |
+| Mean mesh stiffness $k_{mesh_{mean}}$              |           |
+| **Damping**                                        | **Value** |
+| Mass related $\eta$                                |           |
+| Stiffness related $\beta$                          |           |
+| **Wind excitation**                                | **Value** |
+| Blade radius $r_{blade}$                           |           |
+| Air density $\rho_{air}$                           |           |
+| Wind speed (mean) $v_{wind_{mean}}$                |           |
+| Fluctuating external torque $T_{fluct}$            |           |
+| Fluctuating external frequency $f_{fluct}$         |           |
+| Performance efficiency                             |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+|                                                    |           |
+
+# Gearbox simulation
+## Structure
+```
+main.m
+func.m
+    time_sampling
+    inertia
+    gearbox_geometry
+    mesh_stiffness
+    wind_excitation
+    mass_matrix
+    stiffness_matrix
+    stiffness_update
+    damping_matrix
+    forces_matrix
+    initialization
+    newmark
+parameters.m
+display.m
+    disp_xy(t,x,i)
+    speed_xy(t,v,i)
+    acc_xy(t,v,i)
+    disp_theta(t,x,i)
+    speed_theta(t,v,i)
+    acc_theta(t,a,i)
+    mesh_stiffness(t,k_var,k_mean,i)
+    taero(t,T_aero,i)
+    rms_on_x(x)
+    spectral_on_x(x,i,tf,dt)
+    plot_x(t,x,i)
+only_display.m
+display_update.m 
+```
+
+`main.m` is the launcher for the full code compilation
+`func` is the function class, containing every function used in the code
+`parameters` is the model parameters document
+`display` is the displayer for figures
+`only_display` to skip non-needed full compilation
+`display_update` to update a single graph if needed
+
+## How does it work ?
+**For Matlab under windows :** 
+open matlab and launch main.m
+
+> [!WARNING]
+> remove `graphics_toolkit("gnuplot")` which is only useful for Octave visualization
+
+**For Octave/Linux :**
+
+```bash
+cd [path_to_cloned]/code/versions/[current_version]/
+octave
+>>> run main.m
+>>>
+```
+
+## Informations
+- [x] Verified code with base code and paper
+- [x] Implemented wind excitation
+- [x] Implemented Ghorbel parameters
+- [~] Add varying output torque depending on number of tooth in contact ?
+- [~] add brake ?
+- [~] Variation of C ?
+- [~] ODE45 ?
+- [x] Displays from last version (in class)
+- [x] New mesh stiffness variation model (trapezoidal)
+- [x] Print in pdf
+- [ ] Errors/defects implementation
+    - [ ] profile error
+    - [ ] assembly defect (change on gear stiffness)
+- [ ] README
+    - [x] Maths
+    - [ ] Source(s)
+- [ ] Random models implementation
+- [x] Display
+    - [x] time response on a2
+    - [x] speed response
+    - [x] x movement (vs y movement ?)
+    - [x] RMS STD Kurtosis
+- [ ] Tests
+    - [ ] different speed
+    - [ ] different blade radii
+## Maths
+
+The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
+
+### Parameters
+
+
+
+### Energy equations
+
+$$
+\frac{\partial Ep}{\partial q_i}+
+	\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
+	\frac{\partial Ec}{\partial q_i}+
+	\frac{\partial D}{\partial \dot{q}_i}
+	=\frac{W}{\partial q_i} 
+\\
+Ec = 	\frac{1}{2}m_1*\dot{x}_1^2+
+		\frac{1}{2}m_1*\dot{y}_1^2+
+		\frac{1}{2}I_{11}\dot{\theta}_{11}^2+
+		\frac{1}{2}I_{12}\dot{\theta}_{12}^2+
+		\frac{1}{2}m_2*\dot{x_2}^2+
+		\frac{1}{2}m_2*\dot{y_2}^2+
+		\frac{1}{2}I_{22}\dot{\theta}_{22}^2+
+		\frac{1}{2}I_{21}\dot{\theta}_{21}^2
+\\
+Ep =	\frac{1}{2}k_{x_1}x_{1}^2+
+		\frac{1}{2}k_{y_1}y_{1}^2+
+		\frac{1}{2}k_{\theta_1}(\theta_{11}-\theta_{12})^2+
+		\frac{1}{2}k_{x_2}x_{2}^2+
+		\frac{1}{2}k_{y_2}y_{2}^2+
+		\frac{1}{2}k_{\theta_2}(\theta_{21}-\theta_{22})^2+
+		\frac{1}{2}K_e(t)\delta_1^2(t)
+$$
+
+### Equations of movement
+
+$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
+$$
+\begin{cases}
+&\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\
+&\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\
+
+&\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\
+&\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\
+&\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0
+\end{cases}
+$$
+
+With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
+
+### Matrices
+
+
+
+**Equation to solve :**
+$$
+[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
+\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
+$$
+**Mass matrix :**
+$$
+M =
+\begin{bmatrix}
+	m_1	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	m_1	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	I_{11}	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	I_{12}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	m_2	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	m_2	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	I_{22}	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	I_{21}	\\
+\end{bmatrix} \\
+$$
+
+With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
+
+**Variable mesh stiffness matrix**
+$$
+K(t)= K_e(t)
+\begin{bmatrix}
+	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	r_1.sin(\alpha)	&	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	r_2.sin(\alpha)	\\
+	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	r_1.cos(\alpha)	&	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_1.sin(\alpha)	&	r_1.cos(\alpha)	&	0	&	r_1^2	&	-r_1.sin(\alpha)	&	-r_1.cos(\alpha)	&	0	&	r_1.r_2	\\
+	-sin(\alpha)^2	&	-sin(\alpha).cos(\alpha)	&	0	&	-r_1.sin(\alpha)	&	sin(\alpha)^2	&	sin(\alpha).cos(\alpha)	&	0	&	-r_2.sin(\alpha)	\\
+	-sin(\alpha).cos(\alpha)	&	-cos(\alpha)^2	&	0	&	-r_1.cos(\alpha)	&	sin(\alpha).cos(\alpha)	&	cos(\alpha)^2	&	0	&	-r_2.cos(\alpha)	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	r_2.sin(\alpha)	&	r_2.cos(\alpha)	&	0	&	r_1.r_2	&	-r_2.sin(\alpha)	&	-r_2.cos(\alpha)	&	0	&	r_2^2	\\
+\end{bmatrix}
+$$
+
+> [!NOTE]
+>
+> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
+
+**Constant stiffness matrix**
+$$
+Ks=
+\begin{bmatrix}
+	k_{x1}	&	0	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	k_{y1}	&	0	&	0	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	k_{\theta1}&	-k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	-k_{\theta1}	&	k_{\theta1}	&	0	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	k_{x1}	&	0	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	k_{y1}	&	0	&	0	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	k_{\theta2}	&	-k_{\theta2}	\\
+	0	&	0	&	0	&	0	&	0	&	0	&	-k_{\theta2}	&	k_{\theta2}	\\
+\end{bmatrix}
+$$
+
+**Damping matrix**
+$$
+C = \eta M+\beta K_{mean}
+$$
+**Forces**
+$$
+F =
+\begin{bmatrix}
+0	\\	0	\\	T_{in}	\\	0	\\	0\\	0\\	T_{out}\\	0
+\end{bmatrix}
+$$
+
+
+## TO FIX
+
+> [!ATTENTION]
+> ...
+
+## Machine specifications
+Example of computational time : `44sec for fs=2500` / `n_periods = 10`
+```
+                   -`                    
+                  .o+`                   
+                 `ooo/                   OS: Arch Linux x86_64 
+                `+oooo:                  Host: 20EV000UFR ThinkPad E560 
+               `+oooooo:                 Kernel: 6.19.8-arch1-1 
+               -+oooooo+:                Shell: zsh 5.9
+             `/:-:++oooo+:               Terminal: kitty 
+            `/++++/+++++++:              CPU: Intel i5-6200U (4) @ 2.300GHz  
+           `/++++++++++++++:             GPU: Intel Skylake-U GT2 [HD Graphics 520] 
+          `/+++ooooooooooooo/`           Memory: 1703MiB / 3792MiB 
+         ./ooosssso++osssssso+`          
+        .oossssso-````/ossssss+`         
+       -osssssso.      :ssssssso.        
+      :osssssss/        osssso+++.       
+     /ossssssss/        +ssssooo/-       
+   `/ossssso+/:-        -:/+osssso+-     
+  `+sso+:-`                 `.-/+oso:
+ `++:.                           `-/+/                           
+ .`                                 `/                           
+```