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