--- Author: Antoine Foucault-Castelli Date: 19/03/2026 Update: 19/03/2026 Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022 Version: 4.1 --- > [!BUG] > Some equations has been put in latex block code as they didnt show properly on gitlab # 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 error_profile_excitation force_error_profile 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) plot_fep(t,F_ep,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 - [x] 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 | | | Errors | Value | Unit | | Profile error | 50e-6 | m | | Center distance error | 10e-3 | m | ### 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) ```latex $$ \begin{cases} &\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\ &\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\ &\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\ &\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\ &\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\ &\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\ &\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\ &\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0 \end{cases} $$ ``` 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 : ```latex $$ c_1=\frac{1+\frac{1}{Z1}}{\left(\pi cos(\alpha)\left(\frac{sin(\alpha)}{2}+\sqrt{\frac{sin(\alpha)^2}{4}+\frac{1}{Z1^2}+\frac{1}{Z1}}\right)\right)} \\ c_1=\frac{1+\frac{1}{Z2}}{\left(\pi cos(\alpha)\left(\frac{sin(\alpha)}{2}+\sqrt{\frac{sin(\alpha)^2}{4}+\frac{1}{Z2^2}+\frac{1}{Z2}}\right)\right)} \\ c_{12}=c_1+c_2 \approx \varepsilon $$ ``` ## 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: `++:. `-/+/ .` `/ ```