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---
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Author: Antoine Foucault-Castelli
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Date: 23/03/2026
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Update: 31/03/2026
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Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
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Version: 5.0
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---
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> [!BUG]
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> Some equations has been put in latex block code as they didnt show properly on gitlab
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# Gearbox simulation
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## Structure
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```
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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-- main
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main.m
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>> used to launch all run_*.m at once
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main_parameters.m
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>> charge model's parameters
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-- runners
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run_main.m
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>> PURPOSE: launch every class_func functions in the right order to model the system
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run_display.m
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>> PURPOSE: launch every class_displayer functions
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run_print.m
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>> PURPOSE: print the figures made by run_display
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>> OUTPUT: .svg
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-- classes
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class_display.m
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>> displays the inputs and outputs of the model
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[col1,col2,col3,col4,col5,col6,col7,col8,lw1,lw2] = charge_aff()
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[figname,namex,namey,namet,ylabelx,ylabelt,ylabelfft,ylabelfftT] = type_choice(type)
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listOfFig = disp_x(t,x,type,listOfFig)
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listOfFig = excitations_ext(t,T_aero,k_var,k_mean,listOfFig)
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listOfFig = spectral_analysis(x,tf,dt,maxf,type,listOfFig)
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% TODO
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plot_fep(t,Fep,i)
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listOfFig = plotThetaDiff(t,x,listOfFig)
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class_func.m
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>> model functions (explicit cf. comments in code)
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[t, dt, tf, t_mesh, f_mesh, fs] =
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time_sampling(fs_div,n_periods,Z1,N1)
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[Jin, J1, J2, Jout, J_all] =
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inertia(m_all,r_blade,di1,di2)
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[c12, db1, db2] =
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gearbox_geometry(alpha0, Z1, Z2, m0, a)
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[k_var, k_mean] =
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mesh_stiffness(epsilon, t, t_mesh, k_mesh_mean)
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[T_aero,T_out] =
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wind_excitation(t,rho_air,r_blade,v_wind,T_fluct_in,f_ext,omega_in,cp,ratio)
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[M] =
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mass_matrix(m_all, J_all)
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[K_var, K_const, K_mean] =
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stiffness_matrix(db1, db2, alpha0, kx, ky, ktheta, k_mean)
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[K] =
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stiffness_update(K_const,K_var,k)
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[C] =
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damping_matrix(M,K_mean,damp_eta,damp_beta)
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[F] =
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forces_matrix(T_in,T_out)
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[x,v,a] =
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initialization(N1,t,ratio,cp)
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[x,v,a] =
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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)
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[F_ep] =
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error_profile_excitation(e12,f_mesh,t,alpha0,db1,db2,k_var)
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[damp_eta, damp_beta] =
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damping_coeff(M,K,xi1,xiN)
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-- standalone
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_convergence.m
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>> purpose: test the numerical convergence of the model
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>> output: 3 graphs + print/_convergence/*.svg
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_gear-mesh-var.m
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>> purpose: shows the variation of gear mesh with central distance changes
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>> output: 1 graph + print/_gear-mesh-var/*.svg
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_rms-analysis.m
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>> purpose: analysis of RMS, STD and kurtosis for many setups
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>> output: data/rms.svg
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```
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## How does it work ?
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**For Matlab under windows :**
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classic matlab
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> [!WARNING]
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> NOT TESTED UNDER MATLAB
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> remove all occurences of `graphics_toolkit("gnuplot")` in the code which is only useful for Octave visualization
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**For Octave/Linux :**
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*May need a few package installation (follow octave recommandations after first(s) run*
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|
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```baswind_turbine_gearbox/octave/h
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cd wind_turbine_gearbox/octave/
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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 objects)
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- [x] New mesh stiffness variation model (trapezoidal)
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- [x] Print in pdf
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- [x] Errors/defects implementation
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- [x] profile error
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- [x] assembly defect (change on gear stiffness)
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- [x] README
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||||
- [x] Maths
|
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- [~] Images
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- [x] Source(s)
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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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- [x] Printing pdf and svg
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- [x] Convergence verification
|
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- [x] Display effect of defects
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- [ ] Verify convergence for other input
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- [ ] Preparation Monte Carlo Parallel calculation
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- [x] Decide a number of samples per mesh period
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- [ ] Random models implementation
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- [x] Implemented easy disp and print
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- [ ] Add a "peak finder" for FFT
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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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|
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### Parameters
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| Newmark parameters | Value | **Unit** |
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| ------------------------------------------------ | --------- | --------------- |
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| $\gamma$ | 0.7 | |
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| $\beta$ | 0.3 | |
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| **Gears** | **Value** | |
|
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| Pinon 1 tooth number $Z1$ | 72 | |
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| Pinion 2 tooth number $Z2$ | 18 | |
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| Pinion modulus $m_0$ | 0.016 | m |
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| Pinion width $b$ | 0.1 | m |
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| Internal diameter pinion 1 $d_{i1}$ | 0.5 | m |
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| Internal diameter pinion 2 $d_{i_2}$ | 0.3 | m |
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| Rotor radius $r_{rot}$ | 0.5 | m |
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| Input theoretical speed $N_1$ | 17 | m/s |
|
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| Contact angle $\alpha$ | 20 | deg |
|
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| Steel density $\rho_{steel}$ | 7860 | kg/m3 |
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||||
| **Masses** | **Value** | **Unit** |
|
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| Input mass $m_{in}$ (hub + blades) | 54000 | kg |
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| Output mass $m_{out}$ (rotor) | 10000 | kg |
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| **Stiffness** | | |
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||||
| Flexion x : $k_x=k_{x_1}=k_{x_2}$ | 1e8 | N/m |
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| Flexion y : $k_y=k_{y_1}=k_{y_2}$ | 1e8 | N/m |
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| Torsion z : $k_\theta=k_{\theta_1}=k_{\theta_2}$ | 1e6 | N/m |
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| Mean mesh stiffness $k_{mesh_{mean}}$ | 2.8e8 | N/rad |
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||||
| **Damping** | **Value** | **Unit** |
|
||||
| Mass damping ratio | 0.05 | |
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||||
| Stiffness damping ratio | 0.01 | |
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| **Wind excitation** | **Value** | **Unit** |
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| Blade radius $r_{blade}$ | 6 | m |
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| Air density $\rho_{air}$ | 1.225 | kg/m3 |
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| Wind speed (mean) $v_{wind_{mean}}$ | 37.5 | m/s |
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| Fluctuating external torque $T_{fluct}$ | 50 | N |
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| Fluctuating external frequency $f_{fluct}$ | 6 | Hz |
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| Performance efficiency $cp$ | 16/27 | |
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| **Errors** | **Value** | **Unit** |
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| Profile error (approx) | 1e-5 | m |
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| Center distance error (approx) | 1e-2 | m |
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### Energy equations
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$$
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\left[\frac{\partial E_p}{\partial q_i}\right]
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+ \left[\frac{d}{dt}\left(\frac{\partial E_k}{\partial\dot{q}_i}\right)\right]
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+ \frac{\partial C}{\partial\dot{q}_i}
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= \frac{\partial W}{\partial q_i}
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\label{eq:reformlagrange}
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\\
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E_k = \frac{1}{2}
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\left(
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m_1{\dot{x}_1}^2+
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m_1{\dot{y}_1}^2+
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I_{11}{\dot{\theta}_{11}}^2+
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I_{12}{\dot{\theta}_{12}}^2+
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m_2{\dot{x}_2}^2+
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m_2{\dot{y}_2}^2+
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I_{22}{\dot{\theta}_{22}}^2+
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I_{21}{\dot{\theta}_{21}}^2
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\right)
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\\
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E_p = \frac{1}{2}
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\left(
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k_{x_1}{x_{1}}^2+
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k_{y_1}{y_{1}}^2+
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k_{\theta_1}(\theta_{11}-\theta_{12})^2+
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k_{x_2}{x_{2}}^2+
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k_{y_2}{y_{2}}^2+
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k_{\theta_2}(\theta_{21}-\theta_{22})^2+
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K_e(t){\delta}^2
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\right)
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$$
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### Equations of movement
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The equations of movement are given by putting $E_k$ and $E_p$ into the Lagrangian formula.
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$$
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\begin{cases}
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\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}
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= m_1\ddot{x}_1
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+ k_{x_1}x_1
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+ \sin(\alpha)K_e(t)\delta
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+ C_{x_1}\dot{x}_1
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= 0 \\
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\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}
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= m_1\ddot{y}_1
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+ k_{y_1}y_1
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+ \cos(\alpha)K_e(t)\delta
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+ C_{y_1}\dot{y}_1
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= 0 \\
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\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}}
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= I_{11}\ddot{\theta}_{11}
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+ k_{\theta_{11}}(\theta_{11}-\theta_{12})
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+ C_{\theta_{11}}\dot{\theta}_{11}
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= T_{aero} \\
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\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}}
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= I_{12}\ddot{\theta}_{12}
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- k_{\theta_{11}}(\theta_{11}-\theta_{12})
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+ {r_b}_1K_e(t)\delta
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+ C_{\theta_{12}}\dot{\theta}_{12}
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= 0 \\
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\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}}
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= m_2\ddot{x}_2
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+ k_{x_2}x_2
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- \sin(\alpha)K_e(t)\delta
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+ C_{x_2}\dot{x}_2
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= 0 \\
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\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}}
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= m_2\ddot{y}_2
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+ k_{y_2}y_2
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- \cos(\alpha)K_e(t)\delta
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+ C_{y_2}\dot{y}_2
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= 0 \\
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\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}}
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= I_{22}\ddot{\theta}_{22}
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- k_{\theta_{22}}(\theta_{21}-\theta_{22})
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+ C_{x_1}\dot{\theta}_{22}
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= T_{gen} \\
|
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\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}}
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= I_{21}\ddot{\theta}_{21}
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+ k_{\theta_{21}}(\theta_{21}-\theta_{22})
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- {r_b}_2K_e(t)\delta
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+ C_{\theta_{21}}\dot{\theta}_{21}
|
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= 0
|
||||
\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.
|
||||
|
||||
### Matrices
|
||||
|
||||
|
||||
|
||||
**Equation to solve :**
|
||||
|
||||
$$
|
||||
\left[M\right]\{\ddot{q}\} + \left[C\right] \{\dot{q}\}+ \left(\left[K_{var}\right]+\left[K_{cst}\right]\right) \{q\} = \left\{F\right\}
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||||
$$
|
||||
|
||||
**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:
|
||||
`++:. `-/+/
|
||||
.` `/
|
||||
```
|
||||
0
config/Typora/backups/2
Normal file
0
config/Typora/backups/2
Normal file
1
config/Typora/backups/sum
Normal file
1
config/Typora/backups/sum
Normal file
|
|
@ -0,0 +1 @@
|
|||
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|
||||
Loading…
Add table
Add a link
Reference in a new issue