9.3 KiB
Wind turbine gearbox simulator
Wind turbine gearbox simulator is a matlab/octave simulator for a lumped parameters mechanical model of wind turbine gearbox. It implements varying wind profile, defects, varying gear mesh, and compute numerically the dynamical behaviour of the system using a Newmark's scheme.
Related paper: [...]
Dependencies
Development language: Octave 11.1.0 (“some packages” installed following first start requirements, follow the error messages)
Compatibility: Matlab v2026-a (delete all graphics_toolkit("gnuplot") mentions in the code)
Plot engine: Gnuplot 6.0 patchlevel 4
Usage
$ git clone https://gitlab.com/afoucaultc/wind_turbine_gearbox
$ cd wind_turbine_gearbox
$ octave
$ octave:1> run main.m
$ # or run any standalone (starting by a "_")
$ octave:1> >>> run _[standalone].m
File tree
: '
Main process takes the main parameters and compute the
whole process of calculations. It prints every figure
into .svg files, into folder print/run_print/. The
number of periods to calculate are asked as input when
the code is started.
"class_" are static functions used to calculate and plot
see in-code comments for more informations
'
## Main process
├── main.m
├── main_parameters.m
├── run_main.m
│ └── _convergence.m
├── run_display.m
│ └── class_display.m
└── run_print.m
: '
Standalone files are files that do a single specific task
that cannot be done by the main process. These starts by
a "_" and can be ran using ">>> run _file.m"
- _init_speed.m calculates the optimal initial speed to
avoid the transitionnal regime.
- _rms-analysis.m gives a .csv (in /data) containing the
RMS, STD and Kurtosis for different defects conditions
- _animation.m is showing a visual animation of the diff-
-erent shaft rotations and vibrations
- _gear-mesh-var.m plots the difference of mesh stiffness
between different center distance error values
- _convergence.m plots the convergence between a high
sample rate and other sample rates to determine which
one to choose for minimal numerical error and comput-
-ational
time
'
## Standalone functions
├── _init_speed.m
├── _rms-analysis.m
├── _animation.m
├── _gear-mesh-var.m
└── _convergence.m
: '
Rendering folders stores the .csv datasheet and .svg plots
'
## Rendering folders
├── data
│ └── rms.csv
└── print
├── _convergence
├── _gear-mesh-var
└── run_print
## Others
├── octave-workspace
└── README.md
Maths and mechanical model
As showed on the image, the model is based on 2 shaft with 4DDL each (x, y, input and output \theta), as a lumped parameters model. The parameters are listed as a q vector :
{q} = {x_1, y_1,\theta {11},\theta{12},x_2,y_2,\theta_{22},\theta_{21}}
Equations of movement
The equations of movement are given by the Lagrange formulation, using the kinetic and potential energies.
\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 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)
Where I_{ij} inertias for shaft i on its j side; K_e(t) the varying mesh stiffness (cf. Mesh stiffness); \delta=(x_1-x_2)\sin(\alpha)+(y_1-y_2)\cos(\alpha)+r_{b_1}\theta_{12}+r_{b_2}\theta_{21} the tooth deflection.
The derivative of these equations is taken to directly get the movement, using:
\frac{\partial E_p}{\partial q_i}+\frac{d}{dt}\left(\frac{\partial E_k}{\partial \dot{q}_i} \right) = F_i(t)
The different terms (k and m) are regrouped into matrices, to give the following system:
\left[M\right]{\ddot{q}} + \left(\leftK_{mesh}\right+\left[K_{cst}\right]\right) {q} = \left{F(t)\right}
Where K_{cst} represents the constant terms k_{i_j} and K_{mesh} the terms related to the gear mesh stiffness K_e(t).
Mesh stiffness
Mesh stiffness is calculated as a function of mesh period T_{mesh}=60/(Z_1N_1), contact ratio \varepsilon\approx c_{12}=\sum^2_{i=1}\frac{1+Z_i^{-1}}{0.5\sin(\alpha)+\sqrt{0.25\sin(\alpha)^2+(Z_i^{-2})+Z_i^{-1}}} and the one-tooth mesh stiffness k_{mean}. The one-tooth in contact is defined with a period of T_{high} = T_{mesh}(\varepsilon -1) and stiffness k_{max}=k_{mean}\left(1+\frac{2-\varepsilon}{2\varepsilon(\varepsilon-1)}\right); and the two-teeth period is defined by a period T_{low} = T_{mesh}(2-\varepsilon) and stiffness k_{min}=k_{mean}(1-\frac{1}{2\varepsilon}).
Forces
The forces vector depends on:
- wind input (
r_{blade}the blade radius,v_{wind}the mean wind speed,Cpthe performance coefficient of the turbine,\omegathe average rotational speed,T_{fluct}the fluctuation term andf_{ext}its frequency).
T_{in} = \rho_{air}\pi(r_{blade}^2)(v_{wind}^3)Cp/(2\omega) + T_{fluct}\cos(2\pi f_{ext}t)
- output efficiency (
rthe gearbox ratio)
T_{out} = {eff}\times rT_{in}
- a constant brake on 2nd shaft
Then, the force vector is:
\begin{bmatrix}
T_{in} \ 0 \ 0 \ 0 \ 0 \ 0 \ T_{out}+T_{brake} \ 0
\end{bmatrix}
Defects
Two types of defects are studied:
- profile error (which adds a force depending on a multiple cosine profile)
- center distance error (which makes
\alphathe contact angle vary slightly)
Numerical scheme
Numerical computation is done using a Newmark’s scheme with K and F updated for each time step. Mean acceleration parameters are used (\gamma=1/2 and \beta=1/4) for its unconditionally stable properties. Convergence is verified and the usual sample rate fs_{div} (based on mesh period division) is chosen as equal to 11.
Speed is initialized at its permanent regime value, to avoid transient regime.
Parameters
| Gearbox | ||
|---|---|---|
Z1 |
Tooth nbr 1 | 72 |
Z2 |
Tooth nbr 2 | 18 |
m0 |
Gear modulus | 0.016 m |
b |
Gear width | 0.1 m |
N1 |
Shaft 1 speed | 17 RPM |
\alpha |
Contact angle | 20° |
\rho_{steel} |
Steel density | 7850 kg/m3 |
T_{brake} |
Brake torque | 500 Nm |
| Masses | ||
m_{11} |
Blades and input rotor mass | 2000 kg |
m_{22} |
Output rotor mass | 500 kg |
| Shafts | ||
da1 |
Shaft 1 diameter | 0.5 m |
da2 |
Shaft 2 diameter | 0.2 m |
la1 |
Shaft 1 length | 2 m |
la2 |
Shaft 2 length | 1 m |
E_{steel} |
Young modulus of steel | 2.1e11 MPa |
\nu_{steel} |
Poisson ratio of steel | 0.3 |
| Stiffness | ||
k_x |
x translation stiffness |
1e8 N/m |
k_y |
y translation stiffness |
1e8 N/m |
| Wind | ||
r_{blade} |
Blade radius | 6 m |
\rho _{air} |
Air density | 1.225 kg/m3 |
v_{wind} |
Wind mean speed | 12 m/s |
T_{fluct} |
Wind fluctuating torque | 3000 Nm |
f_{fluct} |
Fluctuation frequency | 6 Hz |
Cp |
Performance coefficient | 16/27 |
| Defects | ||
e12_{amp} |
Max profile error amplitude | Around 50$\mu m$ |
gap |
Center distance error | around $\pm$1mm |
Development machine specification
Hardware: Thinkpad E560, CPU Intel(R) Core(TM) i5-6200U(4)@2.8GHz, 3.70GiB RAM OS: Arch Linux 6.19.12