im trying to model the drive train model of wind turbine, i need to get oscillation and the steady state waveform

dwt/dt=1/2Ht(Tt-Tsh)

dwr/dt=1/2Hg(Tsh-Te)

Tsh=k*thetatw+c*dtheta_tw/dt

dtheta_tw/dt=welb(wt-wr)

thetatw=3.35

wt=1

wr=1

Answer 1

[VBBV]

Provide all inputs to the equations used for modeling the drive train,

clc
clear all
close all
a=3.2; %input('Enter the a = ');
b=1.5; %input('Enter the b = ');
c=.1;%input('Enter the c = ');
%initial condition
theta_0=[a,b,c];
t_span=linspace(0,100,100);
[t,results]=ode45(@(t,theta)ode_fun(t,theta),t_span,theta_0);
subplot(211)
plot(t,results(:,2));
subplot(212)
plot(t,results(:,3));
xlabel('t')
ylabel('wt&wg')

function [dtheta_dt]= ode_fun(t,theta)
Tt=1;
Te=0.78;
f=50;
welb=2*pi*f;
Ksh=0.3;
Csh=0.01;
Ht=4;
Hg=0.1*Ht;
wt=theta(1);
wg=theta(2);
thetatw=theta(3);
dthetatw_dt=welb*(wt-wg);


thetatw=3.35;
wt=0.1;
wr=0.01;
c = 0.1;  % give damping constant factor
k = 2.6e8;  % give flexural stiffness of gear train , e.g. shown
dtheta_tw_dt=welb*(wt-wr);
Tsh=k*thetatw+c*dtheta_tw_dt;
dwt_dt=(1/2)*Ht*(Tt-Tsh);  % what are these  for ?  another model ?
dwr_dt=(1/2)*Hg*(Tsh-Te);  %  this one too


Dwt_dt=((Tt/(2*Ht))-((Ksh/(2*Ht))*thetatw)+((Csh/(2*Ht))*(welb*(wt-wg))));  %  check these equations 
Dwg_dt=(((Ksh/(2*Hg))*thetatw)-((Csh/(2*Hg))*(welb*(wt-wg)))+(Te/(2*Hg)));  % check these
dtheta_dt=[dthetatw_dt;Dwt_dt;Dwg_dt];
end

Answer 2

[Sam Chak]

Hi @peddada sandhya rani

In the absence of the effects of the constants and in the original math model, then the system is stable.

If oscillations are desired, then you need to inject some manipulated variables so that the system behavior can be changed. I'm unfamiliar with your system. In physical systems, the manipulated variables can forces, torques, voltage sources, etc.

theta_0 = [0.1, -0.2, 1];
t_span = linspace(0, 10, 10001);
options = odeset('AbsTol', 1e-16, 'RelTol', 1e-12);
[t, theta] = ode45(@(t,theta) ode_fun(t, theta), t_span, theta_0, options);

subplot(311)
plot(t, theta(:,1)), grid on, ylabel('\omega_{t}')
subplot(312)
plot(t, theta(:,2)), grid on, ylabel('\omega_{g}')
subplot(313)
plot(t, theta(:,3)), grid on, ylabel('\theta_{tw}'), xlabel('t')

function [dtheta_dt]= ode_fun(t, theta)
    % Parameters
    Tt   = 1;
    Te   = 0.78;
    f    = 50;
    welb = 2*pi*f;
    Ksh  = 0.3;
    Csh  = 0.01;
    Ht   = 4;
    Hg   = 0.1*Ht;
    
    wt          = theta(1);
    wg          = theta(2);
    thetatw     = theta(3);
    
    % Test 1: without manipulated variables, without constant disturbances
    % dwt_dt      = 0*Tt/(2*Ht) - Ksh/(2*Ht)*thetatw - Csh/(2*Ht)*welb*(wt - wg);
    % dwg_dt      = Ksh/(2*Hg)*thetatw + Csh/(2*Hg)*welb*(wt - wg) - 0*Te/(2*Hg);
    % dthetatw_dt = welb*(wt - wg);
    
    % Test 2: with manipulated variables
    A = [ -pi/8    pi/8 -3/80;
         5*pi/4 -5*pi/4  3/8;
         100*pi -100*pi    0];
    B  = eye(3);
    p = [-1, -2i, +2i];     % desired eigenvalues
    K = place(A, B, p);
    
    % Manipulated variables
    u1 = - K(1,:)*theta - Tt/(2*Ht);    % also cancel out effect of Tt/(2*Ht)
    u2 = - K(2,:)*theta + Te/(2*Hg);    % also cancel out effect of Te/(2*Hg)
    u3 = - K(3,:)*theta;
    
    dwt_dt      = Tt/(2*Ht) - Ksh/(2*Ht)*thetatw - Csh/(2*Ht)*welb*(wt - wg) + u1;
    dwg_dt      = Ksh/(2*Hg)*thetatw + Csh/(2*Hg)*welb*(wt - wg) - Te/(2*Hg) + u2;
    dthetatw_dt = welb*(wt - wg) + u3;
    
    % Reorder the sequence because thetatw = theta(3)
    dtheta_dt   = [dwt_dt;
                   dwg_dt;
                   dthetatw_dt];
end