Practical task

The task to convert following differential equation:

into following form:

taking into account, that at small angles , . We will obtain

Than for simplification of problem we’ve to make substitution

F1 = ,

F2=

F3= . Then

Initial data: ζ =[0.1 0.1 0.1 0.1 0.1],

n_mx=[0.1 4.0 4.0 4.0 4.0],

n_my=[2.0 4.0 6.0 8.0 10.0],

ν =[5.0 5.0 5.0 5.0 5.0],

ε _{x, y}=[0 0],

φ 0=0,

t_max=20,

τ =0.5

Listing of MatLab program:

eta=0.1;

tau=0.5;

tmax=20;

nmx=4;

nmy=2.0;

nu=5.0;

ex=0;

ey=0;

f3=2*eta;

f1=nmx*sin(nu*tau+ex);

f2=nmy*sin(nu*tau+ey);

w=tf([-f1], [1 f3 1+f2]);

figure(1)

step(w), grid on

figure(2)

impulse(w), grid on

x0=[-0.4; 0.1];

[num, den]=tfdata(w, 'v');

[A, B, C, D]=tf2ss(num, den)

disp('teta/T=');

printsys(num, den, 's')

disp('');

t=0: 0.01: tmax;

Tor=sin(pi*t);

[y, x]=lsim(A, B, C, D, Tor, t, x0);

figure(3)

plot(t, y, 'r--'), grid on

xlabel('Time');

ylabel('teta(t)');

title('Pendulum');

hold on;

xk=x0; T=0.05;

for k=0: tmax/T;

xk=(T*A+eye(2, 2))*xk+T*B*sin(pi*k*T);

yk=C*xk;

plot(k*T, yk, '.');

end;

hold;

legend('y(t)', 'y[kT]');

We obtain the following results:

Graphs shown instability of pendulum oscillations.