explanation of ode45 for constant force

I am implementing the shm equation ( m x'' + c x' + k x = F0) in matlab using ode45. The RHS is a constant force (F0). I am still getting an oscillatory response. The steady state response is equal to the static deformation. What is confusing me are the oscillations. What a force with constant magnitude result in an oscillatory response. Is my implementation wrong or my understanding of physics?

The sample code is shown below.

function dydt = sdof_fun(t,y,m,k,c,F)

dydt(1) = y(2);
dydt(2) = (F/m) - (k/m)*y(1) - (c/m)*y(2);

dydt = dydt(:);
end

[t,w] = ode45(@(t,y) sdof_fun(t,y,m,k,c,F),tspan,ic);

[Ajinkya]

Oh, Is that the free vibration response (response of the Homogeneous equation)?

[Torsten]

It depends on the zeros of the characteristic equation

m*x^2 + c*x + k = 0

what kind of response you get (especially the sign of the discriminant disc = c^2 - 4*k*m).

So check your constants.

Answer 1

[Sam Chak]

Hi @Ajinkya

Analysis shows that the steady-state value of of the linear system is given by

.

In your case, I guess that the oscillatory response is observed because . If so, then try extending the simulation time long enough until you see the response converges to .

Another way is to check the value of the system's damping ratio, given by

.

If the damping ratio value is very small, then the oscillations die out very slowly. If , then a critically-damped response will be observed.

The following example shows the convergence, with so that the oscillations die out faster.

m      = 11;    % change to 49/20 to observe a critically-damped response
c      = 7;
k      = 5;
F      = 3;
params = [m c k F];

tspan  = 0:0.01:30;
ic     = [-1 0];
[t, y] = ode45(@(t, y) sdof_fun(t, y, params), tspan, ic);
plot(t, y(:,1), 'linewidth', 1.5), grid on
xlabel('t'), ylabel('y')
title('System Response')

function dydt = sdof_fun(t, y, params)
    m       = params(1);
    c       = params(2);
    k       = params(3);
    F       = params(4);
    dydt(1) = y(2);
    dydt(2) = (F/m) - (k/m)*y(1) - (c/m)*y(2);
    dydt = dydt(:);
end