A damped harmonic oscillator is driven by a force
F external= F sin (omega * t)
where F is a constant, and t is time.
Show that the steady state solution is given by
x(t)= A sin (omega * t - phi)
where A is really A of (omega), the expression for the amplitude where omega is a variable, and phi is the phase shift.
In other words, A of omega is equal to this:
A(omega)= F/ [(k-m omega^2)^2 + c^2omega^2]^1/2
The Attempt at a Solution
Sorry if the equations are not easy to read, but they are pretty standard for harmonic oscillators so I think you can get the general idea.
This seems really simple but there is something I'm not getting. In the text book, the damped harmonic oscillator is solved for an external force= Fcos (omega * t)
by using the substitution:
external force= F e ^i*omega*t
where F is a constant.
And then you can easily solve for A and get the expression for the amplitude that is above, and for the phase shift phi.
But when I try to solve the differential equation for an external force with sin in it, the imaginary part of the expression cancels out of the expression for the force, and there are only imaginary numbers on the left.