How Do You Solve a Damped Harmonic Oscillator Driven by a Sinusoidal Force?

[Absiximab]

Homework Statement

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.

Homework Equations

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 textbook, 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.

 

[Absiximab]

Edit: Trying to be more clear: This is what I tried.
I first plug the steady state solution into the differential equation for the damped harmonic oscillator, with an external force = F sin (omega*t).
I end up with an equation that has imaginary numbers only on the LEFT side of the equation, not on the right side, in the expression for the force (because they got canceled out).
I set up two equations, the real part and the imaginary part.
(So, can I set the imaginary part on the left side of the equation equal to zero? (is that allowed? it seems right.)
I end up with an expression for A that is similar to the one above, as long as I make the assumption that if the imaginary part equals zero, I can just let it drop out. I THINK that that is how I should "show that the steady state solution is given by..." etc.

Is this how I should solve this? Am I allowed to use imaginary numbers this way?

 

[Dr.D]

Don't mess with imaginary numbers at all.

 

[Absiximab]

But how can I assume a solution; the solution is given to me and there is only sin, there is no cos term in it. The magnitude for A that you have given is only true is certain cases I think, such as when there is no external force.
A should be dependent on omega, not on the constants A and B.

 

[Dr.D]

You can always express ...

 

[Absiximab]

Even if phi is not a constant, but dependent on omega? it seems like that would make it even more complicated...how would I do this?
do you mean something like sin u + cos u = sin u + sin (u + pi/2)? how would that help?

 

[Dr.D]

Try it.

 

[Absiximab]

you're saying I should express the solution, which is x=A sin (omega*t - phi) in terms of sin and cos. I would need to start with a solution in the form of sin u + sin (u + pi/2), which is not what you start with, so I don't think that's possible.

Also, by the way, I am pretty sure that the solution involves imaginary numbers. The standard way of solving a driven harmonic oscillator is by assuming there is an external force that varies sinusoidally with time, equal to F cos (omega*t), and then substituting e^(i*omega*t) for cos (omega*t). You then need to work with imaginary numbers to solve the equation for A.

 

[Dr.D]

Well, since you know so much more about this than I do, by all means, carry on. Who am I to get in your way?

It has only been about 45 years since I worked this out for the first time.

 

[Absiximab]

If you have a constructive piece of advice to give, please do so.
If you don't, then admit it rather than wasting both of our time.

 

[Dr.D]

Have fun!