Amplitude of equation of motion

[jimmy42]

Hello,

I have worked out some force diagrams for forced oscillations and ended up with the solution as :

mx_double_dot+rx_dot+kx=Pcos(Ωt)

I am now asked to work out the amplitude. I know all of the variables except frequency(Ω). What equations can I use to find that?

Thanks.

 

[Simon Bridge]

$$m\ddot{x}+r\dot{x}+kx=P\cos(\omega t)$$
... to see how the amplitude behaves, you'll need to solve the equation.
The driving frequency is something you'd normally be given.

I don't see a damping term - what do you think is likely to happen to the amplitude of the oscillations?

 

[jimmy42]

I have worked out that this is a strongly dampered equation, so I expect the amplitude to die down quickly.

So, without frequency, this cannot be done? The question I have is to get the amplitude in order to solve the frequency between certain amplitudes. That last part can be done on the computer.

 

[Simon Bridge]

Oh I misread it - OK. So you have determined the system is overdamped which simplifies things - you need to find the natural frequency and damping ratio.

You don't need the driving frequency to find the amplitude envelope - it's a decaying exponential: compare your equation with the general solutions.
http://en.wikipedia.org/wiki/Harmonic_oscillator#Driven_harmonic_oscillators