Damped Oscillator: Finding Work Rate & Average Power

[DanaBug28]

I am really struggling with this question...

Question: Consider a damped oscillator, with natural frequency w_0 (omega_0) and damping constant B (beta) both fixed, that is driven by a force F(t)= F_0*cos(wt). Find the rate P(t) at which F(t) does work and show that the average < P > over any number of complete cycles is mBw^2*A^2.

any help would be amazing!

 

[Tide]

I think the rules here stipulate that you need to show some of your own work before you can get specific help on homework problems.

 

[DanaBug28]

work

F(t)=m*f_0*cos(wt) in general
long term motion x(t)=A*cos(wt-delta)
delta= arctan((2Bw)/(w_0^2-w^2))
A^2= (f_0^2)/((w_0^2-w^2)^2+4*B^2*w^2)

< P >=mBw^2*A^2
= m*f_0*cos(wt)*distance

mBw^2*A^2 = mBw^2*(f_0^2)/((w_0^2-w^2)^2+4*B^2*w^2) = m*f_0*cos(wt)*distance

cancel stuff...

B*w^2*(f_0)/((w_0^2-w^2)^2+4*B^2*w^2) = cos(wt)*distance

uh...help?

 

[Tide]

To calculuate the average power over a period you need to evaluate the integral

$$<P> = \frac {1}{T} \int_0^T F v dt$$