# Driven Damped Oscillator problem

slam7211

## Homework Statement

Given damping constant b, mass m spring constant k,
in a damped driven oscillation system the average power introduced into the system equals the average power drained out of the system by the damping force, for what values of ω does the instantanious damping power = instantaneous drive power

## Homework Equations

Total power of system =(-kx - b v(t)+ F0Cos(ωt))v(t)
x(t)= A cos(wt + (phi))

## The Attempt at a Solution

Edit: just tried again, I subbed in ma for F and the Still Stuck with amplitudes on one side, and stuck with phi's

Last edited:

Staff Emeritus
Homework Helper
It would help if you showed your actual calculations.

slam7211
x(t) = Acos($\omega$t+$\phi$)=Aei($\omega$t +$\phi$

Force of Driver = -mA$\omega$2 ei$\omega$t

Total force = -kx-bv+ma
total force * x'(t)=x'(t)*(-kx-bv+ma)
divide by m
$\frac{b}{m}$=$\gamma$

$\frac{k}{m}$=$\omega$02

$\omega$02Aei($\omega$t +$\phi$)-A$\gamma$i$\omega$ei($\omega$t +$\phi$)-A$\omega$2ei($\omega$
I think the question is asking for what nonzero ω values is the total instantanious power 0
which means

$\omega$02Aei($\omega$t +$\phi$)-A$\gamma$i$\omega$ei($\omega$t +$\phi$)-A$\omega$2ei($\omega$t =0

Cancel Like Terms

$\omega$02ei($\omega$t +$\phi$)-$\gamma$i$\omega$ei($\omega$t +$\phi$)-$\omega$2ei($\omega$t) =0

if there was a phi in the force term I could cancel all the e's but according to my book there is no phi term, and phi is not a given so I am stuck

Tsunoyukami
Remember that $e^{i(\omega t + \phi)} = e^{i\omega t + i\phi)} = e^{i\omega t} e^{i\phi}$.

Then you can cancel the $e^{i\omega t}$. I'm not sure if this helps too much, but you could also use the vector representation of complex numbers to find $\phi$?

Staff Emeritus