# Question Involving Damped Harmonic Oscillators and Periods

1. Sep 29, 2009

### monkeyboy590

1. The problem statement, all variables and given/known data

Given: The amplitude of a damped harmonic oscillator drops to 1/e of its initial value after n complete cycles. Show that the ratio of period of the oscillation to the period of the same oscillator with no damping is given by

T(sub d)/T(sub o) = (1 + (1/4($$\pi$$^2)(n^2)))^1/2

2. Relevant equations

T(sub d) = (2$$\pi$$)/$$\omega$$$$_{d}$$

T(sub o) = (2$$\pi$$)/((k/m)^1/2)

A/A(sub o) = 1/2 = e^(-t/2$$\tau$$)

n = $$\omega$$$$_{d}$$t/2$$\pi$$

3. The attempt at a solution

I have tried dividing T(sub d) by T(sub o) to find a solution, and subbing known variables into the A/A(sub o) and n equations, to no avail. I have looked ahead to the solution to try to find a way to work towards it, but I don't know where I should start in order to go in the right direction.