# Relationship between period and time in oscillators

## Homework Statement

If the amplitude of a weakly damped oscillator decreased to 1/e of its initial value after n periods, show that the frequency of the oscillator must be approximately [1 − (8π2n2)-1] times the frequency of an undamped oscillator with the same natural frequency.

## Homework Equations

The equation for a weakly damped harmonic oscillator is x(t) =Ae-βtcos(ω1t-δ) where ω1 = sqrt(ωo22)
A = amplitude
β = decay constant
ω1 = period for damped oscillator
ωo = natural frequency
T = period
t = time
δ = phase angle

## The Attempt at a Solution

Since the amplitude was initially Ae-βt and finally 1/e, I solved for t=1/β or β=1/t. I then took the equation ω1 = sqrt(ωo22) and did an expansion to the second term which gave me that ω1o(1-β2/2ωo2). Ridding of β2 for 1/t gives me ω1o(1-1/2ωo2t2). I can then replace ωo with 2πn/T and then I would almost have the correct answer, but for that t= T which doesn't make sense. Also as an aside do I not have to include the phase angle (so I can make it zero) since I'm not measuring the oscillator in reference to another one? Or is it there due to impedance or something?

Last edited:

vela
Staff Emeritus
Homework Helper
You're introducing ##n## in the wrong place. If ##1/\beta## is the time for ##n## periods, you can say ##1/\beta = n(2\pi/\omega_1)##.

• Vitani11
Okay, can you explain why that is?

vela
Staff Emeritus