# Frequency of damped vs. undamped oscillator

1. Mar 13, 2007

### Mindscrape

1. The problem statement, all variables and given/known data
If the amplitude of a damped oscillator decreases to 1/e of its initial value after n periods, show that the frequency of the oscillator must be approximately [1 - (8(π^2)(n^2))^-1] times the frequency of the corresponding undamped oscillator.

2. Relevant equations
Damped
$$m\ddot{x} + b \dot{x} + kx = 0$$

Undamped
$$m\ddot{x} + kx = 0$$

3. The attempt at a solution

Rewrite the damped second order ODE as
$$\ddot{x} + 2\beta \dot{x} + \omega_0^2 x = 0$$

where
$$\beta = \frac{b}{2m}$$
$$\omega_0 = \sqrt{k/m}$$

The undamped first order ODE can be written as
$$\ddot{x} + \omega_0^2 x = 0$$

where
$$\omega_0 = \sqrt{k/m}$$

The solution seems to depend on whether or not the damped oscillator has a complex solution or not, and the general solution will be:

$$x(t) = e^{-\beta t}[A_1 e^{\sqrt{\beta^2 - \omega_0^2}t} + A_2 e^{- \sqrt{\beta^2 - \omega_0^2}t}]$$

I guess I am supposed to assume that the discriminate will be negative and yield complex solutions, since the overdamped and critically damped cases will have at most one period.

Then see how the quasi-frequency of the undamped oscillator relates to the time constant, and further relate it to the angular frequency of the undamped oscillator?