Frequency of damped vs. undamped oscillator

[Mindscrape]

Homework Statement

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.

Homework Equations

Damped
$$m\ddot{x} + b \dot{x} + kx = 0$$

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

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?

 

[vela]

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.

The word is discriminant. The problem statement implies the oscillator is underdamped, so you don't really have to assume anything here.

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?

I'm not sure what a quasi-frequency is.

It sounds like you have the right idea. You can relate $\beta$ to $\omega$ from the information given, which then allows you to solve for $\omega$ in terms of $\omega_0$.

 

[Delta2]

It will be $$e^{-\beta n T}=\frac{1}{e}\iff \beta n T=1$$ where $T$ is the period of the underdamped system, $$T=\frac{2\pi}{\sqrt{|\beta^2-\omega_0^2|}}=\frac{2\pi}{\sqrt{(\frac{2\pi}{T_0})^2-\beta^2}}$$ so its a matter of algebraic manipulations to find the relationship between $T$ and $T_0$ (by replacing $\beta=\frac{1}{nT}$ in the second equation e.t.c.

 

[Delta2]

I am getting the result of the OP only if I assume that the given data is for $n$ periods of the undamped system, that is if $\beta n T_0=1$