A lightly damped harmonic oscillator

[kraigandrews]

Homework Statement

The logarithmic decrement δ of a lightly damped oscillator is defined to be the natural logarithm of the ratio of successive maximum displacements (in the same direction) of a free damped oscillator. That is, δ = ln(An/An+1) where An is the maximum displacement of the n-th cycle. Derive the simple relationship between δ and Q.
Find the spring constant k and damping constant b of a damped oscillator with mass m, frequency of oscillation f and logarithmic decrement δ.
[Data: m = 4.0 kg; f = 0.9 Hz; δ = 0.029.]
First, the spring constant k...

Also, the damping constant b...

Homework Equations

\beta=b/(2m)
\omega_o=(k/m)^1/2
Q=\omega_o/(2\beta)

The Attempt at a Solution

Ok so I do not know where to start, I can solve the equation:

d²x/dt²+2\betadx/dt+\omega_o²x=0

but other than that I have no idea where to go. My best guess is that it is unnecessary to solve it because no initial conditions are given thus one would be unable to find the constant or any initial amplitude. any help is greatly appreciated.
thanks

 

[Mike Pemulis]

You do need to solve the equation. Find a general solution given the parameters in the equation, and go from there. From reading the problem, I think you can assume the oscillator is underdamped.