# Logarithmic decrement of a lightly damped oscillator

by kraigandrews
Tags: damped, decrement, lightly, logarithmic, oscillator
 P: 108 1. The problem statement, all variables and given/known data 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... 2. Relevant equations $\beta$=b/(2m) Q=$\omegao/(2\beta) 3. The attempt at a solution Given the diff eq: d2x/dt2+2[itex]\beta$(dx/dt)+$\omega$o2x=0 I can solve this to find x(t), however I feel this is irrelevant because no initial condition or boundary conditions are given, so I am kinda lost here as to where go or to start at for that matter. Any suggestions are greatly appreciated, Thanks 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution