Problems with Critical Damping and Underdamping

[NewtonsHead]

Homework Statement

The concept of damping is new to me and the problems I have seen have had different known values than I see in the equations I have. Here's two I am working on.

1) An automobile suspension is critically damped, and its period of free oscillation with no
damping is 1s. If the system is initially displaced to a distance x0 and released from rest, find the displacement at t = 1 s.

2) Show that the ratio of two successive maxima in the displacement of an underdamped
harmonic oscillator is constant.

Homework Equations

Critical: x(t) = A e^ (-γ t ) + B e^ (-γ t) where A and B are constants and γ= (2c)/m

Underdamping: x(t) = A₀ e ^ (-γt) cos (ω^'t + θ₀)

I honestly don't know if these are the equations I need to find the answers.

The Attempt at a Solution

1. γ=ω₀
= ω₀ = √k/m
I found this Using the equation for the frequency of a spring (without damping)
∴ γ = 2π
If I plug this into the equation I have, I am still left with those constants... so I must have the wrong approach.

2. I can't think of where to begin on this one... it seems like the variables wouldn't be constant if I set up a ratio like x2/x1

 

[voko]

1. The equation you have for the solution of a critically damped system in incorrect. What is the correct equation?

2. Just do it: set up the ratio for two successive maxima and see where that gets you.