1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Critically Damped Oscillator

  1. Mar 25, 2013 #1
    1. The problem statement, all variables and given/known data
    (A) A damped oscillator is described by the equation
    m x′′ = −b x′− kx .
    What is the condition for critical damping? Assume this condition is satisfied.
    (B) For t < 0 the mass is at rest at x = 0. The mass is set in motion by a sharp impulsive force at t = 0, so that the velocity is v0 at time t = 0. Determine the position x(t) for t > 0.
    (C) Suppose k/m = (2π rad/s)2 and v0=10 m/s. Plot, by hand, an accurate graph of x(t). Use graph paper. Use an appropriate range of t.

    2. Relevant equations
    For critically damped, β2 = w02
    where β = b/(2m) and w0 = √(k/m)

    3. The attempt at a solution
    Ok, for this problem, what I did initially was find the general form of position for a critically damped oscillator, which is:
    x(t) = (A + B*t)*e-β*t

    and the velocity function is:
    v(t) = -Aβe-βt + (Be-βt - Bβte-βt)

    Using the conditions given, I found:
    x(0) = A (obviously) which we don't know x(0)
    B = v0 + Aβ
    and x(t) can be rewritten as:
    x(t) = A(e-βt + βte-βt) + v0te-βt

    This is where I run into a wall. I can't seem to solve for A. I believe that x(0) should also be the max displacement since there is no driver for the impulse force, so A should be the max displacement, but this doesn't seem to get me anywhere. Any help on solving for A? I know how to do the rest other than that.
  2. jcsd
  3. Mar 25, 2013 #2


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    A "sharp impulsive force" is defined such that it instantaneously gives the mass an initial velocity without any displacement of the mass. So the mass is still at x = 0 immediately after the impulse.
  4. Mar 26, 2013 #3
    Ok, that makes sense. Thank you for the help.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted