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: Damped Harmonic Oscillator/ME loss/freq question

  1. May 10, 2012 #1
    1. The problem statement, all variables and given/known data

    A damped harmonic oscillator loses 6.0% of it's mechanical energy per cycle. (a) By what percentage does it's frequency differ from the natural frequency f[itex]_{0} = (\frac{1}{2\pi})\sqrt{\frac{k}{m}}[/itex]? (b) After how many periods will the amplitude have decreased to [itex]\frac{1}{e}[/itex] of it's original value?

    2. Relevant equations

    natural frequency
    [itex]f_{0} = (\frac{1}{2\pi})\sqrt{\frac{k}{m}}[/itex]

    damped frequency
    [itex]f' = \frac{1}{2\pi}\sqrt{\frac{k}{m}-\frac{b^{2}}{4m^{2}}}[/itex]

    displacement for lightly damped harmonic oscillator
    [itex]x = Ae^{(\frac{-b}{2m})t}cos\omega't[/itex]

    Total mechanical energy
    [itex]E = \frac{1}{2}kA^{2} = \frac{1}{2}mv^{2}_{max}[/itex]

    And I know the mean half life, [itex]\frac{2m}{b}[/itex] is the time until oscillations reach 1/e of original.

    3. The attempt at a solution

    I used the A^2 expression for E and the A decay term, [itex]Ae^{(\frac{-b}{2m})t}[/itex] ,said it loses 6% of E when A^2 = .94A^2 (original) or in other words when [itex]Ae^{(\frac{-b}{2m})t}[/itex] = [itex]\sqrt{0.94}[/itex]A
    so, [itex]e^{(\frac{-b}{2m})t}[/itex] = [itex]\sqrt{.94}[/itex]
    [itex]\frac{-b}{2m}t[/itex] = [itex]\frac{1}{2}[/itex]ln(.94)
    t = [itex]\frac{-m}{b}[/itex]ln(.94)
    But this is time and I need it to be one cycle so do I plug the period in for t?
    T = 1/f or 2∏ ω?

    This is where I'm stuck. The answer is (a) -1.21x10^-3 % and (b) 32.3 periods but I don't see how to clear the unknowns with what is given. I could just copy the answer down but I want to know how to solve it. If anybody can give me hint it would be a great result for my first post here.
  2. jcsd
  3. May 10, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Yes, that's essentially what you want to do.

    In terms of angular frequencies, you have
    $$\omega' = 2\pi f' = \sqrt{\omega_0^2-\frac{b^{2}}{4m^{2}}}$$ You also have
    $$-\left(\frac{b}{2m}\right)T' = \log \sqrt{0.94}$$ where T'=1/f' is the period of (damped) oscillation. Use it to eliminate (b/2m) from the first equation. Then you'll be able to solve for ω' in terms of ω0.
  4. May 10, 2012 #3
    Thanks. I beat my head against it a while longer and suddenly, there it was! This is one problem I won't forget after the test.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook