Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Damped Harmonic Oscillator Approximation?

  1. Oct 25, 2004 #1

    cj

    User Avatar

    For a simple damped oscillator...

    [tex] \text {Apparently if } \beta \ll \omega_0 } \text { then ...}[/tex]

    [tex] \omega_d \approx \omega_0[1-\frac {1}{2}(\beta/\omega_0)^2]}[/tex]

    Given that:

    [tex] \beta=R_m/2m \text { (where } R_m= \text {mechanical resistance) } \text { and } \omega _d=\sqrt{(\omega _0^2-\beta ^2)}[/tex]

    How/why is this true? My guess is some kind of
    series approximation is used -- but I'm not sure...
     
  2. jcsd
  3. Oct 25, 2004 #2

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Let's establish the series approximation:
    [tex]\omega_{d}=\sqrt{\omega_{0}^{2}-\beta^{2}}=\omega_{0}\sqrt{1-(\frac{\beta}{\omega_{0}})^{2}}[/tex]

    Now, let [tex]f(\epsilon)=(1+\epsilon)^{m}[/tex]
    When [tex]\epsilon\approx0[/tex]
    we have, by Taylor's theorem:
    [tex]f(\epsilon)\approx{f}(0)+f'(0)\epsilon=1+m\epsilon[/tex]
    Now, recognize:
    [tex]m=\frac{1}{2},\epsilon=-(\frac{\beta}{\omega_{0}})^{2}[/tex]
    and you've got the formula.
     
    Last edited: Oct 25, 2004
  4. Oct 26, 2004 #3

    cj

    User Avatar

    Thanks very much. I've got to somehow get
    more familiar with Taylor series expansions -- they
    seem to be the basis of so many solutions.

    cj


     
  5. Oct 26, 2004 #4

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    You are absolutely correct in this.
    Taylor expansions occur in every branch of physics; for example, they are often used to simplify and approximate difficult non-linear terms occuring in differential equations.
    I'm sure you know this one from the swinging pendulum:
    We simply assume the angle to be small, and approximate the term:
    [tex]\sin\theta(t)\approx\theta(t)[/tex]
    This brings, as you know, the pendulum equation into the form of a simple harmonic oscillator.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Damped Harmonic Oscillator Approximation?
  1. Damped Oscillation (Replies: 5)

Loading...