1. Not finding help here? Sign up for a free 30min 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!

Classical Mechanics: Lightly Damped Oscillator Driven Near Resonance

  1. Sep 18, 2014 #1
    Hello Physics Forum! I have a question:

    The problem: For a lightly damped oscillator being driven near resonance in the steady state, show that the fraction of its energy that is lost per cycle can be approximated by a constant (something like 2pi, which is to be determined) divided by the Q factor (Q is defined as the resonant frequency of a driven damped oscillator divided by 2*β, where β is the damping parameter).


    My professor gave this hint to get us started: ΔE = ∫Fdx. Where E is the energy lost, and F is the force of friction. We are supposed to integrate from 0 to τ(cycle/period), and the professor suggested to change dx to (dx/dt)dt = vdt.

    I have spent much time attempting to figure this out. I think that the frictional force is F=-bv, where b is some positive constant and v is the velocity. I try to use the solution of the differential equation for such motion, which is x(t)=Ae^(-βt)cos(ωt-δ). I take the derivative of this to get v(t). The second term in v(t) can be ignored because the damping is light. So I have:

    ΔE = -b∫v^2dt from 0 to τ, where v(t)≈-Ae^(-βt)ωsin(ωt-δ). This integral makes a nasty mess that doesn't get me anything useful. I think I need to simplify this further by approximation, but I don't know which assumptions to make.

    Any help would be greatly appreciated.

    Thank you!
     
  2. jcsd
  3. Sep 18, 2014 #2

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    You are on the right track! Just do the integral. It might help to rewrite the sine in terms of exps!
     
  4. Sep 18, 2014 #3
    Should I expand sine as a series or use the identity sin(z)=[e^(iz)-e^(-iz)]/2i?
     
  5. Sep 18, 2014 #4

    AlephZero

    User Avatar
    Science Advisor
    Homework Helper

    Use the identity sin(z)=[e^(iz)-e^(-iz)]/2i.

    Or better still, take the general solution as the real part of ##Ce^{st}## where ##s = -b + i\omega## and ##C## is a complex constant.
     
  6. Sep 18, 2014 #5


    As a sanity check, you can work the problem again with Q being defined as reactance over resistance. You should get the same answer. :)
     
  7. Sep 18, 2014 #6
    Thank you everyone for the help. I've been working the problem when I have time. When I put sine in the exp form and integrate, it looks terrible. But a further hint from the professor says that I should be looking for ΔE/E. I think that is the key, but I'm still unsure. I'll be able to work it further when I get home. Again, thanks so much for the help.
     
  8. Sep 18, 2014 #7
    I'm still unable to solve the problem. I can't seem to make this integral manageable enough to continue. I attempted to use Wolfram Alpha to integrate before and after putting sine in exp form, but I'm still unable to move forward. I feel like there must be some terms I should be neglecting because of the light damping.
     
  9. Sep 18, 2014 #8
    Try this:

    Determine the energy dissipation per cycle. (I know, this is cheating, but you'll get some great insights!)

    Eric
     
  10. Sep 19, 2014 #9

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    I still don't see, where there might be a problem. You have an integral of the form
    [tex]\int \mathrm{d} t \exp(-a t) \sin^2(b t)[/tex]
    with constants [itex]a[/itex] and [itex]b[/itex]. Now you write
    [tex]\sin(b t)=\frac{\exp(\mathrm{i} b t)-\exp(-\mathrm{i} b t)}{2 \mathrm{i}},[/tex]
    take the square and multiply it out. Then you end with integrals of the form
    [tex]\int \mathrm{d} t \exp(c t),[/tex]
    and over a constant which are really easy to deal with.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Classical Mechanics: Lightly Damped Oscillator Driven Near Resonance
Loading...