1
$\begingroup$

A forced oscillator can be described by the equation $\ddot{x} + \omega^2x = F(t)$. The solution of this equation will have a free and forced solution, with the free solution being just $x(t) = Ae^{i\omega t + \phi}$ and $A$ and $\phi$ are constants.

I was told that a "slow-start" to the system can "remove" the free oscillation and isolate the forced solution. By "slow-start", I mean $F(t)$ is increased exponentially from zero. I don't understand how this works.

$\endgroup$
4
  • $\begingroup$ @user163104 What I mean is that there will be a complementary component in the general solution, in addition to the particular component. So if there is not forcing, then there will just be the complementary component, the free oscillations. $\endgroup$ Aug 14, 2017 at 17:22
  • $\begingroup$ Apologies, I have not covered this in a while, I am pretty sure this is what you mean : hyperphysics.phy-astr.gsu.edu/hbase/oscdr.html. regarding functions related to transients, and +1 in hopes of you getting an answers $\endgroup$
    – user163104
    Aug 14, 2017 at 18:14
  • $\begingroup$ @user163104 No worries. Yes that's what I meant, and yes adding a damping term would achieve what I want which is to isolate the steady state solution, but unfortunately it's not really impossible for practical reasons. $\endgroup$ Aug 14, 2017 at 19:22
  • $\begingroup$ Glad you said that, I had an answer that relied on the removal of beats because of the short time of damping, lots of mathjax slog saved. Best of luck with it. $\endgroup$
    – user163104
    Aug 14, 2017 at 19:58

1 Answer 1

1
$\begingroup$

I am going to translate your problem into the terminology I am familiar with. Then I will give an example and hope that you can translate it back to your application. When you apply a step function, the signal overshoot its steady state, reverses and undershoots the steady state. This behavior continues for a specified time until the resistance of the circuit dampens the "spurious" oscillations and the circuit reaches steady state. The signal then reverses and the process repeats.
To eliminate (or reduce) the "spurious" oscillations, you can use: higher damping resistance, "slower" application of the signal (not a step function), and filtering.
An example of slowing the signal is to use a "large enough" input capacitor to slow down the leading and trailing edges of the signal.
A mathematical capacitor might be setting F(t) =$ ke^{jw} $ (or something like it).
For filtering, one uses a bandpass filter to eliminate the "spurious" frequency components on the output.
A "mathematical filter" might be one that removes the frequency range of the spurious components.

Hope this helps.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.