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Niner49er52
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Homework Statement
A damped harmonic oscillator originally at rest and in its equilibrium position is subject to a periodic driving force over one period by F(t)=-\tau^2+4t^2 for -\tau/2<t<\tau/2 where \tau =n\pi/\omega
a.) Obtain the Fourier expansion of the function in the integral given and show that it is F(t)=-8\pi^2/3\omega^2+\sum16(-1)^n/(n\omega)^2 cos(n\omegat)
b.)Find the steady state response x_{p}(t) to this driving force when the damped harmonic oscillator satisfies the equation: mx"+bx'+kx=F(t) when m=1kg., b=20kg/s, and k=400N/m. This should be an infinite sum.
Homework Equations
For a driven damped harmonic oscillator with a single diving force, F(t)=F_{0}sin(\omegat) the steady state solution has the form:
x_{p}(t)=(F_{0}/m)/\sqrt{4\beta^2\omega^2}+(\omega^2_{0}-\omega^2)^2 sin(\omegat-\delta)
where \delta=tan^{-1}(2\beta\omega/\omega^2_{0}-\omega^2)
Since the force is an infinite series, you should be able to write the response as an infinite series where each term would have a corresponding term to that above.
Useful Integrals:
\intt^2sin(n\omegat)dt=-t^2 cos(n\omegat)/n\omega +2t(sin(n\omegat)/n\omega)^2 +2cos(n\omegat)/(n\omega)^3
\intt^2cos(n\omegat)dt=t^2sin(n\omegat)/n\omega+2tcos(n\omegat)/(n\omega)^2-2sin(n\omegat)/(n\omega)^3
The Attempt at a Solution
it would take way too long and too much confusing stuff to put everything I've done here, but any helpful suggestions would be welcome. i have it for the most part but keep running into small errors that are throwing me off such as a cosine term ending up a sine term and missing some terms that are associated with \tau. part b.) is where I'm having the most trouble
