Damped Harmonic Oscillator Fourier Expansion

[Niner49er52]

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$$\omega^2$$+$$\sum$$16(-1)^n/(n$$\omega$$)^2 cos(n$$\omega$$t)

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($$\omega$$t) 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($$\omega$$t-$$\delta$$)
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:

$$\int$$t^2sin(n$$\omega$$t)dt=-t^2 cos(n$$\omega$$t)/n$$\omega$$ +2t(sin(n$$\omega$$t)/n$$\omega$$)^2 +2cos(n$$\omega$$t)/(n$$\omega$$)^3

$$\int$$t^2cos(n$$\omega$$t)dt=t^2sin(n$$\omega$$t)/n$$\omega$$+2tcos(n$$\omega$$t)/(n$$\omega$$)^2-2sin(n$$\omega$$t)/(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

 

[Niner49er52]

just realized i made some mistakes when posting, wherever there is an omega as an exponent it should not be there! it is just multiplied through