What form for the particular integral of this 2nd order Diff Eqn

[bugatti79]

Hi Folks,

I have the following forced torsional vibration problem of the form

$\displaystyle J_0 \ddot{\theta}+k_t\theta=\frac{a_0}{2}+\sum_{n=1}^{\infty} (a_n \cos n w t+b_n \sin n w t)$

I assume the solution of the CF is in the form $\theta=A \cos nwt+B\sin nwt$ but I am not sure what to assume for the PI which is a Fourier Harmonic...
Any suggestions?
Regards

 

[da_nang]

In general, in a linear ODE the particular integral of the Fourier series is another Fourier series, except in the case when a summand would result in a linear dependency with any other particular integral or the complementary function. In that case, the specific summand is multiplied by the independent variable until it no longer is linearly dependent (or goes through some other transformation to make it independent).