System of non-homogeneous second order DE’s

For my M.Sc. thesis I'm modeling the response of a bending beam to excitation by an elastically attached point mass with an initial velocity.

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I try to approximate the solution by a summation of eigenmotions, so the motion of the beam is given by:

$w(x,t)= \sum\limits_{n=1}^\infty W_n(x)*p_n(t)$

I have been able to determine the eigenfunctions $W_n$ using the boundary conditions. That leaves the initial value problem for the time functions $p_n$. For example a 3 mode gives following set of DE's:

$\ddot{p}_1+\omega^2p_1=F_1\left(p_1W_1(a)+p_2W_2(a)+p_3W_3(a)-z(t)\right)$ $\ddot{p}_2+\omega^2p_2=F_2\left(p_1W_1(a)+p_2W_2(a)+p_3W_3(a)-z(t)\right)$ $\ddot{p}_3+\omega^2p_3=F_3\left(p_1W_1(a)+p_2W_2(a)+p_3W_3(a)-z(t)\right)$

$M\ddot{z}+k\left(z-p_1W_1(a)-p_2W_2(a)-p_3W_3(a)\right)=0$

IC's:

$w(x,0)=w'(x,0)=z(0)=0$

$z'(0)=v_0$

Tackling this problem gives me some trouble, as I'm not really sure how to begin. If someone could give me some tips, it would be much appreciated.

The problem description above is pretty condensed, I can imagine some things are unclear. Please let me know where further explanation would be helpful.

Thanks and regards, Erik

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