When we studied SDOF systems subject to harmonic loading we found that the response due to cosine loading was:

$u(t)=\frac{P_0}{(k-{}^{2}m)}**(cos(t)-cos({}_n))$

Starting from the equation of motion in modal coordinates $G{M}_r{q}_r^{}+G{K}_{r}q=G{F}_{r}cos(t)$ for a cosine loading with $u(0)=u^{}(0)=\{0\},$ show that the solution in modal coordinates is:

$q_r(t)=\frac{G{F}_r}{(G{K}_r-{}^{2}G{M}_r)}**[cos(t)-cos({}_{r}t)]$