If the system is undamped and subject to a force vector equal to \(\mathbf{F}=\left[\begin{array}{lllll}0 & 0 & \sin 54 t & 0 & 0\end{array}\right]^{T}\), determine the following.

(a) Write the differential equation for the first principal coordinate.

(b) What is the steady-state solution of this differential equation?

(c) Which mode do you expect will have the largest contribution to the response?

(d) What is the relation between the fifth generalized coordinate and the principal coordinates?

Spectral analysis shows that the natural frequencies for a fifthorder system are \(20 \mathrm{rad} / \mathrm{s}, 41 \mathrm{rad} / \mathrm{s}, 55 \mathrm{rad} / \mathrm{s}, 93 \mathrm{rad} / \mathrm{s}\), and \(114 \mathrm{rad} / \mathrm{s}\). Experimental modal analysis is used to determine that its modal matrix is \[ \mathbf{P}=\left[\begin{array}{rcrrr} 1.3 & 1.0 & 0.7 & 0.5 & 0.1 \\ 1.8 & 1.5 & 1.0 & 0.4 & -0.3 \\ 2.4 & 0.5 & -0.4 & -0.3 & 0.2 \\ 2.9 & -0.2 & -0.7 & 0.5 & -0.5 \\ 2.0 & -0.15 & 0.2 & -0.6 & 0.4 \end{array}\right] \]