If the system is damped with modal damping ratios of \(0.3,0.615,0.825,1.395\), and 1.71, and has a forcing 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 \(p_{4}\).

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

(b) Which modes are overdamped and which are underdamped?

(c) What is the constant(s) of proportionality between the damping matrix and the stiffness and mass matrices?

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 \[ \mathbf{P}=\left[\begin{array}{rcrrr} 1.3 & 1.0 & 0.7 & 0.5 & 0.1 \\ 1.8 & 1.5 & 1.0 & 0.4 & -0.3 \\ 2.4 & 0.5 & -0.4 & -0.3 & 0.2 \\ 2.9 & -0.2 & -0.7 & 0.5 & -0.5 \\ 2.0 & -0.15 & 0.2 & -0.6 & 0.4 \end{array}\right] \]