A three degree-of-freedom system with viscous damping that is proportional to the stiffness matrix has natural frequencies of \(10 \mathrm{rad} / \mathrm{s}, 25 \mathrm{rad} / \mathrm{s}\), and \(50 \mathrm{rad} / \mathrm{s}\). The modal damping ratio for the first mode is 0.1 .

(a) What are the modal damping ratios for the higher modes?

(b) Write the differential equations satisfied by the principal coordinates for free vibrations of the system.