Given the two degree of freedom system in Fig. P4.8, complete the following.

(a) Write the equations of motion in matrix form.
(b) Verify that proportional damping exists.
(c) Determine the roots of the characteristic equation. What do these roots represent?
(d) Determine the two mode shapes (normalize to coordinate \(x_{1}\) ).

Fig. P4.9 Three degree of freedom spring-mass-damper model

Fig. P4.8 Two degree of freedom spring-mass system.

k 3c 3m 2k C x1 m 2c x2 2m k X3