The normal modes of a two-degree-of-freedom system are orthogonal if \(X^{(1)^{T}}[m] \vec{X}^{(2)}=0\). Prove that the mode shapes of the system shown in Fig. 5.5 (a) are orthogonal.

Figure 5.5:-

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k1 00000 m1 kx1 m1 C1x1 Spring k under tension for +1 fi(t) x1(t) k2 (a) -X1, X1 k2(x-x1) C2(x-x1) Spring k under tension for +(x-x1) (b) m2 -x2(t) -2(t) k3 -x2, x2 -f2 K3x2 m2 C3X2 Spring k under compression for +x2 FIGURE 5.5 A two-degree-of-freedom spring-mass-damper system.