Using the results of Example 5.1, verify that the mode shapes satisfy the following relations, known as orthogonality relations:

Data From Example 5.1:-

Figure 5.6:-

\(\vec{X}^{(1)^{T}} \vec{X}^{(2)}=0, \quad \vec{X}^{(1)^{T}}[m] \vec{X}^{(2)}=0, \quad \vec{X}^{(1)^{T}}[m] \vec{X}^{(1)}=c_{1}=\) constant \(\vec{X}^{(2)^{T}}[m] \vec{X}^{(2)}=c_{2}=\mathrm{constant}\)
\(\vec{X}^{(1)^{T}}[k] \vec{X}^{(1)}=c_{1} \omega_{1}^{2}, \quad \vec{X}^{(2)^{T}}[k] \vec{X}^{(2)}=c_{2} \omega_{2}^{2}\)

Transcribed Image Text:

Find the natural frequencies and mode shapes of a spring-mass system, shown in Fig. 5.6, which is constrained to move in the vertical direction only. Take n = 1.