A machine tool, having a mass of \(m=1000 \mathrm{~kg}\) and a mass moment of inertia of \(J_{0}=300 \mathrm{~kg}-\mathrm{m}^{2}\), is supported on elastic supports, as shown in Fig. 5.27. If the stiffnesses of the supports are given by \(k_{1}=3000 \mathrm{~N} / \mathrm{mm}\) and \(k_{2}=2000 \mathrm{~N} / \mathrm{mm}\), and the supports are located at \(l_{1}=0.5 \mathrm{~m}\) and \(l_{2}=0.8 \mathrm{~m}\), find the natural frequencies and mode shapes of the machine tool.

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01 41 m 02 12 m2 y2 -x2- FIGURE 5.25 Double pendulum.