A rigid rod of negligible mass and length \(2 l\) is pivoted at the middle point and is constrained to move in the vertical plane by springs and masses, as shown in Fig. 5.42. Find the natural frequencies and mode shapes of the system.

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elle 2m 0(1) 2m 00000 k m x(t) k FIGURE 5.42 Rigid bar connected to masses and springs.