Consider the longitudinally vibrating uniform beam shown in Figure 7.49. The ends of the beam are supported by linear springs with stiffnesses \(k_{1}\) and \(k_{2}\). Derive the frequency equation, the first three natural frequencies, and plot the first three mode shapes for axial vibration. Use \(k_{1} L / E A=0.5\) and \(k_{2} L / E A=2\).

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m, EA, L Figure 7.49: Axially vibrating beam supported by springs at both ends.