Consider the stepped shaft made of the same material of density \(ho\) and modulus \(E\), shown in Figure 7.45. Determine the characteristic equation for the natural frequencies of the system as it undergoes axial motion. Let \(u_{1}(x, t)\) denote the displacement in the shaft with the fixed end, and \(u_{2}(x, t)\) be the displacement in the shaft with the free end.

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p.E -L- Figure 7.45: Stepped shaft undergoing axial motion.