A drum of mass \(m\) is being hoisted by an overhead crane as illustrated in Figure P7.89. The crane is modeled as a simply supported beam with a winch at its midspan. The cable connecting the crane to the drum is of stiffness \(k\). Derive the differential equations governing the motion of the system using four degrees of freedom to model the system, three degrees of freedom for the beam and one for the displacement of the load.

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1x1 Beam of mass m, E, I, L Includes mass of winch m FIGURE P7.89 x4 X3