Determine the natural frequencies of the system of Chapter Problem 7.89. The beam is of length \(5 \mathrm{~m}\), made of a material of elastic modulus \(200 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\), and has a cross-sectional moment of inertia of \(1.4 \times 10^{-5} \mathrm{~m}^{4}\). The total mass of the beam is \(320 \mathrm{~kg}\). The mass of the winch is \(115 \mathrm{~kg}\). The winch cable is made of a material of elastic modulus \(200 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and has a cross-sectional area of \(3.4 \times 10^{-2} \mathrm{~m}^{2}\). The length of the cable is \(5.5 \mathrm{~m}\) and the mass being lifted is \(715 \mathrm{~kg}\).

Data From Chapter Problem 7.89:

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. 

Transcribed Image Text:

FIGURE P7.89 Beam of mass m, E, I, L Includes mass of winch x2 m