An overhead traveling crane can be modeled as indicated in Fig. 5.28. Assuming that the girder has a span of \(40 \mathrm{~m}\), an area moment of inertia \((I)\) of \(0.02 \mathrm{~m}^{4}\), and a modulus of elasticity \((E)\) of \(2.06 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\), the trolley has a mass \(\left(m_{1}\right)\) of \(1000 \mathrm{~kg}\), the load being lifted has a mass \(\left(m_{2}\right)\) of \(5000 \mathrm{~kg}\), and the cable through which the mass \(\left(m_{2}\right)\) is lifted has a stiffness \((k)\) of \(3.0 \times 10^{5} \mathrm{~N} / \mathrm{m}\), determine the natural frequencies and mode shapes of the system.

Figure 5.28:-

Transcribed Image Text:

Girder; kg, E, I Trolley m1 Wire m2 k rope Load lifted 0000 m1 m2 (b) (a) FIGURE 5.28 Overhead traveling crane. k