Design a minimum-weight tubular section for the shaft shown in Fig. 7.3 to achieve a fundamental frequency of vibration of \(0.5 \mathrm{~Hz}\). Assume \(m_{1}=20 \mathrm{~kg}, m_{2}=50 \mathrm{~kg}, m_{3}=40 \mathrm{~kg}\), \(l_{1}=1 \mathrm{~m}, l_{2}=3 \mathrm{~m}, l_{3}=4 \mathrm{~m}, l_{4}=2 \mathrm{~m}\), and \(E=2.07 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\).

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m m2 FIGURE 7.3 Shaft carrying masses. m3