A steel shaft (left(ho=7850 mathrm{~kg} / mathrm{m}^{3}, G=85 times 10^{9} mathrm{~N} / mathrm{m}^{2} ight)) with a inner
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A steel shaft \(\left(ho=7850 \mathrm{~kg} / \mathrm{m}^{3}, G=85 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\right)\) with a inner radius of \(30 \mathrm{~mm}\), outer radius of \(50 \mathrm{~mm}\), and length of \(1.0 \mathrm{~m}\) is fixed at both ends. Determine the three lowest natural frequencies of the shaft.
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