Use a two-element, finite-element model to approximate the two lowest torsional natural frequencies for the system of

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Use a two-element, finite-element model to approximate the two lowest torsional natural frequencies for the system of Figure P11.14.

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\[ \begin{aligned} & J=3.2 \times 10^{-5} \mathrm{~m}^{4} \\ & G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2} \\ & ho=7000 \mathrm{~kg} / \mathrm{m}^{3} \\ & L=4.2 \mathrm{~m} \\ & I=1.5 \mathrm{~kg} \cdot \mathrm{m}^{2} \end{aligned} \]

FIGURE P11.14

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