A gear of mass \(1 \mathrm{~kg}\) is mounted at the center of the shaft in problem 10. Use two elements of equal length to determine how the natural frequencies and mode shapes are affected by the gear.

Data From Problem 10:

Use two elements of equal length to determine the natural frequencies and mode shapes in flexure of a shaft modeled as a simply supported beam. The length of the shaft is \(1 \mathrm{~m}\), \(E=100 \mathrm{GPa}, A=10^{-3} \mathrm{~m}^{2}, I=10^{-7} \mathrm{~m}^{4} ; ho=4000 \mathrm{~kg} / \mathrm{m}^{3}\).