A thin disk attached to the end of an elastic beam has three uncoupled modes of vibration. The longitudinal motion, the transverse motion, and the torsional oscillations are all kinematically independent. Calculate the following for the system of Figure P2.10.

(a) The longitudinal stiffness

(b) The transverse stiffness

(c) The torsional stiffness

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11 FIGURE P 2.10 65 cm U r = 10 mm E = 200 10 N/m G = 80 10 N/m