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and the bending and torsion deformation is represented in terms of a truncated series, such that

;=l where Nw and Ng are the numbers of assumed modes used to represent bending and torsion, respectively; r/, and </>,- are the generalized coordinates associated with bending and torsion, respectively; and and 0, are the assumed mode shapes for bending and torsion, respectively. Determine the potential energy in terms of the generalized coordinates using as assumed modes the uncoupled, clamped-free, free-vibration modes of torsion and bending. For torsion

sin for bending, according to Eq. (2.251), is given as

',• = cosh(a;J) - cos(a/J) - jS/[sinh(Qi/y) - sin(«,y)]

with a, and fr as given in Table 2.1.

12. Rework Problem 11, but for assumed modes, instead of using the expressions given therein, use

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