Question

Torsional vibration of a shaft is govened by the wave equation, dr2 dr2 f twist) along the shaft, x is the distance from the end of the shaft and t is time. For a shaft of length 4x that is supported by frictionless bearings at each end, the boundary conditions are where (x, f) is the angular displacement (angle t 0. 0x (0, )= 0x(4, )= 0, Suppose that the initial angular displacement and angular velocity are x, 0) 2 cos (2x), 0(x, 0) 66 cos(x), 0x 4 respectively. You may use the result that the eigenvalues of the boundary-value problem x'(0) = x'( 4 ) = 0 x"-Ax 0, are An= 2 n0,1,2,3,... with corresponding eigenfunctions cos of variables to find the final solution ex, t). Use the method of separation (It is possible to get half-marks for this question for a good but incomplete solution. However, half marks will still show as wrong with "How did I do?", so check your mark too.) ex, )

Answer 1

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