Question

Torsional vibration of a shaft is governed by the wave equation, = 16 where (x,t) is the angular displacement (angle of twist) along the shaft, ar is the distance from the end of the shaft and t is time. For a shaft of length 2T that supported by frictionless b end, the boundary conditions are 0r(0,t) = 0x(2T, t) = 0, t> 0. Suppose that the initial angular displacement and angular velocity are (x,0) = 6 cos(x), Ot(x,0) =3+2 cos(42), 0 2, respectively You may use the result that the eigenvalues of the boundary-value problem x'(0) = X'(2m) = 0 X"-AX 0, are n = 0,1,2,3,... 22 with corresponding eigenfunctions Xn cos Use the method of separation of variables to find the final solution 0(x, t). (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.) 0(x,t)=

Answer 1

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(0H 3++ 4(t) 16 T-16AT -P u(*+) -2 IAn ces( 2n t)t Ba Sih ( 2ntリ, CoS A2 -6 3-t 2 Cus(4 Br (16) 2

(t6t). Cos4x) 6 Cost ut). Cosx