The twist angle 0 (x, t) of a torsionally vibrating shaft of unit length is determined from: 2 2 820 , 0 < x < 1, t> 0 ae 0 (0,t) = 0, = 0, t> 0 c=1 ae 0(x, 0) = x-x-2, = 0, t> 0 at t=0 Then 0 (x,t) = An cos(a(n + 0.5)t) sin((n+0.5)x) where n=0 1 An S 0 1 S dx dx