________________________________________________________________________ PROBLEM (12.102) A circular shaft of cross-sectional area A and length L with a helical gear (of weight W and radius of gyration r) at left end, is supported by bearings in the pump housing (Fig. P12.102). When turning at a speed ( ω rad/s) the shaft is accidentally stopped because of jamming on a bearing. Verify that the maximum angle of twist φmax and maximum shearing stress produced by the impact in the shaft are φmax = 2L Uk GJ G Uk AL τ max = 2 where U k is the total kinetic energy ( 2ω r 2 2 (P12.101) 2g ) SOLUTION GJ φmax 1 π c4 J= T φmax = U k , where T = L 2 2 Thus, 1 GJ φmax ( )φmax = U k L 2 Therefore 2L Uk φmax = Q.E.D. GJ Similarly, TL τ max J L τ max L φmax = = = GJ c GJ cG Substitute this into Eq. (1): 2 τ max L2 2 LU k = c 2G 2 GJ or 2c 2G 2c 2G (2) 2 Uk = U τ max = LJ π c4 L k Solving 4G Uk τ max = Q.E.D. AL (1) Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.