Analytical and numerical methods for vibration analyses by Jong-Shyong Wu

By Jong-Shyong Wu

''This publication illustrates theories and linked mathematical expressions with numerical examples utilizing a number of tools, resulting in specific recommendations, extra exact effects, and extra computationally effective innovations. It provides the derivations of the equations of movement for all constitution foundations utilizing both the continual version or the discrete version. It discusses purposes for college students taking classes together with vibration mechanics, dynamics of constructions, and finite aspect analyses of buildings, the move matrix strategy, and Jacobi method''

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It is noted that the overdot represents differentiation with respect to time t. , the work done by the external loads respectively, while W per unit time). (c) The Lagrangian equation The equations of motion of a vibrating system may also be determined from the following Lagrangian equations:   d @T @T @V À þ ¼ F i ði ¼ 1; 2; . . 2), while qi , q_ i and F i are the ith generalized displacement, generalized velocity and generalized external load, respectively, t is time and n is the total number of degrees of freedom (DOFs) of the vibrating system.

1Þ sffiffiffiffi sffiffiffiffiffiffiffiffi G G ðr ¼ 0; 1; 2; 3; . . ; 1Þ v r ¼ br ¼ rp r rL2 Qr ðxÞ ¼ Ar cosðrpx=LÞ ðr ¼ 0; 1; 2; 3; . . ; 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 0 ðxÞ ¼ 1=ðrI p LÞ ðr ¼ 0Þ Q ~ r ðxÞ ¼ Q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=ðrI p LÞ cosðrpx=LÞ ðr ¼ 1; 2; 3; . . 88): br L ¼ rp br ¼ rp=L ðr ¼ 1; 2; 3; . . ; 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi vr ¼ br G=r ¼ rp G=ðrL2 Þ ðr ¼ 1; 2; 3; . . ; 1Þ or Qr ðxÞ ¼ Br sin br x ðr ¼ 1; 2; 3; . . ; 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ r ðxÞ ¼ 2=ðrI p LÞ sinðrpx=LÞ ðr ¼ 1; 2; 3; . . 89): rp ðr ¼ 1; 3; 5; .

H. (2001) Torsional vibration analysis of gear-branched systems by finite element method. Journal of Sound and Vibration, 240 (1), 159–182. J. (1965) Axial shaft vibration in large turbine-powered merchant ships. Transactions of the Institute of Marine Engineers, 77, 53–113. R. T. (1949) Longitudinal vibrations of marine propulsion shafting systems. Transactions of the Society of Naval Architects and Marine Engineers (SNAME), 57, 193–252. Panagopulos, E. (1950) Design-stage calculation of torsional, axial and lateral vibrations of marine shafting.

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