A Computationally Efficient Numerical Algorithm for the Transient Response of High-Speed Elastic Linkages

[+] Author and Article Information
A. Midha

Dept. of Mechanical Engineering, The Pennsylvania State University, University Park, Pa. 16802

A. G. Erdman

Department of Mechanical Engineering, University of Minn.

D. A. Frohrib

Design and Control Division, Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minn.

J. Mech. Des 101(1), 138-148 (Jan 01, 1979) (11 pages) doi:10.1115/1.3454012 History: Received June 22, 1978; Online October 21, 2010


A new numerical algorithm, easily adaptable for computer simulation, is developed to approximate the transient response of a single degree-of-freedom vibrating system; governing differential equation is linear and second order with time-dependent and periodic coefficients. This is accomplished by first solving the classical linear single degree-of-freedom problem with constant coefficients. The system is excited by a periodic forcing function possessing a certain degree of smoothness. The integration terms in the solution are systematically expanded into two groups of terms: one consists of non-integral terms while the other contains only integral terms. The final integral terms are bounded. For certain combinations of frequency and damping, within the sub-resonant frequency range, the relative size of the integral terms are demonstrated to be small. The algebraic expansion (non-integral) terms then approximate the solution. The solution to a single degree-of-freedom system with time-dependent and periodic parameters is made possible by discretizing the forcing period into a number of intervals and assuming the system parameters as constant over each interval. The numerical algorithm is then employed to solve an elastic linkage problem via modal superposition. Convergence of the solution is verified by refining the number of intervals of discretization.

Copyright © 1979 by ASME
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