Steady-State Response of Periodically Time-Varying Linear Systems, With Application to an Elastic Mechanism

[+] Author and Article Information
K. Farhang

Department of Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901

A. Midha

Elastic Mechanisms Laboratory, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907

J. Mech. Des 117(4), 633-639 (Dec 01, 1995) (7 pages) doi:10.1115/1.2826732 History: Received March 01, 1994; Revised December 01, 1994; Online December 11, 2007


This paper presents the development of an efficient and direct method for evaluating the steady-state response of periodically time-varying linear systems. The method is general, and its efficacy is demonstrated in its application to a high-speed elastic mechanism. The dynamics of a mechanism comprised of elastic members may be described by a system of coupled, inhomogeneous, nonlinear, second-order partial differential equations with periodically time-varying coefficients. More often than not, these governing equations may be linearized and, facilitated by separation of time and space variables, reduced to a system of linear ordinary differential equations with variable coefficients. Closed-form, numerical expressions for response are derived by dividing the fundamental time period of solution into subintervals, and establishing an equal number of continuity constraints at the intermediate time nodes, and a single periodicity constraint at the end time nodes of the period. The symbolic solution of these constraint equations yields the closed-form numerical expression for the response. The method is exemplified by its application to problems involving a slider-crank mechanism with an elastic coupler link.

Copyright © 1995 by The American Society of Mechanical Engineers
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