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


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.


Mechanical and Aerospace Engineering

Keywords and Phrases

Linear Systems; Steady State; Mechanisms; Equations; Partial Differential Equations; Dynamics (Mechanics); Separation (Technology); Differential Equations

Document Type

Article - Journal

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© 1995 American Society of Mechanical Engineers (ASME), All rights reserved.