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.
Steady-State Response of Periodically Time-Varying Linear Systems, With Application to an Elastic Mechanism
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Farhang, K., and Midha, A. (December 1, 1995). "Steady-State Response of Periodically Time-Varying Linear Systems, With Application to an Elastic Mechanism." ASME. J. Mech. Des. December 1995; 117(4): 633–639. https://doi.org/10.1115/1.2826732
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