A Closed-Form Numerical Algorithm for the Periodic Response of High-Speed Elastic Linkages

Abstract

A numerical closed-form algorithm, easily adaptable for computer simulation, is developed to solve for the periodic solutions of vibrating systems, and in particular, the high-speed elastic linkage. The algorithm is first introduced to solve the single degree-of-freedom mass-dashpot-spring system, the governing differential equation of which is a linear, second-order equation with constant coefficients. This algorithm is utilized as a basic tool and extended to solve a single degree-of-freedom mass-dashpot-spring system whose governing differential equation of motion is a linear, second-order equation with time-dependent and periodic coefficients. The system is excited by a periodic forcing function and solution is made possible by discretizing the forcing time period into a number of time intervals, the system parameters remaining constant over the duration of each interval. During each interval, the solution form is assumed to be that of the differential equation with “constant” coefficients. Constraint equations result from imposing the conditions of “compatibility” of response at the discrete time nodes and the conditions of “periodicity” of response at the end nodes of the time period. Also, the sum of the integration required is over one forcing time period only. This closed-form nature of the computational procedure results in large savings in computer time to acquire the periodic solution. The suggested numerical algorithm is then employed to solve an elastic linkage problem.