The rigid-body dynamic response of most single-degree-of-freedom machine systems are known to be governed by second-order, nonlinear, inhomogeneous, ordinary differential equations with variable coefficients. An earlier effort produced a solution algorithm, by linearizing such an equation and expressing it in the first-order form, to obtain the steady-state rigid-body dynamic response. This paper formulates a “closed-form numerical” algorithm for obtaining this steady-state response. It is demonstrated as leading to significant reductions in computation time. Efficient iterative methods, based on the criterion of balance of energy over the fundamental cycle, are formulated to treat the more practical problems involving known forcing and unknown average speed, and those involving unknown forcing and known average speed. In addition, two numerical procedures are proposed to treat problems involving systems with large coefficients of fluctuations. All methods are shown to be efficient and stable through examples involving a four-stroke reciprocating engine.

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