Method for Calculating the Steady-State Dynamic Response of Rigid-Body Machine Systems

The rigid-body equations of motion for conservative rotating machine systems with position-dependent moments of inertia are found to reduce to a single, second-order, inhomogeneous, nonlinear, ordinary differential equation with variable coefficients. Upon linearization this equation is reduced to first-order form. A rational proportionality between the periods of the variable coefficient and the in-homogeneous term implies that the steady-state rigid-body response will also be periodic. To solve for the steady-state rigid-body response the least common period of the system is divided into an appropriate number of sub-intervals, and the solution over each sub-interval is derived by assuming a constant value of the coefficient during that sub-interval. The final solution is computed by applying appropriate compatiblity and periodicity constraints. The solution algorithm is extended to systems for which the linearization assumptions do not apply through the application of a recursion scheme. Examples are included to illustrate the utility of the algorithm.