The ability to accurately predict the response of rotating machinery to external forces and to assess system-level stability for different modes is crucial from a reliability and preventive maintenance perspective. Geared systems, in particular, contain many complexities, which may lead to instability and even chaotic vibration behavior. No methods for determining the effects of the dynamic meshing forces on the vibrations of complete shaft/bearing systems have been proposed in the literature. Several time-transient and steady-state models for analyzing gear forces and deflections have been proposed, but they focus primarily on the dynamics of the gearbox itself and neglect vibration transmission through the remainder of the drive-train. Models that do incorporate other components of the drive-train propose simplified lumped-parameter models for the shafts and bearings. Recent models have used the finite element method to couple the lateral, torsional, and axial degrees-of-freedom of geared shaft systems to the forces and moments exchanged between the gears via stiffness matrices. Other models in literature capture the backlash non-linearity and the state-varying mesh stiffness and observe the time-transient response of the gearbox and simplified shaft/bearing structure. A finite element formulation of complete geared systems, which couples the axial, lateral, and torsional degrees-of-freedom, is developed in which the shaft is modeled with Timoshenko beam elements and captures the forces and moments due to gyroscopic effects, and rotational accelerations due to start-up. It includes the capability of modeling non-linear contact loss due to backlash clearance and parametric excitations resulting from the state-varying mesh stiffness and solves the time-transient state equations for the displacements and velocities of the shafts using the direct Runge-Kutta method.

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