A Study on State-Dependent Time-Transient Rotor Dynamics of Spur and Helical Geared Rotating Machinery

Predictions of the response of rotating machinery to external forces and assessments of system-level stability for different modes are crucial from a reliability and preventative maintenance perspective. Geared systems, in particular, contain many complexities such that the use of extensive computational effort is required to achieve accurate modeling. Sources of dynamic complexity at the gear mesh include non-linear tooth contact loss due to backlash clearance and parametric excitations from state-varying mesh stiffness effects. Although methods for determining the effects of dynamic meshing forces on the vibrations of rotor-bearing systems are in the literature, the models are either overly simplistic or require immense computational effort. Several time-transient and steady-state models for analyzing gear forces and deflections have been proposed, but those authors have focused primarily on the dynamics of the gearbox instead of vibration transmission through the remainder of the drive-train.

More recent models have used the finite element method to couple the lateral, torsional, and axial motions of the gear and pinion to the mesh forces and moments via element stiffness matrices. A finite element formulation of complete rotor-bearing systems, which couples the axial, lateral, and torsional degrees-of-freedom of geared shafts, is developed in this dissertation. The shaft structure is modeled with linear Timoshenko beam elements, and the non-linear gear mesh forces and moments incorporate effects from gyroscopic moments, shaft rotational speed variations, and includes models for parametric excitations from contact loss due to backlash clearance and state-induced mesh stiffness variations. Time-transient state equations for the displacements and velocities of the shafts are solved using the direct Runge-Kutta method, and the methods are applied to three different geared machines. A parametric study investigating the sensitivity of shaft vibration to several sources of excitation is included and the results yield additional insights into proper modeling techniques.