Stability Analysis Using Predictor‐Corrector Continuation to Develop Ince‐Strutt Diagrams for a Nonlinear Parametric Oscillator

ABSTRACT Time‐varying properties of mechanical systems can lead to parametrically excited oscillations. An example is the time‐varying stiffness of gear teeth meshing in the analysis of gearbox dynamics. A mechanical model can be used to represent the rotational motion of the gear pair. The torsional vibrations occurring around the rigid body rotation can be simplified to a single degree of freedom oscillator. In addition to an assessment of the amplitudes of vibrations, an assessment of the system stability is a crucial aspect to consider. For stability analysis of such a mechanical system, stability charts, known as Ince‐Strutt diagrams, are utilized. These graphical representations of stability regions can be created for various combinations of system parameters to predict whether the system will be stable or unstable. In order to gain an understanding of the stability behavior, a range of methodologies can be employed. The Floquet 's theory serves as the basis for the methodology presented in this study. Stability charts are created by solving the differential equation for discrete parameter values and their combinations. Stability statements are then mapped into the Ince‐Strutt diagram. However, the stability boundaries are of primary interest. This contribution introduces an approach to determine the boundaries between stable and unstable regions without the need to calculate parameter values within the region of interest. Expanding the stability analysis to include a continuation method should make this feasible. It will be shown that the proposed approach is suitable to generate stability charts for the analyzed parametric oscillator by tracing stability boundaries.