Speaker
Description
Time-varying physical properties of mechanical systems can lead to parametrically excited oscillations. A notable example of this is the time-varying stiffness of gear teeth meshing in the analysis of gearbox dynamics. In gear dynamics, the vibration behavior is a key design criterion, as it can lead to increased wear and noise. A mechanical model with two degrees of freedom can be used to mathematically represent the rotational equation of motion of the gear pair. The torsional vibrations occurring around the rigid body rotation can then be simplified to a single-degree-of-freedom parametric oscillator. Mathematically, this results in a description using the ordinary differential equation known as the Hill equation. In addition to an assessment of the amplitudes of vibrations, an assessment of the system stability is a crucial aspect to consider. In the stability analysis of such a mechanical system, a stability chart, also known as an Ince-Strutt diagram, is 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 a comprehensive understanding of the stability behavior, a range of different methodologies can be employed. In the context of periodic time-variable systems, three methods are particularly prevalent: the perturbation method and the two approaches based on Hill’s as well as on Floquet’s theory. The latter serves as the basis for the methodology presented in this contribution. In general, these stability charts are created by solving the differential equation for a large number of discrete parameter values and their combinations. Their stability statements are then mapped into the Ince-Strutt diagram. However, the stability boundaries separating stable and unstable behavior are of primary interest. For this reason, 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. In numerical analysis, continuation methods represent a class of numerical techniques based on predictor-corrector schemes to compute any kind of parametrized curve. This contribution presents the method in general and conducts parameter studies on the above-mentioned application example. It will be shown that the proposed approach is suitable to generate stability charts for the analyzed parametric oscillator by continuation of stability boundaries.
