The 2019 work of Liu, Hong and Parker established that when a spur gear pair is supported on rolling element bearings whose stiffness varies periodically with roller circumferential position (varying-compliance excitation), simultaneous time-varying bearing and tooth mesh stiffness produce additional parametric instabilities and sideband modulation, but the aperiodic and chaotic vibration regimes of this system fell outside that analysis. The works citing the source paper (2020–2026) and the adjacent bifurcation/chaos literature show that later studies either keep the analysis in the periodic-response regime (parametric resonances, jump and saddle-node bifurcations) even when time-varying bearing stiffness is modeled, or analyze chaos with time-varying mesh stiffness while bearings remain constant-stiffness or clearance elements, or treat varying-compliance-induced chaos in the bearing alone without gear mesh coupling. The open problem therefore survives intact: characterize in detail the aperiodic and chaotic regimes of the coupled gear–bearing system, their dependence on initial conditions and structural parameters, and effective parameter-selection strategies to suppress or avoid chaotic behavior.

Background

In a spur gear transmission, the tooth mesh stiffness (the effective stiffness of the contacting tooth pair) varies periodically within each mesh cycle because the number of tooth pairs in contact and the contact positions change as the gears rotate; this time-varying mesh stiffness (TVMS) acts as a parametric excitation on the gear system — a periodic variation of the system's own stiffness coefficients rather than of the external forcing, which can destabilize the response even under constant load. A large classical literature, beginning with Kahraman and Singh (1990), shows that a gear pair with TVMS and nonlinearities such as backlash (the clearance between mating teeth) exhibits rich nonlinear dynamics, including period-doubling cascades, subharmonic and chaotic responses; comparable chaotic behavior has been documented for gear pairs on journal bearings (Chang-Jian and Hsu, 2012) and for planetary gear trains (Li, Wu and Zhang, 2014). In all of this classical work, the rolling element bearings supporting the gear shafts were modeled with constant stiffness, or with oil-film/clearance nonlinearities, so that the bearing contributed no parametric excitation of its own.

Rolling element bearings, however, possess an intrinsic parametric excitation: as the rolling elements orbit with the cage, the number of loaded elements and their circumferential positions change continuously, so the bearing support stiffness varies periodically. This is called varying compliance (VC), and its excitation frequency is the roller (ball) pass frequency, i.e., the number of rolling elements times the cage rotation frequency. VC excitation is known to cause parametric resonances, and in bearing–rotor systems by themselves it can produce period-doubling and chaos, which Zhang, Rui, Yang and Chen (2020) showed can be suppressed by introducing additional stiffness.

Liu, Hong and Parker (2019, Nonlinear Dynamics 97:1403–1424) combined these two parametric excitation sources in one system. They derived the time-varying stiffness of the roller bearings supporting a spur gear pair from a finite element/contact mechanics model of the isolated bearing, built analytical and finite element gear–bearing models with constant and time-varying bearing stiffness, and showed that the time-varying bearing stiffness causes additional resonances (parametric instabilities) and modulation between the mesh frequency and the bearing ball pass frequency in the dynamic response. They explained the parametric instability regions of the system with simultaneous mesh and bearing stiffness fluctuations using perturbation analysis, verified with Floquet theory (a method for determining the stability of systems whose coefficients vary periodically with time). Their analysis, however, concentrated on the parametric instability and periodic response structure; the aperiodic and chaotic vibration regimes of this gear–bearing system, the dependence of these nonlinear dynamics on initial conditions and system structural parameters, and effective strategies to suppress or avoid chaotic behavior through parameter selection were not characterized in that work. Later work has elaborated the modeling side — Wang and Zhu (2023) added bearing load-dependence and varying-compliance excitation to a gear transmission model but studied nonlinear periodic motion states and parametric resonant regions; Yang, Zhang et al. (2023) incorporated time-varying nonlinear ball-bearing support stiffness into a planetary gear set model but analyzed dynamic responses and load sharing rather than chaos; Zhang, Qu et al. (2025) integrated time-varying bearing stiffness and speed-dependent dynamic mesh stiffness into a 6-degree-of-freedom nonlinear gear model with backlash but analyzed vibration amplitudes and their sensitivity to rolling element count and bearing clearance; the Liao–Huang–Zhu group (2025, 2026) analyzed nonlinear periodic responses, resonances and saddle-node bifurcations of gear models with time-varying mesh stiffness, extended tooth contact and time-varying supporting stiffness — but none of these works characterized the chaotic regimes of the coupled system. Chaotic gear–bearing studies that did appear (e.g., Liu, Cui et al. 2025 on a gear–shaft–bearing system with tooth backlash and TVMS, and Liu, Cui and Wang 2026 with fractal mesh stiffness) all retain time-invariant bearing stiffness, so the bearing varying-compliance excitation identified by Liu, Hong and Parker remains outside the chaotic-regime literature. The unresolved question is thus the natural nonlinear successor to the 2019 study: what happens to this doubly parametrically excited, nonlinear gear–bearing system beyond its periodic responses.

Problem Statement

For the spur gear pair supported on rolling element bearings studied by Liu, Hong and Parker (2019) — a gear–bearing system in which the bearing support stiffness varies periodically with roller circumferential position (varying-compliance excitation at the roller pass frequency) simultaneously with the time-varying tooth mesh stiffness — characterize and control the nonlinear dynamics beyond the periodic regime. Throughout, 'aperiodic' response means non-repeating (non-periodic) response, which includes quasiperiodic motion (response combining incommensurate frequencies) as well as chaotic motion, and 'chaotic' means aperiodic response with sensitive dependence on initial conditions, i.e., nearby trajectories separating exponentially fast on average, as quantified by a positive largest Lyapunov exponent. Specifically: (1) explore in detail the aperiodic and chaotic vibration regimes of this system, including their bifurcation structure (the sequence of qualitative changes in the long-time response type — periodic, quasiperiodic, or chaotic — as a control parameter is varied) and the routes by which regular responses lose stability into aperiodic and chaotic motion as operating conditions (e.g., speed and load) vary; (2) establish how these nonlinear dynamics depend on initial conditions (e.g., coexisting attractors — distinct long-time response types reachable from different starts — and their basins of attraction, the sets of initial conditions converging to each attractor) and on the system's structural parameters (e.g., bearing radial clearance, preload, number of rolling elements, damping, and the fluctuation amplitudes of mesh and bearing stiffness); and (3) determine effective strategies to suppress or avoid the chaotic behavior through parameter selection, i.e., identify structural-parameter choices that keep the system in regular bounded operation over its intended operating range. These three parts serve one overarching objective — understanding and controlling the nonlinear dynamics of the simultaneous mesh- and bearing-stiffness parametric excitation in this gear–bearing system, in which both parametric excitation sources are simultaneously present; a model in which either excitation source is removed or frozen is a different system, not the stated one.

References

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