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1. Quasi-Periodic Motions of a Gear Transmission System With Piecewise Linearities by the Incremental Harmonic Balance Method With Two Timescales

   https://doi.org/10.1115/1.4067802  

   Liao, F L; Huang, J L; Zhu, W D (June 2025, Journal of Vibration and Acoustics)

   Abstract Quasi-periodic motions can be numerically found in piecewise-linear systems, however, their characteristics have not been well understood. To illustrate this, an incremental harmonic balance (IHB) method with two timescales is extended in this work to analyze quasi-periodic motions of a non-smooth dynamic system, i.e., a gear transmission system with piecewise linearity stiffness. The gear transmission system is simplified to a four degree-of-freedom nonlinear dynamic model by using a lumped mass method. Nonlinear governing equations of the gear transmission system are formulated by utilizing the Newton’s second law. The IHB method with two timescales applicable to piecewise-linear systems is employed to examine quasi-periodic motions of the gear transmission system whose Fourier spectra display uniformly spaced sideband frequencies around carrier frequencies. The Floquet theory is extended to analyze quasi-periodic solutions of piecewise-linear systems based on introduction of a small perturbation on a steady-state quasi-periodic solution of the gear transmission system with piecewise linearities. Comparison with numerical results calculated using the fourth-order Runge-Kutta method confirms that excellent accuracy of the IHB method with two timescales can be achieved with an appropriate number of harmonic terms. 

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2. Mesh Stiffness Calculation and Error Analysis: Hypoid Gear Pair and System

   https://doi.org/10.1115/1.4069852  

   Wei, Xinqi; Wang, Yawen; Wang, Shuo; Lim, Teik C (December 2025, Journal of Vibration and Acoustics)

   Abstract Accurate mesh stiffness calculation is crucial for developing reliable dynamic models and analyzing vibratory behavior in hypoid gears. Current studies often use a hypoid gear pair, incorporating the misalignment effects of the hypoid gear system model, as a substitute for the full hypoid gear system model to reduce simulation costs. However, this simplification overlooks the boundary problem, particularly the influence of housing on the hypoid gear system. This discrepancy can lead to deviations in mesh stiffness calculation and affect the accuracy of dynamic response predictions. To address this issue, we established a three-dimensional static mesh model to calculate the mesh stiffness under different boundary conditions, i.e., the gear pair model constrained at the base and the gear system model constrained at the housing, based on finite element results. Then, we introduce a 14-degree-of-freedom dynamic model to examine the influence of mesh stiffness differences on system dynamics. Finally, a numerical case study evaluates key factors, including unloaded and loaded transmission error, mesh points, line of action, and static mesh force, to assess their impact on mesh stiffness and the resultant impact on dynamic behavior. The findings provide insights into the selection of an appropriate calculation model for accurate gear design and simulation. 

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3. Study on Dynamic Behaviors of Hypoid Gears Under Variable Tidal Current Energy Harvesting Conditions

   https://doi.org/10.3390/machines13030178  

   Huang, Dequan; Li, Yan; Zheng, Xingyuan; Li, Gang (March 2025, Machines)

   This study investigates dynamic behaviors of hypoid gear rotor systems under variable tidal current energy harvesting conditions through numerical simulations and experimental validation. The study examines dynamic responses of a hypoid gear rotor system induced by cyclical tidal current variations, which generate fluctuating loads and bidirectional rotational speeds in tidal energy conversion systems. Two hypoid gear pairs, modified through precise manufacturing parameters, are evaluated to optimize tooth contact patterns for bidirectional tidal loading conditions. A coupled torsional vibration model is developed, incorporating variable transmission error and mesh stiffness. Experimental validation of dynamic performances of hypoid gear pairs was conducted on a bevel gear testing rig, which can measure both torsional and translational vibrations across diverse tidal speed profiles. The experimental results demonstrate that second-order primary resonances exhibit heightened vibration intensity during flow-reversal phases. This phenomenon has significant implications for system power efficiency and acoustic emissions. The findings extend the current understanding of hypoid gear optimization for tidal energy-harvesting applications. 

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4. Qualitative changes in bifurcation structure for soft vs hard impact models of a vibro-impact energy harvester

   https://doi.org/10.1063/5.0101050  

   Costa, Dimitri; Kuske, Rachel; Yurchenko, Daniil (October 2022, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   Vibro-impact phenomena in engineering systems, considered an adverse effect in some settings, are an intrinsic part of the mechanism in others. In energy harvesting, a vibro-impact component is often intentionally introduced to increase the power output or the system’s bandwidth. The impacts can be treated as “hard” for instantaneous impacts or “soft” for compliant materials. Since both types of models exhibit complex dynamics, a comparison is non-trivial. We develop a soft impact model for a vibro-impact energy harvester, calibrating it with the relevant hard impact model for large stiffness, and systematically compare the different phenomena and dynamics in various compliant regimes. Numerical results are used in two different parametric analyses, considering the bifurcation diagrams in terms of device size and external forcing parameters. Varying the natural frequency of the membranes that form the impact boundaries, we observe shifts in the bifurcation structure that promote period-1 orbits for increased softness parameters, often generating higher power output, but also introducing parameter sensitivities for increased softness. Complementary analytical results reveal unstable periodic orbits and co-existing behaviors, potentially missed by computational methods, that can influence the bifurcation structure and in turn the energy output. A non-dimensional formulation highlights the significance of ratios of external and natural frequencies in delineating soft and hard impact scenarios parametrically. The soft impact model exhibits new symmetry breaking bifurcations related to key quantities that characterize the soft impact dynamics, such as the effective restitution coefficients, the impact phase, and the contact time interval, not captured by hard impact models. 

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5. Nonlinear dynamics of a heaving spar-floater system integrated with inerter pendulum vibration absorber power take-off for wave energy conversion

   https://doi.org/10.1016/j.ymssp.2023.111088  

   Gupta, Aakash; Tai, Wei-Che (March 2024, Mechanical Systems and Signal Processing)

   A power take-off based on the inerter pendulum vibration absorber (called IPVA-PTO) is integrated with a spar-floater system to study its hydrodynamic response suppression and wave energy conversion capabilities in regular waves. The hydrodynamics of the spar-floater system is computed using the boundary element method with linear wave theory. With the wave height and wave frequency as the bifurcation parameters, it is found that the system can undergo two bifurcations: period-doubling bifurcation around the first resonance frequency (spar mode) and secondary Hopf bifurcation around the second resonance frequency (floater mode). The period-doubling bifurcation results in an energy transfer between the spar-floater system and the IPVA-PTO for small electrical damping values. As a result, the IPVA-PTO system simultaneously reduces the maximum response amplitude operator (RAO) of the spar and increases the normalized capture width in comparison with the optimal linear benchmark. Experiments performed on a ‘‘dry’’ single-degree-of-freedom system integrated with the IPVAPTO where the base excitation is substituted for the wave excitation verify the simultaneous performance enhancement due to the period-doubling bifurcation. The system performance beyond the period-doubling bifurcation is also experimentally investigated. On the other hand, as the wave height approaches and passes the secondary Hopf bifurcation, the pendulum responses transition from primary harmonic responses to quasi-periodic responses to rotations. When the rotations occur, the IPVA-PTO system increases the maximum normalized capture width threefold to fivefold compared with the optimal linear benchmark, yet slightly increases the RAO around the second resonance frequency. Nevertheless, the RAO remains smaller than the global maximum RAO of the optimal linear benchmark. Finally, parametric studies are performed to study the effects of parameters on the bifurcations. It is observed that by varying the electrical damping, the wave height required for achieving the period-doubling bifurcation can be changed significantly, which can be exploited to stabilize the spar. 

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