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This study focuses on a class of dynamic behavior observed in spur gear pairs as a parametrically excited system. Stiffness at the gear contact interface that couples the gear bodies is a periodically time-varying due the fluctuation of number of tooth pairs in contact. In presence of such stiffness fluctuation as part of the excitations, various published studies predicted subharmonic resonances near speeds that are integer multiples of the speed at which the primary resonance occurs, with little or no validation. This study presents an extensive experimental investigation to show such subharmonic resonances and various associated period-n motions ( n ≥ 2 ) under both transient and steady-state conditions. A test set-up and a pair of test gears developed for this purpose are described along with solutions to various measurement and data processing challenges. Two different models of the gear pair, a purely torsional discrete model and a deformable-body model, are employed, and their predictions are compared to the measured period-1 and period-n motions to assess their accuracy.

In this study, dynamic behavior of a double-helical gear pair is investigated both experimentally and theoretically. In the experimental side, a new double-helical test set-up consisting of a test machine and test specimens is developed for operating a double-helical gear pair under realistic torque and speed ranges. A test gear pair formed by novel three-piece double-helical gears is developed to allow adjustable right-to-left stagger angles. A measurement system to capture three-dimensional vibratory motions and dynamic motion transmission error under high-speed conditions is implemented. A test matrix that included various combinations of key system parameters is executed under realistic torque values within a wide speed range to establish a database. In view of this experimental data, a linear, time-invariant dynamic model of double-helical gear pair systems including shafts and bearing supports is proposed, also allowing for any stagger angle between the right and left sides of a double-helical gear pair through proper definition of phasing between the two gear mesh excitations. Direct comparisons to measurements are presented to demonstrate the accuracy of the proposed model in predicting three-dimensional gear vibrations. The right-to-left stagger angle is shown to be the most critical parameter impacting dynamic response.

In this study, a tribo-dynamics model for spur gear pairs is proposed. The model couples a mixed elastohydrodynamic lubrication model of a spur gear pair with a transverse–torsional dynamic model. The lubrication model provides the dynamic model with friction forces and moments that couple the vibrations of the gears along the off-line-of-action direction to other gear vibrations. In addition, it predicts damping coefficient at the gear mesh in a physics-based manner from the energy loss associated with viscous shearing across the fluid film. In return, the dynamic model predicts the dynamic tooth forces and surface velocities to be used in the lubrication model. An iterative computational scheme is proposed to implement the lubrication and dynamics models simultaneously to couple tribological and dynamic behaviors of a spur gear pair fully. An example gear pair is analyzed using the proposed model to demonstrate this two-way relationship and quantify the impact of operating conditions, surface roughness and lubrication characteristics on the tribo-dynamics response.

In this study, a nonlinear time-varying dynamic model is proposed to predict modulation sidebands of planetary gear sets. This discrete dynamic model includes periodically time-varying gear mesh stiffnesses and the nonlinearities associated with tooth separations. The model uses forms of gear mesh interface excitations that are amplitude and frequency modulated due to a class of gear manufacturing errors to predict dynamic forces at all sun-planet and ring-planet gear meshes. The predicted gear mesh force spectra are shown to exhibit well-defined modulation sidebands at frequencies associated with the rotational speeds of gears relative to the planet carrier. This model is further combined with a previously developed model that accounts for amplitude modulations due to rotation of the carrier to predict acceleration spectra at a fixed position in the planetary transmission housing. Individual contributions of each gear error in the form of amplitude and frequency modulations are illustrated through an example analysis. Comparisons are made to measured spectra to demonstrate the capability of the model in predicting the sidebands of a planetary gear set with gear manufacturing errors and a rotating carrier.

In this paper, a simplified mathematical model is proposed to describe the mechanisms leading to modulation sidebands of planetary gear sets. The model includes key system parameters such as number of planets, planet position angles, and planet phasing relationships defined by the position angles and the number of teeth of the gears. The model is used to simulate a wide range of gear sets to show that they can be classified in five distinct groups based on their sideband behavior in terms of their frequencies and amplitudes. A special experimental planetary gear set-up is developed and planetary gear sets from of three of these five groups are procured. A methodology is developed to demonstrate modulation sidebands from the ring (internal) gear radial acceleration measurements. For each case, sets of ring gear acceleration measurements at various speed and torque conditions are presented to demonstrate rich sideband activity that agrees well with the model predictions. At the end, based on results of the parametric studies and experiments, general rules on modulation sidebands of planetary gear sets are proposed.

In this study, two different dynamic models, a finite elements-based deformable-body model and a simplified discrete model, and a surface wear model are combined to study the interaction between gear surface wear and gear dynamic response. The proposed dynamic gear wear model includes the influence of worn surface profiles on dynamic tooth forces and transmission error as well as the influence of dynamic tooth forces on wear profiles. This paper first introduces the nonlinear dynamic models that include gear backlash and time-varying gear mesh stiffness, and a wear model separately. It presents a comparison to experiments for validation of the dynamic models. The dynamic models are combined with the wear model to study the interaction of surface wear and dynamic behavior in both linear and nonlinear response regimes. At the end, several sets of simulation results are used to demonstrate the two-way relationship between nonlinear gear dynamics and surface wear.

The subharmonic (period-η, η>1) motions of a piecewise-nonlinear (PN) mechanical oscillator having parametric and external excitations are investigated. The system is formed by a viscously damped, single-degree-of-freedom oscillator subjected to a periodically time-varying, PN stiffness defined by a clearance surrounded by continuous forms of nonlinearity. A multiterm harmonic balance formulation in conjunction with discrete Fourier transforms is used to determine steady-state period-η motions of the system near the parametric instability regions. The accuracy of analytical solutions is verified through a comparison with direct numerical integration results. A parametric study is also presented to demonstrate the combined influence of a clearance and of cubic nonlinearities on period-η motions within typical ranges of system and excitation parameters.

The dynamic behavior of a piecewise-nonlinear mechanical oscillator with parametric and external excitations is investigated. The viscously damped single-degree-of-freedom oscillator is subjected to a periodically time-varying, piecewise-nonlinear restoring function defined by a clearance surrounded by continuously nonlinear (quadratic and cubic) regions. Typical applications represented by this oscillator are highlighted. A multi-term harmonic balance formulation is used in conjunction with discrete Fourier transforms and a parametric continuation scheme to determine steady-state period-1 motions of the system due to both parametric and external excitations. The accuracy of the analytical solutions is demonstrated by comparing them to direct numerical integration solutions. Floquet theory is applied to determine the stability of the steady-state harmonic balance solutions. At the end, detailed parametric studies are presented to quantify the combined influence of clearance, quadratic and cubic nonlinearities within reasonable ranges of all other system parameters. A comparison between time-varying and time-invariant systems is also provided to demonstrate the influence of the parametric and external excitations on a piecewise-nonlinear system.

Planetary gear sets are used commonly by automotive and aerospace industries. Typical applications include jet propulsion systems, rotorcraft transmissions, passenger vehicle automatic transmissions and transfer cases and off-highway vehicle gearboxes. Their high-power-density design combined with their kinematic flexibility in achieving different speed ratios make planetary gears sets often preferable to counter-shaft gear reduction systems. As planetary gear sets possess unique kinematic and geometric properties, they require specialized design knowledge [1]. One type of the key parameters, the rim thickness of the gears, must be defined carefully by the designer in order to meet certain design objectives regarding power density, planet load sharing, noise and durability. From the power density point of view, the rim of the each gear forming the planetary gear set must be as thin as possible in order to minimize mass. Besides reducing mass, added gear flexibility through reduced rim thickness was shown to reduce the influence of a number of internal gear and carrier errors, and piloting inaccuracies [2]. In addition, it was also reported that a flexible internal gear helps improve the load sharing amongst the planets when a number of manufacturing and assembly related gear and carrier errors are present [3–6]. Many of these effects of flexible gear rims were quantified under quasi-static conditions in the absence of any dynamic effects. The effect of rim thickness on gear stresses attracted significant attention in the past. A number of theoretical studies [7–14] modelled mostly a segment of spur gear with a thin rim. In these studies, the gear segment was typically constrained using certain boundary conditions at the cut ends and a point load along the line of action was applied to a single tooth in order to simulate the forces imposed on a sun or an internal gear by the mating planet. This segment of the gear was modelled by using the conventional finite element (FE) method with the same boundary conditions applied in order to simulate the actual support conditions. These models do not

The steady state forced response of a system with clearance subject to commensurate parametric and external forcing citation is investigated analytically by considering a mechanical oscillator with time-varying stiffness and a dead zone type clearance non-linearity. A generalized multiple term harmonic balance procedure is utilized to find the period- η (ηT) solutions with η ≥1. Interactions between periodically time-varying stiffness and forcing functions are characterized in the primary resonance regimes where solutions are mostly of the 1Ttype. Closed form expressions for impact criteria are derived for 1Tsolutions where possible. The effects of system parameters on ηTsubharmonic resonances are examined and experimental results are included which conclusively demonstrate the existence of subharmonic resonances in geared systems.

International audience; A mechanical system exhibiting combined parametric excitation and clearance type non-linearity is examined analytically and experimentally in an effort to explain complex behavior that is commonly observed in the steady state forced response of rotating machines. The specific case of a preloaded mechanical oscillator having a periodically time-varying stiffness function and subject to a symmetric backlash condition is considered. A generalized solution methodology is proposed based on the harmonic balance method. The resulting non-linear algebraic equations are solved by using a direct Newton-Raphson technique, in which a closed form Jacobian matrix is computed using frequency domain methods. Analytical solutions are validated by comparison with numerical integration results and experimental measurements obtained from a gear dynamics test rig.

The paper proposes a reduced-order analytical model of a spur gear pair which consists of two identical spur gears, two identical flexible shafts, and four identical rolling element bearings of a given radial stiffness. The error associated with the undamped eigensolution is estimated by a comparison with a benchmark finite element model.

Frequency response characteristics of a non-linear geared rotor-bearing system with time-varying mesh stiffness k h ( t ) are examined in this paper. First, the single-degree-of-freedom spur gear pair model with backlash is extended to include sinusoidal or periodic mesh stiffness k h ( t ) . Second, a three-degree-of-freedom model with k h ( t ) and clearance non-lineariries associated with gear backlash and rolling element bearings, as excited by the static transmission error e ( t ) under a mean torque load, is developed. The governing equations are solved using digital simulation technique and only the primary resonances are studied. Resonances of the corresponding linear time-varying system associated with parametric and external excitations are identified using the method of multiple scales and digital simulation. Interactions between the mesh stiffness variation and clearance non-linearities have been investigated; a strong interaction between time-varying mesh stiffness k h ( t ) and gear backlash is found, whereas the coupling between k h ( t ) and bearing non-linearities is weak. Finally, our time-varying non-linear formulations yield reasonably good predictions when compared with the benchmark experimental results available in the literature.

The nonlinear frequency response characteristics of a geared rotor-bearing system are examined. A three-DOF dynamic model is developed which includes nonlinearities associated with radial clearances in the radial rolling element bearings and backlash between a spur gear pair; linear time-invariant gear meshing stiffness is assumed. The corresponding linear system problem is also solved, and predicted natural frequencies and modes match with finite element method results. The applicability of both analytical and numerical solution techniques to the multi-DOF nonlinear problem is investigated. Satisfactory agreement is found between the theory presented here and the available experimental data. Nonlinear modal interactions and differences between internal static transmission error excitation and external torque excitation are discussed. In addition, parametric studies are performed to understand the effect of system parameters such as bearing stiffness to gear mesh stiffness ratio, alternating to mean force ratio and radial bearing preload to mean force ratio on the nonlinear dynamic behavior.

The backlash nonlinearity excited primarily by transmission error between spur gear pairs is studied for both external and internal excitations. The digital simulation technique and the method of harmonic balance are used to develop steady state solutions for the internal sinuosidal excitations. The analytic predictions agreed well with available experimental data. Digital simulation is used to observe that at the chaotic and subharmonic resonances may exist in a gear pair depending on the mean or design load, mean to alternating force ratio, damping, and backlash.