Numerical study on reducing the vibration of spur gear pairs with phasing

A new method of reducing gear vibration was analyzed using a simple spur gear pair with phasing. This new method is based on reducing the variation in mesh stiffness by adding another pair of gears with half-pitch phasing. This reduces the variation in the mesh stiffness of the final (phasing) gear, because each gear compensates for the variation in the other's mesh stiffness. A single gear pair model with a time-varying rectangular-type mesh stiffness function and backlash was used, and the dynamic response over a wide range of speeds was obtained by numerical integration. Because of the reduced variation in mesh stiffness and the double frequency, the phasing gear greatly reduced the dynamic response and nonlinear behavior of the normal gears. The results of the analysis indicate the possibility of reducing vibration of spur gear pairs using the proposed method.     

Three-dimensional nonlinear vibration of gear pairs

This work investigates the three-dimensional nonlinear vibration of gear pairs where the nonlinearity is due to portions of gear teeth contact lines losing contact (partial contact loss). The gear contact model tracks partial contact loss using a discretized stiffness network. The nonlinear dynamic response is obtained using the discretized stiffness network, but it is interpreted and discussed with reference to a lumped-parameter gear mesh model named the equivalent stiffness representation. It consists of a translational stiffness acting at a changing center of stiffness location (two parameters) and a twist stiffness. These four parameters, calculated from the dynamic response, change as the gears vibrate, and tracking their behavior as a post-processing tool illuminates the nonlinear gear response. There is a gear mesh twist mode where the twist stiffness is active in addition to the well-known mesh deflection mode where the translational stiffness is active. The twist mode is excited by periodic back and forth axial movement of the center of stiffness in helical gears. The same effect can occur in wide facewidth spur gears if tooth lead modifications or other factors such as shaft and bearing deflections disrupt symmetry about the axial centers of the mating teeth. Resonances of both modes are shown to be nonlinear due to partial and total contact loss. Comparing the numerical results with gear vibration experiments from the literature verifies the model and confirms partial contact loss nonlinearity in experiments.     

Dynamic modelling of spur gear pair and application of empirical mode decomposition-based statistical analysis for early detection of localized tooth defect

Gears are one of the most common and important machine components in many advanced machines. An improved understanding of vibration signal is required for the early detection of incipient gear failure to achieve high reliability. This paper mainly consists of two parts: in the first part, a 6-degree-of-freedom gear dynamic model including localized tooth defect has been developed. The model consists of a spur gear pair, two shafts, two inertias representing load and prime mover and bearings. The model incorporates the effects of time-varying mesh stiffness and damping, backlash, excitation due to gear errors and profile modifications. The second part consists of signal processing of simulated and experimental signals. Empirical mode decomposition (EMD) is a method of breaking down a signal without leaving a time domain. The process is useful for analysing non-stationary and nonlinear signals. EMD decomposes a signal into some individual, nearly monocomponent signals, named as intrinsic mode function (IMF). Crest factor and kurtosis have been calculated of these IMFs. EMD pre-processed kurtosis and crest factor give early detection of pitting as compared to raw signal.     

Random dynamics of a nonlinear spur gear pair in probabilistic domain

This paper investigates the response of a spur gear pair subjected to both deterministic and random loads. Backlash nonlinearity and time-varying mesh stiffness in gear systems are considered in the model. Path integration is adopted to capture the random response in probabilistic domain. In the path integration algorithm, the transition probability density function (PDF) within a short time interval is assumed as Gaussian. Then the mean and variance of the responses are calculated and expressed as closed forms for two different cases in gear systems, which are further used to construct the transition PDF. The simulation results are compared with that from Monte Carlo (MC) simulation and deterministic numerical integration. Good agreements are shown between these results. In addition, the multi-solutions feature characterizing the nonlinear gear system is also captured.     

Dynamic simulation of planetary gear set with flexible spur ring gear

Ring gear is a key element for vibration transmission and noise radiation in the planetary gear system which has been widely employed in different areas, such as wind turbine transmissions. Its flexibility has a great influence on the mesh stiffness of internal gear pair and the dynamic response of the planetary gear system, especially for the thin ring cases. In this paper, the flexibility of the internal ring gear is considered based on the uniformly curved Timoshenko beam theory. The ring deformation is coupled into the mesh stiffness model, which enables the investigation on the effects of the ring flexibility on the mesh stiffness and the dynamic responses of the planetary gear. A method about how to synthesize the total mesh stiffness of the internal gear pairs in multi-tooth region together with the ring deformation and the tooth errors is proposed. Numerical results demonstrate that the ring thickness has a great impact on the shape and magnitude of the mesh stiffness of the internal gear pair. It is noted that the dynamic responses of the planetary gear set with equally spaced supports for the ring gear are modulated due to the cyclic variation of the mesh stiffness resulted from the presence of the supports, which adds more complexity in the frequency structure.     

Nonlinear parametrically excited vibration and active control of gear pair system with time-varying characteristic

In the present work, we investigate the nonlinear parametrically excited vibration and active control of a gear pair system involving backlash, time-varying meshing stiffness and static transmission error. Firstly, a gear pair model is established in a strongly nonlinear form, and its nonlinear vibration characteristics are systematically investigated through different approaches. Several complicated phenomena such as period doubling bifurcation, anti period doubling bifurcation and chaos can be observed under the internal parametric excitation. Then, an active compensation controller is designed to suppress the vibration, including the chaos. Finally, the effectiveness of the proposed controller is verified numerically.     

A method to determine dynamic loads on spur gear teeth and on bearings

A method is described which can be used to calculate dynamic gear tooth force and bearing forces. The model includes elastic bearings. The gear mesh stiffness and the path of contact are determined using the deformations of the gears and the bearings. This gives contact outside the plane-of-action and a time-varying working pressure angle. In a numerical example it is found that the only important vibration mode for the gear contact is the one where the gear tooth deformation is dominant. The bearing force variation, however, will be much more affected by the other vibration modes. The influence of the friction force is also studied. The friction has no dynamic influence on the gear contact force or on the bearing force in the gear mesh line-of-action direction. On the other hand, the changing of sliding directions in the pitch point is a source for critical oscillations of the bearings in the gear tooth frictional direction. These bearing force oscillations in the frictional direction appear unaffected by the dynamic response along the gear mesh line-of-action direction.     

Research on dynamics and fault mechanism of spur gear pair with spalling defect

This study focuses on the nonlinear dynamic and vibration characteristics of spur gear pair with local spalling defect to explore the spalling mechanism. The dynamic model of the gear pair with spalling defect and time-variant mesh stiffness is established to investigate the effect of spalling defect on mesh stiffness and dynamic response. The analytical solutions of the system which is deduced into four different stages of the gear with the time-variant stiffness in a mesh period are obtained. The dynamic responses with the evolvement of sapll are analyzed by using time history, phase contrail, Poincaré section and spectrum analysis. The spalling characteristics are also evaluated by employing statistical techniques, which shows that the spalling failure is suitable to be detected under low velocity and small excitation. The gearbox with spalling defect is designed and the experiments are carried out to get the dynamic characteristics of the spalling vibration signals. The results obtained herein show the good agreement qualitatively with the theoretical analysis, which provides a theoretical basis to spalling fault diagnosis of gearbox.     

PLANETARY GEAR PARAMETRIC INSTABILITY CAUSED BY MESH STIFFNESS VARIATION

Parametric instability is investigated for planetary gears where fluctuating stiffness results from the changing contact conditions at the multiple tooth meshes. The time-varying mesh stiffnesses of the sun-planet and ring-planet meshes are modelled as rectangular waveforms with different contact ratios and mesh phasing. The operating conditions leading to parametric instability are analytically identified. Using the well-defined properties of planetary gear vibration modes, the boundaries separating stable and unstable conditions are obtained as simple expressions in terms of mesh parameters. These expressions allow one to suppress particular instabilities by adjusting the contact ratios and mesh phasing. Tooth separation from parametric instability is numerically simulated to show the strong impact of this non-linearity on the response.     

Nonlinear dynamics of planetary gears using analytical and finite element models

Vibration-induced gear noise and dynamic loads remain key concerns in many transmission applications that use planetary gears. Tooth separations at large vibrations introduce nonlinearity in geared systems. The present work examines the complex, nonlinear dynamic behavior of spur planetary gears using two models: (i) a lumped-parameter model, and (ii) a finite element model. The two-dimensional (2D) lumped-parameter model represents the gears as lumped inertias, the gear meshes as nonlinear springs with tooth contact loss and periodically varying stiffness due to changing tooth contact conditions, and the supports as linear springs. The 2D finite element model is developed from a unique finite element-contact analysis solver specialized for gear dynamics. Mesh stiffness variation excitation, corner contact, and gear tooth contact loss are all intrinsically considered in the finite element analysis. The dynamics of planetary gears show a rich spectrum of nonlinear phenomena. Nonlinear jumps, chaotic motions, and period-doubling bifurcations occur when the mesh frequency or any of its higher harmonics are near a natural frequency of the system. Responses from the dynamic analysis using analytical and finite element models are successfully compared qualitatively and quantitatively. These comparisons validate the effectiveness of the lumped-parameter model to simulate the dynamics of planetary gears. Mesh phasing rules to suppress rotational and translational vibrations in planetary gears are valid even when nonlinearity from tooth contact loss occurs. These mesh phasing rules, however, are not valid in the chaotic and period-doubling regions.     

Identification and control of chaos in nonlinear gear dynamic systems using Melnikov analysis

In this paper, the Melnikov analysis is extended to develop a practical model of gear system to control and eliminate the chaotic behavior. To this end, a nonlinear dynamic model of a spur gear pair with backlash, time-varying stiffness and static transmission error is established. Based on the Melnikov analysis the global homoclinic bifurcation and transition to chaos in this model are predicted. Then non-feedback control method is used to eliminate the chaos by applying an additional control excitation. The regions of the parameter space for the control excitation are obtained analytically. The accuracy of the theoretical predictions and also the performance of the proposed control system are verified by the comparison with the numerical simulations. The simulation results show effectiveness of the proposed control system and present some useful information to analyze and control the gear dynamical systems.     

Determination of time-varying contact length,friction force,torque and forces at the bearings in a helical gear system

This paper deals with determining various time-varying parameters that are instrumental in introducing noise and vibration in a helical gear system. The most important parameter is the contact line variation, which subsequently induces friction force variation, frictional torque variation and variation in the forces at the bearings. The contact line variation will also give rise to gear mesh stiffness and damping variations. All these parameters are simulated for a defect-free and two defective cases of a helical gear system. The defective cases include one tooth missing and two teeth missing in the helical gear. The algorithm formulated in this paper is found to be simple and effective in determining the time-varying parameters.     

A dynamic model to predict modulation sidebands of a planetary gear set having manufacturing errors

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.     

Global behavior of gear system using mixed cell mapping

In some mechanical nonlinear systems, the transient motion will be undergoing a very long process and the attractor-basin boundaries are so complicated that some difficulties occur in analyzing the system global behavior. To solve this problem a mixed cell mapping method based on the point mapping and the principle of simple cell mapping is developed. The algorithm of the mixed cell mapping is studied. A dynamic model of a gear pair is established with the backlash, damping, transmission error and the time-varying stiffness taken into consideration. The global behaviors of this system are analyzed. The coexistence of the system attractors and the respective attractor-basin of each attractor with different parameters are obtained, thus laying a theoretical basis for improvement of the dynamic behaviors of gear system.     

Dynamic analysis for a pair of spur gears with translational motion due to bearing deformation

In this study, the dynamic response of a pair of spur gears is analyzed when the gear set has translational motion due to bearing deformation. A new dynamic model for the gear set, considering translational motion, is proposed, in which the distance between the centers of a pinion and a gear varies with time. Therefore, the proposed model regards the pressure angle and the contact ratio as time-varying variables, while the previous model regards them as constants. After deriving nonlinear equations of motion for the gear set, the dynamic responses are computed by applying the Newmark time integration method. This paper claims that the new model produces more accurate dynamic responses in comparison to those of the previous model. Some dynamic response differences between the new and previous models are demonstrated, and the effects of damping and stiffness upon the dynamic responses are also investigated.     

Global behavior of gear system using mixed cell mapping

In some mechanical nonlinear systems, the transient motion will be undergoing a very long process and the attractor-basin boundaries are so complicated that some difficulties occur in analyzing the system global behavior. To solve this problem a mixed cell mapping method based on the point mapping and the principle of simple celll mapping is developed. The algorithm of the mixed cell mapping is studied. A dynamic model of a gear pair is established with the backlash, damping, transmission error and the time-varying stiffness taken into consideration. The global behaviors of this system are analyzed. The coexistence of the system attractors and the respective attractor-basin of each attractor with different parameters are obtained, thus laying a theoretical basis for improvement of the dynamic behaviors of gear system.

     

CAN: an example of nonclassical acoustic nonlinearity in solids

A new class of nonlinear acoustic phenomena has been observed for acoustic wave interaction with simulated and realistic nonbonded contact interfaces (cracked defects) in solids. "Nonclassical" effects are due to substantially asymmetric stiffness characteristics of the interface for normal stress that results in specific contact acoustic nonlinearity (CAN). The asymmetry in the contact restoring forces causes the stiffness parametric modulation and instability of oscillations, which results in acoustic wave fractional subharmonic generation. The CAN subharmonics and higher harmonics reveal threshold dynamic behaviour, evident hysteresis, and instability effects.     

Analytical investigation on unstable vibrations of single-mesh gear systems with velocity-modulated time-varying stiffness

Time-varying mesh stiffness parametrically excites gear systems and causes severe vibrations and instabilities. Taking speed fluctuations into account, the time-varying mesh stiffness is frequency modulated, and more complex instabilities might arise. Considering two different speed fluctuation models, parametric instability associated with velocity-modulated time-varying stiffness is analytically investigated using a typical single-mesh gear system model. Closed-form approximations are obtained by perturbation analysis, and verified by numerical analysis. The effects of the amplitude of the mesh stiffness variation, the characteristics of speed fluctuations and damping on parametric instability are systematically examined.     

Dynamic analysis for a planetary gear with time-varying pressure angles and contact ratios

In this study, the dynamic responses of a planetary gear are analyzed when component gears have time-varying pressure angles and contact ratios caused by bearing deformations. For this purpose, this study proposes a new dynamic model of the planetary gear, in which the pressure angles and contact ratios change with time. The main difference from previous studies is that the present study regards the pressure angles and contact ratios as time-varying variables, while previous studies regarded them as constants. After nonlinear equations of motion for the planetary gear are derived, the dynamic responses are computed by applying the Newmark time integration method. The time responses for the present and previous studies are compared to show the effects of the time-varying pressure angles and contact ratios on the dynamic behaviors of a planetary gear. In addition, the effects of bearing stiffness on the pressure angles and contact ratios are also analyzed.     

Experimental and numerical analysis of automotive gearbox rattle noise

The aim of this work is to characterize the rattle noise of automotive gearboxes, resulting from impacts between toothed wheels of unselected gear ratios. These stereo-mechanical impacts are modeled by a coefficient of restitution which describes damping during the squeezing of the lubricant film for approaching surfaces, and the elastic deformation of impacting bodies. The dynamic response of the loose gear first depends on the design parameters and the engine operating conditions. The unknown parameters are the drag torque and the coefficient of restitution. They are identified experimentally through implementation of two optical encoders in an actual automotive gearbox and the operation of a specific test bench which replicates the automotive power train. Models of the different drag torque sources are validated from analysis of the free damped response of the driveline. The coefficient of restitution and its probability density function are measured from experiments under stationary operating conditions. A nonlinear model is built. The dynamic response of the loose gear depends on the dimensionless backlash, the coefficient of restitution and a dimensionless parameter proposed to describe the rattle excitation level. Experiments under controlled excitation are performed to validate the assumptions, to confirm the ability of the parameter proposed to describe the rattle noise threshold, and to characterize the dynamic response. The nonlinear model predictions are fitted with the drag torque and coefficient of restitution previously identified. They are compared with measurements to demonstrate the ability of the model to predict gear rattle for any loose gear, any gearbox and any operating condition.