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.     

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.     

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.     

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.     

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.     

Effect of tooth mesh stiffness asymmetric nonlinearity for drive and coast sides on hypoid gear dynamics

A nonlinear time-varying dynamic model of a hypoid gear pair system with time-dependent nonlinear mesh stiffness, mesh damping and backlash properties is formulated to study the effect of mesh stiffness asymmetry for drive and coast sides on dynamic response. The asymmetric characteristic is the result of the inherent curvilinear tooth form and pinion offset in hypoid set. Using the proposed nonlinear time-varying dynamic model, effects of asymmetric mesh stiffness parameters that include mean mesh stiffness ratio, mesh stiffness variation and mesh stiffness phase angle on the dynamic mesh force response and tooth impact regions are examined systematically. Specifically, the dynamic models with only asymmetric mesh stiffness nonlinearity, with only backlash nonlinearity and with both asymmetric mesh stiffness and backlash nonlinearities are analyzed and compared. Using the parameters of a typical hypoid gear set, the extent of the effect of asymmetry in the mesh coupling on gear pair dynamics is quantified numerically. The results show that the increase in the mean mesh stiffness ratio tends to worsen the dynamic response amplitude, and the mesh stiffness parameters for drive side have more effect on dynamic response than those of the coast side one.     

Vibro-acoustic propagation of gear dynamics in a gear-bearing-housing system

This work developed a computational process to predict noise radiation from gearboxes. It developed a system-level vibro-acoustic model of an actual gearbox, including gears, bearings, shafts, and housing structure, and compared the results to experiments. The meshing action of gear teeth causes vibrations to propagate through shafts and bearings to the housing radiating noise. The vibration excitation from the gear mesh and the system response were predicted using finite element and lumped-parameter models. From these results, the radiated noise was calculated using a boundary element model of the housing. Experimental vibration and noise measurements from the gearbox confirmed the computational predictions. The developed tool was used to investigate the influence of standard rolling element and modified journal bearings on gearbox radiated noise.     

Dynamic characteristics of the herringbone planetary gear set during the variable speed process

In this study, a dynamic model for herringbone planetary gears is proposed which can be applied in the dynamic analysis of variable speed processes (including acceleration, deceleration, and large speed fluctuation process, etc.). The dynamic responses of the acceleration process of an example of a herringbone planetary gear set are simulated in cases where the profile error excitations are ignored and included. The phenomenon of tooth separations can be observed as the rotating speed increases in the simulation, and the effect of the profile error excitations on the phenomenon is also investigated. Furthermore, the effects of the profile error excitations on the vibrations and dynamic meshing forces are investigated before and after the appearance of tooth separations. Moreover, the dynamic characteristics of the herringbone planetary gear set are also compared with that of the spur/helical herringbone planetary gear set briefly. Finally, some advice for the design of planetary gear sets is given to avoid the phenomena of tooth separation and tooth back contacts and suppress the vibrations and dynamic meshing forces.     

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.     

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.     

A dynamic model to determine vibrations in involute helical gears

A method to determine the dynamic load between two rotating elastic helical gears is presented. The stiffness of the gear teeth is calculated using the finite element method and includes the contribution from the elliptic distributed tooth load. To make sure that the new incoming contacts which are the main excitation source are properly simulated, the necessary deformation of the gears is determined by using the true geometry and positions of the gears for every time step of the dynamic calculation. This allows the contact to be positioned outside the plane of action. A numerical example is presented with figures that show the behaviour of the dynamic transmission error as well as the variation of the contact pressure due to the dynamic load for different rotational speeds.     

A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic

A vibration isolator consisting of a vertical linear spring and two nonlinear pre-stressed oblique springs is considered in this paper. The system has both geometrical and physical nonlinearity. Firstly, a static analysis is carried out. The softening parameter leading to quasi-zero dynamic stiffness at the equilibrium position is obtained as a function of the initial geometry, pre-stress and the stiffness of the springs. The optimal combination of the system parameters is found that maximises the displacement from the equilibrium position when the prescribed stiffness is equal to that of the vertical spring alone. It also satisfies the condition that the dynamic stiffness only changes slightly in the neighbourhood of the static equilibrium position. For these values, a dynamical analysis of the isolator under asymmetric excitation is performed to quantify the undesirable effects of the nonlinearities. It includes considering the possibilities of the appearance of period-doubling bifurcation and its development into chaotic motion. For this purpose, approximate analytical methods and numerical simulations accompanied with qualitative methods including phase plane plots, Poincaré maps and Lyapunov exponents are used. Finally, the frequency at which the first period-doubling bifurcation appears is found and the effect of damping on this frequency determined.     

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.     

Passive reduction of gear mesh vibration using a periodic drive shaft

In this paper, a passive approach to reduce transmitted vibration generated by gear mesh contact dynamics is presented. The approach utilizes the property of periodic structural components that creates stop band and pass band regions in the frequency spectra. The stop band regions can be tailored to correspond to regions of the frequency spectra that contain harmonics and sub-harmonics of the gear mesh frequency, attenuating the response in those regions. A periodic structural component is comprised of a repeating array of cells, which are themselves an assembly of elements. The elements may have differing material properties as well as geometric variations. For the purpose of this research, only geometric variations are considered and each cell is assumed to be identical. A periodic shaft is designed and machined in order to reduce transmitted vibration of a pair of spur gears. Analytical and experimental results indicate that transmitted vibrations from gear mesh contact to the bearing supports are reduced at a variety of operational speeds under static torque preload.     

Modelling dynamic behaviour and noise generation in gear pumps: Procedure and validation

The paper presents a methodology for noise and vibration analysis of gear pumps and its application to an external gear pump for automotive applications. The methodology addresses the use of a combined numerical model and experimental analyses. The combined model includes a lumped-parameter model, a finite-element model and a boundary-element model. The lumped-parameter (LP) model regards the interior parts of the pump (bearing blocks and gears loaded by the pressure distribution and the driving torque), the finite element (FE) model regards the external parts of the pump (casing and end plates), while the boundary element (BE) model enables the estimation of the emitted noise in operational conditions. Based on experimental evidences, attention has been devoted to the modelling of the pump lubricant oil: the fluid–structure interaction between the oil and pump casing was taken into account. In the case of gear pumps all these important effects have to be considered in the same model in order to take their interactions into account. The model has been assessed using experiments: the experimental accelerations and acoustic pressure measured in operational conditions have been compared with the simulated data coming from the combined LP/FE/BE model. The combined model can be considered a very useful tool for design optimisation.     

Chaotic vibrations of a vocal fold model with a unilateral polyp

A nonlinear model was proposed to study chaotic vibrations of vocal folds with a unilateral vocal polyp. The model study found that the vocal polyp affected glottal closure and caused aperiodic vocal fold vibrations. Using nonlinear dynamic methods, aperiodic vibrations of the vocal fold model with a polyp were attributed to low-dimensional chaos. Bifurcation diagrams showed that vocal polyp size, stiffness, and damping had important effects on vocal fold vibrations. An increase in polyp size tended to induce subharmonic patterns and chaos. This study provides a theoretical basis to model aperiodic vibrations of vocal folds with a laryngeal mass.     

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.     

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.