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.     

A dynamic model to study the influence of planet position errors in planetary gears

An original lumped parameter model of planetary gears is presented which accounts for planet position errors and simulates their contribution to the quasi-static and dynamic load sharing amongst the planets. A unique feature of the model is that instantaneous gear geometry is used which depends on the combination of deflections and errors. The numerical results compare well with experimental evidence from the literature thus validating the modelling methodology. Simulations are then extended to dynamic tooth loads and trajectories for both fixed and rotating carriers. Finally, the advantages and drawbacks of floating members in high-speed planetary gears are illustrated and commented upon.     

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.     

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 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.     

Vertical dynamic response of non-uniform motion of high-speed rails

In this paper, a computational study using the moving element method (MEM) is carried out to investigate the dynamic response of a high-speed rail (HSR) traveling at non-uniform speeds. A new and exact formulation for calculating the generalized mass, damping and stiffness matrices of the moving element is proposed. Two wheel–rail contact models are examined. One is linear and the other nonlinear. A parametric study is carried out to understand the effects of various factors on the dynamic amplification factor (DAF) in contact force between the wheel and rail such as the amplitude of acceleration/deceleration of the train, the severity of railhead roughness and the wheel load. Resonance in the vibration response can possibly occur at various stages of the journey of the HSR when the speed of the train matches the resonance speed. As to be expected, the DAF in contact force peaks when resonance occurs. The effects of the severity of railhead roughness and the wheel load on the occurrence of the jumping wheel phenomenon, which occurs when there is a momentary loss of contact between the wheel and track, are investigated.     

An adaptive hierarchical domain decomposition method for parallel contact dynamics simulations of granular materials

A fully parallel version of the contact dynamics (CD) method is presented in this paper. For large enough systems, 100% efficiency has been demonstrated for up to 256 processors using a hierarchical domain decomposition with dynamic load balancing. The iterative scheme to calculate the contact forces is left domain-wise sequential, with data exchange after each iteration step, which ensures its stability. The number of additional iterations required for convergence by the partially parallel updates at the domain boundaries becomes negligible with increasing number of particles, which allows for an effective parallelization. Compared to the sequential implementation, we found no influence of the parallelization on simulation results.     

Three-dimensional steady state Green function for a layered isotropic plate

The elastodynamic response of a layered isotropic plate to a source point load having an arbitrary direction is studied in this paper. A decomposition technique is developed within each homogeneous isotropic lamina to simplify the general three-dimensional plane-wave propagation problem as a separate plane-strain problem and an anti-plane-wave propagation problem. The accuracy of computation is assured by cross-checking the numerical results by different methods. Results are checked numerically for a vertical point load acting on a homogeneous and a layered plate by using a hybrid method. On the other hand, results are checked for a horizontal point load by using dynamic reciprocal identities. Results are presented for both a homogeneous as well as a layered plate.     

Experiments and numerical results on non-linear vibrations of an impacting Hertzian contact. Part 2: random excitation

Non-linear dynamic behaviour of a normally excited preloaded Hertzian contact (including possible contact losses) is investigated using an experimental test rig. It consists of a double sphere plane contact loaded by the weight of a rigid moving mass. Contact vibrations are generated by a external Gaussian white noise and exhibit vibroimpact responses when the input level is sufficiently high. Spectral contents and statistics of the stationary transmitted normal force are analyzed. A single-degree-of-freedom non-linear oscillator including loss of contact and Hertzian non-linearities is built for modelling the experimental system. Theoretical responses are obtained by using the stationary Fokker-Planck equation and also Monte Carlo simulations. When contact loss occurrence is very occasional, numerical results show a very good agreement with experimental ones. When vibroimpacts occur, results remain in reasonable agreement with experimental ones which justify the modelling and the numerical methods described in this paper.The contact loss non-linearity appears to be rather strong compared to the Hertzian non-linearity. It actually induces a large broadening of the spectral contents of the response. This result is of great importance in noise generation for many systems such as mechanisms using contacts to transform motions and forces (gears, ball-bearings, cam systems, to name a few). It is also of great importance for tribologists preoccupied with preventing surface damage.     

Dynamic photoelastic studies of wave propagation in granular media

The optical method of dynamic photoelasticity is used to visualize the load transfer profiles due to explosive loading in granular aggregates. The granular media are simulated by using arrays of disks fabricated from a brittle polyester material—Homalite 100. Attention is focused on the effect of microstructure or the geometrical packing of the grains on the wave propagation phenomena. The experimental data are analyzed to obtain the stress wave attenuation, wave velocities and contact stresses as a function of time along various directions in the assembly of grains. Dynamic load transfer functions are developed to predict dynamic contact loads in any systematic or random assembly of grains for any given loading. The predicted results are compared with the experimental data. The effect of local inhomogeneities on the wave propagation phenomena is also shown.     

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.     

Nonlinear ultrasonic imaging method for closed cracks using subtraction of responses at different external loads

To improve the selectivity of closed cracks for objects other than cracks in ultrasonic imaging, we propose an extension of a novel imaging method, namely, subharmonic phased array for crack evaluation (SPACE) as well as another approach using the subtraction of responses at different external loads. By applying external static or dynamic loads to closed cracks, the contact state in the cracks varies, resulting in an intensity change of responses at cracks. In contrast, objects other than cracks are independent of external load. Therefore, only cracks can be extracted by subtracting responses at different loads. In this study, we performed fundamental experiments on a closed fatigue crack formed in an aluminum alloy compact tension (CT) specimen using the proposed method. We examined the static load dependence of SPACE images and the dynamic load dependence of linear phased array (PA) images by simulating the external loads with a servohydraulic fatigue testing machine. By subtracting the images at different external loads, we show that this method is useful in extracting only the intensity change of responses related to closed cracks, while canceling the responses of objects other than cracks.     

Dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading

The dynamic instability characteristics of stiffened plates subjected to in-plane partial and concentrated edge loadings are studied using finite element analysis. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The method of Hill's infinite determinants is applied to determine the dynamic instability regions. Numerical results are presented to study the effects of various parameters, such as static load factor, aspect ratio, boundary conditions, stiffening scheme and load parameters on the principal instability regions of stiffened plates using Bolotin's method. The results show that location, size and number of stiffeners have a significant effect on the location of the boundaries of the principal instability region.     

DYNAMIC STABILITY OF CURVED PANELS WITH CUTOUTS

The parametric instability behaviour of curved panels with cutouts subjected to in-plane static and periodic compressive edge loadings are studied using finite element analysis. The first order shear deformation theory is used to model the curved panels, considering the effects of transverse shear deformation and rotary inertia. The theory used is the extension of dynamic, shear deformable theory according to Sanders' first approximation for doubly curved shells, which can be reduced to Love's and Donnell's theories by means of tracers. The effects of static and dynamic load factors, geometry, boundary conditions and the cutout parameters on the principal instability regions of curved panels with cutouts are studied in detail using Bolotin's method. Quantitative results are presented to show the effects of shell geometry and load parameters on the stability boundaries. Results for plates are also presented as special cases and are compared with those available in the literature.     

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.     

Nonlinear damping calculation in cylindrical gear dynamic modeling

The nonlinear dynamic problem posed by cylindrical gear systems has been extensively covered in the literature. Nonetheless, a significant proportion of the mechanisms involved in damping generation remains to be investigated and described. The main objective of this study is to contribute to this task. Overall, damping is assumed to consist of three sources: surrounding element contribution, hysteresis of the teeth, and oil squeeze damping. The first two contributions are considered to be commensurate with the supported load; for its part however, squeeze damping is formulated using expressions developed from the Reynolds equation. A lubricated impact analysis between the teeth is introduced in this study for the minimum film thickness calculation during contact losses. The dynamic transmission error (DTE) obtained from the final model showed close agreement with experimental measurements available in the literature. The nonlinear damping ratio calculated at different mesh frequencies and torque amplitudes presented average values between 5.3 percent and 8 percent, which is comparable to the constant 8 percent ratio used in published numerical simulations of an equivalent gear pair. A close analysis of the oil squeeze damping evidenced the inverse relationship between this damping effect and the applied load.     

A dynamic Lagrangian frequency-time method for the vibration of dry-friction-damped systems

A new frequency-time domain procedure, the dynamic Lagrangian mixed frequency-time method (DLFT), is proposed to calculate the non-linear steady state response to periodic excitation of structural systems subject to dry friction damping. In this formulation, the dynamic Lagrangians are defined as the non-linear contact forces obtained from the equations of motion in the frequency domain, with the adjunction of a penalization on the difference between the interface displacements calculate by the non-linear solver in the frequency domain and those calculated in the time domain from the non-linear contact forces, thus accounting for Coulomb friction and non-penetration conditions. The dynamic Lagrangians allow one to solve for the non-linear forces between two points in contact without using artifacts such as springs. The new DLFT method is thus particularly well suited to handling finite element models of structures in frictional contact, as it does not require a special model for the contact interface. Dynamic Lagrangians are also better suited to frequency-domain friction problems than the traditional time-domain method of augmented Lagrangians. Furthermore, a reduction of the non-linear system to relative interface displacements is introduced to decrease the computation time. The DLFT method is validated for a beam in contact with a flexible dry friction element connected to ground, for frictional constraints that feature two-dimensional relative motion. Results are also obtained for a large-scale structural system with a large number of one-dimensional dry-friction dampers. The DLFT method is shown to be accurate and fast, and it does not suffer from convergence problems, at least in the examples studied.     

Identification and updating of loading in frameworks using dynamic measurements

This paper is concerned with the effect of structural loading on dynamic performance. This topic is recognised as being of importance when validating finite element (FE) models with experimental data. A strategy for including axial load effects in a model updating procedure is developed. The method can be used to identify loading in structural frameworks using measured dynamic data.The effectiveness of the new method is demonstrated by means of case studies involving both simulated and experimental data. The theoretical study allows aspects of the sensitivity of the method to realistic levels of experimental noise to be studied as well as the way in which dynamic load identification can be enhanced with static measurements. The experimental case study proves the practical success of the technique. Updated axial load parameters are compared with static measurements of the same quantities.