Tooth Separations Research Articles

Dynamic analysis of gear pairs with the effects of stick-slip

The instantaneous dynamic contact state analysis is carried out to reveal the process of scuffing failure of the gear tooth pair. A stick-slip dynamic model of a two-gear set is proposed and the coupling effects of time-varying mesh stiffness, tooth separations, friction between the gear teeth surfaces, and potential stick-slip are considered. Dynamic analysis shows that stick contact is an important source of tooth scuffing failure. Additionally, stick contact dramatically increases the vibration amplitudes and causes chaos. Parametric studies show that heavy load and rough tooth surfaces increase the probability of sticking and increase the time of stick state over a single mesh period. This study provides a design guard for avoiding scuffing failure and improving the reliability of gear transmission.

An experimental and theoretical investigation of the influence of backlash on gear train vibro-impacts and rattle noise

Rattling is a perennial gear noise problem observed in various powertrain components ranging from manual transmissions to engine balancers and timing gear trains. Under lightly loaded conditions, gear systems that are subjected to input and/or output torque fluctuations exhibit vibro-impacts as tooth separations and coast-side contacts take place in the presence of backlash. Experimental set-ups that can impose tightly controlled torque fluctuations to single or multi-mesh gear train are used here to determine the sensitivity of the rattling noise levels to the magnitudes of backlash within wide ranges of torque fluctuation parameters. Torsional discrete models of the experimental set-ups were used to simulate these rattling motions and to predict impact velocity-based rattle severity indexes as a function of both excitation parameters and backlash magnitudes. Single-mesh results show that the larger backlash values result in higher noise levels in most of the cases. In case of double-mesh systems, the resultant noise levels and the corresponding rattle indices exhibit different sensitivities to backlash magnitudes depending on the excitation conditions.

Experimental and Theoretical Investigation of Vibro-impact Motions of a Gear Pair Subjected to Torque Fluctuations to Define a Rattle Noise Severity Index

Abstract Vibro-impacts are common in various automotive engine and transmission gear applications. They are known to cause excessive noise levels, often called rattling or hammering. Input and output fluctuations acting on such systems cause tooth separations and sequences of impacts allowed by backlash at the gear mesh interfaces. The fluctuations leading gear rattling have often been studied for specific applications with the excitations produced typically by an internal combustion engine. As such, rattle evaluations have been often empirical and specific to the systems considered. In this study, an experimental test setup of a gear pair is developed to emulate the same torque fluctuations in a laboratory environment. This setup is used to establish an impact velocity-based rattle severity index defined by the measured torsional behavior of the drive train that is shown to correlate well with the measured sound pressure levels. With that, a validated dynamic model of the experimental setup is employed to predict the same index to allow estimation of rattle noise outcome solely from a torsional dynamic model of the drivetrain. Predicted rattle severity indexes are shown to agree well with the measured ones within wide ranges of torque fluctuations and backlash magnitudes, allowing an assessment of rattle performance of a drivetrain solely from a torsional model.

Perturbation analysis of the two-to-one internal resonance of planetary gear trains

In this study, the two-to-one internal resonance between the first two rotational modes of planetary gear trains (PGTs) is investigated. A purely rotational model is applied considering mesh stiffness variations, tooth separations, and tooth profile modifications (TPMs). Semi-analytical solutions for the internal resonance case are obtained using the method of multiple scales (MMS). The solution equations indicate that the mesh stiffness variations and tooth separations are the main factors causing internal resonance. A validation of the MMS was performed by numerical integration (NI). The results from an example analysis indicate that there exists an internal resonance phenomenon in the case of ωN+2 ≈ ω2, where ω2 and ωN+2 are the natural frequencies associated with the rotational modes, and N is the number of planet gears. Internal resonance in PGTs causes chaos, and part of the energy is transmitted from the ring gear to the sun gear through shocks. Proper TPMs that eliminate the tooth separations could suppress the internal resonance. The internal resonance, in turn, affects the optimal areas of the TPM magnitudes.

Influence of indexing errors on dynamic response of spur gear pairs

In this study, a dynamic model of a spur gear pair is employed to investigate the influence of gear tooth indexing errors on the dynamic response. This transverse-torsional dynamic model includes periodically-time varying gear mesh stiffness and nonlinearities caused by tooth separations in resonance regions. With quasi-static transmission error time traces as the primary excitation, the model predicts frequency-domain dynamic mesh force and dynamic transmission error spectra. These long-period quasi-static transmission error time traces are measured using unity-ratio spur gear pairs having certain intentional indexing errors. A special test setup with dedicated instrumentation for the measurement of quasi-static transmission error is employed to perform a number of experiments with gears having deterministic spacing errors at one or two teeth of the test gear only and random spacing errors where all of the test gear teeth have a random distribution of errors as in a typical production gear.

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.

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.

Non-linear dynamic behaviour of compound planetary gear trains: Model formulation and semi-analytical solution

A discrete, non-linear, time-varying, torsional dynamic model of a multi-stage planetary train that is formed by any number of simple planetary stages is proposed in this study. Each planetary stage has a distinct fundamental mesh frequency and any number of planets spaced in any angular positions. The model allows the analysis of the gear train in all possible power flow configurations suitable for various gear drive ratios. It includes periodic variation of gear mesh stiffnesses as well as clearance (backlash) non-linearities that allow tooth separations. Equations of motion for the general case are formulated and solved semi-analytically using a hybrid harmonic balance method (HBM) in conjugate with inverse Fourier transform. Relative mesh displacements along lines of action of individual gear pairs were used as the continuation parameters to pass singular points and ill-conditioned equations in their proximity. At the end, a case study of a two-stage planetary train is used to demonstrate the effectiveness of the model and solution methods. The HBM solutions are compared to those obtained by a direct numerical integration method to assess their accuracy.

Nonlinear dynamics of idler gear systems

This work examines the nonlinear, parametrically excited dynamics of idler gearsets. The two gear tooth meshes provide two interacting parametric excitation sources and two possible tooth separations. The periodic steady state solutions are obtained using analytical and numerical approaches. Asymptotic perturbation analysis gives the solution branches and their stabilities near primary, secondary, and subharmonic resonances. The ratio of mesh stiffness variation to its mean value is the small parameter. The time of tooth separation is assumed to be a small fraction of the mesh period. With these stipulations, the nonsmooth separation function that determines contact loss and the variable mesh stiffness are reformulated into a form suitable for perturbation. Perturbation yields closed-form expressions that expose the impact of key parameters on the nonlinear response. The asymptotic analysis for this strongly nonlinear system compares well to separate harmonic balance/arclength continuation and numerical integration solutions. The expressions in terms of fundamental design quantities have natural practical application.

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 Study of the Relationship Between the Dynamic Factors and the Dynamic Transmission Error of Spur Gear Pairs

In this study, two different dynamic models, a finite-element-based deformable-body model and a simplified discrete model, are developed to predict dynamic behavior of spur gear pairs. Dynamic transmission error (DTE) and dynamic factors (DF) defined based on the gear mesh loads, tooth loads and bending stresses are computed for a number of unmodified and modified spur gears within a wide range of rotational speed for different involute contact ratios and torque values. Although similar models were proposed in the past, they were neither fully validated nor equipped to predict both DTE and different forms of DF. Accordingly, this study focuses on (i) validation of both models through an extensive set of experimental data obtained from a set of tests using spur gear having unmodified and modified tooth profiles, and (ii) establishment of a direct link between DTE and different forms of DF, especially the ones based on tooth forces and the root stresses. The predicted DF and DTE values are related to each other through simplified formulas. Impact of nonlinear behavior, such as tooth separations and jump discontinuities on DF, is also quantified.

Dynamic tooth loads of planetary gear sets having tooth profile wear

A computational model of a planetary gear set is employed to study the influence of surface wear on the dynamic behavior of a typical planetary gear set. The overall computational scheme combines a wear model that defines geometric description of contacting gear tooth surfaces having wear and a deformable-body dynamic model of a planetary gear set. The wear model employs a quasi-static gear contact model to compute contact pressures and Archard's wear model to determine the wear depth distributions. The worn surfaces are input into the dynamic model to quantify the impact of wear on gear tooth and mesh dynamic forces. The results on a planetary gear set having a fixed planet carrier indicates that the dynamic behavior is nonlinear due to tooth separations in its resonance regions. The results for worn gear surfaces indicate that surface wear has a significant influence in off-resonance speed ranges while its influence diminishes near resonance peaks primarily due to tooth separations.

Forces and moments generated at the dental incisors during forceful biting in humans

A miniature load sensor capable of measuring all forces and all moments simultaneously at a single location in space was used to assess the magnitude and direction of loads that affect the dental incisors during forceful, static biting. While prior approaches have not measured all necessary six degrees of freedom during biting, the complete set of loads is needed to serve as realistic boundary conditions for analytical or computational models of mandibular mechanics. Four subjects were asked to perform controlled and repetitive edge-to-edge incisal biting activities. Customized devices were used to rigidly hold the load sensor in place at pre-specified tooth separations of less than 1 mm. The results yielded force resultants with a magnitude range of 24.5 to 28.4 N. This range was intentionally limited in magnitude to avoid damage to the internal strain gauge assembly of the sensor. In all cases, the highest force component was oriented upwards. An additional simultaneous moment resultant (range: 8.9–17.0 N cm) with a main moment component oriented backwards and downwards towards the oral cavity was also detected. These data suggest that in order for the biting loads to be composed of six DOF, the remaining forces acting on the mandibular force system (i.e. muscular and/or articular) may indeed be non-coplanar and nonconcurrent. Although useful for static biting activities, the bulk of the sensor would probably preclude meaningful measurements during dynamic events such as chewing or swallowing.

Load sharing characteristics of planetary transmissions

A nonlinear time-varying dynamic model of a planetary transmission with an arbitrary number of pinions is developed. The model includes several manufacturing errors and assembly variations, and can accommodate for any pinion spacing arrangement, tooth separations and time-varying gear mesh stiffnesses. A dynamic load sharing factor K DLS , which is defined as the ratio of the actual dynamic load at a given gear mesh to the ideal static load, is predicted using the dynamic model. Key design parameters, assembly variations and manufacturing errors are identified, and their effects on K DLS are quantified.

Tooth Separations and Abnormal Noise on Power-Transmission Gears

Some theoretical analyses and experimental investigations are performed to give a solution to the separation of tooth meshing, which is accompanied with alarming noise or occasionally abnormal noise. Tooth separation occurs at the specific speed defined by the amount of transmission error and tangential load. The more transmission errors are or the less the supplied load is, the lower the speed, at which separation occurs, may become. The largest dynamic load and the highest sound intensity level are presented immediately after the separation occurs. Noise spectra are elucidated in comparison with tooth behaviours. It is considered that an unusual subharmonic noise would be caused by abnormal meshing of every second tooth or every third tooth.