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.     

Gear transmission dynamics: Effects of index and run out errors

This work describes a non-linear dynamic model for the study of the vibration signals generated by gear transmissions. The developed model considers both the parametric excitations due to the variable compliance of bearings and gears, can handle changes in the transmitted torque and allows the integration of the dynamic equations quickly and accurately. This model has been developed previously by the authors to assess the profile deviations on the dynamic behavior of gear transmissions and its influence on the transmitted torque. It also includes the presence of gear defects as cracks and pitting during the calculation of meshing forces. In this paper, the model has been enhanced in order to include two common defects such as index errors and run out or eccentricity errors. Index errors occur as a result of a non-uniform angular distribution of the tooth profiles along the pitch circle. Run out appears due to the displacement of the geometric center of the gear with respect to the center of rotation of the shaft on which it is mounted. Although both errors are caused by different reasons, sometimes they have been confused because of their similitudes. The procedure for including both kinds of errors in the model is described and simulations under several transmitted torques are presented. The results are assessed and compared focusing the attention on certain transmission parameters and magnitudes as transmission error, load forces in the tooth flanks and demodulation techniques on the resulting vibratory signals.     

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.     

An investigation into the effect of tooth profile errors on gear rattle

In previous work, experimental data have demonstrated the severity of idling gear rattle depends not only on the amplitude, but also the phase of an external sinusoidal forcing. One possible explanation for this is in small tooth profile errors. In this paper, we investigate this hypothesis, by deriving an equation of motion incorporating an error function and losses at the mesh interface, values of which are obtained from experimental data. By solving the equations of motion, theoretical gear rattle trajectories are obtained. Theoretical and experimental trajectories are then compared, by way of time domain plots as well as via contour plots linking the amplitude of backlash oscillation to the amplitude and phase of input forcing. For most profile error functions, good agreement is achieved between the model and experimental data. In the case where the profile errors are dominated by misalignment between the gear and shaft centres agreement is less good and suggestions of areas of further study required for model refinement are proposed.     

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 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.     

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.     

Dynamics Modeling and Experimental Investigation of Gear Mechanism with Coupled Small Clearances

Internal gear mechanism is widely used in micro-nano satellites due to its compact structure and high precision transmission. However, the vibration coupling caused by the small clearance coupling is more obvious and cannot be ignored under low speed, light load and zero gravity conditions. Based on the geometric relationship between radial clearance and backlash, a coupled model between dynamic backlash and radial clearance of internal meshing gear is established. Based on the conformal contact theory, the radial collision force model of the gear shaft and shaft sleeve considering the small clearances is established. Additionally, a multi-clearance gear rotor system test device is built to measure the vibration acceleration of the internal gear rotor system by an acceleration sensor and transmitted to the industrial computer through a signal collector for data processing. Through the comparison of simulation and experiment, the accuracy of the gear dynamics model is verified. The analysis results show that, compared with the traditional model, the calculation results of the gear mechanism model considering the small clearance coupling is closer to the experimental data.     

Dynamics of regenerative chatter and internal resonance in milling process with structural and cutting force nonlinearities

In this paper, internal resonance and nonlinear dynamics of regenerative chatter in milling process is investigated. An extended dynamic model of the peripheral milling process including both structural and cutting force nonlinearities is presented. Closed form expressions for the nonlinear cutting forces are derived through their Fourier series components. In the presence of the large vibration amplitudes, the loss of contact effect is included in this model. Using the multiple-scales approach, analytical approximate response of the delayed nonlinear system is obtained. Considering the internal resonance dynamics (i.e. mode coupling), the energy transfer between the coupled x–y modes is studied. The results show that during regenerative chatter under specific cutting conditions, one mode can decay. Furthermore, it is possible to adjust the rate at which the x-mode (or y-mode) decays by implementation of the internal resonance. Therefore, under both internal resonance and regenerative chatter conditions, it is possible to suppress the undesirable vibration of one mode (direction) in which accurate surface finish is required. Under the steady-state motion, jump phenomenon is investigated for the process with regenerative chatter under various cutting conditions. Moreover, the effects of structural and cutting force nonlinearities on the stability lobes diagram of the process are investigated.     

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.     

Shield analysis and non-rigid match in positioning the actual interferogram on the tooth flank based on oblique incidence interferometry

When the oblique incidence interferometry is used to measure the form deviation of the tooth flank, because of the anamorphic distortion and image foreshortening, the collected interference fringe pattern (IFP) cannot be positioned on the tooth flank directly. Therefore a simulation tooth image (STI) is employed as a bridge to connect the collected IFP and the tooth flank. The exiting method is good enough to deal with the IFP of spur gear tooth flank, but it is inappropriate for the IFP of helical gear. This paper focuses on two improvements for positioning the IFP of helical gear on the tooth flank: (1) in the process of generating the STI, a mathematical model of helical tooth flank is proposed to judge the shielded situation of tooth flank; (2) in the process of matching the STI with the collected IFP, a non-rigid matching method is proposed to determine the relationship of the two images. The results of an experimental application are given to verify the feasibility and accuracy.     

Feasibility of a phased acoustic array for monitoring acoustic signatures from meshing gear teeth

This paper investigates the feasibility of sensing damage emanating from rotating drivetrain elements such as bearings, gear teeth, and drive shafts via airborne paths. A planar phased acoustic array is evaluated as a potential fault detection scheme for detecting spatially filtered acoustic signatures radiating from gearbox components. Specifically, the use of beam focusing and steering to monitor individual tooth mesh dynamics is analyzed taking into consideration the constraints of the array/gearbox geometry and the spectral content of typical gear noise. Experimental results for a linear array are presented to illustrate the concepts of adaptive beam steering and spatial acoustic filtering. This feasibility study indicates that the planar array can be used to track the acoustic signatures at higher harmonics of the gear mesh frequency.     

Adaptive autoregressive modeling of non-stationary vibration signals under distinct gear states. Part 2: experimental analysis

Parametric time-frequency representation based on parametric models is more desirable for presenting highly precise time-frequency domain information due to its high-resolution property. However, the sensitivity and robustness of parametric models, in particular the parametric models on the basis of advanced adaptive filtering algorithms, has never been investigated for on-line condition monitoring of rotating machinery. Part 1 of this study proposed three adaptive parametric models based on three advanced adaptive filtering algorithms. Part 2 of this study is concerned with the effectiveness of the proposed models under distinct gear states, especially the highly non-stationary conditions accrued from advanced gear faults. Four gear states are considered: healthy state, adjacent gear tooth failure, non-adjacent gear tooth failure and distributed gear tooth failure. The vibration signals used in this study include the time-domain synchronous averaging signal and gear motion residual signal for each considered gear state. The test results demonstrate that the optimum filter behavior can readily be attained and the white Gaussian assumption of innovations can relatively be easily guaranteed for the NAKF-based model under distinct gear states and a wide variety of model initializations. On the other hand, the EKF- and MEKF-based models are capable of generating more accurate time-frequency representations than the NAKF-based model, but in general the optimality condition for white Gaussian assumption cannot be guaranteed for these two advanced models. Therefore, the NAKF-based model is preferred for automatic condition monitoring due to its appealing robustness to distinct gear states and arbitrary model initializations, whereas the EKF- and MEKF-based models are desirable when accurate time-frequency representation is concerned.     

EPSI validation for cylindrical gears

Shape deviations of spur gears obtained by using Electronic Speckle Pattern Interferometry are compared with the results of numerical simulation. The dual illumination method produces a speckle pattern by simultaneous illumination of a sample by two laser waves symmetrically in the direction of observation. Two images obtained, respectively, before and after deformation are subtracted to obtain the field of displacement. Furthermore, a study of meshing stiffness was performed for different types of setting for an internal gear.     

Corrigendum to: Optimum profile modifications of spur gears by means of genetic algorithms

The purpose of the present work is to correct some inaccuracies of the paper “Bonori, G., Barbieri, M., Pellicano, F., 2008, Optimum Profile Modifications of Spur Gears by Means of Genetic Algorithms, Journal of Sound and Vibration, 313, pp. 603–616”; in that work, the aim was the reduction of vibrations in spur gears by means of profile modifications. This goal was achieved by using an ad hoc genetic algorithm, where the objective function was the peak to peak or the harmonic content of the Static Transmission Error (STE) computed by Finite Element calculations.The efficiency in terms of vibration reduction of the optimized profile reliefs was checked using a one degree of freedom dynamic model. This dynamic model considers time varying mesh stiffness, backlash, and profile error. In the original paper the effect of intentional profile modifications was considered as part of the mesh stiffness, thus overestimating their effect in vibration reduction.In the present corrigendum, the dynamic model is updated, keeping into account profile deviations by means of an error function. Finally, the optimal profile modifications found in the original paper are checked using the updated model.     

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.     

渐开线斜齿轮短齿廓修形降噪分析*

为了降低特定工况下减速箱的噪声特性,找到相对最佳的渐开线斜齿轮修形方式,该文基于专业的齿轮修形软件Kisssoft和多学科一体化软件LMS Virtual Lab,根据短齿廓修形可以改变渐开线斜齿轮的端面、轴向重合度的原理,比较了未修形和经过不同短齿廓修形方式达到特定重合度后的渐开线斜齿轮减速箱声功率级的大小。分析了经过短齿廓修形后的减速箱声功率级在转速500~2000 r/min之间显著下降的原因,并对经过短齿廓修形后的减速箱声功率级在2500 r/min和3000 r/min时并未明显下降的原因做了解释。最后根据减速箱声功率级在转速500~2000 r/min之间的平均降幅,认为通过齿形鼓形修形使得渐开线斜齿轮的端面、轴向重合度分别为1.516、0.864时,减速箱声功率级降幅最大,降噪效果最好。     

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.     

APPLICATION OF NON-SMOOTH MODELLING TECHNIQUES TO THE DYNAMICS OF A FLEXIBLE IMPACTING BEAM

Non-smooth modelling techniques have been successfully applied to lumped mass-type structures for modelling phenomena such as vibro-impact and friction oscillators. In this paper, the application of these techniques to continuous elements using the example of a cantilever beam is considered. Employing a Galerkin reduction to form an N -degree-of-freedom modal model, a technique for modelling impact phenomena using a non-smooth dynamics approach is demonstrated. Numerical simulations computed using the non-smooth model are compared with experimentally recorded data for a flexible beam constrained to impact on one side. A method for dealing with sticking motions when numerically simulating the beam motion is presented. In addition, choosing the dimension of the model based on power spectra of experimentally recorded time series is discussed.     

A study of self-organizing processes of nonlinear stochastic variables

A general theory is given for the time evolution of nonlinear stochastic variables a(t) = {a_i(t)} whose statistical distribution is changing due to the self-organization of “macroscopic” order. The dynamics of a(t) is conveniently expressed by self-consistent equations for the ensemble average x(t) = 〈a(t)〉, the supersystem, and for the deviations ξ(t) = a(t)?x(t), the subsystem; the systems are connected to each other by feedback loops in their dynamics. The time dependence of the variance and the correlation function ofξ(t) are studied in terms of relaxation toward local equilibrium underx(t) and dynamical coupling withx(t). A special example shows that the stochastic motions of subsystems are pulled together by the motion of the supersystem through feedback loops, and that this pull-together phenomenon occurs when symmetry-breaking instability exists in nonlinear systems.