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.     

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.     

Gear transmission dynamic: Effects of tooth profile deviations and support flexibility

In this work a non-linear dynamic model of spur gear transmissions previously developed by the authors is extended to include both desired (relief) and undesired (manufacture errors) deviations in the tooth profile. The model uses a hybrid method for the calculation of meshing forces, which combines FE analysis and analytical formulation, so that it enables a very straightforward implementation of the tooth profile deviations. The model approach handles well non-linearity due to the variable meshing stiffness and the clearances involved in gear dynamics, also including the same phenomena linked to bearings. In order to assess the ability of the model to simulate the impact of the deviations on the transmission dynamics, an example is presented including profile deviations under different values of transmitted torque. Several results of this example implementation are presented, showing the model’s effectiveness.     

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 time domain approach to diagnose gearbox fault based on measured vibration signals

Spectral analysis techniques to process vibration measurements have been widely studied to characterize the state of gearboxes. However, in practice, the modulated sidebands resulting from the local gear fault are often difficult to extract accurately from an ambiguous/blurred measured vibration spectrum due to the limited frequency resolution and small fluctuations in the operating speed of the machine that often occurs in an industrial environment. To address this issue, a new time-domain diagnostic algorithm is developed and presented herein for monitoring of gear faults, which shows an improved fault extraction capability from such measured vibration signals. This new time-domain fault detection method combines the fast dynamic time warping (Fast DTW) as well as the correlated kurtosis (CK) techniques to characterize the local gear fault, and identify the corresponding faulty gear and its position. Fast DTW is employed to extract the periodic impulse excitations caused from the faulty gear tooth using an estimated reference signal that has the same frequency as the nominal gear mesh harmonic and is built using vibration characteristics of the gearbox operation under presumed healthy conditions. This technique is beneficial in practical analysis to highlight sideband patterns in situations where data is often contaminated by process/measurement noises and small fluctuations in operating speeds that occur even at otherwise presumed steady-state conditions. The extracted signal is then resampled for subsequent diagnostic analysis using CK technique. CK takes advantages of the periodicity of the geared faults; it is used to identify the position of the local gear fault in the gearbox. Based on simulated gear vibration signals, the Fast DTW and CK based approach is shown to be useful for condition monitoring in both fixed axis as well as epicyclic gearboxes. Finally the effectiveness of the proposed method in fault detection of gears is validated using experimental signals from a planetary gearbox test rig. For fault detection in planetary gear-sets, a window function is introduced to account for the planet motion with respect to the fixed sensor, which is experimentally determined and is later employed for the estimation of reference signal used in Fast DTW algorithm.     

Effects of operating conditions on the Acoustic Emissions (AE) from planetary gearboxes

In the last decade, the use of acoustic emissions has received growing acceptance for its application in machine condition monitoring. This is because it offers good possibilities to diagnose failures at early stages and low rotational speeds. The use of acoustic emissions for condition monitoring of gears, however, is still an active field of research, because several questions remain unanswered. One of these questions is the effect of operating conditions on the AE generated during gear meshing. In this work, the results of experiments carried out on a non-faulty planetary gearbox test bench are presented. A planetary gearbox is considered, because of its usual application on machines subjected to variable operating conditions. The effects of lubricant temperature, load and rotational speed are investigated. The conclusions obtained from the experiments are used for the analysis of the AE measured on the planetary gearbox of a bucket wheel excavator.     

Multi-input multi-output feedforward control of multi-harmonic gearbox vibrations using parallel adaptive notch filters in the principal component space

Rotating mechanical systems such as gearbox are known to generate vibrations at specific frequencies. The generation of asymmetric frequency sidebands around gear meshing frequencies is typical of rotating systems, and often results in amplitude-modulated disturbances. In addition, when the angular speed of such systems exhibits small time variations, a frequency-modulation of the disturbance tones is observed. Such conditions may result in difficulties for feedforward active control systems based on short, finite impulse response control filters, as was observed by previous researchers. This is mainly due to the fact that the reference signal may not provide all the gear meshing frequencies and sidebands found in the disturbance. This work investigates adaptive notch filters in the principal component space to cancel multi-harmonic gearbox vibrations. Experimental results obtained with these adaptive notch filters show good control attenuations at targeted gearbox tones, without amplification of other closely located, unreferenced tones.     

Theoretical modeling and characteristic analysis of moving-train induced ground vibrations

An integrated train-track-subsoil dynamic interaction model of moving-train induced ground vibration is developed on the basis of vehicle dynamics, track dynamics and the Green's functions of subsoil. The model takes account of the vibrations of vehicle components, the quasi-static axle loads and the dynamic excitations between the wheels and track. The analyzed results from an example show that the ground vibration characteristics have a close relationship with train speed and soil properties; the dynamic responses excited by wheel-track irregularity have big influence on the high frequency components of ground vibration; with the increase of distance to the track, the ground acceleration has the tendency of decrease, and the relevance of acceleration curves and train excitation becomes less obvious; the intersections of moving load speed-lines and subsoil dispersion curves are some resonance frequencies that cause the amplification of ground vibrations; there exists a critical speed for moving train that is close to the minimum velocity of the Rayleigh's wave in the subsoil.     

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.     

双面抛光运动的数学建模及轨迹优化

双面抛光运动过程复杂,对光学材料的加工有重要的影响,运动轨迹分布不但影响加工效率,而且影响加工工件的表面质量.通过分析双面抛光加工的运动过程,建立双面抛光中任意一点相对于上抛光盘的运动轨迹的数学模型,改变数学模型中的轨迹运动参数,观察不同参数值对抛光轨迹分布的影响,优化分析得出抛光轨迹分布最佳的运动参数值.研究结果表明...     

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.     

光学镜片平面行星式研磨加工关键技术研究

在平面行星式研磨加工过程中,研磨盘磨损的均匀性与光学镜片的研磨质量有密切关系。通过对光学镜片在平面行星式研磨加工过程中的运动规律进行分析,建立了差动轮系条件下工件运动的轨迹和速度计算模型,开发了Matlab仿真程序。结合生产实际,对行星轮系条件下工件运动的轨迹和速度进行了仿真,确定了研磨机速比对研磨盘磨损均匀性的影响规律:当速比I1-1.5时,镜片的运动轨迹在较短时间内遍布研磨盘各处,研磨盘得到均匀磨损,加工质量和加工效率较高。基于仿真结果,通过优化工艺参数,获得了厚度一致性小于0.002 mm,平行度小于0.002 mm的高质量光学镜片。     

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.     

机械加工塑料齿廓偏差影像法测量及抽样特性

应用影像法,对机械加工的渐开线塑料齿轮齿廓偏差进行测量。由于塑料齿轮在机械加工中受机械应力和热应力影响较大,而导致齿廓应变较大,为精确反映塑料齿轮的齿廓偏差,并判定塑料齿轮齿廓公差的精度等级,对测量中出现的回转偏心误差、回转分度误差、瞄准示值误差和重复性测量误差进行了不确定度的A类和B类综合评定。并通过统计控制方法中的极差控制图进行分析,得到相同的评定结果。利用对塑料齿轮单齿齿廓偏差的检验,结合随机抽样检验的原理,对塑料齿轮齿廓精度等级进行了判定。最终得出机械加工塑料齿轮,齿形误差较大,超出国标中规定的齿轮精度等级的范围,适合实验室模型展示而不适合用作工业齿轮的结论。     

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.     

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.     

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.     

Simulation of ball motion and energy transfer in a planetary ball mill

A kinetic model is proposed for simulating the trajectory of a single milling ball in a planetary ball mill, and a model is also proposed for simulating the local energy transfer during the ball milling process under no-slip conditions. Based on the kinematics of ball motion, the collision frequency and power are described, and the normal impact forces and effective power are derived from analyses of collision geometry. The Hertzian impact theory is applied to formulate these models after having established some relationships among the geometric, dynamic, and thermophysical parameters. Simulation is carried out based on two models, and the effects of the rotation velocity of the planetary disk and the vial-to-disk speed ratio ω/Ω on other kinetic parameters is investigated. As a result, the optimal ratio ω/Ω to obtain high impact energy in the standard operating condition at = 800 rpm is estimated, and is equal to 1.15.     

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.