Research on the Influence of Backlash on Mesh Stiffness and the Nonlinear Dynamics of Spur Gears

In light of ignoring the effect of backlash on mesh stiffness in existing gear dynamic theory, a precise profile equation was established based on the generating processing principle. An improved potential energy method was proposed to calculate the mesh stiffness. The calculation result showed that when compared with the case of ignoring backlash, the mesh stiffness with backlash had an obvious decrease in a mesh cycle and the rate of decline had a trend of decreasing first and then increasing, so a stiffness coefficient was introduced to observe the effect of backlash. The Fourier series expansion was employed to fit the mesh stiffness rather than time-varying mesh stiffness, and the stiffness coefficient was fitted with the same method. The time-varying mesh stiffness was presented in terms of the piecewise function. The single degree of freedom model was employed, and the fourth order Runge–Kutta method was utilized to investigate the effect of backlash on the nonlinear dynamic characteristics with reference to the time history chart, phase diagram, Poincare map, and Fast Fourier Transformation (FFT) spectrogram. The numerical results revealed that the gear system primarily performs a non-harmonic-single-periodic motion. The partially enlarged views indicate that the system also exhibits small-amplitude and low-frequency motion. For different cases of backlash, the low-frequency motion sometimes shows excellent periodicity and stability and sometimes shows chaos. It is of practical guiding significance to know the mechanisms of some unusual noises as well as the design and manufacture of gear backlash.

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Study of the Influences of Transient Crack Propagation in a Pinion on Time-Varying Mesh Stiffness

Shock and Vibration ◽

10.1155/2016/6928686 ◽

2016 ◽

Vol 2016 ◽

pp. 1-13

Author(s):

Youmin Hu ◽

Jikai Fan ◽

Jin Yu

Keyword(s):

Potential Energy ◽

Energy Method ◽

Propagation Model ◽

Time Varying ◽

Mesh Stiffness ◽

Propagation Models ◽

Material Fatigue ◽

Stiffness Variation ◽

Short Time ◽

Potential Energy Method

Cracks in a cracked gear may further propagate by a tiny length in a very short time for several reasons, such as material fatigue and load fluctuations. In this paper, this dynamic process is defined as transient propagation of cracks. This research aims to calculate the time-varying mesh stiffness of gears when transient propagation of cracks arises, which has not been extensively studied in existing literatures. The transient propagation of cracks is modelled. An improved potential energy method is proposed by incorporating the propagation model into the potential energy method. The improved method can also be utilised to calculate the mesh stiffness of gears when transient propagation of cracks arises. Different transient propagation models are considered to simulate the propagation of cracks in a short amount of time. Different deterioration levels of cracks before transient propagation and different lengths and models of transient propagation are also examined. The variation rules of mesh stiffness caused by the transient propagation of cracks are summarised. The influence of the deterioration level of cracks on mesh stiffness variation when transient propagation arises is obtained. Simulation results show that the proposed method accurately calculates time-varying mesh stiffness when transient propagation of cracks arises. Furthermore, the method improves the monitoring of further propagation of cracks in gears from the perspective of time-varying mesh stiffness.

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Evaluating the Time-Varying Mesh Stiffness of a Planetary Gear Set Using the Potential Energy Method

Lecture Notes in Mechanical Engineering - Proceedings of the 7th World Congress on Engineering Asset Management (WCEAM 2012) ◽

10.1007/978-3-319-06966-1_33 ◽

2014 ◽

pp. 365-374 ◽

Cited By ~ 3

Author(s):

Xihui Liang ◽

Ming J. Zuo ◽

Yangming Guo

Keyword(s):

Potential Energy ◽

Energy Method ◽

Planetary Gear ◽

Time Varying ◽

Mesh Stiffness ◽

Potential Energy Method

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Noise Radiation Analysis of Structural-Acoustic Coupling System of a Gear Box

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.37-38.718 ◽

2010 ◽

Vol 37-38 ◽

pp. 718-722

Author(s):

Jian Xing Zhou ◽

Geng Liu ◽

Shang Jun Ma

Keyword(s):

Time History ◽

Noise Spectrum ◽

Time Varying ◽

Mesh Stiffness ◽

Acoustic Coupling ◽

Noise Radiation ◽

Coupling System ◽

Gear Transmission System ◽

Structure Improvement ◽

History Of

The dynamic model of the gear transmission system is built in consideration of the time-varying mesh stiffness and the gear errors. Then the time history of the dynamic load of system is calculated. The gearbox structural-acoustic coupling system is built and its noise radiation is calculated by using BEM. The noise spectrum of the gearbox and the panel contribution are obtained. The influence of the gearbox structure improvement, structure damping and different gear style on the noise radiation is analyzed. The study provides useful theoretical guideline to the design of the gearbox.

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Evaluating the time-varying mesh stiffness of a planetary gear set using the potential energy method

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406213486734 ◽

2013 ◽

Vol 228 (3) ◽

pp. 535-547 ◽

Cited By ~ 47

Author(s):

Xihui Liang ◽

Ming J Zuo ◽

Tejas H Patel

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Potential Energy ◽

Energy Method ◽

Analytical Method ◽

Planetary Gear ◽

Time Varying ◽

The Finite Element Method ◽

Mesh Stiffness ◽

Potential Energy Method

Time-varying mesh stiffness is a periodic function caused by the change in the number of contact tooth pairs and the contact positions of the gear teeth. It is one of the main sources of vibration of a gear transmission system. An efficient and effective way to evaluate the time-varying mesh stiffness is essential to comprehensively understand the dynamic properties of a planetary gear set. According to the literature, there are two ways to evaluate the gear mesh stiffness, the finite element method and the analytical method. The finite element method is time-consuming because one needs to model every meshing gear pair in order to know the mesh stiffness of a range of gear pairs. On the other hand, analytical method can offer a general approach to evaluate the mesh stiffness. In this study, the potential energy method is applied to evaluate the time-varying mesh stiffness of a planetary gear set. Analytical equations are derived without any modification of the gear tooth involute curve. The developed equations are applicable to any transmission structure of a planetary gear set. Detailed discussions are given to three commonly used transmission structures: fixed carrier, fixed ring gear and fixed sun gear.

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Non-probabilistic interval process method for analyzing two-stage straight bevel gear system with uncertain time-varying parameters

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406220967693 ◽

2020 ◽

pp. 095440622096769

Author(s):

Wassim Lafi ◽

Fathi Djemal ◽

Dhouha Tounsi ◽

Ali Akrout ◽

Lassaad Walha ◽

...

Keyword(s):

Bevel Gear ◽

Gear System ◽

Uncertain Parameters ◽

Time Varying ◽

Mesh Stiffness ◽

Time Step ◽

Two Stage ◽

Straight Bevel Gear ◽

Potential Energy Method ◽

Process Method

A two-stage straight bevel gear system is a gear system that can be used in various applications. The straight bevel gear is known for its complex tooth geometry. Due to the variation of the number of pairs of teeth in contact, the mesh stiffness function can be considered as a time-varying function. However, the mesh stiffness for the straight bevel gear is sensitive to measurement and modeling errors. Thus, at each time step, its value can not assigned to deterministic one. Generally, the uncertain parameters are assumed to be time-independent. In this paper, the interval process method has been used to represent the time-varying uncertain parameters, whose bounds are determined through the potential energy method. The lumped parameter model of two-stage straight bevel gear has been proposed. We have considered that the masses of the straight bevel gear system components and bearing stiffnesses along with time-varying mesh stiffnesses are uncertain parameters which can be represented by the interval process model. The Chebyshev polynomial expansion has been used to approximate the response of the two-stage straight bevel gear system with respect to the interval variables. The lower and higher bounds of the eigenvalues of the system have been determined. The bounds of dynamic displacements of the straight bevel gear system have been computed and compared with those computed by the Monte Carlo method.

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Study the Effects From Gear Rim Thickness on the Gear Pair Time-Varying Mesh Stiffness and Dynamic Behaviors

Volume 10: 2017 ASME International Power Transmission and Gearing Conference ◽

10.1115/detc2017-68421 ◽

2017 ◽

Author(s):

Zi Wang ◽

Caichao Zhu ◽

Chaosheng Song

Keyword(s):

Potential Energy ◽

Elastic Deformation ◽

Energy Method ◽

The Body ◽

Time Varying ◽

Gear Pair ◽

Mesh Stiffness ◽

Dynamic Behaviors ◽

Gear Mesh ◽

Potential Energy Method

This work will introduce two methods for calculating the gear mesh stiffness which includes the potential energy method and Finite Element/ Contact Mechanics method. The elastic theory of Muskhelishvili will be used to calculate the elastic deformation from the gear body during one mesh cycle for the gear pair in the potential energy method. Also the involute curve, the geometric and kinematics properties of the gear mesh pair will be taken into account of these two methods. The quasi-static time-varying mesh stiffness considering the deflection of gear rim body is learned in detail. Results from both two methods will show the importance of gear rim body elasticity on the gear pair mesh stiffness and the comparison of the results will reveal the validity and efficiency of the methods. Then lumped-parameter model is presented for studying the whole system dynamic behaviors. The effect from the body elastic deformation from component itself on the macro rigid body motion of the system is investigated. The conclusion from the results shows that the elasticity from gear rim body will take prominent effects on the gear pair dynamic behaviors, which should be regarded as an important factor during the design process.

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Optimization of Compound Planetary Gear Train by Improved Mesh Stiffness Approach

Volume 4A: Dynamics, Vibration, and Control ◽

10.1115/imece2017-70403 ◽

2017 ◽

Cited By ~ 1

Author(s):

Jianwu Zhang ◽

Han Guo ◽

Liang Zou ◽

Haisheng Yu

Keyword(s):

Planetary Gear ◽

Coupling Model ◽

Gear Train ◽

Time Varying ◽

Mesh Stiffness ◽

Planetary Gear Train ◽

Planetary Gear Trains ◽

Hybrid Transmission ◽

Gear Trains ◽

Potential Energy Method

An improved mesh stiffness approach is presented for optimization of vibration and noise performance of the planetary gear trains in a full power split hybrid transmission, in which mesh stiffness time-variability and biaxial gear stiffness couplings in gear pairs are taken into account. For improving accuracy of the mesh stiffness in double teeth-meshing region for spur gear pairs, a simplified solution to the loading gear deformations counting for time-varying mesh stiffness of the helical gear pairs is proposed, based on the integral potential energy method and FEM simulation. By the new biaxial coupling model, effects of gear body and tooth coupled stiffnesses on gear pair vibro-acoustic responses are also investigated and approved to be considerable. Numerical examples with optimal analyses of the specified planetary gear trains for the full hybrid transmission are provided. Numerical solutions of eigen frequencies and vibration modes for the gear pairs with a variety of time-varying mesh stiffnesses are constructed by the biaxial coupling model and Fourier Series. The dynamic parameters optimization of the compound planetary gear train is then conducted. The optimized planetary gear system is applied in the full hybrid transmission and bench tests for its vibro-acoustic performance are also undertaken. Computational predictions and experimental results are shown to be in fairly good agreement.

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