Theoretical and experimental investigation into the influence of the periodic backlash fluctuations on the gear rattle

The paper provides an investigation into the influence of the periodic variation in the boundaries of the dead-zone between the teeth, caused by the Transmission Error, on the gear rattle phenomenon.The study represents an evolution of a theoretical model, which has already been developed by the authors, to analyze lightly loaded gear pairs, accounting for oil squeeze effects between the gear teeth.An experimental test rig and a methodology to estimate the transmission error parameters, starting from an experimental acquisition, are described.A number of experimental-theoretical correlations concerning the influence of transmission errors on gear dynamics are reported in order to validate the theoretical model.     

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.     

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.     

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.     

Analytical modeling of squeeze film damping for rectangular elastic plates using Green's functions

An analytical method for the solution of squeeze film damping based on Green's function to the nonlinear Reynolds equation considering an elastic plate is presented. This allows calculating the stiffness and damping forces rapidly for various boundary conditions. The elastic plate velocity is applied to the nonlinear Reynolds equation as a forcing term. The nonlinear Reynolds equation is divided into multiple linear nonhomogeneous Helmholtz equations, which then can be solvable using the presented approach. Approximate mode shapes of a rectangular elastic plate are used, enabling the calculation of the damping ratio and frequency shift for the linear case, as well as the complex resistant pressure, for both linear and nonlinear cases.     

Rotor drop and following thermal growth simulations using detailed auxiliary bearing and damper models

Catcher bearings (CBs) or auxiliary bearings provide mechanical backup protection in the events of magnetic bearing failure. This paper presents numerical analysis for a rotor drop on CBs and following thermal growths due to their mechanical rub using detailed CB and damper models. The detailed CB model is determined based on its material, geometry, speed and preload using the nonlinear Hertzian load–deflection formula, and the thermal growths of bearing components during the rotor drop are estimated using a 1D thermal model. A finite-element squeeze film damper provides the pressure profile of an annular oil film and the resulting viscous damping force. Numerical simulations of an energy storage flywheel with magnetic suspensions failed reveal that an optimal CB design using the detailed simulation models stabilizes the rotor drop dynamics and lowers the thermal growths while preventing the high-speed backward whirl. Furthermore, CB design guides based on the simulation results are presented.     

Experimental and numerical analysis of automotive gearbox rattle noise

The aim of this work is to characterize the rattle noise of automotive gearboxes, resulting from impacts between toothed wheels of unselected gear ratios. These stereo-mechanical impacts are modeled by a coefficient of restitution which describes damping during the squeezing of the lubricant film for approaching surfaces, and the elastic deformation of impacting bodies. The dynamic response of the loose gear first depends on the design parameters and the engine operating conditions. The unknown parameters are the drag torque and the coefficient of restitution. They are identified experimentally through implementation of two optical encoders in an actual automotive gearbox and the operation of a specific test bench which replicates the automotive power train. Models of the different drag torque sources are validated from analysis of the free damped response of the driveline. The coefficient of restitution and its probability density function are measured from experiments under stationary operating conditions. A nonlinear model is built. The dynamic response of the loose gear depends on the dimensionless backlash, the coefficient of restitution and a dimensionless parameter proposed to describe the rattle excitation level. Experiments under controlled excitation are performed to validate the assumptions, to confirm the ability of the parameter proposed to describe the rattle noise threshold, and to characterize the dynamic response. The nonlinear model predictions are fitted with the drag torque and coefficient of restitution previously identified. They are compared with measurements to demonstrate the ability of the model to predict gear rattle for any loose gear, any gearbox and any operating condition.     

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.     

A dynamic model to determine vibrations in involute helical gears

A method to determine the dynamic load between two rotating elastic helical gears is presented. The stiffness of the gear teeth is calculated using the finite element method and includes the contribution from the elliptic distributed tooth load. To make sure that the new incoming contacts which are the main excitation source are properly simulated, the necessary deformation of the gears is determined by using the true geometry and positions of the gears for every time step of the dynamic calculation. This allows the contact to be positioned outside the plane of action. A numerical example is presented with figures that show the behaviour of the dynamic transmission error as well as the variation of the contact pressure due to the dynamic load for different rotational speeds.     

基于Coulomb摩擦效应的一类非线性相对转动系统的混沌研究

研究一类非线性相对转动系统在负载Coulomb摩擦效应下的混沌运动行为. 根据Lagrange方程建立一类含非线性负载Coulomb摩擦阻尼的两个质量相对转动系统的动力学方程. 利用Cardano公式讨论自治系统的特征值, 在此基础上, 应用待定系数法给出系统同宿轨道的存在性, 并借助Silnikov定理研究了系统的混沌行为. 最后数值模拟了给定参数下系统的混沌运动, 并给出在Coulomb摩擦阻尼变化下系统由周期、倍周期通向混沌的途径, 验证了理论分析的正确性.     

Global behavior of gear system using mixed cell mapping

In some mechanical nonlinear systems, the transient motion will be undergoing a very long process and the attractor-basin boundaries are so complicated that some difficulties occur in analyzing the system global behavior. To solve this problem a mixed cell mapping method based on the point mapping and the principle of simple cell mapping is developed. The algorithm of the mixed cell mapping is studied. A dynamic model of a gear pair is established with the backlash, damping, transmission error and the time-varying stiffness taken into consideration. The global behaviors of this system are analyzed. The coexistence of the system attractors and the respective attractor-basin of each attractor with different parameters are obtained, thus laying a theoretical basis for improvement of the dynamic behaviors of gear system.     

Global behavior of gear system using mixed cell mapping

In some mechanical nonlinear systems, the transient motion will be undergoing a very long process and the attractor-basin boundaries are so complicated that some difficulties occur in analyzing the system global behavior. To solve this problem a mixed cell mapping method based on the point mapping and the principle of simple celll mapping is developed. The algorithm of the mixed cell mapping is studied. A dynamic model of a gear pair is established with the backlash, damping, transmission error and the time-varying stiffness taken into consideration. The global behaviors of this system are analyzed. The coexistence of the system attractors and the respective attractor-basin of each attractor with different parameters are obtained, thus laying a theoretical basis for improvement of the dynamic behaviors of gear system.

     

Determination of time-varying contact length,friction force,torque and forces at the bearings in a helical gear system

This paper deals with determining various time-varying parameters that are instrumental in introducing noise and vibration in a helical gear system. The most important parameter is the contact line variation, which subsequently induces friction force variation, frictional torque variation and variation in the forces at the bearings. The contact line variation will also give rise to gear mesh stiffness and damping variations. All these parameters are simulated for a defect-free and two defective cases of a helical gear system. The defective cases include one tooth missing and two teeth missing in the helical gear. The algorithm formulated in this paper is found to be simple and effective in determining the time-varying parameters.     

Use of modal energy transfer as a new damping device with controllable bandwidth

This paper presents a new design of nonlinear dynamic absorber (NDA) using the phenomenon of modal energy transfer between the symmetric mode and the anti-symmetric mode of a curved beam. It can reduce the resonance vibration of a primary structure with a controllable operational frequency range. The energy transfer is initiated by an autoparametric vibration and the excitation force required is lowest when the ratio of the resonance frequencies of the first symmetric mode (ω₁) and first anti-symmetric mode (ω₂) is close to 2.The resonance frequency of the first anti-symmetric mode (ω₂) can be altered to control the operational frequency range. The autoparametric vibration response can be used to create an energy-dissipative region with a controllable bandwidth. It is also possible to create a non-dissipative region in between two dissipative regions. This is useful for providing damping for a conventional dynamic absorber without adding high damping material. The damping is due to the dissipation of energy to anti-symmetric mode. Numerical calculations indicate that the resonance vibration of a primary structure can be successfully reduced using this approach. The results are verified with experimental data.     

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.     

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.     

Post-buckling longterm dynamics of a forced nonlinear beam: A perturbation approach

The aim of this paper is to determine by a singular perturbation approach the dynamic response of a harmonically forced system experiencing a pitchfork bifurcation. The model of an extensible beam forced by a harmonic excitation and subject to an axial static buckling is space-discretized by a Galerkin approach and studied by the Normal Form Method for different values of equation parameters influencing the nonlinear dynamic behavior like damping coefficient, load amplitude and frequency. A relevant issue in the perturbation methods is the concept of small and zero divisors which are related to the possibility to build a transformation that simplifies the original studied problem, i.e. to obtain the Normal Form, by eliminating as much as possible nonlinearities in the equations. For nonconservative systems, like structural damped systems, there are no conditions in the prior literature that define what “small” means relatively to a divisor. In the present paper some conditions about the order of magnitude of the divisors with respect to the perturbation entity are given and related to some physical parameters in the governing equations in order to estimate the relevance of some nonlinear effects.     

一类弹性和阻尼双分段非线性约束系统周期响应特性研究

考虑动态条件下的两种典型分段非线性约束, 根据广义耗散Lagrange原理建立一类具有弹性和阻尼双分段非线性约束系统动力学模型. 采用平均法求解得到系统在周期激励下的幅频响应关系. 分别比较系统在不同分段非线性约束条件下的时域响应、分岔响应和幅频响应, 得到受分段非线性约束的系统响应特性以及约束条件变化时系统响应的变化规律. 对比两种约束条件下的幅频响应, 研究得到系统稳定性受不同分段非线性因素影响及两种分段非线性约束之间的相互影响规律.     

Scaling laws for spreading of a liquid under pressure

We study squeeze flow of two different fluids (castor oil and ethylene glycol) between a pair of glass plates and a pair of perspex plates, under an applied load. The film thickness is found to vary with time as a power-law, where the exponent increases with load. After a certain time interval the area of fluid-solid contact saturates to a nearly steady value. This saturation area, increases with load at different rates for different fluid-solid combinations.     

Analytic treatment of a system with a vibro-impact nonlinear energy sink

This Communication reports on an approximate analytic procedure for analysis of transient damped response in a system comprising a primary linear oscillator and vibro-impact nonlinear energy sink (NES). Multiple-scale expansion reduces the problem in the first approximation to low-dimensional vibro-impact system with forcing and damping; the latter problem yields to direct analytic approach. The results computed by this way fairly correlate with direct numeric simulations. The analysis takes advantage of small ratio of the NES and the primary masses, but does not require the restitution coefficient to be close to unity. The proposed approach paves the way for efficient design of systems with vibro-impact NESs.