Nonlinear Gear Research Articles

NONLINEAR DYNAMIC RESPONSES OF LOCOMOTIVE EXCITED BY RAIL CORRUGATION AND GEAR TIME-VARYING MESH STIFFNESS

Rail corrugation is usually generated in modern railway transportations, such as high-speed railway, urban railway, and heavy-haul railway. It is one of the major excitations to the wheel–rail dynamic interaction, which will cause extra vibration and noise, failures, or even risk of derailment to the vehicle and its components. A dynamics model of a heavy-haul locomotive considering the traction power from the electric motor to the wheelset through gear transmission is employed to investigate the nonlinear dynamic responses of the locomotive. This dynamics model couples the motions of the vehicle, the track, and the gear transmission together. In this dynamics model, excitations from the rail corrugation, the nonlinear wheel–rail contact, the time-varying mesh stiffness, and the nonlinear gear backlash are considered. Then, numerical simulations are performed to reveal the dynamic responses of the locomotive. The calculated results indicate that different nonlinear phenomenon can be observed under the excitation of the rail corrugation with different amplitude and wavelength. The high frequency vibrations excited by the time-varying mesh stiffness are usually modulated by the low frequency vibrations caused by the rail corrugation. However, this is likely to vanish under the chaotic conditions with some corrugation wavelength. The vibration level of the vehicle and the gear transmission increases generally with the corrugation amplitude. However, some corrugation lengths have been found to be more responsible for the vibration of the dynamics system, which should be concerned greatly during the locomotive operation. Meanwhile, involvement of gear transmission systems will cause different dynamic responses between the wheelsets under rail corrugation and gear mesh excitations.

A PID Control Method Based on Internal Model Control to Suppress Vibration of the Transmission Chain of Wind Power Generation System

Vibrations occur in the wind turbine drivetrain due to the drivetrain’s elasticity and gear clearance. Typically, the PID control method is used to suppress elastic vibration, while the method with the disturbance suppression function is used to suppress nonlinear gear clearance vibration. The purpose of this paper is to propose an equivalent PID control method based on the internal model control (IMC) for suppressing vibration caused by the wind turbine drivetrain’s elasticity and gear clearance. The equivalent PID controller can suppresses elastic vibration; the IMC controller can suppresses gear clearance vibration. First, the vibration principle of the two-mass wind turbine drivetrain with clearance is discussed. After analyzing the nonlinear characteristics of the gear clearance, the nonlinear clearance system is decomposed into a linear unit and a nonlinear bounded disturbance unit. To suppress nonlinear bounded disturbances, a disturbance suppression method based on IMC is proposed; simultaneously, an equivalent PID controller based on IMC is designed to resolve the vibration issue caused by the wind turbine drivetrain’s elasticity. The simulation experimental results show that the clearance vibration is suppressed by the original IMC method. The PID controller obtained by the IMC equivalent transformation can suppress the elastic vibration.

Analysis on Dynamic Contact Performance and Microscopic Trimming Optimization of Pure Electric Bus Transmission Gears

Given the differing structure of transmissions for pure electric vehicles from the conventional ones, how to enhance the transmission performance is the key to improving their dynamic property and economical efficiency. In this work, a two-speed planetary transmission for a self-designed 10–12 m bus is studied. A simulation model of the transmission is created according to the gear meshing principle, and the load spectra are established by simulating the probable loads on the transmission in actual operation. The gear drive process of the transmission is analyzed, as along with the gear safety factors, damage rates and meshing dislocation. Trimming curves are derived based on the principles of axial and tooth profile modification, the trimming amounts are set, and the contact spots and transmission errors of gear meshing obtained before and after trimming are analyzed comparatively using the MASTA’s gear microscopic trimming module, thereby determining the reasonable method and quantity of trimming for the studied transmission gear. Comparison of the contact spots and transmission errors before versus after trimming finds greatly improved contact condition of gear and more stable gear transmission. With the methodology described in this paper, the shortcomings of traditional gear trimming, such as cumbersome process, long duration and nonlinear gear drive system, can be overcome, which improves the operational reliability.

Erratum to “Quadratic Integral Sliding Mode Control for Nonlinear Harmonic Gear Drive Systems with Mismatched Uncertainties”

Erratum to “Quadratic Integral Sliding Mode Control for Nonlinear Harmonic Gear Drive Systems with Mismatched Uncertainties”

Investigation of gear rattle noise including visualization of vibro-impact regimes

This paper presents an experimental study of gear rattle noise induced by vibroimpacts between gear teeth. A specific experimental set-up is designed to analyse the nonlinear dynamic behaviour of a spur gear submitted to input velocity fluctuation. The drag torque, the mean drive gear rotational speed, the velocity fluctuation amplitude and frequency are controlled during experiment. The dynamic transmission error is measured thanks to high resolution optical encoders. The originality of the experimental set-up consists in using a high-speed camera in order to visualize the contact zone and to identify the occurrence of successive impacts between gear teeth. The rattle threshold is identified as a function of velocity fluctuation amplitude and frequency for various operating drag torques and mean rotational speeds. Experiments show very good agreement with the theoretical master curve. Once impacts occur, stationary nonlinear gear dynamic response and rattle noise radiated by the mechanical system are investigated. Most of the time, an almost periodic response is observed with 2 impacts per period. One impact between the active flanks alternates with one impact between the reverse flanks. A contact phase between gear teeth is observed after each impact instead of an instantaneous rebound. The number of successive tooth pairs crossing the meshing zone without any contact between gear teeth varies according to the ratio of the excitation frequency to the rotation frequency. Analytical and numerical works performed using a gear rattle model show good agreement with experiments. Finally, sound pressure emitted from the gear pair is measured. The acoustic power imputable to gear rattle is found to be proportional to the total kinetic energy transferred per second to the system by the successive impacts.

Vibration Control of Automotive Drive System With Nonlinear Gear Backlash

Abstract In automotive drive systems, differential gear backlash degrades the control performance. Specifically, a shock torque, which is generated when the gear runs freely and collides with the backlash, increases the vibration amplitude. Consequently, it is important to develop a vibration control method to suppress the adverse effect of nonlinearity due to backlash. Furthermore, considering implementations on actual vehicles, design at the development site, and mass production, a simple and practical control method is necessary. This paper describes the configuration of a basic experimental device, which abstracts an actual vehicle to focus on the influence due to backlash while reflecting the basic structure of an automotive drive system. Next, a basic controller is designed using a mixed H2/H∞ control theory, and a servo system is constructed to track the target value. A simple control mode switching algorithm is proposed for backlash compensation. This algorithm is suited to practical applications because it uses only an output without a state estimation and it compensates for performance deteriorations due to the nonlinearity by operating a single linear controller. Finally, simulations and experiments verify the effectiveness of the proposed control system.

Investigation of Gear Dynamic Characteristics under Stochastic External Excitations

Engine drive gear systems are widely used in lots of mechanical equipment. In engine drive gear systems, the external excitations, including the driving speed and external load, are generally stochastic due to manufacturing error, energy loss, variation of operating environmental, and so on. The reported works usually considered them as deterministic. In this paper, a stochastic nonlinear gear dynamic model considering the stochastic external excitations is presented. Then, the effects of the stochastic driving speed and external load on gear dynamic characteristics are investigated by Monte Carlo (MC) simulation. By the simulation results, some insightful conclusions, which are important for the gear random dynamics, are obtained.

Effect of strain rate on bending and transmission characteristics of injection molded polyamide 66 spur gears

Polymer material exhibits time-dependent mechanical behavior due to its viscoelastic characteristics. Thus, unlike steel gears, polymer gears exhibit complex behavior when transmitting loads at various rotational speeds. In general, gear tooth surfaces exhibit complex stress due to their nonconformal geometry. Hence, the nonlinear material and nonlinear geometric aspects of polymer gear mesh prompted an investigation of the bending and transmission characteristics of injection molded polyamide 66 gears at various strain rate conditions. The injection molded tensile specimens made from the gear material were subjected to various rates of loading. The stress–strain performance at various rates of loading was evaluated and used to model linear and nonlinear gear materials for the numerical analysis. The numerical investigation was carried out on the steel–polyamide gear pair with the commercial finite element analysis tool ABAQUS® to predict the bending stress and static transmission error. The predicted static transmission error of the gear pair was compared with the experimental results obtained using an in-house developed gear test rig. The bending stress with the linear material models was higher than that of the nonlinear material models. An increase in bending stress with the strain rate was observed in the case of the nonlinear material models. The static transmission error predicted with the nonlinear material model at a higher strain rate was lower for both the single tooth contact and the double teeth contacts.

An analytical study of controlling chaotic dynamics in a spur gear system

In order to design and develop an optimal gear transmission system, it is important to control the occurrence of various types of nonlinear phenomena such as bifurcation and chaotic response. This paper describes a control system for elimination of the chaotic behaviors in a gear dynamic system. Analytical approach concerning the elimination of chaos in a gear system with applying the external control excitation is given by using Melnikov method. The numerical simulations are considered to check the validity of theoretical predictions, and also to investigate the efficiency of the proposed control system to eliminate the homoclinic bifurcation and chaos in nonlinear gear systems.

The nonlinear dynamics response of cracked gear system in a coal cutter taking environmental multi-frequency excitation forces into consideration

The normal operation of gear systems ensures the power transmission in coal cutters. Due to harsh work environment, the gears in coal cutters are prone to crack. By analyzing the gear dynamic response, useful crack detection indicators could be obtained. The literature review indicates that most of the gear dynamic models have been built with single-frequency excitation, but very limited work has considered multi-frequency excitations. The coal cutter gear systems are always subjected to multi-frequency excitations. Hence, a nonlinear gear dynamic model taking multi-frequency excitations into account was established in this work. A meshing stiffness coefficient was defined to model the crack fault influence on the gear model. The amplitude–frequency of the model main resonance was analyzed by means of multi-scale method, and the model dynamic response was calculated by the incremental harmonic balance method. Numerical simulation results show that the dynamic response of the presented gear model provided strong chaotic characteristics and the chaotic degree of the gear dynamics increased with the deterioration of the crack level. Experimental validation was carried out on a real gearbox, and the analysis results were consistent with the simulations. Hence, the simulation and experimental analysis demonstrates that the multi-frequency excitation-based gear dynamic model was more correct than the single-frequency excitation-based model in representing the system dynamics.

Comparisons of gear dynamic responses with rectangular mesh stiffness and its approximate form

The mesh stiffness is close to rectangular stiffness, and the first harmonic approximate term of rectangular stiffness is generally adopted in the nonlinear gear dynamic analysis. The differences between the rectangular stiffness and its approximate form are analyzed in detail. The frequency response and dynamic factor are calculated by a numerical method, to illustrate the dynamic characteristics of the gear nonlinear system with different mesh stiffness forms. The results show that: The trends of frequency response of gear dynamic system with rectangular stiffness and its approximate form are identical. The jump phenomena are detected in both cases. Without the effect of static transmission error, the dynamic factor with rectangular mesh stiffness is larger than that with approximate mesh stiffness. Under design power and speed condition, the result with approximate mesh stiffness function may deduce reasonless suggestions for a designer. The static transmission error will enlarge the vibration amplitude and dynamic factor when the approximate mesh stiffness is adopted, but the effects on the response of gear system with rectangular mesh stiffness are fractional. The mesh stiffness may excite the odd subharmonic resonance, and the static transmission error may excite the even sub-harmonic resonance respectively.

Research on Adaptive Load Sharing Control for Multi-Motor Synchronous Driving System of Shield Machine

This paper presents an adaptable controlling approach on account of the mathematic modeling and the quantitative analysis of experimental data on a ring-coupling controlling structure. It is based on the research of transmission characteristics of non-linear coupling gear rotation in main electric-driving system of cutterhead, and it is aimed at reducing the deviation of system synchronization caused by the mutation load, and improving the compliance capability of multi-motor synchronization system.

Identification and control of chaos in nonlinear gear dynamic systems using Melnikov analysis

In this paper, the Melnikov analysis is extended to develop a practical model of gear system to control and eliminate the chaotic behavior. To this end, a nonlinear dynamic model of a spur gear pair with backlash, time-varying stiffness and static transmission error is established. Based on the Melnikov analysis the global homoclinic bifurcation and transition to chaos in this model are predicted. Then non-feedback control method is used to eliminate the chaos by applying an additional control excitation. The regions of the parameter space for the control excitation are obtained analytically. The accuracy of the theoretical predictions and also the performance of the proposed control system are verified by the comparison with the numerical simulations. The simulation results show effectiveness of the proposed control system and present some useful information to analyze and control the gear dynamical systems.

Spectral Density Prediction for Response of Nonlinear Gear Pairs under Random Excitation

This paper investigates the dynamic response of a gear pair under random excitation from the view point of spectral density. Two methods are used to calculate the response spectral density function (SDF). The first method uses the statistical linearization (SL) technique to find an equivalent linear system to the original nonlinear one, then the response SDF is calculated with the equivalent linear system. While the second method regards the natural frequency of the system as a function of the response amplitude which is a random variable and its probability density function (PDF) is computed through stochastic averaging (SA). Then the response SDF is computed as a probabilistic averaging over the whole range of amplitude. Simulation result shows that both methods can predictthe resonant frequency very well and consistently in the case of weaknonlinearity. But with increasing of nonlinearity, the SDF predicted from the SL technique tends to be narrower than that from method II.

Dynamic Analysis of a Spur Geared Rotor-Bearing System with Nonlinear Gear Mesh Stiffness

The gear mesh stiffnesses have been regarded as constants in most previous models of geared rotor-bearing systems. In this paper, a dynamic analysis of a spur geared rotor-bearing system with nonlinear gear mesh stiffness is presented. The nonlinear gear mesh stiffness is accounted for by bending, fillet-foundation and contact deflections of gear teeth. A finite element model of the geared rotor-bearing system is developed, the equations of motion are obtained by applying Lagrange’s equation, and the dynamic responses are computed by using the fourth-order Runge-Kutta numerical method. Numerical results indicate that the proposed gear mesh stiffness provides a realistic dynamic response for spur geared rotor-bearing system.

Bifurcation and Chaos Prediction in Nonlinear Gear Systems

The homoclinic bifurcation and transition to chaos in gear systems are studied both analytically and numerically. Applying Melnikov analytical method, the threshold values for the occurrence of chaotic motion are obtained. The influence of system parameters on the character of vibration is studied. The numerical simulation of the system including bifurcation diagram, phase plane portraits, Fourier spectra, and time histories is considered to confirm the analytical predictions for the occurrence of homoclinic bifurcation and chaos in nonlinear gear systems.

Chaos and Bifurcation Analysis of Gear Pair with Wear Fault

The spur gear pair’s nonlinear equation of motion including piece-wise backlash and internal error excitation is derived in this research. The worn tooth effect in time-varying mesh stiffness is introduced to do in-depth investigation of the dynamic traits for gear transmission system with wear fault. The internal excitation frequency is selected as a criterion to calculate the bifurcation diagram and the corresponding Lyapunov exponents. Some auxiliary analyzing meanings such as Poincaré maps, phase trajectory, power spectrum and time history curve are utilized to illustrate the system’s nonlinear behaviors with special parameter settings. Different routes to chaos and abundant nonlinear phenomena have been observed in this nonlinear gear transmission system.

Large DOF Coupler/Draft Gear System Models for Rail Vehicles

Aiming at accurate predictions of coupler/daft gear systems' dynamic behaviour. Based on the analysis of structural and kinematic relations of two typical systems, hysteretic nonlinear draft gear models and 9DOF (Degree of freedom) detailed coupler/draft gear system models were developed. Frictional pairs on coupling surfaces and coupler tails along with interactive aligning shoulder models were integrated into system models. Utilizing two detailed locomotive models simulations were performed on a section of special track to validate the rationalities of coupler/draft gear system models. Simulation results indicate that all parts in coupler/draft gear system models can fullfill their roles, and the system models themselves can reproduce the run-time behaviour of coupler/draft gear systems with high reliability and accuracy.

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.

Analysis of Global Character of Single Degree of Freedom Nonlinear Gear System Based on Digraph Cell Mapping Method

The mathematical model of single degree of freedom gear system with time varying stiffness and clearances is set up.The digraph cell mapping method based on graph theory and generalized cell mapping theory is introduced,and accordingly a digraph cell mapping algorithm for solving nonlinear problems is put forward.The global character of nonlinear gear system is analyzed by using the digraph cell mapping algorithm,and a series of the attractors,the domain of attraction,unstable solutions,and the attractive basin boundaries of the gear system are obtained.The result shows: there are one,three or two stable attractors respectively in the system when is 0.45,0.80,or 1.50 and there is also an unstable attraction when is 1.50.Finally,comparative analysis is carried out by using the phase diagram of the system.Initial points are picked up from different domains of attraction and converge at corresponding stable attractors,thus verifying the effectiveness and credibility of the digraph cell mapping algorithm.Using the result of digraph cell mapping method,a good system response can be obtained by rationally controlling the initial conditions in case of bad parameters.