Spur Gear System Research Articles

Bifurcation and Erosion of Safe Basin for a Spur Gear System

Tooth surface contact deformation is part of the main causes for gear system vibration. The safety region of the single-stage spur system vibration is established based on the tooth surface contact deformation. Safe basin and its erosion are calculated numerically according to the safety region as the system control parameters are varied. The basin of attraction in the safe basin is computed by combining the simple cell-to-cell method. Vibration safety and global dynamics of the gear system are investigated. Some bifurcations and mechanisms of the erosion of safe basin are studied by using phase portraits, Poincaré maps, bifurcation diagrams under multi-initial values and top Lyapunov exponents (TLE). The sensitivity of system behavior and bifurcation to initial values are discussed as well. Hidden bifurcating points and attractors are revealed. To get a better understanding of the sensitivity of the system behavior to initial values, a bifurcation dendrogram under multi-initial values is designed. It is found that there is the erosion of safe basin existing in the examining area. Both vibration amplitude changing of coexisting attractors and appearing or disappearing of coexisting attractors are the main causes for the erosion of the safe basin. Bifurcation of the system behavior is selective to initial values. With varying frequency, backlash and comprehensive transmission error, the period-1 response gradually bifurcates under multi-initial conditions and evolves into other new periodic responses that coexist with the period-1 response. The new coexisting response has a key effect on the erosion of the safe basin. The study is helpful to the optimization design of the gear system and the control of the system behavior.

Nonlinear characteristics of a multi-degree-of-freedom spur gear system with bending-torsional coupling vibration

Abstract The paper deals with bending-torsion coupling response of spur gear system due to its complex excitations. Based on the Bartelmus model, a multi-degrees-of-freedom lumped parameter dynamic model of a one-stage gear system, which includes gear mesh stiffness fluctuations, transmission errors, gear backlash and torque loads, is established. Considering the misalignment of gear root circle and base circle, an improved mesh stiffness model is applied to calculate the time-varying mesh stiffness by simplifying the gear tooth as a cantilever beam on the root circle. Based on these equations of gear system, the dynamic orbits of the bending-torsion coupling system are observed using bifurcation diagrams plotted with the frequency ratio and torsional stiffness as control parameters. Meanwhile the dynamic properties of the gear system are identified from the time domain response diagrams, phase plane diagrams, Poincare maps and amplitude-frequency spectrums of the gear system. The numerical results show that the system exhibits a diverse range of periodic, sub-harmonic, and chaotic motion with the coupling effects of bending and torsional vibrations. The results provide us a further understanding about vibration generation mechanisms and work status of the spur gear system.

Effect of number of blades on the dynamic behavior of a Darrieus turbine geared transmission system

Abstract Dynamic behavior of a Darrieus-type vertical axis wind turbine geared transmission system taking into account variable gear mesh stiffness and the external aerodynamic torque excitation is carried out in the present paper. The aerodynamic torque is obtained by a complete campaign of simulations based on Reynolds Averaged Navier-Stokes unsteady calculations for two, three and four-bladed rotor architectures. A three-dimensional model of a one-stage bevel spur gear system is developed. For each proposed rotor configuration, the effect of the number of blades on both aerodynamic performance and dynamic vibration of the Darrieus turbine geared transmission system is discussed at several tip speed ratios in non-stationary operation. The main originality of the present paper is to establish a correlation between the aerodynamic parts and the dynamic vibration of the gearing system by studying the effect of some design parameters on the dynamic vibration of the Darrieus turbine gearing system in non-stationary regime. This new methodology of simulation makes considerable progress in the understanding of the wind turbine dynamic vibration, which is an important issue. The number of the blades is the core problem of wind turbine engineering. In this paper, the numerical results show that the number of blades significantly affects the turbine efficiency and the dynamic response of the studied wind turbine gearing system in non-stationary regime.

Dynamic analysis of nonlinear time-varying spur gear system subjected to multi-frequency excitation

Due to the complex working environment, gear systems always suffer from multiple excitations in actual engineering. This paper concerns the frequency response characteristics of a nonlinear time-varying spur gear system subjected to multi-frequency excitation. Firstly, a single degree-of-freedom gear pair model is established with consideration of the gear backlash, time-varying mesh stiffness and multiple harmonic excitations. Then, using the multiple time scales method, a comprehensive theoretical study is conducted to analyze various resonant cases including primary, parametric and combination resonances. Besides, parametric studies are accomplished to reveal the effects of the multi-frequency excitation on gear dynamics and to provide some useful references for reducing the vibration level. With the help of the fifth-order Runge–Kutta method, the numerical results are obtained to verify the validity of the analytical solutions and to emphasize the significances of the multi-frequency excitation. In addition, a comparison is performed between the numerical results and the published experimental results to validate the proposed gear model. Results show that the presence of the multi-frequency excitation will introduce the interaction between different harmonic excitations, which significantly affects the nonlinear vibration characteristics of a spur gear system. The proposed gear model with multi-frequency excitation could be more reliable and universal than that with single-frequency excitation. In addition, the results of parametric study could provide some suggestions to designers and researchers attempting to obtain desirable dynamic behaviors of a gear system subjected to multi-frequency excitation.

Bifurcation and chaos analysis for a spur gear pair system with friction

Considering time-varying meshing stiffness, gear backlash, static transmission error and tooth face friction, a nonlinear dynamic model for a spur gear pair is proposed to research systematically the dynamic behaviors of system, in which the meshing stiffness of gear pair is deduced and calculated in terms of the extending period method. Meanwhile, the sliding friction force under single-tooth and double-tooth meshing regions is constructed as a function of the meshing principle. Based on the developed model, the bifurcation and chaos characteristics of system under lightly and heavily loaded conditions are studied, respectively, by applied Runge–Kutta numerical method, and the parametric effects of rotational speed, damping ratio and gear backlash on the dynamic behaviors are investigated detailedly. Bifurcation diagram, three-dimensional frequency spectrum, time-domain waveform, frequency plot, phase diagram, Poincare map and dynamic load are used to discuss and determine motion states and dynamic responses of system. The numerical results represent that with the change of control parameters the system undergoes various types of motion states under different loaded conditions. The corresponding meshing contact states of tooth pair are transformed among no impact, single-sided impact and double-sided impact. The research results can provide certain guidance for choosing suitable parameter values to reduce the amplitude of vibration and even avoid the chaotic response in gear system.

Effect of the gear local damage and profile error of the gear on the drivetrain dynamic response

The dynamic modeling of vibration of a drivetrain is used for increasing our information about vibration generating mechanisms, especially in the presence of some kind of gear faults. This paper describes a research work on the automotive driveline modeling, vibration analysis, and the effect of gear defects on the dynamic behavior of the system. Firstly, main drivetrain components including the engine, clutch, single stage spur gearbox and disc brake are modeled, respectively. The nonlinear dynamic model is simulated by a thirteen degrees of freedom (DOF) system and the nonlinear function is due to the dry friction path. Secondly, two types of defects are modeled and introduced into the spur gear system; local damage and profile error. Then, the nonlinear equations of motion are solved by the numerical Runge Kutta method and a comparative study of the dynamic behavior of the system in healthy and defected cases is discussed for each fault type. The influence of the defects on the vibration response is presented in the time and frequency domain. Finally, analysis of the two defects together is presented.

Bifurcation and chaos analysis of a gear pair system with multi-clearance

In order to investigate the characteristics of bifurcation and chaos for a spur gear pair system, a three-degree-of-freedom nonlinear dynamic model with multi-clearance is established, in which time-varying meshing stiffness, static transmission error, gear backlash and bearing clearance are comprehensively taken into account. Through introducing a relative generalized coordinate, the dimensionless dynamic equations of motion of system are derived and then solved by using Runge-Kutta numerical integration method. And the bifurcation and chaos features of gear pair are systematically analyzed and discussed from bifurcation diagrams with meshing frequency, gear backlash, bearing clearance and damping ratio as control parameters under different loaded conditions. Meantime, with the help of Poincaré map and phase diagram, the motion forms of system are accurately identified. The analysis results reveal that as meshing frequency increases, the system shows various types of motion states which contain periodic motion, quasi-periodic motion and chaotic motion. Similarly, with the increasing of gear backlash, the system undergoes complex motion forms under lightly loaded condition, whereas it is only in period-one motion state under heavily loaded condition. Furthermore, the system motion state is gradually switched from chaos to periodic or quasi-periodic motion under lightly loaded condition when bearing clearance changes. However, under heavily loaded condition, the bearing clearance has a weak effect on dynamic behavior of the gear system. Apparently, the system tends to be more stable under heavily loaded condition than that under lightly loaded condition. In addition, the growing damping ratio can effectively suppress the chaotic behavior and control nonlinear vibration of gear system. The research results provide useful guidance for dynamic design and vibration control for gear set.

Study on dynamic characteristics of spur gear transmission system based on different calculation models of time-varying mesh stiffness

Three kinds of gear mesh stiffness calculation models are developed for spur gear transmission system in this paper, namely, the dynamic model of gear transmission system based on the analytical mesh stiffness calculation by “slice method”, the gear transmission system dynamic model based on SIMPACK 225 force element, and the finite element dynamic model of gear transmission system. Based on these dynamic models, the mesh force and dynamic transmission error for the high speed and the low speed operation conditions are analyzed, respectively. The research results indicate that the results obtained by the three dynamic models are in good agreement with each other for both low and high speed operation conditions. The dynamic transmission error calculated by gear transmission system dynamic model based on mesh stiffness calculation with “slice method” are much closer to the results obtained by the finite element dynamic model. Consequently, considering the computation accuracy and computation time, the gear transmission system dynamic model based on the mesh stiffness calculated by “slice method” could be better applied to analyze the dynamic performance of gear transmission system considering the trade off between the computational accuracy and the efficiency.

스퍼 기어계의 진동해석에서 구름 베어링의 강성효과

스퍼 기어계의 진동해석에서 구름 베어링의 강성효과

A nonlinear multi-point meshing model of spur gears for determining the face load factor

The face load factor, which takes into account the influence of the non-uniform distribution of the load over the face width on the resulting stresses, is a significant coefficient of the load capacity calculation of spur gears. In this study, a practical nonlinear multi-point meshing model of spur gears is proposed for the determination of the face load factor. Nonlinear gear meshing conditions over the face width can be simulated precisely and efficiently using the proposed nonlinear multi-point meshing model, which is composed of multiple nonlinear springs along the line of action and rigid bars connected with shaft beam element models. The static equilibrium condition of the entire spur gear system is obtained through an iterative calculation process, and both the load distribution and face load factor are obtained. The effectiveness and advantages of the proposed model are demonstrated through comparisons with different types of contact analysis models presented by previous studies.

Nonlinear dynamic response analysis of two-stage spur gear space driving mechanism with large inertia load

Large inertia load has been widely used in space driving mechanisms, but the research concerning theory is still an unexplored scientific field. Towards the problem of nonlinear disturbance in space driving mechanism with large inertia load, a 14-DOF (Degree of Freedom) nonlinear, time-varying, dynamic model of two-stage spur gear system was established, taking into consideration time-varying stiffness, backlash and transmission error. The dynamic response of the load, under large or small inertia was investigated, basing on the dynamic model. The results indicate that at starting, normal operation and braking, large inertia load system has obvious hysteresis, compared to small inertia. The factors that improve dynamic response speed under large inertia load were studied. The results indicate that improving the stiffness and damping of the output shaft and changing the material of second gear pair to titanium alloy are helpful in improving the dynamic response speed of the system. Some results enrich the research of two-stage spur gear nonlinear model and large inertia load, since they provide important reference for the actual design of the gear system.

Damping models identification of a spur gear pair

In this research paper, estimation damping in spur gear pair system using Integral method and the continuous wavelet transform based approach (CWT) is proposed. In the first method, the motion equations of the model describing the system are reformulated in terms of integrals that give a set of linear equations solved by linear least squares method to allow a new methodology that starts with a constant piecewise model to bootstrap the most suitable model of damping. The constant model is validated using the wavelet demodulation approach that's based on the envelope extraction procedure to obtain the damping ratio associated with structural modes. In order to show the ability to predict the response, damping estimation procedures are tested on a signal obtained from numerical simulation of the spur gear pair system.

An implicit algorithm for the dynamic study of nonlinear vibration of spur gear system with backlash

In this work, we propose some regularization techniques to adapt the implicit high order algorithm based on the coupling of the asymptotic numerical methods (ANM) (Cochelin et al., Méthode Asymptotique Numérique, Hermès-Lavoisier, Paris, 2007; Mottaqui et al., Comput. Methods Appl. Mech. Eng. 199 (2010) 1701–1709; Mottaqui et al., Math. Model. Nat. Phenom. 5 (2010) 16–22) and the implicit Newmark scheme for solving the non-linear problem of dynamic model of a two-stage spur gear system with backlash. The regularization technique is used to overcome the numerical difficulties of singularities existing in the considered problem as in the contact problems (Abichou et al., Comput. Methods Appl. Mech. Eng. 191 (2002) 5795–5810; Aggoune et al., J. Comput. Appl. Math. 168 (2004) 1–9). This algorithm combines a time discretization technique, a homotopy method, Taylor series expansions technique and a continuation method. The performance and effectiveness of this algorithm will be illustrated on two examples of one-stage and two-stage gears with spur teeth. The obtained results are compared with those obtained by the Newton–Raphson method coupled with the implicit Newmark scheme.

Transmission Error Analysis and Disturbance Optimization of Two‐Stage Spur Gear Space Driven Mechanism with Large Inertia Load

For the nonlinear disturbance actual issues of the model space drive mechanism two‐stage spur gear system, a nonlinear dynamic model of 14‐DOF (degree of freedom) two‐stage spur gear with time‐varying stiffness and damping was established. This model has been developed previously by the authors to access the large inertia on the dynamic response of spur gear space driving mechanism, and its effectiveness was proved by a motion simulation experiment. In this paper, the profile error (PE) and the index error (IE) were enhanced in the dynamic model. The effects of profile error, index error, and variable load torque on transmission error (TE) were analyzed, while the optimization was proposed according to the analyzed result. The peak‐to‐peak value of the optimized load transmission error (LTE) was reduced by 60.7%, which improved the transmission accuracy and reduced the phenomenon of disturbance. The research of nonlinear dynamical model of two‐stage spur gear and the TE of the large inertia load were enriched, which provided an important reference for the actual design of the gear system.

Chaos and Stability of Spur Gear Transmission System for Locomotive Based on Energy Method and Floquet Theory

Considering the internal and external excitations such as time‐varying mesh stiffness (TVMS), backlash, transmission error, torque of the traction motor, and load torque of the wheel/rail, a lumped mass model of the spur gear drive system for a railway locomotive is established. Based on Ma models in the relevant literatures, TVMS is calculated by simplifying a gear tooth as a cantilever beam on the root circle, taking into account the effects of extended tooth contact as well as revised foundation stiffness. The bifurcation diagrams and Lyapunov exponent curves of the model parameters are drawn by the numerical method, and the mechanism of chaos evolution of the gear transmission system is analyzed. According to the Floquet theory, variation curves of the maximum Floquet multiplier with pinion speed and support stiffness ratio are drawn by numerical methods. Combined with the bifurcation diagram of the system, the influences of model parameter on the stability of the system are analyzed, and the evolution laws of periodic motion and bifurcation phenomenon are gained. These research results provide the theoretical evidence of model parameter design of the locomotive transmission system.

Volume models for different structures of spur gear

The volume of gear transmission system has been concerned by people. At present, the researchers mainly focus on the optimization algorithm, but there is a lack of research on the volume model of gear transmission system. In the paper, the single-stage spur-gear transmission system is considered and the volume models for different structures of spur gear are built. Firstly, the influence of bottom clearance and the modification coefficient on the spur-gear volume are analysed. The formula of clearance volume is derived. Secondly, volume models of different structure of spur gear are built. In order to illustrate the accuracy of the models, the corresponding three-dimensional models of different structure of spur gear are established. Finally, the design parameters of spur gear are as the optimisation variables, the spur-gears design and transmission requirements are as the constraint conditions, the total minimum volume of single-stage spur-gear transmission system is as the optimisation target, the genetic algorithm (GA) toolbox in Matlab software is adopted in the optimisation. A single-stage spur-gear system is as an example, and the optimisation for minimum volume of single-stage spur-gear transmission is carried out.

Nonlinear dynamic model of a gear-rotor-bearing system considering the flash temperature

The instantaneous flash temperature is an important factor for gears in service. To investigate the effect of the flash temperature of a tooth surface on the dynamics of the spur gear system, a modified nonlinear dynamic model of a gear-rotor-bearing system is established. The factors such as the contact temperature of the tooth surface, time-varying stiffness, tooth surface friction, backlash, the comprehensive transmission error and so on are considered. The flash temperature of a tooth surface of pinion and gear is formulated according to Blok's flash temperature theory. The mathematical expression of the contact temperature of the tooth surface varied with time is derived and the tooth profile deformation caused by the change of the flash temperature of the tooth surface is calculated. The expression of the mesh stiffness varied with the flash temperature of the tooth surface is derived based on Hertz contact theory. The temperature stiffness is proposed and added to the nonlinear dynamic model of the system. The influence of load on the flash temperature of the tooth surface is analyzed in the parameters plane. The variation of the flash temperature of the tooth surface is studied. The numerical results indicate that the calculated method of the flash temperature of the gear tooth surface is effective and it can reflect the rules for the change of gear meshing temperature and sliding of the gear tooth surface. The effects of frequency, backlash, bearing clearance, comprehensive transmission error and time-varying stiffness on the nonlinear dynamics of the system are analyzed according to the bifurcation diagrams, Top Lyapunov Exponent (TLE) spectrums, phase portraits and Poincaré maps. Some nonlinear phenomena such as periodic bifurcation, grazing bifurcation, quasi-periodic bifurcation, chaos and its routes to chaos are investigated and the critical parameters are identified. The results provide an understanding of the system and serve as a useful reference in designing such systems.

Modeling and Analysis of High-Load and High-Speed Involute Spur Gear Systems in Adverse Lubrication

ABSTRACTThis article presents a mathematical model for the meshing of involute spur gears in mixed lubrication. The model is formulated by integrating a gear meshing model with a thermal tribofilm mixed lubrication model. The resulting model can produce a number of important meshing variables from which the gear meshing performance and the tooth surface damage may be assessed. The model is used to analyze the performance of a high-load and high-speed gear system under various degrees of mixed lubrication. The results show a stronger desire to use low-module and high-pressure-angle gears than that suggested in earlier studies under good lubrication conditions. The results also show the high importance of enhanced heat transfer design of the system to prevent or reduce the possibility of the gear system migrating into temperature-induced unfavorable lubrication conditions. Although the model is developed for spur gears and results are obtained around a nominal system, the relations of the gear meshing tribology to the meshing loss and tooth surface damage behind these results should be fundamentally the same as those in other high-load and high-speed gear systems of different sizes and types. Therefore, the results and analysis may serve as qualitative guidance for the design considerations of high-performance gear systems in general.

Bifurcation and Chaos Analysis of the Spur Gear Transmission System for One-Way Clutch, Two-Shaft Assembly

A four-degree-of-freedom nonlinear transverse and torsional vibration model of spur gear transmission system for one-way clutch, two-shaft assembly was developed, in which the one-way clutch was modeled as a piecewise nonlinear spring with discontinuous stiffness, considering the factors such as the time-varying gear mesh stiffness, static transmission error, and nonlinearity backlash. With the help of bifurcation diagrams, time domain response diagrams, phase plane diagrams, and Poincaré maps, the effects of the excitation frequency and the torsional stiffness of one-way clutch on the dynamic behavior of gear transmission system for one-way clutch, two-shaft assembly are investigated in detail by using Runge-Kutta method. Numerical results reveal that the system response involves period-1 motion, multiperiodic motion, bifurcation, and chaotic motion. Large torsional stiffness of one-way clutch can increase the impact and lead to instability in the system. The results can present a useful source of reference for technicians and engineers for dynamic design and vibration control of such system.

Experimental Study on Vibration Reduction Characteristics of Gear Shafts Based on ISFD Installation Position

A novel type of integral squeeze film damper (ISFD) is proposed to reduce and isolate vibration excitations of the gear system through bearing to the foundation. Four ISFD designs were tested experimentally with an open first-grade spur gear system. Vibration reduction characteristics were experimentally studied at different speeds for cases where ISFD elastic damping supports were simultaneously installed on the driving and driven shafts, installed on the driven shaft, or only installed on the driving shaft. Experimental results show that the ISFD elastic damping support can effectively reduce shock vibration of the gear system. Additionally, resonant modulation in gear shafts caused by meshing impact was significantly reduced. Different vibration amplitudes of gear shafts with ISFD installed only on driven or driving shafts were compared. Results indicated that vibration reduction is better when ISFD is only installed on the driven shaft than on the driving shaft.