Dynamic features of planetary gear set with tooth plastic inclination deformation due to tooth root crack

Gear tooth root crack, as one of the popular gear tooth failures, is always caused by the dynamic load or excessive load applied to the tooth. It will devastate the working performance of the gear system, by problems such as vibration and noise, or even lead to a broken tooth, which will stop the normal working process of the gear system. It has attracted wide attention from researchers. However, the previous studies focused their concentration only on the mesh stiffness reduction due to tooth root crack, while the tooth plastic inclination due to tooth bending damages like gear tooth root crack is seldom considered. In this paper, a tooth plastic inclination model for spur gear with tooth root crack is developed by regarding the cracked tooth as a cantilever beam. It influences not only the displacement excitation but also the mesh stiffness and load-sharing factor among tooth pairs in mesh. The simulation results obtained by incorporating the tooth plastic inclination deformation model together with the tooth root crack model into a 21-Degree-of-Freedom planetary gear dynamic model indicate that the tooth plastic inclination has a significant effect on the performance of the gear system rather than the mesh stiffness reduction due to tooth root crack.     

Modeling and analyzing of torsional dynamics for helical gear pair considered double and three teeth drive-side meshing

Helical gears are generally considered to be more stable than spur gears. But rattling of the helical gear transmission is found in the engineering practice. The torsional dynamics equations of helical gear pair in high-speed railway gearbox are established in order to reveal the rattling mechanism of helical gear transmission. Double and three teeth pair drive-side meshing are considered. The multi-state meshing zone, load distribution rate and time-varying stiffness determined by contact ratio are analyzed and calculated. The dynamic characteristic transition process of the system is analyzed according to the bifurcation diagrams and the corresponding top Lyapunov exponent (TLE) diagrams, phase portraits, Poincaré maps and time history spectrums of dynamic meshing force based on the calculation of these parameters. The tooth disengagement, tooth back-side contact and their parameter range are found. This study can provide theoretical basis for rattling suppression and transmission stability improvement of helical gear pair.

     

Dynamic characteristic analysis of spur gear system considering tooth contact state caused by shaft misalignment

The effect of gear contact state change due to shaft misalignment on meshing stiffness is usually neglected in the traditional stiffness calculation model with misalignment error, the further influence mechanism of shaft misalignment on gear dynamic characteristics is also unclear. To address these shortcomings, a new mesh stiffness calculation model with misaligned gear considering the effects of tooth contact state is proposed by combining the improved loaded tooth contact analysis (LTCA) model. Then the effects of tooth contact state changes aroused by shaft misalignment on the meshing stiffness excitation are investigated. Moreover, a dynamic model of the misaligned gear system with 8 degrees of freedom (DOF) is established, and based on which the dynamic characteristics of the gear system are investigated and verified by experiment. The study results indicate that the proposed model can be used to evaluate the stiffness excitation and dynamic characteristics of the misaligned gear system with the tooth contact state taken into consideration. This study can provide a theoretical method for evaluating and identifying shaft misalignment errors.

     

双联行星轮角度偏差对系统均载特性的影响

为了研究双联行星齿轮在实际设计参数下,其相对角度偏差对复合行星传动系统动力学特性的影响,采用集中参数法建立了3K-I型行星齿轮动力学模型,模型中将双联行星齿轮的相对角度偏差转化为啮合副齿侧间隙的变化,考虑了双联齿轮角度偏差、轮齿侧隙和时变啮合刚度等非线性因素,采用龙格库塔法求解了系统的时域响应并计算其均载系数。分析了不同工况、偏差下系统的动态特性。结果表明,存在双联行星轮角度偏差时,轻载下更容易发生齿轮的脱齿与冲击,系统的均载系数随着双联行星轮角度偏差差值及系统的负载降低而增大,各组行星轮角度偏差分布越集中,角度偏差对系统均载特性的影响越小;角度偏差分布同号时,对系统中某一对齿轮的承载影响明显;角度偏差分布异号时,对系统均载特性的影响最大。     

Bifurcation of multi-freedom gear system with spalling defect

This study focuses on the bifurcation characteristics of the four degree-of-freedom gear system with local spalling defect to explore the spalling nonlinear dynamic mechanism. The dynamic model of the gear system with spalling defect, time-variant mesh stiffness, and nonlinear clearance is established to investigate the effect of spalling defect on mesh stiffness and dynamic bifurcation. The primary resonance and internal resonance responses of the spalling model are analyzed by the averaging method, and the bifurcation characteristics with the evolvement of spall and internal excitation are studied by employing the singularity theory for the two-state variable system, which reveal the different bifurcation characteristics caused by the spalling defect. The results obtained herein can provide a theoretical basis to spalling fault diagnosis of gearbox.     

Nonlinear dynamic characteristic of a face gear drive with effect of modification

Face gear drive is one of the main directions of research for aeronautical transmission for its advantages, but the vibration induced gear noise and dynamic load are rarely involved by researchers. The present work examines the complex, nonlinear dynamic behavior of a 6DOF face gear drive system combining with time varying stiffness, backlash, time varying arm of meshing force and supporting stiffness. The mesh pattern of the face gear drive system is analyzed when the modification strategy is applied and the effect of modification on the dynamics response, the time varying arm of meshing force based on the TCA is deduced. The dynamic responses of the face gear drive system show rich nonlinear phenomena. Nonlinear jumps, chaotic motions, period doubling bifurcation and multiple coexisting stable solutions are detected but different from the spur and bevel gear dynamics, which don’t occur near primary and higher harmonic resonance.     

Enhanced model of load distribution along the line of contact for non-standard involute external gears

The presence of undercut at the tooth root, non-equal addendum on pinion and wheel, non-standard tooth height or non-standard center distance may have decisive influence on the load distribution along the line of contact of spur and helical gear teeth. The curve of variation of the meshing stiffness along the path of contact, quite symmetric respect the midpoint of the interval of contact, loses its symmetry for non-standard geometries and operating conditions. As a consequence, the critical contact points for bending and wear calculations may be shifted from their locations for standard gears. In this paper, a non-uniform model of load distribution along the line of contact of standard spur and helical gears, obtained from the minimum elastic potential criterion, has been enhanced to fit with the meshing conditions of the above mentioned non-standard cylindrical gear pairs. The same analytical formulation of the initial model may be used for the non-standard gears by considering appropriate values of a virtual contact ratio, which are also presented in the paper.     

The mathematical phenomenological mapping in non-linear dynamics of spur gear pair and radial ball bearing due to the variable stiffness

In this paper the philosophy of mathematical phenomenological mapping has been applied to the non-linear dynamics of spur gears and radial ball bearings. The spur gear pair dynamics and rolling element bearing dynamics are analyzed separately, but with a tendency to reduce the both of the systems to the same mathematical model. The different reasonable assumptions are taken in every of these analyzes, but they do not have significant influence to the accuracy of the results. The systems are reduced to the single degree of freedom dynamics model. The total gear stiffness and ball bearing stiffness are recognized as the main influent factor of vibration behavior of these machine elements. Therefore, the special attention was paid to the new approach and procedure for stiffness solving and related problems. A single spur gear pair dynamics is solved and the results for total gear stiffness and vibration are shown. The conclusions emphasize the importance of described parallel analyzes in order to reduce the calculation time in solving different phenomena with usage of the principle of mathematical phenomenology.     

磨损与动力学耦合的行星传动齿轮动力学研究

行星齿轮磨损会导致齿轮齿侧间隙非线性增大、传动精度下降、齿面冲击力增大, 进而会导致齿轮传动系统振动加剧, 因此需要对行星齿轮的齿面磨损与动力学耦合特性进行研究. 本文构建了齿轮非线性磨损与考虑齿轮齿侧间隙的非线性动力学耦合计算模型, 对行星传动齿轮磨损动力学特性进行了研究. 首先建立齿轮啮合非线性动力学模型, 获得齿轮运行过程中的非线性啮合力; 进一步将非线性啮合力与齿轮齿面磨损模型相结合, 研究齿轮齿面磨损分布规律; 并根据齿轮磨损后的齿侧间隙对齿面重构, 同时对齿轮动力学模型进行更新; 进而得到行星齿轮传动中动态啮合力和磨损特性的变化趋势, 并获得齿轮传动系统齿轮齿向振动响应. 数值计算结果表明, 行星齿轮磨损导致齿轮在单?双齿交替啮合时产生的冲击增大, 同时太阳轮?行星轮啮合齿对对磨损较为敏感, 齿面啮合条件剧烈恶化, 是造成行星齿轮传动性能退化的主要原因, 本文研究结果为行星齿轮传动系统运行状态评估与可靠性预测提供了理论基础.      

Mesh stiffness modelling and dynamic simulation of helical gears with tooth crack propagation

Meccanica - In order to identify gear crack failure, modelling the mesh stiffness of a cracked tooth is crucial. In this study, an analytical model of the mesh stiffness for cracked helical gears...     

An enhanced multi-term harmonic balance solution for nonlinear period-one dynamic motions in right-angle gear pairs

A nonlinear time-varying dynamic model for right-angle gear pair systems, considering both backlash and asymmetric mesh effects, is formulated. The mesh parameters that are characteristically time-varying and asymmetric include mesh stiffness, directional rotation radius and mesh damping. The period-one dynamic motions are obtained by solving the dimensionless equation of gear motion using an enhanced multi-term harmonic balance method (HBM) with a modified discrete Fourier Transform process and the numerical continuation method. The accuracy of the enhanced HBM solution is verified by comparison of its results to the more computational intensive, direct numerical integration calculations. Also, the Floquet theory is applied to determine the stability of the steady-state harmonic balance solutions. Finally, a set of parametric studies are performed to determine quantitatively the effects of the variation and asymmetry in mesh stiffness and directional rotation radius on the gear dynamic responses.     

Prediction of Gear Dynamics Using Fast Fourier Transform of Static Transmission Error*

ABSTRACT

An analytical computer simulation procedure for dynamic modeling of low-contact-ratio spur gear systems is presented. The procedure computes the gear static transmission error and uses a Fast Fourier Transform (FFT) to generate its frequency spectrum at various tooth profile modifications. The dynamic loading response of an unmodified (perfect involut) gear pair is compared with that of gears with profile modifications. Correlations are found between several profile modifications and the resulting dynamic loads. An effective error, obtained from frequency domain anal     

Theoretical analysis of nonlinear vibration characteristics of gear pair with shafts

The circumferential vibration of a gear pair is a parametric excitation caused by nonlinear tooth stiffness, which fluctuates with meshing. In addition, the vibration characteristics of the gear pair become complicated owing to the tooth profile error and backlash. It is considered that the circumferential vibration of the gear pair is affected by the torsional vibration of the shafts. It is important to understand quantitatively the vibration characteristics of the gear system considering the shafts. Therefore, the purpose of this research was to clarify the nonlinear vibration characteristics of a gear pair considering the influence of the shafts using theoretical methods. To achieve this objective, calculations were performed using equations of motion in which the circumferential vibration of the gear pair and the torsional vibration of the shafts were coupled. The nonlinear tooth stiffness was represented by a sine wave. The influence of tooth separation was considered by defining a nonlinear function using backlash and the tooth profile error. For the numerical calculations, both stable and unstable periodic solutions were obtained by using the shooting method. The effect of the shafts on the gear system vibration were clarified by comparing the results in the cases in which the shaft was not considered, one shaft was considered, and both shafts were considered.     

Using Hob Offset to Balance Dynamic Strength in Spur Gears*

An analytical study of the effect of hob offset on dynamic tooth strength of spur gears is presented. The study was limited to equal and opposite offset values applied to the pinion and gear to maintain the standard operating center distance. The analysis presented is performed using a new version of the NASA gear dynamics code DANST.

The operating speed of a transmission has a significant influence on the amount of hob offset required to equalize dynamic stresses in the pinion and gear. In the transmission studied, at low speeds, the optimum hob offset was found to fluctuate within a range. At higher speeds, the optimum value is constrained by the minimum allowed thickness at the tip of the pinion tooth. For gears that must operate over a range of speeds, an average offset value can be used. Spur gears designed with the procedure presented here can have significant improvements in load capacity.     

Nonlinear behavior analysis of geared rotor bearing system featuring confluence transmission

In this paper, the nonlinear vibration characteristics of geared rotor bearing system and the interactions among gears, shafts, and plain journal bearings were studied. First, with the consideration of backlash, transmission error, time-varying mesh stiffness, and layout parameters, the dynamic model of geared rotor bearing system featuring confluence transmission was proposed. The nonlinear oil-film forces were computed with the Reynolds equation for finite-length journal bearings. Second, the responses of meshing vibration and bearing vibration were discussed. The numerical results revealed that the system exhibited a diverse range of periodic, sub-harmonic, and chaotic behaviors. Under different ranges of rolling frequency, the system got into chaos state through different roads. Moreover, in lower frequency, meshing vibration showed coexist of different periodic motions. Lastly, couplings of nonlinear oil-film force and nonlinear gear mesh force were discussed through a range of rolling frequencies. Gear-bearing dynamic interactions were demonstrated through the analysis of dynamic gear loads and dynamic bearing loads, and the coupling effect behaved different when rolling frequency changed.     

Analytical models of mesh stiffness for cracked spur gears considering gear body deflection and dynamic simulation

Meccanica - Accurate mesh stiffness models for cracked spur gears are critical to obtain gearbox failure characteristics. However, the influence of crack on gear body deflection is often ignored by...     

Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems

The global homoclinic bifurcation and transition to chaotic behavior of a nonlinear gear system are studied by means of Melnikov analytical analysis. It is also an effective approach to analyze homoclinic bifurcation and detect chaotic behavior. A generalized nonlinear time varying (NLTV) dynamic model of a spur gear pair is formulated, where the backlash, time varying stiffness, external excitation, and static transmission error are included. From Melnikov method, the threshold values of the control parameter for the occurrence of homoclinic bifurcation and onset of chaos are predicted. Additionally, the numerical bifurcation analysis and numerical simulation of the system including bifurcation diagrams, phase plane portraits, time histories, power spectras, and Poincare sections are used to confirm the analytical predictions and show the transition to chaos.     

Effect of spalling or tooth breakage on gearmesh stiffness and dynamic response of a one-stage spur gear transmission

Tooth faults affecting gear transmission are always accompanied by a stiffness reduction. In this article an analytical method is proposed to quantify the reduction of gearmesh stiffness due to two common tooth faults: spalling and breakage. Bending, fillet-foundation and contact deflection are taken into account. The dynamic response of a single stage spur gear transmission is computed by using analytical gearmesh issued from analytical modelling and the vibration signatures of each tooth fault is identified.     

A review of space tether in new applications

Gear eccentricities are one of the practical types of the manufacturing errors that affect the dynamic performance of a planetary gear train (PGT). Previous research about the effects of the gear eccentricities is abundant, and many of them focus on the parallel shaft gear set. However, almost none of them have considered the influence of the gear eccentricities on the mesh stiffness. In fact, the existence of the gear eccentricities can change the center distance and the mesh positions of a meshing gear pair, which will directly affect the mesh stiffness. Situation can be even more complex for the PGT with either sun gear eccentricities or planet gear eccentricities or both of them. Based on that, a new dynamic model of a PGT with gear eccentricities is established. The planar motions of the PGT and the mesh stiffness are integrated and solved simultaneously where the mesh stiffness is determined by the actual mesh positions of the meshing gear pair. The mesh stiffness is calculated by the energy potential method. The time-varying center distance caused by the gear eccentricities is also considered, which can result in the change of line of action, pressure angle, contact ratio and mesh positions. The influence of gear eccentricities on the dynamic performance of a 4-planet PGT is studied. Some useful results are derived at last.     

齿轮接触有限元分析

通过接触仿真分析研究了通用接触单元在轮齿变形和接触应力计算中的应用。建立了一对齿轮接触仿真分析的模型，并使用新的接触单元法计算了轮齿变形和接触应力，与赫兹理论比较，同时也计算了摩擦力对接触应力的影响。计算分析了单元离散、几何、边界范围与加载或约束处理方式的误差，建立了一个计算轮齿变形和接触应力的标准，说明了新的接触单元法的精确法、有效性和可靠性。