Free transverse vibration analysis of a toothed gear

Analysis of transverse vibration of the gear found in a high-speed gearbox considered as an annular plate reflecting gear geometry is the subject of this paper. How gear angular velocity affects the deformation of normal modes of transverse vibration of the system under consideration is analysed. Models considered were discretized by the finite elements method. Numerical computations have been performed in the ANSYS environment. The algorithm to identify the proper distorted mode shapes is presented. The Campbell diagram for the system under consideration is elaborated. The problems discussed here can be useful for engineers dealing with dynamics of rotating machine systems.     

Nomographs for synthesis of epicyclic-type automatic transmissions

This paper presents a MATLAB optimization technique for the determination of the gear ratios of epicyclic-type transmission mechanisms for a given set of velocity ratios. First, all of the feasible clutching sequences are enumerated directly without using complicated techniques. Then, for the transmission mechanism with the associated clutching sequence graph, the overall velocity ratios are derived and expressed in terms of the gear ratios of all the mating gears. Next, following the general trend of increased shift stages and a wider range of velocity ratios, the numbers of teeth of all gears are estimated by MATLAB optimization technique in a single run. Ravigneaux gear mechanisms are used as design examples. The methodology can be applied to any transmission mechanism depending on its kinematic and geometric constraints. New five- and six-velocity automatic transmissions are enumerated from the Ravigneaux gear mechanism. It is a major breakthrough to design a completely satisfactory six-speed automatic transmission from the Ravigneaux gear mechanism since it has only eight links. The new design makes use of the benefits of the Ravigneaux gear train and overcomes the previous art difficulties. This structural design has realized a six-speed automatic transmission, while having minimal number of clutches, and brakes. It is also low cost due to adoption of the conventional available Ravigneaux gear train.     

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.      

齿轮系统传动轴受横向冲击的响应分析

将齿轮传动系统的齿轮轴承简化为具有集中质量的固支梁,将齿轮受到啮合齿轮的意外撞击看成是质量块对梁的冲击。给出弯扭组合的Mises屈服条件,指出传动轴受冲击时不能忽略扭矩作用。分析了弯曲和扭转作用下的结构响应,进行了应变率修正,给出特殊情况下弯扭响应的简化分析。算例表明,弯扭冲击下传动轴的横向位移和扭转角都较大,不可忽略应变率效应;传动轴直径是影响横向位移的重要因素。     

An enhanced multi-term harmonic balance solution for nonlinear period-\varvec{\beta } dynamic motions in right-angle gear pairs

A nonlinear, time-varying dynamic model for right-angle gear pair systems is formulated to analyze the existence of sub-harmonics and chaotic motions. This pure torsional gear pair system is characterized by its time-varying excitation, clearance, and asymmetric nonlinearities as well. The period-1 dynamic motions of the same system were 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 presented in another paper by the authors. Here, the sub-harmonics and chaotic motions are studied using the same solution technique. The accuracy of the enhanced multi-term HBM is verified by comparing its results to the solutions obtained using the more computational intensive direct numerical integration method. Due to its inherent features, the enhanced multi-term HBM cannot predict the chaotic motions. However, the frequency ranges where chaotic motions exist can be predicted using the stability analysis of the HBM solutions. Parametric studies reveal that the decrease in drive load or the increase of kinematic transmission error (TE) can result in more complex gear dynamic motions. Finally, the frequency ranges for sub-harmonics and chaotic motions, as a function of TE and drive load, are obtained for an example case.     

On Gear Modelling in Multistage Rotary Vane Engines

A discussion of the dynamics of a multistage rotary vane engine is given in terms of a simplified model for combustion driving torques, power dissipation, and torque transmission. Torque transmission is effected by conjugate gear pairs attached to each unit of the engine. An argument for the design of such pairs is presented so that unwanted torque fluctuations in a flywheel attached to a member of the pair can be significantly attenuated. It is suggested that a variant of simple Cosserat dynamics offers a useful modelling tool for discussing the complex interaction between interacting gear teeth. A quasi-stationary analysis is used to place bounds of a particular choice of conjugate gear coupling in the presence of such interactions. It is concluded that a multistage rotary vane engine with at least two units can be usefully coupled to a single flywheel via a well-defined conjugate gear system that attenuates unwanted torque fluctuations over a broad range of rotary speeds.     

Morphological covering based generalized dimension for gear fault diagnosis

This investigation presents a new generalized dimension estimation method based on morphological covering (MC) technique for characterizing the nonlinearity and complexity of vibration signals measured from gearbox. A synthetic fractal signal is employed to evaluate and compare the proposed MC technique with the traditional box-counting (BC) method and a similar approach developed in literature. Results revealed that the presented MC method is the one providing the most reliable generalized dimension estimation results. Furthermore, we applied this scheme to analyze the vibration signals from a gearbox with three operation states. The estimated general dimensions are used as the input feature vector for classifiers to the gear working states. Experimental results showed that our presented scheme achieves the best performance on discriminating the gear conditions. We also explore the calculational efficiency of the MC method. Results demonstrated that the MC method requires much less computational cost than BC method and seems to be more suitable for on-line condition monitoring of gearboxes.     

Method for detecting fatigue crack in gears

The most undesirable damage that can occur in gear units is crack in the tooth root as it often makes gear unit operation impossible. Monitoring vibrations can be used to detect defects. Time signals are acquired experimentally and afterwards. Different methods can be used to analyse them. The changes in tooth stiffness caused by a fatigue crack in the tooth root are of significance. The dynamic response of a gear unit with a damaged tooth differs from the one of an undamaged tooth. Amplitudes of time signal are, by time–frequency analysis, presented as a function of frequencies in spectrum.     

齿轮有限元强度计算的单元自动生成

本文提出了一种专用于齿轮有限元强度计算的单元自动生成办法,只需输入法截面内一半数据,便可自动生成空间二十节点等参元的全部数据。本文严格地以实际齿面方程所描述的齿廓曲面体为计算对象,并按轮齿法截面划分单元,也适合于其他柱面体和螺旋面体的有限元计算。     

Gas-kinetic numerical method for solving mesoscopic velocity distribution function equation

A gas-kinetic numerical method for directly solving the mesoscopic velocity distribution function equation is presented and applied to the study of three-dimensional complex flows and micro-channel flows covering various flow regimes. The unified velocity distribution function equation describing gas transport phenomena from rarefied transition to continuum flow regimes can be presented on the basis of the kinetic Boltzmann–Shakhov model equation. The gas-kinetic finite-difference schemes for the velocity distribution function are constructed by developing a discrete velocity ordinate method of gas kinetic theory and an unsteady time-splitting technique from computational fluid dynamics. Gas-kinetic boundary conditions and numerical modeling can be established by directly manipulating on the mesoscopic velocity distribution function. A new Gauss-type discrete velocity numerical integration method can be developed and adopted to attack complex flows with different Mach numbers. HPF parallel strategy suitable for the gas-kinetic numerical method is investigated and adopted to solve three-dimensional complex problems. High Mach number flows around three-dimensional bodies are computed preliminarily with massive scale parallel. It is noteworthy and of practical importance that the HPF parallel algorithm for solving three-dimensional complex problems can be effectively developed to cover various flow regimes. On the other hand, the gas-kinetic numerical method is extended and used to study micro-channel gas flows including the classical Couette flow, the Poiseuille- channel flow and pressure-driven gas flows in two-dimensional short micro-channels. The numerical experience shows that the gas-kinetic algorithm may be a powerful tool in the numerical simulation of micro-scale gas flows occuring in the Micro-Electro-Mechanical System (MEMS). The project supported by the National Natural Science Foundation of China (90205009 and 10321002), and the National Parallel Computing Center in Beijing. The English text was polished by Yunming Chen.     

啮入冲击对直齿轮弹流润滑的影响

考虑齿轮啮入冲击载荷,曲率半径、卷吸速度沿啮合线随时间的变化以及温度场的影响,用非牛顿流体的Ree-Erying润滑模型,利用多重网格法模拟了轮齿从啮入到啮出整个时间历程中油膜压力、膜厚和温度分布的变化,对比分析了啮入冲击载荷与平稳载荷对渐开线直齿轮时变非牛顿热弹流润滑结果的影响.数值结果表明,啮入冲击载荷只对啮入初始阶段的油膜压力、膜厚、温度有很大影响,最小膜厚和最大压力都发生在冲击载荷的最大峰值载荷时刻,所以齿轮的啮入冲击对齿轮保持良好的润滑状态是不利的.     

Numerical investigation of powder compaction of gear wheels

A method to judge the porosity distribution within complex powder compacted 3D structures using a dynamic 3D dilatant finite strain finite element program is presented. The method is demonstrated for a gear wheel, using a combined FKM Gurson model with parameters calibrated from experiments to model a ferrous powder. Compaction is pursued until a final average porosity of 3% in the gear. The method is shown successful in judging the influence on local as well as average properties from change in geometrical parameters and compaction speed.     

Numerical continuation analysis of a three-dimensional aircraft main landing gear mechanism

Nonlinear Dynamics - A method of investigating quasi-static landing gear mechanisms is presented and applied to a three-dimensional aircraft main landing gear mechanism model. The model has 19...     

Stochastic optimal control of flexible aircraft taxiing at constant or variable velocity

Active control landing gears are used to alleviate vibration during aircraft taxiing. A nonlinear stochastic dynamics model is established, considering the aircraft body pitch movement and elastic vibration excited by the random runway. The equivalent linearization method is adopted to ensure the model linearity near the balance point, and the Gaussian random process of the runway is generated from the Gaussian white noise using a shape filter. Based on the stochastic optimal control theory, the LQG controller is designed along with weighted quadratic performance index for a better ride comfort, shock absorption, road holding and least energy expenditure. The algebraic Riccati and Lyapunov equations are solved to obtain stationary response while taxiing aircraft at a constant velocity and the differential Riccati and Lyapunov equations are solved to obtain the nonstationary response while taxiing aircraft at a variable velocity. The aircraft dynamic responses are obtained through the runway random process modeled by Monte Carlo method. Simulation results show that active control landing gear can give a better ride comfort, shock absorption and road holding performance no matter whether taxiing is at constant or variable velocity.     

用光电技术测定爆轰参数

本文介绍了一种用光电技术同时测定爆速和爆温、爆压和爆温的方法。用光电比色技术测定炸药爆轰温度的同时,用两个爆轰面上爆轰的时间差计算爆轰波速度;或用透明介质中冲击波速度来反算炸药中的爆轰压力。方法原理可靠,技术简便。     

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.     

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.

     

A solution for the vertical variation of stress,rather than velocity,in a three-dimensional circulation model

A simple technique is presented that allows a numerical solution to be sought for the vertical variation of shear stress as a substitute for the vertical variation of velocity in a three-dimensional hydrodynamic model. In its most general form the direct stress solution (DSS) method depends only upon the validity of an eddy viscosity relation between the shear stress and the vertical gradient of velocity. The rationale for preferring a numerical solution for shear stress to one for velocity is that shear stress tends to vary more slowly over the vertical than velocity, particularly near boundaries. Consequently, a numerical solution can be obtained much more efficiently for shear stress than for velocity. When needed, the velocity profile can be recovered from the stress profile by solving a one-dimensional integral equation over the vertical. For most practical problems this equation can be solved in closed form. Comparisons are presented between the DSS technique, the standard velocity solution technique and analytical solutions for wind-driven circulation in an unstratified, closed, rectangular channel governed by the linear equations of motion. In no case was the computational effort required by the velocity solution competitive with the DSS when a physically realistic boundary layer was included. The DSS technique should be particularly beneficial in numerical models of relatively shallow water bodies in which the bottom and surface boundary layers occupy a significant portion of the water column.     

Simultaneous quantitative flow measurement using PIV on both sides of the air–water interface for breaking waves

An experimental method for simultaneously measuring the velocity fields on the air and water side of unsteady breaking waves is presented. The method includes a novel technique for seeding the air flow such that the air velocity can be resolved in the absence of wind. Low density particles that have large Stokes drag and ability to respond to high-frequency flow fluctuations are used to seed the air flow. Multi-camera, multi-laser particle image velocimetry setups are applied to small-scale unsteady breaking waves, yielding fully time-resolved velocity fields. The surface tension of the fluid is altered and controlled to form spilling breaking waves. Results for the velocity and vorticity fields of representative spilling breakers, which show shedding of an air-side vortex and well-documented generation of water-side vorticity, are presented and discussed.     

Gear tooth stress analysis by the complex potentials method

The complex potentials method of plane elasticity theory is applied to spur gear stress analysis. The concept is that the stress field in a gear tooth loaded by a point force can be obtained using the Boussinesq solution and a transformation function mapping a bell-shaped curve approximating the real tooth shape in the z-plane into the semi-infinite ζ-plane. The main features of an original computer code implementing this formulation are presented along with applications.

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Sommario Dopo un'illustrazione di come lo stato di sollectitazione in un dente di una ruota dentata a denti diritti puo' essere ottenuta mediane il metodo delle trasformazioni conformi e dei potenziali complessi della teoria dell'elasticita' piana, viene presentata la struttura di un codice di calcolo originale che implementa questo approccio ed alcune applicazioni.