Helwan University

Dynamic modeling of the gear vibration is a useful tool to study the vibration response of a geared system under various gear parameters and operating conditions. An improved understanding of vibration signal is required for early detection of incipient gear failure to achieve high reliability. However, the aim of this work is to make use of a 6-degree-of-freedom gear dynamic model including localized tooth defect for early detection of gear failure. The model consists of a 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 modifications. The results indicate that the simulated signal shows that as the defect size increases the amplitude of the acceleration signal increases. The crest factor and kurtosis values of the simulated signal increase as the fault increases. Though the crest factor and kurtosis values give similar trends, kurtosis is a better indicator as compared to crest factor.

KEYWORDS：Vibration acceleration, system modeling, Crest Factor, Kurtosis value, defect size, gear meshing, pinion, gear

，，，   Drive motor, pinion, gear, and load mass moment of inertia

replacement decision in a suitable time.

，       Masses of pinion and gear.

               Driving motor torque.

                Load torque.

，         Friction torque.

，          Viscous damping coefficient of pinion and gear bearing.

              Gear mesh damping.

              Gear mesh stiffness.

，          Pinion and gear shaft stiffness.

              The variance square.

               The number of samples.

               The defect width in face direction.

             Unit width Hertzian stiffness.

，，， Angular displacement of drive motor, pinion, gear and load.

，，， Angular velocity of drive motor, pinion, gear and load.

，，， Angular acceleration of drive motor, pinion, gear and load.

Much of the past research in the dynamic modeling area has concluded that an essential solution to the problem is to use a comprehensive computer modeling and simulation tool to aid the transmission design and experiments. These have been two major obstacles to such an approach: (1) Progress in understanding of the basic gear rattle phenomenon has been limited and slow. This is because the engine-clutch-transmission system involves some strong nonlinearities including gear backlash, multi-valued springs, dry friction, hysteresis, and the like.  (2)The gear rattle is a system problem and not only problem of gear teeth. Even through the research and industrial community has discussed the difficulties in varies stages of the problem, yet no thorough frame work covering the entire investigation process of such problem currently exists. This is largely due o the complexity of the power train system, which may make a computer analysis tool inefficient, in particularly when many different elements and clearances are encountered (e.g., gears, bearings, splines, synchronizers, and clutch) [1-3].

A comprehensive review of mathematical models used in gear dynamics, published before 1986, has been presented by [4]. In this review, gear dynamic models without defects have been discussed. In the past few years, researchers have been working on the gear dynamic models which include defects like pitting, spalling, crack and broken tooth.

A single-degree-of-freedom model is used which include the e4ffects of variable mesh stiffness, damping, gear errors, profile modifications and backlash. The effect of time-varying meshing damping is also included in this case, The solution is obtained by using the harmonic balance methods. A method of calculated the optimum profile modification has been proposed in order to obtain a zero vibration of the gear pair [5-7]. They also proposed a linear approximate equation to mode the gear pair by using a single-degree-of freedom model

Gear rattle vibration is a undesirable vibration for passenger cars and light trucks equipped with manual transmissions. Unlike automatic transmissions, manual transmission do not have the high viscous damping inherent to a hydrodynamic torque converter to suppress the impacting of gear teeth oscillating through their gear backlash. Therefore a significant level of vibration an be produced by the gear rattle and transmitted both inside the passenger compartment and outside the vehicle. Gear rattle, idle shake, and other vibration generated in the automobile driveline have become an important concern to automobile manufactures in their pursuit of an increased level of perception of high vibration quality. The torsional vibration o driveline is a major source of gear rattle vibration. The manual transmission produces gear rattle by the impacting of gear oscillating through their gear backlash. The impact collisions are transmitted to the transmission housing via shafts and bearings [8].

The gear pair dynamic models including defects have been done by [9]. The study suggests that little work has been done on modeling of gear vibration with defect and an accurate analytical procedure to predict gear vibrations in the presence of local tooth fault has yet to be developed.However, the purpose of this paper is to develop a multidegree-of-freedom nonlinear model for a gear pair that can be used to study the effect of lateral-torsional vibration coupling on vibration response in the presence of localized tooth defect. A typical fault signal is assumed to be impulsive in nature because of the way it is generated. The simulation artificially introduced pitting in gears in multi-stage automotive transmission gearbox at different operation conditions (load, speed, etc). The processing of simulated and experimental signals is also introduced.

Among various signal-processing techniques, crest factor and kurtosis analysis have been used for analyzing the whole vibration signal for the early detection of fault. In this section, crest factor and kurtosis value have been explained.

   Helical gears are almost always used in automotive transmissions. The meshing stiffness of a helical tooth pair is time-varying [10], and was modeled as a series of

suggested spur gears so that the simulation techniques for spur gears can be applied. where M is Module (mm), b is Face width (mm),   is pressure angle (deg),  is helix angle (deg) and D1 is pitch diameter (mm). Fig. 2 shows the equivalent gear system in the first gear-shift, where the main parameters for the gear system of Fiat-131 gearbox and the equivalent gear system in the first gear-shift are also shown in the figures.

 

 

 

 

 

 

   Nagwa Abd-elhalim, Nabil Hammed, Magdy Abdel-hady,

                  Shawki Abouel-Seoud and Eid S. Mohamed               

     阿勒旺大学

 摘要

    在研究齿轮系统中各种齿轮参数的振动响应和操作条件时，齿轮振动的动态建模是一个非常有用的工具。对早期的齿轮检测提出了一种改进理解的振动信号，但还没达到高的可靠性。但是，这项工作的目的是利用一个6自由度的齿轮动力学模型对齿轮轮齿缺陷故障的早期检测。该模型包括一对齿轮副、两个轴、两个惯性负载、动力传动装置和轴承。由于齿轮的误差和变动，该模型被采用时受到时变啮合刚度、阻尼、反弹和励磁的影响。模拟信号显示的结果表明，随着缺陷尺寸的增加加速度信号的振幅增加。模拟信号的波峰因素和峰值随着缺陷的增加而增加。虽然波峰因素和峰值做同样的趋势，但和波峰因素相比峰值是一个比较好的指标。

，，，   驱动电机、小齿轮、大齿轮和负载在一定时间内的惯性矩

，            大齿轮、小齿轮的模数

                发动机驱动转矩

                负载力矩

，           摩擦力矩

，            齿轮、轴承的粘滞阻尼系数

                齿轮啮合阻尼

                齿轮啮合刚度

，           齿轮、齿轮轴的刚度

                样本数量

                  宽度方向的缺陷    

                 单位宽度的刚度

，，，     驱动电机、小齿轮、大齿轮和负载的角位移

，，，     驱动电机、小齿轮、大齿轮和负载的角速度

，，，     驱动电机、小齿轮、大齿轮和负载的角加速度

    在大多数过去的动态建模研究领域中，解决问题的重要办法是全面使用计算机建模和仿真工具来辅助变速器的设计和实验。这种方法有两种主要的障碍：（1）对齿轮传动中噪声基本认识的进展是有限的和缓慢的。这是因为发动机离合器传动系统中包括齿轮侧隙、多值弹簧、非线性滞后等等。（2）齿轮发出的噪声是一个系统问题，并不是齿轮的唯一问题。既使是工业研究领域已经讨论了这个问题在不同阶段所出现的不同问题，但并没有彻底覆盖工作的框架，整个研究过程中的问题依然存在。这主要是由于列车电力系统的复杂性，可能导致你的计算机的分析工具效率不高，尤其是工作中遇到许多不同的因素和间隙（例如：齿轮、轴承、花键、同步器和离合器）。

    在1986年出版之前，对齿轮动力学中提出的齿轮动态建模进行了审查。这次审查中，对不存在齿轮缺陷的齿轮动力学模型进行了讨论。在过去的几年里，研究人员对齿轮的动态模型缺陷进行了研究，其中包括点蚀、剥落、裂缝和齿轮折断等。

单自由度系统模型中，对啮合刚度的影响包括4个方面的因素，阻尼、齿轮误差、轮廓变动和齿侧间隙，时变啮合阻尼效应也包含在这种情况中。解决问题的方法是利用谐波平衡的方法。为了实现齿轮副的零振动，提出了一种最优化的计算方法。他们还利用齿轮副单自由度模型提出了一个近似的线性方程模型。

    汽车变速器中的齿轮大都是斜齿圆柱齿轮。被视为一系列齿轮仿真技术适用于螺旋状的轮齿时变啮合刚度。式中m是模数(毫米),b齿面宽(毫米),是压力角(度),是螺旋角(度)，D1是直径(毫米)。图2的数据显示了等效齿轮系统在齿轮变动中变速箱齿轮系统的主要参数。