Effect of shaft misalignment and friction force on time varying mesh stiffness of spur gear pair

Shaft misalignment and sliding friction between meshing teeth are considered as primary excitation to generate vibrations and extra dynamic loads on transmitting gear teeth. Time varying mesh stiffness (TVMS) is an important parameter to understand the dynamics of meshing gear pair. Potential energy method is one of the most acceptable methods to calculate TVMS. This paper proposes a computer simulation based approach to study the effect of shaft misalignment and friction on total effective mesh stiffness for spur gear pair. The results showed clearly that misalignment and friction affect TVMS of gear pair. The effect of misalignment and friction has also been studied for cracked gear pair and results are discussed.     

Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects

Spalling is one of the common tooth surface failures of gear teeth and is defined as the formation of deeper cavities that are mainly developed from subsurface defects. The time varying mesh stiffness (TVMS) of gear pairs, gives significant information about the health of the system. The change indirection of time varying friction on both sides of the pitch line causes the change of gear mesh stiffness. This article proposes a computer simulation based approach to study the effect of time varying friction coefficient on the total effective mesh stiffness for the spur gear pair. An analytical method to calculate the TVMS of the spur gear for different spall shapes, size and location considering sliding friction is also proposed in this study. The results show that spall shape, size and location are very important parameters that need to be considered for calculation of TVMS and subsequently to know the dynamic response of the gear pair in the presence of a spall.     

The effects of spur gear tooth spatial crack propagation on gear mesh stiffness

Due to the presence of non-uniform load distribution, local non-homogeneity of material quality and potential misalignment of gear shafts and bearings, etc., spatial cracks may occur in the fillet region of spur gear teeth. These cracks will eventually propagate in three distinct directions either individually or simultaneously. These directions are the crack depth direction, the tooth width direction and the tooth profile direction. In this paper, an analytical investigation of the influence of spatial crack propagation on the time-varying Gear Mesh Stiffness (GMS) and also the Load Sharing Ratio (LSR) is presented. In order to quantitatively define the spatial crack propagation scenario, the involute spur gear tooth geometry cut with a typical double rounded rack is first determined using two parametric equations. The effects of some gear design parameters and initial crack locations on GMS and LSR are determined and compared with the results from previous papers that used Finite Element Analysis (FEA) in order to verify the proposed analytical model. Finally, a quasi-parabolic crack propagation scenario is assumed, in which 7 propagation cases and 3 typical crack growth paths on the tooth surface are investigated to determine their effect on the maximum reduction of GMS when compared to normal conditions. The results are important for the dynamic simulation of gear transmission behavior, and consequently helpful for the monitoring of gearbox working condition and detection of early crack damage that may exist in gear sets.     

Analytical model for mesh stiffness calculation of spur gear pair with non-uniformly distributed tooth root crack

Gear tooth crack is likely to happen when a gear transmission train is working under excessive and/or long-term dynamic loads. Its appearance will reduce the effective tooth thickness for load carrying, and thus cause a reduction in mesh stiffness and influence the dynamic responses of the gear transmission system, which enables the possibility for gear fault detection from variations of the dynamic features. Accurate mesh stiffness calculation is required for improving the prediction accuracy of the dynamic features with respect to the tooth crack fault. In this paper, an analytical mesh stiffness calculation model for non-uniformly distributed tooth root crack along tooth width is proposed based on previous studies. It enables a good prediction on the mesh stiffness for a spur gear pair with both incipient and larger tooth cracks. This method is verified by comparisons with other analytical models and finite element model (FEM) in previous papers. Finally, a dynamic model of a gear transmission train is developed to simulate the dynamic responses when cracks with different dimensions are seeded in a gear tooth, which could reveal the effect of the tooth root crack on the dynamic responses of the gear transmission system. The results indicate that both the mesh stiffness and the dynamic response results show that the proposed analytical model is an alternative method for mesh stiffness calculation of cracked spur gear pairs with a good accuracy for both small and large cracks.     

Crack behavior in a high contact ratio spur gear tooth and its effect on mesh stiffness

The efficiency of high contact ratio (HCR) gearing can be achieved by proper selection of gear geometry for increased load capacity and smoother operation despite of their high sliding velocities. The prediction of variation in mesh stiffness of HCR gearing is critical as the average number of teeth being in contact is high at a given time as compared to conventional low contact ratio (LCR) gearing. In this paper, linear elastic fracture mechanics (LEFM) based finite element method is used to perform the crack propagation path studies of HCR spur gear having tooth root crack for two gear parameters viz. backup ratio and pressure angle. A total potential energy model has been adopted to analytically estimate the mesh stiffness variation. The results depict the mesh stiffness reduction in the presence of the crack. The percentage change in mesh stiffness with increasing crack length is an important parameter in fault diagnosis of geared transmission. Higher the percentage change in mesh stiffness, easier to detect the fault. Two gear parameters viz. back-up ratio and pressure angle has been studied and the effect of crack length on mesh stiffness have been outlined. With the increase of deterioration level gears having lower back-up ratio fault can be detected at an early stage, similarly, chances for early fault detection is more for gears having higher pressure angle.     

Simulation of crack propagation in spur gear tooth for different gear parameter and its influence on mesh stiffness

Most of the gear dynamic model relies on the analytical measurement of time varying gear mesh stiffness in the presence of a tooth fault. The variation in gear mesh stiffness reflects the severity of tooth damage. This paper proposes a cumulative reduction index (CRI) which uses a variable crack intersection angle to study the effect of different gear parameters on total time varying mesh stiffness. A linear elastic fracture mechanics based two dimensional FRANC (FRacture ANalysis Code) finite element computer program is used to simulate the crack propagation in a single tooth of spur gear at root level. A total potential energy model and variable crack intersection angle approach is adopted to calculate the percentage change in total mesh stiffness using simulated straight line and predicted crack trajectory information. A low contact ratio spur gear pair has been simulated and the effect of crack path on mesh stiffness has been studied under different gear parameters like pressure angle, fillet radius and backup ratio. The percentage reduction of total mesh stiffness for the simulated straight line and predicted crack path is quantified by CRI. The CRI helps in comparing the percentage variation in mesh stiffness for consecutive crack. From the result obtained, it is observed that the proposed method is able to reflect the effect of different gear parameters with increased deterioration level on total gear mesh stiffness values.     

Experimental investigation of spur gear tooth mesh stiffness in the presence of crack using photoelasticity technique

Gear mesh stiffness plays a very important role in gear dynamics and it varies in the presence of gear fault such as crack. The measurement of stress intensity factor can lead to the determination of gear tooth mesh stiffness variation in the presence of crack in a spur gear system. In this paper, the technique of conventional photoelasticity has been revisited to explore the possibility of using it as a supplementary technique to experimentally measure the variation of gear mesh stiffness. An attempt has been made to calculate the variation of mesh stiffness for a pinion having a cracked tooth and a gear tooth with no crack of a spur gear pair. An analytical methodology based on elastic strain energy method in conjunction with total potential energy model has been adopted and implemented within the mesh stiffness calculations. To visualize the state of stress in a structure using finite element and other currently available methods, photoelasticity is considered to be one of the oldest and most developed experimental technique. An experimental methodology based on conventional photo-elasticity technique for computing stress intensity factor (SIF) for cracked spur gear tooth is presented for different single tooth contact position and crack length. The relation between contact position, crack length, crack configuration, SIF and the variation of total effective mesh stiffness have been quantified. Finally, a comparison has been made and the results obtained from finite element method (FEM) based on linear elastic fracture mechanics (LEFM), analytical method and proposed experimental method has been outlined.     

Time-varying mesh stiffness calculation of cracked spur gears

Considering the misalignment of gear root circle and base circle and accurate transition curve, an improved mesh stiffness model for a healthy gear pair is proposed and validated by the finite element method (FEM). Based on the improved method, three mesh stiffness calculation methods (method 1: straight lines for crack path and limiting line; method 2: straight line for crack path and parabolic curve for limiting line proposed in Ref. [1]; method 3: parabolic curves for crack path and limiting line) for cracked gear pair are presented and compared with FEM. The results show that there is a significant difference between method 1 and FEM under large crack condition and the results of methods 2 and 3 are quite close to FEM result, which also shows that the parabolic curve as a limiting line is appropriate. Mesh stiffness of method 2 is very close to that of method 3, which also shows that it is acceptable to assume the crack path to be a straight line.     

Improved time-varying mesh stiffness model of cracked spur gears

Based on our previous work (Ma et al., 2014, Engineering Failure Analysis, 44, 179–194), this paper presents an improved analytical model (IAM) for the time-varying mesh stiffness (TVMS) calculation of cracked spur gears. In the improved analytical model, the calculation error of TVMS under double-tooth engagement due to repeatedly considering the stiffness of the fillet-foundation is revised, and the effects of reduction of fillet-foundation stiffness of cracked gears and extended tooth contact (ETC) are also considered, which have a great influence on TVMS, especially under the condition of large torques and crack levels. Moreover, the comparisons among the IAM, traditional analytical model (TAM) and finite element (FE) model are also carried out under different torques and crack depths. IAM is also verified by comparing TVMS and vibration responses obtained by FE model, which can be considered as a gauge to evaluate the calculation error. The results show that the maximum error of IAM is about 12.04%, however, that of TAM can be up to 32.73%.     

Meshing characteristics of spur gear pair under different crack types

The effects of three different gear crack types such as, for example, the crack along tooth width uniformly and the crack propagating in the depth direction (crack type 1, CT1), the crack along tooth width non-uniformly and the crack propagating in both the depth and the tooth width directions (crack type 2, CT2), and the spatial crack propagating in the depth, the tooth width and the tooth profile directions (crack type 3, CT3) on the time-varying mesh stiffness (TVMS) of spur gear pairs are investigated in this study. Firstly, an analytical model for studying these three types of cracks is established based on potential energy method. A finite element (FE) model of the cracked spur gear pair is also built in the ANSYS software as well. In order to verify the analytical method, the TVMS obtained from analytical method is compared with that obtained from FE method under different crack types. Moreover, the effects of the depth, the length and the height of crack are discussed. The equivalent stress, contact pressure and displacement of tooth are also analyzed under different crack types by using the FE method. The results show that the effect of crack depth on TVMS is the largest, while that of the crack height is the smallest, and the non-penetrating crack for CT2 and CT3 will generate the non-uniform load distribution along tooth width.     

Time-varying mesh stiffness calculation of spur gears with spalling defect

Considering the effects of extended tooth contact (ETC), revised fillet-foundation stiffness under double-tooth engagement region, nonlinear contact stiffness and tooth spalling defect, an analytical model for time-varying mesh stiffness (TVMS) calculation of spur gears is established. In addition, the analytical model is also verified by comparing the TVMS under different spalling widths, lengths and locations with that obtained from finite element method. The results show that gear mesh stiffness decreases sharply with the increase of spalling width, especially during the single-tooth engagement; the spalling length only has an effect on the beginning and ending of gear mesh stiffness reduction; the spalling location can affect the range of gear mesh stiffness reduction, and the range will reduce when the spalling location is close to the addendum. This study can provide a theoretical basis for spalling defect diagnosis.     

裂纹对弧齿锥齿轮扭转啮合刚度的影响分析

运用Pro/E软件建立了无裂纹和含裂纹弧齿锥齿轮完整的三维接触模型,并基于有限元分析软件ANSYS对模型进行了仿真模拟与数值计算,分析了两种运行状态下,不同接触位置上的扭转啮合刚度。仿真结果表明：裂纹使弧齿锥齿轮的扭转啮合刚度变化更加剧烈。     

考虑延长啮合时齿轮参数振动稳定性研究

考虑因轮齿受载弹性变形引起的轮齿延长啮合效应时,将使得啮合刚度随时间变化形式发生改变,而这一变化对于系统的动态特性产生重要影响.在考虑轮齿延长啮合时齿轮参数振动稳定性的变化情况,首先建立齿轮副参数振动分析模型,进而研究了系统稳定性分析方法,主要是状态转移矩阵的确定和稳定性判据;在此基础上,以实际应用的高速重载齿轮副为例讨论了延长啮合与否系统稳定区间的变化情况,同时还考虑了系统阻尼的影响.研究结果表明;轮齿延长啮合作用使得低转速和高转速范围内的各不稳定区减小,而对于中等转速范围内,考虑延长啮合使得不稳定区变大.     

Tribo-dynamics model of a spur gear pair with gyroscopic effect and flexible shaft

Forschung im Ingenieurwesen - This study proposes a tribo-dynamic model of a spur gear by coupling an elastic dynamic model and an isothermal elastohydrodynamic lubrication (EHL)...     

冲击载荷下的齿轮动应力变化规律数值分析

基于虚拟制造方法建立了滚剃工艺的精确圆柱直齿轮三维几何模型,并进行线外啮合冲击动态仿真分析,得到了可靠的齿轮动应力变化规律。仿真结果表明：冲击加载时,不同工况下齿根两侧最大动应力的出现位置呈对称分布;冲击载荷下,齿根最大应力大于ISO标准静强度理论值,最大应力位置比ISO标准确定的最危险截面位置偏高;冲击载荷峰值相同时,冲击时间越短,产生的动应力越大。     

内平动齿轮副啮合综合刚度与系统的分岔特性

用机构反转法将内平动齿轮副转化为定轴齿轮副,然后用有限元方法分析了该内啮合定轴齿轮副的啮合综合刚度,并使用FFT变换得到其频谱特性,进而得到了内平动齿轮副的啮合综合刚度的频谱特性,在此基础上,考虑了齿侧间隙的非线性因素,进一步得到存在多齿接触的时变啮合刚度下内平动齿轮副的运动微分方程。然后,用经典的显式四阶Rouge-Kutta法对系统的各个参数进行了数值计算,得到系统的参数分岔图,并分析了各参数对系统动力学行为的影响。为内平动齿轮副的设计参数选择提供理论依据。     

齿轮传动系统的非线性冲击动力学行为分析

主要研究在考虑时变刚度和摩擦时,轮齿间隙及载荷参数对齿轮系统冲击动力学响应的影响,首次研究了齿轮传动非线性动力学的脱啮冲击、周期解的形成过程及对应的非光滑系统冲击中出现的切分岔行为。含时变参数系统的解析求解非常困难,本文根据间隙函数,将相平面划分为三个区域并构建了与之相对应的Poincare映射,结合谐波平衡法及序列二次规划方法求得了区域 和 内的解,利用数值仿真的方法分析了载荷参数、初始条件等因素对齿轮冲击响应的影响,本文研究表明： 1)随着常量载荷 的增大,脱齿运行的时间减小,且啮合冲击的速度也逐渐减小,齿轮的运动也相对平稳,同时出现切分岔的初始速度的绝对值迅速增大;2) 随着载荷波动分量 的增大,脱齿运行的时间减小,但是啮合冲击的速度也相应的有所增大,轮齿间冲击的冲击比较剧烈&;#61472;。     

Backlash effect on dynamic analysis of a two-stage spur gear system

Gearbox dynamics are characterized by a periodically changing stiffness due to multiple teeth contacts. In real gear systems, a backlash also exists that can lead to a loss in contact between the teeth. Due to this loss of contact, the gear has piecewise linear stiffness characteristics. This paper examines the effect of backlash in the two-stage gear system. A purely torsional gear system is formed by three shafts connected to each other by two spur gear pairs. Using standard methods for nonlinear systems (Newton-Raphson algorithm), the dynamic behavior of a gear system with backlash is examined. Amplitude jumps in systems due to backlash are observed.     

Method for automatic alignment recovery of a spur gear

This paper deals with the development of an online methodology for the automatic control and alignment of spur gears. The proposed methodology works on a set of points acquired on the workpiece surfaces by means of a manufacturing machine equipped with a set of sensors for dimensional measurements. A mathematical algorithm is developed to correct the position of the gear; it constitutes the main part of the developed monitoring and control system: it is able to real-time analyse the collected data of each measure and provide the orientation parameters that minimise the residual positioning errors. The methodology is first tested on simulated data-sets and the final workpiece configuration exhibits a good reproducibility. The tests executed on real data, acquired both through a coordinate measuring machine and the sensor system integrated into the manufacturing machine, confirm this result. The employment of this automation system allows to shorten the time necessary for the alignment process and improves the precision of the final positioning, leading to enhanced quality of the products and higher process flexibility, compared to the currently employed manual operation.     

Investigation of gear performance of MLNGPs as an additive on polyamide 6 spur gear

Molded plastic gears have long provided an alternative to metal gears in lightly loaded drives. They transmit power quietly and often without lubrication in numerous applications, furthermore decrease the quantity of parts and oppose chemicals in numerous applications. Previously, plastic gears were restricted to to 0.25 hp because of varieties in their properties and uncertainties about how they react to natural conditions such as moisture, temperature and chemical. Today, better molding controls combined with design practices that more accurately encompass environmental factors have boosted plastic gear drive capacity to 1.5 hp. Using reinforcement this is standout amongst the most practices to enhance the gear performance.

This study estimated the effects of multilayer graphene nanoplatelets (MLNGPs) as an additive on polyamide 6 (PA6) spur gear performance. These include strength, elastic modulus, thermal stability, dynamic mechanical analysis, moisture absorption, and wear characteristics.The nanocomposite gear was made by melt mixing method and injection moulded into thick flanges. The flanges were machined using CNC milling machine to produce spur gear. The wear experiments were performed at a running speed of 1400 rpm and at torques of 13 and 16 Nm with different concentration 0, 0.1, 0.3 and 0.5 wt% MLNGPs using test rig. The result showed that 0.3% of MLGNPs is the optimum concentration. Young's modulus increased up to 40%, Vickers microhardness value increased up to 25%, storage modulus E’ is increased up to 37% and glass transition temperature is increased up to 14%. On the other hand TGA result shows that the Tonest increased up to 7.5% and Td increased up to 2%, and wear decreased by 35% at 16 Nm and 54% at 13 Nm.