Finite Element Analysis for Time Varying Mesh Stiffness behavior of different shapes of spalling

Time Varying gear mesh stiffness is the variation of stiffness for the one contact period of a gear pair. Spalling on gear tooth is one of the most common defects in gear transmission. The loss of surface materials due to tooth spall reduces the Time-Varying Mesh Stiffness (TVMS) of the gear pair. The time varying mesh stiffness is the change in mesh stiffness for a gear pair in one contact period. The evaluation of the TVMS of the pair of gear tooth under gear tooth spalling conditions may be useful in finding its dynamic behavior or instant change in TVMS behavior may be the signal of fault developed. The potential energy method is used for analytical calculation of time varying mesh stiffness of the gear. The tooth spall may be of any shape and in this paper we have assumed the spall to be circular, triangular and elliptical and conducted analytical calculation to find TVMS and compare with healthy gear pair. The same gear pair with three shape of spall and modelled and verified through finite element analysis. The data may be used for further analysis. Also a number of shape may be modelled in future.


I. Introduction
Gear is a important mechanical element which is used to transmit power from one shaft to another shaft. Gear is used in many industrial applications such as automobile, marine, aviation etc. Spalling occur on the gear face of the tooth profile, due to the insufficient lubrications, high service load, bad operating conditions, high contact speed and hence transmission of power is not uniform. The loss of surface materials due to tooth spall reduces the Time-Varying Mesh Stiffness (TVMS) of the gear pair, and thus modifies the vibration response of the gear transmission. The evaluation of the TVMS of the gear tooth pair under gear tooth spalling conditions plays an important role in gear dynamic Simulation and the corresponding fault feature analysis [1]. Based on the probability distribution a new analytical model for tooth pitting was developed and mesh stiffness was investigated [1]. Xihui Liang, et al. study a circular shape spall having specific location, radius and depth was considered. With the help of potential energy equations mesh stiffness of gears with single and multiple pits are evaluated [2]. Yang Luo, et al Gear tooth spall was studied by curved bottom shape method that better represents the geometries of spall. In this study different shape of spall like circular and ellipsoid are taken and mesh stiffness result were compared with FEA [3]. Ankur Saxena et. al. study the effect of spalling defect on the gear an analytical formulation was developed for the calculations of the TVMS.. The TVMS was calculated for rectangular, circular, ellipsoid shape of spall and observed TVMS reduction when spall in in mesh. H Ma, et al calculated error of TVMS under double tooth engagement by improved analytical method and finite element method l. [5] Yan Ding, F. Reiger In this study spalling is differentiated spalling from pitting. Experimental study in a test rig was done at the contact surfaces about the thickness of 10 microns and spalling appears as a deeper cavityat contact surface about the thickness of 20 to 100 microns. [6] Fakher Chari, Wallid Baccar l given modified the potential energy method for finding the spall and tooth breakage effect on vibratio response [7] Z. Chen, Y. Shao [8] In this study six degree of freedom dynamic spur gear pinion model is taken to examine the gear defects such as tooth root crack along the width and depth. Time varying mesh stiffness is calculated analytically and validated by finite element method. So many research was done to find the spalling defect analytically and its comparison either by FEA or dynamic response. In this paper along with circular and elliptical spall a new triangular error is used for TVMS change response analytically and its FEA was done further verification.

Methodology
For the study of time varying mesh stiffness a tooth is considered cantilever and spalling is assumed symmetrical about pitch line. . The spalling of triangular, Circular and elliptical are taken for study. The shapes are shown in figure 1.
. Fig. 1 Different shapes of spalling for Analysis [4,3] The gear tooth geometry is as shown in fig. 2 spalling is located near the pitch circle. For the analysis purpose the teeth may be divided into a number of strips, the area of strip without spall is given by A=-2yl --------------(1) While for the spalled region it is calculated by As=(y + ys) l -------------(2) Where y, ys, represent the vertical values of the gear tooth contour line with respect to axis of gear teeth in healthy and spalled region.
During the meshing, when teeth contact force F is transmitted from one teeth to another teeth along the line of action as shown in fig4. This force F is having two one is in radial direction and another in tangential direction given by The bending moment of the teeth is also an important data that is necessary for Bending energy calculations. The value of bending moment for the healthy part of the tooth is given by In the similar way the moment of inertial for healthy and spalled region of the teeth are given by equation 7 and 8.

Analytical Calculation:
The potential energy method is used for all the calculations. The Bending stiffness, Shear Stiffness, Axial Stiffness and Contact stiffness were calculates by using the outcome of the following equations. [4,9]. The Hertzian Contact stiffness for faulty gear can be expressed as - Where E, L, μ represents young's modulus, tooth contact width and Poisson's ratio respectively. The total equivalent mesh stiffness of single tooth pair in meshing can be expressed as- Where subscripts 1and 2 denotes the driving gear and driven gear. The total equivalent mesh stiffness of double teeth pair in meshing can be expressed as K total = ∑ 1 1 K hi 3. Elliptical Spall: The parametric function of an ellipsoid is given by X 2 a 2 + y 2 b 2 = 1 If a=b then it will be a circle After modelling of the gear the minor axis will represent the length of spall and the major axis with depth will cause the change in width as width of contact is reduced by width of spall. In this case the major axis were taken as 5mm and the minor axis was 3 mm. . Gear Modeling: For the analytical as well as the finite element analysis the gear-pinion arrangement was modeled in software. Gear parameters used in this study are given below in table 1 [9].

Finite element analysis:
After analytical calculation model was tested in the ansys software for finding its stiffness. The model is imported in preprocessing, meshed and applied by fixing one and moment applied by other. A force of 1000 N is supposed to apply on the gear teeth of pinion by teeth of gear. The deflection of gear is used for its stiffness value. The number of load position used and data were compared with the analytical data. The Figure 4 is showing the deflection in the teeth when the load is applied.

Conclusion
Finite element conducted on the analytical results shows a good agreement. So the method is used for different shape of spall. The data may also be used for dynamic analysis and vibration behavior of the system.