Finite Element Analysis for a New Friction-based Limited-slip Differential Housing

HUANG Xia，DING Jun，QIAO Huili

Chongqing University of Technology，Chongqing 400054，China

1.Introduction

A differential is an important component of a drive axle of a vehicle and ensures that wheels on both sides of the drive axle move at different rotational speeds to meet the requirements of vehicle traction kinematics．At present，the torque of a widely used symmetric style bevel gear differential is evenly distributed because of its less frictional torque．Wheel slip has a significant impact on the passage capacity of vehicles traveling on a road with low friction coefficient or unequal friction coefficients．Friction-based limited-slip differential has been widely used to improve the automatic adjustment of torque allocation problems between driving wheels［1］．In this work，considering the new friction-based limited-slip differential of drive axle imports of heavy vehicles，the current study analyzed the structural strength of a differential housing based on the finite element method under different conditions to improve the design of the differential housing of vehicle drive axles．

2.Structure and working principle of the new friction-based limited-slip differential

Fig．1 shows the structure of the new frictionbased limited-slip differential．The differential housing is designed using a bisection method centered on the sphere．The left housing 1 and the right housing 6 of the differential are integrally formed with bolt connections．

Both ends of the housing assembly are supported in the middle axle housing by a tapered roller bearing．Four pieces of planetary gear 3 without an enclosed cross axle 4 match the sphere in the housing via a spherical gasket 7 at the back．A friction plate system is placed between the half axle gear 2 and the surface of the differential housing，which is composed of a thick friction plate 5，ten driving friction plates arranged at regular intervals，and eight driven friction plates．The four outer teeth of the driving friction plate match the four tooth spaces along the axis distributed in the differential housing．The driven friction plate with a spline bore matches the external splines of the half axle gear．

Fig．1 A new friction-based limited-slip differential

Two half axles have the same speed with torque distributed equally on left and right axles when vehicles are used for straight driving．The axial forces generated by the planetary and the half-axle gears mesh with one another，thereby forcing the half axle at both ends，the thick friction plates，the driving friction plate，and the driven friction plate to tightly press together and to produce frictional torque．This torque can be distributed on the half axle in two ways:① via the cross axle，the planetary gear，and the half-axle gear．② via the active friction and driven the friction plates．

The rotation of the planetary gear produces a differential with unequal rotating speeds of the left and right half-axle gears when the vehicle makes a run or one side of the wheel spins on a slippery road surface．With this speed difference，the driving and driven friction plates produce frictional torque at the time of the slip．The direction of the frictional torque and the direction at which the half axle quickly turned are oriented in opposite directions．By contrast，the direction of the frictional torque is oriented in the same direction as the direction of a slow-turning half axle．The value is proportional to the torque and the number of friction plate transmitted by the differential mechanism．This case is similar to a part of torque transmitted from the quick-turning side to the slow-turning side．Thus，the torque transmitted by the slow-turning half axle is increased to improve the passage capacity of a vehicle［2－4］．

3.Finite element analysis for differential housing

3.1.Establishment of the finite element model of a differential housing

The FEM model for differential housing used as a discrete model of the original structure has a significant influence on the validity and accuracy of FEA．A three-dimensional model of the differential housing is established using CATIA，and which is imported into HyperMesh software to complete the model mesh，the material property definitions，and the load as well as the displacement boundary condition prescription，while solving and postprocess are completed in ABAQUS software．Due to the accuracy and efficiency，the element type used for FEM model is C3D4 in ABAQUS software．It is a four-node linear tetrahedral element，with each node having three translational degrees of freedom，that is，the translational degrees of freedom at x，y，and z directions．Fig．2 shows the mesh configuration for the differential housing with 331，915 elements and a total of 78，557 nodes．The material used is QT500 with elastic modulus is 173 GPa，Poisson’s ratio is 0．3，and yield limit is 310 MPa．

Fig．2 FEM model for differential housing

For the displacement boundary condition constraints，it follows as described．Based on the geometry configuration of differential assemblies in the drive axle housing，the translational DOF along y and z direction are constrained at the nodes located on the outer ring on both ends of the differential housing assembly．While the translational DOF along x direction is also constrained at the nodes on the surface in the right end of the housing．With regard to the nodes located at the cross axle hole of differential housing，the multiple points constraints(MPC)are firstly employed to constrain all DOF of nodes at the hole to the DOF of the reference node at the center of each hole．Then，the rotational DOF along z direction of the reference node for the up-and-down cross axle hole of the differential housing，and the rotational DOF at y direction of the reference node for the front-and-back cross axle hole of the differential housing are constrained，respectively［5 －6］．

3.2.Case study

According to the operational conditions of the limited-slip differential，three cases are to be studied for the differential housing．For the first case，the left half axle and the right half axle evenly distribute the torque when a vehicle is driven at a maximum traction force．In the second case，the torque sustained at the left housing equals to the sum of the torque and the frictional torque in a normal operational mode when the right side of the wheel slips．In the third one，the torque of the right housing is higher than that of the previous one when the left side of the wheel slips．Wherein，the first case is the normal operational mode at a maximum traction，while the second case and the third one are the limiting conditions at a maximum traction．

The load of a differential housing assembly under various cases includes the following conditions．①The right differential housing flange connects the driven gear of the main reducing gear via 16 bolts．The radial force Fr，axial force Fa，and tangential force Ftresulted from the gear meshing are distributed evenly on 16 screws．For the prescription of load applied on each hole in ABAQUS software，the nodes at the inner wall of the hole are firstly coupled to a reference node at the center of the hole using the command*KINEMATIC COUPLING，and the loads are applied at the reference node in the form of concentrated load．② The axial force on the thrust face of the left half-axle gear and the right half-axle gear in the differential housing is equally distributed on the surface of the two sides of the housing．③ The frictional torque of the internal surface of the two sides of the differential housing is necessary．④ The load uniformly distributed along the thrust surface of the planetary gear in the differential housing．

3.3.Results and discussion

According to FEM model and the work case of differential housing described previously，the details for the load are as following tables．Tab．1 and Tab．2 show the calculated force when the driving gear in the main reducer rotates anticlockwise and clockwise，respectively．

Fig．3 The Von-Mises stress distribution for various case of differential housing

Tab．1 The calculated force in anticlockwise rotation for driving gear

Tab．2 The calculated force in clockwise rotation for driving gear

Fig．3 shows the Von-Mises stress distribution for differential housing for six different cases．It shows that，when driving gear rotates anticlockwise in main reducer，the maximum stress value is 229 MPa，which is located near the small hole at the inner end of the right differential housing，as illustrated in Fig．3(a)．When the right side wheel slips，the torque is applied at the inner right end of the housing，illustrated as Fig．3(b)，and the Von-Mises stress attains 235 MPa at the same location，slightly higher than case 1．However，for case 3，as shown in Fig．3(c)，the location occurring the maximum Von-Mises stress value shifts to the left end of the housing，reaching to 251MPa，much smaller than the threshold value when yielding happens．As the driving gear rotates clockwise in the main reducer，in comparison to the values its opposite case，that is，in anticlockwise way，the maximum Von-Mises stress increases to 241 MPa，illustrated asFig．3 (d)， higherthan 229MPa，still nears the small hole at the inner end of the right differential housing．For case 2 and case 3 when clockwise， similar condition happens． The maximum stress values for case 2 and case 3 attain 257 MPa and 272 MPa，as shown in Fig．3(e)and Fig．3(f)，respectively．Likewise，the location at the limiting stress value remains as they are．

4.Conclusions

The structures and the working principles of friction-based limited-slip differential were studied and the finite element analyses for the strength of housing were conducted in this work．The results indicate that the structural strength of the differential housing is sufficient to satisfy the requirements of vehicle driving kinematics，which also provide a theoretical basis for an improved design and structural optimization of differential housing．

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