Modeling and simulation of a spherical bearing mount.

Mihailidis, Athanassios ; Pupaza, Cristina ; Nerantzis, Ioannis 等

1. INTRODUCTION

Contact technology is a powerful functionality implemented in advanced solvers which assists the user to obtain realistic simulation results for complex assemblies and geometries. Formula SAE race cars components such as spherical bearing mounts are special machine elements requiring performance, reliability and safety. These assemblies are used more than 70 times in a race car design. Modeling and simulation are necessary in order to verify the suitability to the applied load and the possibility of using smaller size mounts for future releases of the race cars. Although recent structural reports regarding the mechanical systems of race cars are available (Hill et al., 2001) (Johnson, 2003), no information about spherical bearing mounts and bolted highly-prestressed joints are included. Finite element analysis for bolted joints (Kim et al., 2007), taking into account the pretension and friction with different types of contact elements has been reported, but simplified models were included in the assembly in order to reduce the dimension of the model. The paper deals with modern contact technology procedures applied on a spherical bearing mount, emphasizing the efficiency of this functionality. Remarks regarding possible links with optimization procedures integrated in solvers are also included.

2. SPHERICAL BEARING MOUNT

The A-arms, the pushrods, the shock absorbers and various levers of modern formula SAE race cars are connected to the chassis, bell cranks or wheel uprights by ball joints, also known as spherical bearings. The spherical bearing mount considered in this study is supplied by Askubal which provides the requiring technical data (Askubal, 2008).

Figure 1 shows the bearing mount. The outer ring is press-fitted to a carrying ring which is welded to the carrying part (i.e. the A-arm or the pushrod etc.). It is secured axially by deforming the lips, which have been manufactured at both faces of the carrying ring. This design has proved to be compact, light, reliable and cost effective. No further investigation is needed. The inner ring is typically attached in a U-holder or between two thin metal sheets with a thickness of 1,5-3 mm, by using two identical spacers and a bolt. The cylindrical portions of the spacers have a thickness of just 1 mm and are press-fitted, 0,015 mm oversize, in the inner ring.

[FIGURE 1 OMITTED]

Due to their small thickness, their effect on the clearance of the spherical bearing can be neglected. The load is transmitted from the spherical bearing to the sheets entirely by the friction on the faces A1, A2 and A3, A4. The required normal force is applied by the bolt, which is mounted through the sheets and the spacers. The racing versions of spherical bearing mounts are made of hardened steel and they operate without clearance. Their attachment has to fulfill the following design requirements: the connection must be completely clearance free, the A-arms, the push-rods and the shock absorbers should be easy to disassemble and reassemble.

3. MODELING AND SIMULATION

3.1 CAD and FEM models

The CAD model was completed using Autodesk's Mechanical Desktop and Inventor, modifications were done in SolidWorks 2007 and solved with ANSYS. Friction coefficients with a value of 0,12 were included in the model for the faces A1, A2 and A3, A4 and between the outer and inner ring a 0,04 value was considered. Corresponding materials were properly assigned, as follows: the spacer--30CrNiMo6 with [R.sub.p0,2]=1050N/[mm.sup.2], and the inner ring--100Cr6 hardened steel.

[FIGURE 2 OMITTED]

Several load cases were performed. Finally, the bolt pretension was 20 KN, and the force acting on the carrying part was 2kN, when the carrying part is 10[degrees] right positioned. Figure 2 shows the FEM model which contains 144490 nodes and 36868 elements. Contact sizing, refinement and hexahedral dominant mesh options were chosen in order to obtain a sufficient refined mesh. The final error estimation energy, which gives information for optimizing the mesh density was also processed and controlled. No penetration between parts was detected.

3.2 Contact nonlinearities and FEM procedure

Contact conditions in assemblies are severe form of nonlinearities. The algorithm used for solving boundary or contact nonlinearities is the Newton-Raphson procedure. The finite element discretization yields a set of simultaneous equations:

[K ]{u} = {[F.sup.a]} (1)

where [K]--coefficient matrix; {u}--vector of unknown DOF values, {[F.sup.a]}--vector of applied loads. Equation (1) is nonlinear because [K] is a function of unknown degrees of freedom values and can be written (Bathe, 1996):

[[K.sup.T.sub.i]]{[DELTA][u.sub.i]} = {[F.sup.a]} - {[F.sub.i.sup.nr]} (2)

{[u.sub.i+1]} = {[u.sub.i] + {[DELTA][u.sub.i]} (3)

where [[K.sup.T.sub.i]]--Jacobian matrix (tangent matrix); i--subscript representing the current equilibrium iteration; {[F.sup.nr.sub.i]}--vector of restoring loads corresponding to the element internal loads. The solver options were: iterative, contact stiffness updating after each iteration, weak springs included to facilitate solution, preventing numerical instability, force convergence value, contact interface treatment: add offset value, no ramping.

3.3. Results

Figure 3 contains a plot of the equivalent von-Mises stress, on the pre-tensioned bolted joint. When processing the results, the sheet bracket was hidden in order to show the contact stress distribution. The maximum von Misses stress was 473 MPa and the maximum normal stress 137 MPa. The highest value of the stress acts on spacers, parts manufactured from materials with excellent strength characteristics: tensile yield strength [[sigma].sub.yield]=789 MPa, and tensile ultimate strength [[sigma].sub.u]=1050 MPa.

[FIGURE 3 OMITTED]

3.4. Verification

Because the big number of spherical mounts included in the SAE race cars and the high levels of the stress obtained multiple checks were performed. The theoretical value for the displacement along Z axis was 0,0246 mm and the value obtained through simulation was 0,0241 mm (Fig. 4.). The error can be considered even lower taking into account that the pre-tensioned bolt assembly was modeled as a bar under tension.

[FIGURE 4 OMITTED]

This allows the reduction of the spherical bearing mount size without an increased potential for failure, but additional checks regarding the dynamical behavior of the assembly are necessary, especially when the load is applied with shock.

4. COUPLING CONTACT ANALYSIS WITH OPTIMIZATION PROCEDURES

Recent contact features implemented in solvers allow the user to rapidly obtain stress evaluation in complicated assemblies, but handling and maintaining the parameters definition is not easy. Usually, when transferring data between CAD-CAE systems the definition of the parameters in a text format is not available anymore. The DS option allows the definition of geometrical parameters in a late stage of the design without returning to the CAD system. This is an important advantage because parametrical optimization procedures can be accessed.

5. CONCLUSION

The research revealed the possibility of using smaller size bearing mounts for future releases of the race cars, which means the reduction of the total weight. Contact technology supports the designer in rapidly evaluating solutions, and allows the reduction of the total weight of a product in cases were reliability, safety and integrity are required. The procedure has the following advantages: it is simple and fast; no assumptions have to be done regarding the load input and no empirical factors have to be used. Mesh quality in the contact region is controlled, model parameterization is possible and the user can edit the solver input file. Future development of the procedure in optimization loops has also been investigated. The possibility of applying the same procedure to internal spur gears was already done and promising results are in progress.

6. REFERENCES

Askubal (2008). Rod ends and spherical bearings. Available from: http://www.askubal.de. Accessed: 2008-04-12

Bathe, J.K. (1996). Finite Elements Procedures, Prentice-Hall Inc., ISBN 0-13-301458-4, New Jersey, USA

Hill, J.; Binderup, A.; McBride, H.; Nimmergut, B. (2001). American Solar Challenge, PrISUm Odyssey Structural Report, Available from: http://xnet.rrc.mb.ca/solacar/ downloads/structuralReport.pdf. Accessed: 2008-03-06

Johnson, D. (2003). American Solar Challenge 2003 Structural Report. Team Lxu, Yale University 10 Marston Hall, Ames, Iowa 50011. Available from: http://www.eng.yale.edu/TeamLux/TLProject_Web/docs/4-JL_Structural_Report.pdf. Accessed: 2008-03-10

Kim, J.; Yoon, J. C. & Kang, B-S. (2007). Finite element analysis and modeling of structure with bolted joints. Applied Mathematical Modeling, Vol. 31, Issue 5, May 2007, p. 895-911. Elsevier Inc., Available from: http://www.sciencedirect.com/. Accessed: 2008-04-1