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Finite Elements in Analysis and Design 35 (2000) 213225 Stress analysis of the ring in continuously variable transmission mechanism Serdar Tumkor Design and Manufacturing Institute, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA Abstract During the service performance of a friction ring in a continuously variable transmission (CVT) mecha- nism, the most commonly observed failure mode is fatigue failure. The main purpose of this study is to determine the reason of this failure and to improve the shape of the ring. The ring is modeled by the “nite elementmethod(FEM) andalso checkedbyanalyticalformulasgeneratedfor circularrings.The calculations show that high interior stresses are present on the contact surfaces. The shape of the ring is modi“ed to “nd the optimum shape and dimensions of the ring. By turning a groove inside of the ring, the surface stress is decreased, but the stresses near the groove is increased based on a FEM analysis depending on the groove radii. After analysis and optimization of the groove radius, the dimension R“1.95 mm is calculated and tested. Fatigue failure is observed to decrease by testing the modi“ed ring. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Continuously variable transmission; Shape optimization; Parametric “nite element model 1. Introduction In Industry continuously varying speeds are needed for many applications concerning mecha- nisms. These types of mechanisms, which have a variable velocity and torque ratio, are called continuously variable transmission (CVT) mechanisms. Di!erent types of CVT-drives have been successfully used in di!erent industrial applications for many years. Variable metal belt and chain drives are very well known. In the scope of this study, a conical and steel ring assembled traction drive is investigated. A CVT-drive with a steel ring is inexpensive because of the low-cost simple ring part used. But high torque cannot be transmitted by frictional drives. Depending on time, a type of fatigue failure called pitting can be observed due to the normal force, which is needed for the frictional bound between the cones and ring. The literature on CVT is very scarce. A performance characteristic of a variable metal V-belt drive, which has a similar assembly and performance characteristic, is given by Sun 1. Speed 0168-874X/00/$-see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X( 9 9 ) 0 0 0 6 5 - 7 Nomenclature Aarea of the cross section (mm2) b1width (mm) ddiameter (mm) Emodulus of elasticity (Youngs modulus) (Pa) FNnormal force (N) hdistance from the centroidal (mm) kconstant Mbending moment (N mm) Rgroove radius (mm) rradius (mm), radial position of the stress rnneutral axis radius(mm) Rrcentroidal axis radius(mm) x,y,zcoordinates kcoe$cient of friction lpoisson ratio p.!9maximum stress (MPa) phcircumferential normal stress (MPa) ratio, transmitted torque, and friction coe$cient e!ect for the variable metal V-belt drive are analyzed by Karam 2. The CVT mechanism of power transmitting has been studied by Kuwabaraet al. 3,4.Dynamicof the frictionalchain drivesis simulated to getrealisticpredictions about the system behavior 5,6. The nonlinear dynamics of the chain drive CVT are also studied by Pausch and Pfei!er 7 in automotive drive train systems. Another study on CVT is performed by Takemoto et al. 8 about the noise problem caused by metal-to-metal contact scheme. Based on the “nite element method (FEM), new optimization procedures have been developed to transfer biological optimization mechanisms to mechanical engineering 9. It means that any biological load carrier tends to achieve a constant stress at least in a time average. Unacceptable high stresses due to stress concentrationswould cause failure, while the underloaded zones are wasted materials. Therefore, the principle of lightweight design is the main criterion for the shape of natural structures. In some cases the geometrical constraints should be considered for the functioning of the mechanisms, therefore the geometry becomes more critical than the weight considerations. In those cases principles of the computer aided optimization (CAO) Method can be used however, certain geometric constraints must be considered. This method was developed by Mattheck and Burkhardt 10 to improve the shape of highly loaded components with respect to a reduction and homogenization of stresses on the surface. In the “rst FEM run the stresses are calculated with a “nite element model, in which actual load cases and boundary conditions are used. This stress distributionwilllead to improve the shape.AfteranotherFEM run of improvedstructurethe stress peaks will be reducedand homogenized.Kasper discusses some optimization methods in combina- tion with general numeric “eld computation methods like FEM 11. 214S. Tumkor / Finite Elements in Analysis and Design 35 (2000) 213225 Fig. 1. The assembly of CVT mechanism. In this paper a CVT mechanism is reported, which has a fatigue damage problem because of the high stress on its metal ring. An approximate analytical solution for the circumferential stress is calculated. To “nd out the stress distribution, FEM analysis is done. Modi“cations of the ring design are achieved using a hybrid of the CAO and curve “tting procedures together. The modi“ed cases are tested and the fatigue failure is seen to decrease. 2. Description of the mechanism In Fig. 1 there are two friction cones on the motor shaft of the mechanism forming a pulley. b1 cones are “xed and b2 cones are free in the axial direction. b2 cones are allowed to move within the prescribed range. Both of these cones move the same distance since they are connected to each other. An Isometric view and the picture of the CVT-mechanism are shown in Figs. 2 and 3, respectively. Velocity ratio varies between the minimum and maximum values depending on the location of the ring relative to the pulley. Torque is transmitted through line contacts between the pulley formed by the cones and the steel ring. Friction interface is satis“ed by the applied normal force due to given prestress to connection pointsof the steel ring andcones. For rolling withoutsliding, the normalforceand the frictionforce are related by the Coulombs model as follows: FN*F numerical analysis of forces acting on a belt at steady state, JSAE 19 (1998) 117122. 5 J. Srnik, F. Pfei!er, Dynamik von CVT-Kettengetrieben; Modellbildung und-Veri“kation, VDI Berichte 1285 (1996) 441. 6 M. Pausch, F. Pfei!er, Simulation of a chain drive CVT as a mechatronic system, Proceedings of COC97, International Conference On Control of Oscillations and Chaos, Vol. 3, St. Petersburg, 1997, pp. 391394. 7 M. Pausch, F. Pfei!er, Nonlinear dynamics of a chain drive CVT, Proceedings of ICNM98, International Conference On Nonlinear Mechanics, University of Shanghai, China, 1998, pp. 336341. 8 Y. Takemoto, A. Shozaki, K. Imamura, Y. Yoneda, M. Hiraoka, Y. Shiina, Noise reduction on contiuosly variable transmission, Proceedings of the International Modal Analysis Conference, 1998, pp. 855861. 9 C. Mattheck, A. Baumgartner, F. Walther, Optimization procedures by use of the “nite element method, Eng. Systems Des. Anal. 4 (1994) 2732. 10 C. Mattheck, S. Burkhardt, A new method of structural shape optimization based on biological growth, Int. J. Fatigue 12 (1990) 185190. 11 M. Kasper, Optimization of FEM models by stochastic methods, Int. J. Appl. Electromagn. Mater.