Analysis of Bending Stress at Root of the Spur Gear by Finite Element Method

Anil PatilLecturer in Department of Mechanical Engineering, Shantiniketan Polytechnic

Maharashtra, India__________________________________________________________________________

Abstract The basic task of this paper is determining and analyzing bending stress in gear tooth .In this paper, the bending stresses are determined by using Lewis cantilever beam theory and finite element analysis by considering only one gear tooth pair in contact. This paper gives detailed procedure of gear model generation, meshing, loading conditions and post processing of results. In determining bending stress by finite element analysis, in one case only tangential force acting on gear tooth is considered while in another case both tangential force as well as radial force are considered. The bending stress obtained by finite element analysis method is compared with bending stress obtained by Lewis equation.

Key words: - Gear tooth, bending stress, finite element analysis, tangential force, radial force___________________________________________________________________________

Introduction Gears are means bywhich power is transmitted from source to application. Gearing and geared transmissions drive the machines of modern industry. Yet many engineers and researchers continue to develop into areas where improvements are necessary, seeking to quantify, establish and codify methods to make gears to meet the ever-widening needs of advancing technology. Many advanced technology applications have a general requirement that the power to the transmission weight ratio should be high. Therefore it is necessary to determine accurate stresses induced in gear tooth, while transmitting the power. The failure of gear tooth occurs due to pitting of gear tooth surface or and bending stress at the root of gear tooth. There are several approaches to determine stresses and deflections in gear tooth. One of the modes of failure of a gear tooth results from breakage in the root fillet area. The most commonly used method to determine bending stress is Lewis cantilever beam theory. The most powerful method to determine bending stress and deflection is finite element

method. The purpose of this paper is to apply finite element analysis method for bending stress calculation of gear tooth.

Nomenclature

σb Bending Stress

P Diametrical pitch

F Face width

Y Lewis Factor

W Load Applied at Tip of Tooth

Wt Tangential Component of Tooth Load

Ka Application Factor

Ks Size Factor

Km Load Distribution Factor

Kv Dynamic Load Factor

J Geometry Factor

t Thickness of gear tooth

T Number of Teeth

M Module

Wr Radial component of tooth Load

Lewis Equation for Gear Strength:-

Fig No.1 (a) Gear Tooth Load.

Fig. No.1 (b) Gear tooth modeled as a cantilever beam

Historically, the first equation used to determine the bending stress at the root of the gear tooth was derived by Lewis. In this equation the gear tooth is considered as a simple cantilever beam as shown above. Lewis equation to calculate bending stress is as given below,

σb = Wt*P/FY, The Lewis equation is based on following assumptions

1) The effect of radial load is ignored.2) The effect of stress concentration at

the fillet is ignored.3) It is assumed that at any time only

one pair of teeth is in contact and takes total load.

Modified equation by AGMA

A more comprehensive equation has been defined by AGMA as follows,

σb = Wt * P* Ka* Ks * Km /F * J * Kv

AGMA equation for bending stress calculation takes into account radial load as well as stress concentration at root of gear tooth.

Values of Key Parameters used for analysis are as given below

Number of teeth (T) 25

Module (M) 4 mm

Pressure Angle 20 degree

Addendum(mm) 1.00 M

Dedendum (mm) 1.25 M

Face width 15 mm

Modulus of Elasticity 2*105 N/mm2

Poisson’s Ratio 0.27

Hub Diameter 25 mm

Finite Element Analysis:- Fatigue and yielding of a gear tooth due to excessive bending stresses is two important gear design considerations. In order to predict fatigue and yielding, the maximum stresses on the tensile and compressive sides of the tooth, respectively, are required. In the past, the bending stress sensitivity of a gear tooth has been calculated using photo elasticity or relatively coarse FEM meshes. However, with present computerdevelopments we can make significant improvements for more accurate FEM simulations

Finite Element Model of Spur Gear Tooth The most time consuming step in finite element analysis is creating the finite element model. After determining type of analysis required and the characteristics of operatingenvironment, a finite element model must be produced with appropriate analysis parameters, such as load, constraints, element choice and a suitable mesh. The model can be generated by using Different3D modeling commercial softwares such Ideas, Solidworks, Catia etc are available in market. In this study 3D model of the gear is developed by using Catia. After generating model in Catia it is imported in Ansys and meshed by using brick45 element. After meshing,loading conditions and boundary conditions are applied on gear model. The root stresses and the deflection of one tooth of a spur gear is calculated. Fig No. 2 shows loading condition of gear tooth considering only tangential component of load 850 N.

Fig No. 2 FEA model of Gear Tooth with Tangential Load

In first case, only tangential component of the force is considered and force is varied from 850 N to 1277 N. In this case, stresses induced on tensile side and on compressive side of root of tooth have same magnitude.

Fig No. 3 Von Misses stresses at the root of tooth

Fig. No. 3 shows the bending stress

induced at root of gear tooth when load of 850 N is applied at tip of the single tooth and its only tangential component is considered. In above case following results are obtained.

Load Applied Stress Induced

850 N 83.94 N/mm2

958 N 94.53 N/mm2

1065 N 105.02 N/mm2

1170 N 115.53 N/mm2

1277 N 126.33 N/mm2

In second case, tangential component as well as radial component of the force is considered with same boundary condition as above and also load is varied from 850 N to 1277 N.

Fig No. 3 FEA model of Gear Tooth with Tangential and Radial Load

Fig No. 2 shows loading condition of gear tooth considering tangential component as well as radial component of force 850 N. The results obtained in this case shows different values of stresses on tensile side and on compressive side of tooth.

Fig No. 5 Von Misses stresses on the root of tooth on Tensile Side of Tooth

Fig No. 6 Von Misses stresses at the root of tooth on Compressive Side of Tooth

Fig No. 5 shows stress induced at root of tooth on tensile side while fig no.6 shows stress induced at root of tooth on compressive side by applying load of 850 N. The stresses induced on tensile side and compressive side by considering tangential component as well as radial component of load is given below,

Load Applied

Stress on Tensile Side

Stress on Compressive

side850 N 72.36 N/mm2 81.40 N/mm2

958 N 81.95 N/mm2 91.15 N/mm2

1065 N 90.38 N/mm2 101.67 N/mm2

1170 N 99.42 N/mm2 111.85 N/mm2

1277 N 108.8 N/mm2 122.02 N/mm2

Conclusion The objective of current work is to study analytic and finite element method for determination of bending stresses in gear tooth. By analytic method it is possible to calculate stress by applying load only at one position on gear tooth. While, in finite element it is possible to apply load at various positions on tooth profile and determine stresses for particular loading conditions. In this method we can apply tangential force as well as radial force on true shape of gear profile. The accurate results of finite element method can be used in fatigue life estimations of gear tooth structure. The result shows that the bending stresses determined by analytical method have same magnitude of stress on tensile side as well as compressive side of the tooth. In case of finite element analysis, if only tangential component of force is considered then result shows that tensile stress and compressive stress have same magnitude. But, in finite element analysis, if we consider both tangential and radial component of force then result shows that compressive stress has more magnitude compared to tensile stress. Also result shows that if radial force is considered in analysis, the maximum value of the induced stress is also decreases compared to stress induced by considering only tangential load.

References:-1) John J. Coy, Dennis P. Townsend, Erwin V. Zeretsky. “Gearing”, NASA Reference Publication 1985.2) Nice Carter, “The Involute Curve, Drafting a Gear in CAD and Applications”.3) S. M. Mahbub Murshed, M Arifeen Wahed, M. A. Salam Akanda, “Parametric Solution of Spur Gear Tooth under Tip Load”. Dhaka University.4) C. E. Passerello and L. W. Shuey, “An Investigation of Dynamic Response of Spur Gear Teeth With Moving Load.”, Michigan Technological University, Michigan, 1987.5) Niemann G., “Machine Elements Volume 2”, Springer – Verlag, Berlin6) Gitin Mitra, “Handbook of Gear Design”, Tata McGraw Hill International Edition.