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Research Article

International Journal of Current Engineering and Technology ISSN 2277 - 4106

2013 INPRESSCO. All Rights Reserved.

Available at http://inpressco.com/category/ijcet

Modeling and Finite Element Analysis of Spur Gear

Vivek Karaveer*

, Ashish Mogrekar and T. Preman Reynold Joseph

SMBS Department, VIT University, Chennai, Tamilnadu, India

Accepted 25 December 2013, Available online 30 December 2013, Vol.3, No.5 (December 2013)

Abstract

The contact stress in the mating gears is the key parameter in gear design. This paper presents the stress analysis of

mating teeth of spur gear to find maximum contact stress in the gear teeth. The results obtained from Finite Element

Analysis (FEA) are compared with theoretical Hertzian equation values. For the analysis, steel and grey cast iron are

used as the materials of spur gear. The spur gears are sketched, modeled and assembled in ANSYS DesignModeler. As

Finite Element Method (FEM) is the easy and accurate technique for stress analysis, FEA is done in finite element

software ANSYS 14.5. Also deformation for steel and grey cast iron is obtained as efficiency of the gear depends on its

deformation. The results show that the difference between maximum contact stresses obtained from Hertz equation and

Finite Element Analysis is very less and it is acceptable. The deformation patterns of steel and grey cast iron gears depict

that the difference in their deformation is negligible.

Keywords: ANSYS, Contact Stress, Finite Element Analysis, Spur Gear.

1. Introduction

1 Gear is a rotating cylindrical wheel having teeth cut on it

and which meshes with another toothed part to transmit

the torque or power. Spur gear is a simplest type of gear

having its teeth cut parallel to the axis of shaft on which

the gear is mounted. Spur gears are used to transmit the

power between parallel shafts. The spur gear has 98-99%

operating efficiency (T. Shoba Rani et al 2013). They are

usually employed to achieve constant drive ratio. There

are several stresses present in the teeth of rotating gears

but out of all the stresses, root bending stress and surface

contact stress calculation is the basic of stress analysis

(Sushil Kumar Tiwari et al 2012). Theoretically, for the

calculation of contact stress at the surface of mating teeth,

Hertz equation is used and for determining bending stress

at the root of meshing gears, Lewis formula is used. In

detail study of the contact stress produced in the mating

gears is the most important task in design of gears as it is

the deciding parameter in finding the dimensions of gear

(Mr. Bharat Gupta et al 2012). Also the module of a gear

plays an important role in transmitting the power between

two shafts. The spur gear with higher module is the best

choice for transmitting large power between the parallel

shafts (Mr. Bharat Gupta et al 2012).

T. Shoba Rani et al. (T. Shoba Rani et al 2013) have

used cast iron, nylon and polycarbonate as the materials of

spur gear for finite element analysis. They concluded that

the deflection of cast iron is more as compared to nylon

and

*Corresponding author: vivek karaveer

Polycarbonate. Therefore cast iron spur gear can be

replaced with nylon gear whenever necessary to get the

good efficiency, life and less noise. Sushil Kumar Tiwari

et al. (Sushil Kumar Tiwari et al 2012) found out the

contact stress and bending stress for involute spur gear

teeth in meshing by finite element method and the results

are compared with those obtained by Lewis formula, Hurtz

equation and AGMA/ANSI equations. They concluded

that Hertz theory is the basic theory of contact stress

calculation and for determining bending stress in a pair of

gear, Lewis formula is used. Mr. Bharat Gupta et al. (Mr.

Bharat Gupta et al 2012) have done the finite element

analysis of spur gear using ANSYS 13 to obtain contact

stress between two spur gears and compared with the

results from Hertzian theoretical equation. Based on the

obtained results, they came to the conclusion that the

hardness of the gear tooth profile can be improved to resist

pitting failure. Also the maximum contact stress decreases

with increase in module. If contact stress minimization is

the basic concern and if the large power has to be

transmitted then a spur gear with higher module is

preferred. Ali Raad Hassan (Ali Raad Hassan 2009) has

developed a program to plot paired teeth in contact. This

program was run each 30

of rotation of pinion from the

first location of contact to the last one to create 10 cases.

The program gave graphic results for the teeth of different

finite element models and stress analysis was carried out

in ANSYS. The results obtained from the finite element

analysis were compared with the results determined from

the theoretical calculations, wherever available. Mrs.

Shinde S.P. et al. have performed the bending stress

Vivek Karaveer et al International Journal of Current Engineering and Technology, Vol3., No.5 (December 2013)

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analysis of spur gear in ANSYS software and the results

are compared with the calculated theoretical values. They

found that the values of stress obtained numerically were

in good agreement with theoretical results. Vera Nikolic-

Stanojevic et al. (Vera Nikolic-Stanojevic et al 2003)

described the procedure which is used to determine the

maximum value of active stress and stress fields on tooth

flanks during the period of meshing. Finite Element

Method (FEM) was used for modeling the contact of tooth

flanks. From the results of equivalent stress fields in

contact areas, they concluded that FEM is the suitable

numerical method for different analyses of contact stresses

on mating tooth flanks. The standard model of gear

without the modification of linear tip relief profile has

been meshed and analyzed by Kristina Markovic et al.

(Kristina Markovic et al 2011) using Finite Element

Method to compare the stresses obtained from Hertz

theory on tooth profile.

In this paper, steel and grey cast iron are used as the

spur gear materials. The material properties of steel and

grey cast iron are given in Table 1.

Table 1 Material Property of Steel and Grey Cast Iron

Material Property

Unit Steel Grey Cast Iron

Density Kg/m3 7850 7100

Coefficient of

Thermal Expansion

K-1 1.2e-5 10.5e-6

Poisson Ratio - 0.3 0.26

Youngs Modulus MPa 2e5 165e3

Tensile Yield

Strength

MPa 250 250

Tensile Ultimate

Strength

MPa 460 350

2. Modeling of Spur Gear

In this study, maximum contact stress is determined,

during the transmission of torque of 15000 lb-in or

1694.7725 Nm (Huei-Huang Lee 2012) by steel and grey

cast iron spur gears, using finite element analysis. The

spur gear is sketched and modeled in the ANSYS

DesignModeler. The dimensions of the gears (Huei-Huang

Lee 2012) are given in Table 2.

Table 2 Dimensions of Spur Gear

Dimension Unit Symbol

Value (For

both gears in

assembly)

Number of Teeth - Z 20

Pitch Circle

Diameter Mm D 127

Pressure Angle 0 20

Addendum Radius Mm RA 69.85

Dedendum Radius Mm RD 55.88

Face Width Mm B 25.4

Shaft Radius Mm Rs 31.75

3. Theoretical Calculation of Contact Stresses by

Analytical Method (Hertz equation)

Hertz equation is used to determine the contact stresses in

the mating teeth of gear. Hertzian equation for contact

stress in the teeth of mating gears is given by,

(

)

(

)

(1)

Fig. 1 2D Sketch in ANSYS DesignModeler

Fig. 2 Assembly of Spur Gears

Where is the contact stress in mating teeth of spur gear, is the force, and are pitch radii of two mating gears, is the face width of gears, is the pressure angle, , are the Poisson ratios and , are the modulii of elasticity of two gears in mesh.

Allowable maximum stress is given by,

=

(2)

Here is the factor of safety which can be taken from the ANSYS results or other tables. Equation (2) gives the allowable maximum contact stress in the mating gears.

In this paper, factor of safety is taken from ANSYS

results. But minimum factor of safety from the ANSYS

results is preferred in order to get accurate allowable

maximum contact stress at the point of contact of gear

teeth (Mr. Bharat Gupta et al 2012).

Now, Torque is given by,

Torque = Force ( ) * Shaft Radius (Rs)

1694.7725e3 (N-mm) = * 31.75 (mm)

Vivek Karaveer et al International Journal of Current Engineering and Technology, Vol3., No.5 (December 2013)

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= 53378.6614 N

Case I- Steel Spur Gear:

For steel gear, equation (1) becomes,

= 2601.7984 MPa

From Fig. 6, factor of safety for steel gear is 1.1538.

From (2),

=

= 2254.9821 MPa

Case II- Grey Cast Iron Spur Gear:

For grey cast iron gear, equation (1) becomes,

= 2334.6414 MPa

From Fig. 9, factor of safety for grey cast gear is 1.

From (2),

=

= 2334.6414 MPa

4. Finite Element Analysis

Fig. 3 Meshing

Finite Element Method is the easy technique as compared

to the theoretical methods to find out the stress developed

in a pair of gears. Therefore FEM is widely used for the

stress analysis of mating gears. In this paper, finite

element analysis is carried out in ANSYS Workbench 14.5

to determine the maximum contact stresses for steel and

grey cast iron gears. Also the deformation is found out for

both the gears.

4.1 Meshing

Fine meshing is done to get the accurate results of contact

stress.

4.2 Boundary Condition

Fixed support is applied on inner rim of the lower gear.

Frictionless support is applied on the inner rim of upper

gear to allow its tangential rotation but restrict from radial

translation. Moment of 15000 lb-in or 1694.7725 N-m is

applied on the inner rim of upper gear in clockwise

direction as a driving torque.

Fig. 4 Boundary Condition

5. Results and Discussion

Fig. 5 Stress distribution in Steel gear

Fig. 6 Safety Factor for Steel Gear

Vivek Karaveer et al International Journal of Current Engineering and Technology, Vol3., No.5 (December 2013)

2107

Fig. 7 Deformation pattern for Steel gear

Fig. 8 Stress distribution in Grey Cast Iron gear

Fig. 9 Safety Factor for Grey Cast Iron Gear

Fig. 10 Deformation pattern for Grey Cast Iron gear

Comparison of maximum contact stresses, for both steel

and grey cast iron, obtained from Hertz equation and

ANSYS 14.5 is given in Table 3.

Table 3 Comparison of maximum contact stress obtained

from Hertz equation and ANSYS 14.5

Gear (Hertz)

(MPa)

(ANSYS) (MPa)

Difference

(%)

Steel 2254.9821 2261.2052 0.28

Grey CI 2334.6414 2365.1782 1.29

Conclusions

Here the theoretical maximum contact stress is calculated

by Hertz equation. Also the finite element analysis of spur

gear is done to determine the maximum contact stress by

ANSYS 14.5. It was found that the results from both Hertz

equation and Finite Element Analysis are comparable.

From the deformation pattern of steel and grey cast iron, it

could be concluded that difference between the maximum

values of steel and grey CI gear deformation is very less.

Acknowledgements

The authors wish to acknowledge the technical advice of

Mr. Sumedh Suryawanshi. This work was carried out at

Vellore Institute of Technology, Chennai, PG students of

School of Mechanical and Building Sciences (SMBS)

Department, VIT University, Tamilnadu, India.

References

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Huei-Huang Lee (2012), Finite Element Simulations with

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Sushil Kumar Tiwari, Upendra Kumar Joshi (2012), Stress

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