International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 55

(ICCOMIM - 2012), 11-13 July, 2012

ISBN 978-93-82338-03-1 | © 2012 Bonfring

Abstract--- In spite of the number of investigations devoted to gear research and analysis there still remains to

be developed, a general numerical approach capable of predicting the effects of variations in gear geometry,

contact and bending stresses, and transmission errors. One of the primary objectives is to use a numerical approach

to develop theoretical models of the behavior of spur gears in mesh, to help to predict the effect of gear tooth

stresses and transmission error.

The main focus is to develop and to determine appropriate models of contact elements, to calculate contact

stresses using ANSYS and compare the results with Hertzian theory and also to generate the profile of spur gear

teeth and to predict the effect of gear bending using a three dimensional model and two dimensional model and

compare the results with those of the Lewis equation and also determines the static transmission errors of whole

gear bodies in mesh.

This work investigates characteristics of an involute gear system using Finite Element (FE) Methods. Bending

and contact stresses are evaluated using nonlinear FE methods and then compared against AGMA standards to

establish an accurate design procedure. The study started with evaluation of contact stress using ANSYS code for

simulating a pair of cylinders in contact. The results obtained are in good agreement with Hertz’s equation. A single

tooth model was then analyzed for arriving at the bending stress. Forces were applied at different radii of the tooth

and peak stresses obtained at the root were compared with AGMA standard evolved out of basic Lewis formula with

several corrections taken into account.

The results of 2D and 3D FE models of complete pinion and gear are also presented. It is shown that AGMA

standards provide a very conservative approach with a single tooth analysis and FE approach provides a more

accurate result for the bending and contact stress. Finally, transmission error arising from deformations in the

pinion and gear due to variations in stiffness in one meshing period is evaluated. Different positions within the

meshing cycle are analyzed and investigated and the results are reported.

I. INTRODUCTION

EARING is one of the most critical components in a mechanical power transmission system, and in most

industrial rotating machinery. It is possible that gears will predominate as the most effective means of

transmitting power in future machines due to their high degree of reliability and compactness. In addition, the rapid

shift in the industry from heavy industries such as shipbuilding, automobile manufacture and office automation tools

will necessitate a refined application of gear technology.

Gears analyses in the past were performed using analytical methods, which required a number of assumptions

and simplifications. In general, gear analyses are multidisciplinary, including calculations related to the tooth

stresses and to tribological failures such as like wear or scoring. In this thesis, static contact and bending stress

analyses are performed, while trying to design spur gears to resist bending failure and pitting of the teeth, as both

affect transmission error.

The prime source of vibration and noise in a gear system is the transmission error between meshing gears.

Transmission error is a term used to describe or is defined as the differences between the theoretical and actual

positions between a pinion (driving gear) and a driven gear. It has been recognized as a main source for mesh

frequency excited noise and vibration. With prior knowledge of the operating conditions of the gear set, it is possible

S. Puttaswamaiah, Assistant Professor and Research Scholar, Department of Mechanical Engineering, EWIT, Bangalore

Dr.J.N. Prakash, Professor, Department of Mechanical Engineering, EWIT, Bangalore K.B. Kiran, Research Scholar, Department of Mechanical Engineering, EWIT, Bangalore

PAPER ID: MED10

Finite Element Stress Analysis of Spur Gear and

Evaluation of Transmission Error S. Puttaswamaiah, Dr. J.N. Prakash and K.B. Kiran

G

International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 56

(ICCOMIM - 2012), 11-13 July, 2012

ISBN 978-93-82338-03-1 | © 2012 Bonfring

to design the gears such that the vibration and noise is minimized.

Under normal operating conditions, the main source of vibration excitation is from the periodic changes in tooth

stiffness due to non-uniform load distributions from the double to single contact zone and then from the single to

double contact zone in each meshing cycle of the mating teeth. This indicates that the variation in mesh stiffness can

produce considerable vibration and dynamic loading of gears with teeth, in mesh. The torsional stiffness of two spur

gears in mesh varied within the meshing cycle as the number of teeth in mesh changed from two to one pair of teeth

in contact. Usually the torsional stiffness increased as the meshing of the teeth changed from one pair to two pairs in

contact. The theoretical changes in the torsional mesh stiffness throughout the mesh cycle are generated by using

finite element analysis. Even though the transmission error is relatively small, these slight variations can cause noise

at a frequency which matches a resonance of the shafts or the gear housing, causing the noise to be enhanced. This

phenomenon has been actively studied in order to minimize the amount of transmission error in gears. The purpose

of this thesis is to study and predict the transmission error, torsional mesh stiffness, bending and contact stresses of

gears in mesh using the ANSYS R10 software package based on numerical method. Suggestions to reduce the

transmission error in the gears, and thereby reduce the amount of noise generated are made.

II. SPUR GEARS

Spur gears are the most commonly used gear type. They are characterized by teeth which are perpendicular to

the face of the gear. Spur gears are by far the most commonly available, and are generally the least expensive. The

basic descriptive geometry for a spur gear is shown in the figure below.

Limitations: Spur gears generally cannot be used when a direction change between the two shafts is required.

2.1. Terminology and Definitions

Figure 1: Gear Terminology

Pitch surface: The surface of the imaginary rolling cylinder (cone, etc.) that the toothed gear may be considered

to replace.

Pinion: The smallest of any pair of mating gears. The larger of the pair is called simply the gear.

Pitch circle: A right section of the pitch surface.

Addendum circle: A circle bounding the ends of the teeth, in a right section of the gear.

Root (or dedendum) circle: The circle bounding the spaces between the teeth, in a right section of the gear.

Addendum: The radial distance between the pitch circle and the addendum circle.

Dedendum: The radial distance between the pitch circle and the root circle. Clearance: The difference between

the dedendum of one gear and the addendum of the mating gear.

Face of a tooth: That part of the tooth surface lying outside the pitch surface.

International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 57

(ICCOMIM - 2012), 11-13 July, 2012

ISBN 978-93-82338-03-1 | © 2012 Bonfring

Flank of a tooth: The part of the tooth surface lying inside the pitch surface.

Module, m: Pitch diameter divided by number of teeth. The pitch diameter is usually specified in inches or

millimeters; in the former case the module is the inverse of diametral pitch.

Tooth space: The distance between adjacent teeth measured on the pitch circle.

Backlash: The difference between the circle thickness of one gear and the tooth space of the mating gear.

Circular pitch, p: The width of a tooth and a space, measured on the pitch circle.

2.2. Basic Law of Gearing

Figure 2: Law of Gearing

A common normal (the line of action) to the tooth profiles at their point of contact must, in all positions of the

contacting teeth, pass through a fixed point on the line-of-centers called the pitch point. Any two curves or profiles

engaging each other and satisfying the law of gearing are conjugate curves, and the relative rotation speed of the

gears will be constant.

III. DESIGN PARAMETERS OF SPUR GEARS

Figure 3: Generated Spur Gear Train Model

Line of action

passing through

pitch point P

International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 58

(ICCOMIM - 2012), 11-13 July, 2012

ISBN 978-93-82338-03-1 | © 2012 Bonfring

Table 1: Pinion Model Details

Gear profile Involute type

Module 3 mm

No. of teeth 32

Pitch circle diameter 96 mm

Pressure Angle 20º

Tooth thickness 4.7124 mm

Fillet radius 1.2 mm

Addendum 3 mm

Dedendum 3.75 mm

Tooth depth 6.75 mm

Power transmitted 28 kW

Speed 2500 rpm

Table 2: Pinion Model Details

Gear profile Involute type

Module 3 mm

No. of teeth 50

Pitch circle diameter 150 mm

Pressure Angle 20º

Tooth thickness 4.7124 mm

Fillet radius 1.2 mm

Addendum 3 mm

Dedendum 3.75 mm

Tooth depth 6.75 mm

IV. GEAR TOOTH STRESS ANALYSIS

This analysis is carried out to investigate the bending strength of gear tooth. The gear tooth is assumed as a

International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 59

(ICCOMIM - 2012), 11-13 July, 2012

ISBN 978-93-82338-03-1 | © 2012 Bonfring

simple cantilever beam and the load is applied at the tip of the tooth beam. The gear tooth strength is calculated by

Lewis equation. Bending stress in a single tooth is analyzed for a load applied at different radii. The peak stresses

are expected to occur in the root fillets and a fine mesh is generated to capture these stresses. The stresses found are

then compared with computed results using AGMA standard. For this analysis, the pinion tooth is taken as shown is

below Figure 4.

Figure 4: Pinion Single Tooth Figure 5: FE Model of Single Tooth

4.1. Load Application on the Tooth at Different Contact Points Calculated ( based on radius)

Figure 6: Load Application on the Single Tooth Model at Different Radii

The single tooth model is assumed as a simple cantilever beam. The normal load is applied at the tip of the tooth

along the pressure angle. Bottom of the tooth nodes are constrained at all DOF. The tooth side nodes are constrained

in tangential direction. Here, the maximum bending stress occurs at the root of the gear tooth. To capture the stress

at the root, very fine mesh is used in that region. The meshed model is analyzed for load applied at different radii.

The obtained results are given in section 5.

4.2. One Pair of Teeth in Contact

A gear tooth pair is analyzed by simulating contacts over the involute profile. Bending stress in the roots and

contact stress at the gear meshing region are determined at different contact positions. Only a sector of model with

plane elements is considered for analysis.

International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 60

(ICCOMIM - 2012), 11-13 July, 2012

ISBN 978-93-82338-03-1 | © 2012 Bonfring

Figure 7: One Pair of Teeth in Contact Figure 8: FE Model of One Pair Model

4.3. Loads and Boundary Conditions

Figure 9: Load and Boundary Conditions of One Pair Tooth Model

Table 3: Element Details

Type of element

Plane 42 (Plane Stress with Thickness)

Contact element

CONTA171 2-D surface to surface

Target element

TARGE169

Link elements Link 2D spar

No. of elements 7764

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(ICCOMIM - 2012), 11-13 July, 2012

ISBN 978-93-82338-03-1 | © 2012 Bonfring

Torque is simulated by application of force in tangential direction on the Pinion hub nodes. To avoid rigid body

motion pinion hub nodes are connected with center of pinion by Link elements and the center of the pinion is

constrained in all DOF to avoid rigid body motion. Gear hub nodes are constrained in all DOF and side nodes are

constrained at tangential direction to avoid sliding.

Contacts are made between the tooth by creating the contact elements where the contact region. Here, Gear is

considered as a Target region. Pinion is considered as a contact region.

V. RESULTS AND DISCUSSION

5.1. Contact Stress Simulation of Two Cylinders

In this analysis Hertz’s stress is calculated and compared with FE results. The results obtained are plotted.

It represents the Contact stress over the contact region. In which we infer that due to the compressive force,

contact established results in a stress distribution dying out in a small region. Maximum compressive stress of

552.042 MPa occurs at the center of the contact width and reduces when moving along the contact width region and

its radial direction.

The contact conditions are sensitive to the geometry of the contacting surfaces, which means that the finite

element mesh near the contact zone needs to be highly refined

It is recommended not to have a fine mesh everywhere in the model to reduce the computational

requirements

It gives us an idea about the contact element selection and solve the non-linearities by increasing number of

iterations

The results of the two dimensional FEM analysis from ANSYS and the Hertz’s stress values are compared

and found to be close

VI. RESULTS

6.1. Contact Stress Simulation of Two Cylinders

Figure 10: Contact Stress over the Contact Area

International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 62

(ICCOMIM - 2012), 11-13 July, 2012

ISBN 978-93-82338-03-1 | © 2012 Bonfring

Table 4: Hertz’s stress in MPa

By Calculation By ANSYS

574.54 552

VII. CONCLUSION

2D FE model of two cylinders was used to estimate the contact stress. The results of the two dimensional

FEM analysis from ANSYS and the Hertz’s stress values are compared and found to be close.

In the single gear tooth bending stress analysis, the load is applied at various radii. The maximum bending

stress occurs when the load is applied at the tooth tip. It gradually decreases as the load applied shifts

towards to the root. The FE value and AGMA calculated value are in close agreement when the load is

applied at the tip.

Analysis of a pair of gear tooth gave the bending stress values which are lesser than the AGMA standard.

The reasons may be approximations involved in the model such as a single tooth carrying the load, model

being simple cantilever beam with load applied tangentially at the tip.

2D and 3D whole gear train FE models were used to estimate the contact stress and bending stress values

which were lower than the results obtained through AGMA standards.

For one complete mesh cycle, the transmission error is evaluated and the results are plotted. By using these

results, torsional mesh stiffness of the gears is found and the results obtained are plotted.

REFERENCES

[1] Shigley, J.E., and Mischke, L.D., 1983, “Mechanical Engineering Design ". McGraw-Hill.

[2] Rao, J.S., “Turbine Blade Life Estimation”

[3] Buckingham, E., 1949, “Analytical Mechanics of Gears”, McGraw-Hill, New York.

[4] Tsay, C.B., 1988, “Helical Gears with Involute Shaped Teeth: Geometry, Computer Simulation, Tooth

Contact Analysis, and Stress Analysis”, Trans, J. Mechanisms, Transmissions, and Automation in Design.

[5] O’Donnell, W. J., 1974, “Stress and Deflection of Built-in Beams”, ASME Paper No. 62-WA-16.

[6] Gitin M. Maitra, 1984, “Hand Book of Gear Design”, Second Edition, Tata McGraw-Hill.

[7] Klenz, S. R., 1999, “Finite Element Analyses of A Spur Gear Set”, M.Sc. Thesis, Dept. of Mechanical

Engineering, University of Saskatchewan.

[8] Harris, S. L., 1958, “Dynamic load on the teeth of spur gears”, Proc. Instn Mech. Engrs, 172, 87-112.

[9] Mark, W. D., 1978, “Analysis of the vibratory excitation of gear system: Basic theory”, J. Acoust. Soc.

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[10] Kubo, A., et al., 1991, “Estimation of transmission error of cylindrical involute gears by tooth contact

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