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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
International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 61
(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.
Am., 63, 1409-1430.
[10] Kubo, A., et al., 1991, “Estimation of transmission error of cylindrical involute gears by tooth contact
pattern”, JSME int. J., Ser. III, 34(2), 252-259.
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