Static Analysis of Gear Tooth Stress Using FEM

International Journal of Engineering and Techniques - Volume 2 Issue 3, May – June 2016
ISSN: 2395-1303 http://www.ijetjournal.org Page 39
Static Structural analysis of gear tooth
Pravin B. Sonawane1
, P.G.Damle2
1
B.E. Mechanical, SSBT’s COE Bambhori, Jalgaon
2(Associate Prof.) Department. of Mechanical Engineering ,COE , Bambhori, Jalgaon.
INTRODUCTION:
Spur gears are the most common means
of transmitting power in the modern mechanical
world. They vary from tiny size used in the watches
to the large gears used in marine speed reducers;
bridge lifting mechanism and railroad turn table
drivers. They for vital elements of main and
ancillary mechanism in many machine tools,
rolling mills, hoisting and transmitting machinery
and marine engines etc.
The four major failure modes in gear
systems are tooth bending fatigue, contact fatigue,
surface wear and scoring. Two kinds of teeth
damage can occur on gears under repeated loading
due to fatigue; namely the pitting of gear teeth
flanks and tooth breakage in the tooth root. Tooth
breakage is clearly the worst case of damage, since
the gear could have seriously hampered operating
condition. Because of this, the stress in the tooth
should always be carefully studied in all practical
gear application. The fatigue process leading to
tooth breakage is divided into crack initiation and
crack propagation period. However, the crack
initiation period generally account for the most of
service life, especially in high cycle fatigue.
Spur gears are very useful in numerous
applications. Not only can they transfer velocity
and torque one shaft to other shaft, but, by using
different size gears, they can alter the ratio between
velocity and torque as they transfer them; a gear
with many teeth driving a gear with fewer teeth will
RESEARCH ARTICLE OPEN ACCESS
Abstract:
This study investigate the characteristics of a gear system including contact stresses, bending
stresses, and the transmission errors of gears in mesh. Gearing is one of most critical component in
mechanical power transmission systems. The bending stresses in the tooth root were examined using 3D
model.
Current methods of calculating gear contact stresses use Hertz’s equations and Lewis Equation
which were originally derived for contact between two cylinders. To enable the investigation of contact
problems with FEM, the stiffness relationship between the two contact areas is usually established
through a spring placed between the two contacting areas. This can be achieve by inserting a contact
element in between the two areas where contact occurs. The results of the three dimensional FEM analysis
from ANSYS are presented. These stresses were compared with theoretical values. Both results agree very
well. This indicates that the FEM model is accurate.
This report also considers the variations of the whole body stiffness arising from gear body
rotation due to bending deflection, shearing displacement and contact deformation. Many different
positions within the meshing cycle were investigated. Investigation of contact and bending stress
characteristics of spur gears continues to be immense attention to both engineers and researchers in spite
of many studies in the past.
This is because of the advances in the engineering technology that demands for gears with ever
increasing load capacities and speeds with high reliability, the designers need to be able to accurately
predict the stresses experienced the stresses by the loaded gears.
Keywords — Spur Gear Tooth, Ansys 14.5 Workbench and classic, Lewis Equation.

have less torque, but greater velocity and vice
versa. Unfortunately, spur gears require a very
specific shape for their teeth to work smoothly.
Even a simple mockup gear would require a
complex surface in order to function properly.
Without the calculations required to create these
surfaces, two gears would not mesh together
smoothly, making it difficult to test the gears. But
mockup does not need to work for a long period of
time; it can be made of lighter, easily cut materials.
These lighter materials could be handle by a laser
cutter if suitable instructions were developed so
that the proper shape of teeth could be computed.
The tooth of spur gear is based on a
mathematical shape as an involute. Since each
tooth can be described by a series of mathematical
equations, it is possible to define a gear in terms of
a few key parameters, such as the number of teeth
and diameteral pitch. These parameters make it
easy to tell if two gears can mesh together.
Similarly by specifying the parameters first, it
would be simple to design a gear for any given
applications from scratch.
Because spur gear is essentially two
dimensional shapes they could be cut out quickly
using a laser cutter. Laser cutters use a laser beam
to slice two dimensional shapes out of flat material,
so the silhouette of a spur gear would be easy to
make using a laser cutter. Unfortunately, most laser
cutter do not have built in software’s to cut gears;
They are driven by a series of simple move-to and
draw-to commands, tracing out straight lines or
elliptical arcs across the material. Therefore, an
involute must first be converted into
approximations using these simple commands in
order for the cutter to understand it. If a laser cutter
is to be useful in creating mockups, it must be able
to cut these gear designs quickly and easily.
Figure 1 Gear teeth meshing.[1]
SPUR GEAR TERMINOLOGY:
Figure 3.1 Gear terminology [1]
DEFINITIONS:-
1) Module: - It is defined as the ratio of diameter to
the number of teeth.
2) Face width (b):- It is the width along the contact
surface between the gears.
3) Addendum: - The radial distance between the pitch
circle and the top land of the gear is called the
addendum.
4) Dedendum: - The radial distance between the pitch
circle and the bottom land of the gear is called
dedendum.

5) Pitch circle: - It is an imaginary circle which by
pure rolling action would give the same motion as
the actual gear.
6) Root circle: - It is the circle drawn through the
bottom of the tooth.
7) Pressure angle: - It is the angle between the
common normal to two gear teeth at the point of
contact and the common tangent at the pitch point.
The standard pressure angles are 14.5 and 20.
8) Diametral pitch: - it is the ratio of number of teeth
to the pitch circle diameter.
DESIGN PARAMETERS:
= Velocity factor
= Structural stiffness
U = Displacement vector
F =Applied load vector
Pmax = Maximum contact stress
Dp = Pinion pitch diameter
Dg = Gear pitch diameter
Fi =Load per unit width
Φ =Pressure angle
E = Young‘s modulus
b = face width
pd = Diametric pitch
Y = Lewis form factor
Ks = Size factor
Ft = Normal tangential load
MATHEMATICAL CALCULATIONS:
PROBLEM:-
Calculate the power that can be transmitted safely
by a pair of spur gears with the data given below.
Calculate the power and the bending stress induced
in the two wheels when the pair transmits that
power. [1]
The theoretical design calculations are performed
using the input parameters given below
No. of Teeth in the Pinion = 20
No. of Teeth in the Gear = 80
Module m = 4 mm
Width of Teeth = 60 mm
Speed of Pinion N = 400 rpm
Tooth profile = 20 degree involute
Service factor = Cs = 0.8
Velocity factor = Kv = 3
Allowable bending strength of material
= 200 Mpa, for pinion
= 160 Mpa, for Gear
SOLUTION:-
Lewis form factor = 0
Velocity factor
Dp = m * Tp = 4 * 20 = 80 mm
Dg = m * Tg = 4 * 80 = 320 mm
Addendum = m = 4 mm
Dedendum = 1.25 * m = 1.25* 4 = 5 mm
Minimum clearance = 0.25 * m = 0.25 * 4 = 1 mm
Tip circle dia. Of pinion = Dp + (2*m) = 88 mm
Root circle dia. Of Pinion = Dp –(2 * m) = 72 mm
Tip circle dia. Of Gear = Dg +(2 * m) = 328 mm
Root circle dia. Of Gear = Dg – ( 2*m) = 312 mm
V = 3.14* Dp*Np/ (60*1000)
= 3.14*80*400/(60*1000)
= 1.67 m/s
Velocity factor Cv = 3 / (3+ 1.67)
= 0.642
Lewis form factor for pinion =
Yp = 0.154 – (0.912/20)
= 0.1084
Lewis form factor for Gear =
Yg = 0.154 – (0.912/80)
= 0.1426
Therefore tangential force transmitted is given by
Ft =
= 200*0.642*3.14*60*4*0.1426
= 8382.40 N
Now power transmitted for the given force is
P = Ft *1.67 = 13998.60
=13.99 kw
For calculating bending stress for gear, the Lewis
equation is
σ t = Ft/(Kv*m*b*Yp)
σ t = 8382.40/(3*4*60*0.1084)
σ t = 107.40 Mpa
And bending stress for pinion is calculated using

σ t = Ft/(Kv*m*b*Yg)
= 8382.4/(3*4*60*0.1420)
= 81.98 Mpa
MATERIAL PROPERTIES:-
Properties Gear
Material Name Structural Steel
Youngs Modulus 2E5 Mpa
Yield strength 250 Mpa
Poissons Ratio 0.3
Density 7850 kg/m3
FINITE ELEMENT ANALYSIS:
The basic concept in FEA is that the
body or structure may be divided into smaller
elements of finite dimensions called “Finite
elements”. The original body or the structure is
then considered as an assemblage of these elements
connected at a finite number of joints called
“Nodes” or “Nodal Points”. Simple functions are
chosen to approximate the displacement over each
finite element. Such assumed functions are called
“shape functions”. This will represent the
displacement within the element in terms of the
displacement at the nodes of the element.
The FEM is a mathematical tool for
solving ordinary and partial differential equations.
Because it is a numerical tool, it has the ability to
solve the complex problem that can be represented
in differential equations form. The applications of
FEM are listless as regards the solution of practical
design problems. Due to high cost of computing
power of years gone by, FEA has a history of being
used to solve complex and cost critical problems.
Classical methods alone usually cannot provide
adequate information to determine the safe
working limits of a major civil engineering
construction or an automobile or aircraft.
In the recent years, FEA has been
universally used to solve structural engineering
problems. The departments, which are heavily
relied on this technology, are the automotive and
aerospace industry. Due to the need to meet the
extreme demands for faster, stronger, efficient and
lightweight automobiles and aircraft,
manufacturers have to reply on this technique to
stay competitive.
FEA has been used routinely in high volume
production and manufacturing industries for many
years, as to get a product design wrong would be
detrimental. For example, if a large manufacturer
had to recall one model alone due to a hand brake
design fault, they would end up having to replace
up to few millions of hand brakes. This will cause a
heavier loss to the company.
STATIC ANALYSIS:-
There are two types of static analysis
1) Linear static analysis:
Linear means straight line. In this analysis
the ratio stress to strain is linear or straight line. But
in real life after crossing yield point material
follows nonlinear curve but software follows same
straight line. Component brake into two separate
pieces after crossing ultimate stress but software
based analysis never shows failure in this fashion.
It shows single unbroken part with red colour zone
at the location of failure. Analyst has to conclude
whether the component is safe or failed by
comparing the maximum stress value with yield or
ultimate stress.
Fig. 5.1 Stress and Strain curve
There are two conditions for static analysis
Force is static. i.e. no variation with respect to
time(dead weight).
Equilibrium condition: - summation of forces and
moments in x,y and z direction is zero. FE model
fulfils this condition at each and every nodes.

Fig. 5.2 static force
2) Non-linear static analysis:
In nonlinear analysis stress Vs strain curve
is nonlinear.
Fig.5.3 stress and strain curve of metals and nonmetals
WHY FEA?
FEA used in problems
where analytical solutions are not easily obtained.
Mathematical expressions required for solutions
not simple because of complex geometry, loading
and material properties.
Analytical methods involve solving for
entire system in one operation. FEA involving
defining equations for each element and combining
to obtain system solution.
BASIC STEPS INVOLVED IN FEA:
Basic steps: -
Discretization of domain:-
All real life objects are continuous.
Means there is no physical gap between any two
consecutive particles. As per material science any
object is made of small particles, particles of
molecules, molecules of atom and so on. Hence the
task is to divide the continuous object into number
of subdivisions called Element. Based on
continuum it can be divided into line or area or
volume elements.
Application of Boundary conditions:-
From the physics of the problem we
have to apply the field conditions i. e. loads and
constraints, which will help us in solving for the
unknowns.
Assembling the system equations:-
This involves the formulations of
respective characteristics (stiffness in case of
structural) equation of matrices and assembly.
Solution for system equation:-
Solving for the equations to know the
unknowns. This is basically the system of matrices
which are nothing but a set of simultaneous
equations are solved.
Viewing the results:-
After the completion of the solution we have to
review the required results. The first two steps of
the above said process is known as pre-processing
stage, third and fourth is the processing stage and
final stage is known as post-processing stage.
What is an element?
Element is an entity, into which a system
under study can be divided into. An element
definition can be specified by nodes. The shape
(area, length, and volume) of the element depends
upon the nodes with which it is made up of.
 What are Nodes?
Nodes are the corner points of the
element. Nodes are independent entities in the
space. These are similar to points in geometry. By
moving a node in space an element shape can be
changed.
 TYPES OF FINITE ELEMENTS:
1-D Element: -( Line Elements: )
2-D Element:-
3-D Element: -
CREATING A SOLID MODEL:
Modeling provides the design engineer
comfortable modeling techniques such as
sketching, feature based modeling, and dimension
driven editing. An excellent way to begin a design
concept is with a sketch. When you use a sketch, a
rough idea of the part becomes represented and
constrained, based on the fit and function
requirements of your design. In this way, your
design intent is captured. This ensures that when
the design is passed down to the next level of
engineering, the basic requirements are not lost
when the design is edited.

PARAMETRIC MODELING OF GEAR USING CATIA
V5
Figure 7.1 3D model of gear
The strategy you use to create and edit
your model to form the desired object depends on
the form and complexity of the object. You will
likely use several different methods during a work
session. The next several figures illustrate one
example the design process, starting with a sketch
and ending with a finished model. First, you can
create a sketch "outline" of curves.
Then you can sweep or rotate these curves to create
a complex portion of your design.
SPUR GEAR ANALYSIS:
Now, the 3D which was created in
CATIA is imported in ANSYS workbench 12 for
stress analysis. It is done by saving drawing in STP
or IGS file format in CATIA. After the model is
imported in ANSYS workbench 12, as the both
teeth are already in contact, our main purpose is to
find the root bending stress. This is done by using
following steps in ANSYS workbench 12.
The objective of the analysis is to perform
structural static analysis on the gear by applying
tangential load and examine the deflections and
stresses.
The 3-D model of the spur gear in CATIA
converted it into “iges” and “step” file. And then
iges file is then imported into ansys workbench.
3-DIMENSIONAL ANALYSIS OF SPUR GEAR: - For
imported 3-dimensional geometry, firstly we select
3-D and static structural analysis from menu and
connecting the geometry to the analysis tab. Then
next we enter Young’s modulus and Poisson’s ratio
of the material.
MESH GENERATION AND BOUNDARY
CONDITION:- (supports and load):- A tetrahedron
solid elements is used in mesh generations.
Boundary condition refers to the external load on
the border of the structure. We assumed gear is
with fixed support and pinion is subjected to a
moment or torque along its axis with frictionless
support.
MESHING OF GEAR:-
Modeling CATIA
Start
Get the input data
required
Draw the parts as per
given
Assemble all the
individual parts
Save the files in iges
/step format
Stop

BOUNDARY CONDITION OF GEAR:- Figure 8.1 Boundary Conditions
RESULT:
MAXIMUM VONMISSES STRESS:-

DEFLECTION :-
CONCLUSION:
The above study shows that the Finite Element
Method can be used for bending stress analysis in a
pair of gear. Bending Stress calculation is play
more significant role in the design of gear. This
study is shows that Lewis formula is use for
calculating bending stress in a pair of gear.
Theoretically result obtained by Lewis formula are
comparable with Finite Element Analysis of spur
gear.
Maximum von mises stress observed in structural
steel gear is 105 Mpa on Ansys and 107 Mpa by
theoretically.
Maximum deflection of 0.035mm observed in the
gear along x-direction.
ACKNOWLEDGEMENT:
It gives me a great pleasure in expressing
my thanks and profound gratitude to Dr. D. S.
Deshmukh HOD, Mechanical Engineering
Department, COE, Bambhori,Jalgaon for their
valuable guidance and encouragement throughout.
I am heartily thankful to him for his continuous
suggestions and clarifying the concepts of the topic
that helped me a lot during this study.
REFERENCE:
1) R.S.Khurmi “ Machine Design”.
2) Nitin S. Gokhale “Practical
Finite Element Analysis”.
3) Sushilkumar Tiwari and
Upendra kumar Joshi “Stress
Analysis Of Mating Involute Spur
Gear”IJERT 2012.
4) Ansys 14.5
5) Catia v5 R20
I Am Pravin B. Sonawane , B.E. Mechanical from
MSS’s CET,Nagewadi,Jalna,2010.
Now Pursuing M.E. Mechanical (Machine Design)
from SSBT’s COE Bambhori, Jalgaon.
Guide- P.G.Damle, (Associate Prof.) Department.
of Mechanical Engineering ,COE , Bambhori,
Jalgaon.

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