International Journal of Engineering Trends and Technology (IJETT) – Volume 5 Number 6 - Nov 2013
Parametric Modelling and Static Structural Analysis
of Four Speed, Constant Mesh, Manual Gear Box
Santosh Ukamnal#1, Ketan Nalawade#2, P. Baskar*3
#
P G Scholar, M. Tech in Automotive Engineering
VIT University, Vellore, India
*
Assistant Professor, School of Mechanical and Building Sciences
VIT University, Vellore, India
Abstract— Gears are one of the important part in the power
transmission systems. Good design of transmission enebles better
engine performance. Gear box is most widely used in automotive
sector. This paper deals with the modeling in Pro Engineer and
static structural analysis in Ansys Workbench 12.1 of four speed
constant mesh gear box for 305cc engine, in order to conclude the
stresses induced in the gears while transmitting power.
involute is always at the point of tangency of the string with
the base circle. A tangent to the involute is always normal to
the string, which is the instantaneous radius of curvature of
the involute curve.
Keywords— Pro Engineer, Von Mises Stress, Ansys Workbench
12.1
I. INTRODUCTION
Power transmission has always been of high importance.
Gears are used in past, present and they will be used in future
automobiles, due to their high reability, torque handling
capacity and rapid shifting of gears. As the world moves on
there is always new demands, which are to be fulfilled. People
now prefer cars with low weight and low engine sound. This
opens up a demand for quite power transmission. Higher
reliability and lighter weight gears are necessary to make
automobile light in weight as lighter automobiles continue to
be in demand.
Spur gears are the most commonly used gears to transmit
power and rotational motion between parallel shafts. The
tooth of gear is cut parallel to the shaft. The simplest motion
of two external spur gears can be seen by an example of two
external rotating cylinders, if sufficient friction is present at
the rolling interface.
Fig 1. Involute profile generated on the base circle [1]
Figure 1 shows the way in which involute curve is formed.
These represent gear teeth. The cylinders from which the
strings are unwrapped are called the base circles of the
respective gears. The base circles should always be smaller
than the pitch circles, which are at radii of the original rolling
cylinders.
C. Gear Mesh Geometry
A. Law of Gearing
Fundamental law of gearing states that “angular velocity
ratio between the gears of a gear set must remain constant
throughout the mesh”. The angular velocity ratio m equals to
the ratio of the pitch radius of the input gear to that of the
output gear.
Mv=
=
The surfaces of the rolling cylinders will become the pitch
circles, and their diameters become the pitch diameters of the
gears.
B. Involute Profile Forms
The involute of a circle is a curve that can be generated by
unwrapping a taut string from a cylinder. The string is always
tangent to the base circle. The centre of the curvature of the
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Fig 2. Gear mesh geometry [1]
Gear mesh geometry is the most complicated part of gears,
because most of the loss in power transmission occurs at this
point. Figure 2 shows a pair of involute tooth coming in
contact and also leaving contact. From the figure 2 it can be
seen that common normal for both the contact points passes
from the same pitch point. Mesh between the gear and the
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International Journal of Engineering Trends and Technology (IJETT) – Volume 5 Number 6 - Nov 2013
pinion, is defined by the points of beginning and leaving
contact. The distance along the line of action between the two
contact points in mesh is called length of action.
After defining and relating all the gear parameters we can
then as needed in sketched features. The figure below shows
the screen shot from Pro/E after the completion of this step.
II. MODELLING OF GEAR
First and the foremost step in this is to model a spur gear.
The most complicated part in spur gear is the involute profile
of its teeth. There are number of ways of creating involute
profile of a spur gear. In this the spur gear model was
designed in Pro Engineer design modeler.
A. Model and Relation
The word Relation and Equation itself gives the idea about
relating the feature with the help of equations. Relations are
used to express dependencies between the dimensions of a
feature. Following are steps involved in this modelling
procedure.
First Step while modelling is define the parameters in PRO-E.
The Parameters are
N = No. of teeth
Phi = Pressure Angle
D = Pitch circle diameter
P= Diametrical Pitch
As shown in figure
Fig 4. Image showing relations
Next step is extrude feature. After defining the gear
parameters, next step is to extrude circular disc having a
diameter equal to addendum diameter and thickness equal to
face width. To do this a circle is to be drawn with centre on
the sketch references and then go to Tools / Relations menu to
define relations between the sketch dimensions and the part
parameters. After defining these relations, the circle should
have a diameter equal to the addendum diameter of the gear
blank.
Next step is to define a datum curve showing the involute
profile for radius Ri ≤ R≤ Ro. Go to the Insert => Model
Datum => Curve menu. Select from Equation, when prompted
for a co-ordinate system for a part “PART_CSYS_DEF”. Set
the co-ordinate system to cylindrical system. At this point, a
notepad window (shown below in the figure 5) will pop up,
Fig 3. Image showing parameters
For the remaining parameters, which are to be related to basic where one can enter all the equations for the datum curve.
parameters, relations can be defined in Tools / Relations menu. Enter the equations listed above for R, θ and Z in terms of t, (a
Formulas for the remaining parameters are written below.
parametric variable ranging from 0 to 1) and other part
Tools => Relations [2]
parameters. For the involute profile, the equations will be [2]:
Z = D/m
Gamma = t*sqrt(Ro^2-Ri^2) (mm)
ha = m
R = sqrt(gamma*gamma+Ri*Ri)
hf = 1.16 * m
Theta = theta0+((gamma/Ri)*(180/pi))-atan(gamma/Ri)
Ro = ( D / 2 ) + ha (mm)
Z=0
Rd = ( D / 2 ) - hf (mm)
Ri = ( D / 2 ) * cos ( PHI ) (mm)
g_c = sqrt(( D ^ 2 ) / 4 - Ri ^ 2)
theta_0 = (360 / (4 * Z)) - ((g_c / Ri) * (180 / pi)) + atan(g_c/
Ri) (degrees)
Where,
Z = Number of teeth
Hf = dedendum height
Ha = addendum height
Ro= Radius of addendum circle
Rd = Radius of dedendum circle
Phi = Pressure Angle
Fig 5. Image of equation of involute curve in notpad
D = pitch diameter
When all the datum curve equations are entered, close
notepad and click preview, a curve on the flat surface of the
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International Journal of Engineering Trends and Technology (IJETT) – Volume 5 Number 6 - Nov 2013
gear blank should be seen like the one shown in the figure 6
below.
Fig 6: Involute corve on the flat surfce of the gear blank
After this, define the root circle and make the cut in such
away that shown in figure 7 below.
Fig 9: Generated gear model in Pro-E
The gear model, with involute pfofile teeth, which is
generated in Pro Engineer by using relations and parameters
option is shown in the figure 9. Like above mentioned
methods, all the gears of the gear box are generated. The next
step is assembly.
B. Assembly
In this section, the generated gears will be assembled to
defined mesh geometry.
1) Defining of Axis: Two axes, i.e. output and input gear axes
must be generated by providing the distance between them
before proceeding to the next step which is shown in figure 10.
The distance between toe axes of this gear box is is 70mm.
Fig 7: Image showing the cut section of the gear
After a single tooth space is generated, it has to be patterned
along the centre axis of the gear blank. Using the pattern
option in Pro/E and by selecting the following option, one can
easily pattern the tooth space, as needed depending on the
number of teeth.
Pattern Type => Axis Pattern: Axis, which goes through the
centre of the gear blank => Number of copies
Figure 8 shows the preview and the spur gear model created
after patterning the tooth space.
Fig 10: Axes with distance between two shafts
2) Adding gears and constraints: The gears that are modelled
are added to the assembly by Assemble option. Once the gear
is added, change the constraint to Pin and align the axis of the
gear to one of the generated axes which is as shown in the
figure 11.
Fig 8: Paterning of the geat teeth
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International Journal of Engineering Trends and Technology (IJETT) – Volume 5 Number 6 - Nov 2013
Fig 14: Standard view of the assembled gear box
Fig 11: Input gear with given constraints
Once the constraint to input gear is over, again using the
option Assemble, the corresponding gear on the output shaft is
also added. The constraint chosen is pin and mate the lateral
surfaces of the gear as shown in the figure 12.
III. ANALYSIS OF GEARS
The analysis of meshing gears at different gear ratios
present in the gear box is done in ANSYS Workbench 12.1.
The assembled model in Pro-E is saved in STEP format,
which has to be imported in the Ansys. After the importing is
done, following steps has to be accomplished:
A. Selection of Material
The material that is chosen for gear must have high strength,
high hardness and low ductility and should offer high
resistance to surface fatigue and wear. To satisfy both core
and surface properties, the gear material is generally subjected
surface hardening process. Most commonly used gear
materials which have application in the automotive industry
are 8620steel and 4320 steel [3].
Fig 12: Geometric meshing of two gears
TABLE I
PROPERTIES OF 4320 STEEL
4320 Steel
Hardness
Young’s Modulus
0.2% Yield Strength
UTS
654 BHN
201 GPa
1250 MPa
1912 MPa
TABLE III
PROPERTIES OF 4320 STEEL
620 Steel
Hardness
615 BHN
Young’s Modulus
208 GPa
0.2% Yield Strength
1125 MPa
UTS
1869 MPa
Before proceeding further with the analysis, the properties of
the selected materials is updated in the library. Here the 4320
Steel is chosen as the gear material because of its higher Yield
Strength.
Fig 13: Final assembly of gear box
In the above said manner, remaining gears of the input and
output shafts, input shaft and output shaft are assembled with
gear shifter as shown in the figure 13. The standard view of
the total assemble model is shown in the figure 14. As this
gear box is constant mesh gear box, all the gears are
constantly meshing at all times.
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B. Defining Contact
Type of contact surface has to be defined in order to carry
forward with the analysis. When the two gears are mating the
power is transferred by rolling and sliding mechanism. Here
we choose frictionless contact, between two mating gears,
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assuming that there is no friction between gear teeth [4]. This
can be provided by adding lubricant in the gear box.
C. Meshing the Model
The meshing of model is done before proceeding to next
step. The meshed gear pair is shown in figure 15. In this step
the gear model is divided into number of small elements, and
these elements are further analysed.
Fig 17: Firctionless support provided for input gear
E. Defining Loads
To define the loads which is in the form of torque, engine
specifications has to be noted. Specifications of the engine is
as follows:
TABLE III
SPECIFICATION OF ENGINE
Fig 15: Meshing of gears
D. Defining Supports
1) Fixed Support: One of the gear must be completely fixed in
order to apply the loads. For this purpose the fixed support is
given to the inner surface of the driver gear which is located
on the output shaft as shown in figure.
Type of Engine
Manufacturer’s Name
Max Torque
Capacity
Four stroke, Gasoline Engine
Briggs and Stratton
20 Nm at 3600 rpm
305cc
The torque produced by the engine is transmitted to the
wheels by transmission unit, which consists of clutch, gear
box, final drive and differential. The gear ratio for the top gear
in the gear box is taken as unity (1:1) and the ratio of the final
drive is taken as top gear transmission ratio (7.35:1). If we
divide the transmission ratio at each gear by top gear
transmission ratio, we will arrive at the gear box gear ratios at
each gear [5].
Multiplying the torque produced by the engine with the
respective gear box gear ratio, torque transmitted at each gear
can be calculated. The transmission ratio, gear box ratio and
torque transmitted is given in the table below.
The load distribution along the face width is assumed
uniform and plane strain method is adopted [6].
TABLE IV
TORQUE TRANSMITTED BY EACH GEAR IN THE GEAR BOX
Fig 16: Fixed support provided for output gear
2) Frictionless Support: The input gear must rotate on the
application of the torque. Hence a friction less support is
given on the inner surface of the driving gear which is on
input shaft as shown in figure.
Gear
Number
Transmission
Gear Ratio
Torque
Transmitted
31.48:1
18.7:1
Gear Box
Gear
Ratio
4.2829:1
2.5442:1
First Gear
Second
Gear
Third Gear
Fourth
Gear
11.4:1
7.35:1
1.551:1
1:1
31.02 Nm
20 Nm
85.65 Nm
50.884 Nm
F. Solving the Problem
The effect of case hardness, case depth and the oil film
thickness is neglected. The surface asperities and waviness is
neglected [7].
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The meshed and applied boundary conditioned model is
solver for the results by clicking Solve option present in the
Ansys Workbench.
This process can be very helpful in contact problems as it
needs model with high accuracy. It also decreases the lead
times and improves overall engineering productivity.
G. Generating Equivalent Von-Mises Stress and Factor of
Safety
In the solution part of the analysis, the option for
Equivalent Stress and Stress Tool will be present. Once
selected, the results have to be evaluated. Thus the results are
displayed on the screen.
Which shows that the gear box so designed can be able to
transmit the power from engine to drive shaft.
IV. RESULTS AND CONCLUSION
The Von-Mises stress of the gears in mesh for power
transmission is as shown in the figure.
V. FUTURE SCOPE
While analysisng the gear box, the dynimic loads must also
be taken into account. The analysis of the gear box can be
further extended to the dynamic loads as well. Vibration
effects on each gear pair can also be analysed.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Mechanical Engineering, Sighley’s Mechanical Engineering Design,
McGraw Hill Publications, Eighth Edition, pp 569.661.
Shreyash D. Patel, Finite Element Analysis of Stresses in Involute Spur
and Helical Gear, Master of Science in Mechanical Engineering, The
University of Texas at Arlington, December 2010.
Hong Lin, Robert R. Binoniemi, Gregory A. Fett, Thomas Woodard
and Mick Deis, “Mechanical Properties of Gear Steels and Other
Perspective Light Weight Materials for Gear Applications”, SAE
Technical Paper Series, ISSN 0148-7191, November 2006.
Walton, D., Tessema, A.A., Hooke, C.J., and Shippen, J. Load sharing
in metallic and non-metallic gears. Proc. Instn Mech. Engrs, Journal of
Mechanical Engineering Science, 1994, 208 (C2), 81-87.
Harald Naunheimer, Bernd Bertsche, Joachim Ryborz, Wolfgang
Novak Automotive Transmissions, ISBN 978-3-642-16213-8, pp 101105, Second Edition.
Cornell R.W., Compliance and stress sensitivity of spur gear teeth,
ASME Trans., J. Mech. Des., April 1981.
M. Rameshkumar, G. Venkatesan and P. Sivakumar, Finite Element
Analysis of High Contact Ratio Gear, AGMA Technical Paper, ISBN:
978-1-55589-981-3, October 2010.
Fig 18: Equivalent Von-Mises stress for fourth gear 354.63MPa
The maximum stresses induced in each gear ratio i.e. each
gear of the gear box is as given in the table.
TABLE V
EQUIVALENT VON-MISES STRESS AND FACTOR OF SAFETY FOR EACH GEAR
IN THE GEAR B OX
Gear Number
First Gear
Second Gear
Third Gear
Fourth Gear
Equivalent VonMises Stress (MPa)
670.17
251
829.6
354.62
Factor Of Safety
1.8652
4.9801
1.5067
3.5249
In this study, a three dimensional deformable-body of spur
gear was developed in Pro-E modelling and analysed in Ansys
Workbench 12.1 software.From the results obtained,
following results can be drawn:
1) It is easy and more accurate to design the complicated
involute profiles by using relational equation modelling in Pro
Engineer.
2) The laod can be applied in terms of torque or moment
instead of line loads on a single tooth of the gear.
3) From the results of Von-Mises stress induced in each gear
of the gear box, it can be observed that the maximum strress is
below the yield stress of the material which is 1250 MPa.
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