International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
Parametric Modelling of Straight Bevel Gearing
system and Analyze the Forces and Stresses by
Analytical Approach
A.V.Ramana Rao1, CH.Bhanu Prakash2, M.N.V.S.A.Sivaram.K3
1
Asst.Prof., Mechanical Engineering Department, Vishnu Institute of Technology, Bhimavaram, A.P, INDIA
Asst.Prof., Mechanical Engineering Department, Vishnu Institute of Technology, Bhimavaram, A.P, INDIA
3
Asst.Prof., Mechanical Engineering Department, Vishnu Institute of Technology, Bhimavaram, A.P, INDIA
2
Abstract— The three-dimensional parametric modelling of
straight bevel gear pair using CATIA V5 software system. Gears
modelling are based on geometric and perspective
transformation. Parameter modelling application makes possible
the control of created 3D gear model through previous defined
parameter which is based on relations and geometric constraints.
The final shape of gear changes by changing of values of
parameters, and that makes model creation in short time and
analysis of forces and stress by analytical approach. Modelling
methodology and functionality that is described by selected
example can be used in cone crusher.
Keywords— Bevel gears, parametric modelling, CATIA V5.
I. INTRODUCTION
A bevel gear is one of the most fundamental types of gear; it
is widely used in aircrafts, automobiles and heavy engineering
machines, etc... Two important concepts in gearing are pitch
surface and pitch angle. The pitch surface of a gear is the
imaginary toothless surface that you would have by averaging
out the peaks and valleys of the individual teeth. The pitch
surface of an ordinary gear is the shape of a cylinder. The
pitch angle of a gear is the angle between the face of the pitch
surface and the axis. The most familiar kinds of bevel gears
have pitch angles of less than 90 degrees and therefore are
cone-shaped. This type of bevel gear is called external
because the gear teeth point outward. The pitch surfaces of
meshed external bevel gears are coaxial with the gear shafts;
the apexes of the two surfaces are at the point of intersection
of the shaft axes. Bevel gears that have pitch angles of greater
than ninety degrees have teeth that point inward and are
called internal bevel gears.Bevel gears that have pitch angles
of exactly 90 degrees have teeth that point outward parallel
with the axis and resemble the points on a crown. That's why
this type of bevel gear is called a crown gear. Straight bevel
gears are the most economical of the various bevel gears,
owing to their ease of manufacture. The drive unit for cone
crusher is affected by the frequent failures of bevel gear pair
at higher loads. So these gears are again redesigned but the
surface profile of the gear teeth is relatively difficult to draw
accurately. . Either of these will require an exact CAD model
ISSN: 2231-5381
of the tooth geometry, and approximate methods will not
always meet the requirements.
II. GEAR TERMINOLOGY
(1) Pitch circle: Theoretical circle upon which all gear
calculations are usually based. Pitch circles of mating gears
are tangent to one another.
(2) Pitch diameter: The diameter of the pitch circle.
(3) Number of teeth: The number of teeth on the gear.
(4) Diametric pitch: The number of teeth of a gear per inch of
its pitch diameter.
(5) Module: The ratio of pitch diameter to the number of teeth.
It is reciprocal of the diametric pitch. The pitch diameter is
specified either in inches or millimeters.
(6) Pressure angle: The angle through which forces are
transmitted between meshing gears. It is either 14.5°or 20°. It
defines the geometry of the gear tooth and also determines the
diameter of the base circle.
(7) Addendum: The amount of tooth that protrudes above the
pitch circle (from top land to pitch circle)
(8) Dedendum: The Radial distance from the pitch circle to
the bottom of the tooth space.
(9) Clearance circle: the circle tangent to the addendum circle
of the mating gear.
(10) Clearance: the distance between the tooth top surface and
the bottom surface of a mating gear.
Fig.1 Geometry of straight bevel gear
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
III.GEOMETRY OF STRAIGHT BEVEL GEARS
The geometry of bevel gears is shown in Fig.2. They have
teeth that are straight and tapered, if extended inward, the
teeth would intersect at a common point. The shaft angle is
900, which is the sum of the pitch cone angle of gear and
pinion. In order to secure uniform bearing along the tooth, the
face width is generally not made longer than one-third of the
pitch cone length R, usually b=φR.R. The gear ratio is
i=n1/n2, 1270/400 = 3.175
(1)
Fig. 4 Involute curve
An involute curve is the loci of a point on the generating line,
as the line rolls without slipping along a base circle, see Fig.4.
Gear teeth are cut in the shape of an involute curve between
the base and the addendum circles, while the part of the tooth
between the base and dedendum circles is generally a radial
line. In Cartesian coordinate system, the involute curve is
expressed as follows:
x = rb(cosθ + θsinθ)
y = rb(sinθ - θcosθ)
Where rb is radius of the base circle, θ is the spread angle.
Fig.2 Geometry of straight bevel gear
IV.PARAMETRIC DESIGN OF STRAIGHT BEVEL
GEARING SYSTEM
Determination of Coordinates control points:
Before cearing parametric design, some coordinates points
must be determined, according to the design variables. Thus, a
Cartesian coordinate system was set up, as shown in Fig.5,
whose origin is at the apex of the back cone. In this coordinate
system all the control points were determined, referring to
Fig.5 and Table-1.
Fig. 3 Back cone and shape of the teeth
As shown in Fig.2 and Fig.3, the size and shape of the straight
bevel gear’s teeth are defined at the large end, on the back
cone, which have standard involute profiles.
re = d/2 cosδ
(2)
The length of back cone has relationship with the pitch
diameter as equation -2; it is equal to the pitch circle radius of
bevel gear’s virtual spur gear.
ISSN: 2231-5381
Fig. 5 Cross section of a bevel gear
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
TABLE-1
Control points coordinates determination
Points
x
O’
O
A
B
C
0
x0 = 0
xa = df2 / 2
xb = da2 / 2
xc = (1-φR) * cosδ2 + xd
y
0
y0 = R/cos δ2
ya = df2 * i /2
yb = da2 * i/2
yc = (1-φR) * h * sinδ2
+ yd
D
xd = (1-φR) * df / 2
yd = ye
E
xe = 0
ye = (y0 – ya) * φR + ya
F
xf = 0
yf = yg
G
xg = (1)*
yg = (1)
xa
* ya
According to the above parametric equations, design systems
for straight bevel gears were achieved. The cone crusher drive
unit straight bevel gear and straight bevel gear pinion were
modelled as Shown in Fig. 6&7.
Fig. 6 Parametric modelling of Straight bevel pinion gear and
straight bevel gear wheel
Fig. 7 Isometric views of straight bevel gear pair
ISSN: 2231-5381
V.DESIGN OF STRAIGHT BEVEL GEAR PAIR BY
ANALITICALLY
In straight bevel gears the teeth are straight and parallel to the
generators of the cone. This is the simplest form of bevel gear.
It resembles a spur gear, only conical rather than cylindrical.
TABLE-2
Design data for straight bevel gear pair
S.No
Parameter
pinion
Gear
1
No. of teeth (z)
17
54
2
Module (m)
15.5613 mm
3
Pressure angle (α)
200
4
Shaft angle
∑ = δ1 + δ2 = 900
5
Face width (b)
180 mm
6
Speed (n)
7
Power (P)
1270 rpm
400 rpm
200 kW
Design of straight bevel pinion gear:
In the present cone crusher, the drive system consists of
straight bevel gear pair in which the pinion gear acts as a drive
gear, transmits power of 200kW with speed 1270 rpm, having
17 teeth, normal module of 15.5613 mm and spiral angle is 00.
Calculations of pinion gear:
Transmission ratio (i) = 3.175
Tooth ratio ratio (u) = 3.176
Pitch cone angle (δ1) = 17.4750
Middle pitch cone distance (Rm) = 440.478 mm
Outer pitch cone distance (Ra) = 530.478 mm
Inner pitch cone distance (Ri) = 350.478 mm
Outer transverse module (mto) = 18.740 mm
Middle transverse module (mtm) = 15.5613 mm
Inner transverse module (mti) = 12.381 mm
Outer normal module (mno ) = 18.7408 mm
Middle normal module (mnm) = 15.5613 mm
Inner normal module (mni) = 12.3818 mm
Outer pitch diameter (d1) = 318.593 mm
Middle pitch diameter (dm1) = 264.542 mm
Inner pitch diameter (di1) = 210.49 mm
Addendum (ha1) = 21.926 mm
Similarly addendum (middle and inner) = 18.263 mm, 15.164
mm respectively.
Dedendum (hf1) = 13.455 mm
Similarly dedendum (middle and inner) = 11.172 mm, 9.276
mm respectively.
Outer tip circle diameter (da1) = 365.786 mm
Middle tip circle diameter (dam1) = 303.370 mm
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
Inner tip circle diameter (dai1) = 240.1 mm
Base circle radius of the involute spiral (Br) = 440.478 mm
Middle transverse pitch (ptm) = 48.887 mm
Diameteral pitch (pd) = 0.0533 mm
Circular pitch (pc) = 58.876 mm
Tooth thickness on outer PCD (tpo) = 34.590 mm
Tooth thickness on inner PCD (tpi) = 22.853 mm
Clearance (c) = 2.925 mm
Virtual number of tooth (zv1) = 17.822
The correction factor of the virtual toothing (xv1) = +0.3200
Design of straight bevel gear wheel:
In the present cone crusher, the drive system consists of
straight sided bevel gear pair in which the gear wheel acts as a
drive gear, transmits power of 200kW with speed 400 rpm,
having 54 teeth, normal module of 15.5613 mm and spiral
angle of 00.
Calculations of gear wheel:
Transmission ratio (i) = 3.175
Tooth ratio ratio (u) = 3.176
Pitch cone angle (δ1) = 17.4750
Middle pitch cone distance (Rm) = 440.478 mm
Outer pitch cone distance (Ra) = 530.478 mm
Inner pitch cone distance (Ri) = 350.484 mm
Outer transverse module (mto ) = 18.740 mm
Middle transverse module (mtm) = 15.5613 mm
Inner transverse module (mti) = 12.381 mm
Outer normal module (mno) = 18.7408 mm
Middle normal module (mnm) = 15.5613 mm
Inner normal module (mni) = 12.3818 mm
Outer pitch diameter (d2) = 1012.003 mm
Middle pitch diameter (dm2) = 840.310 mm
Inner pitch diameter (di2) = 668.617 mm
Addendum (ha2) = 9.9326 mm
Similarly addendum (middle and inner) = 8.247 mm, 6.848
mm respectively.
Dedendum (hf2) = 25.450 mm
Similarly dedendum (middle and inner) = 21.132 mm, 17.547
mm respectively.
Outer tip circle diameter (da2) = 1019.657 mm
Middle tip circle diameter (dam2) = 846.25 mm
Inner tip circle diameter (dai2) = 673.674 mm
Base circle radius of the involute spiral (Br) = 440.478 mm
Middle transverse pitch (ptm) = 48.887 mm
Diameteral pitch (pd) = 0.0533 mm
Circular pitch (pc) = 58.876 mm
Tooth thickness on outer PCD (tpo) = 24.590 mm
Tooth thickness on inner PCD (tpi) = 16.045 mm
Clearance (c) = 2.925 mm
Virtual number of tooth (zv1) = 179.826
The correction factor of the virtual toothing (xv1) = -0.3200
Forces and stress analysis of the straight bevel gear pair:
For force analysis of a pair of mating straight bevel gears, it is
assumed that the total force FN acting on straight bevel gear
teeth is resolved into three mutually perpendicular
components, they are:
1. Tangential tooth load (ft)
2. Axial thrust (fa)
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3. Radial separating force (fr)
The tangential tooth load (ft):
ft = (2T1/dm1) x 1000 N
Here T is torque on bevel gear pair.
 9550  P1 

T  
n1


The axial thrust (fa):
 ft
f a  
 cos  m

  tan  sin   sin  m cos  

The radial separating force (fr):
 ft 
  tan  cos   sin  m sin  
f r  
 cos  m 
Calculation of stresses in straight bevel gear pair:
In general, for contact stress calculations in case of straight
bevel gear pair, it is sufficient to check the stress at the pitch
point P to ensure the Hertzian pressure at the pitch point is
within the allowable limit.


   ym  y p 


 ft 

 
 b  dm 
(u 2  1)
u




In the above equation ym is the material coefficient; yp is the
pitch point coefficient.
VI .RESULTS & DISCUSSION
TABLE-3
Determined forces and stresses of straight bevel gear pair
S.No
Parameter
Pinion
Gear
Units
1
Torque
1503.937
4775
N-m
2
Tangential force
11370.11
N
3
Normal force
12099.82
N
4
Axial force
1242.70
3947.39
N
5
Radial force
3947.39
1242.70
N
6
Contact stresses
134.5
75.5
MPa
In this study, structural features of straight bevel gear were
analyzed; design variables were specified, and then the
interactive interface was designed; Further, a Cartesian
coordinate system was set up and coordinates of control points
were determined according to its geometric relationship. The
parametric design of straight bevel gear has been implemented,
running results showed that the parametric modeling of bevel
gear was speedy and accurate, which will facilitate the 3D
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
modeling of straight bevel gear pair in CATIA and the
appropriate forces were found which are acting on straight
bevel gear pair and analyze the contact stresses by analytical
approach.
However, the methodology proposed in this paper is mainly
addressed for straight bevel gears which intersect at right
angle.
VII .CONCLUSION
[12] Karam F., Kleismit C.: Using Catia V5, Thomson
Learning, 2004.
[13] Amirouche, F.: Principles of Computer-Aided Design and
Manufacturing, 2nd edition, Prentice Hall, Upper Saddle
River, New Jersey, 2004.
In geometric gear modelling, we do not have to create shape
directly, but, instead of that, we can put parameters integrated
in geometric and/or dimensional constraints. We get resulted
solid straight bevel gear model by parameters changing. In
this way, designer can create more alternative designing
samples. Time used for creating more designing samples is
reduced for 50%, by parameter modelling. These results can
be seen in reducing the production cost, and the production
also increases. In that way, better profit and price of products
are lower. This type of functionality makes possible flexibility
and it could be used in combination with traditional geometric
modelling approach.
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