Engineering, 2010, 2, 740-749
doi:10.4236/eng.2010.29096 Published Online September 2010 (
Copyright © 2010 SciRes. ENG
Profile Modification for Increasing the Tooth Strength in
Spur Gear Using CAD
Shanmugasundaram Sankar1, Maasanamuthu Sundar Raj2, Muthusamy Nataraj2
1Research Scholar, Anna University, Coimbatore, India
2Department of Mechanical Engineering, Government College of Technology, Coimbatore, India
E-mail: {shanmugasundaramsankar, m_natanuragct}
Received July 13, 2010; revised August 5, 2010; accepted August 18, 2010
This paper examines the tooth failure in spur gears. Corrective measures are taken to avoid tooth damage by
introducing profile modification in root fillet. In general, spur gear with less than 17 numbers of teeth had the
problem of undercutting during gear manufacturing process which minimizes the strength of gear at root. In
this study, a novel design method, namely circular root fillet instead of the standard trochoidal root fillet is
introduced in spur gear and analyzed using ANSYS version 11.0 software. The strength of these modified
teeth is studied in comparison with the standard design. The analysis demonstrates that the novel design ex-
hibit higher bending strength over the standard trochoidal root fillet gear. The result reveals that the circular
root fillet design is particularly suitable for lesser number of teeth in pinion and where as the trochoidal root
fillet gear is more opt for higher number of teeth.
Keywords: Bending Stress, Circular Root Fillet, Deflection, Profile Modification, Spur Gear, Trochoidal
Root Fillet
1. Introduction
The objective of the gear drive is to transmit power with
comparatively smaller dimensions, runs reasonably free
of noise and vibration with least manufacturing and
maintenance cost. There is a growing need for higher
load carrying capacity and increased fatigue life in the
field of gear transmissions. Spitas and Costopoulos [1]
have introduced one–sided involute asymmetric spur
gear teeth to increase load carrying capacity and combine
the meshing properties. Tesfahunegn and Rosa [2] inves-
tigated the influence of the shape of profile modifications
on transmission error, root stress and contact pressure
through non linear finite element approach. Spitas and
Costopoulos [3] expressed that the circular fillet design
is particularly suitable in gears with small number of
teeth (pinion). Fredette and Brown [4] discussed the pos-
sibility of reducing gear tooth root stresses by adding
internal stress relief features. Ciavarella and Demeliio [5]
concluded that the fatigue life is lower on gears with a
lesser number of teeth. Hebbal and Math [6] have re-
duced the root fillet stress in spur gear using internal
stress relieving feature of different shapes. Senthilvelan
and gnanamoorthy [7] studied the effect of gear tooth
fillet radius on the performance of injection moulded
nylon 6/6 gears. Tae Hyong Chong and Jae Hyong
Myong [8] conducted a study to calculate simultaneously
the optimum amounts of tooth profile modification for
minimization of vibration and noise.
Beghini et al. [9] proposed a simple method to reduce
the transmission error for a given spur gear at the nomi-
nal torque by means of the profile modification parame-
ters. Researchers focused either on the development of
advanced materials or new heat treatment methods or
designing the gears with stronger tooth profiles. Gears
having standard involute with smaller number of teeth
(i.e., less than 17 teeth) had the problem of undercutting.
In gear manufacturing process the tooth root fillet is
generated as the tip of the cutter removes material from
the involute profile resulting teeth that have less thick-
ness at root. This reduces the tooth strength and leads to
the crack initiation and propagation at root fillet area. To
improve the gear tooth strength many works have been
done but all mostly employed positive profile shifting
[10-13]. These contributions exhibit lower pitting and
scoring resistance with lesser contact ratio resulting in
more noise and vibration during the power transmission
Copyright © 2010 SciRes. ENG
2. Gear Geometry
The involute spur gear with circular root fillet is illus-
trated in Figure 1. The point ´O´ is the center of the gear,
‘Oy’ is the axis of symmetry of the tooth and ‘B’ is the
point where the involute profile starts from the form cir-
cle rs.
‘A’ is the point of tangency of the circular fillet with
the root circle rf. ‘D’ laying on (ε2) = ‘OA’ represents the
center of the circular fillet. Line (ε3) is tangent to the root
circle at A and intersects with line (ε1) at C. The fillet is
tangent to the line (ε1) at point E. Since it is always rs > rf,
the proposed circular fillet can be implemented without
exceptions on all spur gears irrelevant of number of teeth
or other manufacturing parameters. A comparison of the
geometrical shape of a tooth of circular fillet with that of
standard fillet is presented in Figure 2.
The geometry of the circular fillet coordinates (points
A and B) in Figure 1 is obtained using the following
XA = rf sin(ζ + s), YA = rf cos(ζ + s)
XB = rf sins, YB = rf coss
XD = (rf + AD) sin(ζ + s), YD = (rf + AD cos(ζ + s)
XE = (OC + CE) sins , YE = (OC + CE) coss
2.1. Part Modeling
In actual practice, trochoidal root fillet is present in spur
gear having large number of teeth (more than 17) and
exhibits less bending stress for higher number of teeth.
The circular root fillet is preferable for gears with
smaller number of teeth (less than 17) depending on the
tip radius of the hob. The proposed teeth are composed
of a standard involute working profile from the outer to
the form circle of the gear and of a circular fillet profile
from the form circle to the root circle of the gear replac-
ing the conventional trochoidal fillet profile.
Table 1 gives the parametric specification of 15 teeth
and 16 teeth spur gear. These design specifications have
been arrived from KISS soft an application software for
the given centre distance. The virtual model of the spur
gear with 15 teeth and 16 teeth having Circular and Tro-
choidal root fillet are modeled in Pro-E wildfire version
3.0 software and are presented in the following Figure 3
and Figure 4.
3. Force Analysis
The load transmitting capability of gear tooth is analyzed
and checked for designing a gear system. The effective
circumferential force on the tooth at the pitch circle of
the gear while in meshing is estimated. Two kinds of
stresses are induced in gear pair during the power trans-
mission from one shaft to another shaft. They are: 1)
Bending stress – Induced on gear teeth due to tangential
force developed by the power and 2) Surface contact
stress or Compressive stress. The load is assumed to be
uniformly distributed along the face width of the tooth.
3.1. Components of Forces
When the mating gears are engaged the line of contact
starts from bottom of the tooth to tip of the tooth along
Figure 1. Geometry of the circular fillet.
Figure 2. Superposition of circular fillet on a standard tooth.
Table 1. Specification of gear.
Gear tooth type :Standard involute full depth
Number of teeth ( Z) :15 and 16
Normal module (mn) :4 mm
Pressure angle (α) :20
Helix angle (β) :0°
Tooth root fillet :Trochoidal and Circular (proposed)
Copyright © 2010 SciRes. ENG
Circular fillet Trochoidal fillet
Figure 3. Gear with 15 teeth.
Circular fillet Trochoidal fillet
Figure 4. Gear with 16 teeth.
tooth-profile for the pinion and tip to bottom for the gear.
While the force is acting at the tip of tooth, the long dis-
tance of action from root cause maximum bending stress
at the bottom of tooth. Hence the force at this position
(i.e., at tip) is considered for analysis.
The normal force (Fn) to the tip of the gear is depicted
in Figure 5. This force (Fn) is at an angle with the com-
mon tangent to pitch circle (i.e., pressure angle) is re-
solved into two components:
1) Tangential Force (Ft)
2) RadialForce (Fr).
The tangential force ‘Ft’ or transmitting load can be
derived from the following standard equation;
Ft = [2000T]/d
where, T = 9550 P/n
Irrespective of the value of the contact ratio, the gear
forces are effective on a single pair of teeth in mesh. Re-
ferring to Figure 5, the normal force (Fn) acts along the
pressure line. The normal force produces an equal and
opposite reaction at the gear tooth. Since the gear is
mounted on the shaft, the radial force Fr acts at the centre
of the shaft and is equal in magnitude but opposite in
direction to the normal force Fn.
As far as the transmission power is concerned, the
component of forces Fn and Fr plays no role and the
driving component is tangential force Ft. The tangential
force Ft constitutes a couple which produces the torque
on the pinion which in turn drives the mating gears. The
tangential force bends the tooth and the radial force
compresses it. The magnitudes of the components of the
normal force Fn are given by:
Ft = Fn.cosα
Fr = Fn.sinα
Forces are calculated based on power transmission
(power is equal to 20 kW), the speed of the gear are 1000
rpm,1500 rpm and 2000 rpm respectively for which the
components of forces are calculated for 15 teeth and 16
teeth and are given in Table 2 and Table 3.
4. Finite Element Analysis
A finite element model with a single tooth is considered
for analysis. Gear material strength is a major considera-
tion for the operational loading and environment. Gener-
ally, cast iron is used in normal loading and higher wear
resisting conditions. In modern practice, the heat treated
alloy steels are used to overcome the wear resistance.
ANSYS version 11.0 software is used for analysis. In
this work, heat treated alloy is taken for analysis. The
gear tooth is meshed in 3-dimensional (3-D) solid 20
nodes 92 elements with fine mesh. SOLID92 has a
quadratic displacement behavior and is well suited to
model irregular meshes. The material properties chosen
for analysis are presented in Table 4.
Figure 6 illustrates a single tooth of 2-dimensional
(2-D) Circular fillet roots and Figure 7 shows a single
tooth of 2-dimensional Trochoidal fillet roots. Figure 8
shows the FEM meshed model of single tooth of Circular
fillet roots. Similarly, Figure 9 shows the FEM meshed
model of single tooth Trochoidal fillet roots.
Figure 5. Tooth forces in spur gear.
Table 2. Force components for 15 teeth.
Force Components (Newton)
(N-mm) Ft F
n F
1000 191000 6366.67 6775.27 2317.28
1500 127330 4244.44 4516.84 1544.85
2000 95500 3638.10 3871.58 1324.16
Copyright © 2010 SciRes. ENG
Table 3. Force components for 16 teeth.
Force Components
(N-mm) Ft F
n F
1000 191000 5968.75 6351.81 2172.45
1500 127330 3979.17 4234.54 1448.30
2000 95500 3410.71 3629.61 1241.40
Table 4. Material properties.
Gear material : Alloy structural steel
Density : 7870 kg/m3
Young’s modulus : 206 GPa
Poisons ratio : 0.3
Yield strength : 637 MPa
Figure 6. 2-D Circular root fillet tooth.
Figure 7. 2-D Trochidal root fillet tooth.
4.1. Displacement and Loading
In order to facilitate the finite element analysis, the gear
tooth was considered as a cantilever beam. All the de-
Figure 8. Meshed model of circular root fillet tooth.
Figure 9. Meshed model of trochidal root fillet tooth.
grees of freedom were constrained at the root circle but
for analysis purpose the constrained degrees of freedoms
are transferred to gear hub surface. The revolutions of
the gears are limited to 2000 rpm. In nonlinear contact
analysis the tooth forces are applied on tip of the tooth
5. Results and Discussion
The deflection and bending stress analysis were carried
out for the spur gear with 15 teeth and 16 teeth. The in-
duced bending stress and obtained deflection values are
presented in Table 5.
The investigation reveals that the deflection value of
both circular and trochoidal root fillet gears are identical.
But, looking in to bending stress the 15T gear generated
with circular root fillet have lesser stress (609.654 N/
mm2) at 1000 rpm when compared with trochoidal fillet
gear (626.699 N/mm2).
Correspondingly, the induced bending stress for 16 T
circular root fillet gear at 2000 rpm was 348.374 N/mm2
Copyright © 2010 SciRes. ENG
where as it was noticed as 358.114 N/mm2 for 16 T tro-
choidal root fillet gear. The bending stress and deflection
values taken from FEA results for 15 teeth gear with
circular and trochoidal root fillet are depicted in Figure
Similarly, the bending stress and deflection values
taken from FEA results for 16 teeth gear with circular
and trochoidal root fillet are depicted in Figure 11. It is
observed from ANSYS study that the 16T gear generated
with circular root fillet have lesser stress (328.381
N/mm2) at 1000 rpm when compared with trochoidal
fillet gear (558.287 N/mm2). Also, the bending stress
Table 5. Deflection and bending stress result.
Deflection(mm) Bending Stress (N/mm2)
15 Teeth 16 Teeth 15 Teeth 16 Teeth
Trochoidal Circular Trochoidal Circular Trochoidal Circular Trochoidal Circular
1000 0.013343 0.013747 0.013909 0.012407 626.699 609.654 558.287 328.381
1500 0.008895 0.009164 0.009272 0.008271 417.798 406.435 372.192 218.921
2000 0.007624 0.007855 0.007948 0.007090 358.114 348.374 319.021 187.646
15 Teeth - Deflection
Trochoidal Fillet Circular Fillet
at 1000 rpm
Copyright © 2010 SciRes. ENG
at 1500 rpm
Copyright © 2010 SciRes. ENG
at 2000 rpm
Figure 10. FEA results for 15 teeth gear.
16 Teeth - Deflection
Trochoidal root fillet Circular root fillet
Copyright © 2010 SciRes. ENG
at 1000 rpm
at 1500 rpm
Copyright © 2010 SciRes. ENG
at 2000 rpm
Figure 11. FEA results for 16 teeth gear.
(187.646 N/mm2) was least for 16 T circular root fillet
gear at 2000 rpm when compared with trochoidal root
fillet gear (319.021 N/mm2). In sort, the results obtained
from ANSYS result shows that the bending stress and
deflection values are lesser for Circular root fillet gear
irrespective of speed than Trochoidal root fillet gear.
6. Conclusions
The investigation result infers that the deflection in cir-
cular root fillet is almost same comparing to the trochoi-
dal root fillet gear tooth. However, there is appreciable
reduction in bending stress value for circular root fillet
design in comparison to that of bending stress value in
trochoidal root fillet design.
From the foregoing analysis it is also found that the
circular fillet design is more opt for lesser number of
teeth in pinion and trochoidal fillet design is more suit-
able for higher number of teeth in gear (more than 17
teeth) and whatever may be the pinion speed. In addition
to that the ANSYS results indicates that the gears with
circular root fillet design will result in better strength,
reduced bending stress and also improve the fatigue life
of gear material. Further work shall be done to ascertain
the stiffness and rigidity of gear tooth in the circular root
fillet design so that the feasibility of this particular de-
sign can be useful to put in practical application in fu-
7. References
[1] T. Costopoulos and V. Spitas, “Reduction of Gear Fillet
Stresses by Using One Sided Involute Asymmetric
Teeth,” Mechanism and Machine Theory, Vol. 44, No. 8,
2009, pp. 1524-1534.
[2] Y. A. Tesfahunegn and F. Rosa, “The Effects of the
Shape of Tooth Profile Modification on the Transmission
Error Bending and Contact Stress of Spur Gears,” Journal
of Mechanical Engineering Science, Vol. 224, No. 8,
2010, pp. 1749-1758.
[3] V. Spitas, T. Costopoulos and C. Spitas, “Increasing the
Strength of Standard Involute Gear Teeth with Novel
Circular Root Fillet Design,” American Journal of Ap-
plied Sciences, Vol. 2, No. 6, 2005, pp. 1058-1064.
[4] L. Fredette and M. Brown, “Gear Stress Reduction Using
Internal Stress Relief Features,” Journal of Mechanical
Design, Vol. 119, No. 4, 1997, pp. 518-521.
[5] M. Ciavarella and G. Demelio, “Numerical Methods for
the Optimization of Specific Sliding Stress Concentration
and Fatigue Life of Gears,” International Journal of fa-
tigue, Vol. 21, No. 5, 1999, pp. 465-474.
[6] M. S. Hebbal, V. B. Math and B. G. Sheeparamatti, “A
Study on Reducing the Root Fillet Stress in Spur Gear
Using Internal Stress Relieving Feature of Different
Shapes,” International Journal of RTE, Vol. 1, No. 5,
May 2009, pp. 163-165.
[7] S. Senthilvelan and R. Gnanamoorthy, “Effects of Gear
Tooth Fillet Radius on the Performance of Injection
Moulded Nylon 6/6 Gears,” Materials and Design, Vol.
27, No. 8, 2005, pp. 632-639.
[8] T. H. Chong, T. H. Myong and K. T. Kim, “Tooth Modi-
fication of Helical Gears for Minimization of Vibration
and Noise,” International Journal of KSPE, Vol. 2, No. 4,
2001, pp. 5-11.
[9] M. Beghini, F. Presicce and C. Santus, “A Method to
Define Profile Modification of Spur Gear and Minimize
the Transmission Error,” AGMA Fall Technical Meeting,
Milwaukee, Wisconsin, October 2004, pp. 1-28.
[10] ISO, 6336-3, “Calculation of the Load Capacity of Spur
Copyright © 2010 SciRes. ENG
and Helical Gears-Part 3,” Calculation of Bending
Strength, 1996.
[11] AGMA, 2101-C95, “Fundamental Rating Factors and
Calculation Methods for Involute Spur and Helical Gear
(Metric Version),” American Gear Manufacturers Asso-
ciation, 1995.
[12] H. H. Mabie, C. A. Rogers and C. F. Reinholtz, “Design
of Nonstandard Spur Gears Cut by a Hob,” Mechanism
and Machine Theory, Vol. 25, No. 6, 1990, pp. 635-644.
[13] C. A. Rogers, H. H. Mabie and C. F. Reinholtz, “Design
of Spur Gears Generated with Pinion Cutters,” Mecha-
nism and Machine Theory, Vol. 25, No. 6, 1990, pp.
[14] G. Niemann,“Maschinenelemente,” Band 2, Springer,
Verlag, 1965.
Fn – Normal force
P – Rated power
Ft – Tangential force
d – PCD of gear
R – Reaction of shaft
n – Speed of the gear in rpm
rf – Root Circle of gear
T – Transmitted Torque
Ss Arc length at BCD
rs – Form Circle of gear
FEA – Finite Element Analysis