Stress basically used to determine the bending stress,

Stress analysis on Functionally graded spur gear

, S. Adharsh1,  D.Prakash1*, K.Babu2

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1,School of
Mechanical Engineering, SASTRA Deemed to be University, Thanjavur, India

{[email protected]}

of Mechanical Engineering,  SSN College
of Engineering

{[email protected]}


is a vital power transmitting element
used in a wide range of applications and
among the many types of gears, spur gears
with the involute profile is less laborious
to design and manufacture. However, the failure in such gears predominant at the gear root portion as a consequence of  bending stress. Many researches are in progress
in replacing the gear materials to reduce the stress
and to improve the load carrying capacity. In this context, this paper employs
a functionally graded material for the
gear tooth, and the respective stress analysis is made through
finite element analysis technique. The finite element analysis is verified for
grid independence and validated with benchmark problem. FGM materials namely, Al-Sic, Al-SiN , Al- Al2o3 , Al-Steel
, Steel-zirconia are included in this research,
and the variation in the material property is  along the radial direction as Exponential ,
Linear , Elliptical and power law equations. The variation of stress, strain,
and displacement for various FGM materials and the best equation for the variation in the material property is identified
under uniform and varying face width value of gear tooth.


Keywords: Spur gear, Functionally graded material, Finite
element analysis.

1 Introduction

is the most commonly and widely used element in power transmissions since the
design is simple, high reliability and compactness with positive drive requirement. Spur gears  generally fails due to bending and contact
stresses at root portion. Lewis equation is basically
used to determine the bending stress, in
which the spur gear teeth is considered as a parabolic beam 1 . The effect of
radial stress, stress concentration is
neglected and assumed the tangential component of stress is distributed uniformly for a pair of tooth at contact at
any time. Timokhuko 2 used the photoelastic method to determine bending stress
at the root portion and observed that it was higher than the values obtained by Lewis
equation. However, the success of numerical simulation technique in the
accurate analysis for strength of gear
made it popular. Proveer used numerical
simulation – FEA techniques to estimate the fatigue life along with the stress
at the root portion of gear 3. Pawar
and Abhey 4 analysed the composite gear
using ANSYS software and reported that composite gear provides improved properties than alloy steels. Timoty 1
determined the root  bending stress through FEA technique and the strain
gauge measurement and compared with ISO 6336:2006, AGMA 2001-DO4 method.

metal matrix composite materials are emerging in the manufacture of many
engineering components 5. MMC has
unique advantages like lightweight,
higher stability, higher strength and are corrosion resistant 6. Pawar and
Abhay 4 prepared Aluminium silicate composite with 18% off Sic and improved
the hardness, Tensile strength over base metal and also observed that 60% less
weight in comparisson with steel gears, for the same power rating.  Anand mohan
and senthilvelan improved the bending
load carrying capacity by adding 20% of glass fibre
reinforced polypropylene materials for gear.
Imbaby and Jiang fabricated the stainless steel- titania composite micro spur
gear and reported that adding off titania increases the microhardness and decreases the sintered density and linear
shrinkage by varying the percentage of Titania as 2.5, 5,7.5 and 10%. However,
in these gears, the physical and the mechanical property is discontinuous at
the interface of two different layer of material13. In this context, FGMs are
widely employed in the engineering components, since their material property
changes gradually. Functionally graded materials are mordern engineering
materials designed to acheive specific tailored properties. This is achieved by
providing gradually changing compositions,
microstructures and properties 9. The gradual changes in volume fraction of the constituents and
non-homogenous structure offer continuous
graded macroscopic properties, such as
hardness, wear resistance, corrosion resistivity, thermal conductivity,
specific heat and mass density that are critical for thermal barrier coating
(TBC) as well as thermal protection of the re-entry capsule, furnace liners,
body armour, piezoelectric actuators and electromagnetic sensors 10-12.
Many researches are in progress in the stress analysis of functionally graded rotating
disk 15-18. Since ,stress analysis on rotating disk is a critical issue in turbojet engines, rotors,brake
disks, flywheels,jet engines, pumps automobiles and turbines 14.  Even tough,
many research works were done in the FGM rotating disk, a specific study on FGM gears is very limited, and hence in this research, an initial attempt is made to investigate
the stresses on the Functionally Graded spur gear tooth. 



2 Spur Gear Design and Model

spur gear was designed using Lewis method for the power of 1500 watt and speed
of 1400 rpm. The determined geometric parameter values are tabulated below in table 1. The geometry of the spur gear was
modelled using ANSYS APDL module for the determined
values and shown in the figure below (Fig.1).Single
tooth from the gear is modelled by
considering it as a cantilever beam.


1. Design values of the spur gear tooth

Geometric parameter




No of Teeth


Pressure angle






Pitch circle diameter


Tooth thickness


Whole depth


Face width


Fig.1 Geometry of
the spur gear tooth


3 Numerical Simulation Procedure

The gear model is
created as a 2-dimensional geometry and divided into
20 radial segments as shown in the figure 2 a. The geometry is meshed with PLANE
183 element having 8 nodes with 2 degree of freedom on each node. The two
degree of freedom are  translation in x
and y-direction. This element has a capability to analyze the
behavior of plasticity, creep , hyperelasticity,
stress stiffening, large strain
capabilities, and large deflections 19 The geometry is solved
under plane stress with thickness consideration option. The thickness of the
gear tooth is specified through real constant option.The geometry is meshed as shown
in the figure 2b.

Fig. 2 FEA model
of gear

The gear edges A,B and C are constrained in both x
and y-direction, and a tangential load of
115N is applied at the tooth tip. Material combinations like Al-Sic, Al-SiN,  Steel-Zr and Al-Al2O3
materials are included in this work and its mechanical properties are mentioned
in table 2.


Table 2.
Material properties






















As an outcome of the analysis, the stress and the
deflection are determined for the gear domain,
and the results are shown in figure 3 for a sample case of steel-Zr material
and the physical  property is  varied  linearly. 

Structural analysis contour plots


For the above gear, the bending stress and the deflection
are determined theoretically from the equation 1 and 2.



maximum the bending stress and the deflection are calculated as 2.1×106
Nm-2 and 0.12×10-6m and the determined values are in good
agreement with FEA results.


Results and Discussions

The above gear
is analysed for a different pair of FGMs
such as Aluminium- silicon carbide, Aluminium -silicon nitride, Aluminium-
aluminium oxide and steel- zirconium. Also, the variation of material
properties is governed by the liner, power law, exponential and elliptical equation.
The gear is analysed for uniform face width and varying face width value. For
the examined cases, the displacement and
stress along the gear involute and stress at the teeth root are predicted from
finite FEA and submitted for comparative
discussion.The material property is varied along the radial direction of the gear as linear,
power law, exponential and elliptical equation as mentioned in table 3.


Table 3. Material property variation








 ?(r)= ?0 eµr








(r)= ?0(r/b) ?3




(r) =(1-r/a22)×b22


?0,?0are the mechanicalproperties, subscript o the
materials at the outer end.m,c,m1,c1,m2,c2
are the slope and the constant variable respectively, ?1,?2,?3
are the gradient indices of Young’smodulus,
density and the Poisson ratio. a,b,a1,b1,a2,b2
are the semi-major and the Semi-minor axis of the Ellipse.

the first study, the material employed is
steel zirconium with constant face width value analysed for different material
gradient equation and the displacement variation along the gear involute is shown in figure 4a. The deflection is
gradually increasing from the gear root to the gear tip for all the equation.
The variation of deflection along the involute is almost same for all the
equations except elliptical equation. The elliptical equation shows low
defection along the involute. In the second study, the FGM pairs are varied, and the deflection along the involute
is shown in figure 4b. From this figure, it is observed that the deflection is
comparatively less for the FGM pair Al-Sic and Al-Al2O3,
Al-Sin and steel-Zr stands next. In the third study, the face width of  gear is varied in the radial direction in
accordance with the equation 3 and its
influence on deflection is shown in figure 4c.


where ho
is the gear thickness at r=b and m is the geometric parameter index. The
geometric parameter index is varied as 0.3, 0.6, 0.9 and 1.2 and the gear tooths
are analysed for the material steel-Zr
with an elliptical pattern of material
variation. From this figure, it is observed
that increasing the geometric parameter index (m), decreases the deflection and
however the difference is insignificant.

4 Variation of deflection

For the above-analysed
cases, the average stress at the gear root portion is determined and shown in
figure 5.



















Fig. 5 Root stress



The stress at the gear root potion is
comparatively minimum for the geometric parameter index, m= 1.2 and increases
significantly by decreasing m. While changing material
property equation, the stresses at  root
portion is almost same for the linear, exponential and the power law equation
and a significant increase is noticed for the elliptical equation.Also,
steel -Zr shows low stress among the other analysed materials . Finally, the influence of speed of rotation is
also investigated for a uniform and varying the face
width of  gear teeth. The speed of rotation is varied as 250rpm, 500 rpm,
750 rpm and 1000 rpm and the deflection and the stress along the gear involute are shown in figure 6. In figure 6, the deflection and the stress is
influenced significantly by the speed of rotation. Also, variable thickness gear index m=1.2 indicates comparatively low deflection
and stress values than uniform thickness gear. For all the cases, the stress is maximum at the gear root portion, and deflection is maximum at the gear tip portion.


Fig.6 Variation of deflection and


this work, the gear made of the functionally
graded material is analysed for the stress and the deflection through FEA
methods. The gear tooth is modelled as 2-dimensional geometry in ANSYS APDL.15
software and treated as cantilever beam. The
FGM pair such as  Al-Sic and Al-Al2O3,
Al-Sin and steel-Zr are included in this study, and its material behaviour is varied
linearly, exponentially, power law and elliptically. Among the analysed materials, steel-Zr shows reduced stress at the gear
root portion, and material property variation by elliptical equation shows less
deflection comparatively.Also, it is noticed that gear of variable face
width creates less stress and deflections in comparison with uniform thickness
, hence increases the load carrying capability. Especially at the gear root
portion, the induced stress by variable face width gear (m=1.2) is almost 20%
lesser than uniformgear. Finally, the influence of speed on the stress and the
deflection are studied and noticed that variable face width gear shows
comparatively less stress and deflection for all speeds of rotation in comparison with uniform face width gear.




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