Stress analysis on Functionally graded spur gear

V.Aravind1

, S. Adharsh1, D.Prakash1*, K.Babu2

1,School of

Mechanical Engineering, SASTRA Deemed to be University, Thanjavur, India

2,

Department

of Mechanical Engineering, SSN College

of Engineering

Abstract.

Gear

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

Gear

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.

Now-a-days

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

The

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.

Table

1. Design values of the spur gear tooth

Geometric parameter

Values

Module

10mm

No of Teeth

18

Pressure angle

20°

Addendum

10mm

Dedendum

11.57mm

Pitch circle diameter

180mm

Tooth thickness

15.71mm

Whole depth

22.5mm

Face width

100mm

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

Material

Young’s

Modulus

Density

Poisson

ratio

Al-SiC

EA=68.9

Gpa

EB=410.47

Gpa

?A=2700

Kg/m3

?B=3210

Kg/m3

µA=0.33

µB=0.183

Al-SiN

EA=68.9

Gpa

EB=310

Gpa

?A=2700

Kg/m3

?B=3440

Kg/m3

µA=0.33

µB=0.27

Al-Al2O3

EA=68.9

Gpa

EB=353.1

Gpa

?A=2700

Kg/m3

?B=3950

Kg/m3

µA=0.33

µB=0.21

Steel-Zr

EA=200

Gpa

EB=250

Gpa

?A=8000

Kg/m3

?B=5680

Kg/m3

µA=0.3

µB=0.22

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.

Fig.3

Structural analysis contour plots

For the above gear, the bending stress and the deflection

are determined theoretically from the equation 1 and 2.

(1)

(2)

The

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.

4

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

equations

Laws

Young’s

Modulus

Density

Poisson

Ratio

Exponential

E(r)=E0e?r

?(r)=

?0e?r

?(r)= ?0 eµr

Linear

E(r)=(m×r)+c

?(r)=(m1×r)+c1

?

(r)=(m2×r)+c2

Power

E(r)=

E0(r/b)?1

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

?2

?

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

Elliptical

E(r)=(1-r/a2)×b2

?(r)

=(1-r/a12)×b12

?

(r) =(1-r/a22)×b22

E0,

?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.

In

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.

(3)

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.

Fig.

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

stress

5.Conclusion

In

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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