ANALYSIS OF STRESSES AND DEFLECTIONS IN SPUR GEAR · 2017-05-06 · 2.11. GRAPH SHOWING RELATIONSHIP BETWEEN LEWIS FORM FACTOR, PRESSURE ANGLE AND NUMBER OF TEETH [3] 3. CAD MODEL

http://www.iaeme.com/IJMET/index.asp 461 [email protected]

International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 4, April 2017, pp. 461–473 Article ID: IJMET_08_04_050

Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=4

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

ANALYSIS OF STRESSES AND DEFLECTIONS IN

SPUR GEAR

Joginder Singh

Department of Mechanical Engineering, MRU,

Faridabad, India

Dr. M R Tyagi

Department of Mechanical Engineering, MRU,

Faridabad, India

ABSTRACT

Power is generated by various methods. Then it has to be transmitted from one point

to another point in a mechanical system or machine by various methods. Gear system is

one of the most efficient methods for transmitting power. Gears are used in applications

from tiny toys to giant machineries like earth movers. They constitute prominent part of

power transmission in automotive vehicles. The efficient and reliable performance of an

automotive vehicle is very much dependent on the quality of gears and their stress bearing

capability. The present work is an attempt to analyze the stresses and total deformation in

a spur gear. The analysis is done for the gears with different materials. The torque

specifications and dimensions of the gear of three existing models of commercial cars from

the Indian market are taken for the study. The analysis is made using ANSYS software. The

results of this study are presented in this paper.

Key words: Spur Gear, Involute Profile, Bending Stresses, Deflection and Finite Element

Analysis.

Cite this Article: Joginder Singh and Dr. M R Tyagi, Analysis of Stresses and Deflections

In Spur Gear, International Journal of Mechanical Engineering and Technology, 8(4),

2017, pp. 461-473.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=8&IType=4

1. INTRODUCTION

Understanding of stresses and deflection/deformations in mechanical components is the key to

their efficient design. Power transmission system is very important component contributing to the

performance and efficiency in an automotive vehicle. The efficient and reliable performance of an

automotive vehicle is very much dependent on the quality of gears and their stress bearing

Analysis of Stresses and Deflections In Spur Gear


capability. Therefore, we have undertaken studies on the analysis of stresses and deflections in

gears. The present work is an attempt to analyze the stresses and total deformation in a spur gear.

The same will be utilized for detailed studies on the subject with an ultimate objective of

developing machine components such as gears with unconventional material. The analysis is done

for the gears with structural steel material. The torque specifications and dimensions of the gear of

three existing models of commercial cars from the Indian market are taken for the study. The

analysis is made using ANSYS 17.2 software. The results of this study are presented in this paper.

Gears are defined as toothed wheels which transmits power and motion from one shaft to

another by means of successive engagement of the teeth. It is a positive drive and the velocity

ration remains constant. The center distance between the shafts is relatively small, which results

in compact construction. It can transmit very large power, which is beyond the range of belt or

chain drives. It can transmit motion at very low velocity, which is not possible with belt drives.

The efficiency of gears drives is very high, even up to 99 percent in case of spur gears. A provision

can be made in the gearbox for gear shifting, thus changing the velocity ratio over a wide range.

The different types of gears are spur gears, helical gears, herringbone gear, bevel gears and

worm gears. The fundamental law of gearing states that the common normal to the tooth profile at

the point of contact should always pass through a fixed point called the pitch point in order to

obtain a constant velocity ratio. It has been found that only involute and cycloidal curves satisfy

the law of gearing. An involute is a curve traced by appoint on a line as the line rolls without

slipping on a circle. A cycloid is a curve traced by a point on the circumference of a generating

circle as it rolls without slipping along the inside and outside of another circle. The cycloid profile

consists of two curves, namely, epicycloid and hypocycloid [1].

2. MATHEMATICAL FORMULATION

2.1. SYMBOLS AND NOMENCLATURES OF SPUR GEAR [1]

d' = Pitch Circle Diameter/ Pitch Diameter (mm) fs = Factor of Safety

Mb = Torque Transmitted by gears (N-mm) PN = Resultant Force (N)

np = Speed of rotation of pinion (rpm) Pt = Tangential Component (N)

ng = Speed of rotation of gear (rpm) Pr = Radial Component (N)

zp = Number of teeth on pinion zg = Number of teeth on gear

I = Moment of inertia about the neutral axis h = height (mm)

i = Velocity/Speed Ratio i' = Transmission Ratio ha = Addendum (mm)

hf = Dedendum (mm) c = Clearance (mm) b = Face width (mm)

hk = Working Depth (mm) h = Whole Depth (mm) α= Pressure Angle (°)

mp = Contact Ratio p = Circular Pitch P = Diametral Pitch

m = Module (mm) Y = Lewis Form Factor s = Tooth Thickness (mm)

Joginder Singh and Dr. M R Tyagi


2.2. GIVEN SPECIFICATION OF SPUR GEAR [2]

S. No. PARAMETER VALUE

1 d’ 180 mm

2 m 10 mm

3 zp 18

4 b 54 mm

5 α 20°

6 Y 0.308

7

Torque and Speed(N-m @ rpm)

Car ‘A’ - 132 @ 3000

Car ‘B’ - 190 @ 2000

Car ‘C’ -225 @ 4000

2.3. CALCULATED PARAMETERS FOR 20° FULL DEPTH SYSTEM [1]

S. No. PARAMETERS FORMULA VALUE

1 Circular Pitch (mm) p = π * d’/zp p = 31.4 mm

2 Circular Tooth Thickness (mm) p/2 15.7 mm

3 Diametral Pitch P = zp/d’ P = 0.1Tooth per mm

4 Module m = d’/zp m = 10 mm

5 Addendum (mm) ha = m ha = 10 mm

6 Dedendum (mm) hf = 1.25 * m hf = 12.5 mm

7 Clearance (mm) c = 0.25 * m c = 2.5 mm

8 Working Depth (mm) hk = 2 * m hk = 20 mm

9 Whole Depth (mm) h = 2.25 * m h = 22.5 mm

10 Tooth Thickness (mm) s = 1.5708 * m s = 15.708 mm

11 Tooth Space (mm) 1.5708 * m 15.708 mm

12 Fillet Radius (mm) 0.4 * m 4 mm

13 Crowning (mm) 0.0003 * b 0.0162 mm

2.4. GIVEN SPECIFICATION OF GEAR MATERIAL [2]

Gear Material which we are using is Structural Steel.

MECHANICAL PROPERTIES OF STRUCTURAL STEEL

S. No. PARAMETER VALUE

1 Young’s modulus ‘E’ 2.1 x 105 MPa

2 Ultimate Tensile Strength 460 MPa

3 Poisson’s Ratio 0.3

2.5. FACTOR OF SAFETY [1]

While designing a component, it is necessary to provide sufficient reserve strength in case of an

accident. This is achieved by taking a suitable factor of safety (fs). The magnitude of factor of

safety depends on effect of failure, type of load, degree of accuracy in force analysis, material of

component, reliability of component, cost of component, testing of machine element, service

conditions and quality of manufacture. The design of certain components such as cams and

followers, gears, rolling contact bearing or rail and wheel is based on the calculation of contact

stresses by Hertz’ theory. Failure of such components is usually in the form of small pits on the

surface of the component. Pitting is surface fatigue failure, which occurs when contact stress



exceeds the surface endurance limit. The damage due to pitting is local and does not put the

component out of operation. The surface endurance limit can be improved by increasing the

surface hardness. The recommended factor of safety for such components is 1.8 to 2.5 based on

surface endurance limit. We are considering factor of safety as 2.5.

2.6. BEAM STRENGTH OF GEAR TOOTH [1]

The analysis of bending stresses in gear tooth was done by Lewis approach. In the Lewis analysis,

the gear tooth is treated as a cantilever beam as shown in figure.

GEAR TOOTH AS CANTILEVER

GEAR TOOTH AS PARABOLIC BEAM

The tangential component (Pt) causes the bending moment about the base of the tooth. The Lewis

equation is based on the following assumptions:

• The effect of the radial component (Pr), which induces compressive stresses, is neglected.

• It is assumed that the tangential component (Pt) is uniformly distributed over the face width of the

gear. This is possible when the gears are rigid and accurately machined.

• The effect of stress concentration is neglected.

• It is assumed that at any time, only one pair of teeth is in contact and takes the total load.

It is observed that the cross-section of the tooth varies from the free end to the fixed end.

Therefore, a parabola is constructed within the tooth profile and shown by a dotted line in figure

above. The advantage of parabolic outline is that it is a beam of uniform strength. For this beam,

the stress at any cross-section is uniform or same. The weakest section of the gear tooth is at the

section XX, where the parabola is tangent to the tooth profile.



At the section XX,

Mb= Pt * h

The bending stresses (σb) are given by,

Pt (Tangential Component) = (2/d’) * Torque

σb = (Mb * y)/I = (Pt * h *t/2)/ {(b * t3)/12} or

σb = Pt/ (m * b * Y)

Deflection = (16 *Pt* h3)/ (E x b x t3)

Permissible σall = 0.5 * UTS/ fs = 0.5 * 460/2.5

= 92 MPa

t = 15.68 mm from CAD Model TOOTH GEOMETRY FOR HEIGHTS

2.7. CALCULATION FOR h, y and I [2]

S.No. PARAMETERS FORMULA VALUE

1 Lewis Height ‘h’ (mm) Y = t2/ (6*h*m) h = 13.3 mm

2 y (mm) y = t/2 y = 7.84 mm

3 I I = (b * t3)/12 I = 17348.05 mm4

2.8. CALCULATION OF σb AND DEFLECTION FOR CAR ‘A’ (132 @ 3000) [2]

S.No. Height (mm) Pt(N) σb(MPa) Deflection(mm)

1 9.1 1467 6.032 0.00040

2 13.3 1467 8.816 0.00126

3 19.4 1467 12.86 0.00392

2.9. CALCULATION OF σb AND DEFLECTION FOR CAR ‘B’ (190 @ 2000) [2]

S.No. Height (mm) Pt(N) σb(MPa) Deflection(mm)

1 9.1 2134 8.682 0.00058

2 13.3 2134 12.69 0.00182

3 19.4 2134 18.51 0.00564

2.10. CALCULATION OF σb AND DEFLECTION FOR CAR ‘C’ (225 @ 4000) [2]

S.No. Height(mm) Pt(N) σb(MPa) Deflection(mm)

1 9.1 2500 10.281 0.00069

2 13.3 2500 15.206 0.00215

3 19.4 2500 21.918 0.00668



2.11. GRAPH SHOWING RELATIONSHIP BETWEEN LEWIS FORM FACTOR,

PRESSURE ANGLE AND NUMBER OF TEETH [3]

3. CAD MODEL OF SPUR GEAR

3.1. THE INVOLUTE TOOTH FORM [4]

The involute of a circle is a curve that can be generated by unwrapping a taut string from a cylinder

as shown in below figure.

DEVELOPMENT OF THE INVOLUTE OF A CIRCLE



The involute profile of spur gear is generated in 3D CAD software with the help of CATIA

software. 3D Model designed strictly as per gear specification like circular pitch, module, number

of teeth, pitch diameter, etc. Involute profile starts from base circle and extends up to top tip of the

gear tooth. Each point is taken after every 1° so that generated profile will be the best. Note the

following about this involute curve:

• The string is always tangent to the base circle.

• The center of curvature of the 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.

3D MODEL OF SPUR GEAR BY CATIA SOFTWARE

3.2. VIRTUAL ANALYSIS

Once the 3D Model generated in CAD software then the work of CAE started. CAE software can

be ANSYS, NASTRAN, HYPERWORKS, etc. Here we are using ANSYS 17.2 for the virtual

analysis. In virtual analysis, there are three steps i.e. Pre-processing, Solver and Post-processing

3.3. PRE-PROCESSING IN ANSYS 17.2 [6]

Pre-processing involves the following steps:

• IMPORT THE CAD DATA: We have to convert it into *.stp format.

• MESHING: 3D CAD model meshed into 3D elements for

• Discretization so that number of equations can be finite.

• MATERIALS: We have defined the materials by Poisson’s ratio and Young’s modulus.

• BOUNDARY CONDITION: Gear is fixed at the center

• with the help of key and keyway on the shaft



• LOADING CONDITION: We have considered on Pt (Tangential Component) and Pr (Radial

Component) Force at different heights like at top land, Lewis height and Pitch point/pitch circle.

TABLE OF DIFFERENT LOADING CONDITIONS

S. No. Loads Height (9.1 mm) Height (13.3 mm) Height (19.4 mm)

1 Pt

2 Pt + Pr

3.4. SOLVER

The problem solved for the bending stresses and deflections.

3.5. POST-PROCESSING

We can check the bending stresses and deflection in spur gear. Bending stresses should not exceed

the permissible bending stress.

1467 N

1467 N

1467 N

1467 N

534 N

1467 N

534 N

1467 N

534 N



4. RESULTS BY ANSYS

4.1. σb AND DEFLECTION FOR CAR ‘A’ (132 @ 3000) [2]

S.No. Height (mm) Pt(N) σb (Bending Stress)MPa Deflection(mm)

1 9.1 1467

σb = 12.02 0.0020

2 13.3 1467

σb = 15.36 0.0026

3 19.4 1467

σb = 20.79 0.0041

4.2. σb AND DEFLECTION FOR CAR ‘B’ (190 @ 2000) [2]

S.No. Height (mm) Pt (N) σb (Bending Stress)MPa Deflection(mm)

1 9.1 2134

σb = 17.48 0.0029

2 13.3 2134

σb = 22.34 0.0038

3 19.4 2134

σb = 30.24 0.0059



4.3. σb AND DEFLECTION FOR CAR ‘C’ (225 @ 4000) [2]

S.No. Height (mm) Pt (N) σb (Bending Stress)MPa Deflection(mm)

1 9.1 2500

σb = 20.48 0.0034

2 13.3 2500

σb = 26.18 0.0045

3 19.4 2500

σb = 34.76 0.0070

Graph of Height vs Stress Graph of Height vs Deflection

4.4. σb AND DEFLECTION FOR CAR ‘A’ (132 @ 3000) [2]

S.No. Height (mm) Force (N) σb (Bending Stress)MPa Deflection(mm)

1 9.1 Pt = 1467

Pr = 534

σb = 8.46 0.0016

2 13.3 Pt = 1467

Pr = 534

σb = 12.33 0.0023

3 19.4

Pt = 1467

Pr = 534

σb = 18.90 0.0039



4.5. σb AND DEFLECTION FOR CAR ‘B’ (190 @ 2000) [2]

S.

No.

Height (mm) Force (N) σb (Bending Stress)MPa Deflection(mm)

1 9.1 Pt = 2134

Pr = 777

σb = 12.30 0.0024

2 13.3 Pt = 2134

Pr = 777

σb = 17.93 0.0034

3 19.4

Pt = 2134

Pr = 777 σb = 27.50 0.0057

4.6. σb AND DEFLECTION FOR CAR ‘C’ (225 @ 4000) [2]

S.No. Height (mm) Force (N) σb (Bending Stress)MPa Deflection(mm)

1 9.1 Pt = 2500

Pr = 910

σb = 14.41 0.0028

2 13.3 Pt = 2500

Pr = 910 σb = 21.01 0.0045

3 19.4

Pt = 2500

Pr = 910

σb = 32.21 0.0066

Graph of Height vs Stress Graph of Height vs Deflection



It is observed that stresses and deflections increases as the location of forces and deflections

increases from dedendum to addendum circle of the spur gear. It is trying to follow a linear path

in both conditions i.e. stresses and deflections. And we saw we got a linear variation which is what

we expected.

TABLE OF SUMMARY OF BENDING STRESSES AND DEFLECTIONS FOR

DIFFERENT LOADING CONDITIONS

S.No. Torque@RPM Height (mm) Force (N) σb(MPa) Deflection(mm)

1

132@3000

9.1

Pt = 1467

12.02 0.0020

2 13.3 15.36 0.0026

3 19.4 20.79 0.0041

4

190@2000

9.1

Pt = 2134

17.48 0.0029

5 13.3 22.34 0.0038

6 19.4 30.24 0.0059

7

225@4000

9.1

Pt = 2500

20.48 0.0034

8 13.3 26.18 0.0045

9 19.4 34.76 0.0070

10

132@3000

9.1

Pt = 1467, Pr = 534

8.46 0.0016

11 13.3 12.33 0.0023

12 19.4 18.90 0.0039

13

190@2000

9.1

Pt = 2134, Pr = 777

12.30 0.0024

14 13.3 17.93 0.0034

15 19.4 27.50 0.0057

16

225@4000

9.1

Pt = 2500, Pr = 910

14.41 0.0028

17 13.3 21.01 0.0045

18 19.4 32.21 0.0066

We have done the analysis with both Pt (Tangential Component) and Pr (Radial Component)

Force. Stresses and deflections reduces if we considered both components. If we want safer design

of gears we can neglect radial component.

5. CONCLUSION

The bending stresses and the deflections in a gear tooth have been obtained thru FEM analysis in

Ansys software. There are differences in the theoretical values and the values obtained from

analysis for both the parameters. There could be several possible reasons for these differences

which will be investigated and the FEM analysis would be further refined. It is preliminary exercise

on the analysis of stress and deflection of spur gear tooth. Future work is to extend the developed

methodology for the analysis of gears made of new lighter and stronger materials like fiber

composites.



REFERENCES

[1] V.B. Bhandari, Design of Machine Elements, 3rd ed. New Delhi, India, McGraw Hill

Education, ISBN: 0-07-068179-1, 2010, Ch. 17, sec. 17.17, pp. 646-693.

[2] Sushovan Ghosh, Rohit Ghosh, Bhuwaneshwar Patel, Tanuj Srivastava, Dr. Rabindra Nath

Barman, Structural Analysis Of Spur Gear Using Ansys Workbench 14.5, International Journal

of Mechanical Engineering and Technology (IJMET), ISSN Online: 0976-6359, pp.132–141.

[3] http://www.engineersedge.com/gears/lewis-factor.htm.

[4] P.S. Gill, Machine Drawing, 17th ed. New Delhi, India, S.K. Kataria & Sons, ISBN: 81-85749-

79-5, pp. 625-645.

[5] https://www.edx.org/course?course=all.

[6] https://www.mae.cornell.edu/people/profile.cfm?netid=rb88.

[7] Pinaknath Dewanji, Design and Analysis of Spur Gear. International Journal of Mechanical

Engineering and Technology, 7(5), 2016, pp. 209–220.

[8] Shubham A. Badkas and Nimish Ajmera, Static and Dynamic Analysis of Spur Gear.

International Journal of Mechanical Engineering and Technology, 7(4), 2016, pp. 8–21.

[9] Devendra Singh, Structural Analysis of Spur Gear Using FEM. International Journal of

Mechanical Engineering and Technology, 7(6), 2016, pp. 01–08.

Documents

ANALYSIS OF STRESSES AND DEFLECTIONS IN SPUR GEAR · 2017-05-06 · 2.11. GRAPH SHOWING RELATIONSHIP BETWEEN LEWIS FORM FACTOR, PRESSURE ANGLE AND NUMBER OF TEETH [3] 3. CAD MODEL