ANALYSIS OF STRESSES AND DEFLECTIONS IN SPUR GEAR

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 http://www.iaeme.com/ijmet/index.asp 461 editor@iaeme.com

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) f s = Factor of Safety M b = Torque Transmitted by gears (N-mm) P N = 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 h a = Addendum (mm) h f = Dedendum (mm) c = Clearance (mm) b = Face width (mm) h k = Working Depth (mm) h = Whole Depth (mm) α= Pressure Angle ( ) m p = Contact Ratio p = Circular Pitch P = Diametral Pitch m = Module (mm) Y = Lewis Form Factor s = Tooth Thickness (mm) http://www.iaeme.com/ijmet/index.asp 462 editor@iaeme.com

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 z p 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 /z p p = 31.4 mm 2 Circular Tooth Thickness (mm) p/2 15.7 mm 3 Diametral Pitch P = z p/d P = 0.1Tooth per mm 4 Module m = d /z p m = 10 mm 5 Addendum (mm) h a = m h a = 10 mm 6 Dedendum (mm) h f = 1.25 * m h f = 12.5 mm 7 Clearance (mm) c = 0.25 * m c = 2.5 mm 8 Working Depth (mm) h k = 2 * m h k = 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 10 5 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 http://www.iaeme.com/ijmet/index.asp 463 editor@iaeme.com

Analysis of Stresses and Deflections In Spur Gear 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 (P r), which induces compressive stresses, is neglected. It is assumed that the tangential component (P t) 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. http://www.iaeme.com/ijmet/index.asp 464 editor@iaeme.com

Joginder Singh and Dr. M R Tyagi 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 * t 3 )/12} or σb = Pt/ (m * b * Y) Deflection = (16 *Pt* h 3 )/ (E x b x t 3 ) 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 = t 2 / (6*h*m) h = 13.3 mm 2 y (mm) y = t/2 y = 7.84 mm 3 I I = (b * t 3 )/12 I = 17348.05 mm 4 2.8. CALCULATION OF σb AND DEFLECTION FOR CAR A (132 @ 3000) [2] S.No. Height (mm) P t(n) σ b(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) P t(n) σ b(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) P t(n) σ b(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 http://www.iaeme.com/ijmet/index.asp 465 editor@iaeme.com

Analysis of Stresses and Deflections In Spur Gear 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 http://www.iaeme.com/ijmet/index.asp 466 editor@iaeme.com

Joginder Singh and Dr. M R Tyagi 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 http://www.iaeme.com/ijmet/index.asp 467 editor@iaeme.com

Analysis of Stresses and Deflections In Spur Gear 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 P t 1467 N 1467 N 1467 N 2 P t + P r 1467 N 534 N 1467 N 534 N 1467 N 534 N 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. http://www.iaeme.com/ijmet/index.asp 468 editor@iaeme.com

Joginder Singh and Dr. M R Tyagi 4. RESULTS BY ANSYS 4.1. σb AND DEFLECTION FOR CAR A (132 @ 3000) [2] S.No. Height (mm) P t(n) σ b (Bending Stress)MPa Deflection(mm) 1 9.1 1467 σb = 12.02 0.0020 2 13.3 1467 3 19.4 1467 σ b = 15.36 0.0026 4.2. σb AND DEFLECTION FOR CAR B (190 @ 2000) [2] σb = 20.79 0.0041 S.No. Height (mm) P t (N) σ b (Bending Stress)MPa Deflection(mm) 1 9.1 2134 2 13.3 2134 σb = 17.48 0.0029 3 19.4 2134 σb = 22.34 0.0038 σ b = 30.24 0.0059 http://www.iaeme.com/ijmet/index.asp 469 editor@iaeme.com

Analysis of Stresses and Deflections In Spur Gear 4.3. σb AND DEFLECTION FOR CAR C (225 @ 4000) [2] S.No. Height (mm) P t (N) σ b (Bending Stress)MPa Deflection(mm) 1 9.1 2500 σ b = 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) σ b (Bending Stress)MPa Deflection(mm) 1 9.1 2 13.3 3 19.4 P t = 1467 Pr = 534 P t = 1467 P r = 534 P t = 1467 P r = 534 σb = 8.46 0.0016 σ b = 12.33 0.0023 σb = 18.90 0.0039 http://www.iaeme.com/ijmet/index.asp 470 editor@iaeme.com

Joginder Singh and Dr. M R Tyagi 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 2 13.3 3 19.4 Pt = 2134 Pr = 777 Pt = 2134 Pr = 777 Pt = 2134 Pr = 777 σb = 12.30 0.0024 σb = 17.93 0.0034 σb = 27.50 0.0057 4.6. σb AND DEFLECTION FOR CAR C (225 @ 4000) [2] S.No. Height (mm) Force (N) σ b (Bending Stress)MPa Deflection(mm) 1 9.1 2 13.3 P t = 2500 Pr = 910 P t = 2500 P r = 910 σb = 14.41 0.0028 σ b = 21.01 0.0045 3 19.4 P t = 2500 P r = 910 σ b = 32.21 0.0066 Graph of Height vs Stress Graph of Height vs Deflection http://www.iaeme.com/ijmet/index.asp 471 editor@iaeme.com

Analysis of Stresses and Deflections In Spur Gear 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) σ b(mpa) Deflection(mm) 1 9.1 12.02 0.0020 2 132@3000 13.3 P t = 1467 15.36 0.0026 3 19.4 20.79 0.0041 4 9.1 17.48 0.0029 5 190@2000 13.3 P t = 2134 22.34 0.0038 6 19.4 30.24 0.0059 7 9.1 20.48 0.0034 8 225@4000 13.3 Pt = 2500 26.18 0.0045 9 19.4 34.76 0.0070 10 9.1 8.46 0.0016 11 132@3000 13.3 P t = 1467, P r = 534 12.33 0.0023 12 19.4 18.90 0.0039 13 9.1 12.30 0.0024 14 190@2000 13.3 P t = 2134, P r = 777 17.93 0.0034 15 19.4 27.50 0.0057 16 9.1 14.41 0.0028 17 225@4000 13.3 P t = 2500, P r = 910 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. http://www.iaeme.com/ijmet/index.asp 472 editor@iaeme.com

Joginder Singh and Dr. M R Tyagi 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. http://www.iaeme.com/ijmet/index.asp 473 editor@iaeme.com