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Preliminary bending fatigue performance evaluation of asymmetric composite gears
Mechanism and Machine Theory 78 (2014) 92–104
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Short communication
Preliminary bending fatigue performance evaluation of asymmetric composite gears N. Anand Mohan 1, S. Senthilvelan ⁎ Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India
a r t i c l e
i n f o
Article history: Received 20 September 2013 Received in revised form 26 February 2014 Accepted 11 March 2014 Available online xxxx Keywords: Gear Asymmetric Fatigue Molding Composite
a b s t r a c t In this work, symmetric gears 20°/20° and asymmetric gears 20°/34° were injection molded using unreinforced and 20% glass fiber reinforced polypropylene materials. To evaluate the bending fatigue performance of symmetric and asymmetric gears, a test rig was specially designed and developed and integrated with the servo hydraulic fatigue test facility. Gear teeth were subjected to constant, completely reversed deflection and the load required to maintain constant gear tooth deflection was continuously measured. Addition of glass fibers improved the bending load carrying capacity of both symmetric and asymmetric gears. Test gears exhibited superior performance at higher loading frequencies due to the strain rate sensitiveness of polymer based materials. The effective cantilever length of the gear tooth contributed to the improvement of the load carrying capacity of 34°/20° asymmetric gears over 20°/34° asymmetric gears as well as symmetric gears. The load carrying capacity of 20°/34° asymmetric gears were found to be superior to that of 20°/20° symmetric gears due to the increased gear tooth width at the root region. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Symmetric gears (same pressure angle at both the drive and coast sides of the tooth profile) are used in bi-directional load/motion transmissions. Asymmetric gears (different pressure angles at the drive and coast sides) are more suitable than conventional symmetric gears for uni-directional load transmission applications. However, the utilization of asymmetric gears for the engineering application is limited due to the difficulty in manufacturing asymmetric gears through conventional techniques such as milling, shaping and hobbing. Since powder metallurgy and the injection molding manufacturing process necessitate only a single/few die(s) for mass production, asymmetric gears can be manufactured economically as symmetric gears. Litvin et al. [1] modified asymmetric gear geometry to reduce localized loading and to stabilize the bearing contact for the significant reduction in transmission error. Karpat et al. [2] optimized the asymmetric gear teeth design to minimize dynamic loads. Pedersen [3] observed significant reduction of bending stress at drive side than coast side. Walsange [4] developed a finite element model of asymmetric gears to understand the bending and contact stress behavior for the various positions of gear meshing. Senthilkumar et al. [5] optimized the asymmetric gear for non standard rack cutters. Prior experimental and numerical research work on polymer based gears under dry conditions confirmed that contact fatigue performance was less significant than bending fatigue performance [6,7]. Due to the relatively low modulus of polymer gears, the possibility of pitting failure due to contact stress was very small. In spite of low sub-surface shear stress, the depth at which the maximum shear stress occurs from the surface is high due low material modulus. Bending fatigue performance evaluation of
⁎ Corresponding author. Tel.: +91 3612582671; fax: +91 3612582699. E-mail address: [email protected] (S. Senthilvelan). 1 Tel.: +91 3612582671; fax: +91 3612582699.
http://dx.doi.org/10.1016/j.mechmachtheory.2014.03.006 0094-114X /© 2014 Elsevier Ltd. All rights reserved.
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single teeth was effective over power absorption and power recirculation test facilities. Considerable effort has been attempted to evaluate the fatigue performance of polymeric symmetric gears [8–12]. Injection molded polymer and polymer composite symmetric gears were injection molded and tested in a specially designed power recirculation gear test rig. These test facilities enable the evaluation of failure due to bending as well as contact fatigue. Gear wear due to contact fatigue was given more importance than that due to bending fatigue in these research works. However, no experimental investigations have been carried out on the performance evaluation of asymmetry gears. Akata et al. [13] proposed the three point bending test fixture for the stand alone tests using a servo hydraulic fatigue testing machine to evaluate the bending fatigue performance. Bending load was applied only on the highest point single tooth contact of individual gear teeth during the tests. Daniewicz and Moore [14] observed the improved fatigue life of gears by introducing beneficial compressive residual stress. A single tooth bending fixture was used for the studies, where fluctuating bending loads were applied only on the highest point single tooth contact of individual gear teeth using a servo hydraulic fatigue testing machine. The number of test trials was carried out on the same gear by re-fixing another tooth for the tests. Costopoulos and Spitas [15] introduced the concept of the asymmetric half-involute gear teeth and studied feasibility using finite element analysis. It is found that the increase in load carrying capacity can reach up to 28% compared to the standard 20° involute teeth. Kramberger et al. [16] presented a computational model for the determination of service life of gears with regard to bending fatigue in a gear tooth root. A failure analysis of micro straight bevel gears was carried out by Dayi et al. [17]. Mechanics analyses through the finite element method were carried out to give a more comprehensive analysis. Osman and Velex [18] developed dynamic contact fatigue models to account for crack initiation and propagation. Three characteristic points on a tooth profile are analyzed and it is observed that contact fatigue on spur gears clearly depends on dynamic phenomena. From the prior literature it is observed that asymmetric gear researches were limited to computational work and there is a need for experimental investigation. In this work, symmetric and asymmetric gears were injection molded. A test rig was developed to evaluate bending fatigue performance. Gears were subjected to constant completely reversed deflection and the load required to maintain constant gear tooth deflection was measured. Asymmetric gears were tested to understand the effect of drive side pressure angle on bending fatigue performance. The effect of gear loading frequency was evaluated to understand the time dependence behavior of unreinforced and reinforced gears.
2. Gear details and testing conditions Unreinforced and 20% glass fiber reinforced polypropylene materials were considered for injection molding symmetric and asymmetric gears. Table 1[19] lists the mechanical properties of the selected gear materials. Gear materials were preheated to 80 °C for 3 h to remove moisture content. Injection molding of test gears were carried out by maintaining barrel temperature at 225, 230 and 235 °C and at an injection pressure of 9.31 MPa. The spur gear of a 3 mm module, 18 numbers of teeth and a 20° pressure angle were considered for this work. For asymmetric gears, the pressure angle at the other side was fixed as 34° to get maximum advantage. A fillet radius of 1.14 mm (0.38 times the module) was selected for the chosen asymmetric gear [20]. These critical gear parameters were decided after comprehensive finite element analysis [4,21,22]. Fig. 1 and Table 2 show the parameters of symmetric and asymmetric test gears. Fig. 2a and b shows the tooth of an injection molded glass fiber reinforced asymmetric gear and an unreinforced symmetric gear. Gear tooth bending fatigue evaluation of molded gears was carried out with the aid of a servo hydraulic test facility (Instron 8801) of 100 kN capacity with a stroke length of ± 75 mm. In this test facility, various forms of cyclic load can be applied at a frequency range of 0.1–10 Hz. To simulate the actual gear meshing, a test rig was specially designed and developed to integrate with the fatigue test machine to convert axial displacement of ram to torsional movement of standard test gears. Fig. 3a shows the model of a developed test rig, where a test gear is meshed with a standard stainless steel gear. This standard stainless steel gear (SS 316) was manufactured by a wire cut electrical discharge machining process. The test rig consists of two shafts which are mounted in a block at the standard center distance of the gear pair. The steel gear is sufficiently supported by bearings and made to rotate at the desired magnitude by converting the axial ram displacement of a servo hydraulic test facility through a crank and connecting rod mechanism. As the test gear mounted shaft is rigidly fixed, the angular rotation of the steel gear is constrained, thereby exerting torque on the test gear tooth. By flipping horizontally asymmetric gears 20°/34° in the shaft of test rig, tests were also carried out on 34°/20° asymmetric gears. Fig. 3b shows the view of the developed test rig; in this the applied load is not a point load as in the case of conventional single tooth loading test rigs. Fig. 4 shows the schematic of the test rig, where load is applied through the meshing gear similar to the actual gear transmitting system. The test rig consists of two shafts; the test gear shaft and the loading gear shaft which are mounted in a block at the standard center distance of the gear pair. The loading gear shaft is mounted in a block such that it can
Table 1 Mechanical properties of test gear materials. Material
Young's modulus (MPa)
Tensile strength (MPa)
Density (g/cm3)
20% glass fiber reinforced polypropylene Unreinforced polypropylene
4500 1700
135 27
1.04 0.91
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Fig. 1. a. Symmetric test gear details; b. Asymmetric test gear details.
rotate about its axis with a pair of supporting bearings. A loading arm is connected rigidly to the loading gear shaft. A ram of fatigue testing machine is directly connected to this loading arm. When the fatigue testing machine provides a known amount of axial displacement, the loading gear shaft carrying the steel gear tends to rotate. Since the steel gear is meshing with the test gear, a known amount of torque acts on the test gear. Linear motion (1, 1.5, 2, 2.5, 3 mm) of the fatigue machine ram was converted to angular motion (1.7°, 2.5°, 3.4°, 4.2°, 5°) of the standard stainless steel gear. The axial force of the ram was converted into torsional force of the standard gear and tangential load acted on the pitch of the test gear as shown in Fig. 4. Fig. 5a shows the ram displacement (3 mm) of the servo hydraulic fatigue testing machine at 1 Hz frequency and the corresponding angular motion of the standard stainless steel gear is shown in Fig. 5b. Fig. 5c shows applied torque on the test gear; since test gears (symmetric, asymmetric, unreinforced and glass fiber reinforced) were subjected to constant gear tooth deflection, corresponding torque values were different due to the different load carrying capacities of test gears. Gear teeth were subjected to constant deflection mode at 1, 1,5 and 2 Hz (Corresponds to 60, 90 and 120 rpm) and the load required to maintain constant gear tooth deflection was measured during testing. In practice, gears are subjected to pulsating bending load and flank
Table 2 Test gear parameters. Symmetric gear Module Number of teeth Pressure angle Root circle diameter Face width Addendum Fillet radius
Asymmetric gear
3 mm 18 20° 20° and 34° 47 mm (Mean of the base circle for 20° and 34° pressure angle involute profile teeth) 8 mm 3 mm 1.14 mm (0.38 times the module)
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Fig. 2. a. Injection molded glass fiber reinforced asymmetric gear tooth profile; b. Injection molded unreinforced symmetric gear tooth profile.
Fig. 3. a. Model of test rig for the bending fatigue performance evaluation of gear tooth; b. View of developed test rig.
surfaces are subjected to contact stresses along with sliding. In the present work, completely reversed bending fatigue loads were applied on the test gear to understand the effect of pressure angle on the drive side of asymmetric gears. Axial load (measured with 0.01 N accuracy) responsible for the gear tooth load and ram displacement (measured with 0.01 mm accuracy) was acquired at the rate of 10–20 Hz. All the tests were conducted up to the tooth fracture and at least three trials were carried out at the same loading conditions and the average number of cycles at the failure was used for further analysis.
Fig. 4. Schematic of test and standard gears.
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Fig. 5. a. Ram displacement of the fatigue testing machine; b. Angular motion of the standard SS gear; c. Applied torque on test gear.
3. Results and discussion 3.1. Bending fatigue performance of unreinforced symmetric gears Fig. 6a shows the bending fatigue performance of unreinforced polypropylene symmetric gear tested at 1 Hz, 3 mm ram displacement. As the number of cycles increases, gear teeth soften due to the repeated bending fatigue load. This behavior is due to the fact that plastics are viscoelastic and a large amount of internal friction is generated during mechanical deformation [23]. This was confirmed with the drop in load (30–40% from initial value) required to maintain the same gear tooth deflection. This
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Fig. 6. a. Bending fatigue performance of unreinforced polypropylene symmetric gear tested at 3 mm displacement and 1 Hz; b. Cyclic load version of unreinforced polypropylene gear at various ram displacements.
drop was significant at the beginning of the test and it gradually stabilizes. Cyclic load variation at various numbers of cycles was plotted (Fig. 6b) to confirm the drop in load required. The maximum load required in a cycle to maintain the same gear tooth deflection for the service life of test gears was extracted and plotted in Fig. 7. To understand the effect of load on gear teeth, ram displacement was varied from 2 mm to 3 mm and tested at the same 1 Hz frequency. Fig. 7 shows the maximum load required in a cycle to maintain unreinforced symmetric gear tooth deflection corresponding to 2, 2.5 and 3 mm ram displacements at 1 Hz frequency. As the ram displacement increases, the load required to maintain constant gear tooth deflection increases.
Fig. 7. Load carrying capacity of unreinforced polypropylene tested at various displacements, 1 Hz.
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Fig. 8. Failure morphology of symmetric unreinforced polypropylene gear tested at 1 Hz frequency at (a) 2 mm (b) 2.5 and (c) 3 m ram displacements.
Failure morphology of test gears subjected to load corresponding to 2, 2.5 and 3 mm ram displacements are shown in Fig. 8(a–c). The failure morphology of test gears confirms the gear tooth bending fatigue failure with the crack at the tensile surface of the gear tooth root region. The life of the unreinforced symmetric gear corresponding to 2, 2.5 and 3 mm displacements were 5158, 1556, and 1129 cycles, respectively. To understand the effect of frequency, gears were tested at 3 mm ram displacement and at 1, 1.5 and 2 Hz frequencies and the results were plotted in Fig. 9. When the gear tooth is subjected to higher rates of loading, the load required for the same amount of gear tooth deflection increases. This is due to the fact that gear material strength is sensitive to the loading rate, when the time available to respond is shorter; a material tends to resist deformation and exhibit superior mechanical performance [23]. To maintain gear tooth deflection corresponding to 3 mm ram displacement, the load required at 1, 1.5, and 2 Hz rates were 0.77, 0.95, and 2.93 kN respectively. As the cycle progresses, the load requirement also follows the same trend.
3.2. Bending fatigue performance of glass fiber reinforced polypropylene symmetric gear The bending performance of glass fiber reinforced polypropylene symmetric gears was also evaluated as unreinforced gears. Fig. 10a shows the bending fatigue performance of reinforced gears tested at 1 Hz, 2.5 mm ram displacement. Similar to unreinforced gears; as the number of cycle increases, gear teeth soften due to the repeated bending fatigue load. The load drop is only 20–30% from the initial load condition; cyclic load variation at various ram displacements (1.5–3 mm) at 2 Hz frequency was plotted (Fig. 10b). The maximum load required in a cycle to maintain the same gear tooth deflection (corresponding to 2 and 3 mm ram displacements) at 1 Hz test conditions was extracted and plotted in Fig. 11. As the ram displacement increases, the load required to maintain constant tooth deflection increases. Fig. 12 (a–c) shows the failure morphology of test gears at loads corresponds to 2, 2.5 and 3 mm ram displacements. The failure morphology of the test gears confirms the gear tooth bending fatigue failure with the crack at the tensile surface of gear tooth root region. A scanning electron micrograph of reinforced gear tooth failure (Fig. 13(a–b)) confirms good fiber matrix binding and no fiber pull out failure from matrix or fiber fracture failure. The life of the reinforced gear corresponding to 2, 2.5 and
Fig. 9. Frequency effect on unreinforced polypropylene gear at 3 mm ram displacement.
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Fig. 10. a. Bending fatigue performance of glass fiber reinforced polypropylene symmetric gear tested at 2.5 mm displacement and 1 Hz; b. Cyclic load version of glass fiber reinforced polypropylene gear at various displacements.
3 mm ram displacements are 3953, 1556, and 53 cycles respectively. It is to be noted that load required for gear tooth deflection of reinforced gears were considerably higher than that of unreinforced test gears (Figs. 7 and 11). To understand the effect of frequency on reinforced gears, gears were tested at 3 mm ram displacement and at 1, 1.5 and 2 Hz frequencies as shown in Fig. 14. When the gear teeth were subjected to higher rates of loading, the load required for the same gear tooth deflection increases. Thus reinforced gear material also exhibits time-dependent mechanical performance, i.e. superior performance is exhibited at higher rates of loading. For the same 3 mm displacement of gears, the load required at 1, 1.5, and 2 Hz frequencies are 0.8, 1.3, and 2.9 kN, respectively. Fig. 15 shows the effect of reinforcement on symmetric gears tested at 1.5 mm
Fig. 11. Load carrying capacity of glass fiber reinforced polypropylene gear at 1 Hz and at various displacements.
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Fig. 12. Failure morphology of symmetric glass fiber reinforced polypropylene gear teeth loaded at 1 Hz frequency (a) 2 mm (b) 2.5 mm and (c) 3 mm displacement conditions.
Fig. 13. (a–b) Failure mechanism of reinforced symmetric gear at 1 Hz, 2 mm.
displacement and 2 Hz. Reinforced gears exhibit an improvement of load requirement by 68–74% compared to that of unreinforced gears. As the cycle progresses, the load requirement also follows the same trend.
Fig. 14. Frequency effect on glass reinforced symmetric gear tested at 3 mm displacement.
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Fig. 15. Effect of reinforcement on symmetric gears tested at 1.5 mm displacement and 2 Hz.
Fig. 16. Load carrying capacity of asymmetric (20°/34°) gear at various displacements.
3.3. Performance of asymmetric gear Asymmetric gears of 20°/34° and 34°/20° were considered to understand the performance of asymmetric gears. Since the effect of reinforcement and frequency were already investigated in detail with symmetric gears, only reinforced asymmetric gears tested at 1 Hz was considered for this investigation. Asymmetric gears of 20°/34° were tested at 1.5, 2, 2.5 mm ram displacements up to fracture. Fig. 16 shows the maximum amount of load experienced during cyclic loading corresponding to various displacements of ram. Life of the asymmetric gear at
Fig. 17. Failure morphology of reinforced asymmetric gear (20°/34°) at (a) 1.5 mm (b) 2 mm and (c) 2.5 mm displacements.
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Fig. 18. Load carrying capacity of 34°/20° asymmetric gear at various displacements.
Fig. 19. Failure morphology of 34°/20° reinforced asymmetric gear tested at (a) 1.5 mm (b) 2 mm and (c) 2.5 mm displacements.
1.5, 2, and 2.5 mm displacements were 20566, 117, and 79 cycles, respectively. Fig. 17 shows the failure morphology of 20°/34° asymmetric gears tested at 1.5, 2 and 2.5 mm displacements. Asymmetric gears of 34°/20° were loaded with the same loading condition (displacement) as asymmetric gears of 20°/34°. Similar to symmetric gears, asymmetric gears of 20°/34° exhibited a load drop as the number of cycles increases to maintain the same gear tooth deflection. However the load required for the same amount of gear tooth deflection was found to be more than that of 20°/34° asymmetric gears. Fig. 18 shows the maximum load taken for the various displacements of 34°/20° asymmetric
Fig. 20. Schematic loading condition of 20°/34° and 34°/20° asymmetric gears.
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Fig. 21. Load carrying capacity of symmetric 20°/20°, asymmetric 20°/34° and asymmetric 34°/20° gears.
gears. The failure morphology of 34°/20° asymmetric gears at 1.5, 2 and 2.5 displacements is shown in Fig. 19. The reason for superior bending fatigue performance of 34°/20° asymmetric gears over 20°/34° asymmetric gears is explained below. Fig. 20 shows the schematic loading condition of asymmetric gear. When the asymmetric gear is loaded on a 20° pressure angle side, the effective cantilever length of gear tooth observed is BA [24,25]. However when the same asymmetric gear is loaded from a 34° pressure angle side, then the effective cantilever length of gear tooth is reduced to BA′. This reduced cantilever length contributes to the reduction of gear tooth bending moment. Thus asymmetric gears of 34°/20° are subjected to less bending moment than of asymmetric gears of 20°/34°. Hence the bending fatigue performance of 34°/20° gears was found to be superior to that of 20°/34° symmetric gear. Similarly the effective cantilever length of 20°/20° symmetric gears is more than that of 34°/20° asymmetric gears, so 34/20 asymmetric gears are subjected to less bending moment than symmetric gears. The effective cantilever length of 20°/20° symmetric gears and 20°/34° asymmetric gears are the same due to the fact that load is acting at the 20° pressure angle side. However the superior performance of 20°/34° asymmetric gears over 20°/20° symmetric gears was due to increased thickness at the weakest region at the gear tooth root which resist bending moment. In the case of 20°/ 34° asymmetric gears, the thickness of the weakest region at the gear tooth root is RP, however for the 20°/20° symmetric gear, it is only QR. The bending fatigue performance of 20°/34° and 34°/20° asymmetric gears as well as 20°/20° symmetric gears (Fig. 21) confirms the above facts. 4. Conclusions Unreinforced and 20% glass fiber reinforced polypropylene gears were injection molded into symmetric 20°/20° and asymmetric gears 20°/34°. A test rig for evaluating the bending fatigue performance of symmetric and asymmetric gears was developed and used along with a servo hydraulic fatigue test facility. Tests were carried out till the gear tooth fracture and the load required to maintain constant gear tooth deflection was continuously monitored. Reinforced and unreinforced symmetric and asymmetric test gears exhibit gradual load drop for constant gear tooth deflection as the number of cycles increases. The failure morphology of all the test gear confirms the gear tooth bending fatigue failure with the crack at the tensile surface of the gear tooth root region. An increase in the loading frequency of polymeric test gears was found to improve the load carrying capacity due to the less available time for the material response. Reinforced gears tend to carry higher loads than unreinforced gears for the same amount of gear tooth deflection and the drop in load to maintain the same gear tooth deflection was also found to be less. The fracture surface of reinforced gear teeth confirms good fiber matrix bonding. A reduced effective cantilever length contributes to the superior performance of 34°/20° asymmetric gears over 20°/34° asymmetric gears and 20°/20° symmetric gears. Pedersen [3] also observed bending stress reduction when the pressure angle is higher at the drive side than the coast side. The increased width available to resist bending moment contributes to the superior performance of 20°/34° asymmetric gears over 20°/20° symmetric gears. Acknowledgments We wish to thank the Department of Science and Technology (DST) (SR/FTP/ETP-53/2007), Government of India for supporting this project. Author also wish to thank Mr. M. Kodeeswaran, VSSC and Mr. Johnny Mertens, IIT Guwahati for their help. References [1] L.F. Litvin, L. Qiming, L.A. Kapelevich, Asymmetric modified spur gear drives: Reduction of noise, localization of contact, simulation of meshing and stress analysis, Comput. Methods Appl. Mech. Eng. 188 (2000) 363–390. [2] F. Karpat, O.S. Ekwaro, K. Cavdar, F.C. Babalik, Dynamic analysis of involute spur gears with asymmetric teeth, Int. J. Mech. Sci. 50 (2008) 1598–1610. [3] N.L. Pedersen, Improving bending stress in spur gears using asymmetric gears and shape optimization, Mech. Mach. Theory 45 (2010) 1707–1720.
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[4] G.D. Walsange, Evaluation of Bending and Contact Stresses of Asymmetric Spur Gear Teeth Along the Path of Action, Department of Mechanical Engineering, Indian Institute of Technology, Guwahati, 2007. (MTech Thesis). [5] V. Senthilkumar, D.V. Muni, G. Muthuveerappan, Optimization of asymmetric spur gear drives to improve the bending load capacity, Mech. Mach. Theory 43 (2008) 829–858. [6] S. Senthilvelan, R. Gnanamoorthy, Effect of rotational speed on the performance of unreinforced and glass fiber reinforced Nylon 6 spur gears, Mater. Des. 28 (2007) 765–772. [7] S. Senthilvelan, R. Gnanamoorthy, Effect of gear tooth fillet radius on the performance of injection molded NYLON6/6 gears, Mater. Des. 27 (2006) 632–639. [8] K. Mao, W. Li, C.J. Hooke, D. Walton, Polymer gear surface thermal wear and its performance prediction, Tribol. Int. 43 (2010) 433–439. [9] C.J. Hooke, S.N. Kukureka, P. Liao, M. Rao, Y.K. Chen, The friction and wear of polymer in non-conformal contacts, Wear 185 (1966) 83–94. [10] D. Walton, Y.W. Shi, Comparison of rating for plastic gears, Proc. IMechE Part C, J. Mech. Eng. Sci. 203 (1989) 31–38. [11] K. Mao, W. Li, C.J. Hooke, D. Walton, Friction and wear behaviour of acetal and nylon gears, Wear 267 (2008) 639–645. [12] N.A. Wright, S.N. Kukureka, Wear testing and measurement techniques for polymer composite gears, Wear 251 (2001) 1567–1578. [13] E. Akata, M.T. Altınbalık, Y. Can, Three point load application in single teeth bending fatigue test for evaluation of gear blank manufacturing methods, Int. J. Fatigue 26 (2003) 785–789. [14] S.R. Daniewicz, D.H. Moore, Increasing the bending fatigue resistance of spur gear teeth using a presetting process, Int. J. Fatigue 20 (1998) 537–542. [15] T. Costopoulos, V. Spitas, Reduction of gear fillet stresses by using one-sided involute asymmetric teeth, Mech. Mach. Theory 44 (8) (2009) 1524–1534. [16] J. Kramberger, M. Sraml, S. Glodez, J. Flasker, I. Potrc, Computational model for the analysis of bending fatigue in gears, Comput. Struct. 82 (23–26) (2004) 2261–2269. [17] Z. Dayi, L. Shuguo, L. Baolong, H. Jie, Investigation on bending fatigue failure of a micro-gear through finite element analysis, Eng. Fail. Anal. 31 (2013) 225–235. [18] T. Osman, P. Velex, A model for the simulation of the interactions between dynamic tooth loads and contact fatigue in spur gears, Tribol. Int. 46 (1) (2012) 84–96. [19] Twintex Product Data Sheet, Long Fiber Thermoplastic Pellets, Saint Gobain Vetrotex America Inc., 2005 [20] G.M. Maitra, Handbook of Gear Design, Tata McGraw Hill, New Delhi, 1994. [21] V. Singh, S. Senthilvelan, Computer aided design of asymmetric gear, 13th National Conference on Mechanisms and Machines (NaCoMM), Indian Institute of Science, Bangalore, India, Dec 12–13, 2007. [22] K. Prateek, S. Mandal, Dynamic analysis of asymmetric spur gear, Bachelor of Technology Thesis, Indian Institute of Technology, Guwahati, 2011. [23] D.V. Rosato, D.V. Rosato, M.G. Rosato, Plastics Deign Handbook, Springer (India) Private Limited, New Delhi, 2005. [24] J.E. Shigley, C.R. Mischke, R.G. Budynas, J.K. Nisbett, Mechanical Engineering Design, 8th ed., Tata McGraw, Delhi, 2008. [25] D.W. Dudley, Handbook of Practical Gear Design, CRC press, New York, 2004.
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Short communication
Preliminary bending fatigue performance evaluation of asymmetric composite gears N. Anand Mohan 1, S. Senthilvelan ⁎ Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India
a r t i c l e
i n f o
Article history: Received 20 September 2013 Received in revised form 26 February 2014 Accepted 11 March 2014 Available online xxxx Keywords: Gear Asymmetric Fatigue Molding Composite
a b s t r a c t In this work, symmetric gears 20°/20° and asymmetric gears 20°/34° were injection molded using unreinforced and 20% glass fiber reinforced polypropylene materials. To evaluate the bending fatigue performance of symmetric and asymmetric gears, a test rig was specially designed and developed and integrated with the servo hydraulic fatigue test facility. Gear teeth were subjected to constant, completely reversed deflection and the load required to maintain constant gear tooth deflection was continuously measured. Addition of glass fibers improved the bending load carrying capacity of both symmetric and asymmetric gears. Test gears exhibited superior performance at higher loading frequencies due to the strain rate sensitiveness of polymer based materials. The effective cantilever length of the gear tooth contributed to the improvement of the load carrying capacity of 34°/20° asymmetric gears over 20°/34° asymmetric gears as well as symmetric gears. The load carrying capacity of 20°/34° asymmetric gears were found to be superior to that of 20°/20° symmetric gears due to the increased gear tooth width at the root region. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Symmetric gears (same pressure angle at both the drive and coast sides of the tooth profile) are used in bi-directional load/motion transmissions. Asymmetric gears (different pressure angles at the drive and coast sides) are more suitable than conventional symmetric gears for uni-directional load transmission applications. However, the utilization of asymmetric gears for the engineering application is limited due to the difficulty in manufacturing asymmetric gears through conventional techniques such as milling, shaping and hobbing. Since powder metallurgy and the injection molding manufacturing process necessitate only a single/few die(s) for mass production, asymmetric gears can be manufactured economically as symmetric gears. Litvin et al. [1] modified asymmetric gear geometry to reduce localized loading and to stabilize the bearing contact for the significant reduction in transmission error. Karpat et al. [2] optimized the asymmetric gear teeth design to minimize dynamic loads. Pedersen [3] observed significant reduction of bending stress at drive side than coast side. Walsange [4] developed a finite element model of asymmetric gears to understand the bending and contact stress behavior for the various positions of gear meshing. Senthilkumar et al. [5] optimized the asymmetric gear for non standard rack cutters. Prior experimental and numerical research work on polymer based gears under dry conditions confirmed that contact fatigue performance was less significant than bending fatigue performance [6,7]. Due to the relatively low modulus of polymer gears, the possibility of pitting failure due to contact stress was very small. In spite of low sub-surface shear stress, the depth at which the maximum shear stress occurs from the surface is high due low material modulus. Bending fatigue performance evaluation of
⁎ Corresponding author. Tel.: +91 3612582671; fax: +91 3612582699. E-mail address: [email protected] (S. Senthilvelan). 1 Tel.: +91 3612582671; fax: +91 3612582699.
http://dx.doi.org/10.1016/j.mechmachtheory.2014.03.006 0094-114X /© 2014 Elsevier Ltd. All rights reserved.
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single teeth was effective over power absorption and power recirculation test facilities. Considerable effort has been attempted to evaluate the fatigue performance of polymeric symmetric gears [8–12]. Injection molded polymer and polymer composite symmetric gears were injection molded and tested in a specially designed power recirculation gear test rig. These test facilities enable the evaluation of failure due to bending as well as contact fatigue. Gear wear due to contact fatigue was given more importance than that due to bending fatigue in these research works. However, no experimental investigations have been carried out on the performance evaluation of asymmetry gears. Akata et al. [13] proposed the three point bending test fixture for the stand alone tests using a servo hydraulic fatigue testing machine to evaluate the bending fatigue performance. Bending load was applied only on the highest point single tooth contact of individual gear teeth during the tests. Daniewicz and Moore [14] observed the improved fatigue life of gears by introducing beneficial compressive residual stress. A single tooth bending fixture was used for the studies, where fluctuating bending loads were applied only on the highest point single tooth contact of individual gear teeth using a servo hydraulic fatigue testing machine. The number of test trials was carried out on the same gear by re-fixing another tooth for the tests. Costopoulos and Spitas [15] introduced the concept of the asymmetric half-involute gear teeth and studied feasibility using finite element analysis. It is found that the increase in load carrying capacity can reach up to 28% compared to the standard 20° involute teeth. Kramberger et al. [16] presented a computational model for the determination of service life of gears with regard to bending fatigue in a gear tooth root. A failure analysis of micro straight bevel gears was carried out by Dayi et al. [17]. Mechanics analyses through the finite element method were carried out to give a more comprehensive analysis. Osman and Velex [18] developed dynamic contact fatigue models to account for crack initiation and propagation. Three characteristic points on a tooth profile are analyzed and it is observed that contact fatigue on spur gears clearly depends on dynamic phenomena. From the prior literature it is observed that asymmetric gear researches were limited to computational work and there is a need for experimental investigation. In this work, symmetric and asymmetric gears were injection molded. A test rig was developed to evaluate bending fatigue performance. Gears were subjected to constant completely reversed deflection and the load required to maintain constant gear tooth deflection was measured. Asymmetric gears were tested to understand the effect of drive side pressure angle on bending fatigue performance. The effect of gear loading frequency was evaluated to understand the time dependence behavior of unreinforced and reinforced gears.
2. Gear details and testing conditions Unreinforced and 20% glass fiber reinforced polypropylene materials were considered for injection molding symmetric and asymmetric gears. Table 1[19] lists the mechanical properties of the selected gear materials. Gear materials were preheated to 80 °C for 3 h to remove moisture content. Injection molding of test gears were carried out by maintaining barrel temperature at 225, 230 and 235 °C and at an injection pressure of 9.31 MPa. The spur gear of a 3 mm module, 18 numbers of teeth and a 20° pressure angle were considered for this work. For asymmetric gears, the pressure angle at the other side was fixed as 34° to get maximum advantage. A fillet radius of 1.14 mm (0.38 times the module) was selected for the chosen asymmetric gear [20]. These critical gear parameters were decided after comprehensive finite element analysis [4,21,22]. Fig. 1 and Table 2 show the parameters of symmetric and asymmetric test gears. Fig. 2a and b shows the tooth of an injection molded glass fiber reinforced asymmetric gear and an unreinforced symmetric gear. Gear tooth bending fatigue evaluation of molded gears was carried out with the aid of a servo hydraulic test facility (Instron 8801) of 100 kN capacity with a stroke length of ± 75 mm. In this test facility, various forms of cyclic load can be applied at a frequency range of 0.1–10 Hz. To simulate the actual gear meshing, a test rig was specially designed and developed to integrate with the fatigue test machine to convert axial displacement of ram to torsional movement of standard test gears. Fig. 3a shows the model of a developed test rig, where a test gear is meshed with a standard stainless steel gear. This standard stainless steel gear (SS 316) was manufactured by a wire cut electrical discharge machining process. The test rig consists of two shafts which are mounted in a block at the standard center distance of the gear pair. The steel gear is sufficiently supported by bearings and made to rotate at the desired magnitude by converting the axial ram displacement of a servo hydraulic test facility through a crank and connecting rod mechanism. As the test gear mounted shaft is rigidly fixed, the angular rotation of the steel gear is constrained, thereby exerting torque on the test gear tooth. By flipping horizontally asymmetric gears 20°/34° in the shaft of test rig, tests were also carried out on 34°/20° asymmetric gears. Fig. 3b shows the view of the developed test rig; in this the applied load is not a point load as in the case of conventional single tooth loading test rigs. Fig. 4 shows the schematic of the test rig, where load is applied through the meshing gear similar to the actual gear transmitting system. The test rig consists of two shafts; the test gear shaft and the loading gear shaft which are mounted in a block at the standard center distance of the gear pair. The loading gear shaft is mounted in a block such that it can
Table 1 Mechanical properties of test gear materials. Material
Young's modulus (MPa)
Tensile strength (MPa)
Density (g/cm3)
20% glass fiber reinforced polypropylene Unreinforced polypropylene
4500 1700
135 27
1.04 0.91
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Fig. 1. a. Symmetric test gear details; b. Asymmetric test gear details.
rotate about its axis with a pair of supporting bearings. A loading arm is connected rigidly to the loading gear shaft. A ram of fatigue testing machine is directly connected to this loading arm. When the fatigue testing machine provides a known amount of axial displacement, the loading gear shaft carrying the steel gear tends to rotate. Since the steel gear is meshing with the test gear, a known amount of torque acts on the test gear. Linear motion (1, 1.5, 2, 2.5, 3 mm) of the fatigue machine ram was converted to angular motion (1.7°, 2.5°, 3.4°, 4.2°, 5°) of the standard stainless steel gear. The axial force of the ram was converted into torsional force of the standard gear and tangential load acted on the pitch of the test gear as shown in Fig. 4. Fig. 5a shows the ram displacement (3 mm) of the servo hydraulic fatigue testing machine at 1 Hz frequency and the corresponding angular motion of the standard stainless steel gear is shown in Fig. 5b. Fig. 5c shows applied torque on the test gear; since test gears (symmetric, asymmetric, unreinforced and glass fiber reinforced) were subjected to constant gear tooth deflection, corresponding torque values were different due to the different load carrying capacities of test gears. Gear teeth were subjected to constant deflection mode at 1, 1,5 and 2 Hz (Corresponds to 60, 90 and 120 rpm) and the load required to maintain constant gear tooth deflection was measured during testing. In practice, gears are subjected to pulsating bending load and flank
Table 2 Test gear parameters. Symmetric gear Module Number of teeth Pressure angle Root circle diameter Face width Addendum Fillet radius
Asymmetric gear
3 mm 18 20° 20° and 34° 47 mm (Mean of the base circle for 20° and 34° pressure angle involute profile teeth) 8 mm 3 mm 1.14 mm (0.38 times the module)
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Fig. 2. a. Injection molded glass fiber reinforced asymmetric gear tooth profile; b. Injection molded unreinforced symmetric gear tooth profile.
Fig. 3. a. Model of test rig for the bending fatigue performance evaluation of gear tooth; b. View of developed test rig.
surfaces are subjected to contact stresses along with sliding. In the present work, completely reversed bending fatigue loads were applied on the test gear to understand the effect of pressure angle on the drive side of asymmetric gears. Axial load (measured with 0.01 N accuracy) responsible for the gear tooth load and ram displacement (measured with 0.01 mm accuracy) was acquired at the rate of 10–20 Hz. All the tests were conducted up to the tooth fracture and at least three trials were carried out at the same loading conditions and the average number of cycles at the failure was used for further analysis.
Fig. 4. Schematic of test and standard gears.
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Fig. 5. a. Ram displacement of the fatigue testing machine; b. Angular motion of the standard SS gear; c. Applied torque on test gear.
3. Results and discussion 3.1. Bending fatigue performance of unreinforced symmetric gears Fig. 6a shows the bending fatigue performance of unreinforced polypropylene symmetric gear tested at 1 Hz, 3 mm ram displacement. As the number of cycles increases, gear teeth soften due to the repeated bending fatigue load. This behavior is due to the fact that plastics are viscoelastic and a large amount of internal friction is generated during mechanical deformation [23]. This was confirmed with the drop in load (30–40% from initial value) required to maintain the same gear tooth deflection. This
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Fig. 6. a. Bending fatigue performance of unreinforced polypropylene symmetric gear tested at 3 mm displacement and 1 Hz; b. Cyclic load version of unreinforced polypropylene gear at various ram displacements.
drop was significant at the beginning of the test and it gradually stabilizes. Cyclic load variation at various numbers of cycles was plotted (Fig. 6b) to confirm the drop in load required. The maximum load required in a cycle to maintain the same gear tooth deflection for the service life of test gears was extracted and plotted in Fig. 7. To understand the effect of load on gear teeth, ram displacement was varied from 2 mm to 3 mm and tested at the same 1 Hz frequency. Fig. 7 shows the maximum load required in a cycle to maintain unreinforced symmetric gear tooth deflection corresponding to 2, 2.5 and 3 mm ram displacements at 1 Hz frequency. As the ram displacement increases, the load required to maintain constant gear tooth deflection increases.
Fig. 7. Load carrying capacity of unreinforced polypropylene tested at various displacements, 1 Hz.
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Fig. 8. Failure morphology of symmetric unreinforced polypropylene gear tested at 1 Hz frequency at (a) 2 mm (b) 2.5 and (c) 3 m ram displacements.
Failure morphology of test gears subjected to load corresponding to 2, 2.5 and 3 mm ram displacements are shown in Fig. 8(a–c). The failure morphology of test gears confirms the gear tooth bending fatigue failure with the crack at the tensile surface of the gear tooth root region. The life of the unreinforced symmetric gear corresponding to 2, 2.5 and 3 mm displacements were 5158, 1556, and 1129 cycles, respectively. To understand the effect of frequency, gears were tested at 3 mm ram displacement and at 1, 1.5 and 2 Hz frequencies and the results were plotted in Fig. 9. When the gear tooth is subjected to higher rates of loading, the load required for the same amount of gear tooth deflection increases. This is due to the fact that gear material strength is sensitive to the loading rate, when the time available to respond is shorter; a material tends to resist deformation and exhibit superior mechanical performance [23]. To maintain gear tooth deflection corresponding to 3 mm ram displacement, the load required at 1, 1.5, and 2 Hz rates were 0.77, 0.95, and 2.93 kN respectively. As the cycle progresses, the load requirement also follows the same trend.
3.2. Bending fatigue performance of glass fiber reinforced polypropylene symmetric gear The bending performance of glass fiber reinforced polypropylene symmetric gears was also evaluated as unreinforced gears. Fig. 10a shows the bending fatigue performance of reinforced gears tested at 1 Hz, 2.5 mm ram displacement. Similar to unreinforced gears; as the number of cycle increases, gear teeth soften due to the repeated bending fatigue load. The load drop is only 20–30% from the initial load condition; cyclic load variation at various ram displacements (1.5–3 mm) at 2 Hz frequency was plotted (Fig. 10b). The maximum load required in a cycle to maintain the same gear tooth deflection (corresponding to 2 and 3 mm ram displacements) at 1 Hz test conditions was extracted and plotted in Fig. 11. As the ram displacement increases, the load required to maintain constant tooth deflection increases. Fig. 12 (a–c) shows the failure morphology of test gears at loads corresponds to 2, 2.5 and 3 mm ram displacements. The failure morphology of the test gears confirms the gear tooth bending fatigue failure with the crack at the tensile surface of gear tooth root region. A scanning electron micrograph of reinforced gear tooth failure (Fig. 13(a–b)) confirms good fiber matrix binding and no fiber pull out failure from matrix or fiber fracture failure. The life of the reinforced gear corresponding to 2, 2.5 and
Fig. 9. Frequency effect on unreinforced polypropylene gear at 3 mm ram displacement.
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Fig. 10. a. Bending fatigue performance of glass fiber reinforced polypropylene symmetric gear tested at 2.5 mm displacement and 1 Hz; b. Cyclic load version of glass fiber reinforced polypropylene gear at various displacements.
3 mm ram displacements are 3953, 1556, and 53 cycles respectively. It is to be noted that load required for gear tooth deflection of reinforced gears were considerably higher than that of unreinforced test gears (Figs. 7 and 11). To understand the effect of frequency on reinforced gears, gears were tested at 3 mm ram displacement and at 1, 1.5 and 2 Hz frequencies as shown in Fig. 14. When the gear teeth were subjected to higher rates of loading, the load required for the same gear tooth deflection increases. Thus reinforced gear material also exhibits time-dependent mechanical performance, i.e. superior performance is exhibited at higher rates of loading. For the same 3 mm displacement of gears, the load required at 1, 1.5, and 2 Hz frequencies are 0.8, 1.3, and 2.9 kN, respectively. Fig. 15 shows the effect of reinforcement on symmetric gears tested at 1.5 mm
Fig. 11. Load carrying capacity of glass fiber reinforced polypropylene gear at 1 Hz and at various displacements.
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Fig. 12. Failure morphology of symmetric glass fiber reinforced polypropylene gear teeth loaded at 1 Hz frequency (a) 2 mm (b) 2.5 mm and (c) 3 mm displacement conditions.
Fig. 13. (a–b) Failure mechanism of reinforced symmetric gear at 1 Hz, 2 mm.
displacement and 2 Hz. Reinforced gears exhibit an improvement of load requirement by 68–74% compared to that of unreinforced gears. As the cycle progresses, the load requirement also follows the same trend.
Fig. 14. Frequency effect on glass reinforced symmetric gear tested at 3 mm displacement.
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Fig. 15. Effect of reinforcement on symmetric gears tested at 1.5 mm displacement and 2 Hz.
Fig. 16. Load carrying capacity of asymmetric (20°/34°) gear at various displacements.
3.3. Performance of asymmetric gear Asymmetric gears of 20°/34° and 34°/20° were considered to understand the performance of asymmetric gears. Since the effect of reinforcement and frequency were already investigated in detail with symmetric gears, only reinforced asymmetric gears tested at 1 Hz was considered for this investigation. Asymmetric gears of 20°/34° were tested at 1.5, 2, 2.5 mm ram displacements up to fracture. Fig. 16 shows the maximum amount of load experienced during cyclic loading corresponding to various displacements of ram. Life of the asymmetric gear at
Fig. 17. Failure morphology of reinforced asymmetric gear (20°/34°) at (a) 1.5 mm (b) 2 mm and (c) 2.5 mm displacements.
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Fig. 18. Load carrying capacity of 34°/20° asymmetric gear at various displacements.
Fig. 19. Failure morphology of 34°/20° reinforced asymmetric gear tested at (a) 1.5 mm (b) 2 mm and (c) 2.5 mm displacements.
1.5, 2, and 2.5 mm displacements were 20566, 117, and 79 cycles, respectively. Fig. 17 shows the failure morphology of 20°/34° asymmetric gears tested at 1.5, 2 and 2.5 mm displacements. Asymmetric gears of 34°/20° were loaded with the same loading condition (displacement) as asymmetric gears of 20°/34°. Similar to symmetric gears, asymmetric gears of 20°/34° exhibited a load drop as the number of cycles increases to maintain the same gear tooth deflection. However the load required for the same amount of gear tooth deflection was found to be more than that of 20°/34° asymmetric gears. Fig. 18 shows the maximum load taken for the various displacements of 34°/20° asymmetric
Fig. 20. Schematic loading condition of 20°/34° and 34°/20° asymmetric gears.
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Fig. 21. Load carrying capacity of symmetric 20°/20°, asymmetric 20°/34° and asymmetric 34°/20° gears.
gears. The failure morphology of 34°/20° asymmetric gears at 1.5, 2 and 2.5 displacements is shown in Fig. 19. The reason for superior bending fatigue performance of 34°/20° asymmetric gears over 20°/34° asymmetric gears is explained below. Fig. 20 shows the schematic loading condition of asymmetric gear. When the asymmetric gear is loaded on a 20° pressure angle side, the effective cantilever length of gear tooth observed is BA [24,25]. However when the same asymmetric gear is loaded from a 34° pressure angle side, then the effective cantilever length of gear tooth is reduced to BA′. This reduced cantilever length contributes to the reduction of gear tooth bending moment. Thus asymmetric gears of 34°/20° are subjected to less bending moment than of asymmetric gears of 20°/34°. Hence the bending fatigue performance of 34°/20° gears was found to be superior to that of 20°/34° symmetric gear. Similarly the effective cantilever length of 20°/20° symmetric gears is more than that of 34°/20° asymmetric gears, so 34/20 asymmetric gears are subjected to less bending moment than symmetric gears. The effective cantilever length of 20°/20° symmetric gears and 20°/34° asymmetric gears are the same due to the fact that load is acting at the 20° pressure angle side. However the superior performance of 20°/34° asymmetric gears over 20°/20° symmetric gears was due to increased thickness at the weakest region at the gear tooth root which resist bending moment. In the case of 20°/ 34° asymmetric gears, the thickness of the weakest region at the gear tooth root is RP, however for the 20°/20° symmetric gear, it is only QR. The bending fatigue performance of 20°/34° and 34°/20° asymmetric gears as well as 20°/20° symmetric gears (Fig. 21) confirms the above facts. 4. Conclusions Unreinforced and 20% glass fiber reinforced polypropylene gears were injection molded into symmetric 20°/20° and asymmetric gears 20°/34°. A test rig for evaluating the bending fatigue performance of symmetric and asymmetric gears was developed and used along with a servo hydraulic fatigue test facility. Tests were carried out till the gear tooth fracture and the load required to maintain constant gear tooth deflection was continuously monitored. Reinforced and unreinforced symmetric and asymmetric test gears exhibit gradual load drop for constant gear tooth deflection as the number of cycles increases. The failure morphology of all the test gear confirms the gear tooth bending fatigue failure with the crack at the tensile surface of the gear tooth root region. An increase in the loading frequency of polymeric test gears was found to improve the load carrying capacity due to the less available time for the material response. Reinforced gears tend to carry higher loads than unreinforced gears for the same amount of gear tooth deflection and the drop in load to maintain the same gear tooth deflection was also found to be less. The fracture surface of reinforced gear teeth confirms good fiber matrix bonding. A reduced effective cantilever length contributes to the superior performance of 34°/20° asymmetric gears over 20°/34° asymmetric gears and 20°/20° symmetric gears. Pedersen [3] also observed bending stress reduction when the pressure angle is higher at the drive side than the coast side. The increased width available to resist bending moment contributes to the superior performance of 20°/34° asymmetric gears over 20°/20° symmetric gears. Acknowledgments We wish to thank the Department of Science and Technology (DST) (SR/FTP/ETP-53/2007), Government of India for supporting this project. Author also wish to thank Mr. M. Kodeeswaran, VSSC and Mr. Johnny Mertens, IIT Guwahati for their help. References [1] L.F. Litvin, L. Qiming, L.A. Kapelevich, Asymmetric modified spur gear drives: Reduction of noise, localization of contact, simulation of meshing and stress analysis, Comput. Methods Appl. Mech. Eng. 188 (2000) 363–390. [2] F. Karpat, O.S. Ekwaro, K. Cavdar, F.C. Babalik, Dynamic analysis of involute spur gears with asymmetric teeth, Int. J. Mech. Sci. 50 (2008) 1598–1610. [3] N.L. Pedersen, Improving bending stress in spur gears using asymmetric gears and shape optimization, Mech. Mach. Theory 45 (2010) 1707–1720.
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