LUBRICATION SCIENCE Lubrication Science 2017; 29:43–58 Published online 21 July 2016 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ls.1348
Sliding velocity in profile-corrected gears H. K. Sachidananda1,*,†, K. Raghunandana2 and Joseph Gonsalvis3 1
Department of Mechanical Engineering, School of Engineering and Information Technology, Manipal University, Dubai, UAE 2 Department of Mechatronics Engineering, Manipal Institute of Technology, Manipal University, Manipal, India 3 St Joseph Engineering College, Mangalore, India
ABSTRACT The effects of sliding velocity in a spur gear for a tooth sum of 100 (altered by ±4%) with constant centre distance have been studied. The product of contact stress and sliding velocity (σ Vs) factor is calculated for varied profile shift. The effect of Hertzian compression, bending and shear deflection on sliding velocity at points along length of the path of contact has been computed. The scanning electron micrographs at pitch point C show that negative alteration in tooth-sum gears is better compared with standard and positive alterations in tooth sum. Copyright © 2016 John Wiley & Sons, Ltd. Received 20 April 2015; Revised 21 June 2016; Accepted 26 June 2016 KEY WORDS: gears; spur gear; contact stress; sliding velocity; surface contact
NOMENCLATURE b m E F G H Kh P Pd Rb R1 R2 T
face width (mm) module (mm) Young’s modulus (GPa) load (N) shear modulus (MPa) tooth height (mm) stiffness (N mm1) power (kW) diametrical pitch (mm) base circle radius (mm) radius of curvature of pinion (mm) radius of curvature of gear (mm) torque (N mm)
*Correspondence to: H. K. Sachidananda, Department of Mechanical Engineering, School of Engineering and Information Technology, Manipal University, Dubai, UAE. † E-mail:
[email protected] Copyright © 2016 John Wiley & Sons, Ltd.
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Z1 Z2 σ Vs σ
H. K. SACHIDANANDA, K. RAGHUNANDANA AND J. GONSALVIS
number of teeth on pinion (—) number of teeth on gear (—) Pressure = (Contact stress × Sliding velocity) (MPa ms1) contact stress(MPa)
INTRODUCTION Gears are the most widely used machine elements for transmission of motion and power.1–4 They are designed using profile modifications to improve strength and performance.5 Gears with addendum modifications are widely used in the field of machinery industries, and selection of addendum modification coefficients is one of the most important research orientations in the design of gears.6 In a gear pair, the pinion is generated with positive profile shift as the radius of curvature of mating gears is increased, which in turn reduces contact stresses. Also, the selection of proper positive profile shift coefficients will help to retard pit initiation. Thus, the contact fatigue life of the gears is increased considerably by locating the pitch point in the region of double tooth contact. The increased radius of curvature due to positive profile shift increases the sliding velocity and can result in squeezing out the lubricant causing metal-to-metal contact and consequent scoring. Thus, scoring limits the magnitude of positive profile shift. The moving of contact points of mating gears in mesh is considered in a kinematic analysis as complex due to profiles rolling and sliding in relation to one another during the contact period.7 Apart from rolling and sliding, application of load, compressive stress and frictional forces cause a change in the region of contact stresses from line contact along the width of the gear into rectangular contact.8 This change from line contact to rectangular contact leads to a large difference in theoretical contact stress and actual contact stresses.9 In the theoretical approach, the maximum contact stress is on pitch point,2 but in case of altered tooth-sum gear train, the position of pitch point depends on profile shift of the meshing gears. The maximum contact stress and sliding velocity at point of contact are an important design consideration during gear tooth mesh. The specific wear rate of gear surface depends on sliding velocity, contact pressure (contact stress) and load.10 The stress developed in gear material due to Hertzian contact influences on wear rate. Also, the product of contact stress and sliding velocity (σ Vs) value is crucial at the beginning and end of path of contact in wear prediction of gear tooth. Gear teeth deddendum have negative sliding, i.e. direction of rolling velocity is opposite to the sliding velocity. It is considered as one of the significant factors in gear design. This promotes Hertzian fatigue by allowing oil to enter surface cracks, where it accelerates crack growth by the hydraulic pressure propagation mechanism.11 Hence, the critical factor (σ Vs) and also separately contact stress and sliding velocity have been considered and analysed by many researchers in gear design. Using Hertz equation, the magnitude and location of maximum contact stress have been analysed by Yangang Wei.12,13 The dynamic characteristics and static stress distribution during meshing of gears were analysed by F. R. M. Romlay.14 Numerical methods for optimisation of modern tools considering kinematic optimisation, static stress analysis and crack propagation to estimate fatigue life of gear tooth have been analysed by Michele Ciavarella.15 Song he7 analysed the interfacial friction force formulations during spur gear meshing. A method of determining the variation of the relative speed and the specific sliding during a gear mesh for asymmetric gears has been analysed by Flavia Chira.16 The influence of non-standard geometry of plastic gear on sliding velocities on curved face width has been analysed using solid modelling by Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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Laurentia Andrei.17 It is seen that most of the research studies show the study of contact stress and sliding velocity on standard gears, whereas, research literature relating to contact stress and sliding velocity influence on altered tooth-sum gear performance is rather scarce. In this work, an attempt was made to study the contact stress and sliding velocity along the path of contact of altered tooth-sum gearing.
METHODOLOGY Estimation of contact stress The geometry of contact is theoretically a line in pair of meshed spur gears. This line of contact due to mutual compressive pressure develops into a band of certain width along length of the teeth. This developed band is also continuously moving along path of contact as tooth surfaces move each other with a combination of rolling and sliding.18 The pitting on gear tooth surface mostly occurs in the vicinity of the pitch line near start and end of path of contact. This happens due to the change of direction of sliding velocity (Vs); it makes compressive stress more critical along the region (Figures 1 and 2). In general, when two cylindrical shapes come in contact against each other, the area of contact is theoretically a rectangle. The stress distribution is represented by a semi-elliptical prism. The maximum contact pressure (Pc) is given by Equation 1.2 PC ¼
2F π bc L
(1)
where F is the normal force applied, L is the length, and bc is the half contact width given by Equation 2. n bc ¼
2F
1ϑ 1 2 E1
þ
π L D11 þ
1ϑ 2 2 E2
1 D2
o
(2)
In the equation mentioned in the preceding texts, υ, E and D are the Poisson’s ratio, modulus of elasticity and diameters of the contact cylinders respectively. In this work, both meshing gears are made of the same material (C-40 steel of υ = 0.3 and E = 200 GPa). Hence, considering the same material and combining the general Hertztian Equation 1 and 2, the maximum contact stress σ max is given by Equation 3.2
σ max
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u F R11 þ R12 u ¼ t0:35 L E11 þ E12
(3)
In this context, the authors have shown the shifting of maximum contact stresses along the length of contact for various profile shifts19–21 for a tooth sum of 100 (altered by ±4% with a standard centre distance and gear ratio of 1 : 1), but the effect of sliding velocity has not been taken into consideration Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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H. K. SACHIDANANDA, K. RAGHUNANDANA AND J. GONSALVIS
Figure 1. Product of σ Vs at contact points for higher range of pressure angle: (a) 96 tooth sum versus X1, (b) 100 tooth sum versus X1 and (c) 104 tooth sum versus X1. Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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Figure 2. Product of σ Vs at contact points for lower range of pressure angle: (a) 96 tooth sum versus X1, (b) 100 tooth sum versus X1 and (c) 104 tooth sum versus X1. Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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H. K. SACHIDANANDA, K. RAGHUNANDANA AND J. GONSALVIS
in previous studies. In view of this research gap, the (σ Vs) factor has been analysed in this investigation for tooth sum for various profile shifts mentioned in the preceding texts. Contact ratio Contact ratio is defined as the maximum number of teeth in mesh during the course of action between two gears along the path of contact.22 When gears transmit power, the gear teeth in mesh transmit loads normal to their surfaces of contact. Along the path of contact, the number of teeth in mesh transmitting this load depends on the contact ratio of the gear pair in mesh. Hence, if the contact ratio is 1.0, a single pair of teeth transmits the entire load from beginning of contact till end of contact. If the contact ratio is between 1 and 2, for some duration of time along the path of contact, two pairs of teeth transmit this load, and for the rest, a single pair of teeth transmits this load. Figure 3 shows the variation of contact ratio, with profile shift for a spur gear system having following gear parameters: type: spur gear, tooth sum: 100 (altered by ±4%), module: 2 mm, centre-to-centre distance: 100 mm and face width: 20 mm. Estimation of sliding velocity It is known that when profiles of two meshing gear teeth contact at the pitch point ‘C’, the motion is pure rolling without slippage. As contact moves up or down, the motion is combination of rolling and sliding. In profile shift gears, the Vs remains zero at pitch point and continuously changes due to variation in radius of curvature for various profile shifts along different points of contact. Also, due to alteration in tooth sum between a fixed centre distance, the distance of pitch point C from centres changes for various profile shifts. This leads to increase in Vs, which results in squeezing of the lubricant, causing metal-to-metal contact. Further, the positive addendum modifications to the pinion and negative addendum modification to gear produce a reduction in length of approach action and increase
Figure 3. Variation of contact ratio for profile shift (tooth sum 100 teeth). Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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in length of recess action, but the Vs, according to theory of gearing for a meshed pair of standard involute tooth flank geometry, is given by Equation 4.17 V s ¼ ðω1 þ ω2 Þ Distance of that point from pitch point:
(4)
where ω1 and ω2 are the angular velocities of the pinion and gear respectively.Based on the literature, it is found that few researchers estimated the influence of Hertzian deflection, sliding deflection and shear deflection of gear tooth mesh on sliding velocity. Weber22 estimated the influence of Hertzian deflection on sliding velocity. According to Weber, the Hertzian compression modifies the tooth geometry, changing the involute profile and extends down to a pressure line as follows: 2 δH ¼
T 2 Rb K h
(5)
where T is the transmitted torque, and Kh is Hertzian contact stiffness. The Kh resulting from the tooth surface contact was first approximated by Yang,23 as mentioned in Equation 6: πEB Kh ¼ 4 1 ϑ2
(6)
where E, B and υ are the Young’s modulus, face width and Poisson’s ratio respectively. This Hertzian deflection increases the sliding velocity by an amount as mentioned in Equation 7: VH S ¼ ðω1 þ ω2 Þ 2δH
(7)
According to Walton,24 the influence of sliding deflection and shear deflection on gear tooth mesh follows the relation mentioned in Equation 8: δb ¼
32 T h3 π 3 b E m3 Rb
(8)
where h is the tooth height measured from deddendum circle to the point of contact. The increase in sliding velocity due to bending deflection varies along the tooth flank, as mentioned in Equation 9: V BS ¼ ðω1 þ ω2 Þδb
(9)
The influence of shear deflection is expressed as follows. δS ¼
2T h π b G m Rb
(10)
where G is the modulus of rigidity. The increase in sliding velocity varies along the tooth flank as follows: Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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H. K. SACHIDANANDA, K. RAGHUNANDANA AND J. GONSALVIS
V SS ¼ ðω1 þ ω2 ÞδS
(11)
From the research review mentioned in the preceding texts, the real sliding velocity by taking into consideration of all the factors mentioned in the preceding texts can be calculated as follows: B S V S ðRealÞ ¼ V S þ V H S þ VS þ VS
(12)
Estimation of gear tooth deflection The gear tooth deflection was estimated by Yallamti et al.25 using Equation 13. δ¼
15:12 P Pd ðh1 h2 ÞZ 1 Z 2 bE
h1 h1 h1 3 1 þ 2log h2 h2 h2
(13)
where h1 = 2 m (0.7854 tanφ), h2 = 2 m (1.25tanφ + 0.7854). Parameters considered for computation To estimate the (σ Vs) factor, the following parameters are considered in addition to the parameters mentioned in the preceding texts in section contact ratio. The material considered was steel C40, while the modulus of elasticity is 200 GPa. The magnitude of tangential load is 10 N mm1 of face width, and the angular velocity is 104 rad s1. Importance of PV Contact stress and sliding velocity are the critical factors in limiting the power-transmitting capacity of the gear train. Because of high contact stress and sliding velocity in meshing gears, failure of gear tooth occurs due to fatigue, pitting, abrasion, scuffing and scoring. Hence, while designing gears, the contact stress and sliding velocity are the important points to be considered. In this connection, Kasuba26 has analysed static and dynamic loads in normal and high contact ratio spur gearing and has found that high contact ratio gearing has lower dynamic loads and lower Hertzian stresses than normal contact ratio gears. Figure 3 shows the plot of contact ratio versus profile shift for altered tooth-sum gearing (100 tooth ± 4%). The higher contact ratio in positive-altered tooth-sum gearing due to increase of pressure angle and path of contact as compared with negative-altered tooth-sum gearing is observed. As per Kasuba,26 in high contact ratio gearing, although the maximum Herztian stresses (contact stress) are lower, the product σ Vs (Hertzian contact stress × sliding velocity) is higher because of the large sliding velocity. Hence, large sliding velocity more effectively contributes to enhance the σ Vs. Results and discussions based on theoretical computation In our investigation, it is observed from the results tabulated in Table I the low sliding velocities in negative-altered tooth-sum gearing due to smaller length of approach as compared with positivealtered tooth-sum gearing. As the length of approach increases due to altered tooth-sum gearing, the sliding velocity also increases, whereas no remarkable change in length of recess. Hence, there is no change of sliding velocity for both negative and positive-altered tooth-sum gearing in recess section. Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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Table I. Sliding velocity (m/s) at maximum contact stress points along path of contact for high range of pressure angle. Tooth sum 96 97 98 99 100 101 102 103 104
X1
Point A
Point B
Point B2
Point C
Point D1
Point D2
Point E
1.6 1.7 1.3 1.4 0.9 1 0.7 0.8 0.3 0.4 0 0.1 0.3 0.2 0.5 0.4 0.8 0.7
0.328 0.247 0.426 0.343 0.52 0.436 0.54 0.453 0.651 0.561 0.767 0.676 0.804 0.711 0.842 0.742 0.896 0.792
0.064 0.008 0.058 0.016 0.056 0.02 0 0.079 0.024 0.056 0.076 0.006 0.062 0.027 0.058 0.033 0.085 0.012
0.064 0.008 0.058 0.016 0.056 0.02 0 0.079 0.024 0.056 0.076 0.006 0.062 0.027 0.058 0.033 0.085 0.012
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.856 0.936 0.757 0.84 0.663 0.748 0.644 0.732 0.534 0.621 0.416 0.507 0.378 0.472 0.343 0.44 0.289 0.391
0.856 0.936 0.757 0.84 0.663 0.748 0.644 0.732 0.534 0.621 0.416 0.507 0.378 0.472 0.343 0.44 0.289 0.391
1.119 1.193 1.125 1.2 1.127 1.204 1.185 1.262 1.158 1.241 1.106 1.191 1.123 1.21 1.125 1.216 1.098 1.196
From this, it may be noticed that lower sliding velocity at the approach reduces the squeezing action of the lubricant on the surface of meshing gear tooth, which in turn protects the gear teeth. Further, we investigated the σ Vs for altered tooth-sum gearing and plotted, as shown in Figures 1 and 2. Figure 1a, b and c shows the plot of σ Vs (pressure × sliding velocity) versus profile shift for negative-altered tooth-sum, standard and positive-altered tooth-sum (higher range of α) gearing respectively. In this case, for the selected range of altered tooth-sum gearing, the negative-altered tooth-sum gearing shows the lower range of σ Vs (approximately maximum 200 MPa m s1) as compared with standard and positive-altered tooth-sum gearing (approximately maximum 320 MPa m s1). This is because of the combined effect of low contact stresses (i.e. due to high radius of curvature of tooth that leads to low stress concentration) and lower sliding velocity. In case of positive-altered tooth-sum gearing, the tooth profile has a very small radius of curvature, which leads to high stress concentration. Due to this, the contact stress becomes higher at the beginning and end point of contact. However, the contact stresses are fairly lower than the standard gears at other point of contact along the length of path of contact. Influence of sliding velocity and contact stress on σV for altered tooth-sum gearing It is observed in Figure 1a that the products of σ Vs for 96 tooth sums at initial point of contact and end point of contact for profile shift X1 = 0.3 are 168.68 and 14.11 MPa ms1 respectively. The value of σ Vs is remarkably more (approximately 12 times) at initial point of contact compared with end point of contact (128.32 MPa at initial point of contact as compared with 180.02 MPa at end point of contact). Hence, in this case, sliding velocity influences σ Vs. Due to higher σ Vs, the arc of approach is more critical compared with arc of recess. Also, observed for the same profile shift, the σ Vs at highest point of single tooth contact (HPSTC) and lowest point of single tooth contact (LPSTC) are Copyright © 2016 John Wiley & Sons, Ltd.
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H. K. SACHIDANANDA, K. RAGHUNANDANA AND J. GONSALVIS
137.06 and 23.21 MPa ms1 respectively. The σ Vs at HPSTC is remarkably more (six times) than that at LPSTC. The σ Vs for 100 tooth sums at initial point of contact and end point of contact for profile shift X1 = 1 are 244.01 and 0.00 MPa ms1 respectively. From this, it is observed that the σ Vs is critical at initial point of contact, as compared with at end point of contact. This causes higher sliding friction due to compression of meshing teeth during initial point of contact and tends to elongate the teeth during the end point of contact. Also, it is seen that the σ Vs at HPSTC and LPSTC are 162.71 and 100.61 MPa ms1 respectively, but it is found that they are 1.6 times more at HPSTC compared with those at LPSTC. From Figure 1c, it is observed that the σ Vs for positive-altered tooth-sum gearing at initial point of contact and end point of contact for profile shift X1 = 1.8 are 315.07 and 0.30 MPa ms1 respectively. In this case, the radius of curvature is less at initial point of contact, and in turn, the contact stresses are high (171.55 MPa), and σ Vs will be much more. In this case, the contact stress is more influencing on σ Vs. In this case, the σ Vs at HPSTC and LPSTC are 184.13 and 138.29 MPa ms1 respectively, which is in a higher range in both HPSTC and LPSTC. From the preceding texts, it is noticed that the σ Vs at critical point (at initial point of contact ) is in the range of 168 MPa, which is less in negative-altered tooth-sum gearing as compared with the σ Vs of 244 and 315 MPa at critical point (at initial point of contact) of standard and positive-altered tooth-sum gears. Also, it is found that in the case of positive-altered tooth-sum gearing, the value of σ Vs is 315 MPa, which is higher than the σ Vs of 244 MPa of standard gear. In addition to the preceding texts, it also noticed that in the case of positive-altered tooth-sum gearing and standard gearing, both HPSTC and at LPSTC are critical due to higher σ Vs, whereas in negative-altered tooth-sum gearing, the σ Vs not only has lower range, but it is also 23.21 MPa ms1, which is very low at LPSTC.22 Similarly to the discussion in the preceding texts, σ Vs versus profile shift was plotted for negative, standard and positive-altered tooth-sum, with a lower pressure angle range as shown in Figure 2a, b and c. A similar kind of influence of sliding velocity on σ Vs is observed. Effect of pressure angle on σV in altered tooth-sum gearing In this study, 25° and 20° pressure angles were considered for standard gear. However, the pressure angle changes due to alteration in tooth sum; the change in pressure angle from standard gear to negative alteration and positive alteration was observed. Hence, in this paper, the range of pressure angle obtained from altered-tooth-sum gearing for standard gear pressure angle of 25° is referred as higher range of pressure angle. Similarly for range of pressure angle obtained from altered-tooth-sum gearing for standard gear, a pressure angle of 20° is referred as lower range of pressure angle. The trends of σ Vs when point of contact moves from LPSTC to pitch point in a meshed gear tooth with a lower and higher range of pressure angle, negative-altered, positive-altered and standard toothsum gear set are tabulated, as shown in Table II. It is observed from the tabulated results that for lower pressure angle, the σ Vs on negative alteration in tooth-sum gear tooth is lesser (49.68 MPa ms1) compared with standard (148.12 MPa ms1) and positive-altered tooth-sum (260.29 MPa ms1). A similar kind of trend is observed in case of higher range of pressure angle, but the effect of pressure angle on σ Vs is not much significant in negative-altered tooth-sum gearing. The increase of pressure angle from lower to higher range (standard gear from 20° to 25°) influences on a remarkable reduction of σ Vs in positive-altered tooth-sum gearing is noticed. Also, it is observed that the value of σ Vs is lower in higher range pressure angle as compared with lower range pressure angle. This is due to increased area Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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Table II. σ Vs factor (Mpa ms1): maximum contact stress shifts from point ‘B’ to point ‘C’. Lower range of pressure angle Tooth sum 96 100 104
Higher range of pressure angle
X1
Point A
Point E
Point A
Point E
1.6 1.7 0.1 0.2 0.9 0.8
49.68 37.48 148.12 132.16 260.29 223.8
165.76 178.56 187.23 205 214.08 249.8
41.71 31.44 111.78 99.75 153.09 136.46
150.49 161.41 140.86 153.66 158.4 175.88
Note: The σ Vs have been taken because gear ratio, torque, velocity, module and centre distance are the common factors for each gear pair.
resistance to compression because of increased length of path of contact. Hence, when design objective demands high load-carrying capacity with smooth power transmission, it is better to choose higher pressure angle for compact gear systems. Table III shows the Hertzian, bending and shearing deflections and sliding velocity for lower range of pressure angle. It is observed that at initial point of contact, the sliding velocity is higher in case of positive-altered tooth-sum gearing. It is almost 50% less in case of negative-altered tooth-sum gearing. It is also noticed that the sliding velocity along the path of contact at various point is lower in case of negative-altered tooth-sum gearing and positive-altered tooth-sum gearing as compared with standard gearing. But overall, the sliding velocity is lower side in case of negative-altered tooth-sum gearing as compared with standard and positive-altered tooth-sum gearing. This contributes in reduction of σ Vs in case of negative-altered tooth-sum gearing. From this, it is concluded that the results obtained by this investigation in line with the results obtained by Kasuba.26 Based on the result analysis mentioned in the preceding texts, the σ Vs influenced by sliding velocity in case of negative-altered tooth-sum gearing is found, but it is influenced by contact pressure in case of positive-altered tooth sum and standard gearing. Also, it is found that the pressure angle inversely affects the σ Vs factor. In conventional gear sets at lower pressure angle, gear teeth are weak, contact pressure is higher, and it transmits high power. But in our design, the pressure angle depends on profile shift, and it increased with negative alteration. This helps the gear tooth profile to withstand higher loads in addition to high power transmission. Hence, to investigate the effect of sliding velocity and contact pressure on gear tooth, failure morphology was taken for lower range of pressure angle sets for experimentation. B S Considering Hertzian deflection (VH S ), bending (VS ) and shear deflection (VS), the real sliding velocity (VS[Rea])) is computed and tabulated with gear tooth deflection for standard gear, negative and positive-altered tooth-sum gears and tabulated in Table IV.
Result and discussion based on experimental investigation Based on the result obtained by theoretical computation, the absolute (σ Vs) values are inversely proportional to pressure angle. From this, it came to know that lower range of pressure angle is more critical, which needs to be analysed by experimentation. Hence, the experiments were conducted using FZG gear testing machine only for lower pressure angle gears. Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
Hertzian deflection
Bending deflection
Initial point of contact (point A) 8.495 × 105 96 41.053 × 103 1.920 × 105 100 1.011 × 103 3 3.720× 104 104 9.720 × 10 Lowest point of single tooth contact (point B) 5.350 × 104 96 1.053 × 103 2.483 × 103 100 1.011 × 103 3 2.200× 103 104 9.720 × 10 Pitch point (point C) 2.490 × 103 96 1.053 × 103 3.360 × 103 100 1.011 × 103 3 2.050 × 103 104 9.720 × 10 Highest point of single tooth contact (point D) 7.388 × 103 96 1.053 × 103 4.484 × 103 100 1.011 × 103 3 1.917 × 103 104 9.720 × 10 End of point of contact (point E) 0.014 96 1.053 × 103 0.0197 100 1.011 × 103 0.0114 104 9.720 × 103
Tooth sum
Copyright © 2016 John Wiley & Sons, Ltd.
0 0 0 0.44 0.14 0.04
3.110 × 105 3.350 × 105 2.760 × 105 4.470 × 105 3.690 × 105 2.690 × 105
0.78 1.07 1.27
0.44 0.15 0.04
1.860 × 105 3.030 × 105 2.820 × 105
5.540 × 105 6.050 × 105 4.880 × 105
0.78 1.07 1.27
Vs
1.010 × 105 1.290 × 105 1.560 × 105
Shear deflection
0.0294 0.21 0.202
0.0294 0.21 0.202
0.0294 0.21 0.202
0.0294 0.21 0.202
0.0294 0.21 0.202
VH S
VSS
2.91 4.1 2.37
1.536 0.932 0.398
0.517 0.698 0.426
0.111 0.516 0.457
0.011 0.012 0.01
9.29 × 103 7.67 × 103 5.59 × 103
6.46 × 103 6.96 × 103 5.74 × 103
3.86 × 103 6.30 × 103 5.86 × 103
0.017 2.10 × 103 3.99 × 103 2.68 × 103 0.077 3.24 × 103
VBS
3.73 5.39 3.85
2.01 1.29 0.65
0.55 0.91 0.63
0.59 0.88 0.47
0.82 1.29 1.55
VS(Real)
Table III. Shows the Hertzian deflection, bending deflection, shearing deflection and sliding velocity for lower range of pressure angle.
54 H. K. SACHIDANANDA, K. RAGHUNANDANA AND J. GONSALVIS
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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Table IV. Gear tooth deflection versus VS(Real) for 96 tooth sum, 100 tooth sum and 104 tooth sum for lower range of pressure angle. VS(Real) (ms1) Tooth sum
Gear tooth deflection (mm) 7
1.36 × 10 1.59 × 107 1.81 × 107
96 100 104
A
B
C
D
E
0.82 1.29 1.55
0.59 0.88 0.47
0.55 0.91 0.63
2.01 1.29 0.65
3.73 5.39 3.85
For the experimental investigation, gear sets consisting of spur gear with a module of 2 mm and a gear ratio 1 : 1 were manufactured by C40 (steel), as per the details given in Table V. A tangential tooth load of 200 N mm1 of face width is applied through experimentation. The gears have been tested for a total life of 107 cycles. The morphological study of the tested gears was conducted using scanning electron microscope (SEM) (JEOL-JSM-6380LA, Japan). Morphological investigation of tested gear tooth surface After being subjected to fatigue loading of gears by FZG recirculating test rig,27 the samples were prepared at point B (LPSTC), point D (HPSTC) and at the pitch circle for negative-altered, standard and positive-altered tooth-sum gearing using precision wire EDM process. Then, the samples were cleaned by using compressed air. SEM was used to investigate the morphology of the gear tooth sample. The scanning electron microphotographs are taken at pitch circle area. The morphological results of points B (LPSTC) and D (HPSTC) were studied and published earlier by the authors.21 According to that, as per the consideration of only contact stress, points B (LPSTC) and D (HPSTC) are critical. At these points, it was also found that the negative-altered tooth-sum gear was better as compared with standard and positive-altered tooth-sum gears. Further, considering the effect of the product of contact stress and sliding velocity factor, it changes from a lower to a higher value from (LPSTC) point B to pitch point C. Similarly, it increases from pitch point C to point D (HPSTC). Therefore, it needs further investigation as a point approaches to pitch point. Hence, pitch point has been considered for morphological investigation in this paper and reported, as mentioned in the succeeding texts. Figure 4a shows the SEM image of negative-altered tooth-sum gear tooth spot at pitch point. Due to the continuous rolling and sliding action along the pitch line, the interface of gear tooth surface causes surface fatigue, which resulted to cracks on the surface,28 because further rolling and sliding action Table V. Gear specification for altered tooth-sum gearing, Z = 100 teeth (X1 = X2) for lower range of pressure angle.
Z1–Z2
Face Profile Pressure width Contact shift angle (mm) ratio
48–48 50–50 52–52
2.27 0 1.65
25.56 20 12.24
12 12 12
1.26 1.75 2.08
Copyright © 2016 John Wiley & Sons, Ltd.
Addendum Deddendum circle circle Whole diameter diameter Addendum Deddendum depth (mm) (mm) (mm) (mm) (mm) 103.45 104 103.31
95.54 95.5 95.68
3.72 2 0.34
0.22 2.5 4.16
3.94 4.5 3.81
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
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H. K. SACHIDANANDA, K. RAGHUNANDANA AND J. GONSALVIS
Figure 4. Gear tooth surface morphology at pitch circle area for lower range of pressure angle: (a) 96 tooth sum, (b) 100 tooth sum and (c) 104 tooth sum. Copyright © 2016 John Wiley & Sons, Ltd.
Lubrication Science 2017; 29:43–58 DOI: 10.1002/ls
SLIDING VELOCITY IN PROFILE-CORRECTED GEARS
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enhances the growth of open to surface and subsurface cracks. When multiple cracks joined together, it causes detachment of material from the surface develops pits (Figure 4a). It is noticed from Figure 4b and c that at the point of contact, there are changes from line contact to rectangular contact along the face width due to critical contact stress and higher σ Vs in case of positive-altered tooth-sum gearing, and standard gearing leads to scratches, abrasion and pits along the flank of profile.29 In these gear teeth, the path of contact extends beyond the theoretical path, and load distribution between the mating gear tooth changes from line contact into rectangular contact even though they are perfectly meshed, which in turn alters the sliding velocities.30–32 The observed plastic deformation, cracks, pits and debris due to higher contact stress and sliding velocity and the tooth surface subjected to plastic deformation are shown in Figure 4b. Debris on the surface is due to abrasion, while sliding creates ploughing. Very high contact stress and high sliding velocity cause severe fatigue, and being led to longer length subsurface directional cracks creates longitudinal pits, as shown in Figure 4c. It is observed from the theoretical analysis that the σ Vs is less in negative-altered tooth-sum gearing as compared with those in standard gearing and positive-altered tooth-sum gearing. Morphological analysis also reveals that the surface damage of negative-altered tooth sum is significantly mild as compared with those of standard and positive-altered tooth-sum gearing. This results in line with the theoretical analysis.
CONCLUSION From the theoretical and experimental investigations of altered tooth-sum gearing, it is found that: • The profile shift method of the design of gears changes the geometry of the gear and intern sliding velocity. • Contact ratio in gearing is an important parameter which determines the maximum number of teeth in mesh along the path of contact. The higher the number of teeth in mesh, the lower is the tooth load. For positive alteration in tooth sum, the higher the length of path of contact, the higher the sliding velocities. But for negative-altered tooth sum, the sliding velocities are lower. • In case of negative-altered tooth-sum gearing, lower sliding velocity at the approach reduces the squeezing action of the lubricant on the surface of meshing gear tooth, which in turn protects the gear teeth. • The factor σ Vs plays an important role in the surface wear of gear tooth. The velocity is the dominating factor in σ Vs in case of altered tooth-sum gearing. • Negative-altered tooth-sum gears performed better compared with the standard and positive-altered tooth-sum gearing. REFERENCES 1. Niemann G. Machine Elements, Vol. II. Allied Publishers, New delhi 1981. 2. Maitra G. Handbook of Gear Design (3rd edn). Pitman, London 2006. 3. Moldovean G, Velicy D. On the maximal contact stress point for cylindrical gears, 12th IFToMM World Congress, Besancon (France), June 18–21, 2007. 4. Feng X. Analysis of field of stress and displacement in process of meshing gears. International Journal of Digital Content technology and its applications 2011; 5(6):345–357. 5. Maag M. Maag Gear Book. Maag gear wheel co ltd, Zurich 1987.
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6. Singh D, Suhaib M. Kinematic considerations in gear drives — a review. International Journal of Innovative Research in Science, Engineering and Technology 2014; 3(1):8204–8214. 7. He S, Cho S, Rajendra S. Prediction of dynamic friction forces in spur gears using alternate sliding friction formulations. Journal of Sound and Vibration 2008; 309(3–5):843–851. 8. Dearn KD, Hoskins TJ, Andrei L, Walton D. Lubrication regimes in high-performance polymer spur gears. Advances in Tribology, Hindawi Publishing Corporation 2013; 2013:1–9. 9. Ivanovic L, Josifovic D, Ilic A, Stojanovic B. Tribological aspect of the kinematical analysis at trochoidal gearing in contact. Journal of the Balcon Tribological Association 2011; 17(1):37–47. 10. Shivamurthy B, Bhat KU, Anandhan S. Mechanical and sliding wear properties of multi-layered laminates from glass fabric/graphite/epoxy composites. Materials & Design 2013; 44:136–143. 11. He S, Gunda R, Singh R. Effect of sliding friction on the dynamics of spur gear pair with realistic time-varying stiffness. Journal of Sound and VibrationElsevier 2007; 301:927–949. 12. Wei Y, Zhang X, Liu Y. Theoretical research on the maximum contact stress of involute spur cylindrical gear pairs in the external meshing process. Mechanic Automation and Control Engineering 2010:491–495 Issn 978-1-4244-7737-1. 13. Li X, Jiang S, Qingliang Z. Optimal selection of addendum modification coefficients of involute cylindrical gears. Journal of computers 2013; 8(8):2156–2162. 14. Romlay FRM. Modeling of a surface contact stress for spur gear mechanism using static and transient finite element method. Journal of Structural Durability and Health Monitoring 2008; 4(1):19–27. 15. Ciavarella M, Demelio G. Numerical methods for the optimisation of specific sliding, stress concentration and fatigue life of gears. International Journal of Fatigue 1999; 21:465–474. 16. F Chira, Mihai B. On the slide between the teeth flanks of the cylindrical gears with asymmetric teeth. International conference of the cartathian euro-region specialists in industrial systems, 7th edn), 2008; 22:81–86. ISSN1224-3264 17. Andrei L, Walton D, Andrei G, Mereuta E. Influence of a non-standard geometry on plastic gear on sliding velocities. The Annals of University “dunarea de jos” of Galati Fascicle Tribology 2004. 18. Ristivojević M, Lazović T, Vencl A. Studying the load carrying capacity of spur gear tooth flanks. Mechanism and Machine Theory 2013; 59:125–137. 19. Sachidananda HK, Gonsalvis J, Prakash HR. Analysis of contact stresses in altered tooth-sum spur gearing. Applied mechanical Engineering 2012; 1(1):1–5. 20. Sachidananda HK, Gonsalvis J, Prakash HR. Experimental investigation of fatigue behavior of spur gear in altered toothsum gearing. Frontiers of Mechanical Engineering 2012; 7(3):268–278. 21. Sachidananda HK, Raghunandana K, Gonsalvis J. Design of spur gears using profile modification. Tribology Transactions 2015; 58(4):737–744. 22. Weber C. The deformation of loaded gears and the effect of their load carrying capacity. British Science and Industrial Research. 1949; 3. 23. Yang DC, Lin IY. Hertzian damping, tooth friction and bending elasticity in gear impact dynamics. Journal of Mechanical Transmission and Automation in Design, c, “No Title” 1987; 109:189–196. 24. Walton D, Cropper AB, Weale DJ, Klein MP. The efficiency of plastic cylindrical gears: part 1 — influence of materials. In IMechE, part J: Journal of Engineering Tribology 2002; 216(2):75–78. 25. Yallamti MM, Seshaiah T. Spur gear optimization by using genetic algorithm. International Journal of Engineering Research and Applications (IJERA) 2012; 2(1):311–318. 26. Kasuba R. Dynamic loads in normal and high contact ratio spur gearing. In International symposium on gearing and power transmission, Tokyo, 1981: 49–54. 27. Yakut R, Düzcükoglu H, Demirci MT. The load capacity of PC/ABS spur gears and investigation of gear damage. Archieves of Material Science and Engineering 2009; 40(1):41–46. 28. Amarnath M, Sujatha C, Swarnamani S. Detection and diagnosis of gear tooth wear through metallurgical and oil analysis. Tribology Online, Japenese Society of. Tribologist 2010; 5(10):102–110. 29. Xu H, Li H, Hu J, Wang S. A study on contact fatigue performance of nitride and TiN coated gears. Advances in Materials Science and Engineering 2013; 13(7article id) 1–7. 30. Hoskins TJ, Dearn KD, Chen YK, Kukereka SN. The wear of PEEK in rolling-sliding contact — simulation of polymer gear applications. Wear 2014; 309(1–2):35–42. 31. Van Melick IHGH. Tooth bending effects in plastic spur gears influence on load sharing, stresses and wear. Gear Technology 2007; 3(10):58–66. 32. Errichello RL. Morphology of micropitting. Gear technology 2012; 4:74–81.
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