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Design of asymmetric normal contact ratio spur gear drive through direct design to enhance the load carrying capacity
Mechanism and Machine Theory 95 (2016) 22–34
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Design of asymmetric normal contact ratio spur gear drive through direct design to enhance the load carrying capacity P. Marimuthu, G. Muthuveerappan ⁎ Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 36, India
a r t i c l e
i n f o
Article history: Received 21 February 2015 Received in revised form 7 July 2015 Accepted 19 August 2015 Available online xxxx Keywords: Asymmetric spur gear Fillet stress Contact stress Direct design Load sharing ratio Non-dimensional stresses
a b s t r a c t Maximum fillet and contact stresses of asymmetric normal contact ratio spur gears designed by direct design method are evaluated based on the load sharing ratio, using finite element method. For a direct gear design, area of existence diagrams are developed for known input gear parameters such as number of teeth, coefficient of asymmetry, top land thickness coefficient and drive side contact ratio. A unique Ansys parametric design language code is developed to find the load sharing ratio, maximum fillet and contact stresses. The fillet stress is calculated in terms of non-dimensional stress. The influence of gear drive parameters such as drive and coast side pressure angles, top-land thickness coefficients, contact ratio, coefficient of asymmetry, gear ratio and teeth number on load carrying capacity has been studied extensively on non-dimensional fillet stress and maximum contact stress and compared with that of the conventionally designed gears. Through parametric study, suitable suggestions are made for the design of asymmetric gear drive for an enhanced load carrying capacity with constant contact ratio and constant drive side pressure angle separately. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Conventional gear design is done based on a standard rack cutter, whereas in a Direct Gear Design® approach the gear tooth profile is defined without using the rack cutter parameters like module, addendum and pressure angle. The load carrying capacity of gear drives is limited by fillet and contact strength, but there are many possible ways to increase it. One such solution is modification of gear geometry with asymmetric profiles, which find wider applications as in wind mill, helicopter main gearbox and turbo-prop engine drives. In the early development of asymmetric gear drives, low pressure angle side was considered as drive side and high pressure angle side as coast side. But, the measured contact stress was found to be the same as in standard gear drives [1]. According to Kapelevich [2], tool characteristics and manufacturing operations are secondary for direct design, which makes this approach a feasible one. Deng et al. [3] estimated load sharing ratio (LSR) of asymmetric gear for different pressure angles based on bending stiffness method. Litvin et al. [4] proposed a modified gear geometry with crowned pinion capable of reduced transmission error, and bending and contact stresses for asymmetric gears with load applied at high pressure angle side. Muni et al. [5] conducted stress analysis and found the optimum profile shift for balanced fillet strength of asymmetric normal contact ratio spur gear through direct gear design. Cavdar et al. [6] developed a computer program based on ISO method C to find the form and stress concentration factor for asymmetric teeth and justified that, fillet stress decreases due to increase in pressure angle on drive side. Spitas et al. [7] normalized the maximum fillet stress in terms of non-dimensional stress. Costopoulos and Spitas [8] and Pedersen [9] developed asymmetric gear to enhance the load carrying capacity. Li [10] investigated the influence of addendum height on contact and bending stresses and reported that ⁎ Corresponding author. Tel.: +91 94434 07580; fax: +91 44 2257 4652. E-mail addresses: [email protected] (P. Marimuthu), [email protected] (G. Muthuveerappan).
http://dx.doi.org/10.1016/j.mechmachtheory.2015.08.013 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
P. Marimuthu, G. Muthuveerappan / Mechanism and Machine Theory 95 (2016) 22–34
Nomenclature center distance (mm) working center distance (mm) cutter tip radius (mm) face width (mm) base circle diameter (mm) Young's modulus (GPa) normal load (N) gear ratio coefficient of asymmetry module (mm) module of rack cutter (mm) base pitch root circle radius (mm) working circle radius (mm) top land thickness coefficient Sa thickness of rack cutter (mm) Twr T1 and T2 interference limiting points coast side tip angle (degree) vc drive side tip angle (degree) vd x addendum modification factor (mm) X distance between a contact point and the pitch point at any instance (mm) z number of teeth addendum pressure angle of pinion (degree) αad1 addendum pressure angle of gear (degree) αad2 pressure angle at limiting circle (degree) αl pressure angle at base circle (degree) αb pressure angle (degree) αo working pressure angle (degree) αw coast side contact ratio εc drive side contact ratio εd γ Poisson's ratio (σt)max maximum fillet stress (MPa) (σH)max maximum contact stress (MPa) non-dimensional fillet stress σfu ao aw A b db E Fn i k m mr pb rf rw
Subscripts c coast side d drive side 1 pinion 2 gear Abbreviations AGMA American Gear Manufacturing Association DIN Deutsches Institut für Normung (German Institute for Standardization) ISO International Standard Organization FE finite element HPTC highest point of tooth contact HPSTC highest point of single tooth contact LPC length of path of contact LPTC lowest point of tooth contact LPSTC lowest point of single tooth contact MPCM multi pair contact model NCR normal contact ratio HCR high contact ratio APDL Ansys parametric design language 2D two dimentional 3D three dimentional
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Fig. 1. Area of existence diagrams. (a) Area of existence diagram for constant contact ratio. (b) Area of existence diagram for constant drive side pressure angle.
increase in addendum results in an increase in bending stress for normal contact ratio spur gear, whereas it is reduced in high contact ratio gears. Karpat [11] developed a dynamic model for asymmetric gear to reduce the dynamic load. Gears fail due to inadequate fillet strength at root and lower pitting resistance at contact surfaces. Hence, a reasonably accurate method is required to estimate the fillet and contact strength for high quality gears. The existing gear standards such as AGMA, ISO and DIN are applicable only for symmetric gears. It is also found that only a few analyses have been made in the past for fillet and contact stresses based on LSR for asymmetric spur gears. So, in the present study, a LSR based stress analysis on asymmetric NCR spur gears designed through direct and conventional method has been attempted. Further, to handle the gear and gear drive parameters in the asymmetric gear drive design for an enhanced load capacity, influence of parameters such as drive and coast side pressure angle, coefficient of asymmetry (k), gear ratio (i) number of teeth in the pinion and gear (z1, z2), top land thickness coefficient (sa) and drive side contact ratio (εd) were studied. The influence of these parameters on LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max in asymmetric NCR spur gears has also been investigated and suitable suggestions are made accordingly. 2. Area of existence for asymmetric normal contact ratio spur gears Two typical area of existence diagrams [2] for asymmetric NCR gear are shown in Fig. 1a and b, one for constant drive side contact ratio and the other for constant drive side pressure angle, from which a number of possible solutions of asymmetric gear pairs may be selected. These diagrams consist of five isograms for known input values are z1, z2, sa1, sa2, k and εd. Isogram Ai in Fig. 1a described by Eq. (3) is a line for constant contact ratio on drive side. Similarly isogram D in Fig. 1b is made for a constant pressure angle for drive side, from which the existence of numerous design solutions relating the addendum pressure angles of the pinion and gear (αad1, αad2) can be found. These are used respectively to develop a rack dimension either for a required constant contact ratio (Fig. 1a) or for a constant working pressure angle (Fig. 1b). Isograms for this study on asymmetric gear profiles are by fixing the following parameters viz. z1 = 40, z2 = 40, i = 1, k = 1.1, sa1 = 0.4 / z1, sa2 = 0.4 / z2. Isograms Ai (in Fig. 1a) are drawn such that, A1 for εd = 1.0, A2 for
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εd = 1.2, and A3 for εd = 1.4. Isograms Bi are drawn for the pinion and gear such that B1 for αfc1 = 0 and B2 for αfc2 = 0. Isograms Ci are drawn for the pinion and gear such that C1 for αad1 = αd, and C2 for αad2 = αd and isograms Di (in Fig. 1b) are drawn such that D1 for αd = 30°, D2 for αd = 35°, and D3 for αd = 40°. Isograms Ai, Bi, Ci and Di have been accomplished for the minimum requirements of gear mesh conditions such that 1. Isograms Ai represent the drive side contact ratio (εd). For continuous contact it should be greater than or equal to one (i.e., εd ≥ 1). 2. Isograms Bi are representing the interference free conditions. To avoid interference at the fillet, pressure angle at limiting circles of pinion and gear should be greater than or equal to zero (i.e., B1 for αlc1 ≥ 0 and αlc2 ≥ 0). 3. Isograms Ci are drawn based on minimum addendum condition. To meet this, addendum pressure angle of the pinion or gear should be greater than or equal to drive side pressure angle at pitch point (i.e., αad1 ≥ αd and αad2 ≥ αd). 4. Isograms Di are drawn for the working pressure angle at drive side as desired. Coefficient of asymmetry (k), top land thickness coefficient (sa), drive and coast side contact ratio (εd & εc) are found through Eqs. (1)–(4) [1]. k¼
dbc cos νc cos αbc ¼ ¼ ; and dbd cos νd cos αbd
sa ¼
ð1Þ
ð inv νd þ inv νc − inv αad − inv αac Þ 2 cos αad
εd ¼ z1 εc ¼ z1
ð2Þ
tan αad1 þ i tan αad2 −ð1 þ iÞ tan αod 2π tan αac1 þ i tan αac2 −ð1 þ iÞ tan αoc 2π
ð3Þ
ð4Þ
In order to avoid interference, Eqs. (5) and (6) should be satisfied [1]. tan αlc1 ¼ ð1 þ iÞ tan αoc −i tan αac2 ≥ 0
tan αlc2 ¼
ð1 þ iÞ tan αoc tan αac1 − ≥0 i i
ð5Þ
ð6Þ
Addendum modification factor (x) and cutter tip radii (A) are given by Eqs. (7) and (8). [1]
x¼
πmr −Twr 2 tanαwd þ tanαwc
ð7Þ
Fig. 2. Critical contact position on pinion asymmetric NCR spur gear pair.
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Fig. 3. Finite element model with boundary conditions for MPCM (z1 = 40, m = 1, i = 1, αad1 = 37.46°, αad2 = 37.4°, k = 1.1, sa1 = 0.4 / z1, sa2 = 0.4 / z2, εd = 1.4).
A¼
πmr 2 : 1 1 ð tanαwd þ tanαwc Þ− − cosαwd cosαwc
ð tanαwd þ tanαwc Þ ðrw −r f þ xÞ−
ð8Þ
2.1. Critical contact positions The critical contact positions are shown in Fig. 2. Using these, relevant radii rA, rB, rC and rD are derived from the gear geometry (Eqs. (9)–(13)). Critical loading point for NCR spur gear on maximum fillet stress is the highest point of single tooth contact (HPSTC). rA ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT1 AÞ2 þ rbd1 2
ð9Þ
rB ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT1 BÞ2 þ rbd1 2
ð10Þ
rC ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT1 CÞ2 þ rbd1 2
ð11Þ
Table. 1 Gear parameters for conventional and direct gear design methods. Parameters
Conventional design
Direct design
Module (m) mm Pinion teeth number (z1) Gear ratio (i) Working center distance (aw) mm Pressure angle (αo) degree Drive side contact ratio (εd) Addendum pressure angle of pinion (αad1) degree Addendum pressure angle of gear (αad2) degree Addendum (ha) mm Dedendum (hf) mm Top land thickness coefficient (ma) Asymmetric factor (k) Face width (b) mm Normal Force (Fn) N
1 40 1 40 25° 1.5 – – 1.00 1.25 – – 1.0 10
1 40 1 40 28.03° 1.5 33.04° 33.14° – – 0.25/z1 1.0 1.0 10
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Fig. 4. Comparison of LSR, non-dimensional fillet stress and maximum contact stress on gears of conventional and direct gear design.
rD ¼ ra1
ð12Þ
T1 T2 ¼ ao sinðαod Þ
ð13Þ
T1 A ¼ T1 D−AD
ð14Þ
T1 B ¼ T1 D−pbd
ð15Þ
T1 C ¼ T1 A þ pbd
ð16Þ
T1 D ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra1 2 −rbd1 2
ð17Þ
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Fig. 5. Effect of drive side pressure angle on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 40, i = 1, k = 1.1, αd = 33.24°, αd = 23.07°).
AD ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra1 2 −rbd1 2 þ ra2 2 −rbd2 2 −ao sinαod
ð18Þ
3. Finite element model of asymmetric NCR spur gear A finite element based multi pair contact model (MPCM) is one of the models that accounts contact stiffness and adjacent tooth effect while estimating the contact stress. Because of that, other models like single point loaded model (SPLM) and multi point loaded model (MPLM) are not preferred, even though it takes less computational time compared to that of MPCM. Hence, in this present work MPCM is used to estimate LSR, LSR based fillet and contact stresses using the two-dimensional (2D) three teeth full rim model (Fig. 3). A unique Ansys parametric design language (APDL) [12] code is developed to generate this 2D three teeth full rim model with following assumptions. 1. Plain strain condition is assumed in this study. 2. Zero backlashes between the pinion and gear model. 3. The gear tooth profile generated by APDL is free from errors like tooth profile error, radial run out error, tooth thickness error and pitch error. 4. Friction is neglected in the development of contact model.
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Fig. 6. Effect of coefficient of asymmetry on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 40, i = 1, εd = 1.4, k = 1.0, 1.1 and 1.2).
The material property used in this study are linear elastic, isotropic and homogeneous with an elastic constant, E = 210 GPa and Poisson's ratio, γ = 0.3. A six nodded triangular element (2D-PLANE82 [12]) with two degrees of freedom at each node is used to discretize the asymmetric NCR spur gear. Contact element (CONTA172) and target element (TARG169) are used to establish surface to surface contact between pinion and gear. As far as the boundary conditions are concerned, inner periphery of the gear is radially restrained in all directions and for a normal force of 10 N an equivalent torque is applied at inner periphery of the pinion. Augmented Lagrangian contact algorithm is used for contact analysis. The element size of fillet (0.02 mm) and contact (0.002 mm) region is fixed based on the convergence study. Rotation angle is calculated for one complete mesh cycle. A contact force (Fni) at any instantaneous position is obtained from nonlinear contact analysis for a normal force (Fn). After estimating a contact force, LSR is calculated at every contact position using Eq. (19). LSR ¼
Fni Fn
ð19Þ
4. Non-dimensional fillet stress The LSR based maximum fillet stress ((σt)max) and contact stress ((σH)max) are estimated using finite element analysis for a normal force (Fn) of 10 N with unit face width (b = 1 mm). The fillet stress can also be calculated in non-dimensional with normal loading and related to the maximum fillet stress using Eq. (20) [7].
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Fig. 7. Effect of gear ratio on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 20, i = 1, 2, 3, εd = 1.4, i = 1, 2, 3).
ðσ Þ σ fu ¼ t max Fn bm
ð20Þ
where, σfu is the non-dimensional fillet stress (no units). 5. Comparison between conventional and direct design symmetric NCR spur gears A comparison is made between gears of same size designed through conventional and direct gear design method. The details of parameters used in gear design are given in Table. 1. The values of LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max are plotted against contact position as shown in Fig. 4a–c. Fig. 4a shows the effect of LSR on the conventional and direct gear design approach. The values of LSR for conventional and direct gear design method are almost same. However, other parameters influence more on reduction of σfu. σfu for a load at HPSTC is lesser by 8.4% and (σH)max for a contact at a pitch point is lesser by 5.49% in the direct design NCR spur gear compared with that of the conventional NCR spur gear. Hence, this parametric study is focused on gears of direct design.
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Fig. 8. Effect of teeth number in pinion and gear on LSR, non-dimensional fillet and maximum contact stresses (m = 1, z1/z2 = 20/20, 60/60, 100/100, i = 1, k = 1.1, εd = 1.4).
6. Parametric study of direct design asymmetric NCR spur gear drives 6.1. Comparison between load at low and high pressure angles The effects on LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max for a load at low and high pressure angle sides are shown in Fig. 5. Due to the load at HPSTC, σfu for load at high pressure angle side is approximately 8.23% lower than that of load at a low pressure angle for any selected pair of gear and pinion (Fig. 5b). Also (σH)max is approximately 15.23% lower for load at a high pressure angle compared to that of the lower pressure angle (Fig. 5c). Hence, in this parametric analysis, high pressure angle side is considered as drive side.
6.2. Influence of coefficient of asymmetry The influence of coefficient of asymmetry on LSR, non-dimensional fillet stress (σfu) and maximum contact stress, (σH)max are plotted against the contact position (for gear parameters m = 1, z1 = 40, i = 1, sa1 = 0.4 / z1, sa2 = 0.4 / z2, k = 1.0, 1.1 and 1.2) are shown in Fig. 6a–c. σfu and (σH)max for a load at critical loading point HPSTC and pitch point decreases with increase in coefficient of asymmetry respectively (Fig. 6b and c).
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P. Marimuthu, G. Muthuveerappan / Mechanism and Machine Theory 95 (2016) 22–34
(a)
vs. contact position
(c)
(b)
H)max
fu vs.
contact position
vs. contact position
Fig. 9. Effect of top land thickness coefficient on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 40, i = 1, k = 1.1, εd = 1.4).
6.3. Influence of gear ratio The effect of change in gear ratio on LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max are shown in Fig. 7. For this study z1 = 20 is kept constant and z2 is varied for different gear ratios. With an increase in gear ratio of the pinion, the moment arm decreases resulting in a decrease in σfu (Fig. 7b). (σH)max decreases due to an increase in gear ratio. This is because an increase in gear ratio will increase the radius of curvature. However, as the gear ratio increases, LPSTC of the pinion moves away from pitch point and as a result equivalent radius of curvature decreases, hence the contact stress is higher at LPSTC as compared to HPSTC (Fig. 7c).
6.4. Influence of teeth number The respective LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max are estimated for different numbers of teeth in the pinion and gear (z1 = z2 = 20, 60, 100 and module remain same) are shown in Fig. 8. σfu decreases marginally for a load at HPSTC (Fig. 8b). But (σH)max reduces drastically as the radius of curvature increases at pitch point due to increase in teeth number (Fig. 8c).
6.5. Influence of top land thickness coefficient Fig. 9 shows the influence of top land thickness coefficient (sa) on LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max. σfu decreases with an increase in top land thickness coefficient (Fig. 9b) for a load at HPSTC. However, (σH)max for contact at pitch point is the same for different top land thickness coefficients (Fig. 9c).
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Fig. 10. Effect of drive side contact ratio on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 40, i = 1, k = 1.1, εd = 1.4, 1.6, 1.8).
6.6. Influence of contact ratio The effects of contact ratio on LSR, non-dimensional fillet stress (σfu) and maximum contact stress, (σH)max are shown in Fig. 10. σfu and (σH)max increases due to an increase in drive side contact ratio (Fig. 10a–c) for a load at critical loading point HPSTC. It is attributed to decrease in pressure angle which is an effect of increase in drive side contact ratio.
7. Conclusions In this study, LSR, maximum fillet and contact stresses have been estimated. Non-dimensional fillet stress and maximum contact stresses are determined for a one mesh cycle. The following observations are made from this parametric study. It is also concluded from this parametric study on such asymmetric NCR gear drive that for a given drive side contact ratio (εd), as an increase in i) load side pressure angle (αd), ii) coefficient of asymmetry (k), iii) gear ratio (i), iv) teeth number (z) and v) top land thickness coefficient (sa) which has resulted in reduction of stresses (non-dimensional fillet stress, σfu and maximum contact stress, (σH)max). The pinion and gear are designed accordingly to make use of these parameters, to enhance the load carrying capacity of gear drive. It is also preferred to have higher pressure angle at the load side of asymmetric teeth. However, smaller contact ratio gear drives (εd mim = 1.1) are preferred than the higher one, to enhance the load carrying capacity.
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In a similar way, for a design of such gear drive with constant drive side pressure angle, outcome of the parametric study can be accordingly considered to suit the required applications and to enhance the load carrying capacity.
References [1] D.G. Francesco, S. Marini, Structural analysis of asymmetrical teeth: reduction of size and weight, Gear Technol. 14 (1997) 47–51. [2] A.L. Kapelevich, Geometry and design of involute spur gears with asymmetric teeth, Mech. Mach. Theory 35 (2000) 117–130. [3] G. Deng, T. Nakanishi, K. Inoue, Fillet load capacity enhancement using and asymmetric tooth profile (1st report, influences of pressure angle on tooth root stress and fillet stiffness), JSME 46 (2003) 1171–1177. [4] F.L. Litvin, Q. Lian, A.L. 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. [5] D.V. Muni, V. Senthil Kumar, G. Muthuveerappan, Optimization of asymmetric spur gear derives for maximum fillet strength using direct gear design method, Mech. Based Des. Struct. Mach. 35 (2007) 127–145. [6] K. Cavdar, F. Karpat, F.C. Babalik, Computer aided analysis of fillet strength of involute spur gears with asymmetric profile, ASME J Mech. Des. 127 (2005) 477–484. [7] V.A. Spitas, T.N. Costopoulos, C.A. Spitas, Optimum gear tooth geometry for minimum fillet stresses using BEM and experimental verification with photoelasticity, J. Mech. Des. 128 (2006) 1159–1164. [8] T. Costopoulos, V. Spitas, Reduction of gear fillet stresses by using one-sided involute asymmetric teeth, Mech. Mach. Theory 44 (2009) 1524–1534. [9] N.L. Pedersen, Improving bending stress in spur gears using asymmetric gears and shape optimization, Mech. Mach. Theory 45 (2010) 1707–1720. [10] L. Shuting, Effect of addendum on contact strength, bending strength and basic performance parameters of a pair of spur gears, Mech. Mach. Theory 43 (2008) 1557–1584. [11] F. Karpat, S. Ekwaro-Osire, K. Cavdar, F.C. Babalik, Dynamic analysis of involute spur gears with asymmetric teeth, Int. J. Mech. Sci. 50 (2008) 1598–1610. [12] ANSYS 12.1 User Manual.
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Design of asymmetric normal contact ratio spur gear drive through direct design to enhance the load carrying capacity P. Marimuthu, G. Muthuveerappan ⁎ Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 36, India
a r t i c l e
i n f o
Article history: Received 21 February 2015 Received in revised form 7 July 2015 Accepted 19 August 2015 Available online xxxx Keywords: Asymmetric spur gear Fillet stress Contact stress Direct design Load sharing ratio Non-dimensional stresses
a b s t r a c t Maximum fillet and contact stresses of asymmetric normal contact ratio spur gears designed by direct design method are evaluated based on the load sharing ratio, using finite element method. For a direct gear design, area of existence diagrams are developed for known input gear parameters such as number of teeth, coefficient of asymmetry, top land thickness coefficient and drive side contact ratio. A unique Ansys parametric design language code is developed to find the load sharing ratio, maximum fillet and contact stresses. The fillet stress is calculated in terms of non-dimensional stress. The influence of gear drive parameters such as drive and coast side pressure angles, top-land thickness coefficients, contact ratio, coefficient of asymmetry, gear ratio and teeth number on load carrying capacity has been studied extensively on non-dimensional fillet stress and maximum contact stress and compared with that of the conventionally designed gears. Through parametric study, suitable suggestions are made for the design of asymmetric gear drive for an enhanced load carrying capacity with constant contact ratio and constant drive side pressure angle separately. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Conventional gear design is done based on a standard rack cutter, whereas in a Direct Gear Design® approach the gear tooth profile is defined without using the rack cutter parameters like module, addendum and pressure angle. The load carrying capacity of gear drives is limited by fillet and contact strength, but there are many possible ways to increase it. One such solution is modification of gear geometry with asymmetric profiles, which find wider applications as in wind mill, helicopter main gearbox and turbo-prop engine drives. In the early development of asymmetric gear drives, low pressure angle side was considered as drive side and high pressure angle side as coast side. But, the measured contact stress was found to be the same as in standard gear drives [1]. According to Kapelevich [2], tool characteristics and manufacturing operations are secondary for direct design, which makes this approach a feasible one. Deng et al. [3] estimated load sharing ratio (LSR) of asymmetric gear for different pressure angles based on bending stiffness method. Litvin et al. [4] proposed a modified gear geometry with crowned pinion capable of reduced transmission error, and bending and contact stresses for asymmetric gears with load applied at high pressure angle side. Muni et al. [5] conducted stress analysis and found the optimum profile shift for balanced fillet strength of asymmetric normal contact ratio spur gear through direct gear design. Cavdar et al. [6] developed a computer program based on ISO method C to find the form and stress concentration factor for asymmetric teeth and justified that, fillet stress decreases due to increase in pressure angle on drive side. Spitas et al. [7] normalized the maximum fillet stress in terms of non-dimensional stress. Costopoulos and Spitas [8] and Pedersen [9] developed asymmetric gear to enhance the load carrying capacity. Li [10] investigated the influence of addendum height on contact and bending stresses and reported that ⁎ Corresponding author. Tel.: +91 94434 07580; fax: +91 44 2257 4652. E-mail addresses: [email protected] (P. Marimuthu), [email protected] (G. Muthuveerappan).
http://dx.doi.org/10.1016/j.mechmachtheory.2015.08.013 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
P. Marimuthu, G. Muthuveerappan / Mechanism and Machine Theory 95 (2016) 22–34
Nomenclature center distance (mm) working center distance (mm) cutter tip radius (mm) face width (mm) base circle diameter (mm) Young's modulus (GPa) normal load (N) gear ratio coefficient of asymmetry module (mm) module of rack cutter (mm) base pitch root circle radius (mm) working circle radius (mm) top land thickness coefficient Sa thickness of rack cutter (mm) Twr T1 and T2 interference limiting points coast side tip angle (degree) vc drive side tip angle (degree) vd x addendum modification factor (mm) X distance between a contact point and the pitch point at any instance (mm) z number of teeth addendum pressure angle of pinion (degree) αad1 addendum pressure angle of gear (degree) αad2 pressure angle at limiting circle (degree) αl pressure angle at base circle (degree) αb pressure angle (degree) αo working pressure angle (degree) αw coast side contact ratio εc drive side contact ratio εd γ Poisson's ratio (σt)max maximum fillet stress (MPa) (σH)max maximum contact stress (MPa) non-dimensional fillet stress σfu ao aw A b db E Fn i k m mr pb rf rw
Subscripts c coast side d drive side 1 pinion 2 gear Abbreviations AGMA American Gear Manufacturing Association DIN Deutsches Institut für Normung (German Institute for Standardization) ISO International Standard Organization FE finite element HPTC highest point of tooth contact HPSTC highest point of single tooth contact LPC length of path of contact LPTC lowest point of tooth contact LPSTC lowest point of single tooth contact MPCM multi pair contact model NCR normal contact ratio HCR high contact ratio APDL Ansys parametric design language 2D two dimentional 3D three dimentional
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Fig. 1. Area of existence diagrams. (a) Area of existence diagram for constant contact ratio. (b) Area of existence diagram for constant drive side pressure angle.
increase in addendum results in an increase in bending stress for normal contact ratio spur gear, whereas it is reduced in high contact ratio gears. Karpat [11] developed a dynamic model for asymmetric gear to reduce the dynamic load. Gears fail due to inadequate fillet strength at root and lower pitting resistance at contact surfaces. Hence, a reasonably accurate method is required to estimate the fillet and contact strength for high quality gears. The existing gear standards such as AGMA, ISO and DIN are applicable only for symmetric gears. It is also found that only a few analyses have been made in the past for fillet and contact stresses based on LSR for asymmetric spur gears. So, in the present study, a LSR based stress analysis on asymmetric NCR spur gears designed through direct and conventional method has been attempted. Further, to handle the gear and gear drive parameters in the asymmetric gear drive design for an enhanced load capacity, influence of parameters such as drive and coast side pressure angle, coefficient of asymmetry (k), gear ratio (i) number of teeth in the pinion and gear (z1, z2), top land thickness coefficient (sa) and drive side contact ratio (εd) were studied. The influence of these parameters on LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max in asymmetric NCR spur gears has also been investigated and suitable suggestions are made accordingly. 2. Area of existence for asymmetric normal contact ratio spur gears Two typical area of existence diagrams [2] for asymmetric NCR gear are shown in Fig. 1a and b, one for constant drive side contact ratio and the other for constant drive side pressure angle, from which a number of possible solutions of asymmetric gear pairs may be selected. These diagrams consist of five isograms for known input values are z1, z2, sa1, sa2, k and εd. Isogram Ai in Fig. 1a described by Eq. (3) is a line for constant contact ratio on drive side. Similarly isogram D in Fig. 1b is made for a constant pressure angle for drive side, from which the existence of numerous design solutions relating the addendum pressure angles of the pinion and gear (αad1, αad2) can be found. These are used respectively to develop a rack dimension either for a required constant contact ratio (Fig. 1a) or for a constant working pressure angle (Fig. 1b). Isograms for this study on asymmetric gear profiles are by fixing the following parameters viz. z1 = 40, z2 = 40, i = 1, k = 1.1, sa1 = 0.4 / z1, sa2 = 0.4 / z2. Isograms Ai (in Fig. 1a) are drawn such that, A1 for εd = 1.0, A2 for
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εd = 1.2, and A3 for εd = 1.4. Isograms Bi are drawn for the pinion and gear such that B1 for αfc1 = 0 and B2 for αfc2 = 0. Isograms Ci are drawn for the pinion and gear such that C1 for αad1 = αd, and C2 for αad2 = αd and isograms Di (in Fig. 1b) are drawn such that D1 for αd = 30°, D2 for αd = 35°, and D3 for αd = 40°. Isograms Ai, Bi, Ci and Di have been accomplished for the minimum requirements of gear mesh conditions such that 1. Isograms Ai represent the drive side contact ratio (εd). For continuous contact it should be greater than or equal to one (i.e., εd ≥ 1). 2. Isograms Bi are representing the interference free conditions. To avoid interference at the fillet, pressure angle at limiting circles of pinion and gear should be greater than or equal to zero (i.e., B1 for αlc1 ≥ 0 and αlc2 ≥ 0). 3. Isograms Ci are drawn based on minimum addendum condition. To meet this, addendum pressure angle of the pinion or gear should be greater than or equal to drive side pressure angle at pitch point (i.e., αad1 ≥ αd and αad2 ≥ αd). 4. Isograms Di are drawn for the working pressure angle at drive side as desired. Coefficient of asymmetry (k), top land thickness coefficient (sa), drive and coast side contact ratio (εd & εc) are found through Eqs. (1)–(4) [1]. k¼
dbc cos νc cos αbc ¼ ¼ ; and dbd cos νd cos αbd
sa ¼
ð1Þ
ð inv νd þ inv νc − inv αad − inv αac Þ 2 cos αad
εd ¼ z1 εc ¼ z1
ð2Þ
tan αad1 þ i tan αad2 −ð1 þ iÞ tan αod 2π tan αac1 þ i tan αac2 −ð1 þ iÞ tan αoc 2π
ð3Þ
ð4Þ
In order to avoid interference, Eqs. (5) and (6) should be satisfied [1]. tan αlc1 ¼ ð1 þ iÞ tan αoc −i tan αac2 ≥ 0
tan αlc2 ¼
ð1 þ iÞ tan αoc tan αac1 − ≥0 i i
ð5Þ
ð6Þ
Addendum modification factor (x) and cutter tip radii (A) are given by Eqs. (7) and (8). [1]
x¼
πmr −Twr 2 tanαwd þ tanαwc
ð7Þ
Fig. 2. Critical contact position on pinion asymmetric NCR spur gear pair.
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Fig. 3. Finite element model with boundary conditions for MPCM (z1 = 40, m = 1, i = 1, αad1 = 37.46°, αad2 = 37.4°, k = 1.1, sa1 = 0.4 / z1, sa2 = 0.4 / z2, εd = 1.4).
A¼
πmr 2 : 1 1 ð tanαwd þ tanαwc Þ− − cosαwd cosαwc
ð tanαwd þ tanαwc Þ ðrw −r f þ xÞ−
ð8Þ
2.1. Critical contact positions The critical contact positions are shown in Fig. 2. Using these, relevant radii rA, rB, rC and rD are derived from the gear geometry (Eqs. (9)–(13)). Critical loading point for NCR spur gear on maximum fillet stress is the highest point of single tooth contact (HPSTC). rA ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT1 AÞ2 þ rbd1 2
ð9Þ
rB ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT1 BÞ2 þ rbd1 2
ð10Þ
rC ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT1 CÞ2 þ rbd1 2
ð11Þ
Table. 1 Gear parameters for conventional and direct gear design methods. Parameters
Conventional design
Direct design
Module (m) mm Pinion teeth number (z1) Gear ratio (i) Working center distance (aw) mm Pressure angle (αo) degree Drive side contact ratio (εd) Addendum pressure angle of pinion (αad1) degree Addendum pressure angle of gear (αad2) degree Addendum (ha) mm Dedendum (hf) mm Top land thickness coefficient (ma) Asymmetric factor (k) Face width (b) mm Normal Force (Fn) N
1 40 1 40 25° 1.5 – – 1.00 1.25 – – 1.0 10
1 40 1 40 28.03° 1.5 33.04° 33.14° – – 0.25/z1 1.0 1.0 10
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Fig. 4. Comparison of LSR, non-dimensional fillet stress and maximum contact stress on gears of conventional and direct gear design.
rD ¼ ra1
ð12Þ
T1 T2 ¼ ao sinðαod Þ
ð13Þ
T1 A ¼ T1 D−AD
ð14Þ
T1 B ¼ T1 D−pbd
ð15Þ
T1 C ¼ T1 A þ pbd
ð16Þ
T1 D ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra1 2 −rbd1 2
ð17Þ
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Fig. 5. Effect of drive side pressure angle on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 40, i = 1, k = 1.1, αd = 33.24°, αd = 23.07°).
AD ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra1 2 −rbd1 2 þ ra2 2 −rbd2 2 −ao sinαod
ð18Þ
3. Finite element model of asymmetric NCR spur gear A finite element based multi pair contact model (MPCM) is one of the models that accounts contact stiffness and adjacent tooth effect while estimating the contact stress. Because of that, other models like single point loaded model (SPLM) and multi point loaded model (MPLM) are not preferred, even though it takes less computational time compared to that of MPCM. Hence, in this present work MPCM is used to estimate LSR, LSR based fillet and contact stresses using the two-dimensional (2D) three teeth full rim model (Fig. 3). A unique Ansys parametric design language (APDL) [12] code is developed to generate this 2D three teeth full rim model with following assumptions. 1. Plain strain condition is assumed in this study. 2. Zero backlashes between the pinion and gear model. 3. The gear tooth profile generated by APDL is free from errors like tooth profile error, radial run out error, tooth thickness error and pitch error. 4. Friction is neglected in the development of contact model.
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Fig. 6. Effect of coefficient of asymmetry on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 40, i = 1, εd = 1.4, k = 1.0, 1.1 and 1.2).
The material property used in this study are linear elastic, isotropic and homogeneous with an elastic constant, E = 210 GPa and Poisson's ratio, γ = 0.3. A six nodded triangular element (2D-PLANE82 [12]) with two degrees of freedom at each node is used to discretize the asymmetric NCR spur gear. Contact element (CONTA172) and target element (TARG169) are used to establish surface to surface contact between pinion and gear. As far as the boundary conditions are concerned, inner periphery of the gear is radially restrained in all directions and for a normal force of 10 N an equivalent torque is applied at inner periphery of the pinion. Augmented Lagrangian contact algorithm is used for contact analysis. The element size of fillet (0.02 mm) and contact (0.002 mm) region is fixed based on the convergence study. Rotation angle is calculated for one complete mesh cycle. A contact force (Fni) at any instantaneous position is obtained from nonlinear contact analysis for a normal force (Fn). After estimating a contact force, LSR is calculated at every contact position using Eq. (19). LSR ¼
Fni Fn
ð19Þ
4. Non-dimensional fillet stress The LSR based maximum fillet stress ((σt)max) and contact stress ((σH)max) are estimated using finite element analysis for a normal force (Fn) of 10 N with unit face width (b = 1 mm). The fillet stress can also be calculated in non-dimensional with normal loading and related to the maximum fillet stress using Eq. (20) [7].
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Fig. 7. Effect of gear ratio on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 20, i = 1, 2, 3, εd = 1.4, i = 1, 2, 3).
ðσ Þ σ fu ¼ t max Fn bm
ð20Þ
where, σfu is the non-dimensional fillet stress (no units). 5. Comparison between conventional and direct design symmetric NCR spur gears A comparison is made between gears of same size designed through conventional and direct gear design method. The details of parameters used in gear design are given in Table. 1. The values of LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max are plotted against contact position as shown in Fig. 4a–c. Fig. 4a shows the effect of LSR on the conventional and direct gear design approach. The values of LSR for conventional and direct gear design method are almost same. However, other parameters influence more on reduction of σfu. σfu for a load at HPSTC is lesser by 8.4% and (σH)max for a contact at a pitch point is lesser by 5.49% in the direct design NCR spur gear compared with that of the conventional NCR spur gear. Hence, this parametric study is focused on gears of direct design.
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Fig. 8. Effect of teeth number in pinion and gear on LSR, non-dimensional fillet and maximum contact stresses (m = 1, z1/z2 = 20/20, 60/60, 100/100, i = 1, k = 1.1, εd = 1.4).
6. Parametric study of direct design asymmetric NCR spur gear drives 6.1. Comparison between load at low and high pressure angles The effects on LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max for a load at low and high pressure angle sides are shown in Fig. 5. Due to the load at HPSTC, σfu for load at high pressure angle side is approximately 8.23% lower than that of load at a low pressure angle for any selected pair of gear and pinion (Fig. 5b). Also (σH)max is approximately 15.23% lower for load at a high pressure angle compared to that of the lower pressure angle (Fig. 5c). Hence, in this parametric analysis, high pressure angle side is considered as drive side.
6.2. Influence of coefficient of asymmetry The influence of coefficient of asymmetry on LSR, non-dimensional fillet stress (σfu) and maximum contact stress, (σH)max are plotted against the contact position (for gear parameters m = 1, z1 = 40, i = 1, sa1 = 0.4 / z1, sa2 = 0.4 / z2, k = 1.0, 1.1 and 1.2) are shown in Fig. 6a–c. σfu and (σH)max for a load at critical loading point HPSTC and pitch point decreases with increase in coefficient of asymmetry respectively (Fig. 6b and c).
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(a)
vs. contact position
(c)
(b)
H)max
fu vs.
contact position
vs. contact position
Fig. 9. Effect of top land thickness coefficient on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 40, i = 1, k = 1.1, εd = 1.4).
6.3. Influence of gear ratio The effect of change in gear ratio on LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max are shown in Fig. 7. For this study z1 = 20 is kept constant and z2 is varied for different gear ratios. With an increase in gear ratio of the pinion, the moment arm decreases resulting in a decrease in σfu (Fig. 7b). (σH)max decreases due to an increase in gear ratio. This is because an increase in gear ratio will increase the radius of curvature. However, as the gear ratio increases, LPSTC of the pinion moves away from pitch point and as a result equivalent radius of curvature decreases, hence the contact stress is higher at LPSTC as compared to HPSTC (Fig. 7c).
6.4. Influence of teeth number The respective LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max are estimated for different numbers of teeth in the pinion and gear (z1 = z2 = 20, 60, 100 and module remain same) are shown in Fig. 8. σfu decreases marginally for a load at HPSTC (Fig. 8b). But (σH)max reduces drastically as the radius of curvature increases at pitch point due to increase in teeth number (Fig. 8c).
6.5. Influence of top land thickness coefficient Fig. 9 shows the influence of top land thickness coefficient (sa) on LSR, non-dimensional fillet stress (σfu) and maximum contact stress (σH)max. σfu decreases with an increase in top land thickness coefficient (Fig. 9b) for a load at HPSTC. However, (σH)max for contact at pitch point is the same for different top land thickness coefficients (Fig. 9c).
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Fig. 10. Effect of drive side contact ratio on LSR, non-dimensional fillet stress and maximum contact stress (m = 1, z1 = 40, i = 1, k = 1.1, εd = 1.4, 1.6, 1.8).
6.6. Influence of contact ratio The effects of contact ratio on LSR, non-dimensional fillet stress (σfu) and maximum contact stress, (σH)max are shown in Fig. 10. σfu and (σH)max increases due to an increase in drive side contact ratio (Fig. 10a–c) for a load at critical loading point HPSTC. It is attributed to decrease in pressure angle which is an effect of increase in drive side contact ratio.
7. Conclusions In this study, LSR, maximum fillet and contact stresses have been estimated. Non-dimensional fillet stress and maximum contact stresses are determined for a one mesh cycle. The following observations are made from this parametric study. It is also concluded from this parametric study on such asymmetric NCR gear drive that for a given drive side contact ratio (εd), as an increase in i) load side pressure angle (αd), ii) coefficient of asymmetry (k), iii) gear ratio (i), iv) teeth number (z) and v) top land thickness coefficient (sa) which has resulted in reduction of stresses (non-dimensional fillet stress, σfu and maximum contact stress, (σH)max). The pinion and gear are designed accordingly to make use of these parameters, to enhance the load carrying capacity of gear drive. It is also preferred to have higher pressure angle at the load side of asymmetric teeth. However, smaller contact ratio gear drives (εd mim = 1.1) are preferred than the higher one, to enhance the load carrying capacity.
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In a similar way, for a design of such gear drive with constant drive side pressure angle, outcome of the parametric study can be accordingly considered to suit the required applications and to enhance the load carrying capacity.
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