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Investigation on the influence of work holding equipment errors on contact characteristics of face-hobbed hypoid gear
Mechanism and Machine Theory 138 (2019) 95–111
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
Investigation on the influence of work holding equipment errors on contact characteristics of face-hobbed hypoid gear Siyuan Liu, Chaosheng Song∗, Caichao Zhu, Chengcheng Liang, Xingyu Yang The State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400030, China
a r t i c l e
i n f o
Article history: Received 19 January 2019 Revised 23 March 2019 Accepted 29 March 2019
Keyword: Face-hobbed hypoid gear Work holding equipment error Geometric morphology Mesh characteristics
a b s t r a c t This paper proposed the accurate mathematical model considering work holding equipment errors based on the three-face cutter differing from typical commercial software. The simulated flow included manufacturing issues from industry applications have been established. Quasi-static loaded tooth contact analysis was used to discuss the contact characteristic and gear geometry with existence of radial and angular eccentric errors. In addition, the correspondence between transmission error and gear geometry was established. The method of separating the long wave (low frequency) and short wave (high frequency) from total transmission error has been proposed. Results show that the effect of angular eccentric error on gear geometry is more sensitive than that of radial eccentric error. The gear geometry deviations present periodic changes because of variation of pitch error. And radial eccentric error cannot influence the position and shape of contact pattern. On the contrary, with existence of angular eccentric error, the contact area moves to the heel side and then backward. Furthermore, the length of contact pattern is firstly decreased and then increased in one period. The transmission error is sensitivity for both two eccentric errors and the peak to peak value of transmission error increase obviously. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Hypoid gear has a wide utilization in the spatial angle transmission including axle transmission in the advantage of high contact ratio, high efficiency. With the rapid development of automobile industry, the demand of hypoid gear included the weight, smoothness of transmission, load capacity and life time were much higher than before. Currently, there have been many researches on deduction of processing mathematical model and tooth contact analysis for hypoid gear because of the complexity of gear geometric morphology [1–8]. But very little investigations have been recorded about discussing the impact of work holding equipment error on the mesh behavior of hypoid gear. Recent years, numerous researchers presented many studies about generated process of face-hobbed hypoid gear. Zhang, Litvin et al. [9] proposed a technique for computerized simulation and tangency of gears provided with real tooth surfaces based on the superimposition of theoretical tooth surface and deviations from heating treatment and lapping. Fan [10,11] established the precise analytical mesh model of face-hobbed hypoid gear pair to enhance tooth contact analysis (TCA) algorithm based on the method of Gleason face hobbing generator and proposed a technological process included designing, manufacturing and analysis of hypoid gear. Shih et al. [12–15] proposed a mathematical model of Oerlikon hypoid gear processing base on the computer numerical control (CNC) machine and transformed the mechanical processing setting ∗
Corresponding author. E-mail address: [email protected] (C. Song).
https://doi.org/10.1016/j.mechmachtheory.2019.03.042 0094-114X/© 2019 Elsevier Ltd. All rights reserved.
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into rotation and movement of three axes with Taylor polynomials, the mathematical model and manufacturing process was verified by experiment. Vimercati [16] aimed at an accurate geometrical hypoid gear model with face-hobbed method to propose an analytical description of the cutting process and generated cut based on the complex blade shapes included protuberance (TOPREM). Tsay et al. [17,18] presented an optimization procedure to find corrective machine-tool setting to minimize surface deviation of gear flank according to the mathematical model of hypoid gear. K. Kawasaki [19] proposed a method for remanufacturing pinion member of large-sized skew bevel gears using a CNC machining center and respecting an existing gear member. The effect of assembly errors on mesh behavior of hypoid gear is also a research hotspot. Lelkes et al. [19] developed the theoretical contact identification program to discuss the influence of each cutting parameter on longitudinal settings of contact patterns based on Klingelnberg’s Cyclo-Palloid System. Lim et al. [20] applied a quasi-static gear tooth contact analysis to study the effects of assembly errors on hypoid gear mesh behavior and predicted the assembly errors to optimize the gear dynamic response. Zhuo et al. [21] investigated the impact of assembly of hypoid gear with coaxially deviation on location of contact path, length of contact ellipse and long wave of transmission error based on the quasi-static multi tooth contact analysis (MTCA). G. Ignacio [22] aimed at getting favorable conditions of meshing to proposed a computerized generation model of spiral bevel gears based on the cyclopalloid system to optimize the basic machine-tool settings. K. Kawasaki [23] based on simultaneous generations of tooth surfaces to investigate the effect of cutting edge profile on meshing and contact of spiral bevel gears in the Klingelnberg cyclopalloid system analytically. V. Simon [24] applied computer aided tooth contact analysis for the investigation of the influence of misalignments of the mating members and tooth errors on mesh behavior. J. Tang [25] focused on low noise and high strength from industry applications to propose a new multi-step integrated analytical identification approach considered misalignments for hypoid gear. The above papers mostly focus on the deduction of hypoid gear machining model and tooth contact analysis of hypoid gear. Less of studies investigated the work holding equipment of machine. The presented commercial software cannot simulate the mesh behavior of hypoid gear with manufacturing issues. Therefore, discussing the impacts of work holding equipment error on the mesh behavior become more essential. This study mainly focused on the simulated flow of face-hobbed hypoid gear with radial and angular work holding equipment (WHE) errors based on the three-face cutter. The mathematical model and quasi-static finite element analysis model of the hypoid gear with WHE errors were proposed according to the actual manufacturing process. Then, the influences of WHE errors on gear flank geometrical morphology and mesh characteristics included the contact pattern, transmission error were investigated. Finally, the relationship between transmission error and gear geometry included pitch error and gear flank deviation has been established. 2. Mathematical model of face-hobbed hypoid gear with work holding equipment errors Traditional cradle machine tool include all manufacturing structures is shown in Fig. 1. The red dotted line represented the position of gear blank with work holding equipment errors. The WHE errors can be divided into radial eccentric and angular eccentric errors. The pinion can be considered as a shaft part, so it is possible for pinion to have radial eccentricity error. For wheel, it is common for radial eccentricity existing in manufacturing process because of the round dish structure. CE and δ ANG represent the quantity of the radial eccentric and angular eccentric errors, respectively. The eccentricity exists in the process of hypoid gear manufacturing more or less. Therefore, the mathematical model of hypoid gear with eccentric error from manufacturing perspective should be established for research. The mathematical model of manufacturing can be divided into two parts: three-face cutter and machine tool generation process. The mathematical model of three-face cutter head from manufacturing cutter structure is shown in the Fig. 2. The transfer matrixes of working segment of cutter can be represented as follows
f (Mc1 ) = Map M pq Mqr Mrs Msu
(1)
The transfer matrixes of arc segment of cutter can be represented as follows
f (Ma1 ) = Map M pq Mqr Mrs Msu Muv Mvw
(2)
where Map , Mpq , Mqr , Mrs , Msu , Muv , Mvw are the transfer matrixes from coordinate system Ow to Oa . α 0 represent the tool angle, RBHV and rBHV are the spheric radius of cutter working segment and arc segment, respectively. ζ is the regrind angle, ϑA is the rake angle. E0 is the radius of roll circle. Rac and Rav are the radius of pitch points in concave and convex flank, respectively. Rfc and Rfv are the radius of reference points in the concave and convex flank, respectively. δ A is the cutter direction angle and can be obtained by δA = asin(E0 /Ra(c,v) ). Based on the face-hobbed manufacturing process and method [26,27], the coordinate systems of gear manufacturing include WHE errors and tilt structure were proposed in Fig. 3. The transfer matrixes of whole gear cutting process can be obtained by
f (M2 ) = Mml Mlk Mk j M ji Mih Mhg Mg f M f e Med Mdc Mcb Mba
(3)
where Mml , Mlk , Mkj , Mji , Mih , Mhg , Mgf , Mfe , Med , Mdc , Mcb , Mba are the transfer matrixes from coordinate system Oa to Om . These transfer matrixes also contain eight parameters of machine tool processing. i and j represent the tile angle and swivel angle, respectively. Sr is the radial setting and q is the initial cradle angle. The vertical offset and machine root angle are
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Fig. 1. Traditional cradle machine tool with work holding equipment errors for hypoid gear.
Fig. 2. Mathematical model of three-face cutter.
represented by Em and γ m . A is synthetical horizontal (the distance between coordinate systems Oi and Oj ) and B is the sliding base. The transfer matrixes of radial eccentricity can be represented as follows
⎡
Mg f
1 ⎢0 =⎣ 0 0
0 1 0 0
0 0 1 0
⎤
0 CE ⎥ 0 ⎦ 1
(4)
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Fig. 3. Coordinate system for face-hobbed generator with WHE error.
The transfer matrixes of angular eccentricity are shown as follows
⎡
Mk j
cosδANG ⎢ sinδANG =⎣ 0 0
⎡
1 ⎢0 Mlk = ⎣ 0 0
0 1 0 0
−sinδANG cosδANG 0 0 0 0 1 0
0 0 1 0
⎤
0 0⎥ 0⎦ 1
(5)
⎤
Hd 0⎥ 0⎦ 1
(6)
The physical meaning of work holding equipment errors can be represented as Fig. 4. Where MC represents crossed point (CP) which means the intersection point between common vertical line and rotary axis. Hd is the distance between fixed point of pinion and crossed point (CP).
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Fig. 4. The relationship between work holding equipment with and without errors.
The representation of working segment is shown by
Ric (u, β , ϕ1 ) = f (M2 ) × f (Mc1 ) × Rc (u )
(7)
The expression of arc segment is given by
Ria (θ , β , ϕ1 ) = f (M2 ) × f (Ma1 ) × Ra (θ )
(8)
The working segment and arc segment are represented as follows
⎡
⎤
2RBHV sin(u/2RBHV )sin(α0 − u/2RBHV ) 0 ⎢ ⎥
Rc ( u ) = ⎢
⎥ ⎣2RBHV sin(u/2RBHV )cos(α0 − u/2RBHV )⎦ ⎡
⎤
(9)
1
rBHV cosθ 0 ⎥ ⎦ BHV sinθ 1
⎢ Ra ( θ ) = ⎣ r
(10)
The normal vector of gear flank can be obtained by
n pc (u, β , ϕ1 ) =
∂ Ric (u, β , ϕ1 ) ∂ Ric (u, β , ϕ1 ) ∂ Ric (u, β , ϕ1 ) ∂ Ric (u, β , ϕ1 ) × × ∂u ∂β ∂u ∂β
(11)
The normal vector of arc segment can be obtained by
n pa (θ , β , ϕ1 ) =
∂ Ria (θ , β , ϕ1 ) ∂ Ria (θ , β , ϕ1 ) ∂ Ria (θ , β , ϕ1 ) ∂ Ria (θ , β , ϕ1 ) × × ∂θ ∂β ∂θ ∂β
(12)
where u and θ are the variables of working segment and arc segment in the transverse section of cutter blade, respectively. β is rotation angle of cutter head. The mesh equation of working segment is shown as follows
f (u, β , ϕ1 ) = n pc × ν pc = n pc × (ϕ˙ 1 × ∂ r pc /∂ ϕ1 ) = 0
(13)
The mesh equation of arc segment is shown as follows
f (θ , β , ϕ1 ) = n pa × ν pa = n pa × (ϕ˙ 1 × ∂ r pa /∂ ϕ1 ) = 0
(14)
The relationship of these angles can be represented as follows
ϕc1 = z0 /z p × β
(15)
ϕc2 = 1/Ra × ϕ1
(16)
where ϕ 1 is the rotation angle of gear blank and can be solved by mesh equation. ϕ c1 is the angle of epicycloidal cradle rotation, ϕ c2 is the angle of generating cradle rotation, z0 is the number of blade groups, zp is the tooth number of generating gear, Ra is the roll ratio. The hypoid gear manufacturing can be regarded as the space trajectory of cutter tool acted on gear blank to cut the material, then the remainder of gear flank is the hypoid gear. As shown in Fig. 5, the red part is the space trajectory of
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Fig. 5. Mathematical model of hypoid gear with work holding equipment error.
Fig. 6. Flow chart of hypoid gear computerized modeling.
cutter tool. The mathematical and analytical models of hypoid gear pair with WHE errors can be established by moving the rotation center of gear blank with eccentricity. The flow chart of hypoid gear computerized modeling is shown in Fig. 6. The point sets of pinion and wheel can be obtained by MATLAB through the mesh equation. Inputting the point sets into the CREO, the space trajectory of cutter tool can be simulated by surface fitting. Furthermore, the manufacturing process can be simulated by Boolean operation of gear blank with WHE errors. Finally, the hypoid gear pair can be obtained by assembling the models of pinion and wheel according to the assembly parameters. 3. Hypoid gear mesh model The major parameters of hypoid gear and cutter tool are shown in Table. 1. The assembly and eccentric parameters are shown in Table. 2. The value of machine setting are shown in Table. 3. According to the parameters, the mesh model and finite element model of hypoid gear with WHE errors were established. The mesh model with pinion angular eccentric error and gear angular eccentric error are shown in the Fig. 7(a)-(b), respectively. There are four assembly parameters including offset E, shaft angle , axial position of pinion Dxp and axial position of gear Dxw for hypoid gear. The impacts of eccentricity on gear geometry and mesh behavior are periodic, the selected teeth should be enough for one period in mesh model. Therefore, the teeth number of wheel with radial and angular eccentricities are different.
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Fig. 7. Mesh model of hypoid gear.
Table 1 Parameter of hypoid gear and cutter.
Number of teeth Face width (mm) Mean normal module (mm) Mean spiral angle (°) Pitch angle (°) Number of blade groups Convex tool angle (°) Concave tool angle (°) Convex pitch point radius (mm) Concave pitch point radius (mm) Roll circle radius (mm)
Pinion
Wheel
11 38.6 3.4884 51.8225 23.6383 17 22.3059 20.1933 75.9341 76.0660 30.023
47 31.75 3.4884 29.2833 64.6445 18.9931 23.5060 75.9194 76.0806
101
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Fig. 8. Finite element mesh model of hypoid gear.
Fig. 9. Contact pattern (ABAQUS).
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Fig. 10. Contact pattern (KIMOS).
Fig. 11. Transmission error.
Table 2 Parameter of assembly and eccentric errors. Shaft angle (°)
90
Hypoid offset (mm) Axial position of pinion (mm) Axial position of wheel (mm) Radial eccentricity of gear (mm) Angular eccentricity of pinion (°)
38.1 109.52 60.55 0.02 0.05
For the eccentricity of pinion, the whole pinion and 11 teeth for wheel was selected in order to investigate the mesh characteristic in one period of pinion rotation. But for wheel, the whole pinion and 45 teeth of wheel were selected for one cycle of wheel rotation. The finite element mesh models in Fig. 8 were developed to investigate the influences of eccentric errors on mesh characteristics. For the simulation, the material used in gear pair is 17CrNiMo6 and the property is isotropic. The young’s modulus
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Fig. 12. Gear flank geometric morphology deviations for wheel with radial eccentric error.
and Poisson’s ratio of gear pair are 2090 0 0 Mpa and 0.298, respectively. The interaction constraint (tooth contact) was established between concave flank of pinion and convex flank of wheel. The kinematic couplings were created between the rotation points and the inner surfaces of gear blank to set boundary constraints. The torsional displacement was applied to pinion as input for the loaded tooth contact analysis and torque load 450 Nm was applied to wheel as output. In order to save the computational cost in whole process, the grid seeds of working gear flank are denser than those of gear blank. Because of the complexity of pinion flank, the tetrahedral mesh was selected for pinion and hexahedron mesh for gear, respectively.
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Fig. 13. Gear flank geometric morphology deviations for pinion with angular eccentric error. Table 3 Value of machine setting. Machine Setting Tile angle i Swivel angle j Radial setting Sr Initial cradle angle q Vertical offset Em Machine root angle γ m Horizontal A Sliding base B
Pinion
Wheel ◦
25.3716 −131.8117◦ 116.5702 52.1131◦ 40.5991 0.3457◦ 1.3084 28.7681
0◦ 0◦ 121.8758 38.3013◦ 0 64.6445◦ 0 0
For the model with angular eccentricity, 11 teeth were selected for gear in Fig. 8(a). The analysis of step length is 0.02, and there are 163,246 elements for the whole model. But for the model with radial eccentricity is different, in Fig. 8(b), 45 teeth were selected. The analysis of step length is 0.002 to guarantee the accuracy of calculation, and there are 547,295 elements for hypoid gear pair. The hypoid gear without eccentric error called master gear. The contact pattern without eccentric error from loaded tooth contact analysis (ABAQUS) and KIMOS are shown in Figs. 9 and 10. The shape and location of contact pattern from ABAQUS are close to KIMOS. From the Fig. 11(a) and (b), the shape and peak to peak value of transmission error under
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Fig. 14. Contact pattern of wheel with radial eccentric error.
LTCA (41 μm) are closed to KIMOS (60.9 μm). In general, the correctness of mathematical model and finite element model of hypoid gear pair were validated from the KIMOS. Using the proposed mathematical model and finite element model of the hypoid gear with WHE error, the influence of gear pairs with radial eccentric error and angular eccentric error on mesh behavior and gear flank geometry can be studied easily. 4. Gear flank geometric morphology deviation analysis Case 1: Radial eccentricity of wheel The hypoid gears are made by generation cutting processing, so the WHE error will influence the gear geometry directly. The comparison of deviations between wheel with WHE error and original wheel surface can be expressed in Fig. 12. In one cycle of wheel, the deviation value of toe side firstly decreased from positive to negative and then increased back to
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Fig. 15. Transmission error with wheel angular eccentric error.
Fig. 16. The comparison of transmission error between wheel with radial eccentric error and master gear.
positive. The radial error only influences the deviation value, furthermore, the pitch error has been changed. And the shape of gear flank is not effected too much. Case 2: Angular eccentricity of pinion The comparison of deviations between pinion with WHE error and original pinion surface can be represented in Fig. 13. In one cycle of pinion, the deviation of toe side firstly increase from −10.8μm (−11.8μm) to 41.6μm (50.8μm) and then decrease to 22.9μm (28.6μm). Furthermore, the deviation of heel side firstly increase from 17.7μm (23.9μm) to 63.2μm (52.0μm) and then decreased to −11.0μm (−15.3μm). Compared with radial eccentric error, the gear flank geometric morphology deviation and pitch error are more sensitive to angular eccentric error. 5. Contact behavior with loaded tooth contact analysis Case 1: Radial eccentricity of wheel As depicted in Fig. 14, 12 teeth (every four teeth) of wheel were selected to investigate the influence of wheel radial eccentricity on contact pattern and transmission error of hypoid gear. From the results, the location of contact pattern is in the heel side and top side of gear flank in one period. Compared with master gear pair, the location and shape have not changed. In Fig. 15, the red dash-dotted line represents the total transmission error with eccentricity in one meshing period of wheel. There are two components included high frequency and low frequency in the total transmission error. The low frequency component can be fitted by three orders Fourier series, the fitting curve called long wave and the separated high frequency called short wave. The total transmission error and long wave presented the periodic variation like the sinusoidal function. Compared with transmission error of standard gear pair in Fig. 16 and enlarged drawing Fig. 17, the short wave of transmission error shows the irregular fluctuations in one cycle because each geometry of tooth is different. The peak to peak value of transmission error increased with eccentric error according to Fig. 18. The first three orders of transmission error harmonics are given in Fig. 19. The transmission error harmonics decreased with the increasing of orders gradually. Case 2: Angular eccentricity of pinion In Fig. 20(a)–(f), 6 teeth (every two teeth) were selected to investigate the effect of angular eccentric error of pinion on the contact pattern. From the results, the location of contact pattern firstly moved to the heel side and then back to the original position in one cycle. Furthermore, the length of the contact pattern firstly decreased and then increased with the increasing of pinion rolling angle.
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Fig. 17. Enlarged drawing.
Fig. 18. The comparison of peak to peak value of transmission error between wheel with radial eccentric error and standard processing.
Fig. 19. First three orders of transmission error harmonics with wheel radial eccentric error.
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Fig. 20. Contact pattern of wheel with angular eccentric error.
Fig. 21. Transmission error with pinion angular eccentric error.
Fig. 22. The comparison of transmission error between pinion with angular eccentric error and master gear.
As shown in Fig. 21, the total transmission error and long wave firstly increased and then decreased in one cycle of pinion. The high frequency part of total transmission error vibrated along the short frequency. The comparison of transmission error (short wave) between pinion with angular eccentricity and master gear pair was depicted in the Fig. 22. Each single waveform of short wave was different because the distinctness of wheel geometry. The peak to peak value of transmission error was shown in Fig. 23, the peak to peak value increased with existence of angular eccentric error. The transmission error harmonics decreased with the orders increasing gradually in Fig. 24. According to the above results, the quasi-static loaded tooth contact analysis can be used to evaluate the gear geometry. As shown in Fig. 25, the transmission error contains whole information of gear geometry characteristic. The low frequency component can be fitted by Fourier series, the rest is the high frequency part. The low frequency (long wave) and high
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Fig. 23. The comparison of peak to peak value of transmission error between pinion with radial eccentric error and master gear.
Fig. 24. First three order of transmission error harmonics with pinion angular eccentric error.
Fig. 25. Flow chart of transmission error analysis.
frequency (short wave) represented the gear blank and gear flank geometry, respectively. If short wave is out of tolerance, there must exist the design or manufacturing issues on gear blank such as pitch error, the same as those of long wave. Therefore, the transmission error analysis can guide the design and manufacturing of hypoid gear. The above process discussed the drive side. This flow chart of analysis is suitable for both side. And for the coast side, the method of analysis is the same except the rotation direction of gear. 6. Conclusion In this paper, based on the three-face cutter, the accurate mathematical model and analytical mesh model of face-hobbed hypoid gear considered work holding equipment errors have been established. Differing from the typical commercial software, the simulated flow of hypoid gear included manufacturing issues from industry applications was proposed. This study also use the quasi-static loaded tooth contact analysis to investigate the impact of eccentric error on contact behavior and geometry deviation. The method which divided the transmission error into long wave (low frequency) and short wave (high frequency) was proposed. The correspondence between long/short wave and gear blank/flank was established. From the above, four specific findings can be obtained as follows
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(1) With existence of radial eccentric error, the deviation of toe side firstly decreases from positive value to negative value and then increases to positive value, and final back to original position in one rotation cycle. (2) The effect of angular eccentric error on the gear geometry is more sensitive than that of radial eccentricity. With angular eccentricity, the deviations of toe side firstly increase and then decrease, furthermore, the deviation of heel side firstly increase and then decrease. (3) Radial eccentricity of wheel does not affect the position and shape of contact pattern. And the eccentricity has strong effect on transmission error, the peak to peak value of transmission error is increasing. Compared with master gear, the shape of transmission error is the short wave (high frequency part) vibrates along the long wave (low frequency part) which is closed to sinusoidal function in one mesh cycle. (4) The position of contact pattern firstly moves to the heel side and then back to the original position with existence of angular eccentricity. Furthermore, the length of the contact pattern firstly decreased and then increased in one meshing period. With increasing of pinion rolling angle, the transmission error firstly increased and then decreased. The peak to peak value of transmission error is increasing. Acknowledgements The authors would like to thank the National Natural Science Foundation of China (No. 51775061 and No. 51575060), Fundamental Research Funds for the Central Universities (2018CDQYJX0012) and Key Research and Development Project of Chongqing Science and Technology Program (cstc2018jszx-cyztzxX0038). References [1] F.L. Litvin, Y. Gutman, Methods of synthesis and analysis for hypoid gear-drives of formate and helixform-Part 1, calculations for machine settings for member gear manufacture of the formate and helixform hypoid gears, J. Mech. 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Research paper
Investigation on the influence of work holding equipment errors on contact characteristics of face-hobbed hypoid gear Siyuan Liu, Chaosheng Song∗, Caichao Zhu, Chengcheng Liang, Xingyu Yang The State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400030, China
a r t i c l e
i n f o
Article history: Received 19 January 2019 Revised 23 March 2019 Accepted 29 March 2019
Keyword: Face-hobbed hypoid gear Work holding equipment error Geometric morphology Mesh characteristics
a b s t r a c t This paper proposed the accurate mathematical model considering work holding equipment errors based on the three-face cutter differing from typical commercial software. The simulated flow included manufacturing issues from industry applications have been established. Quasi-static loaded tooth contact analysis was used to discuss the contact characteristic and gear geometry with existence of radial and angular eccentric errors. In addition, the correspondence between transmission error and gear geometry was established. The method of separating the long wave (low frequency) and short wave (high frequency) from total transmission error has been proposed. Results show that the effect of angular eccentric error on gear geometry is more sensitive than that of radial eccentric error. The gear geometry deviations present periodic changes because of variation of pitch error. And radial eccentric error cannot influence the position and shape of contact pattern. On the contrary, with existence of angular eccentric error, the contact area moves to the heel side and then backward. Furthermore, the length of contact pattern is firstly decreased and then increased in one period. The transmission error is sensitivity for both two eccentric errors and the peak to peak value of transmission error increase obviously. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Hypoid gear has a wide utilization in the spatial angle transmission including axle transmission in the advantage of high contact ratio, high efficiency. With the rapid development of automobile industry, the demand of hypoid gear included the weight, smoothness of transmission, load capacity and life time were much higher than before. Currently, there have been many researches on deduction of processing mathematical model and tooth contact analysis for hypoid gear because of the complexity of gear geometric morphology [1–8]. But very little investigations have been recorded about discussing the impact of work holding equipment error on the mesh behavior of hypoid gear. Recent years, numerous researchers presented many studies about generated process of face-hobbed hypoid gear. Zhang, Litvin et al. [9] proposed a technique for computerized simulation and tangency of gears provided with real tooth surfaces based on the superimposition of theoretical tooth surface and deviations from heating treatment and lapping. Fan [10,11] established the precise analytical mesh model of face-hobbed hypoid gear pair to enhance tooth contact analysis (TCA) algorithm based on the method of Gleason face hobbing generator and proposed a technological process included designing, manufacturing and analysis of hypoid gear. Shih et al. [12–15] proposed a mathematical model of Oerlikon hypoid gear processing base on the computer numerical control (CNC) machine and transformed the mechanical processing setting ∗
Corresponding author. E-mail address: [email protected] (C. Song).
https://doi.org/10.1016/j.mechmachtheory.2019.03.042 0094-114X/© 2019 Elsevier Ltd. All rights reserved.
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into rotation and movement of three axes with Taylor polynomials, the mathematical model and manufacturing process was verified by experiment. Vimercati [16] aimed at an accurate geometrical hypoid gear model with face-hobbed method to propose an analytical description of the cutting process and generated cut based on the complex blade shapes included protuberance (TOPREM). Tsay et al. [17,18] presented an optimization procedure to find corrective machine-tool setting to minimize surface deviation of gear flank according to the mathematical model of hypoid gear. K. Kawasaki [19] proposed a method for remanufacturing pinion member of large-sized skew bevel gears using a CNC machining center and respecting an existing gear member. The effect of assembly errors on mesh behavior of hypoid gear is also a research hotspot. Lelkes et al. [19] developed the theoretical contact identification program to discuss the influence of each cutting parameter on longitudinal settings of contact patterns based on Klingelnberg’s Cyclo-Palloid System. Lim et al. [20] applied a quasi-static gear tooth contact analysis to study the effects of assembly errors on hypoid gear mesh behavior and predicted the assembly errors to optimize the gear dynamic response. Zhuo et al. [21] investigated the impact of assembly of hypoid gear with coaxially deviation on location of contact path, length of contact ellipse and long wave of transmission error based on the quasi-static multi tooth contact analysis (MTCA). G. Ignacio [22] aimed at getting favorable conditions of meshing to proposed a computerized generation model of spiral bevel gears based on the cyclopalloid system to optimize the basic machine-tool settings. K. Kawasaki [23] based on simultaneous generations of tooth surfaces to investigate the effect of cutting edge profile on meshing and contact of spiral bevel gears in the Klingelnberg cyclopalloid system analytically. V. Simon [24] applied computer aided tooth contact analysis for the investigation of the influence of misalignments of the mating members and tooth errors on mesh behavior. J. Tang [25] focused on low noise and high strength from industry applications to propose a new multi-step integrated analytical identification approach considered misalignments for hypoid gear. The above papers mostly focus on the deduction of hypoid gear machining model and tooth contact analysis of hypoid gear. Less of studies investigated the work holding equipment of machine. The presented commercial software cannot simulate the mesh behavior of hypoid gear with manufacturing issues. Therefore, discussing the impacts of work holding equipment error on the mesh behavior become more essential. This study mainly focused on the simulated flow of face-hobbed hypoid gear with radial and angular work holding equipment (WHE) errors based on the three-face cutter. The mathematical model and quasi-static finite element analysis model of the hypoid gear with WHE errors were proposed according to the actual manufacturing process. Then, the influences of WHE errors on gear flank geometrical morphology and mesh characteristics included the contact pattern, transmission error were investigated. Finally, the relationship between transmission error and gear geometry included pitch error and gear flank deviation has been established. 2. Mathematical model of face-hobbed hypoid gear with work holding equipment errors Traditional cradle machine tool include all manufacturing structures is shown in Fig. 1. The red dotted line represented the position of gear blank with work holding equipment errors. The WHE errors can be divided into radial eccentric and angular eccentric errors. The pinion can be considered as a shaft part, so it is possible for pinion to have radial eccentricity error. For wheel, it is common for radial eccentricity existing in manufacturing process because of the round dish structure. CE and δ ANG represent the quantity of the radial eccentric and angular eccentric errors, respectively. The eccentricity exists in the process of hypoid gear manufacturing more or less. Therefore, the mathematical model of hypoid gear with eccentric error from manufacturing perspective should be established for research. The mathematical model of manufacturing can be divided into two parts: three-face cutter and machine tool generation process. The mathematical model of three-face cutter head from manufacturing cutter structure is shown in the Fig. 2. The transfer matrixes of working segment of cutter can be represented as follows
f (Mc1 ) = Map M pq Mqr Mrs Msu
(1)
The transfer matrixes of arc segment of cutter can be represented as follows
f (Ma1 ) = Map M pq Mqr Mrs Msu Muv Mvw
(2)
where Map , Mpq , Mqr , Mrs , Msu , Muv , Mvw are the transfer matrixes from coordinate system Ow to Oa . α 0 represent the tool angle, RBHV and rBHV are the spheric radius of cutter working segment and arc segment, respectively. ζ is the regrind angle, ϑA is the rake angle. E0 is the radius of roll circle. Rac and Rav are the radius of pitch points in concave and convex flank, respectively. Rfc and Rfv are the radius of reference points in the concave and convex flank, respectively. δ A is the cutter direction angle and can be obtained by δA = asin(E0 /Ra(c,v) ). Based on the face-hobbed manufacturing process and method [26,27], the coordinate systems of gear manufacturing include WHE errors and tilt structure were proposed in Fig. 3. The transfer matrixes of whole gear cutting process can be obtained by
f (M2 ) = Mml Mlk Mk j M ji Mih Mhg Mg f M f e Med Mdc Mcb Mba
(3)
where Mml , Mlk , Mkj , Mji , Mih , Mhg , Mgf , Mfe , Med , Mdc , Mcb , Mba are the transfer matrixes from coordinate system Oa to Om . These transfer matrixes also contain eight parameters of machine tool processing. i and j represent the tile angle and swivel angle, respectively. Sr is the radial setting and q is the initial cradle angle. The vertical offset and machine root angle are
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Fig. 1. Traditional cradle machine tool with work holding equipment errors for hypoid gear.
Fig. 2. Mathematical model of three-face cutter.
represented by Em and γ m . A is synthetical horizontal (the distance between coordinate systems Oi and Oj ) and B is the sliding base. The transfer matrixes of radial eccentricity can be represented as follows
⎡
Mg f
1 ⎢0 =⎣ 0 0
0 1 0 0
0 0 1 0
⎤
0 CE ⎥ 0 ⎦ 1
(4)
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Fig. 3. Coordinate system for face-hobbed generator with WHE error.
The transfer matrixes of angular eccentricity are shown as follows
⎡
Mk j
cosδANG ⎢ sinδANG =⎣ 0 0
⎡
1 ⎢0 Mlk = ⎣ 0 0
0 1 0 0
−sinδANG cosδANG 0 0 0 0 1 0
0 0 1 0
⎤
0 0⎥ 0⎦ 1
(5)
⎤
Hd 0⎥ 0⎦ 1
(6)
The physical meaning of work holding equipment errors can be represented as Fig. 4. Where MC represents crossed point (CP) which means the intersection point between common vertical line and rotary axis. Hd is the distance between fixed point of pinion and crossed point (CP).
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Fig. 4. The relationship between work holding equipment with and without errors.
The representation of working segment is shown by
Ric (u, β , ϕ1 ) = f (M2 ) × f (Mc1 ) × Rc (u )
(7)
The expression of arc segment is given by
Ria (θ , β , ϕ1 ) = f (M2 ) × f (Ma1 ) × Ra (θ )
(8)
The working segment and arc segment are represented as follows
⎡
⎤
2RBHV sin(u/2RBHV )sin(α0 − u/2RBHV ) 0 ⎢ ⎥
Rc ( u ) = ⎢
⎥ ⎣2RBHV sin(u/2RBHV )cos(α0 − u/2RBHV )⎦ ⎡
⎤
(9)
1
rBHV cosθ 0 ⎥ ⎦ BHV sinθ 1
⎢ Ra ( θ ) = ⎣ r
(10)
The normal vector of gear flank can be obtained by
n pc (u, β , ϕ1 ) =
∂ Ric (u, β , ϕ1 ) ∂ Ric (u, β , ϕ1 ) ∂ Ric (u, β , ϕ1 ) ∂ Ric (u, β , ϕ1 ) × × ∂u ∂β ∂u ∂β
(11)
The normal vector of arc segment can be obtained by
n pa (θ , β , ϕ1 ) =
∂ Ria (θ , β , ϕ1 ) ∂ Ria (θ , β , ϕ1 ) ∂ Ria (θ , β , ϕ1 ) ∂ Ria (θ , β , ϕ1 ) × × ∂θ ∂β ∂θ ∂β
(12)
where u and θ are the variables of working segment and arc segment in the transverse section of cutter blade, respectively. β is rotation angle of cutter head. The mesh equation of working segment is shown as follows
f (u, β , ϕ1 ) = n pc × ν pc = n pc × (ϕ˙ 1 × ∂ r pc /∂ ϕ1 ) = 0
(13)
The mesh equation of arc segment is shown as follows
f (θ , β , ϕ1 ) = n pa × ν pa = n pa × (ϕ˙ 1 × ∂ r pa /∂ ϕ1 ) = 0
(14)
The relationship of these angles can be represented as follows
ϕc1 = z0 /z p × β
(15)
ϕc2 = 1/Ra × ϕ1
(16)
where ϕ 1 is the rotation angle of gear blank and can be solved by mesh equation. ϕ c1 is the angle of epicycloidal cradle rotation, ϕ c2 is the angle of generating cradle rotation, z0 is the number of blade groups, zp is the tooth number of generating gear, Ra is the roll ratio. The hypoid gear manufacturing can be regarded as the space trajectory of cutter tool acted on gear blank to cut the material, then the remainder of gear flank is the hypoid gear. As shown in Fig. 5, the red part is the space trajectory of
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Fig. 5. Mathematical model of hypoid gear with work holding equipment error.
Fig. 6. Flow chart of hypoid gear computerized modeling.
cutter tool. The mathematical and analytical models of hypoid gear pair with WHE errors can be established by moving the rotation center of gear blank with eccentricity. The flow chart of hypoid gear computerized modeling is shown in Fig. 6. The point sets of pinion and wheel can be obtained by MATLAB through the mesh equation. Inputting the point sets into the CREO, the space trajectory of cutter tool can be simulated by surface fitting. Furthermore, the manufacturing process can be simulated by Boolean operation of gear blank with WHE errors. Finally, the hypoid gear pair can be obtained by assembling the models of pinion and wheel according to the assembly parameters. 3. Hypoid gear mesh model The major parameters of hypoid gear and cutter tool are shown in Table. 1. The assembly and eccentric parameters are shown in Table. 2. The value of machine setting are shown in Table. 3. According to the parameters, the mesh model and finite element model of hypoid gear with WHE errors were established. The mesh model with pinion angular eccentric error and gear angular eccentric error are shown in the Fig. 7(a)-(b), respectively. There are four assembly parameters including offset E, shaft angle , axial position of pinion Dxp and axial position of gear Dxw for hypoid gear. The impacts of eccentricity on gear geometry and mesh behavior are periodic, the selected teeth should be enough for one period in mesh model. Therefore, the teeth number of wheel with radial and angular eccentricities are different.
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Fig. 7. Mesh model of hypoid gear.
Table 1 Parameter of hypoid gear and cutter.
Number of teeth Face width (mm) Mean normal module (mm) Mean spiral angle (°) Pitch angle (°) Number of blade groups Convex tool angle (°) Concave tool angle (°) Convex pitch point radius (mm) Concave pitch point radius (mm) Roll circle radius (mm)
Pinion
Wheel
11 38.6 3.4884 51.8225 23.6383 17 22.3059 20.1933 75.9341 76.0660 30.023
47 31.75 3.4884 29.2833 64.6445 18.9931 23.5060 75.9194 76.0806
101
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Fig. 8. Finite element mesh model of hypoid gear.
Fig. 9. Contact pattern (ABAQUS).
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Fig. 10. Contact pattern (KIMOS).
Fig. 11. Transmission error.
Table 2 Parameter of assembly and eccentric errors. Shaft angle (°)
90
Hypoid offset (mm) Axial position of pinion (mm) Axial position of wheel (mm) Radial eccentricity of gear (mm) Angular eccentricity of pinion (°)
38.1 109.52 60.55 0.02 0.05
For the eccentricity of pinion, the whole pinion and 11 teeth for wheel was selected in order to investigate the mesh characteristic in one period of pinion rotation. But for wheel, the whole pinion and 45 teeth of wheel were selected for one cycle of wheel rotation. The finite element mesh models in Fig. 8 were developed to investigate the influences of eccentric errors on mesh characteristics. For the simulation, the material used in gear pair is 17CrNiMo6 and the property is isotropic. The young’s modulus
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Fig. 12. Gear flank geometric morphology deviations for wheel with radial eccentric error.
and Poisson’s ratio of gear pair are 2090 0 0 Mpa and 0.298, respectively. The interaction constraint (tooth contact) was established between concave flank of pinion and convex flank of wheel. The kinematic couplings were created between the rotation points and the inner surfaces of gear blank to set boundary constraints. The torsional displacement was applied to pinion as input for the loaded tooth contact analysis and torque load 450 Nm was applied to wheel as output. In order to save the computational cost in whole process, the grid seeds of working gear flank are denser than those of gear blank. Because of the complexity of pinion flank, the tetrahedral mesh was selected for pinion and hexahedron mesh for gear, respectively.
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Fig. 13. Gear flank geometric morphology deviations for pinion with angular eccentric error. Table 3 Value of machine setting. Machine Setting Tile angle i Swivel angle j Radial setting Sr Initial cradle angle q Vertical offset Em Machine root angle γ m Horizontal A Sliding base B
Pinion
Wheel ◦
25.3716 −131.8117◦ 116.5702 52.1131◦ 40.5991 0.3457◦ 1.3084 28.7681
0◦ 0◦ 121.8758 38.3013◦ 0 64.6445◦ 0 0
For the model with angular eccentricity, 11 teeth were selected for gear in Fig. 8(a). The analysis of step length is 0.02, and there are 163,246 elements for the whole model. But for the model with radial eccentricity is different, in Fig. 8(b), 45 teeth were selected. The analysis of step length is 0.002 to guarantee the accuracy of calculation, and there are 547,295 elements for hypoid gear pair. The hypoid gear without eccentric error called master gear. The contact pattern without eccentric error from loaded tooth contact analysis (ABAQUS) and KIMOS are shown in Figs. 9 and 10. The shape and location of contact pattern from ABAQUS are close to KIMOS. From the Fig. 11(a) and (b), the shape and peak to peak value of transmission error under
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Fig. 14. Contact pattern of wheel with radial eccentric error.
LTCA (41 μm) are closed to KIMOS (60.9 μm). In general, the correctness of mathematical model and finite element model of hypoid gear pair were validated from the KIMOS. Using the proposed mathematical model and finite element model of the hypoid gear with WHE error, the influence of gear pairs with radial eccentric error and angular eccentric error on mesh behavior and gear flank geometry can be studied easily. 4. Gear flank geometric morphology deviation analysis Case 1: Radial eccentricity of wheel The hypoid gears are made by generation cutting processing, so the WHE error will influence the gear geometry directly. The comparison of deviations between wheel with WHE error and original wheel surface can be expressed in Fig. 12. In one cycle of wheel, the deviation value of toe side firstly decreased from positive to negative and then increased back to
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Fig. 15. Transmission error with wheel angular eccentric error.
Fig. 16. The comparison of transmission error between wheel with radial eccentric error and master gear.
positive. The radial error only influences the deviation value, furthermore, the pitch error has been changed. And the shape of gear flank is not effected too much. Case 2: Angular eccentricity of pinion The comparison of deviations between pinion with WHE error and original pinion surface can be represented in Fig. 13. In one cycle of pinion, the deviation of toe side firstly increase from −10.8μm (−11.8μm) to 41.6μm (50.8μm) and then decrease to 22.9μm (28.6μm). Furthermore, the deviation of heel side firstly increase from 17.7μm (23.9μm) to 63.2μm (52.0μm) and then decreased to −11.0μm (−15.3μm). Compared with radial eccentric error, the gear flank geometric morphology deviation and pitch error are more sensitive to angular eccentric error. 5. Contact behavior with loaded tooth contact analysis Case 1: Radial eccentricity of wheel As depicted in Fig. 14, 12 teeth (every four teeth) of wheel were selected to investigate the influence of wheel radial eccentricity on contact pattern and transmission error of hypoid gear. From the results, the location of contact pattern is in the heel side and top side of gear flank in one period. Compared with master gear pair, the location and shape have not changed. In Fig. 15, the red dash-dotted line represents the total transmission error with eccentricity in one meshing period of wheel. There are two components included high frequency and low frequency in the total transmission error. The low frequency component can be fitted by three orders Fourier series, the fitting curve called long wave and the separated high frequency called short wave. The total transmission error and long wave presented the periodic variation like the sinusoidal function. Compared with transmission error of standard gear pair in Fig. 16 and enlarged drawing Fig. 17, the short wave of transmission error shows the irregular fluctuations in one cycle because each geometry of tooth is different. The peak to peak value of transmission error increased with eccentric error according to Fig. 18. The first three orders of transmission error harmonics are given in Fig. 19. The transmission error harmonics decreased with the increasing of orders gradually. Case 2: Angular eccentricity of pinion In Fig. 20(a)–(f), 6 teeth (every two teeth) were selected to investigate the effect of angular eccentric error of pinion on the contact pattern. From the results, the location of contact pattern firstly moved to the heel side and then back to the original position in one cycle. Furthermore, the length of the contact pattern firstly decreased and then increased with the increasing of pinion rolling angle.
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Fig. 17. Enlarged drawing.
Fig. 18. The comparison of peak to peak value of transmission error between wheel with radial eccentric error and standard processing.
Fig. 19. First three orders of transmission error harmonics with wheel radial eccentric error.
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Fig. 20. Contact pattern of wheel with angular eccentric error.
Fig. 21. Transmission error with pinion angular eccentric error.
Fig. 22. The comparison of transmission error between pinion with angular eccentric error and master gear.
As shown in Fig. 21, the total transmission error and long wave firstly increased and then decreased in one cycle of pinion. The high frequency part of total transmission error vibrated along the short frequency. The comparison of transmission error (short wave) between pinion with angular eccentricity and master gear pair was depicted in the Fig. 22. Each single waveform of short wave was different because the distinctness of wheel geometry. The peak to peak value of transmission error was shown in Fig. 23, the peak to peak value increased with existence of angular eccentric error. The transmission error harmonics decreased with the orders increasing gradually in Fig. 24. According to the above results, the quasi-static loaded tooth contact analysis can be used to evaluate the gear geometry. As shown in Fig. 25, the transmission error contains whole information of gear geometry characteristic. The low frequency component can be fitted by Fourier series, the rest is the high frequency part. The low frequency (long wave) and high
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Fig. 23. The comparison of peak to peak value of transmission error between pinion with radial eccentric error and master gear.
Fig. 24. First three order of transmission error harmonics with pinion angular eccentric error.
Fig. 25. Flow chart of transmission error analysis.
frequency (short wave) represented the gear blank and gear flank geometry, respectively. If short wave is out of tolerance, there must exist the design or manufacturing issues on gear blank such as pitch error, the same as those of long wave. Therefore, the transmission error analysis can guide the design and manufacturing of hypoid gear. The above process discussed the drive side. This flow chart of analysis is suitable for both side. And for the coast side, the method of analysis is the same except the rotation direction of gear. 6. Conclusion In this paper, based on the three-face cutter, the accurate mathematical model and analytical mesh model of face-hobbed hypoid gear considered work holding equipment errors have been established. Differing from the typical commercial software, the simulated flow of hypoid gear included manufacturing issues from industry applications was proposed. This study also use the quasi-static loaded tooth contact analysis to investigate the impact of eccentric error on contact behavior and geometry deviation. The method which divided the transmission error into long wave (low frequency) and short wave (high frequency) was proposed. The correspondence between long/short wave and gear blank/flank was established. From the above, four specific findings can be obtained as follows
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(1) With existence of radial eccentric error, the deviation of toe side firstly decreases from positive value to negative value and then increases to positive value, and final back to original position in one rotation cycle. (2) The effect of angular eccentric error on the gear geometry is more sensitive than that of radial eccentricity. With angular eccentricity, the deviations of toe side firstly increase and then decrease, furthermore, the deviation of heel side firstly increase and then decrease. (3) Radial eccentricity of wheel does not affect the position and shape of contact pattern. And the eccentricity has strong effect on transmission error, the peak to peak value of transmission error is increasing. Compared with master gear, the shape of transmission error is the short wave (high frequency part) vibrates along the long wave (low frequency part) which is closed to sinusoidal function in one mesh cycle. (4) The position of contact pattern firstly moves to the heel side and then back to the original position with existence of angular eccentricity. Furthermore, the length of the contact pattern firstly decreased and then increased in one meshing period. With increasing of pinion rolling angle, the transmission error firstly increased and then decreased. The peak to peak value of transmission error is increasing. Acknowledgements The authors would like to thank the National Natural Science Foundation of China (No. 51775061 and No. 51575060), Fundamental Research Funds for the Central Universities (2018CDQYJX0012) and Key Research and Development Project of Chongqing Science and Technology Program (cstc2018jszx-cyztzxX0038). References [1] F.L. Litvin, Y. Gutman, Methods of synthesis and analysis for hypoid gear-drives of formate and helixform-Part 1, calculations for machine settings for member gear manufacture of the formate and helixform hypoid gears, J. Mech. 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