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Investigation of the effects with linear, circular and polynomial blades on contact characteristics for face-hobbed hypoid gears
Mechanism and Machine Theory 146 (2020) 103739
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
Investigation of the effects with linear, circular and polynomial blades on contact characteristics for face-hobbed hypoid gears Chengcheng Liang, Chaosheng Song∗, Caichao Zhu, Siyuan Liu, 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 26 October 2019 Revised 4 December 2019 Accepted 4 December 2019
Keywords: Face-hobbed hypoid gears Blade sections Finite element mesh model Mesh characteristics
a b s t r a c t On this research, the effects of the three different types of blade sections including linear, circular and polynomial on mesh characteristics for face-hobbed hypoid gears are investigated. Firstly, with the equations of three blade sections and the transformation matrixes based on the processing method derived, the mathematical model of gear flank and loaded finite element mesh model are proposed. Then, the impacts of different blade sections on tooth surface deviation and mesh characteristics are analyzed. Results show that the effects are almost identical when the parameters choose reasonable for the three blades. However, with the circular radius reducing, the peak-to-peak value of transmission error decreases but the contact pattern area shrinks and the location of maximum bending stress will change. For the polynomial blade, the effects the design parameters on mesh characteristics are obvious. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction The shape of blade section, the structure of cutter plate and the motion of processing machine are important factors of the manufacturing tooth surface for face-hobbed hypoid gears. Meanwhile, the machine and cutter plate are regulation and the development of cutting blade have a tendency from simple to complex due to the improvement of the technical level in actual cutting process. The blade design of more details becomes an important direction to optimize the tooth surface topography. However, computerized modeling and analyzing of hypoid gears are related to the linear and circular blade section in many researches. And the polynomial blade section adopts a more advanced curve and the curve is not applied in hypoid gears currently. Thus, it is very significant to research the tendency and effects of the linear, circular and polynomial blade section on mesh characteristics for face-hobbed hypoid gears. Theoretically, the mathematical model for hypoid gears is established by simulating the cutting process of gear blank by cutter plate. An accurate simulation model is a precondition of face-hobbed hypoid gears for some research. Litvin and Zhang proposed the local synthesis method of the geometry model in order to build face-milling hypoid gears [1]. Fan proposed an enhanced approach to improve the nonlinear iterations and iteration convergence for solving the mathematical tooth flank equation conveniently [2]. Du and Fang obtained a real tooth surface equation by measuring the real tooth surface of gear and pinion [3]. Liu et al. researched the impacts of work holding equipment errors on contact characteristics and ∗
Corresponding author. E-mail address: [email protected] (C. Song).
https://doi.org/10.1016/j.mechmachtheory.2019.103739 0094-114X/© 2019 Elsevier Ltd. All rights reserved.
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C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739
Fig. 1. Blade sections of wheel and pinion.
gear flank geometry of face-hobbed hypoid gears based on the tooth surface model [4]. Shao et al. proposed a data-driven optimization model to calculate the machine settings and established a double-curved shell finite element model of hypoid gear [5]. Some researches in order to improve the modeling accuracy of hypoid gears by description of cutting edge in detail. Guo et al. introduced a multi-segment cutter blade profile to obtain the ideal load and contact stress distribution [6]. Vimercati derived the expressions of blade section included the major cutting edge and fillet segment and the mathematical representation of gear tooth surfaces by using theory of gearing [7]. Du and Fang proposed an active design method of tooth flank based on the one time of correction for pinion tooth surface [8]. Zhou et al. developed to automatically generating the 3D models with general spur pinions or the involute helical pinion [9]. The design and choose of blade section in details can be a method to optimize the mesh condition for reducing vibration noise. In generally, the blade section including linear blade, circular blade and polynomial blade for hypoid gear was introduced in many research papers [10–12]. And many researchers notice the point of the cutter geometric. Xie proposed a genuine face milling cutter geometric model for spiral bevel and hypoid gears and proves that the genuine cutter geometric model was correct and essential [13]. Chen et al. introduced a method to generate gear pairs by asymmetric parabolic profiles and the mathematical model was obtained using the coordinate transformation matrix [14]. Fuentes-Aznar et al. investigated the dual interlocking circular cutters was developed for straight bevel gears manufactured and the impact of the radius for the cutting disks on the maximum bending stresses [15]. Guo et al. proposed a novel skiving tool with multiple subclades on the rake face with the traditional tool to save the cost of skiving tool [16]. Wang and Fong proposed a new type of double-crowned helical gear and the circular-arc cutter blades be used in gears model [17]. However, there are very few researches to notice the effects of different blade sections and polynomial blade on mesh characteristics and it is significant on choose different blade section to improve the meshing quality and to reduce vibration noise for hypoid gears. This paper mainly focused on the advantages and disadvantages and difference of three blade sections on mesh characteristics for face-hobbed hypoid gears. Firstly, the accuracy mathematical model and finite element model with the three kinds of blade sections for hypoid gears are proposed. Meanwhile, the pinion is established with linear, circular and polynomial blade and the wheel use circular blade. Then, the effects of different blades on mesh characteristics are investigated. Finally, the advantages and disadvantages of gear pair with linear, circular and polynomial blade are analyzed. 2. Mathematical and mesh model for face-hobbed hypoid gears The tooth surface topography is related to blade section, tool geometry and machine parameters. The forming method is used to manufacturing wheel and the generating method for pinion. The tooth surface mathematical model is established by the equation and transformation matrix of blade section, cutter plate and machine. 2.1. Depiction of blade sections From above-mentioned, the blade section can be divided into linear blade, circular blade and polynomial blade. The details of blade section are described in Fig. 1. The N–M segment is the major cutting edge and M–F segment is fillet. The symbols oq and of are the reference points of major cutting edge and center of fillet, respectively. The coordinate system Sq is established reference point. For the pinion, the location and pressure angle of point oq are identical. The descriptions of cutting edge for wheel and pinion can be obtained by the derivation of mathematical equations and transformation matrixes [18]. Firstly, the blade section expression of wheel in coordinate system Sq can be deduced according to blade section of wheel in Fig. 1(a). The expression can be represented as follow
C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739
N–M:
⎡
⎤
⎡
⎤
xqv 2RBH v sin (u/(2RBH v ) ) sin (α0v − u/(2RBH v ) ) 0 ⎢y q v ⎥ ⎢ ⎥ rqv ( u ) = ⎣ ⎦ = ⎣ zqv 2RBH v sin (u/(2RBH v ) ) cos (α0v − u/(2RBH v ) )⎦ 1 1 M–F:
⎡
⎤
3
⎡
(1)
⎤
xqv rbhv cosθ ⎢y ⎥ ⎢ 0 ⎥ r q v θ = ⎣ q v ⎦ = M q f v ∗⎣ zqv rbhv sinθ ⎦ 1 1
(2)
where RBHv and rbhv are the circular radius of major cutting edge and arc segment radius of wheel convex machining inner blade, respectively. The parameters u and θ are the variable for the major cutting edge and arc segment in coordinate system Sq , respectively. α 0v is the pressure angle of inner blade reference point for wheel. The symbol Mqf is transformation matrix from of to oq of inner blade. Then, the value of pressure angle and location of reference point oq are defined to be consistent for three kinds of blade sections. Finally, the expressions of linear, circular and polynomial blade sections can be calculated according to the description of blade section of pinion in Fig. 1(b). The linear blade expression of pinion can be represented as follow N–M:
⎡
⎤
⎡
⎤
xqc −usinα0c 0 ⎢yqc ⎥ ⎢ ⎥ rqc (u ) = ⎣ ⎦ = ⎣ zqc ucosα0c ⎦ 1 1 M–F:
⎡
⎤
⎡
(3)
⎤
−wm + rbhc sin α0c + θ − sinα0c xqc ⎢yqc ⎥ ⎢ 0 rqc θ = ⎣ ⎦ = ⎣ zqc hm + rbhc cos α0c + θ − cosα0c 1 1
⎥ ⎦
(4)
where the symbols α 0c is outer blade pressure angle and rbhc is arc segment radius of pinion. wm and hm are the horizontal and vertical coordinate value of point M, respectively. Similarly, the circular blade expression of pinion can be represented as follow N–M:
⎡
⎤
⎡
⎤
xqc 2RBHc sin (u/(2RBHc ) ) sin (α0c − u/(2RBHc ) ) 0 ⎢yqc ⎥ ⎢ ⎥ rqc (u ) = ⎣ ⎦ = ⎣ zqc 2RBHc sin (u/(2RBHc ) ) cos (α0c − u/(2RBHc ) )⎦ 1 1 M–F:
⎡
⎤
⎡
(5)
⎤
xqc rbhc cosθ ⎢yqc ⎥ ⎢ 0 ⎥ r q v θ = ⎣ ⎦ = M q f c ∗⎣ zqc rbhc sinθ ⎦ 1 1
(6)
where RBHc and rbhc are the circular radius of major cutting edge and arc segment radius of pinion concave machining outer blade, respectively. The symbol α 0c is outer blade pressure angle of reference point for pinion. The symbol Mqf is transformation matrix form of to oq of outer blade [18]. The polynomial blade expression of pinion can be represented as follow O–M:
⎡
⎤
⎡
⎤
xqc −u1 ∗ sinα0c − c1 ∗ u21 ∗ cosα0c 0 ⎢yqc ⎥ ⎢ ⎥ rqc (u ) = ⎣ ⎦ = ⎣ zqc u1 ∗ cosα0c − c1 ∗ u21 ∗ sinα0c ⎦ 1 1 O–N:
⎡
⎤
⎡
(7)
⎤
xqc u2 ∗ sinα0c − c2 ∗ u22 ∗ cosα0c 0 ⎢yqc ⎥ ⎢ ⎥ rqc (u ) = ⎣ ⎦ = ⎣ zqc −u2 ∗ cosα0c − c2 ∗ u22 ∗ sinα0c ⎦ 1 1
(8)
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Fig 2. The mathematical model of cutter plate.
M–F:
⎡
⎤
⎡
⎤
−wm + rbhc sin α0c + θ − sinα0c xqc ⎢yqc ⎥ ⎢ 0 rqc θ = ⎣ ⎦ = ⎣ zqc hm + rbhc cos α0c + θ − cosα0c 1 1
⎥ ⎦
(9)
where the parameters u1 and u2 are the variable for the tangent of O–M cutting edge and O–N cutting edge in coordinate system Sq , respectively. The symbols c1 and c2 are the polynomial coefficient for the O–M segment and O–N segment, respectively. By similar deduction of major cutting edge and fillet for concave tooth surface of wheel and convex tooth surface, the equations for blade section of hypoid gears can be obtained. 2.2. Tooth surface equation for face-hobbed hypoid gears Tooth surface can be considered as the cutting track of the blade section in gear blank. With computing of the tooth surface equation, the expressions of blade section for wheel and pinion in coordinate system Sq and proper coordinate transformation for three-faced blade, cutter plate and machine are essential based on actual manufacturing process and the theory of gear meshing. The expressions of blade section have been calculated by above section. Fig. 2 reports a three-faced blade and face-hobbed cutter plate. There are three parameters related to the tooth surface according to the geometry and the benchmark of blade for three-faced blade, including rake angle, regrind angle and cutting side relief angle. The cutter plate have four parameters relating to tooth surface in the light of the location of three-face blade at cutter plate and the rotate vector, including offset angle of cutter blade, cutter radius, the initial assemble angle of inner blade and outer blade and the rotate angle of cutter plate. As mentioned, the transformation matrix of three-faced blade and cutter plate from Sq to St can be obtained by
Mtq = Mts (βv , Rc , δs ; β ) ∗ Msq (γs , ε , γa )
(10)
where Msq is transformation matrix from Sq to Ss based on the three-faced blade and Mts is transformation matrix from Ss to St based on the cutter plate. The symbols γ a , ɛ and γ s are rake angle, regrind angle and cutting side relief angle, respectively. Moreover, the parameters δ s , R, β v /(β c ) and β are offset angle of cutter blade, cutter plate radius, the initial assembly angle of outer blade/(inner blade) and the rotate angle of cutter plate, respectively. The initial assemble angle of outer blade and inner blade is fixed. In this paper, the initial assemble angles of outer blade and inner blade are 10.5882◦ and 21.1765◦ , respectively. Further, the transform matrix of machine from St to Sp are proposed in order to get the cutting track of blade on the gear blank. Fig. 3 represents the machine coordinate system from the center of cutter plate to gear blank for hypoid gears. The generator coordinates have nine parameters including the tilt angle i, swivel angle j, radical distance SR , cradle angle q0 , ϕ c (q0 is initial cradle angle and ϕ c is cradle rotation angle), work offset E, sliding base B, machine root angle γ m and horizontal A. The transformation matrixes of manufacturing machine from St to Sp can be obtained as follow
M pt = M pk (ϕ1 ) ∗ Mkh ∗ Mhg ∗ Mgm ∗ Mmc (ϕc ) ∗ Mce ∗ M f e ∗ Met
(11)
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Fig 3. Manufacturing machine coordinate system. Table 1 Parameters of cutter plate. Parameter
Pinion Convex
Number of cutter heads Teeth number Offset angle (◦ ) Regrind angle (◦ ) Cutting side relief angle (◦ ) Rake angle (◦ ) Cutter plate radius (mm)
z0 z
σ0
ɛ
γa γs Rc, v
Gear Concave
Convex
Concave
17 22.9309 4.3674 13.3885 9.5870 75.934
11 22.9309 4.7856 12.6729 10.3059 76.066
22.9309 5.0309 13.3823 12.6781 75.9193
47 22.9309 4.1101 12.6781 9.6107 76.0807
Thus, the expression of cutting track for wheel and pinion in the gear blank can be calculated by
r pv (ϕ1 , β ) = M pt ∗ Mtq ∗ rqv r pc (ϕ1 , ϕc , β ) = M pt ∗ Mtq ∗ rqc
(12)
In order to solve this equation, the processing methods and the principle of gear meshing must be applied. For the forming processing, the cutting track is equal to tooth surface due to the cradle fixed. Therefore, the equation of tooth surface can be solved according to relational about cutter plate and gear blank as follow
ϕ1 =
z ∗β z0
(13)
For the pinion with generating method, it is necessity to solve tooth surface for meshing equation [19]. The relationship of rotation angle of gear blank with cradle angle and the rotate angle of cutter plate.
f = n pv p = 0
ϕ1 = igp ∗ ϕc +
(14) z0 β z
(15)
where np is unit normal vector and vp is direction of relative velocity in coordinate system Sp . The parameter igp is ratio of roll. The parameters z and z0 are the number of processing gear and blade groups, respectively. 2.3. Finite element mesh model The main parameters of cutter plate are shown in Table 1. The machine processing parameters are shown in Table 2. By means of the mathematical model of wheel and pinion and these parameters, the finite element model is established in ABAQUS. The details of the finite element model are demonstrated in Fig. 4. Firstly, the material property is isotropic with young’s modulus 210,0 0 0 MPa and Poisson’s ratio 0.3. The reference point P1 and P2 are established on pinion and wheel rotation axis and the surface 1 and surface 2 are coupled in reference points
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C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739 Table 2 Parameters of machine processing. Parameter
Pinion Convex
Radial distance (mm) Initial cradle angle (◦ ) Work offset (mm) Sliding base (mm) Horizontal (mm) Machine root angle (◦ ) Tilt angle (◦ ) Swivel angle (◦ ) Ratio of roll
SR q Em B A
γm i j igp
Gear Concave
Convex
116.6202 52.1131 40.5991 28.7681 0.4666 0.3457 4.348969 131.8117 4.348969
Concave
122.009 −29.7048 0 0 13.9199 64.7995 0 0 0
Table 3 Blade section parameters of wheel convex. Parameters Pressure angle (◦ ) Cutter plate radius (mm) Arc segment radius (mm) Circular radius (mm)
C-B
α 0v Rc rbhv RBHv
18.9931 75.9193 0.9 1047
Fig. 4. Finite element model of hypoid gears.
P1 and P2 , respectively. Then, the contact surface be set to pinion concave and wheel convex and the boundary conditions are divide into pre-contact, applying load and mesh process. The torque load is applied on the wheel. Finally, the mesh of contact surface becomes more intensive by changing the size of local seed. The meshing of pinion are 176,328 elements and 89,400 elements for wheel. 3. Results and discussion According to different matching method for wheel and pinion, the results of mesh characteristics are analyzed. The torque is set to 450 Nm. The wheel model is all the same, but the pinion is replaced by choosing different types and parameters of blade section. Thus, this paper analyzed four situations from following. The details parameters of wheel convex are shown in Table 3. 3.1. Three kinds of blade section Firstly, the effects of the different blade sections for linear blade (L-B), circular blade (C-B) and polynomial blade (P-B) must be researched. The parameters of blade section of pinion concave are shown in Table 4. The tooth surface deviation of circular blade and polynomial blade compare to linear blade is shown in Fig. 5.
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Table 4 Parameters of three different types of blade section. Parameters ◦
Pressure angle ( ) Cutter plate radius (mm) Arc segment radius (mm) Circular radius (mm) Polynomial coefficient
L-B
C-B
P-B
– – –
20.1933 76.066 0.9 1200 – –
– −0.00005 −0.00005
α 0c Rv rbhc RBHc c1 c2
Fig. 5. Tooth surface deviation for the three kinds of blade.
Fig. 6. Contact pattern for the three kinds of blade sections.
The circular blade removes more materials in top and root of tooth surface and the change of tooth surface geometry in pitch line have less influence compared to linear blade. For the polynomial blade, the tooth surface geometry is related to the value and direction of polynomial coefficient. When the polynomial coefficient selection is minimum, the deviation of tooth surface is slight. The influences of different blade section on contact pattern are shown in Fig. 6. The shape and location of contact pattern are almost similar among the three kinds of blade sections when the circular radius choose large and the polynomial coefficient is small. The area radio of contact pattern and the maximum value of contact stress have difference due to the deviation of tooth surface geometry as shown Fig. 5. The influences of different blade sections on tooth root stress are shown in Fig. 7. The tendency and the maximum root stress are consistent. The influences of different blade sections on transmission error are shown in Fig. 8. The angular transmission error and the value of peak-to-peak (P–P) of transmission error are almost identical. Thus, there is less change in the mesh characteristics of three kinds of blade section when the circular radius chooses very large and the polynomial coefficient is small. 3.2. Different circular radius In order to obtain the influence tendency for C-B, the effect analysis of changing the circular radius is significance. In this section, the circular radius select 200 mm, 400 mm, 900 mm and 1200 mm and the details are shown in Table 5.
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Fig. 7. Tooth root stress for the three kinds of blade sections.
Fig. 8. Transmission error for the three kinds of blade sections. Table 5 Parameters of circular blade. Parameters
Circular blade ◦
Pressure angle ( ) Cutter plate radius (mm) Arc segment radius (mm) Circular radius (mm)
α 0c Rv rbhc RBHc
200
20.1933 76.066 0.9 400
900
1200
The results of contact pattern are depicted in Fig. 9. The location of contact pattern is similar with the circular radius changing. However, with the circular radius reducing, the area of contact pattern gradually decreases due to the reduction of length axis of contact ellipse and the maximum value of contact stress gradually getting stronger. Moreover, the high stress zone will become bigger if the circular radius is smaller. The effects of circular radius on tooth root stress are shown in Fig. 10. With the circular radius increasing, the tendency of tooth root stress is similar but the maximum value for tooth root bending stress is reduced and the location point gradually moves from toe side to heel side. The influences of circular radius on transmission error are shown in Fig. 11. It is obvious effect on transmission error with the changing of circular radius. The tendency of angular time-varying transmission error is analogous while the circular radius changes from 200 mm to 1200 mm. However, the circular radius is increasing from small to large, the angular timevarying transmission error gradually become larger and the P–P value of transmission error is gradually rising. Particularly, the changing is very small when the circular radius increase to quite large. 3.3. The effect of the different magnitude coefficient In order to obtain the influence law for P-B, the analysis of the magnitude coefficient and the different coefficient direction of O–M and O–N segment is essential. In this section, the magnitude coefficient choose negative values referencing to circular blade. The parameters of magnitude coefficient are shown in Table 6. The contact pattern of different magnitude coefficient are shown in Fig. 12.
C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739
Fig. 9. Contact pattern for different circular radius.
Fig. 10. Tooth root stress for different circular radius.
Fig. 11. Transmission error for different circular radius.
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C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739 Table 6 Parameters of magnitude coefficient. Parameters
Polynomial blade ◦
Pressure angle ( ) Cutter plate radius (mm) Arc segment radius (mm) Polynomial coefficient
α 0c Rv rbhc c1 c2
−0.00005 −0.00005
20.1933 76.066 0.9 −0.0005 −0.0005
−0.005 −0.005
Fig. 12. Contact pattern for different magnitude coefficient.
Fig. 13. Tooth root stress for different magnitude coefficient.
When the coefficient is −0.0 0 0 05, the contact pattern is closed to linear blade, but the contact pattern is more closed to the result when the circular blade radius is 200 mm and the coefficient is −0.005. In addition, the maximum contact stress of tooth surface is gradually decreasing with the magnitude coefficient increasing. The effects of different magnitude coefficient on tooth root stress are shown in Fig. 13. The tendency of tooth root stress is also similar with the changing of magnitude coefficient. The value of the maximum tooth root stress is slightly increasing and the location move from heel side to toe side with the value increasing. The influences of different magnitude coefficient on transmission error are shown in Fig. 14. It is obvious influence of the different magnitude coefficient on transmission error and the tendency of angular time-varying transmission error is similar. The angular time-varying transmission error gradually decrease and the P–P value of transmission error is gradually reduced with the magnitude coefficient increasing. 3.4. The effects of the coefficient direction for polynomial blade In this section, the effects of the different coefficient direction on tooth surface deviation and mesh characteristics are analyzed. The direction of O–M and O–N segment have four types that the coefficients are negative with negative, negative with positive, positive with negative and positive with positive and the parameters are shown in Table 7.
C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739
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Fig. 14. Transmission error for different magnitude coefficient.
Fig. 15. Tooth surface deviation for the four coefficient direction. Table 7 Parameters of coefficient direction. Parameters Pressure angle (◦ ) Cutter plate radius (mm) Arc segment radius (mm) Polynomial coefficient
Polynomial blade
α 0c Rv rbhc c1 c2
−0.0005 −0.0005
20.1933 76.066 0.9 −0.0005 +0.0005
+0.0005 −0.0005
+0.0005 +0.0005
The deviation of tooth surface topography for different coefficient direction is shown in Fig. 15. The deviation is small at pitch line and the top and root side is large. The cutting amount is different while the direction of polynomial coefficient is opposite. The results of different coefficient direction on contact pattern are shown in Fig. 16. When the polynomial coefficient is negative with positive and positive with negative, the area of contact pattern is increased and the maximum contact stress of tooth surface is decreased compared to the situation that the coefficient is negative with negative. The phenomenon of edge contact will appear while the coefficient is positive with positive. The different coefficients can slight change the location for contact pattern and it can be a method to optimize tooth surface
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Fig. 16. Contact pattern for the four kinds of coefficient direction.
Fig. 17. Tooth root stress for the four coefficient direction.
Fig. 18. Transmission error for the four kinds of coefficient direction.
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design. The effects of different coefficient direction on tooth root stress are shown in Fig. 17. The tendency of tooth root stress is also similarly with different coefficient directions. The value of the maximum tooth root stress is almost identical without edge contact. The location will slight moving to heel side while the coefficient is negative with positive. The influences of different coefficient direction on transmission error are shown in Fig. 18. The tendency of angular timevarying transmission error is also similar but the change and amplitude of transmission error is bigger while arising edge contact. In addition, with the coefficient choose positive with negative, the P–P value of transmission error is the lowest. 4. Conclusion In this paper, the expressions of linear blade, circular blade and polynomial blade and an accurate tooth surface mathematical model based on the three-faced blade and simulating the cutting process have been presented. The effects of three kinds of blade sections for pinion and circular blade for wheel on mesh characteristics are analyzed by the finite element method. There are four findings as following (1) While the circular radius is very big of circular blade and the coefficient is small of polynomial blade, the effects are almost analogous on mesh characteristics with linear blade. (2) The circular radius has an adjustable parameter compared to linear blade. The P–P value transmission error can be decreased by reducing circular radius but the contact pattern area shrinks, the maximum value of contact stress will increase and the location of maximum tooth root bending will move from toe to heel side. (3) With the polynomial blade use more advanced curve, the adjustable parameters have more choices. The effects of different magnitude polynomial coefficient (negative with negative) on mesh characteristics are similar to linear and circular blade by changing the magnitude polynomial coefficients. (4) The mesh characteristics are the best that can reduce transmission error without losing contact area and decreasing tooth root bending stress when the coefficient is positive with negative and the effects are the worst due to arise edge contact when the coefficient is positive with positive among all types of polynomial coefficients. In addition, the situation of negative with positive can slightly moving the location of contact pattern, which can provide a design method by changing blade section. Declaration of Competing Interest None. Acknowledgments The authors would like to thank the National Natural Science Foundation of China (51775061), Science and Technology Major Project of Guangxi (Guike AA19182001), Research and Development Plans in Key Areas of Guangdong (2019B090917002) and National Key Research and Development Plan of China (2019YFB20 0470 0). References [1] F.L. Litvin, Y. 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[17] W.S. Wang, Z.H. Fong, Tooth contact analysis of longitudinal cycloidal-crowned helical gears with circular arc teeth, J. Mech. Des. 132 (3) (2010) pp. 031008 (9). [18] C.C. Liang, C.S. Song, C.C. Zhu, Y.W. Wang, S.Y. Liu, R.H. Sun, Investigation of tool errors and their influences on tooth surface topography for face-hobbed hypoid gears, J. Mech. Des. 142 (2020) 1–11. [19] Litvin, F.L., Theory of Gearing. NASA Reference Publication 1212, (1989).
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Research paper
Investigation of the effects with linear, circular and polynomial blades on contact characteristics for face-hobbed hypoid gears Chengcheng Liang, Chaosheng Song∗, Caichao Zhu, Siyuan Liu, 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 26 October 2019 Revised 4 December 2019 Accepted 4 December 2019
Keywords: Face-hobbed hypoid gears Blade sections Finite element mesh model Mesh characteristics
a b s t r a c t On this research, the effects of the three different types of blade sections including linear, circular and polynomial on mesh characteristics for face-hobbed hypoid gears are investigated. Firstly, with the equations of three blade sections and the transformation matrixes based on the processing method derived, the mathematical model of gear flank and loaded finite element mesh model are proposed. Then, the impacts of different blade sections on tooth surface deviation and mesh characteristics are analyzed. Results show that the effects are almost identical when the parameters choose reasonable for the three blades. However, with the circular radius reducing, the peak-to-peak value of transmission error decreases but the contact pattern area shrinks and the location of maximum bending stress will change. For the polynomial blade, the effects the design parameters on mesh characteristics are obvious. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction The shape of blade section, the structure of cutter plate and the motion of processing machine are important factors of the manufacturing tooth surface for face-hobbed hypoid gears. Meanwhile, the machine and cutter plate are regulation and the development of cutting blade have a tendency from simple to complex due to the improvement of the technical level in actual cutting process. The blade design of more details becomes an important direction to optimize the tooth surface topography. However, computerized modeling and analyzing of hypoid gears are related to the linear and circular blade section in many researches. And the polynomial blade section adopts a more advanced curve and the curve is not applied in hypoid gears currently. Thus, it is very significant to research the tendency and effects of the linear, circular and polynomial blade section on mesh characteristics for face-hobbed hypoid gears. Theoretically, the mathematical model for hypoid gears is established by simulating the cutting process of gear blank by cutter plate. An accurate simulation model is a precondition of face-hobbed hypoid gears for some research. Litvin and Zhang proposed the local synthesis method of the geometry model in order to build face-milling hypoid gears [1]. Fan proposed an enhanced approach to improve the nonlinear iterations and iteration convergence for solving the mathematical tooth flank equation conveniently [2]. Du and Fang obtained a real tooth surface equation by measuring the real tooth surface of gear and pinion [3]. Liu et al. researched the impacts of work holding equipment errors on contact characteristics and ∗
Corresponding author. E-mail address: [email protected] (C. Song).
https://doi.org/10.1016/j.mechmachtheory.2019.103739 0094-114X/© 2019 Elsevier Ltd. All rights reserved.
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C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739
Fig. 1. Blade sections of wheel and pinion.
gear flank geometry of face-hobbed hypoid gears based on the tooth surface model [4]. Shao et al. proposed a data-driven optimization model to calculate the machine settings and established a double-curved shell finite element model of hypoid gear [5]. Some researches in order to improve the modeling accuracy of hypoid gears by description of cutting edge in detail. Guo et al. introduced a multi-segment cutter blade profile to obtain the ideal load and contact stress distribution [6]. Vimercati derived the expressions of blade section included the major cutting edge and fillet segment and the mathematical representation of gear tooth surfaces by using theory of gearing [7]. Du and Fang proposed an active design method of tooth flank based on the one time of correction for pinion tooth surface [8]. Zhou et al. developed to automatically generating the 3D models with general spur pinions or the involute helical pinion [9]. The design and choose of blade section in details can be a method to optimize the mesh condition for reducing vibration noise. In generally, the blade section including linear blade, circular blade and polynomial blade for hypoid gear was introduced in many research papers [10–12]. And many researchers notice the point of the cutter geometric. Xie proposed a genuine face milling cutter geometric model for spiral bevel and hypoid gears and proves that the genuine cutter geometric model was correct and essential [13]. Chen et al. introduced a method to generate gear pairs by asymmetric parabolic profiles and the mathematical model was obtained using the coordinate transformation matrix [14]. Fuentes-Aznar et al. investigated the dual interlocking circular cutters was developed for straight bevel gears manufactured and the impact of the radius for the cutting disks on the maximum bending stresses [15]. Guo et al. proposed a novel skiving tool with multiple subclades on the rake face with the traditional tool to save the cost of skiving tool [16]. Wang and Fong proposed a new type of double-crowned helical gear and the circular-arc cutter blades be used in gears model [17]. However, there are very few researches to notice the effects of different blade sections and polynomial blade on mesh characteristics and it is significant on choose different blade section to improve the meshing quality and to reduce vibration noise for hypoid gears. This paper mainly focused on the advantages and disadvantages and difference of three blade sections on mesh characteristics for face-hobbed hypoid gears. Firstly, the accuracy mathematical model and finite element model with the three kinds of blade sections for hypoid gears are proposed. Meanwhile, the pinion is established with linear, circular and polynomial blade and the wheel use circular blade. Then, the effects of different blades on mesh characteristics are investigated. Finally, the advantages and disadvantages of gear pair with linear, circular and polynomial blade are analyzed. 2. Mathematical and mesh model for face-hobbed hypoid gears The tooth surface topography is related to blade section, tool geometry and machine parameters. The forming method is used to manufacturing wheel and the generating method for pinion. The tooth surface mathematical model is established by the equation and transformation matrix of blade section, cutter plate and machine. 2.1. Depiction of blade sections From above-mentioned, the blade section can be divided into linear blade, circular blade and polynomial blade. The details of blade section are described in Fig. 1. The N–M segment is the major cutting edge and M–F segment is fillet. The symbols oq and of are the reference points of major cutting edge and center of fillet, respectively. The coordinate system Sq is established reference point. For the pinion, the location and pressure angle of point oq are identical. The descriptions of cutting edge for wheel and pinion can be obtained by the derivation of mathematical equations and transformation matrixes [18]. Firstly, the blade section expression of wheel in coordinate system Sq can be deduced according to blade section of wheel in Fig. 1(a). The expression can be represented as follow
C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739
N–M:
⎡
⎤
⎡
⎤
xqv 2RBH v sin (u/(2RBH v ) ) sin (α0v − u/(2RBH v ) ) 0 ⎢y q v ⎥ ⎢ ⎥ rqv ( u ) = ⎣ ⎦ = ⎣ zqv 2RBH v sin (u/(2RBH v ) ) cos (α0v − u/(2RBH v ) )⎦ 1 1 M–F:
⎡
⎤
3
⎡
(1)
⎤
xqv rbhv cosθ ⎢y ⎥ ⎢ 0 ⎥ r q v θ = ⎣ q v ⎦ = M q f v ∗⎣ zqv rbhv sinθ ⎦ 1 1
(2)
where RBHv and rbhv are the circular radius of major cutting edge and arc segment radius of wheel convex machining inner blade, respectively. The parameters u and θ are the variable for the major cutting edge and arc segment in coordinate system Sq , respectively. α 0v is the pressure angle of inner blade reference point for wheel. The symbol Mqf is transformation matrix from of to oq of inner blade. Then, the value of pressure angle and location of reference point oq are defined to be consistent for three kinds of blade sections. Finally, the expressions of linear, circular and polynomial blade sections can be calculated according to the description of blade section of pinion in Fig. 1(b). The linear blade expression of pinion can be represented as follow N–M:
⎡
⎤
⎡
⎤
xqc −usinα0c 0 ⎢yqc ⎥ ⎢ ⎥ rqc (u ) = ⎣ ⎦ = ⎣ zqc ucosα0c ⎦ 1 1 M–F:
⎡
⎤
⎡
(3)
⎤
−wm + rbhc sin α0c + θ − sinα0c xqc ⎢yqc ⎥ ⎢ 0 rqc θ = ⎣ ⎦ = ⎣ zqc hm + rbhc cos α0c + θ − cosα0c 1 1
⎥ ⎦
(4)
where the symbols α 0c is outer blade pressure angle and rbhc is arc segment radius of pinion. wm and hm are the horizontal and vertical coordinate value of point M, respectively. Similarly, the circular blade expression of pinion can be represented as follow N–M:
⎡
⎤
⎡
⎤
xqc 2RBHc sin (u/(2RBHc ) ) sin (α0c − u/(2RBHc ) ) 0 ⎢yqc ⎥ ⎢ ⎥ rqc (u ) = ⎣ ⎦ = ⎣ zqc 2RBHc sin (u/(2RBHc ) ) cos (α0c − u/(2RBHc ) )⎦ 1 1 M–F:
⎡
⎤
⎡
(5)
⎤
xqc rbhc cosθ ⎢yqc ⎥ ⎢ 0 ⎥ r q v θ = ⎣ ⎦ = M q f c ∗⎣ zqc rbhc sinθ ⎦ 1 1
(6)
where RBHc and rbhc are the circular radius of major cutting edge and arc segment radius of pinion concave machining outer blade, respectively. The symbol α 0c is outer blade pressure angle of reference point for pinion. The symbol Mqf is transformation matrix form of to oq of outer blade [18]. The polynomial blade expression of pinion can be represented as follow O–M:
⎡
⎤
⎡
⎤
xqc −u1 ∗ sinα0c − c1 ∗ u21 ∗ cosα0c 0 ⎢yqc ⎥ ⎢ ⎥ rqc (u ) = ⎣ ⎦ = ⎣ zqc u1 ∗ cosα0c − c1 ∗ u21 ∗ sinα0c ⎦ 1 1 O–N:
⎡
⎤
⎡
(7)
⎤
xqc u2 ∗ sinα0c − c2 ∗ u22 ∗ cosα0c 0 ⎢yqc ⎥ ⎢ ⎥ rqc (u ) = ⎣ ⎦ = ⎣ zqc −u2 ∗ cosα0c − c2 ∗ u22 ∗ sinα0c ⎦ 1 1
(8)
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Fig 2. The mathematical model of cutter plate.
M–F:
⎡
⎤
⎡
⎤
−wm + rbhc sin α0c + θ − sinα0c xqc ⎢yqc ⎥ ⎢ 0 rqc θ = ⎣ ⎦ = ⎣ zqc hm + rbhc cos α0c + θ − cosα0c 1 1
⎥ ⎦
(9)
where the parameters u1 and u2 are the variable for the tangent of O–M cutting edge and O–N cutting edge in coordinate system Sq , respectively. The symbols c1 and c2 are the polynomial coefficient for the O–M segment and O–N segment, respectively. By similar deduction of major cutting edge and fillet for concave tooth surface of wheel and convex tooth surface, the equations for blade section of hypoid gears can be obtained. 2.2. Tooth surface equation for face-hobbed hypoid gears Tooth surface can be considered as the cutting track of the blade section in gear blank. With computing of the tooth surface equation, the expressions of blade section for wheel and pinion in coordinate system Sq and proper coordinate transformation for three-faced blade, cutter plate and machine are essential based on actual manufacturing process and the theory of gear meshing. The expressions of blade section have been calculated by above section. Fig. 2 reports a three-faced blade and face-hobbed cutter plate. There are three parameters related to the tooth surface according to the geometry and the benchmark of blade for three-faced blade, including rake angle, regrind angle and cutting side relief angle. The cutter plate have four parameters relating to tooth surface in the light of the location of three-face blade at cutter plate and the rotate vector, including offset angle of cutter blade, cutter radius, the initial assemble angle of inner blade and outer blade and the rotate angle of cutter plate. As mentioned, the transformation matrix of three-faced blade and cutter plate from Sq to St can be obtained by
Mtq = Mts (βv , Rc , δs ; β ) ∗ Msq (γs , ε , γa )
(10)
where Msq is transformation matrix from Sq to Ss based on the three-faced blade and Mts is transformation matrix from Ss to St based on the cutter plate. The symbols γ a , ɛ and γ s are rake angle, regrind angle and cutting side relief angle, respectively. Moreover, the parameters δ s , R, β v /(β c ) and β are offset angle of cutter blade, cutter plate radius, the initial assembly angle of outer blade/(inner blade) and the rotate angle of cutter plate, respectively. The initial assemble angle of outer blade and inner blade is fixed. In this paper, the initial assemble angles of outer blade and inner blade are 10.5882◦ and 21.1765◦ , respectively. Further, the transform matrix of machine from St to Sp are proposed in order to get the cutting track of blade on the gear blank. Fig. 3 represents the machine coordinate system from the center of cutter plate to gear blank for hypoid gears. The generator coordinates have nine parameters including the tilt angle i, swivel angle j, radical distance SR , cradle angle q0 , ϕ c (q0 is initial cradle angle and ϕ c is cradle rotation angle), work offset E, sliding base B, machine root angle γ m and horizontal A. The transformation matrixes of manufacturing machine from St to Sp can be obtained as follow
M pt = M pk (ϕ1 ) ∗ Mkh ∗ Mhg ∗ Mgm ∗ Mmc (ϕc ) ∗ Mce ∗ M f e ∗ Met
(11)
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Fig 3. Manufacturing machine coordinate system. Table 1 Parameters of cutter plate. Parameter
Pinion Convex
Number of cutter heads Teeth number Offset angle (◦ ) Regrind angle (◦ ) Cutting side relief angle (◦ ) Rake angle (◦ ) Cutter plate radius (mm)
z0 z
σ0
ɛ
γa γs Rc, v
Gear Concave
Convex
Concave
17 22.9309 4.3674 13.3885 9.5870 75.934
11 22.9309 4.7856 12.6729 10.3059 76.066
22.9309 5.0309 13.3823 12.6781 75.9193
47 22.9309 4.1101 12.6781 9.6107 76.0807
Thus, the expression of cutting track for wheel and pinion in the gear blank can be calculated by
r pv (ϕ1 , β ) = M pt ∗ Mtq ∗ rqv r pc (ϕ1 , ϕc , β ) = M pt ∗ Mtq ∗ rqc
(12)
In order to solve this equation, the processing methods and the principle of gear meshing must be applied. For the forming processing, the cutting track is equal to tooth surface due to the cradle fixed. Therefore, the equation of tooth surface can be solved according to relational about cutter plate and gear blank as follow
ϕ1 =
z ∗β z0
(13)
For the pinion with generating method, it is necessity to solve tooth surface for meshing equation [19]. The relationship of rotation angle of gear blank with cradle angle and the rotate angle of cutter plate.
f = n pv p = 0
ϕ1 = igp ∗ ϕc +
(14) z0 β z
(15)
where np is unit normal vector and vp is direction of relative velocity in coordinate system Sp . The parameter igp is ratio of roll. The parameters z and z0 are the number of processing gear and blade groups, respectively. 2.3. Finite element mesh model The main parameters of cutter plate are shown in Table 1. The machine processing parameters are shown in Table 2. By means of the mathematical model of wheel and pinion and these parameters, the finite element model is established in ABAQUS. The details of the finite element model are demonstrated in Fig. 4. Firstly, the material property is isotropic with young’s modulus 210,0 0 0 MPa and Poisson’s ratio 0.3. The reference point P1 and P2 are established on pinion and wheel rotation axis and the surface 1 and surface 2 are coupled in reference points
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C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739 Table 2 Parameters of machine processing. Parameter
Pinion Convex
Radial distance (mm) Initial cradle angle (◦ ) Work offset (mm) Sliding base (mm) Horizontal (mm) Machine root angle (◦ ) Tilt angle (◦ ) Swivel angle (◦ ) Ratio of roll
SR q Em B A
γm i j igp
Gear Concave
Convex
116.6202 52.1131 40.5991 28.7681 0.4666 0.3457 4.348969 131.8117 4.348969
Concave
122.009 −29.7048 0 0 13.9199 64.7995 0 0 0
Table 3 Blade section parameters of wheel convex. Parameters Pressure angle (◦ ) Cutter plate radius (mm) Arc segment radius (mm) Circular radius (mm)
C-B
α 0v Rc rbhv RBHv
18.9931 75.9193 0.9 1047
Fig. 4. Finite element model of hypoid gears.
P1 and P2 , respectively. Then, the contact surface be set to pinion concave and wheel convex and the boundary conditions are divide into pre-contact, applying load and mesh process. The torque load is applied on the wheel. Finally, the mesh of contact surface becomes more intensive by changing the size of local seed. The meshing of pinion are 176,328 elements and 89,400 elements for wheel. 3. Results and discussion According to different matching method for wheel and pinion, the results of mesh characteristics are analyzed. The torque is set to 450 Nm. The wheel model is all the same, but the pinion is replaced by choosing different types and parameters of blade section. Thus, this paper analyzed four situations from following. The details parameters of wheel convex are shown in Table 3. 3.1. Three kinds of blade section Firstly, the effects of the different blade sections for linear blade (L-B), circular blade (C-B) and polynomial blade (P-B) must be researched. The parameters of blade section of pinion concave are shown in Table 4. The tooth surface deviation of circular blade and polynomial blade compare to linear blade is shown in Fig. 5.
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Table 4 Parameters of three different types of blade section. Parameters ◦
Pressure angle ( ) Cutter plate radius (mm) Arc segment radius (mm) Circular radius (mm) Polynomial coefficient
L-B
C-B
P-B
– – –
20.1933 76.066 0.9 1200 – –
– −0.00005 −0.00005
α 0c Rv rbhc RBHc c1 c2
Fig. 5. Tooth surface deviation for the three kinds of blade.
Fig. 6. Contact pattern for the three kinds of blade sections.
The circular blade removes more materials in top and root of tooth surface and the change of tooth surface geometry in pitch line have less influence compared to linear blade. For the polynomial blade, the tooth surface geometry is related to the value and direction of polynomial coefficient. When the polynomial coefficient selection is minimum, the deviation of tooth surface is slight. The influences of different blade section on contact pattern are shown in Fig. 6. The shape and location of contact pattern are almost similar among the three kinds of blade sections when the circular radius choose large and the polynomial coefficient is small. The area radio of contact pattern and the maximum value of contact stress have difference due to the deviation of tooth surface geometry as shown Fig. 5. The influences of different blade sections on tooth root stress are shown in Fig. 7. The tendency and the maximum root stress are consistent. The influences of different blade sections on transmission error are shown in Fig. 8. The angular transmission error and the value of peak-to-peak (P–P) of transmission error are almost identical. Thus, there is less change in the mesh characteristics of three kinds of blade section when the circular radius chooses very large and the polynomial coefficient is small. 3.2. Different circular radius In order to obtain the influence tendency for C-B, the effect analysis of changing the circular radius is significance. In this section, the circular radius select 200 mm, 400 mm, 900 mm and 1200 mm and the details are shown in Table 5.
8
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Fig. 7. Tooth root stress for the three kinds of blade sections.
Fig. 8. Transmission error for the three kinds of blade sections. Table 5 Parameters of circular blade. Parameters
Circular blade ◦
Pressure angle ( ) Cutter plate radius (mm) Arc segment radius (mm) Circular radius (mm)
α 0c Rv rbhc RBHc
200
20.1933 76.066 0.9 400
900
1200
The results of contact pattern are depicted in Fig. 9. The location of contact pattern is similar with the circular radius changing. However, with the circular radius reducing, the area of contact pattern gradually decreases due to the reduction of length axis of contact ellipse and the maximum value of contact stress gradually getting stronger. Moreover, the high stress zone will become bigger if the circular radius is smaller. The effects of circular radius on tooth root stress are shown in Fig. 10. With the circular radius increasing, the tendency of tooth root stress is similar but the maximum value for tooth root bending stress is reduced and the location point gradually moves from toe side to heel side. The influences of circular radius on transmission error are shown in Fig. 11. It is obvious effect on transmission error with the changing of circular radius. The tendency of angular time-varying transmission error is analogous while the circular radius changes from 200 mm to 1200 mm. However, the circular radius is increasing from small to large, the angular timevarying transmission error gradually become larger and the P–P value of transmission error is gradually rising. Particularly, the changing is very small when the circular radius increase to quite large. 3.3. The effect of the different magnitude coefficient In order to obtain the influence law for P-B, the analysis of the magnitude coefficient and the different coefficient direction of O–M and O–N segment is essential. In this section, the magnitude coefficient choose negative values referencing to circular blade. The parameters of magnitude coefficient are shown in Table 6. The contact pattern of different magnitude coefficient are shown in Fig. 12.
C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739
Fig. 9. Contact pattern for different circular radius.
Fig. 10. Tooth root stress for different circular radius.
Fig. 11. Transmission error for different circular radius.
9
10
C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739 Table 6 Parameters of magnitude coefficient. Parameters
Polynomial blade ◦
Pressure angle ( ) Cutter plate radius (mm) Arc segment radius (mm) Polynomial coefficient
α 0c Rv rbhc c1 c2
−0.00005 −0.00005
20.1933 76.066 0.9 −0.0005 −0.0005
−0.005 −0.005
Fig. 12. Contact pattern for different magnitude coefficient.
Fig. 13. Tooth root stress for different magnitude coefficient.
When the coefficient is −0.0 0 0 05, the contact pattern is closed to linear blade, but the contact pattern is more closed to the result when the circular blade radius is 200 mm and the coefficient is −0.005. In addition, the maximum contact stress of tooth surface is gradually decreasing with the magnitude coefficient increasing. The effects of different magnitude coefficient on tooth root stress are shown in Fig. 13. The tendency of tooth root stress is also similar with the changing of magnitude coefficient. The value of the maximum tooth root stress is slightly increasing and the location move from heel side to toe side with the value increasing. The influences of different magnitude coefficient on transmission error are shown in Fig. 14. It is obvious influence of the different magnitude coefficient on transmission error and the tendency of angular time-varying transmission error is similar. The angular time-varying transmission error gradually decrease and the P–P value of transmission error is gradually reduced with the magnitude coefficient increasing. 3.4. The effects of the coefficient direction for polynomial blade In this section, the effects of the different coefficient direction on tooth surface deviation and mesh characteristics are analyzed. The direction of O–M and O–N segment have four types that the coefficients are negative with negative, negative with positive, positive with negative and positive with positive and the parameters are shown in Table 7.
C. Liang, C. Song and C. Zhu et al. / Mechanism and Machine Theory 146 (2020) 103739
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Fig. 14. Transmission error for different magnitude coefficient.
Fig. 15. Tooth surface deviation for the four coefficient direction. Table 7 Parameters of coefficient direction. Parameters Pressure angle (◦ ) Cutter plate radius (mm) Arc segment radius (mm) Polynomial coefficient
Polynomial blade
α 0c Rv rbhc c1 c2
−0.0005 −0.0005
20.1933 76.066 0.9 −0.0005 +0.0005
+0.0005 −0.0005
+0.0005 +0.0005
The deviation of tooth surface topography for different coefficient direction is shown in Fig. 15. The deviation is small at pitch line and the top and root side is large. The cutting amount is different while the direction of polynomial coefficient is opposite. The results of different coefficient direction on contact pattern are shown in Fig. 16. When the polynomial coefficient is negative with positive and positive with negative, the area of contact pattern is increased and the maximum contact stress of tooth surface is decreased compared to the situation that the coefficient is negative with negative. The phenomenon of edge contact will appear while the coefficient is positive with positive. The different coefficients can slight change the location for contact pattern and it can be a method to optimize tooth surface
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Fig. 16. Contact pattern for the four kinds of coefficient direction.
Fig. 17. Tooth root stress for the four coefficient direction.
Fig. 18. Transmission error for the four kinds of coefficient direction.
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design. The effects of different coefficient direction on tooth root stress are shown in Fig. 17. The tendency of tooth root stress is also similarly with different coefficient directions. The value of the maximum tooth root stress is almost identical without edge contact. The location will slight moving to heel side while the coefficient is negative with positive. The influences of different coefficient direction on transmission error are shown in Fig. 18. The tendency of angular timevarying transmission error is also similar but the change and amplitude of transmission error is bigger while arising edge contact. In addition, with the coefficient choose positive with negative, the P–P value of transmission error is the lowest. 4. Conclusion In this paper, the expressions of linear blade, circular blade and polynomial blade and an accurate tooth surface mathematical model based on the three-faced blade and simulating the cutting process have been presented. The effects of three kinds of blade sections for pinion and circular blade for wheel on mesh characteristics are analyzed by the finite element method. There are four findings as following (1) While the circular radius is very big of circular blade and the coefficient is small of polynomial blade, the effects are almost analogous on mesh characteristics with linear blade. (2) The circular radius has an adjustable parameter compared to linear blade. The P–P value transmission error can be decreased by reducing circular radius but the contact pattern area shrinks, the maximum value of contact stress will increase and the location of maximum tooth root bending will move from toe to heel side. (3) With the polynomial blade use more advanced curve, the adjustable parameters have more choices. The effects of different magnitude polynomial coefficient (negative with negative) on mesh characteristics are similar to linear and circular blade by changing the magnitude polynomial coefficients. (4) The mesh characteristics are the best that can reduce transmission error without losing contact area and decreasing tooth root bending stress when the coefficient is positive with negative and the effects are the worst due to arise edge contact when the coefficient is positive with positive among all types of polynomial coefficients. In addition, the situation of negative with positive can slightly moving the location of contact pattern, which can provide a design method by changing blade section. Declaration of Competing Interest None. Acknowledgments The authors would like to thank the National Natural Science Foundation of China (51775061), Science and Technology Major Project of Guangxi (Guike AA19182001), Research and Development Plans in Key Areas of Guangdong (2019B090917002) and National Key Research and Development Plan of China (2019YFB20 0470 0). References [1] F.L. Litvin, Y. 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