A CNC tooth grinding method for formate face hobbed hypoid gears

Mechanism and Machine Theory 144 (2020) 103628

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Research paper

A CNC tooth grinding method for formate face hobbed hypoid gears Weiqing Zhang a,∗, Xiaodong Guo a, Yawen Wang b, Yikai Wen a, Chia-Ching Lin b, Teik C. Lim b a b

School of Mechanical Engineering, Chongqing University of Technology, 400054, China Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, 76019, USA

a r t i c l e

i n f o

Article history: Received 1 May 2019 Revised 19 July 2019 Accepted 22 September 2019

Keywords: Tooth grinding Hypoid gears Face hobbing Tool location Interference checking

a b s t r a c t Face hobbing is an efficient process for hypoid gear cutting. However, the continuous indexing process renders it inapplicable for tooth grinding, which limits the further improvement of gear accuracy and transmission performance. In this paper, a tooth grinding method using large diameter conical grinding wheel for Formate® face hobbed hypoid gears (FFHHG) was proposed. The generator of conical grinding wheel was used to replace the cutter blade of the face hobbing process. By controlling the conical grinding wheel, the generator of the grinding wheel can be tangentially in contact with the tooth profile curve of the FFHHG. The sweeping of the generator of the grinding wheel along the extended epicycloidal tooth curve can grind one tooth surface accurately without theoretical deviation. Firstly, the tooth surface was formulated and the geometric parameters of the working side of the grinding wheel were determined to avoid the curvature interference between the grinding wheel and the ground tooth surface. Then, a five-axis tooth grinding tool location model was established to position the grinding wheel and the tooth surface correctly. The interference of the non-working side of grinding wheel was checked and if necessary the grinding wheel axis can be tilted to avoid interference. Finally, the above processes were verified by case analysis and grinding simulations. This study can be employed to improve the pitch accuracy of FFHHG and reduce the tooth profile deviation. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The face hobbing process of hypoid gears is more efficient than face milling process due to the principle of continuous indexing, in which both concave and convex surfaces can be machined in one cutting cycle with one cutter. Therefore, this process has been widely employed by the gear industry. The fundamental tooth surface generation and cutting process were studied extensively by many researchers. Litvin and Fan [1-2] analyzed the kinematics for generation of face-hobbed hypoid gears and derived the tooth surface equations. Shih [3-4] developed a universal mathematical model for face-hobbed spiral bevel and hypoid gears, which can be used to simulate Klingelnberg’s and Gleason’s face hobbing processes. She also proposed a flank modification methodology for face-hobbed hypoid gears based on ease-off topography. Zhang [56] proposed a machine setting calculation method for face hobbed hypoid gears, which can be used to designate the position ∗

Corresponding author. E-mail address: zwqcool20 0 [email protected] (W. Zhang).

https://doi.org/10.1016/j.mechmachtheory.2019.103628 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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W. Zhang, X. Guo and Y. Wang et al. / Mechanism and Machine Theory 144 (2020) 103628

of contact reference points on the tooth surface and simultaneously control the contact on both sides of the tooth. He also provided a motion control solution for face-hobbing on the computer numerically controlled (CNC) hypoid generator. Habibi [7] researched the instantaneous undeformed chip geometry and predicted the cutting forces during face-hobbing process. Fan [8] introduced a method of correcting tooth surface errors for spiral bevel and hypoid gears generated by face-hobbing process using CNC hypoid generators. Simon [9] investigated the influence of tooth modifications induced by machine tool setting and head-cutter profile variations on tooth contact characteristics in face-hobbed spiral bevel gears. Vimercati [10] derived the accurate mathematical representation of hypoid gears tooth surface with face-hobbing method, which is used as the input for loaded tooth contact analysis. However, the face hobbing process has its disadvantages. The principle of continuous indexing requires the cutter blades to be divided into several groups like the teeth of the cutter. In addition, the tooth surface of its generating gear is not a revolution surface, so the tooth grinding can hardly be applied for this gear type. Most of the current research efforts on tooth grinding are aimed at the hypoid gears with face milling process. McVea [11] described a machine process for rapid hard finish grinding of large non-generated bevel gears and their mating pinions. A specially designed flaring cup shaped wheel was used for the gear member to grind the non-generated profile without burning. Kimmet [12] developed a process for high production rate finish grinding of hardened spiral bevel and hypoid gears using cubic boron nitride (CBN) grinding wheels. Liu [13] studied the volumetric error compensation of the computer numerical control spiral bevel gear grinding machine. Zhang [14] built the machining errors model of CNC spiral bevel gear grinding machine, which is used to correct tooth form deviation. Wang [15] proposed a mathematical model of surface roughness for ground spiral bevel gears. Ding [16] proposed a new grinding machine settings modification methodology considering the spatial geometric errors for grinding hypoid gears with high accuracy. Despite of the aforementioned studies on grinding methods for face-milled hypoid gears, there are very few research publications on the grinding of face-hobbed hypoid gears. Wiener [17] established a semi-completing process in which the cup-shaped grinding wheel replaces the epicycloidal flank curve. However, this method leads to non-uniform stock removal and large tooth deviation. Deng and Álvarez [18,19] proposed a machining method of spiral bevel gears on universal 5-axis milling machines with generic cylindrical or disk tools. This method can be potentially used for grinding face-hobbed hypoid gears, but is not suitable for mass production due to low efficiency. The driven member of hypoid gear pair for automobile applications are usually machined by Formate® method. When Formate® method is used, there is only continuous indexing motion and no tooth generating motion. Due to the lack of effective means for grinding face-hobbed hypoid gears, the gear tooth distortion after heat treatment can hardly be corrected, which limits the further improvement of gear accuracy and transmission performance [20]. In this paper, a tooth grinding method for Formate® face hobbed hypoid gears (FFHHG) using large diameter conical grinding wheel is proposed. First, the tooth surface equation of the FFHHG is deduced, and the geometric parameters of the working side of the grinding wheel are determined by analyzing the curvature characteristics of the tooth surface. Then a tool location calculation model for tooth grinding is established to ensure accurate tooth profile in the traditional five-axis hypoid grinding machine. Next, the interference is checked to avoid the interference of non-working side of grinding wheel. The proposed tooth grinding method does not generate theoretical tooth deviation and is highly efficient, which is suitable for mass production. 2. Tooth grinding In the face hobbing process, the cutter and the gear rotate according to a certain transmission ratio, which achieves continuous gear indexing motion. At the same time, the blade forms an extended epicycloidal flank trace on the generating gear in this relative motion. When the Formate® method is adopted, there is only continuous indexing motion and no tooth generating motion during face hobbing process. Therefore, the generating gear and the machined gear have same tooth surface, which is the sweeping surface of the cutter blade along the extended epicycloid. When the blade is straight, the tooth surface is a ruled surface which is relatively easy to be reconstructed. The tooth grinding method for FFHHG without tooth deviation can be achieved according to the flowchart shown in Fig. 1. The procedures of tooth grinding are as follows: (1) Calculate the tooth surface equation by machine setting parameters; (2) Select geometry parameters of conical grinding wheel according to the geometrical characteristics of the tooth surface. It needs to be ensured that the grinding wheel can be placed in the tooth slot and curvature interference does not occur on the working side of the grinding wheel; (3) Replace the cutter blade with the generator of conical grinding wheel. By controlling the position of the conical grinding wheel, the generator of the grinding wheel can be tangentially in contact with the tooth profile curve of the FFHHG; (4) Sweep the generator of the grinding wheel along the root curve of the tooth to calculate the tool locations of tooth grinding; (5) Check interference for no-working side of grinding wheel; (6) Avoid interference by tilting the axis of the grinding wheel; and (7) Update the grinding wheel parameters and the tool locations of tooth grinding at the same time. 3. Tooth surface equation The machine setting of Formate® face hobbing process is shown in Fig. 2. Firstly, the coordinate systems are established for cradle, cutter and gear separately. The coordinate system r = {O, ir , jr , kr } is rigidly connected with virtual cradle, in

W. Zhang, X. Guo and Y. Wang et al. / Mechanism and Machine Theory 144 (2020) 103628

Fig. 1. The flowchart of tooth grinding process for FFHHG.

Fig. 2. The machine setting model of Formate® face hobbing process.

3

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which O is the center of cradle, ir Ojr is the plane of cradle, kr is the axis of cradle, and its positive direction points to inner of cradle. The coordinate system c = {Oc , ic , jc , kc } is rigidly connected to the cutter, in which Oc is the center of cutter, ic Oc jc is the plane where the blade pitch point is located, kc is the axis of cutter whose positive direction opposites to tip of cutter, and jc is the projection vector of kc on the cradle plane. The coordinate system c = {Oc , ic , jc , kc } is for cutter installation. Its origin coincides with that of c , such that ic points to cutter center from cradle center, and kc and kr are in the same direction. The coordinate system g = {Og , ig , jg , kg } is rigidly connected to the gear, in which Og is the cross point of gear and the direction of its coordinate axes is the same as that of r . Assuming that p is an arbitrary point on the blade, then the position vector of p in the coordinate system c can be expressed as:



 



Vp = rc jc + bM jc , η M kc , δ tc tc = [0, −sinα , −cosα ]

T

(For outside of blade) (1) (For inside of blade) where α is the pressure angle of blade, δ is the offset angle of blade, η is the hook angle of blade, rc is the cutter radius at the pitch point of blade, and b is the distance from p to the pitch point of blade. The expression M(u, τ ) is the transformation matrix in which a vector is rotated by angle τ around unit vector u. The normal vector of blade at point p can be T tc = [0, sinα , −cosα ]

expressed as:



 



nc = M jc , η M kc , δ nc0 nc0 = [0, cosα , −sinα ] (For outside of blade) T

nc0 = [0, −cosα , −sinα ] (For inside of blade) T

(2)

When the cutter is titled, the axis vector of the cutter is no longer parallel to the axis of the cradle, which can be expressed in the coordinate system c :

Vc = M ( jc , J )M (ic , I ) kc

(3)

where I is tilt angle and J is swivel angle. Assuming that θ is the rotation angle of the cutter, the normal and position vectors of point p after cutter titled can be transformed to the coordinate system c :

nc = M (Vc , θ ) M ( jc , J )M (ic , I ) nc

(4)

Vp = M (Vc , θ ) M ( jc , J )M (ic , I ) Vp

(5)

The above vectors are transformed into the cradle coordinate system r

respectively:

nrc = M (kr , q )nc

(6)

Vpr = M (kr , q )Vp + Vsr

(7)

Vsr = S[cosq, −sinq, 0]

T

(8) Vsr

where q is the cradle angle, S is radial distance, and is the vector form cradle center to cutter center. When the cutter rotates the angle θ , the gear should also rotate in proportion to angle ϕ due to continuous indexing motion.

ϕ=

zs θ z

(9)

The normal vector

ngc

= M (Vg , ϕ )

g nc

and position vector

nrc

g Vl

of point p can be expressed in gear coordinate system g :

(10)

where

Vg = [−cosγ , −sinγ , 0]

T

Vlg = M (Vg , ϕ )Vlr

(11) (12)

where

Vlr = V rp + Vdr

(13)

Vdr = X pVg + Xb kr + Xe kr

(14)

in which Xp is machine center to back, Xb is sliding base, Xe is work offset, γ is machine root angle, Vg is the vector of gear g axis, and Vdr is the vector from gear cross point to cradle center. The vector function Vl has two variables θ and b, which can be used to express the tooth surface equation of the FFHHG.

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4. Grinding wheel parameters In the process of tooth grinding using conical grinding wheel, it is necessary to avoid curvature interference at the contact point while the conical surface of the grinding wheel is in tangent contact with the ground tooth surface. Therefore, the curvature characteristics of the tooth surface need to be analyzed to determine the curvature of the grinding wheel. Partial g derivatives of vector function Vl are taken with respect to variables θ and b, then coefficients of the first fundamental form of tooth surface can be obtained:

∂ Vlg ∂b ∂ Vlg F = ∂b ∂ Vlg G= ∂θ E =

∂ Vlg =0 ∂b g ∂ Vl · ∂θ ∂ Vlg · ∂θ ·

(15)

Then partial derivatives of second-order are taken to get coefficients of the second fundamental form of tooth surface:

∂ 2Vlg ∂ b2 2 g ∂ Vl M = ngc · ∂ b∂ θ ∂ 2Vlg N = ngc · ∂θ2 L = ngc ·

(16)

According to the differential geometry theory, the following two equations can be constructed using these coefficients:

  κn E − L   κn F − M   Edb + Fdθ   Ldb + Mdθ

 κn F − M  =0 κn G − N    Fdb + Gdθ =0 Mdb + Ndθ 

(17)

(18)

where κn (n = 1, 2) is the principal curvature of tooth surface, db and dθ are the derivatives of tooth surface equation along two parametric curves, which can represent the tangent direction of two parametric curves, as shown in Fig. 3. The two principal curvature can be obtained by solving Eq. (17). Solving the Eq. (18), two ratio values of db/dθ can be obtained, which can represent two principal directions g1 and g2 . Since the b curve of the tooth surface is a set of straight lines, the tangent direction of b curve is one of the principal directions of the tooth surface. Assuming this direction is g1 , then the principal curvature κ1 = 0. The two principal curvatures of the grinding wheel are set to be κ p and κ w respectively, in which κ p is the curvature in the tp (tangent direction of generator of the grinding wheel), and κ w is the curvature in the tw (circumferential direction

Fig. 3. The geometry of tooth surface and grinding wheel.

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of the grinding wheel). In order to make the grinding wheel contact with the tooth surface along a curve during grinding, then

κ p = κ1 = 0

(19)

t p = g1

(20)

In this case, another principal direction tw of the grinding wheel coincides with g2 the principal direction of the tooth surface along the tooth width direction. In order to avoid the curvature interference in the other directions, The following requirements need to be met simultaneously for the relative curvatures κ n1 , κ n2 and relative geodesic torsion τ g1 between tooth surface and grinding wheel along g1 and g2 :

κn1 = sgn∗(κ p − κ1 ) ≥ 0

(21)

κn2 = sgn∗(κw − κ2 ) ≥ 0

(22)

 2 κn1 κn2 − τg1 ≥ 0

(23)

where sgn = 1 for concave side and sgn = −1 for convex side. Because the geodesic torsion in the principal direction is zero, τg1 = 0. From Eq. (21), κn1 = 0 can be derived. Therefore, the Eqs. (21) and (23) can be satisfied. In order to make Eq. (22) to be also satisfied, then

κw ≥ κ2 (For concave)

(24)

κw ≤ κ2 (For convex )

(25)

The principal curvature κ 2 varies with the position of the tooth surface, and so it is necessary to find the extremum of κ 2 in order to determine the extremum of κ w . According to the above analysis, the tooth surface is discretized into grids, and the variation regularity of κ 2 can be analyzed as shown in Figs. 4 and 5. The curves L1~L9 are principal curvature radius (1/κ 2 ) of tooth surface along b curves from the heel to the toe respectively. It can be seen that the 1/κ 2 increases gradually from the toe to the heel in the tooth width direction. In addition, by analyzing the variation pattern of 1/κ 2 along the tooth profile direction (tangent direction of b curve), it shows that 1/κ 2 increases linearly from the root to the top of the tooth for the concave side, and it decreases linearly from the root to the top of the tooth for the convex surface. The slope angle η of each curve is calculated separately:

η = tan−1 [(1/κ2 )/b]

(26)

In Fig. 6, it can be seen that the slope of curves increases gradually from toe to heel (L9 to L1) for the concave side. For convex side, the slope of curves increases gradually from heel to toe (L1-L9). Since the tip of grinding wheel is in contact with the root of the gear, the principal curvature κ2 of any point on the tooth surface along the direction of tooth profile

Fig. 4. The radius of principal curvature of concave (L1–L9 are profile curves from heel to toe).

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Fig. 5. The radius of principal curvature of convex (L1–L9 are profile curves from heel to toe).

Fig. 6. The slope angle of the curve of principal curvature radius:

concave side, and

convex side.

is determined by reference to the principal curvature κ 2r at root curve:

κ2 =

1 κ2r ± h ∗ tan η 1

(27)

According to the geometrical characteristics of the conical surface, the principal curvature of the corresponding points on the generator of grinding wheel can be represented as follows:

κw =

rw cosμ

1 ± h∗tanμ

(28)

where rw is the radius of the top of the grinding wheel, μ is the pressure angle of the grinding wheel. Comparing Eq. (27) and Eq. (28), the principal curvature of the root position of the tooth toe along the tooth width direction should be used to determine the principal curvature of the grinding wheel along the circumferential direction for the concave side. Let

μ ≤ |ηt |

(29)

rw ≤ cosμ/κ2tr

(30)

then the curvature of grinding wheel satisfies Eq. (24). For the convex side, the principal curvature of the grinding wheel along the circumferential direction can be determined by the principal curvature of the root position of the tooth heel along

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the tooth width direction. The following two equations should be satisfied by the parameters of grinding wheel:

μ ≤ | ηh |

(31)

rw ≥ cosμ/κ2hr

(32)

where ηt is the slope angle of principal curvature curve corresponding to the b curve at the tooth toe of concave side, ηh is the slope angle of principal curvature curve corresponding to the b curve at the tooth heel of convex side. 5. Tool location of tooth grinding After obtaining the geometric parameters of the working side of grinding wheel, the position of the grinding wheel can be determined according to the contact state between the grinding wheel and the tooth surface, and then the CNC tool location of tooth grinding can be calculated. As shown in Fig. 3, while the generator of grinding wheel is tangent to the b curve of the tooth surface, the tip of the grinding wheel should also be tangent to the root cone of the gear to ensure the whole tooth surface being ground. Therefore, the tooth root curve can be regarded as the guide curve during tool path planning. According to the Section 3, the tooth surface equation of the FFHHG can be expressed as a vector function containing two variables θ and b. The root curve of the tooth is the θ curve corresponding to the tip of the cutter blade. Then the position and normal vectors of the root curve can be expressed as:

Vrg (θ ) = Vlg (θ , ba ),

θ ∈ (θt , θh )

(33)

ngr (θ ) = ngc (θ , ba )

(34)

ba = ha /cosα

(35)

where ba is the distance from pitch point of the blade to the tip point, ha is the height from the pitch point of the blade to the top plane of the cutter, and θ t , θ h are the θ parameter values of the root curve at the toe and heel of the tooth. They can be obtained by solving the following equation:

−Vrg (θt ) · Vg = lt − dl ∗ cosγr

(36)

−Vrg (θh ) · Vg = lh + dl ∗ cosγr

(37)

where lt and lh are the axial distances from the toe and heel of the gear root cone to the cross point of the gear pair respectively, γ r is the root angle, and dl is the extension value of the root curve to both ends of the actual tooth surface range. The position vector of grinding wheel top center in gear coordinate system g can be expressed as follows:

Vwg = Vrg + rw ∗ M (g2 , μ )ngr

(38)

The axis vector of grinding wheel kw in gear coordinate system g can be expressed as follows: g kw = M (g2 , μ )g1

(39)

Another coordinate system g = {Og , ig , jg , kg } is established to rigidly connect with gear at the cross point of gear pair, in which ig is in the same direction as the axis vector of the gear Vg , and jg is in the opposite direction with jg . The motion schematic diagram of a typical CNC hypoid grinding machine is shown in Fig. 7, which is a five-axis machine tool. The grinding wheel is located on the left side of the machine tool and is driven up and down by Y-axis and back and forth by Z-axis. The gear is located on the right side of the machine tool. A-axis is located on the B-axis, and the gear rotation is driven by the A axis, which can swing around the Y-axis. The B axis is located on the X-axis, which can be driven by it to move left and right longitudinally. Let machine tool coordinate system m = {Om , im , jm , km } have the same axis direction as g , and the coordinate origin is located at Om which is located on the A-axis (workpiece spindle), and its distance to the end of shaft A is R0 . The axis vector of grinding wheel is transformed into machine tool coordinate system m : g m kw = Mm kw

⎡

−cosγ

M m = ⎣0

sinγ

(40) 0

sinγ

−1

0

0

cosγ

⎤ ⎦

(41)

The position vector of grinding wheel top center is also transformed into machine tool coordinate system m :

Vwm = MmVwg + (R0 + LB + LM )im

(42)

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Fig. 7. The motion schematic diagram of a typical CNC hypoid grinding machine.

where LB and LM are arbor length and mounting distance of gear. Since the grinding wheel axis is in the direction of km and cannot be adjusted, the angle of the grinding wheel axis relative to the gear axis needs to be obtained by adjusting the gear axis. As shown in Fig. 7, the grinding wheel axis can be transformed to the direction of km by two steps. First, the original grinding wheel axis is rotated angle ψ 1 around the coordinate axis im of machine tool, so that it can be located in the im Om km plane. Then, it is rotated by angle ψ 2 around the coordinate axis jm of machine tool. m km = M ( jm , ψ2 )M (im , ψ1 )kw

(43)

ψ1 = tan−1 (kwm · jm /kwm · km )

(44)

ψ2 = −tan−1 (kwm · km /kwm · im )

(45)

In order to keep the relative position of the gear and grinding wheel, both the gear axis vector and the position vector of the grinding wheel top center should be transformed at the same time. The new vectors after transformation can be expressed as:

Vgm = M ( jm , ψ2 )M (im , ψ1 )im

(46)

Vtlm = M ( jm , ψ2 )M (im , ψ1 )Vwm

(47)

The tool location of tooth grinding for the five-axis hypoid grinding machine are as follows:

⎧ X = −Vtlm · im ⎪ ⎪ ⎪ ⎪Y = Vtlm · jm ⎨ Z = V m · km

tl ⎪ ⎪ ⎪ A = ψ1 ⎪ ⎩ B = ψ2

(48)

6. Interference check As mentioned in Section 3, the interference between the working side of grinding wheel and the machined tooth surface can be avoided by reasonably selecting the curvature radius and pressure angle. According to the pressure angle of the gear and the slot width, the geometric dimensions of the non-working side of the grinding wheel can be selected preliminarily. However, the shape of the grinding wheel determined in this way cannot ensure that there is no interference between the non-working side of the grinding wheel and the other side of the gear slot, so the interference of the non-working side of the grinding wheel must be checked. Because the curvature of tooth surface of the FFHHG varies monotonously along the tooth flank, the distance between grinding wheel and the other end of the tooth surface is the greatest when

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Fig. 8. The interference checking model.

grinding any end of the tooth surface, which is easy to interfere with the opposite tooth surface. Therefore, the interference of non-working side should be checked in the case of grinding the both ends of tooth surface.Taking the concave side as an example, when grinding the heel of the tooth surface with the outside of grinding wheel, the inside of grinding wheel is easy to interfere with the toe of convex side. The situation of grinding the heel of concave side is shown in Fig. 8. Let the position vector of the root point at the heel g g g of concave side in the gear coordinate system be Vrh . The position vector Vwh and axis vector kwh of the grinding wheel in grinding the heel of the gear concave side can be calculated by the method described in Section 5. According to the method described in Section 3, the tooth surface equation and normal vector equation of gear convex side can be obtained. g g Let them be Vv (θ , b) and nv (θ , b) respectively. According to the parameters of the gear blank, the coordinates of two points at the toe of gear teeth on the cross section along the axis direction can be obtained. If the two points are (Rt , Lt ), (Rr , Lr ), the coordinates of any point (R, L) on the boundary curve of the toe of the tooth on the cross section can be expressed as follows:

R = Rr + u(Rt − Rr ), L = Lr + u(Lt − Lr ) u ∈ (0, 1 )

(49)

Substituting them into the following equations, the tooth surface parameters (θ , b) corresponding to the boundary curve of the toe can be solved:

−Vvg (θ , b) · Vg = L

(50)

|Vvg (θ , b) × Vg | = R

(51) g Vvt (θ , b),

g nvt (θ , b)

The position and normal vectors of the point pgt on the toe of the convex side can be obtained by putting tooth surface parameters to the tooth surface equation. They are transformed into the grinding wheel coordinate system w = {Ow , iw , jw , kw } in order to analyze the relative position with the inside of the grinding wheel:





g g Vvwt = Mw Vvgt − Vwh , nw vt = Mw nvt

(52)

where Mw is the coordinate transformation matrix, which can be expressed as:

Mw = igwh T , j gwh T , kgwh T







(53)



g g g g g  jwh = Vrh − Vwh /Vrh − Vwh

(54)

g g igwh = jwh × kwh

(55)

The phase angle β of the point pwt on the grinding wheel corresponding to the toe boundary of the gear convex side can be expressed as:

β = cos−1 [(Vvwt − ht kw ) · jw ]

(56)

W. Zhang, X. Guo and Y. Wang et al. / Mechanism and Machine Theory 144 (2020) 103628

ht = Vvwt · kw

11

(57)

where ht is the distance from the point pgt to the top plane of the grinding wheel along the axis direction. Then the position vector of point pwt can be expressed as: w Vwt = rwt M (kw , β ) jw + ht kw

(58)

rwt = rw − dw − ht sinμn

(59)

where rwt is the tip radius of the non-working side of the grinding wheel, dw is the tip width of the grinding wheel, and μn is the pressure angle of the non-working side of the grinding wheel. Then the distance ρ between pwt and pgt can be expressed as: w ρ = [Vwt − (Vvwt − ht kw )] · [M (kw , β ) jw ]

(60)

If ρ is negative, it indicates interference between the inside of the grinding wheel and the convex of the gear, and if it is positive, it indicates that there is a gap. If ht is negative, the point does not need to be checked for interference, because it is below the top of the grinding wheel. Similarly, the interference of the heel of the convex side can be checked when grinding the toe of the concave side, and the interference between the grinding wheel and the concave side can be checked when grinding the convex side. When there exists interference between non-working side of grinding wheel and gear, it can be eliminated by reducing the tip width of grinding wheel and the pressure angle of non-working side of grinding wheel. However, it only works when the interference is insignificant, because the reduction of these two parameters is usually very limited in order to ensure the grinding wheel strength. In this case, the non-working side interference of grinding wheel can be avoided by tilting the axis of the grinding wheel. As shown in Fig. 9, if the grinding wheel axis is tilted to an angle It towards the grinding position of the tooth, the projection of the grinding wheel tip circle at the other end of the tooth surface will move away from the root to the top of the tooth, which helps increase the gap between the no-working side of the grinding wheel and the opposite tooth surface. After the axis of the grinding wheel is tilted, the pressure angle of the grinding wheel working side needs to be modified in order to keep the working side of grinding wheel in tangent contact with the ground tooth surface. As shown in Fig. 9(a), when grinding the concave side, the pressure angle of grinding wheel working side (outside) decreases after the tilting of the axis of grinding wheel. If the pressure angle of non-working side remains unchanged, it

Fig. 9. Interference avoidance by tilting the axis of the grinding wheel.

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will move away from the opposite tooth surface, which helps reduce the interference of non-working side. However, this may lead to insufficient strength of grinding wheel. Thus, one can consider to increase the pressure angle of non-working side in an appropriate amount. As shown in Fig. 9(b), when grinding gear convex side, the pressure angle of grinding wheel working side (inside) increases after tilts of the axis of grinding wheel. If the pressure angle of non-working side remains unchanged, the non-working side will approach to the opposite tooth surface, which may increase the interference of nonworking side in the grinding position. One can consider to reduce the pressure angle of non-working side accordingly.

μ = μ − It , μn = μn + It (For concave)

(61)

μ = μ + It , μn = μn − It (For convex )

(62)

The initial pressure angle of the grinding wheel non-working side should be determined according to the following equation:

μn = 2αa − μ − dμ

(63)

where α a is the average pressure angle of gear pair, dμ is the reduction of pressure angle of non-working side. According to the analysis in Section 4, when grinding gear concave side, the pressure angle of grinding wheel working side decreases after the tilting of the axis of grinding wheel, which can satisfy Eq. (24) and still ensure that there is no curvature interference in the direction of tooth profile. However, when grinding gear convex side, the pressure angle of grinding wheel working side increases after the axis of grinding wheel tilts. The curvature reduction rate of grinding wheel will exceed that of tooth surface in the direction of tooth profile, which cannot satisfy Eq. (25). It is necessary to increase the radius of curvature at the top of the grinding wheel:



rw ≥ cosμ /κ2hr + hmax tanμn − tan ηh



(64)

Therefore, the interference of non-working side of grinding wheel can be avoided theoretically by tilting the axis of the grinding wheel. The geometric shape of grinding wheel is changed after increasing the axis tilt angle of grinding wheel. It is necessary to re-check the interference of non-working side according to the above method until there is a reasonable gap between the non-working side of grinding wheel and the opposite tooth surface. 7. Grinding process In order to verify the above methods, a hypoid gear set, shown in Table 1, is used as an example for grinding simulation process. The gear is machined by Formate® face hobbing process. The machine setting parameters of gear cutting are shown in Table 2. The mesh coordinates of both sides of the tooth surface can be obtained by solving the tooth surface equation in Section 3, and the 3D model of the gear is shown in Fig. 10. The calculated mesh coordinates of the tooth surface are compared with the coordinates calculated by the hypoid gears design and machine setting calculation program [21]. Taking the concave side as an example, the maximum tooth deviation (shown in Fig. 11) is less than 0.25 μm, which may be due to rounding errors of input parameters. It can be considered that the two tooth surfaces are consistent. According to the method described in Section 4, the curvature characteristics of concave and convex sides are analyzed. The principal curvature radius of the tooth surface is shown in Fig. 4–6. The minimum principal curvature radius of the tooth root of concave side is 57.87 mm, and the maximum principal curvature radius of the tooth root of convex side is 80.56 mm. The slope angle of the principal curvature corresponding to the b curve at the toe of the concave side is 26.65° The slope angle of the principal curvature corresponding to the b curve at the heel of the convex side is 10.48° These two angles can be used as initial values of grinding wheel pressure angle. According to Eqs. (30) and (32), the top radius of the grinding wheel is 51.72 mm for concave side and 79.22 mm for convex side, which ensures that the working side of the grinding wheel does not interfere with the ground tooth surface. After reserving 0.5 mm allowance, the initial parameters of grinding wheel are obtained as shown in Table 3.

Table 1 The basic parameters of gear pair. Item

Pinion

Number of teeth Module(mm) Shaft angle Offset (mm) Face width(mm) Spiral angle Hand of spiral Average pressure angle

14

Gear 47 5.106 90° 30

42.19 44° LH

21°15

38 27°59 RH

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Table 2 The machine setting parameters of the gear. Item

Value

Machine root angle Machine Center to Cross point (mm) Work offset(mm) Sliding base (mm) Horizontal(mm) Vertical (mm) Number of blade group Reference radius of outside blade (mm) Pressure angle of outside blade Eccentric angle of outside blade Reference radius of inside blade (mm) Pressure angle of inside blade Eccentric angle of outside blade

63.67° 10.6474 0 0 100.8647 74.7623 13 76.252 22°25 18.657° 75.688 19° 6 17.161°

Fig. 10. 3D model of the gear.

Fig. 11. The tooth deviation of concave side.

Table 3 The initial parameters of grinding wheel. Item

Concave

Convex

Pressure angle of working side Pressure angle of non-working side Tip radius of working side (mm) Tip radius of no-working side (mm)

26.65° 15.85° 51.22 50.22

10.48° 35.98° 79.72 80.72

13

14

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Fig. 12. The interference status at different grinding positions.

Based on the initial parameters, the tool path of gear grinding can be planned, and the interference test of non-working side is carried out. It can be seen that the non-working side of the grinding wheel has obvious interference with the other end of the opposite tooth surface when grinding both ends. The interference status between the non-working side of the grinding wheel and the opposite tooth surface at different grinding positions are shown in Fig. 12, and the detail interference values are shown in Fig. 13. It can be seen that the interference is more serious when grinding the heel of concave side and the toe of convex side. According to the method described in Section 6, such a large amount of interference in the non-working side can only be avoided by tilting the axis of the grinding wheel. The tilt angle of grinding wheel axis increases gradually. Then the pressure angle and the top radius of the grinding wheel are modified according to Eqs. (61)–(64), and the maximum interference values between the non-working side of the grinding wheel and the opposite tooth surface are calculated. As shown in Fig. 14, with the increase of the tilt angle, the interference of the non-working side of the grinding wheel decreases gradually. The interference of the non-working side of the grinding wheel will be eliminated completely when the tilt angle increases to 19.64° for grinding concave side, and when the tilt angle increases to 18.27° for grinding convex side. The tilt angle of the grinding wheel is set to 22° for concave grinding and 20° in convex grinding with some margin considered. At the same time, the pressure angle of the non-working side of the grinding wheel is reduced by 10° Then the final grinding wheel parameters can be obtained as shown in Table 4. The final parameters of the grinding wheel are used to check the interference of the non-working side of the grinding wheel again as shown in Fig. 15. It can be seen that there is no interference, and the minimum gap for concave side grinding is 0.449 mm, the minimum gap for convex side grinding is 0.217 mm. Both locate at the top of the grinding wheel. The distance increases gradually from the top of grinding wheel to the top of tooth. According to the motion principle of CNC hypoid grinding machine shown in Fig. 7, a five-axis tooth grinding simulation platform for the FFHHG is constructed, as shown in Fig. 16. The distance R0 is 350 mm, LB and LM are 102.538 mm and 65 mm, respectively. Then the grinding wheel location of tooth grinding can be calculated according to the method described

W. Zhang, X. Guo and Y. Wang et al. / Mechanism and Machine Theory 144 (2020) 103628

Fig. 13. The interference values at different grinding positions: convex.

heel of concave,

Fig. 14. The max interference at different tilt angles:

toe of concave,

concave side, and

15

heel of convex, and

toe of

convex side.

Table 4 The final parameters of grinding wheel. Item

Concave

Convex

Pressure angle of working side Pressure angle of non-working side Tip radius of working side (mm) Tip radius of no-working side (mm)

4.65° 27.85° 57.68 56.68

30.48° 5.98° 73.84 74.84

in Section 5, which are shown in Figs. 17 and 18. At the same time, the grinding wheel model (the green part) is built according to Table 4, and the imported gear model (the orange part) is shown in Fig. 10. There is no feeding during the first grinding, and the grinding wheel should be tangent to the tooth surface theoretically. The actual ground tooth surfaces are shown in Fig. 19(a), which shows that the grinding wheel and the tooth surface are in intermittent contact state, and the ground area extends from the top of the tooth to the root of the tooth evenly at the contact positions. This is because the material removal simulation of machining simulation software is based on the discrete grid model, but not on Boolean operation of accurate 3D model. The intermittent grinding state can be considered as tangent contact between the grinding wheel and the tooth surface along the b curve, and there is no curvature interference. Then tooth grinding is carried out again after adjusting the gear rotational position (corresponding to 0.02 mm grinding feed), and the ground tooth surfaces are shown in Fig. 19(b). It can be seen that a layer of material is removed evenly for both sides, which proves the correctness of the grinding path.

16

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Fig. 15. The gap between non-working side of grinding wheel and the gear.

heel of concave, and

Fig. 16. The tooth grinding simulation platform.

Fig. 17. The grinding wheel location of grinding concave side.

toe of convex.

W. Zhang, X. Guo and Y. Wang et al. / Mechanism and Machine Theory 144 (2020) 103628

17

Fig. 18. The grinding wheel location of grinding convex side.

Fig. 19. The ground tooth surfaces.

8. Conclusion This paper presents a new method for face-hobbed hypoid gear tooth grinding using large diameter conical grinding wheel. The analysis shows that the proposed approach can efficiently and accurately grind the FFHHG without theoretical tooth deviation. The main conclusions are summarized below: (1) In order to avoid curvature interference, the parameters of the grinding wheel working side for concave grinding should be determined by the principal curvature radius and its change rate at the root position of the concave toe. Moreover, for convex grinding, it should be determined by the principal curvature radius and its change rate at the root position of the convex heel. (2) The key to realize the accurate calculation of grinding wheel location without theoretical deviation is to control the grinding wheel position so that its generator is tangentially contacted with the profile curve of the tooth surface, and the tooth root curve is used as the guide curve to ensure the processed tooth depth. (3) The non-working side of the grinding wheel is prone to interfere with the opposite side of the ground tooth surface. The interference is more substantial in grinding the heel of the concave surface and the toe of the convex surface. Tilting the grinding wheel axis can effectively avoid the interference. At present, the proposed method can only be applied to non-generated (Formate) hypoid gears. The grinding method for the generated member of hypoid gears will be considered in a future study. As a next step, we will study the tooth grinding method for the pinion member by substituting the epicycloidal tooth surface of generating gear with revolution surface, and correcting the substituted tooth deviation using the high order motion of the CNC hypoid grinding machine.

18

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Acknowledgments We would like to thank the financial aid and support from the National Natural Science Foundation of China (No. 51775073), the National Key R&D Program of China (No. 2018YFB2001700), Key R&D Program of Chongqing Technology Innovation and Application Demonstration (No. cstc2018jszx-cyzd0661) and China Scholarship Council (No. 20180850 0 044). References [1] F.L. Litvin, W.S. Chaing, C. Kuan, M. Lundy, W.J. Tsung, Generation and geometry of hypoid gear-member with face-hobbed teeth of uniform depth, Int. J. Mach. Tools Manuf. 31 (2) (1991) 167–181. [2] Q. Fan, Computerized modeling and simulation of spiral bevel and hypoid gears manufactured by Gleason face hobbing process, J. Mech. Des. 128 (6) (2006) 1315–1327. [3] Y.P. Shih, Z.H. Fong, G.C. Lin, Mathematical model for a universal face hobbing hypoid gear generator, J. Mech. Des. 129 (1) (2007) 38–47. [4] Y.P. Shih, Z.H. Fong, Flank modification methodology for face-hobbing hypoid gears based on ease-off topography, J. Mech. Des. 129 (12) (2007) 1294–1302. [5] Z. Weiqing, G. Xiaodong, Z. Mingde, Machine setting calculation and contact characteristics control method of face hobbing hypoid gear, J. Mech. Eng. 54 (19) (2018) 49–57. [6] W. Zhang, B. Cheng, X. Guo, M. Zhang, Y. Xing, A motion control method for face hobbing on CNC hypoid generator, Mech. Mach. Theory 92 (2015) 127–143. [7] M. Habibi, Z.C. Chen, An accurate and efficient approach to undeformed chip geometry in face-hobbing and its application in cutting force prediction, J. Mech. Des. 138 (2) (2016) 023302. [8] Q. Fan, Tooth surface error correction for face-hobbed hypoid gears, J. Mech. Des. 132 (1) (2010) 011004. [9] V.V. Simon, Influence of tooth modifications on tooth contact in face-hobbed spiral bevel gears, Mech. Mach. Theory 46 (12) (2011) 1980–1998. [10] M. Vimercati, Mathematical model for tooth surfaces representation of face-hobbed hypoid gears and its application to contact analysis and stress calculation, Mech. Mach. Theory 42 (6) (2007) 668–690. [11] W.R. McVea, Flaring cup grinding FORMATE® bevel and hypoid gears (No. 891929), SAE Technical Paper, 1989. [12] G.J. Kimmet, H.D. Dodd, CBN finish grinding of hardened spiral bevel and hypoid gears, SAE Trans. (1985) 799–808. [13] H. Liu, Z. Wang, S. Yu, Research on machining precision of CNC spiral bevel gear grinding machine, in: 2011 Second International Conference on Mechanic Automation and Control Engineering, 2011, July, pp. 1549–1552. IEEE. [14] W.Q. Zhang, X.D. Guo, M.D. Zhang, The tooth form deviation correction of cnc spiral bevel gears grinding machine, in: Applied Mechanics and Materials, 86, Trans Tech Publications, 2011, pp. 454–457. [15] Y. Wang, Y. Chen, G. Zhou, Q. Lv, Z. Zhang, W. Tang, Y. Liu, Roughness model for tooth surfaces of spiral bevel gears under grinding, Mech. Mach. Theory 104 (2016) 17–30. [16] H. Ding, J. Tang, J. Zhong, Accurate nonlinear modeling and computing of grinding machine settings modification considering spatial geometric errors for hypoid gears, Mech. Mach. Theory 99 (2016) 155–175. [17] Wiener, Dieter. Method of grinding the teeth of spiral-toothed bevel gear wheels. U.S. Patent 6,050,883, issued April 18, 20 0 0. [18] X.Z. Deng, G.G. Li, B.Y. Wei, J. Deng, Face-milling spiral bevel gear tooth surfaces by application of 5-axis CNC machine tool, Int. J. Adv. Manuf. Technol. 71 (5–8) (2014) 1049–1057. [19] A. Álvarez, L.L. de Lacalle, A. Olaiz, A. Rivero, Large spiral bevel gears on universal 5-axis milling machines: a complete process, Procedia Eng. 132 (2015) 397–404. [20] B. Karpuschewski, H.J. Knoche, M. Hipke, Gear finishing by abrasive processes, CIRP Ann. 57 (2) (2008) 621–640. [21] Gleason Works, Bevel Gear Design CAGE4Win User’s Manual. Rochester, N.Y, 2005.