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Modified worm gear hobbing for symmetric longitudinal crowning in high lead cylindrical worm gear drives
Mechanism and Machine Theory 117 (2017) 133–147
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Research paper
Modified worm gear hobbing for symmetric longitudinal crowning in high lead cylindrical worm gear drives Jonghyeon Sohn∗, Nogill Park Department of Mechanical Engineering, Pusan National University, 2 Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan, South Korea
a r t i c l e
i n f o
Article history: Received 31 October 2016 Revised 5 March 2017 Accepted 4 July 2017 Available online 12 July 2017 Keywords: Worm gear Crowning Edge contact Tooth modification Tooth contact analysis
a b s t r a c t A single enveloping worm gear of cylindrical worm gear drive is cut by an oversized hob. The oversize provides a longitudinal crowning on the worm gear tooth surface. However, when a lead angle of the worm becomes higher, the longitudinal crowning becomes more asymmetric. The asymmetrically crowned worm gear surface makes the gear contact more sensitive to the certain direction of misalignment. To change the longitudinal crowning from asymmetric to symmetric, a method of modified worm gear hobbing is proposed. The modification uses a modified machine tool setting and a modified hob specification. A new hobbing mechanism or hob geometry is not required in this method. Results of the proposed modification are confirmed by using a surface separation topology based on differential geometry, and contact pressure analysis based on finite element method. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In cylindrical worm gear drives, the single enveloping worm gear is cut by an oversized hob [1] which has larger pitch diameter than the worm. The oversized hob makes the worm gear tooth surface longitudinally crowned, making the instantaneous line contact of a fully conjugated set into the instantaneous point contact [2]. Consequently, the contact between the worm and worm gear becomes localized, reducing sensitivity to misalignment of the worm gear set. The influence of the hob oversize and the shaft misalignment on the contact have been studied by many researchers [3–10]. An edge contact problem caused by elastic deformation and the positive parabolic function of transmission errors was also studied with corresponding solutions [11,12]. The edge contact mainly occurs at a tip of the worm and worm gear tooth. The proposed solutions for the edge contact were a type of profile crowning. However, an edge contact can also occur at a side corner of the tip of the worm gear surface. Such edge contact easily occurs when a high worm lead angle is applied because the higher lead angle makes the longitudinal crowning more asymmetric. Fig. 1 shows the worm gear surface of the three cases in Table 1: Case 1 (Fig. 1(a)), Case 2 (Fig. 1(b)) and Case 3 (Fig. 1(c)). The three cases have identical pitch radii of the worm and worm gear. In particular, Case 3 has the highest lead angle and Case 1 has the lowest one. To examine a characteristic of the longitudinal crowning by the hob oversize, the surfaces Sgos0 , generated by 0% oversized hobs, are presented along with the surfaces Sgos4 , generated by 4% oversized hobs. In Case 1, the clearances between Sgos4 and Sgos0 at the left and right corner of the tip of the worm gear surface are 48 μm and 23 μm, respectively. In Case 2, the clearances at the left and right are 60 μm and 9 μm, respectively. In both Case 1 and 2, the clearances of left and right side are different to each other, and the case with higher lead angle shows a greater difference.
∗
Corresponding author. E-mail address: [email protected] (J. Sohn).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.07.004 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
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Nomenclature Lc Li Zg Zh Zw d0 dc dh lh , lw snc uc0 uh , θ h uw , θ w
αn γ λh , λw φw, φg ψ h, ψ g
path of the instantaneous contact points boundary of interference area tooth number of the worm gear thread number of the hob thread number of the worm center distance between the worm and the worm gear center distance between the hob (or worm) and the grinding wheel center distance between the hob and the worm gear leads of the hob and the worm normal tooth thickness of the grinding wheel pitch radius of the grinding wheel hob surface parameters worm surface parameters normal pressure angle crossing angle error of the worm and the worm gear lead angles of the hob and the worm rotational angles of the worm and the worm gear in contact rotational angles of the hob and the worm gear in hobbing process
Such difference in the clearances means that the worm gear surface is asymmetrically crowned in the longitudinal direction. Because of such asymmetric longitudinal crowning, worm gear drives with high lead angle have greater sensitivity to a certain direction of misalignment. For example, −0.3° of the crossing angle error may not be a problem, but 0.1° of the crossing angle error can cause an edge contact. Moreover, the edge contact can occur without misalignment. As shown in Fig. 1(c), the clearance at the left and right are 74 μm and −6 μm, respectively. Sgos4 and Sgos0 intersect near the right corner, and the clearance at the right corner shows a negative value, indicating that the hob oversize fails to provide the longitudinal crowning at the right corner. This phenomenon is geometric interference studied by Sohn and Park [13,14]. The interference can be avoided by cutting out the corner of the worm gear tooth. Nonetheless, the sensitivity to certain direction of misalignment remains high because the asymmetrically crowned geometry is unchanged. This study proposes a modified hobbing method to make a symmetric longitudinal crowning on the worm gear surface of cylindrical worm gear drives. The method includes a modified hobbing machine tool setting and a modified hob specification. The proposed modification method can be utilized without any development of a new hobbing machine or a hob cutter with a special geometry because the method employs the existing hobbing mechanism and the hob geometry. Results of the modification are confirmed by the surface separation topology based on differential geometry, and the contact pressure analysis based on finite element method. In particular, an initial guessing method for the tooth contact analysis (TCA) [15], which is required to determine the surface separation topology, is presented. The initial guessing method utilizes the line of action of the involute crossed helical gears [16] to estimate the instantaneous contact points. Converged solution of the TCA shows that the initial guess can provide a good convergence to the TCA of the single enveloping worm gear drives. 2. Theoretical background 2.1. Surfaces of worm, hob and worm gear The characteristic that the longitudinal crowning becomes asymmetric when the lead angle becomes higher is commonly found in ZA, ZN, ZK and ZI type of cylindrical worm gear drives. Also, the proposed modification can be used for these types. In this section, the ZK type is presented for an illustration. As shown in Fig. 2, the ZK hob surface Sh can be represented in the hob coordinate system Ch [xh , yh , zh ] as follows:
rh (uh , θh ) = [xh
yh
zh
1]T
θh cosuh + (uc − uc0 )tanαn − s2nc cosθh sinλh − dc sinθh yh = uc sinθh cosλh sinuh − uc cos θh cosuh + (uc − uc0 )tanαn − s2nc sinθh sinλh + dc cosθh zh = −uc sinλh sinuh + (uc − uc0 )tanαn − s2nc cosλh + θ2hπlh xh = uc cosθh cosλh sinuh + uc sin
where
uc =
− 2lπh
cotλh
cosuh
+ tanαn tanuh +
dc cos uh
+ tanαn
Snc
1 + tan2 αn
2
(1)
− dc cotλh tanuh + tan2 αn uc0
(2)
J. Sohn, N. Park / Mechanism and Machine Theory 117 (2017) 133–147
135
Fig. 1. Worm gear surfaces cut by both 0% and 4% oversized hobs; (a) Case 1, (b) Case 2 and (c) Case 3.
Table 1 Design parameters of worm gear set for three cases. Design data
Case 1
Profile type Thread number of worm Tooth number of worm gear Normal module Normal pressure angle Lead angle of the worm (right-hand helix) Pitch diameter of worm Pitch diameter of worm gear Pitch diameter of grinding wheel of worm and hob Normal tooth thickness of hob Normal tooth thickness of worm Oversize of hob Addendum of worm Dedendum of worm Addendum of worm gear Dedendum of worm gear Worm gear tooth width
ZK 1 50 2.3829 20° 6.8428° 20 mm 120 mm 330 mm 3.12 3.00 4% 2.4 mm 3.0 mm 2.4 mm 3.0 mm 14 mm
Case 2
Case 3
2 49 2.3787
3 47 2.3843
13.7608°
20.9558°
The ZK worm surface Sw can be represented by substituting the subscript of all parameters of Sh from h to w. As shown in Fig. 3, the worm gear surface Sg can be represented in the worm gear coordinate system Cg [xg , yg , zg ] as follows:
rg (uh , θh , ψh ) = Mgh (ψh )rh (uh , θh )
(3)
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J. Sohn, N. Park / Mechanism and Machine Theory 117 (2017) 133–147
Fig. 2. Relation between the coordinate systems of the hob and the grinding wheel.
Fig. 3. Relation between the coordinate systems of the worm gear and the hob.
f h ( u h , θh , ψ h ) = where ⎡ Mgh
cos ψg ⎢− sin ψg =⎣ 0 0
∂ rg ∗ ∂ rg ∗ × ∂ uh ∂ θh
sin ψg cos ψg 0 0
0 0 1 0
⎤⎡
·
0 0 1 0 ⎥⎢ 0 0 ⎦⎣1 0 0 1 0 0
∂ rg ∗ =0 ∂ ψh ⎤⎡ ⎤⎡ 0 −dh cos 0 sin 0 cos ψh 1 0 ⎥⎢ 0 1 0 0⎥⎢ sin ψh ⎦⎣− sin 0 cos 0⎦⎣ 0 0 0 0 1 0 0 0 1 0
(4)
− sin ψh cos ψh 0 0
0 0 1 0
⎤
0 0⎥ ⎦ 0 1
where
ψg =
Zh ψh and = λw − λh . Zg
rg ∗ in Eq. (4) is the vector with the last element of rg removed. Here, to modify the hobbing machine tool setting, and dh are set as follows:
=
+
d h = d h + d h
(5) (6)
The modification uses and dh instead of and dh . 2.2. Tooth contact analysis Tooth contact analysis (TCA) is performed to determine instantaneous contact points and transmission errors. A position vector and a unit normal vector of the worm surface Sw can be represented in the frame coordinate system Cf [xf , yf , zf ]
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137
Fig. 4. Relation among the coordinate systems of the frame, the worm, and the worm gear.
(Fig. 4) as follows:
rwf (uw , θw , φw ) = M f w (φw )rw (uw , θw )
(7)
nwf (uw , θw , φw ) = M∗fw (φw )nw (uw , θw )
(8)
where
⎡
cos φw
⎢− sin φ w ⎢ Mfw = ⎢ ⎣ 0
⎤
sin φw
0
cos φw
0
0
1
⎥, M∗fw = ⎣− sin φw 0 ⎦
0
0
1
0
0
⎡
−d0 ⎥ ⎥
cos φw
sin φw cos φw
0
0
0
⎤
0⎦, nw = 1
Nw , N = |Nw | w
∂ rw ∗ ∂ rw ∗ × ∂ uw ∂θw
A position vector and a unit normal vector of the worm gear surface Sg can be represented in the frame coordinate system Cf as follows:
rgf (uh , θh , ψh ,
φg ) = M f g (φg )rg (uh , θh , ψh )
(9)
ngf (uh , θh , ψh ,
φg ) = M∗fg (φg )ng (uh , θh , ψh )
(10)
f h ( u h , θh , ψ h ) = 0 where
⎡ Mfg =
⎢1 ⎢ ⎢ ⎣0 ⎡
M∗fg
0
0
1
0
0
(11) 0
⎤⎡
1
⎢ 0⎥ ⎥⎢0
1
0
⎥⎢ 0⎦⎣0
0
0
0
1
0
⎤⎡
1
0
0
cos γ
− sin γ
sin γ
cos γ
0
0
0
0
1
= ⎣1
0
0⎦⎣0
cos γ
0
1
0
sin γ
0
0
⎤⎡
0
⎤⎡
⎢ 0⎥ ⎥⎢− sin φg
cos γ
sin φg
0
cos φg
0
0
⎤
⎥⎢ 0⎦⎣
0
0
1
0⎥ ⎥ ⎥, 0⎦
1
0
0
0
1
cos φg ⎦ ⎣ − sin γ − sin φg 0
cos φg
0
sin φg cos φg 0
0
⎤
0⎦ , n g = 1
Ng , N = |Ng | g
∂ rg ∗ ∂ rg ∗ × ∂ uh ∂θh
The Eq. (11) is identical to the Eq. (4). The condition that the worm and the worm gear surfaces are in point contact can be represented as follows:
rwf (uw , θw , φw ) = rgf (uh , θh , ψh ,
φg )
(12)
nwf (uw , θw , φw ) = ngf (uh , θh , ψh ,
φg )
(13)
f h ( u h , θh , ψ h ) = 0
(14)
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J. Sohn, N. Park / Mechanism and Machine Theory 117 (2017) 133–147
Fig. 5. Illustration of the pitch coordinate system Cp , a line of action (LOA) and a plane .
ˆ from (a) the normal view of the zp xp plane and (b) the normal view of the plane . Fig. 6. An Illustration of an estimated instantaneous contact point K
The Eq. (14) is identical to the Eq. (4). The Eqs. (12)–(14) are six independent scalar equations in seven unknowns, uw ,
θ w , φ w , uh , θ h , ψ h , and φ g . Newton-Rhapson method is applied to solve these equations. However, the numerical solutions
of the TCA of worm gear drives can be easily diverged. To obtain a good convergence, it is essential to select an initial guess carefully. Here, the initial guess is determined by using an estimated line of action (LOA). The estimated LOA is identical to the LOA of aligned involute crossed helical gears [16] where the LOA is a straight line. This estimation is reliable because the instantaneous contact point of the single enveloping worm gear drives is located closely to the instantaneous contact point of the involute crossed helical gears. Fig. 5 shows the location of the estimated LOA on the plane . The angle between the plane and the yp zp plane is λw . The origin Op of the pitch coordinate system Cp [xp , yp , zp ] is on the plane . Aligned axes of the hob, the worm, the worm gear, and the grinding wheels are presented together. zc is the axis of the grinding wheel, Oc is the center of the grinding wheel, and ν c is the translational displacement of the grinding wheel. The superscript w and h indicate the case of the worm grinding and the hob grinding, respectively. The unit vector eLOA of LOA represented in the pitch coordinate system Cp can be calculated by using the unit vector of zp axis as follows:
eLOA =
exLOA eyLOA ezLOA
=
cosλw 0 −sinλw
0 sinλw 1 0 0 cosλw
1 0 0 cosαn 0 −sinαn
0 sinαn cosαn
0 0 1
(15)
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139
E Fig. 7. Illustration of a point P1 on the Sw and a point P2 on the Sg .
Fig. 8. The path of the instantaneous contact points Lc and the boundary of the interference Li on the worm gear surface, according to the variation of and dh .
Fig. 9. Illustration of the main idea of the modification using shifts of the Lc and Li .
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Fig. 10. The surface separation and the transmission errors (a) before and (b) after the modification, in Case 2.
Fig. 11. The surface separation and the transmission errors (a) before and (b) after the modification, in Case 3.
The equation of the LOA can be represented in the pitch coordinate system Cp as follows:
xp yp zp = = = tLOA exLOA eyLOA ezLOA
(16)
ˆ on Fig. 6 shows the normal views of the zp xp plane and the plane . Here, an estimated instantaneous contact point K ˆ along the LOA from the origin point Op . It is assumed the LOA is considered. The value tLOA indicates a displacement of the K that the reference condition has all zero value of the worm surface parameter θ w , the worm rotational angle φ w , the hob surface parameter θ h , the hob rotational angle ψ h , and the worm gear rotational angle φ g . Under the reference condition, h Ow c and Oc will be located at the same position of Op in the normal view of the zp xp plane (Fig. 6(a)). If we assume that
J. Sohn, N. Park / Mechanism and Machine Theory 117 (2017) 133–147
141
Fig. 12. The surface separation before applying the modification in Case 2: (a) with the crossing angle error γ = −0.1◦ and (b) with the crossing angle error γ = 0.1◦
the grinding wheel surface is equal to the common rack surface of the involute crossed helical gears, the estimated instan because of the tooth thickness snc of the grinding wheel (Fig. 6(b)). Therefore, a taneous contact point will be located at K 0 to K ˆ along the LOA can be represented as follows: displacement of the estimated instantaneous contact point from K 0
DKˆ = tLOA + 2π Zw
snc cos αn 2
(17)
A surface normal pitch of the common rack is π mn cos α n and a worm rotational angle, φ w , corresponding to a pitch is ˆ , φ w can be estimated as follows: . Therefore, when the estimated instantaneous contact point is located at K
φw =
2π Zw
DKˆ π mn cos αn
=
2π Zw
tLOA + s2nc cos αn π mn cos αn
Similarly, ψ h can be estimated as follows:
=− ψ h
2π Zh
DKˆ π mn cos αn
=
2π Zh
tLOA + s2nc cos αn π mn cos αn
(18)
(19)
as follows: The worm gear rotational angle φ g can be estimated by using ψ h
=− φ g
Zh ψ Zg h
(20)
The minus sign of Eq. (20) is caused by opposite direction of ψ g and φ g as shown in Figs. 3 and 4. The worm surface parameter θ w has the same value to the worm rotational angle φ w . Similarly, the hob surface parameter θ h has the same magnitude to the hob rotational angle ψ h , but it has an opposite sign. Therefore, the estimated θ w and θ h can be represented as follows:
θw = φw
(21)
θh = −ψ h
(22)
uw and uh are the rotational angle of the grinding wheel coordinate system Cc [xc , yc , zc ] (Fig. 2) in the generation of the worm and hob, respectively. In Fig. 6(a), the distance ρ w between zcw axis and LOA can be calculated as follows:
ρw = νcw sin λw
(23)
where
νcw =
θw l w 2π
(24)
ˆ and zc xc0 plane (Fig. 5) is uc0 − y p where y p = tLOA eyLOA . Thus, an estimated uw can be calculated The distance between K as follows:
−1 u w = tan
ρw
= tan−1
uc0 − y p
θw lw sin λw 2π (uc0 − y p )
(25)
Similarly, an estimated uh can be calculated as follows:
uh = tan−1
ρh uc0 − y p
= tan−1
θh lh sin λh 2π (uc0 − y p )
(26)
, θ , ψ , and φ , of the seven unknowns, u , θ , φ , u , θ , ψ , and φ , Therefore, the initial guess, u w , θw , φw , u g w w w g h h h h h h ˆ and the converged solution X of several can be defined by the value of tLOA . Table 2 shows the calculated initial guess X
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J. Sohn, N. Park / Mechanism and Machine Theory 117 (2017) 133–147 Table 2 ˆ ) calculated by estimated line of action and its converged solutions (X) of Case 2 with no Initial guess (X misalignment. tLOA [mm] ˆ X X ˆ X
−5.7038 −2.8159
2.9598
X ˆ X X ˆ X
5.8477
X ˆ X
0.0720
X
uw [°]
θ w [°]
φ w [°]
uh [°]
θ h [°]
ψ h [°]
φ g [°]
−0.3222 −0.3231 −0.0643 −0.0647 0.1967 0.1968 0.4609 0.4615 0.7282 0.7293
−92.3355 −96.9350 −18.3107 −19.4302 55.7140 59.0123 129.7387 138.4096 203.7635 218.7795
−92.3355 −97.2346 −18.3107 −19.4955 55.7140 59.1844 129.7387 138.8224 203.7635 219.4362
−0.3139 −0.3139 −0.0665 −0.0665 0.1838 0.1838 0.4372 0.4372 0.6936 0.6936
−93.7807 −98.5082 −19.7560 −20.8785 54.2688 57.6934 128.2935 137.2250 202.3182 217.7340
93.7807 98.7988 19.7560 20.9456 −54.2688 −57.8529 −128.2940 −137.6140 −202.3180 −218.3560
−3.8278 −4.0277 −0.8064 −0.8547 2.2151 2.3567 5.2365 5.6074 8.2579 8.8981
Table 3 ˆ ) calculated by estimated line of action and its converged solutions (X) of Case 2 with crossing Initial guess (X angle error γ = 0.1◦ . tLOA [mm]
−2.8159
ˆ X X ˆ X
0.0720
X ˆ X
−5.7038
X ˆ X X ˆ X
2.9598 5.8477
X
uw [°]
θ w [°]
φ w [°]
uh [°]
θ h [°]
ψ h [°]
φ g [°]
−0.3222 −0.3217 −0.0643 −0.0632 0.1967 0.1984 0.4609 0.4632 0.7282 0.7311
−92.3355 −108.4217 −18.3107 −31.3154 55.7140 46.6809 129.7387 125.5735 203.7635 205.3648
−92.3355 −99.2889 −18.3107 −21.7619 55.7140 56.6749 129.7387 136.0314 203.7635 216.3145
−0.3139 −0.3139 −0.0665 −0.0665 0.1838 0.1838 0.4372 0.4372 0.6936 0.6936
−93.7807 −105.1722 −19.7560 −27.8522 54.2688 50.3700 128.2935 129.5026 202.3182 209.5501
93.7807 96.3513 19.7560 18.6266 −54.2688 −60.0199 −128.2940 −139.5992 −202.3180 −220.1206
−3.8278 −4.1103 −0.8064 −0.9460 2.2151 2.2556 5.2365 5.4948 8.2579 8.7719
ˆ and X of several instantaneous contact points instantaneous contact points of aligned Case 2 in Table 1. Table 3 shows the X of the Case 2 with crossing angle error γ = 0.1◦ . Here, uh is used as a fixed parameter. The converged solutions are close enough to the initial guess, and thus a good convergence of the numerical method can be obtained. This method of initial guessing is also effective in the TCA of other single enveloping worm gear drives such as ZA, ZN and ZI, whether those are either aligned or moderately misaligned. Corresponding transmission error is calculated by following equation:
T ransmission error = φg −
Zw φw Zg
(27)
2.3. Calculation of the minimum surface separation The minimum surface separation between the worm and worm gear can be represented by using the envelope to the family of the worm surface on the worm gear surface. As shown in Fig. 4, the envelope to the family of the worm surface E can be represented in the worm gear coordinate system C as follows: Sw g E rw (uw , θw , φw ) = Mgw (φw , φg (φw ) )rw (uw , θw )
f w ( u w , θ w , φw ) =
∗
∗
∂ rEw ∂ rEw × ∂ uw ∂ θw
(28)
∗
·
∂ rEw =0 ∂ φw
(29)
where ⎡ ⎤⎡ ⎤⎡ ⎤⎡ ⎤ cos φg − sin φg 0 0 1 0 0 0 0 1 0 0 cos φw sin φw 0 0 ⎢ sin φg cos φg 0 0⎥⎢0 cos γ sin γ 0⎥⎢0 0 1 0⎥⎢− sin φw cos φw 0 −d0 ⎥ Mgw = ⎣ ⎦⎣0 − sin γ cos γ 0⎦⎣1 0 0 0⎦⎣ 0 ⎦ 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 E∗ in Eq. (29) is the vector with the last element of r E removed. Here, the transmission function φ (φ ) is fitted to a rw g w w fourth-order polynomial function by using the numerical solutions obtained by TCA in the Section 2.2. Fig. 7 shows points E , and P is on S . P and P P1 (xp1 , yp1 , zp1 ) and P2 (xp2 , yp2 , zp2 ), represented in the coordinate system Cg [xg , yg , zg ]. P1 is on Sw g 2 1 2 are both located at (R, t). R is the distance between P1 and zg axis, or P2 and zg axis. t is the zg coordinate of P1 or P2 . An angle P1 Otg P2 is χ1 − χ2 . Therefore, the length ζ of the arc P1 P2 can be calculated as follows:
ζ = R(χ1 − χ2 ) = R tan
−1
y p1 x p1
− tan
−1
y p2 x p2
(30)
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143
The coordinates of P1 (xp1 , yp1 , zp1 ) can be represented as follows:
x2p1 + y2p1 = R2
(31)
z p1 = t
(32)
f w ( u w , θ w , φw ) = where
∗
∗
∂ rEw ∂ rEw × ∂ uw ∂ θw
∗
·
∂ rEw =0 ∂ φw
(33)
T
E rw (uw , θw , φw ) = x p1 y p1 z p1 1
The Eqs. (31) to (33) are three independent scalar equations in three unknowns, uw , θ w and φ w . These equations can be solved by using Newton-Raphson method. Similarly, the coordinates of P2 (xp2 , yp2 , zp2 ) can be represented as follows:
x2p2 + y2p2 = R2
(34)
z p2 = t
(35)
f h ( u h , θh , ψ h ) = where
∂ rg ∗ ∂ rg ∗ × ∂ uh ∂ θh
·
∂ rg ∗ =0 ∂ ψh
(36)
T
rg (uh , θh , ψh ) = x p2 y p2 z p2 1
The Eqs. (34)–(36) are three independent scalar equations in three unknowns, uh , θ h and ψ h . By using the calculated value of ζ for a specified range of R and t, we obtained the contour of the minimum separation between Sw and Sg . In normal case, ζ is zero at Lc , the path of the instantaneous contact points, and has positive value at the rest of the area. However, ζ can also be zero at the Li , a boundary of interference area. The interference area has a negative value of ζ as E is drawn into S . If the interference exists, the interference area appears around the right corner of the tip of the worm Sw g gear surface in a case with a right-hand helix. In a case with a left-hand helix, the interference area appears around the left corner of the tip of the worm gear surface. This method is fully described in Sohn and Park [13]. 3. Main idea of modification The main idea of the modification is obtained by the study that examines a characteristic that Lc and Li on the worm gear surface shift with respect to and dh . Fig. 8(a) shows Lc and Li in three different cases; dh = 0 mm in the all three cases, and = −0.05◦ , = 0◦ and = 0.05◦ in each case. Likewise, Fig. 8(b) presents Lc and Li in three different cases; = 0◦ in the all three cases, and dh = 0.025 mm, dh = 0 mm and dh = −0.025 mm in each case. Lc is depicted as a line with dots and Li as a solid line. Fig. 8 is based on the design data of Case 3 on the Table 1. Wider range of the worm gear surface is presented to illustrate clearly the shift of Lc and Li . In Fig. 8(a), Lc shifts to the left side and Li shifts to the right side when is a negative value. On the other hand, Lc shifts to the right side and Li shifts to the left side when is a positive value. The direction of shift with respect to dh in Fig. 8(b) is opposite to the direction of shift with respect to in Fig. 8(a). A comparison between Fig. 8(a) and Fig. 8(b) shows that the magnitudes of Lc shift in both cases are similar. However, the magnitude of Li shifts are dissimilar; the magnitude of Li shift in Fig. 8(a) is greater than that of Li shift in Fig. 8(b). In other words, when the shift of Lc with respect to has similar magnitude to the shifts of Lc with respect to dh , the shift of Li with respect to shows greater magnitude than the shift of Li with respect to dh . Such characteristic of the shifts of Lc and Li with respect to and dh appears in similar manner in other types of worm gear drives; ZA, ZN, and ZI. Using such characteristic, the simultaneous application of and dh is the main idea of the modification as follows (see Fig. 9): (i) LAc is the initial position of Lc and LAi is the initial position of Li when = 0 and dh = 0. (ii) When we apply that = 0 where 0 < 0, LAi is shifted to LBi that is placed outside of the boundary of the worm gear surface, and LAc is shifted to LBc simultaneously. (iii) When applying additionally that dh = dh0 where dh0 < 0, LBc is shifted to LCc , which is similar position of LAc . In parallel, LBi is shifted to LCi and still remains outside of the boundary of the worm gear surface. The separation value of the right corner becomes greater as Li goes further from the boundary of the worm gear surface. Therefore, adjustment of and dh can make the separation value of the left and right corner similar while making Lc
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Fig. 13. The surface separation after applying the modification in Case 2: (a) with the crossing angle error γ = −0.1◦ and (b) with the crossing angle error γ = 0.1◦
Fig. 14. (a) Finite element model of the worm and worm gear and (b) boundary conditions of the finite element analysis.
Fig. 15. Distribution of the contact pressure on the worm gear surface (a) before and (b) after the modification.
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Fig. 16. A graph of the maximum values of contact pressure on a worm gear tooth with respect to the rotational angle of the worm in unmodified and modified cases.
Fig. 17. Tooth contact pressure distribution on the worm gear for 45° of worm rotational angle (a) before and (b) after the modification.
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J. Sohn, N. Park / Mechanism and Machine Theory 117 (2017) 133–147 Table 4 Processing parameters and resultant surface separation on the left and right side corner of the worm gear before and after modification. Parameter
[°] [°] dh [mm] dh [mm] snh [mm] ha0 [mm] ζ [μm] Left corner Right corner
Case 2
Case 3
Unmodified
Modified
Unmodified
Modified
0.5391 0 70.4 0 3.12 3.00 60 9
0.1111 −0.428 70.078 −0.322 2.8856 2.6780 24 24
0.8416 0 70.4 0 3.12 3.00 74 −6
0.3146 −0.527 70.144 −0.256 2.9336 2.7440 46 46
remain at the center of the worm gear surface. The application of dh changes a tooth thickness of the worm gear. Thus, a normal tooth thickness snh of the hob should be changed as follows:
snh = snh + 2dh tan αn
(37)
In addition, the addendum of the hob ha0 should be changed as follows:
ha0 = ha0 + dh
(38)
In this way, application of and dh in the hobbing machine tool setting, and the adjustment of hob tooth thickness and hob addendum are the modification for the symmetric longitudinal crowning of the worm gear. 4. Results The results of the proposed modification are confirmed by the surface separation topology and contact pressure analysis. The results are similar in the four types of worm gear drives: ZA, ZN, ZK, and ZI. In this paper, we have taken the ZK type as a sample type of worm gear drive and presented the results of the modification applied in this ZK type. In Figs. 10– 13, the surface separation value ζ in Eq. (30) is illustrated by a contour graph, the path of the contact points Lc by a line with dots, and the interference area by inverted triangles. A unit of separation value of the contour graph is millimeter. The processing parameters of the worm gear before and after modification, and the resultant minimum surface separation on the left and right side corner are shown in Table 4. Case 2 in Table 1 is illustrated in Fig. 10, showing the surface separation and the transmission errors before and after the modification is applied. Before applying the modification, the separation value is 60 μm on left side corner and 9 μm on the right side corner. After applying the modification, both the separation values on the left and right side corner are turned into 24 μm symmetrically. Although peak to peak of transmission errors increases from 1.64 to 4.58 , it remains in an allowable range. Case 3 in Table 1 is illustrated in Fig. 11. Before the modification applied, the left side separation value is 74 μm. However, on the right side, the separation value is −6 μm and an interference area appears. After the modification applied, both the separation values on the left and right side corner are turned into 46 μm symmetrically, and the interference area on the right side is removed. Peak to peak of transmission errors increases from 4.24 to 10.17 , but again it remains in an allowable range. The path of contact maintains at the center of the worm gear surface in both Case 2 and Case 3. Figs. 12 and 13 presents how the proposed modification mitigates the characteristic of the high sensitivity to a certain direction of misalignment. Case 2 in Table 1 is used as a design data and shaft crossing angle errors γ = −0.1◦ and γ = 0.1◦ are used as misalignment. Fig. 12 shows the surface separation before the modification. When γ = −0.1◦ , as shown in Fig. 12(a), the left and right side separation values are 48 μm and 31 μm, respectively, and no interference occurs. However, when γ = 0.1◦ as shown in Fig. 12(b), the left side separation value is 75 μm, but the separation value on the right side is −10 μm, indicating that interference occurs on the right corner. Fig. 13 presents the surface separation after the modification. As shown in Fig. 13(a), when γ = −0.1◦ , the left and right side separation values are 12 μm and 46 μm, respectively, and no interference occurs. As shown in Fig. 13(b), when γ = 0.1◦ , the separation value on the left corner is 40 μm and the value on the right corner is 6 μm, exhibiting no interference unlike the Fig. 12(b). By using the finite element method, distributions of contact pressure in the cases used in Figs. 12(b) and 13(b) are compared. ANSYS, a commercial program, is used for the finite element analysis. The finite element model of the worm and worm gear is obtained from the three-dimensional surface model of the worm and worm gear, which is drawn from the coordinates of Sw and Sg . Fig. 14(a) and (b) shows the finite element model and boundary conditions, respectively. A rigid surface 1 on the inside of the worm is rigidly connected to ground by a revolute joint which rotates 4.5° per one step at the rotational axis of the worm. A rigid surface 2 on the worm gear rim is rigidly connected to ground by a revolute joint, where 100Nm torque is applied, at the rotational axis of the worm gear. Element size on the contacting surfaces is set to 0.2 mm and otherwise 0.5 mm. Augmented Lagrange [17] is used for the contact formulation. Frictional force is not considered in this analysis. The material of the worm is steel with Young’s Modulus E = 200 GPa and Poisson’s ratio 0.29. The material of the worm gear is bronze alloy with Young’s Modulus E = 105 GPa and Poisson’s ratio 0.3.
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Fig. 15(a) and (b) show the distribution of contact pressure before the modification and after the modification, respectively. Before the modification, the contact pressure distributes not only around the instantaneous contact points, but also around right corner, indicating the interference area as depicted in Fig. 12(b). Moreover, an edge contact occurs at a tip of the right corner with high contact pressure. However, after the modification, the contact pressure distributes only around the instantaneous contact points and no edge contact occurs at a tip of the right corner. Fig. 16 presents the maximum values of contact pressure on a worm gear tooth according to rotational angle of the worm in unmodified and modified cases. The line of dashes indicates the unmodified case and the solid line indicates the modified case. In the unmodified case, the maximum contact pressure around 45° of worm rotational angle shows a significantly high level of pressure. The highest pressure is 1090.5 MPa. In the modified case, however, the highest value of maximum contact pressure is reduced to 660.68 MPa as the edge contact at the tip of the right corner disappears. Fig. 17 shows the magnified view of the tooth contact pressure distribution on the worm gear for 45° of worm rotational angle where the highest pressure appears in the unmodified case. The unmodified case (Fig. 17(a)) shows an edge contact with over 700 MPa contact pressure at the right corner, while the modified case (Fig. 17(b)) shows less than 350 MPa contact pressure around the right corner. As a result, the advantage of the proposed modification is that such modification allows the asymmetric longitudinal crowning to be symmetric, mitigating the characteristic of the high sensitivity to a certain direction of misalignment. 5. Conclusions (1) This paper has introduced an asymmetric longitudinal crowning of the worm gear in cylindrical worm gear drives. The longitudinal crowning becomes more asymmetric when a higher lead angle is used; minimum separation between the worm and worm gear surface at the both corners of the worm gear tip differs significantly as the lead angle is increased. (2) The modification method to change the longitudinal crowning from asymmetric to symmetric is proposed. The modification utilizes the existing hobbing mechanism and the hob geometry. Any development of a new hobbing machine or a hob cutter with a special geometry is not required. The modification only requires minor adjustments to (i) the crossing angle between the hob and the worm gear axes, (ii) the shortest distance between the hob and the worm gear, (iii) the hob tooth thickness, and (iv) the hob addendum. (3) The results of the modification are confirmed by surface separation topology and contact pressure analysis. The asymmetrically crowned worm gear surface of existing design shows the following characteristics. First, the sensitivity to a certain direction of misalignment between the worm and worm gear is significantly higher than the sensitivity to the other direction of misalignment, making an edge contact easy in the certain direction. Second, as a lead angle is increased, the edge contact becomes more likely to occur even without misalignment. After the modification applied however, the minimum separation values between the worm and worm gear surface at the both corners of the worm gear tip become almost identical. In addition, the sensitivity to the both directions of misalignment also come to be similar. The modification increases transmission errors of the gear set, but the errors remain in an acceptable range. Based on these results, this paper has proved that the proposed modification can be a feasible solution to make the longitudinal crowning from asymmetric to symmetric in high lead cylindrical worm gear drives. References [1] J.R. Colbourne, The use of oversize hobs to cut worm gears, AGMA Technical Paper 89FTM8 (1989) [2] F.L. Litvin, V. Kin, Computerized simulation of meshing and bearing contact for single-enveloping worm-gear drives, J. Mech. Des. 114 (2) (1992) 313–316. [3] I.H. Seol, F.L. Litvin, Computerized design, generation and simulation of meshing and contact of modified involute, Klingelnberg and Flender type worm-gear drives, J. Mech. Des. 118 (1996) 551–555. [4] A. Narayan, D. Houser, S. Vijayakar, Study of Effect of Machinging Parameters on Performance of Worm Gears, AGMA 95FTM14, Alexandria, VA, 1995. [5] H.S. Fang, C.B. Tsay, Mathematical model and bearing contacts of the ZK-type worm gear set cut by oversize hob cutters, Mech. Mach. Theory 31 (1996) 271–282. [6] H.S. Fang, C.B. Tsay, Mathematical model and bearing contacts of the ZN-type worm gear set cut by oversize hob cutters, Mech. Mach. Theory 35 (20 0 0) 1689–1708. [7] V. Simon, Load distribution in cylindrical worm gears, J. Mech. Des. 125 (2) (2003) 356–364. [8] V. Simon, Computer aided loaded tooth contact analysis in cylindrical worm gears, J. Mech. Des. 127 (5) (2005) 973–981. [9] V. Simon, Influence of tooth errors and shaft misalignments on loaded tooth contact in cylindrical worm gears, Mech. Mach. Theory 41 (2006) 707–724. [10] V. Simon, The influence of gear hobbing on worm gear characteristics, J. Manuf. Sci. Eng. 129 (5) (2007) 919–925. [11] F.L. Litvin, K. Yukishima, K. Hayasaka, I. Gonzalez-Perez, A. Fuentes, Geometry and investigation of Klingelnberg-type worm gear drive, J. Mech. Des. 129 (2007) 17–22. [12] F.L. Litvin, I. Gonzalez-Perez, K. Yukishima, A. Fuentes, K. Hayasaka, Design, simulation of meshing, and contact stresses for an improved worm gear drive, Mech. Mach. Theory 42 (2007) 940–959. [13] J. Sohn, N. Park, Geometric interference in cylindrical worm gear drives using oversized hob to cut worm gears, Mech. Mach. Theory 100 (2016) 83–103. [14] J. Sohn, N. Park, Study on the influence of gear hobbing and shaft misalignments on the geometric interference of cylindrical worm gear set, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. (2016) Published online before print September 28, doi:10.1177/0954406216671543. [15] F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory, second ed., Cambridge University Press, New York, USA, 2004. [16] F.L. Litvin, I. Gonzalez-Perez, A. Fuentes, D. Vecchiato, T.M. Sep, Generalized concept of meshing and contact of involute crossed helical gears and its application, Comput. Methods Appl. Mech. Eng. 194 (2005) 3710–3745. [17] ANSYS mechanical structural nonlinearities, ANSYS Customer Training Material, ANSYS, inc., 2010
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
Modified worm gear hobbing for symmetric longitudinal crowning in high lead cylindrical worm gear drives Jonghyeon Sohn∗, Nogill Park Department of Mechanical Engineering, Pusan National University, 2 Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan, South Korea
a r t i c l e
i n f o
Article history: Received 31 October 2016 Revised 5 March 2017 Accepted 4 July 2017 Available online 12 July 2017 Keywords: Worm gear Crowning Edge contact Tooth modification Tooth contact analysis
a b s t r a c t A single enveloping worm gear of cylindrical worm gear drive is cut by an oversized hob. The oversize provides a longitudinal crowning on the worm gear tooth surface. However, when a lead angle of the worm becomes higher, the longitudinal crowning becomes more asymmetric. The asymmetrically crowned worm gear surface makes the gear contact more sensitive to the certain direction of misalignment. To change the longitudinal crowning from asymmetric to symmetric, a method of modified worm gear hobbing is proposed. The modification uses a modified machine tool setting and a modified hob specification. A new hobbing mechanism or hob geometry is not required in this method. Results of the proposed modification are confirmed by using a surface separation topology based on differential geometry, and contact pressure analysis based on finite element method. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In cylindrical worm gear drives, the single enveloping worm gear is cut by an oversized hob [1] which has larger pitch diameter than the worm. The oversized hob makes the worm gear tooth surface longitudinally crowned, making the instantaneous line contact of a fully conjugated set into the instantaneous point contact [2]. Consequently, the contact between the worm and worm gear becomes localized, reducing sensitivity to misalignment of the worm gear set. The influence of the hob oversize and the shaft misalignment on the contact have been studied by many researchers [3–10]. An edge contact problem caused by elastic deformation and the positive parabolic function of transmission errors was also studied with corresponding solutions [11,12]. The edge contact mainly occurs at a tip of the worm and worm gear tooth. The proposed solutions for the edge contact were a type of profile crowning. However, an edge contact can also occur at a side corner of the tip of the worm gear surface. Such edge contact easily occurs when a high worm lead angle is applied because the higher lead angle makes the longitudinal crowning more asymmetric. Fig. 1 shows the worm gear surface of the three cases in Table 1: Case 1 (Fig. 1(a)), Case 2 (Fig. 1(b)) and Case 3 (Fig. 1(c)). The three cases have identical pitch radii of the worm and worm gear. In particular, Case 3 has the highest lead angle and Case 1 has the lowest one. To examine a characteristic of the longitudinal crowning by the hob oversize, the surfaces Sgos0 , generated by 0% oversized hobs, are presented along with the surfaces Sgos4 , generated by 4% oversized hobs. In Case 1, the clearances between Sgos4 and Sgos0 at the left and right corner of the tip of the worm gear surface are 48 μm and 23 μm, respectively. In Case 2, the clearances at the left and right are 60 μm and 9 μm, respectively. In both Case 1 and 2, the clearances of left and right side are different to each other, and the case with higher lead angle shows a greater difference.
∗
Corresponding author. E-mail address: [email protected] (J. Sohn).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.07.004 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
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Nomenclature Lc Li Zg Zh Zw d0 dc dh lh , lw snc uc0 uh , θ h uw , θ w
αn γ λh , λw φw, φg ψ h, ψ g
path of the instantaneous contact points boundary of interference area tooth number of the worm gear thread number of the hob thread number of the worm center distance between the worm and the worm gear center distance between the hob (or worm) and the grinding wheel center distance between the hob and the worm gear leads of the hob and the worm normal tooth thickness of the grinding wheel pitch radius of the grinding wheel hob surface parameters worm surface parameters normal pressure angle crossing angle error of the worm and the worm gear lead angles of the hob and the worm rotational angles of the worm and the worm gear in contact rotational angles of the hob and the worm gear in hobbing process
Such difference in the clearances means that the worm gear surface is asymmetrically crowned in the longitudinal direction. Because of such asymmetric longitudinal crowning, worm gear drives with high lead angle have greater sensitivity to a certain direction of misalignment. For example, −0.3° of the crossing angle error may not be a problem, but 0.1° of the crossing angle error can cause an edge contact. Moreover, the edge contact can occur without misalignment. As shown in Fig. 1(c), the clearance at the left and right are 74 μm and −6 μm, respectively. Sgos4 and Sgos0 intersect near the right corner, and the clearance at the right corner shows a negative value, indicating that the hob oversize fails to provide the longitudinal crowning at the right corner. This phenomenon is geometric interference studied by Sohn and Park [13,14]. The interference can be avoided by cutting out the corner of the worm gear tooth. Nonetheless, the sensitivity to certain direction of misalignment remains high because the asymmetrically crowned geometry is unchanged. This study proposes a modified hobbing method to make a symmetric longitudinal crowning on the worm gear surface of cylindrical worm gear drives. The method includes a modified hobbing machine tool setting and a modified hob specification. The proposed modification method can be utilized without any development of a new hobbing machine or a hob cutter with a special geometry because the method employs the existing hobbing mechanism and the hob geometry. Results of the modification are confirmed by the surface separation topology based on differential geometry, and the contact pressure analysis based on finite element method. In particular, an initial guessing method for the tooth contact analysis (TCA) [15], which is required to determine the surface separation topology, is presented. The initial guessing method utilizes the line of action of the involute crossed helical gears [16] to estimate the instantaneous contact points. Converged solution of the TCA shows that the initial guess can provide a good convergence to the TCA of the single enveloping worm gear drives. 2. Theoretical background 2.1. Surfaces of worm, hob and worm gear The characteristic that the longitudinal crowning becomes asymmetric when the lead angle becomes higher is commonly found in ZA, ZN, ZK and ZI type of cylindrical worm gear drives. Also, the proposed modification can be used for these types. In this section, the ZK type is presented for an illustration. As shown in Fig. 2, the ZK hob surface Sh can be represented in the hob coordinate system Ch [xh , yh , zh ] as follows:
rh (uh , θh ) = [xh
yh
zh
1]T
θh cosuh + (uc − uc0 )tanαn − s2nc cosθh sinλh − dc sinθh yh = uc sinθh cosλh sinuh − uc cos θh cosuh + (uc − uc0 )tanαn − s2nc sinθh sinλh + dc cosθh zh = −uc sinλh sinuh + (uc − uc0 )tanαn − s2nc cosλh + θ2hπlh xh = uc cosθh cosλh sinuh + uc sin
where
uc =
− 2lπh
cotλh
cosuh
+ tanαn tanuh +
dc cos uh
+ tanαn
Snc
1 + tan2 αn
2
(1)
− dc cotλh tanuh + tan2 αn uc0
(2)
J. Sohn, N. Park / Mechanism and Machine Theory 117 (2017) 133–147
135
Fig. 1. Worm gear surfaces cut by both 0% and 4% oversized hobs; (a) Case 1, (b) Case 2 and (c) Case 3.
Table 1 Design parameters of worm gear set for three cases. Design data
Case 1
Profile type Thread number of worm Tooth number of worm gear Normal module Normal pressure angle Lead angle of the worm (right-hand helix) Pitch diameter of worm Pitch diameter of worm gear Pitch diameter of grinding wheel of worm and hob Normal tooth thickness of hob Normal tooth thickness of worm Oversize of hob Addendum of worm Dedendum of worm Addendum of worm gear Dedendum of worm gear Worm gear tooth width
ZK 1 50 2.3829 20° 6.8428° 20 mm 120 mm 330 mm 3.12 3.00 4% 2.4 mm 3.0 mm 2.4 mm 3.0 mm 14 mm
Case 2
Case 3
2 49 2.3787
3 47 2.3843
13.7608°
20.9558°
The ZK worm surface Sw can be represented by substituting the subscript of all parameters of Sh from h to w. As shown in Fig. 3, the worm gear surface Sg can be represented in the worm gear coordinate system Cg [xg , yg , zg ] as follows:
rg (uh , θh , ψh ) = Mgh (ψh )rh (uh , θh )
(3)
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Fig. 2. Relation between the coordinate systems of the hob and the grinding wheel.
Fig. 3. Relation between the coordinate systems of the worm gear and the hob.
f h ( u h , θh , ψ h ) = where ⎡ Mgh
cos ψg ⎢− sin ψg =⎣ 0 0
∂ rg ∗ ∂ rg ∗ × ∂ uh ∂ θh
sin ψg cos ψg 0 0
0 0 1 0
⎤⎡
·
0 0 1 0 ⎥⎢ 0 0 ⎦⎣1 0 0 1 0 0
∂ rg ∗ =0 ∂ ψh ⎤⎡ ⎤⎡ 0 −dh cos 0 sin 0 cos ψh 1 0 ⎥⎢ 0 1 0 0⎥⎢ sin ψh ⎦⎣− sin 0 cos 0⎦⎣ 0 0 0 0 1 0 0 0 1 0
(4)
− sin ψh cos ψh 0 0
0 0 1 0
⎤
0 0⎥ ⎦ 0 1
where
ψg =
Zh ψh and = λw − λh . Zg
rg ∗ in Eq. (4) is the vector with the last element of rg removed. Here, to modify the hobbing machine tool setting, and dh are set as follows:
=
+
d h = d h + d h
(5) (6)
The modification uses and dh instead of and dh . 2.2. Tooth contact analysis Tooth contact analysis (TCA) is performed to determine instantaneous contact points and transmission errors. A position vector and a unit normal vector of the worm surface Sw can be represented in the frame coordinate system Cf [xf , yf , zf ]
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137
Fig. 4. Relation among the coordinate systems of the frame, the worm, and the worm gear.
(Fig. 4) as follows:
rwf (uw , θw , φw ) = M f w (φw )rw (uw , θw )
(7)
nwf (uw , θw , φw ) = M∗fw (φw )nw (uw , θw )
(8)
where
⎡
cos φw
⎢− sin φ w ⎢ Mfw = ⎢ ⎣ 0
⎤
sin φw
0
cos φw
0
0
1
⎥, M∗fw = ⎣− sin φw 0 ⎦
0
0
1
0
0
⎡
−d0 ⎥ ⎥
cos φw
sin φw cos φw
0
0
0
⎤
0⎦, nw = 1
Nw , N = |Nw | w
∂ rw ∗ ∂ rw ∗ × ∂ uw ∂θw
A position vector and a unit normal vector of the worm gear surface Sg can be represented in the frame coordinate system Cf as follows:
rgf (uh , θh , ψh ,
φg ) = M f g (φg )rg (uh , θh , ψh )
(9)
ngf (uh , θh , ψh ,
φg ) = M∗fg (φg )ng (uh , θh , ψh )
(10)
f h ( u h , θh , ψ h ) = 0 where
⎡ Mfg =
⎢1 ⎢ ⎢ ⎣0 ⎡
M∗fg
0
0
1
0
0
(11) 0
⎤⎡
1
⎢ 0⎥ ⎥⎢0
1
0
⎥⎢ 0⎦⎣0
0
0
0
1
0
⎤⎡
1
0
0
cos γ
− sin γ
sin γ
cos γ
0
0
0
0
1
= ⎣1
0
0⎦⎣0
cos γ
0
1
0
sin γ
0
0
⎤⎡
0
⎤⎡
⎢ 0⎥ ⎥⎢− sin φg
cos γ
sin φg
0
cos φg
0
0
⎤
⎥⎢ 0⎦⎣
0
0
1
0⎥ ⎥ ⎥, 0⎦
1
0
0
0
1
cos φg ⎦ ⎣ − sin γ − sin φg 0
cos φg
0
sin φg cos φg 0
0
⎤
0⎦ , n g = 1
Ng , N = |Ng | g
∂ rg ∗ ∂ rg ∗ × ∂ uh ∂θh
The Eq. (11) is identical to the Eq. (4). The condition that the worm and the worm gear surfaces are in point contact can be represented as follows:
rwf (uw , θw , φw ) = rgf (uh , θh , ψh ,
φg )
(12)
nwf (uw , θw , φw ) = ngf (uh , θh , ψh ,
φg )
(13)
f h ( u h , θh , ψ h ) = 0
(14)
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Fig. 5. Illustration of the pitch coordinate system Cp , a line of action (LOA) and a plane .
ˆ from (a) the normal view of the zp xp plane and (b) the normal view of the plane . Fig. 6. An Illustration of an estimated instantaneous contact point K
The Eq. (14) is identical to the Eq. (4). The Eqs. (12)–(14) are six independent scalar equations in seven unknowns, uw ,
θ w , φ w , uh , θ h , ψ h , and φ g . Newton-Rhapson method is applied to solve these equations. However, the numerical solutions
of the TCA of worm gear drives can be easily diverged. To obtain a good convergence, it is essential to select an initial guess carefully. Here, the initial guess is determined by using an estimated line of action (LOA). The estimated LOA is identical to the LOA of aligned involute crossed helical gears [16] where the LOA is a straight line. This estimation is reliable because the instantaneous contact point of the single enveloping worm gear drives is located closely to the instantaneous contact point of the involute crossed helical gears. Fig. 5 shows the location of the estimated LOA on the plane . The angle between the plane and the yp zp plane is λw . The origin Op of the pitch coordinate system Cp [xp , yp , zp ] is on the plane . Aligned axes of the hob, the worm, the worm gear, and the grinding wheels are presented together. zc is the axis of the grinding wheel, Oc is the center of the grinding wheel, and ν c is the translational displacement of the grinding wheel. The superscript w and h indicate the case of the worm grinding and the hob grinding, respectively. The unit vector eLOA of LOA represented in the pitch coordinate system Cp can be calculated by using the unit vector of zp axis as follows:
eLOA =
exLOA eyLOA ezLOA
=
cosλw 0 −sinλw
0 sinλw 1 0 0 cosλw
1 0 0 cosαn 0 −sinαn
0 sinαn cosαn
0 0 1
(15)
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E Fig. 7. Illustration of a point P1 on the Sw and a point P2 on the Sg .
Fig. 8. The path of the instantaneous contact points Lc and the boundary of the interference Li on the worm gear surface, according to the variation of and dh .
Fig. 9. Illustration of the main idea of the modification using shifts of the Lc and Li .
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Fig. 10. The surface separation and the transmission errors (a) before and (b) after the modification, in Case 2.
Fig. 11. The surface separation and the transmission errors (a) before and (b) after the modification, in Case 3.
The equation of the LOA can be represented in the pitch coordinate system Cp as follows:
xp yp zp = = = tLOA exLOA eyLOA ezLOA
(16)
ˆ on Fig. 6 shows the normal views of the zp xp plane and the plane . Here, an estimated instantaneous contact point K ˆ along the LOA from the origin point Op . It is assumed the LOA is considered. The value tLOA indicates a displacement of the K that the reference condition has all zero value of the worm surface parameter θ w , the worm rotational angle φ w , the hob surface parameter θ h , the hob rotational angle ψ h , and the worm gear rotational angle φ g . Under the reference condition, h Ow c and Oc will be located at the same position of Op in the normal view of the zp xp plane (Fig. 6(a)). If we assume that
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Fig. 12. The surface separation before applying the modification in Case 2: (a) with the crossing angle error γ = −0.1◦ and (b) with the crossing angle error γ = 0.1◦
the grinding wheel surface is equal to the common rack surface of the involute crossed helical gears, the estimated instan because of the tooth thickness snc of the grinding wheel (Fig. 6(b)). Therefore, a taneous contact point will be located at K 0 to K ˆ along the LOA can be represented as follows: displacement of the estimated instantaneous contact point from K 0
DKˆ = tLOA + 2π Zw
snc cos αn 2
(17)
A surface normal pitch of the common rack is π mn cos α n and a worm rotational angle, φ w , corresponding to a pitch is ˆ , φ w can be estimated as follows: . Therefore, when the estimated instantaneous contact point is located at K
φw =
2π Zw
DKˆ π mn cos αn
=
2π Zw
tLOA + s2nc cos αn π mn cos αn
Similarly, ψ h can be estimated as follows:
=− ψ h
2π Zh
DKˆ π mn cos αn
=
2π Zh
tLOA + s2nc cos αn π mn cos αn
(18)
(19)
as follows: The worm gear rotational angle φ g can be estimated by using ψ h
=− φ g
Zh ψ Zg h
(20)
The minus sign of Eq. (20) is caused by opposite direction of ψ g and φ g as shown in Figs. 3 and 4. The worm surface parameter θ w has the same value to the worm rotational angle φ w . Similarly, the hob surface parameter θ h has the same magnitude to the hob rotational angle ψ h , but it has an opposite sign. Therefore, the estimated θ w and θ h can be represented as follows:
θw = φw
(21)
θh = −ψ h
(22)
uw and uh are the rotational angle of the grinding wheel coordinate system Cc [xc , yc , zc ] (Fig. 2) in the generation of the worm and hob, respectively. In Fig. 6(a), the distance ρ w between zcw axis and LOA can be calculated as follows:
ρw = νcw sin λw
(23)
where
νcw =
θw l w 2π
(24)
ˆ and zc xc0 plane (Fig. 5) is uc0 − y p where y p = tLOA eyLOA . Thus, an estimated uw can be calculated The distance between K as follows:
−1 u w = tan
ρw
= tan−1
uc0 − y p
θw lw sin λw 2π (uc0 − y p )
(25)
Similarly, an estimated uh can be calculated as follows:
uh = tan−1
ρh uc0 − y p
= tan−1
θh lh sin λh 2π (uc0 − y p )
(26)
, θ , ψ , and φ , of the seven unknowns, u , θ , φ , u , θ , ψ , and φ , Therefore, the initial guess, u w , θw , φw , u g w w w g h h h h h h ˆ and the converged solution X of several can be defined by the value of tLOA . Table 2 shows the calculated initial guess X
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J. Sohn, N. Park / Mechanism and Machine Theory 117 (2017) 133–147 Table 2 ˆ ) calculated by estimated line of action and its converged solutions (X) of Case 2 with no Initial guess (X misalignment. tLOA [mm] ˆ X X ˆ X
−5.7038 −2.8159
2.9598
X ˆ X X ˆ X
5.8477
X ˆ X
0.0720
X
uw [°]
θ w [°]
φ w [°]
uh [°]
θ h [°]
ψ h [°]
φ g [°]
−0.3222 −0.3231 −0.0643 −0.0647 0.1967 0.1968 0.4609 0.4615 0.7282 0.7293
−92.3355 −96.9350 −18.3107 −19.4302 55.7140 59.0123 129.7387 138.4096 203.7635 218.7795
−92.3355 −97.2346 −18.3107 −19.4955 55.7140 59.1844 129.7387 138.8224 203.7635 219.4362
−0.3139 −0.3139 −0.0665 −0.0665 0.1838 0.1838 0.4372 0.4372 0.6936 0.6936
−93.7807 −98.5082 −19.7560 −20.8785 54.2688 57.6934 128.2935 137.2250 202.3182 217.7340
93.7807 98.7988 19.7560 20.9456 −54.2688 −57.8529 −128.2940 −137.6140 −202.3180 −218.3560
−3.8278 −4.0277 −0.8064 −0.8547 2.2151 2.3567 5.2365 5.6074 8.2579 8.8981
Table 3 ˆ ) calculated by estimated line of action and its converged solutions (X) of Case 2 with crossing Initial guess (X angle error γ = 0.1◦ . tLOA [mm]
−2.8159
ˆ X X ˆ X
0.0720
X ˆ X
−5.7038
X ˆ X X ˆ X
2.9598 5.8477
X
uw [°]
θ w [°]
φ w [°]
uh [°]
θ h [°]
ψ h [°]
φ g [°]
−0.3222 −0.3217 −0.0643 −0.0632 0.1967 0.1984 0.4609 0.4632 0.7282 0.7311
−92.3355 −108.4217 −18.3107 −31.3154 55.7140 46.6809 129.7387 125.5735 203.7635 205.3648
−92.3355 −99.2889 −18.3107 −21.7619 55.7140 56.6749 129.7387 136.0314 203.7635 216.3145
−0.3139 −0.3139 −0.0665 −0.0665 0.1838 0.1838 0.4372 0.4372 0.6936 0.6936
−93.7807 −105.1722 −19.7560 −27.8522 54.2688 50.3700 128.2935 129.5026 202.3182 209.5501
93.7807 96.3513 19.7560 18.6266 −54.2688 −60.0199 −128.2940 −139.5992 −202.3180 −220.1206
−3.8278 −4.1103 −0.8064 −0.9460 2.2151 2.2556 5.2365 5.4948 8.2579 8.7719
ˆ and X of several instantaneous contact points instantaneous contact points of aligned Case 2 in Table 1. Table 3 shows the X of the Case 2 with crossing angle error γ = 0.1◦ . Here, uh is used as a fixed parameter. The converged solutions are close enough to the initial guess, and thus a good convergence of the numerical method can be obtained. This method of initial guessing is also effective in the TCA of other single enveloping worm gear drives such as ZA, ZN and ZI, whether those are either aligned or moderately misaligned. Corresponding transmission error is calculated by following equation:
T ransmission error = φg −
Zw φw Zg
(27)
2.3. Calculation of the minimum surface separation The minimum surface separation between the worm and worm gear can be represented by using the envelope to the family of the worm surface on the worm gear surface. As shown in Fig. 4, the envelope to the family of the worm surface E can be represented in the worm gear coordinate system C as follows: Sw g E rw (uw , θw , φw ) = Mgw (φw , φg (φw ) )rw (uw , θw )
f w ( u w , θ w , φw ) =
∗
∗
∂ rEw ∂ rEw × ∂ uw ∂ θw
(28)
∗
·
∂ rEw =0 ∂ φw
(29)
where ⎡ ⎤⎡ ⎤⎡ ⎤⎡ ⎤ cos φg − sin φg 0 0 1 0 0 0 0 1 0 0 cos φw sin φw 0 0 ⎢ sin φg cos φg 0 0⎥⎢0 cos γ sin γ 0⎥⎢0 0 1 0⎥⎢− sin φw cos φw 0 −d0 ⎥ Mgw = ⎣ ⎦⎣0 − sin γ cos γ 0⎦⎣1 0 0 0⎦⎣ 0 ⎦ 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 E∗ in Eq. (29) is the vector with the last element of r E removed. Here, the transmission function φ (φ ) is fitted to a rw g w w fourth-order polynomial function by using the numerical solutions obtained by TCA in the Section 2.2. Fig. 7 shows points E , and P is on S . P and P P1 (xp1 , yp1 , zp1 ) and P2 (xp2 , yp2 , zp2 ), represented in the coordinate system Cg [xg , yg , zg ]. P1 is on Sw g 2 1 2 are both located at (R, t). R is the distance between P1 and zg axis, or P2 and zg axis. t is the zg coordinate of P1 or P2 . An angle P1 Otg P2 is χ1 − χ2 . Therefore, the length ζ of the arc P1 P2 can be calculated as follows:
ζ = R(χ1 − χ2 ) = R tan
−1
y p1 x p1
− tan
−1
y p2 x p2
(30)
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The coordinates of P1 (xp1 , yp1 , zp1 ) can be represented as follows:
x2p1 + y2p1 = R2
(31)
z p1 = t
(32)
f w ( u w , θ w , φw ) = where
∗
∗
∂ rEw ∂ rEw × ∂ uw ∂ θw
∗
·
∂ rEw =0 ∂ φw
(33)
T
E rw (uw , θw , φw ) = x p1 y p1 z p1 1
The Eqs. (31) to (33) are three independent scalar equations in three unknowns, uw , θ w and φ w . These equations can be solved by using Newton-Raphson method. Similarly, the coordinates of P2 (xp2 , yp2 , zp2 ) can be represented as follows:
x2p2 + y2p2 = R2
(34)
z p2 = t
(35)
f h ( u h , θh , ψ h ) = where
∂ rg ∗ ∂ rg ∗ × ∂ uh ∂ θh
·
∂ rg ∗ =0 ∂ ψh
(36)
T
rg (uh , θh , ψh ) = x p2 y p2 z p2 1
The Eqs. (34)–(36) are three independent scalar equations in three unknowns, uh , θ h and ψ h . By using the calculated value of ζ for a specified range of R and t, we obtained the contour of the minimum separation between Sw and Sg . In normal case, ζ is zero at Lc , the path of the instantaneous contact points, and has positive value at the rest of the area. However, ζ can also be zero at the Li , a boundary of interference area. The interference area has a negative value of ζ as E is drawn into S . If the interference exists, the interference area appears around the right corner of the tip of the worm Sw g gear surface in a case with a right-hand helix. In a case with a left-hand helix, the interference area appears around the left corner of the tip of the worm gear surface. This method is fully described in Sohn and Park [13]. 3. Main idea of modification The main idea of the modification is obtained by the study that examines a characteristic that Lc and Li on the worm gear surface shift with respect to and dh . Fig. 8(a) shows Lc and Li in three different cases; dh = 0 mm in the all three cases, and = −0.05◦ , = 0◦ and = 0.05◦ in each case. Likewise, Fig. 8(b) presents Lc and Li in three different cases; = 0◦ in the all three cases, and dh = 0.025 mm, dh = 0 mm and dh = −0.025 mm in each case. Lc is depicted as a line with dots and Li as a solid line. Fig. 8 is based on the design data of Case 3 on the Table 1. Wider range of the worm gear surface is presented to illustrate clearly the shift of Lc and Li . In Fig. 8(a), Lc shifts to the left side and Li shifts to the right side when is a negative value. On the other hand, Lc shifts to the right side and Li shifts to the left side when is a positive value. The direction of shift with respect to dh in Fig. 8(b) is opposite to the direction of shift with respect to in Fig. 8(a). A comparison between Fig. 8(a) and Fig. 8(b) shows that the magnitudes of Lc shift in both cases are similar. However, the magnitude of Li shifts are dissimilar; the magnitude of Li shift in Fig. 8(a) is greater than that of Li shift in Fig. 8(b). In other words, when the shift of Lc with respect to has similar magnitude to the shifts of Lc with respect to dh , the shift of Li with respect to shows greater magnitude than the shift of Li with respect to dh . Such characteristic of the shifts of Lc and Li with respect to and dh appears in similar manner in other types of worm gear drives; ZA, ZN, and ZI. Using such characteristic, the simultaneous application of and dh is the main idea of the modification as follows (see Fig. 9): (i) LAc is the initial position of Lc and LAi is the initial position of Li when = 0 and dh = 0. (ii) When we apply that = 0 where 0 < 0, LAi is shifted to LBi that is placed outside of the boundary of the worm gear surface, and LAc is shifted to LBc simultaneously. (iii) When applying additionally that dh = dh0 where dh0 < 0, LBc is shifted to LCc , which is similar position of LAc . In parallel, LBi is shifted to LCi and still remains outside of the boundary of the worm gear surface. The separation value of the right corner becomes greater as Li goes further from the boundary of the worm gear surface. Therefore, adjustment of and dh can make the separation value of the left and right corner similar while making Lc
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Fig. 13. The surface separation after applying the modification in Case 2: (a) with the crossing angle error γ = −0.1◦ and (b) with the crossing angle error γ = 0.1◦
Fig. 14. (a) Finite element model of the worm and worm gear and (b) boundary conditions of the finite element analysis.
Fig. 15. Distribution of the contact pressure on the worm gear surface (a) before and (b) after the modification.
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Fig. 16. A graph of the maximum values of contact pressure on a worm gear tooth with respect to the rotational angle of the worm in unmodified and modified cases.
Fig. 17. Tooth contact pressure distribution on the worm gear for 45° of worm rotational angle (a) before and (b) after the modification.
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J. Sohn, N. Park / Mechanism and Machine Theory 117 (2017) 133–147 Table 4 Processing parameters and resultant surface separation on the left and right side corner of the worm gear before and after modification. Parameter
[°] [°] dh [mm] dh [mm] snh [mm] ha0 [mm] ζ [μm] Left corner Right corner
Case 2
Case 3
Unmodified
Modified
Unmodified
Modified
0.5391 0 70.4 0 3.12 3.00 60 9
0.1111 −0.428 70.078 −0.322 2.8856 2.6780 24 24
0.8416 0 70.4 0 3.12 3.00 74 −6
0.3146 −0.527 70.144 −0.256 2.9336 2.7440 46 46
remain at the center of the worm gear surface. The application of dh changes a tooth thickness of the worm gear. Thus, a normal tooth thickness snh of the hob should be changed as follows:
snh = snh + 2dh tan αn
(37)
In addition, the addendum of the hob ha0 should be changed as follows:
ha0 = ha0 + dh
(38)
In this way, application of and dh in the hobbing machine tool setting, and the adjustment of hob tooth thickness and hob addendum are the modification for the symmetric longitudinal crowning of the worm gear. 4. Results The results of the proposed modification are confirmed by the surface separation topology and contact pressure analysis. The results are similar in the four types of worm gear drives: ZA, ZN, ZK, and ZI. In this paper, we have taken the ZK type as a sample type of worm gear drive and presented the results of the modification applied in this ZK type. In Figs. 10– 13, the surface separation value ζ in Eq. (30) is illustrated by a contour graph, the path of the contact points Lc by a line with dots, and the interference area by inverted triangles. A unit of separation value of the contour graph is millimeter. The processing parameters of the worm gear before and after modification, and the resultant minimum surface separation on the left and right side corner are shown in Table 4. Case 2 in Table 1 is illustrated in Fig. 10, showing the surface separation and the transmission errors before and after the modification is applied. Before applying the modification, the separation value is 60 μm on left side corner and 9 μm on the right side corner. After applying the modification, both the separation values on the left and right side corner are turned into 24 μm symmetrically. Although peak to peak of transmission errors increases from 1.64 to 4.58 , it remains in an allowable range. Case 3 in Table 1 is illustrated in Fig. 11. Before the modification applied, the left side separation value is 74 μm. However, on the right side, the separation value is −6 μm and an interference area appears. After the modification applied, both the separation values on the left and right side corner are turned into 46 μm symmetrically, and the interference area on the right side is removed. Peak to peak of transmission errors increases from 4.24 to 10.17 , but again it remains in an allowable range. The path of contact maintains at the center of the worm gear surface in both Case 2 and Case 3. Figs. 12 and 13 presents how the proposed modification mitigates the characteristic of the high sensitivity to a certain direction of misalignment. Case 2 in Table 1 is used as a design data and shaft crossing angle errors γ = −0.1◦ and γ = 0.1◦ are used as misalignment. Fig. 12 shows the surface separation before the modification. When γ = −0.1◦ , as shown in Fig. 12(a), the left and right side separation values are 48 μm and 31 μm, respectively, and no interference occurs. However, when γ = 0.1◦ as shown in Fig. 12(b), the left side separation value is 75 μm, but the separation value on the right side is −10 μm, indicating that interference occurs on the right corner. Fig. 13 presents the surface separation after the modification. As shown in Fig. 13(a), when γ = −0.1◦ , the left and right side separation values are 12 μm and 46 μm, respectively, and no interference occurs. As shown in Fig. 13(b), when γ = 0.1◦ , the separation value on the left corner is 40 μm and the value on the right corner is 6 μm, exhibiting no interference unlike the Fig. 12(b). By using the finite element method, distributions of contact pressure in the cases used in Figs. 12(b) and 13(b) are compared. ANSYS, a commercial program, is used for the finite element analysis. The finite element model of the worm and worm gear is obtained from the three-dimensional surface model of the worm and worm gear, which is drawn from the coordinates of Sw and Sg . Fig. 14(a) and (b) shows the finite element model and boundary conditions, respectively. A rigid surface 1 on the inside of the worm is rigidly connected to ground by a revolute joint which rotates 4.5° per one step at the rotational axis of the worm. A rigid surface 2 on the worm gear rim is rigidly connected to ground by a revolute joint, where 100Nm torque is applied, at the rotational axis of the worm gear. Element size on the contacting surfaces is set to 0.2 mm and otherwise 0.5 mm. Augmented Lagrange [17] is used for the contact formulation. Frictional force is not considered in this analysis. The material of the worm is steel with Young’s Modulus E = 200 GPa and Poisson’s ratio 0.29. The material of the worm gear is bronze alloy with Young’s Modulus E = 105 GPa and Poisson’s ratio 0.3.
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Fig. 15(a) and (b) show the distribution of contact pressure before the modification and after the modification, respectively. Before the modification, the contact pressure distributes not only around the instantaneous contact points, but also around right corner, indicating the interference area as depicted in Fig. 12(b). Moreover, an edge contact occurs at a tip of the right corner with high contact pressure. However, after the modification, the contact pressure distributes only around the instantaneous contact points and no edge contact occurs at a tip of the right corner. Fig. 16 presents the maximum values of contact pressure on a worm gear tooth according to rotational angle of the worm in unmodified and modified cases. The line of dashes indicates the unmodified case and the solid line indicates the modified case. In the unmodified case, the maximum contact pressure around 45° of worm rotational angle shows a significantly high level of pressure. The highest pressure is 1090.5 MPa. In the modified case, however, the highest value of maximum contact pressure is reduced to 660.68 MPa as the edge contact at the tip of the right corner disappears. Fig. 17 shows the magnified view of the tooth contact pressure distribution on the worm gear for 45° of worm rotational angle where the highest pressure appears in the unmodified case. The unmodified case (Fig. 17(a)) shows an edge contact with over 700 MPa contact pressure at the right corner, while the modified case (Fig. 17(b)) shows less than 350 MPa contact pressure around the right corner. As a result, the advantage of the proposed modification is that such modification allows the asymmetric longitudinal crowning to be symmetric, mitigating the characteristic of the high sensitivity to a certain direction of misalignment. 5. Conclusions (1) This paper has introduced an asymmetric longitudinal crowning of the worm gear in cylindrical worm gear drives. The longitudinal crowning becomes more asymmetric when a higher lead angle is used; minimum separation between the worm and worm gear surface at the both corners of the worm gear tip differs significantly as the lead angle is increased. (2) The modification method to change the longitudinal crowning from asymmetric to symmetric is proposed. The modification utilizes the existing hobbing mechanism and the hob geometry. Any development of a new hobbing machine or a hob cutter with a special geometry is not required. The modification only requires minor adjustments to (i) the crossing angle between the hob and the worm gear axes, (ii) the shortest distance between the hob and the worm gear, (iii) the hob tooth thickness, and (iv) the hob addendum. (3) The results of the modification are confirmed by surface separation topology and contact pressure analysis. The asymmetrically crowned worm gear surface of existing design shows the following characteristics. First, the sensitivity to a certain direction of misalignment between the worm and worm gear is significantly higher than the sensitivity to the other direction of misalignment, making an edge contact easy in the certain direction. Second, as a lead angle is increased, the edge contact becomes more likely to occur even without misalignment. After the modification applied however, the minimum separation values between the worm and worm gear surface at the both corners of the worm gear tip become almost identical. In addition, the sensitivity to the both directions of misalignment also come to be similar. The modification increases transmission errors of the gear set, but the errors remain in an acceptable range. Based on these results, this paper has proved that the proposed modification can be a feasible solution to make the longitudinal crowning from asymmetric to symmetric in high lead cylindrical worm gear drives. References [1] J.R. 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Park, Study on the influence of gear hobbing and shaft misalignments on the geometric interference of cylindrical worm gear set, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. (2016) Published online before print September 28, doi:10.1177/0954406216671543. [15] F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory, second ed., Cambridge University Press, New York, USA, 2004. [16] F.L. Litvin, I. Gonzalez-Perez, A. Fuentes, D. Vecchiato, T.M. Sep, Generalized concept of meshing and contact of involute crossed helical gears and its application, Comput. Methods Appl. Mech. Eng. 194 (2005) 3710–3745. [17] ANSYS mechanical structural nonlinearities, ANSYS Customer Training Material, ANSYS, inc., 2010