A linkage model and applications of hobbing non-circular helical gears with axial shift of hob

Mechanism and Machine Theory 70 (2013) 32–44

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A linkage model and applications of hobbing non-circular helical gears with axial shift of hob Lian Xia a,⁎, Youyu Liu a,b, Dazhu Li a, Jiang Han a a b

CIMS Institute, Hefei University of Technology, Hefei 230009, PR China School of Mechanical and Automotive Engineering, Anhui Polytechnic University, Wuhu 241000, PR China

a r t i c l e

i n f o

Article history: Received 17 December 2012 Received in revised form 1 July 2013 Accepted 3 July 2013 Available online 31 July 2013 Keywords: Non-circular helical gears Axial shift of hob Hobbing Linkage model Hob life

a b s t r a c t The four-axis linkage hobbing without axial shift of hob has some disadvantages: narrow operating range, uneven load and wear among hob teeth. To solve these problems, a strategy of hobbing non-circular helical gears and its linkage model had been developed based on a method with axial shift of hob. The method includes two schemes: one is meshing point on hob fixed, and the other is meshing point moving at a constant velocity. Moreover, six possible operating modes are provided. The linkage model under every mode was verified to be valid by a virtual hobbing, and then their profile accuracies under the six modes have been compared and analyzed. We analyzed and compared their dynamics performances under every operating mode, and obtained their dynamic qualities of hobbing. Three practical applications of the linkage model were offered. The linkage model and the practical applications had been verified to be correct and feasible by a hobbing testing, which manifests that the hobbing method that meshing point on hob moving at a constant velocity has the longest hob life. Crown Copyright © 2013 Published by Elsevier Ltd. All rights reserved.

1. Introduction Non-circular gears are important, which are used in agricultural machineries such as transplanters and fertilizer distributors [1]. They are also widely employed in other stepless shift components for instruments or vehicles [2]. Non-circular helical gears can be applied in low speed and heavy load fields for their stable transmission, high carrying capacity, compact bodies, etc. Until now, the research on non-circular helical gears is far from enough for their various categories, complicated shape, complex calculation in design, especially their difficult manufacturing. Many researchers paid more attention to the design of non-circular gears and their applications in mechanisms [3,4], but little to hobbing process. Tan et al. [5] had built a basic mathematical model for hobbing non-circular spur gears. Hu et al. [6] had also built one for hobbing non-circular helical gears. However, they had provided no processing program for those models. We had constructed some available hobbing schemes and linkage models based on a four-axis linkage [7] and a five-axis linkage [8] respectively, and had singled out two excellent strategies with their linkage models progressively. Past studies have achieved non-circular spur or helical gears hobbing, which have improved the efficiency comparing to conventional wire-electrode cutting. Furthermore, the wire-electrode cutting cannot manufacture non-circular helical gear. However, the hobs in all of the hobbing methods mentioned have no shift along their axes, which leads to such disadvantages that hob shaft should be long enough and severely uneven abrasion on hob teeth, even breakage [9]. To solve those problems, on the basis of the method of four-axis linkage [7], we make hob continuous move along its axis while it is processing. Some hobbing linkage models and applications for non-circular helical gears with the hob axial shift will be studied in this paper.

⁎ Corresponding author. Tel.: +86 13956999895; fax: +86 551 2901632. E-mail address: [email protected] (L. Xia). 0094-114X/$ – see front matter. Crown Copyright © 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2013.07.002

L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

33

2. Problem statement As shown in Fig. 1, the four-axis linkage with no axial shift of hob is as follows [7]: hob rotation ωb generates a meshing movement with workpiece rotation ωc, and both of them keep a strict transmission ratio; workpiece moves along y-axis, vy, to form the pitch curve of non-circular helical gear; hob moves along z-axis, vz, with an additional movement Δωc, to cut a full-depth tooth with a helical angle βc; workpiece is fixed along x-axis. The four-axis linkage can reduce a linkage axis with respect to the five-axis linkage [8], having a motion vx along x-axis. Thus, the four-axis linkage can reduce an error source and enhance the stiffness of process system. The kinematic relation in cross-section of the workpiece for hobbing non-circular helical gears is shown in Fig. 2. The equation of pitch curve of high-order elliptic helical gears, a kind of non-circular helical gear, is given as follows [10]:
 2 r ¼ a 1−e =ð1−e cos n θÞ

ð1Þ

where r is polar radius, θ is polar angle, a is semi-major axis, e is eccentricity ratio, and n is order of non-circular helical gear. As shown in Fig. 2, w is position range of meshing point p on the pitch line of rack in cross-section of workpiece. μ is a angle between polar radius and the tangent of pitch curve of workpiece [11]. μ ¼ arctan½r=ðdr=dθÞ

ð0≤ μ b πÞ

ð2Þ

From Figs. 1 and 2, ne sinðnθÞ w ¼ 2rcos  1=2 : 1 þ e2 cos2 ðnθÞ−2e cosðnθÞ þ n2 e2 sin2 ðnθÞ

ð3Þ

As shown in Fig. 3, for an elliptical gear with a = 100 mm, e = 0.4, n = 1, wmax is 86.2 mm, and then it can be hobbed because the overall length of the hob usually less than 150 mm. For an oval gear with a = 300 mm, e = 0.3, n = 2, wmax is 336.6 mm, and then it cannot be hobbed because wmax is beyond the overall length of the hob. One critical defect of the four-axis linkage with no hob axial movement is that the meshing point P has a large displacement in the pitch line of hob, which is usually beyond the overall length of hobs thus limits the operating range of machine tools. 3. Hobbing strategy with axial shift of hob As shown in Fig. 2, S(o-xyz) is a machine coordinate system, in which the y-axis passes through the turning axis of the workpiece, and the z-axis and x-axis are independently parallel to the turning axis of the workpiece and the pitch line of the hob. Sb(ob-xbybzb) is a cutting tool coordinate system, moved with the helical rack, in which each axis is parallel with that of S(o-xyz). At the beginning, Sb(ob-xbybzb) and S(o-xyz) are kept in superposition. Sc(oc-xcyczc) is a workpiece coordinate system, revolving with the workpiece, in which the zc-axis is coincident with the turning axis of the workpiece. At the beginning, xc-axis is parallel with y-axis, and yc-axis is parallel with x-axis. In addition, we build a polar coordinate system which has a pole oc and a polar axis xc. The pitch curve equation of the non-circular helical gears is r = r(θ). The polar-angular velocity of non-circular helical gears is ω. The forward direction of θ, φc, and ω⁎c (resultant angular velocity of workpiece) are as shown in Fig. 2. There are two kinds of strategies in hobbing non-circular helical gears with axial shift of hob.

ωc Workpiece

z

Δω c

Hob

y O

vd vy b vz ω

x Fig. 1. Diagram of a four-axis hobbing process.

34

L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

vb1

vd1x

vd2x

vb2 ωb

y w

s µ

o

P

Pitch Curve of Gear

yc

ob x xb

P

r ϕc

vy

yb

l

A θ xc

Pitch Line of Rack

oc ω c*

Fig. 2. Kinematic relation in cross-section of workpiece.

3.1. Strategy of fixed meshing point on hob Based on the above strategy, we set continuous hob movement along its axis. That is to say, the meshing point of pitch curve of the gear is located on a fixed point P of pitch line of hob by adding an oblique feed vd1 along hob axis. In this way, the length of the hob is only determined by the contact ratio of helical rack with helical gear, and then hob can be shortened by the most. Consequently, the rigidity of processing system is enhanced and hobbing accuracy is improved. As shown in Fig. 2, the horizontal component of the velocity vd1 is vxd1. The meshing point on hob moves from point O to point P t x in time t. Obviously, ∫ vd1 = l − s. Then, 0

x

vd1 ¼

dð−r cos μ Þ dμ dr dθ ¼ r sin μ − cos μ : dt dt dθ dt

ð4Þ

From Eq. (2), we can infer the following. tan μ ¼ r=ðdr=dθÞ:

ð5Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < sin μ ¼ r= r 2 þ ðdr=dθÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : : cos μ ¼ ðdr=dθÞ= r 2 þ ðdr=dθÞ2

ð6Þ

Thus,

Fig. 3 shows following. θ ¼ φc −μ þ π=2

ð7Þ 500

Displacement (mm)

400

α=100

e=0.4

n=1

α=300

e=0.3

n=2

300

200

100

0

0

0.5π

π

1.5π

Polar angle θ (rad) Fig. 3. Position range of meshing point.

2π

L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

35

From Eq. (5),
 2 2 2 dμ ðdr=dθÞ −r d r=dθ dθ ¼  : dt dt r 2 þ ðdr=dθÞ2

ð8Þ

Thus, the polar-angular velocity ω is as follows. ω¼

dθ r 2 þ ðdr=dθÞ2  ω : ¼ 2 dt r þ 2ðdr=dθÞ2 −r d2 r=dθ2 c

ð9Þ

Substituting Eqs. (6), (7) and (8) into Eq. (4), then,

x vd1

¼−

r

3




 2 2 4 d r=dθ þ ðdr=dθÞ ω  2  3=2 r þ ðdr=dθÞ2

ð10Þ


 3 2 2 4 x r d r=dθ þ ðdr=dθÞ vd1 ¼ − ¼ ω  3=2 cosðλb −βc Þ r 2 þ ðdr=dθÞ2 cosðλb −βc Þ

vd1

ð11Þ

where λb is the lead angle of hob, and βc is the helical angle of the gear. As shown in Fig. 4, the direction of vd1 on hob axis conforms to the “right-hand corkscrew” rule with ωb while vd1 ≥ 0; otherwise, it conforms to the “left-hand corkscrew” rule [12]. vb1 is a velocity of helical rack caused by hob rotation, vb⁎ is a resultant velocity of the helical rack. 

x

vb ¼ vb1 þ vd1

ð12Þ

As shown in Fig. 2, as the pitch curve of the workpiece makes a pure rolling at point P with the pitch line of rack, the translational distance s of the pitch line of rack should be equal to the turned arc length AP of the pitch curve of the workpiece. s¼

Z θ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ ðdr=dθÞ2 dθ

ð13Þ

0

and then, vb1 ¼

ds ¼ dt

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ ðdr=dθÞ2 ω

t

ωb

ð14Þ

ω b Δω c ωc

Δω c ωc n

v

v

vb1 v

n

v vn

v

v

n t

P

βc

vn

v

λb

x d1

λb

d1

* z

vz 1

vz*

vt t

z d1

P

v

vb1

v*b

v

βc

d1

vt a)

d1

n t

z d1

x d1

*n b

v

v *nb

v

d1

* b

n

t

n

vz 1 t

b)

Fig. 4. Schematic diagram of hobbing in common tangent plan. a) Hobbing left-hand helical gears. b) Hobbing right-hand helical gears.

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L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

Eqs. (10) and (14) are substituted in Eq. (12), 4

 vb

r −r

¼

3


 2 2 2 2 d r=dθ þ 2r ðdr=dθÞ ω:  2 3=2 r þ ðdr=dθÞ2

ð15Þ

The pure rolling requires that the tangent velocity of workpiece at point P should be equal to the translational speed vb⁎ of the rack. As shown in Fig. 4, the component of the velocity vd1, vb⁎ and vt (linear velocity of pitch circle of hob) at normal direction are separately vnd1, vb⁎ n and vnt at the meshing point P. The normal velocity of hob should be equal to that of workpiece at the meshing point P [13]. n

n

n

vt þ vd1 ¼ vb :

ð16Þ

Moreover, vt ¼ Kmt ωb = 2 ¼ Kmn ωb =ð2 sin λb Þ

ð17Þ

where mt is the transverse module, mn is the normal module, and K is the lobe of hob. According to Fig. 4 and Eqs. (15), (16) and (17), h ω¼

i

2 3=2

2

r þ ðdr=dθÞ

cosðλb −βc ÞKmn

2ξ

ωb

ð18Þ

where ξ = [r3(d2r/dθ2) + (dr/dθ)4]cos λb + [r4 − r3(d2r/dθ2) + 2r2(dr/dθ)2]cos βc cos(λb − βc). Eq. (9) is substituted in Eq. (18), "

dr r þ2 dθ 2

ωc ¼

!2

d2 r −r 2 dθ

ffi #sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dr 2 cosðλb −βc ÞKmn ωb r þ dθ 2ξ

:

ð19Þ

According to Eqs. (9), (11) and (19),

vd1 ¼ −


 r3 d2 r=dθ2 þ ðdr=dθÞ4 2ξ

Kmn ωb :

ð20Þ

The velocity of workpiece along the y-axis is as follows, vy ¼

dðr sin μ Þ dr dθ dμ ¼ sin μ þ r cos μ : dt dt dθ dt

ð21Þ

Eqs. (6), (8) and (9)are substituted in Eq. (21), r vy ¼

" # 2 dr 2 dr d2 r r þ2 −r 2 cosðλb −βc ÞKmn ωb dθ dθ dθ 2ξ

:

ð22Þ

vzd1 is the vertical component of the velocity vd1, which forms the resultant velocity v⁎z with the original vertical movement vz1 of hob. Moreover, vz⁎ = vz1 + vzd1. Non-circular helical gears are divided into two categories: left-hand gears (marked as “L”) and right-hand gears (mark “R”). As shown in Fig. 4, z

vd1 ¼ vd1 sinðλb  βc Þ

ð23Þ

where the “–” is adopted for left-hand helical gears; the “+” is adopted for right-hand helical gears. Thus, 

vz ¼ vz1 þ vd1 sinðλb  βc Þ

ð24Þ

L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

37

We can set a rational and constant vz⁎ to ensure axial precision being uniform, and then vz1 is as follows, 

vz1 ¼ vz ‐vd1 sinðλb  βc Þ:

ð25Þ

As for non-circular helical gears, workpiece should rotate a cycle additionally while v⁎z moves a screw lead along its axle [14]. t t That is to say, ∫ rΔωc dt = ∫ tan βcvz⁎dt. 0 0 Then, 

Δωc ¼ vz tan βc =r:

ð26Þ

The resultant velocity ω⁎c of workpiece is as follows, 



ωc ¼ ωc1  vz tan βc =r:

ð27Þ

where the “+” is adopted while the direction of helix of hob is in accordance with that of gears; otherwise, the “–” is adopted. According to Eqs. (19), (20), (22), (25) and (27), we can achieve a hobbing linkage model by use of a method of equal arc length for workpiece [5]. In this linkage model, the revolving speed of hob is constant, ωb and v⁎z are two independent fundamental frequencies with constant amplitude, other axes move with them as strict transmission ratio. The linkage model obtained is as follows. " # 8 2 2 > dr 2 dr d r > > r r þ 2 −r cosðλb −βc ÞKmn ωb > > dθ dθ > dθ2 > > > v ¼ > y > 2ξ >
 > > > > r 3 d2 r=dθ2 þ ðdr=dθÞ4 < Kmn ωb vd1 ¼ − 2ξ > # > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi" >     > d2 r 2 > 2 dr 2 dr 2 > þ þ 2 −r r r cosðλb −βc ÞKmn ωb > dθ dθ  > > dθ2 v > > ω ¼ >  z tan βc c > > r 2ξ > :  vz1 ¼ vz −vd1 sinðλb  βc Þ:

ð28Þ

The strategy of meshing point on hob fixed can overcome the drawback of four-axis linkage without axial shift, for the meshing point is fixed on one point on the hob and the operating range of machine tools is expanded. However, just partial cutters participate in hobbing, which leads to severely uneven load and wear among hob teeth. Thus hob life is lower. 3.2. Strategy of moving meshing point on hob at a constant velocity Based on the strategy of fixed meshing point on hob, another shift vd2 along hob shaft is imposed to it, which can let all teeth participate in hobbing uniformly. Different from four-axis linkage without axial shift, the speed and displacement of meshing point P on the pitch line of hob can be controlled. Accordingly, both the operating range of machine tools and hob life can be taken into account. As shown in Fig. 2, vxd2 is the horizontal component of the velocity vd2 . vb2 is the velocity of rack caused by hob rotation. Order: x

vb1 ¼ vb2 þ vd2 :

ð29Þ

For the method of equal arc length for workpiece, both vb1 and vb2 are constant. Thus, vxd2 (vd2) is constant, and then every tooth participates in hobbing uniformly. The direction of vd2 conforms to the “right-hand corkscrew” rule with ωb (marked as “I”), and also can conform to the “left-hand corkscrew” rule with it (marked as “II”). Then, 

x

x

vb ¼ vb2 þ κvd2 þ vd1

ð30Þ

where κ is symbol coefficient. For scheme “I”, κ = 1; for scheme “II”, κ = − 1; vxd2 = vd2 cos(λb − βc). Eqs. (12) and (30) are substituted in Eq. (15),

 vb


 r 4 ‐r 3 d2 r=dθ2 þ 2r 2 ðdr=dθÞ2 ¼ ω:  2 3=2 r þ ðdr=dθÞ2

ð31Þ

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L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

According to Eqs. (12) and (16), n

n

n

n

vb ¼ vt þ vd2 þ vd1 :

ð32Þ

According to Eqs. (9), (17), (31) and (32), " 2

r þ2 ωc ¼

dr dθ

!2 ‐r

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi d2 r dr 2  r þ cosðλb ‐βc Þ dθ dθ2 ξ

ζ

ð33Þ

where ζ = Kmnωb/2 + κvd2 cos λb. The velocity vy of workpiece along the y-axis is as follows. vy ¼ rωc cos μ:

ð34Þ

Eqs. (6) and (33) are substituted in Eq. (34),

vy ¼

" # 2 dr 2 dr d2 r r þ2 r −r 2 cosðλb −βc Þ dθ dθ dθ ξ

ζ

ð35Þ

vzd2 is the vertical component of the velocity vd2, which forms the resultant velocity vz⁎ with the original vertical velocity vz2 and vzd1 of hob. Moreover, vz⁎ = vz2 + vzd1 + vzd2. Thus, 

vz ¼ vz2 þ ðvd1 þ κvd2 Þsinðλb  βc Þ

ð36Þ

where the “−” is adopted for left-hand helical gears; the “+” is adopted for right-hand ones. For the same sake, we can set vz⁎ to be rational and constant. Then, vz2 can be determined according to Eq. (37); the resultant angular velocity ω⁎c of workpiece can be determined according to Eq. (27). 

vz2 ¼ vz ‐ðvd1 þ κvd2 Þsinðλb  βc Þ

ð37Þ

According to Eqs. (20), (27), (33), (35) and (37), we can achieve another hobbing linkage model by use of the method of equal arc length for workpiece. ωb, vd2 and vz⁎ are independent fundamental frequencies with constant amplitude, other axes move with them as strict transmission ratio. The linkage model obtained is as follows. " # 8 2 2 > dr 2 dr d r > > r þ2 r −r 2 cosðλb −βc Þ > > dθ dθ > dθ > > > vy ¼ ζ > > ξ >
 > > 3 2 2 4 > > r d r=dθ þ ðdr=dθÞ < Kmn ωb vd1 ¼ − 2ξ # > " > > q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > dr 2 dr 2ffi > d r 2 > 2 > r þ 2 −r þ cosðλb −βc Þ r > dθ dθ > > dθ2 v >  > > ζ  z tan βc ω ¼ c > > r ξ > :  vz2 ¼ vz −ðvd1 þ κvd2 Þsinðλb  βc Þ:

ð38Þ

If κ = 0 (marked as “III”), Eq. (38) can be changed into Eq. (28). Thus, the model for non-circular helical gears with axial shift of hob can be unified as model (38), which includes three schemes: “I” (κ = 1), “II” (K = − 1) and “III” (κ = 0). There are altogether C13C12 = 6 operating modes for the “L” gears or “R” gears. The operating modes are as follows: IL, IR, IIL, IIR, IIIL and IIIR. 4. Verification by virtual hobbing simulation Similar to non-circular helical gears with random shape, polar radius r of pitch curve of an ovoid helical gear (two-order elliptic helical gear) changes with different polar angle θ, which has the essential features of any non-circular helical gears [15].

L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

39

The conclusions drawn based on an ovoid helical gear can be expanded to other non-circular helical gears with random pitch curve. Eq. (1) is the pitch curve equation of ovoid helical gear. Its principal parameters are as follows: a = 300 mm; e = 0.3; n = 2; mn = 12 mm; βc = 11°51′; number of teeth Z = 51; tooth width b = 50 mm; K = 1; λb = 3°19′. The non-circular helical gear is hobbed using virtual technology according to Eq. (38) by use of MATLAB [16]. For each operating mode, the processing time of hobbing one cycle along its pitch curve is set as 30 s, and that of whole process needs 600 s. The processing parameters are shown in Table 1. The tool paths are shown in Fig. 5, which shows that the non-circular helical gear can be hobbed correctly under every kind of operating mode. As shown in Fig. 2, according to the principle of pure rolling, the linear velocity of pitch curve of non-circular helical gear at the point P is constant while ωb and vd2 (or vxd2) is constant, which still meets the principle of equal arc length for the workpiece. Consequently, the tool paths of six operating modes shown in Fig. 5 are distributed uniformly along their pitch curves, and then the profile precision of each tooth is uniform in every operating mode. Since vz2 is controlled by an independent fundamental frequency and do not make linkage with other axes, the velocity vz⁎, synthesized by vz2, vd1 and vd2 and determined by Eq. (38), is − 0.083 mm in every operating mode, which conforms to the requirements in Table 1. Therefore, the axial precision of each tooth is also uniform in every operating mode. Obviously, a non-circular helical gear of kind of “L” or kind of “R” can be hobbed by use of any scheme of “I”, “II” or “III”. In theory, there is no difference in hobbing precision for those schemes. 5. Dynamics analysis for six operating modes To compare with the dynamic performance of the six operating modes above, the velocity or angular velocity of each axis and acceleration or angular acceleration of that are studied separately based on the linkage model (38). We will still work with the ovoid helical gear mentioned, and still adopt the value of parameter ωb, v⁎z and κvd2 listed in Table 1. 5.1. Performance analysis of velocities or angular velocities As shown in Fig. 6, the curves ω⁎c , vd1, vy and vz2 of six operating modes are drawn while the polar angle of the pitch curve changes from 0 to 2π by MATLAB. The maximum difference of ω⁎c between the operating mode “IL” (see Fig. 6a) and that of “IR” (see Fig. 6b) is only about 4.91%. Similarly, the changing trends and the amplitudes of ω⁎c between the operating mode “IIL” (see Fig. 6c) and that of “IIR” (see Fig. 6d) are essentially identical, the same as between the operating mode “IIIL” (see Fig. 6e) and that of “IIIR” (see Fig. 6f). However, the range of ω⁎c in scheme “I” (“IL” and “IR”) is about 1.37 times as much as that in scheme “II” (“IIL” and “IIR”), and is about 1.18 times as much as that in scheme “III” (“IIIL” and “IIIR”). Therefore, there is a little difference of the changing trends of curves ω⁎c when the rotating direction of teeth is different. On the other hand, with different κ, there is a remarkable difference of the changing trends of curves ω⁎c among the three schemes. The maximum of that is scheme “I”, and the minimum is scheme “II”. As shown in Fig. 6, there is a little difference of the changing trends of curves vd1 when the rotating direction of teeth is different. Moreover, the curves vd1 of the left-hand gears are almost perfectly consistent under every operating mode (“IL”, “IIL”, “IIIL”). The curves vd1 of the right-hand gears are almost perfectly consistent under every operating mode (“IR”, “IIR”, “IIIR”). Therefore, we can draw a conclusion that the curves vd1 only depend on the rotating direction of teeth, but nothing to do with the value of κ. As shown in Fig. 6, the curve of vy under the operating mode “IL” is completely consistent with the one under the operating mode “IR”, the same as between the operating mode “IIL” and that of “IIR” and between the operating mode “IIIL” and that of “IIIR”. The reason is that the shape of pitch curve is totally unrelated to the different rotating direction of teeth, and then the different rotating direction of teeth has nothing to do with vy. However, the range of vy in scheme “I” is 1.37 times as much as that in scheme “II”, and 1.16 times as much as that in scheme “III”. Obviously, the range of vy of the three schemes with different κ is greatly different. the range of vy in scheme “I” is the greatest, and that in scheme “II” is the least. As shown in Fig. 6, with the same κ, there is a remarkable difference of the changing trends of curves vz2 when the rotating direction of teeth is different. Nevertheless, the changing trends of curves vz2 with the same rotating direction of teeth do not vary hugely. For the left-hand gear, the peak value of curve vz2 under the operating modes “IL” is the maximum, and under the Table 1 Processing parameters of virtual hobbing under the six kinds of operating modes. Operating modes

IL IR IIL IIR IIIL IIIR

v⁎z

κvd2

(rad/s)

(mm/s)

(mm/s)

10.681 10.681 10.681 10.681 10.681 10.681

−0.083 −0.083 −0.083 −0.083 −0.083 −0.083

10.000 10.000 −10.000 −10.000 0.000 0.000

ωb

40

L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

a)

b)

c)

d)

e)

f)

Fig. 5. Hobbing ovoid helical gear with axial shift of hob using virtual technology. a) Operating mode “IL”. b) Operating mode “IR”. c) Operating mode “IIL”. d) Operating mode “IIR”. e) Operating mode “IIIL”. f) Operating mode “IIIR”.

operating modes “IIL” is the minimum. For the right-hand gear, the peak value of curve vz2 under the operating modes “IR” is the maximum, and under the operating modes “IIR” is the minimum. 5.2. Performance analysis of accelerations or angular accelerations According to Fig. 4 and Eq. (32), Z θ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kmn ωb þ 2ðvd1 þ κvd2 Þ cosλb t: r 2 þ ðdr=dθÞ2 dθ ¼ 2cos βc 0

ð39Þ

According to Eq. (39), the horizontal ordinate θ in Fig. 6 can be transformed into corresponding hobbing time t. Thus, all the curves v(ω)–θ in Fig. 6 can be transformed into v(ω)–t. All the curves v(ω)–t represented by discrete points can be fitted by some polynomials using the function p = polyfit (t,v(ω), n) of MATLAB [17]. Then, we take the derivative on the polynomials obtained using the function pp = polyder (p) [17], and gain the differential value of curves at the points of time series using the function a(α) = polyval(pp, t) [17]. The curves a(α)–t obtained are shown in Fig. 7. As shown in Fig. 6, the velocity or angle velocity curves are discontinuous at the points that θ = 0 or θ = 2π, which leads to the differential curves shown in Fig. 7 having no practical effect in the neighborhoods of 0 and 30. Those neighborhoods are not taken as analysis scopes. As shown in Fig. 7, the changing trends and the amplitudes of α⁎c with the same κ are essentially identical. For example, the maximum difference of α⁎c between the operating mode “IL” (see Fig. 7a) and that of “IR” (see Fig. 6b) is only about 6.40%. However, the range of α⁎c in scheme “I” (“IL” and “IR”) is about 1.45 times as much as that in scheme “II” (“IIL” and “IIR”), and is about 1.20 times as much as that in scheme “III” (“IIIL” and “IIIR”). Consequently, there is a little difference of the curves α⁎c between the left-hand gears and the right-hand gears. On the other hand, there is a remarkable difference of the curves α⁎c among the three schemes with different κ. The maximum of that is scheme “I”, and the minimum is scheme “II”. Similar to curves vd1, the curves ad1 shown in Fig. 7 only depend on the rotating direction of the teeth, but have nothing to do with the value of κ. Moreover, there is a little difference of the curves ad1 between the left-hand gears and the right-hand gears. As shown in Fig. 7, similar to curves vy, the curves ay of the left-hand gears are completely consistent with that of the right-hand gears with the same κ. the range of ay in scheme “I” is 1.40 times as much as that in scheme “II”, and 1.16 times as much as that in scheme “III”. Obviously, the ranges of ay of the three schemes with different κ are greatly different. The ranges of ay in scheme “I” is the greatest, and that in scheme “II” is the least. As shown in Fig. 7, there is a remarkable difference of the changing trends of curves az2 between the left-hand gears and the right-hand gears with the same κ. However, the changing trends of curves az2 with the same rotating direction of teeth do not vary hugely. In conclusion, the range of ω⁎c , vy, vz2 and α⁎c , vy, vz2 in scheme “II” are the least, which helps to control various velocities (or angular velocities) and accelerations (or angular accelerations) while high speed hobbing, and can ensure that the system has better dynamic quality. Every velocity or angular velocity mentioned in scheme “I” is about 1.37 times as much as that in scheme

L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

60

0.3

30

0.2

0

0.1

-30

0

-60 2π

π

0.5π

1.5π

0.4

60

0.3

30

0.2

0

0.1

-30

0

-60 2π

0

Polar angle θ (rad)

vz2

60

0.3

30

0.2

0

0.1

-30 0

vz2

120 90

0.4

60

0.3

30

0.2

0

0.1

-30 π

0.5π

1.5π

Angular velocity (rad/s)

vy

0.5

0.6

vd1 vy

0.5

vz2

120 90

0.4

60

0.3

30

0.2

0

0.1

-30 0

π

0.5π

Polar angle θ (rad)

0.6

vd1 vy

0.5

vz2

120 90

0.4

60

0.3

30

0.2

0

0.1

-30

0

-60 2π

1.5π

150 ωc*

0

π

0.5π

Polar angle θ (rad)

d)

-60 2π

1.5π

0.7

150 ωc*

0

-60 2π

π

0.5π

Polar angle θ (rad)

0.7

velocity (mm/s)

0.6

90

c)

velocity (mm/s)

150 vd1

ωc*

120

0.4

b)

0.7

Angular velocity (rad/s)

1.5π

vy

Polar angle θ (rad)

a)

0 0

π

0.5π

vd1

0.5

0

Angular velocity (rad/s)

0

vz2

90

150 ωc*

0.6

velocity (mm/s)

0.4

vy

0.5

0.7

120

velocity (mm/s)

90

vd1

velocity (mm/s)

0.5

vz2

150 ωc*

0.6

Angular velocity (rad/s)

vy

120

Angular velocity (rad/s)

vd1

ωc*

0.6

velocity (mm/s)

Angular velocity (rad/s)

0.7

150

0.7

41

-60 2π

1.5π

Polar angle θ (rad)

e)

f)

0

30 0

-0.1

-30

-0.2

-60 6

12

18

24

-90 30

v z2

0.1

-30

-0.2

-60

-0.3

0

6

12

18

24

2

-90 30

0.2 0.1 0

v z2

0.1 0

60 30 0

-0.1

-30

-0.2

-60 6

12

18

24

-90 30

a d1

αc*

0.2

ay v z2

0.1

90 60 30 0

0 -0.1

-30

-0.2

-60 0

6

Process time (s)

30 0

-0.1

-30

-0.2

-60

-0.3

0

6

12

18

12

18

24

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0.3

24

a d1

αc*

0.2

ay v z2

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-90 30

0

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6

12

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e)

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Process time (s)

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0.3

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c) Angular acceleration (rad/s )

ay

Angular acceleration (rad/s )

0.2

90

2

a d1

αc*

ay v z2

Process time (s)

0.3

a d1

αc*

b) Acceleration (mm/s2)

Angular acceleration (rad/s2)

30 0

0

Process time (s)

d)

60

-0.1

a)

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ay

Acceleration (mm/s2)

-0.3 0

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0.1

60

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Acceleration (mm/s2)

v z2

a d1

αc*

Angular acceleration (rad/s )

ay

0.3

Acceleration (mm/s2)

0.2

90

Angular acceleration (rad/s )

a d1

αc*

2

0.3

Acceleration (mm/s2)

Angular acceleration (rad/s2)

Fig. 6. Graphs of v(ω)–θ. a) Operating mode “IL”. b) Operating mode “IR”. c) Operating mode “IIL”. d) Operating mode “IIR”. e) Operating mode “IIIL”. f) Operating mode “IIIR”.

f)

Fig. 7. Graphs of a(α)–t. a)Operating mode “IL”. b) Operating mode “IR”. c) Operating mode “IIL”. d)Operating mode “IIR”. e) Operating mode “IIIL”. f) Operating mode “IIIR”.

42

L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

“II”. Every acceleration or angular acceleration mentioned in scheme “I” is about 1.40 times as much as that in scheme “II”. Scheme “I” can also be adopted in medium-low speed hobbing for its control performance do not behave hardly. However, scheme “I” should not be adopted in high speed hobbing for the excessive hobbing velocities or accelerations will reduce the control precision and the dynamic qualities of the system. The amplitudes of various velocities (or angular velocities) and accelerations (or angular accelerations) in scheme “III” fall in between those in scheme “I” and scheme “II”. Scheme “III” can also be adopted while using a hob with shorter overall length. The amplitudes of vd1 and ad1 are independent to any schemes adopted. Every velocity and acceleration of the three schemes is not sensitive to the different rotating direction of teeth. 6. Applications of linkage model The six operating modes above are acceptable while hobbing at low acceleration and speed. Moreover, scheme “II”, including “IIL” and “IIR”, can be adopted in priority. To enhance the stiffness of process system, the overall length of a hob should not be too long. Consequently, a non-circular helical gear with too long pitch curve or too long tooth width cannot be hobbed directly using scheme “I” or “II”, in which hob moves along its shaft in only one direction. As for high speed hobbing, we can apply a method called axial repeating shift to solve the problem, in which scheme “II” is just adopted. Hob exits along its shaft (or perpendicular to its shaft) and goes back with non-cutting stroke while the meshing point P moves to the end of hob, and then the hob cuts in along its shaft (or perpendicular to its shaft) again. Then, the meshing point P goes back to the beginning, and so forth. However, this method will increase the time of idle travel. It is also very easy to generate connective marks on teeth-profile surfaces because of secondary feed. As for medium-low speed hobbing, we can apply a method called axial reciprocating shift to solve the above-mentioned issue. In this method, scheme “II” is adopted firstly. Then, scheme “I” is applied while the meshing point P moves to the end of hob, in which the meshing point P moves reversely from the end of hob to the beginning. Scheme “II” is adopted again while the meshing point P reaches the beginning, and so forth. This method should be adaptively applied in priority because there is no idle travel of hob or connective marks on teeth-profile surfaces in hobbing. Scheme “III” can be adaptively applied while hob shaft is very short, for the meshing point on hob is fixed. 7. Experimental analyses 7.1. Hobbing verification A manufacturing module for non-circular helical gears is developed according to the linkage model (38) based on the hobbing NC platform of ARM & DSP & FPGA [18], which includes functions of four-axis linkage with no axial shift of hob, meshing point on hob fixed, and meshing point on hob moving at a constant velocity. The module is currently deployed on a hobbing machine named STAR-2010G. To compare and analyze their hob lives, three hobbing tests had been operated with the above three methods, respectively. The principal parameters of the non-circular helical gear hobbed are as follows: a = 100 mm, e = 0.3, n = 2, mn = 4 mm, βc = 11°51′, Z = 51, b = 25 mm; the principal parameters of hob are as follows: K = 1, λb = 3°19′; processing parameters are as follows: ωb = 10.681 rad/s, vz⁎ = − 0.083 mm/s, vd2 = 4.000 mm/s. The machining process and an ovoid helical gear obtained by the method called axial reciprocating shift, one of the above three methods, are shown in Fig. 8. Our experiment shows that the ovoid helical gear can be hobbed correctly using the method. 7.2. Analysis of hob life The workpiece is steel 45. High-speed steel hobs coated TiN are applied to the three tests. In each experiment, 76 gears are hobbed using a same hob. The hobbing route lm of each gear is 150 m, in 600 s. The wear standard of hob is the permissible value

Fig. 8. Machining process and ovoid helical gear obtained.

L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

43

[VB] of wear width on flank face at normal wear stage [19]. Under this conditions, the wear morphology of hob on flank face in the three experiments are shown in Fig. 9. The experimental result of the four-axis linkage with no axial shift of hob is shown in Fig. 9a, which shows that all teeth are abraded. However, the wear mass loss of each tooth varies in size and has no obvious regularity. Their maximum wear value VBmax is 0.536 mm. Their average wear value VBav is 0.342 mm. The reason is that non-circular pitch curve leads to the velocity and the acceleration of each axis vary sharply, and the hobbing conditions vary in different tooth. The experimental result of the strategy of meshing point on hob fixed is shown in Fig. 9b, which shows that there are only five teeth participating in hobbing and serious wearing. VBmax = 1.187 mm; VBav = 0.896 mm. As the rest of the teeth do not participate in hobbing and are not worn, the utilization efficiency of hob is low. The experimental result of the strategy of meshing point on hob moving at a constant velocity is shown in Fig. 9c, which shows that there is a little wear in every tooth. VBmax = 0.272 mm; VBav = 0.253 mm. Each of the teeth participates in hobbing at a constant speed because of the extra movement vd2. The wear mass loss of each tooth is basically identical because of their same hobbing conditions. VBmax of every tooth is substituted in Eq. (40), a formula of hob life [20], from which we can see that the hob life of meshing point on hob moving at a constant velocity is 1.97 times as much as that of four-axis linkage with no axial shift of hob, and is 4.36 times as much as that of meshing point on hob fixed.

Τ¼

½VB lm vt VB

ð40Þ

where T is hob life; vt is the hobbing velocity and its value is the same for the three schemes. 8. Conclusions 1) A strategy of hobbing non-circular helical gears has been built based on the method with axial shift of hob, which includes two schemes: one is the fixed meshing point on hob, and the other is the moving meshing point at a constant velocity. Moreover, six possible operating modes are provided. The solution presented in this article can expand the operating range of machine tools by comparing to the four-axis linkage without axial shift, and overcome the drawback that only micro-sized or small-eccentricity gears can be hobbed by the latter. 2) The results of virtual hobbing verify the correctness and effectiveness of the linkage model under the six operating modes. The tool paths show that the profile precision of each tooth is uniform in every operating mode.

a)

b)

c) Fig. 9. Wear morphology in flank faces of hob. a) Four-axis linkage with no axial shift of hob. b) Strategy of meshing point on hob fixed. c) Strategy of meshing point on hob moving at a constant velocity.

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L. Xia et al. / Mechanism and Machine Theory 70 (2013) 32–44

3) The dynamics analysis on six operating modes comes to the conclusion that the range of various velocities (or angular velocities) and accelerations (or angular accelerations) in scheme “I” is greatest, followed by that in scheme “III”, and that in scheme “II” is least. As for high speed hobbing, we can adopt the method called axial repeating shift, for scheme “II” has better dynamic quality. As for medium-low speed hobbing, we can adopt scheme “I” with scheme “II” together, namely the method called axial reciprocating shift. Scheme “III” can be adaptively applied while hob shaft is very short. 4) Hobbing experiment shows that the linkage model and the practical applications are correct and feasible, and also shows that the hob life of meshing point on hob moving at a constant velocity is the highest, and that of meshing point on hob fixed is the lowest. Overall, the method called axial repeating shift applying scheme “II” and the method called axial reciprocating shift applying scheme “I” with scheme “II” together are available. Not only can those methods expand the operating range of machine tools, the methods can also make the load and wear on the hob teeth even. Acknowledgment This work supported by National Natural Science Foundation of China (Grant No. 51275147); and Natural Science Foundation of Anhui Province (Grant No. 1308085ME78). Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.mechmachtheory.2013.07.002. References [1] B.L. Ye, G.H. Yu, Z.W. Chen, Y. Zhao, Kinematics modeling and parameters optimization of seedling pick-up mechanism of planetary gear train with eccentric gear and non-circular gear, Transactions of the Chinese Society of Agricultural Engineering 27 (2011) 7–12, (in Chinese). [2] F.L. Litvin, I. Gonzalez-Perez, A. Fuentes, K. Hayasaka, Design and investigation of gear drives with non-circular gears applied for speed variation and generation of functions, Computer Methods in Applied Mechanics and Engineering 197 (2008) 3783–3802. [3] D. Mundo, G. Gatti, D.B. Dooner, Optimized five-bar linkages with non-circular gears for exact path generation, Mechanism and Machine Theory 44 (2009) 751–760. [4] K.H. Modler, E.C. Lovasz, G.F. Bär, R. Neumann, D. Perju, M. Perner, D. Mărgineanu, General method for the synthesis of geared linkages with non-circular gears, Mechanism and Machine Theory 44 (2009) 726–738. [5] W.M. Tan, C.B. Hu, W.J. Xian, Y. Qu, Concise mathematical model for hobbing non-circular gear and its graphic simulation, Chinese Journal of Mechanical Engineering 37 (2001) 26–29, (in Chinese). [6] C.B. Hu, H.Y. Ding, K.M. Yan, Z.X. Wu, Simultaneous-control methods of CNC for hobbing noncircular helical gears, China Mechanical Engineering 15 (2004) 2175–2178, (in Chinese). [7] Y.Y. Liu, J. Han, L. Xia, X.Q. Tian, Hobbing strategy and performance analyses of linkage models for non-circular helical gears based on four-axis linkage, Strojniski Vestnik, Journal of Mechanical Engineering 58 (2012) 701–708. [8] Y.Y. Liu, J. Han, L. Xia, G.Z. Zhang, Hobbing process strategy for non-circular helical gears and performance analyses for functional models, Transactions of the Chinese Society for Agricultural Machinery 44 (2013) 281–287. [9] S. Stein, M. Lechthaler, S. Krassnitzer, K. Albrecht, A. Schindler, M. Arndt, Gear hobbing: a contribution to analogy testing and its wear mechanisms, Procedia CIRP 1 (2012) 220–225. [10] Y.Y. Liu, J. Han, G.Z. Zhang, Research on hobbing process principles and linkage models for higher-order elliptic gear, Advanced Materials Research 479–481 (2012) 2343–2346. [11] L.G. Meredith, M. Radestock, A reflective higher-order calculus, Electronic Notes in Theoretical Computer Science 141 (2005) 49–67. [12] C.M. John, Statistical physics, Modern Physics for Scientists and Engineers, Elsevier Inc., Atlanta, 2010. [13] B. Bair, M. Sung, J. Wang, C. Chen, Tooth profile generation and analysis of oval gears with circular-arc teeth, Mechanism and Machine Theory 44 (2009) 1306–1317. [14] R. Hsu, Z. Fong, Novel variable-tooth-thickness hob for longitudinal crowning in the gear-hobbing process, Mechanism and Machine Theory 46 (2011) 1084–1096. [15] N.G. Farzane, L.M. Lifshits, Increasing the life of liquid meters with oval gears, Measurement Techniques 18 (1975) 1645–1646. [16] Y.Y. Liu, G.Z. Zhang, J. Han, Graphic simulation of hobbing process for higher-order elliptic gear, Advanced Materials Research 482–484 (2012) 466–469. [17] J.E.D.S. Pereira, A.J. Strieder, J.P. Amador, J.L.S.D. Silva, L.L.V.D. Filho, A heuristic algorithm for pattern identification in large multivariate analysis of geophysical data sets, Computers and Geosciences 36 (2010) 83–90. [18] J. Fei, R. Deng, Z. Zhang, M. Zhou, Research on embedded CNC device based on ARM and FPGA, Procedia Engineering 16 (2011) 818–824. [19] J. Gerth, M. Werner, M. Larsson, U. Wiklund, Reproducing wear mechanisms in gear hobbing—evaluation of a single insert milling test, Wear 267 (2009) 2257–2268. [20] C. Claudin, J. Rech, Development of a new rapid characterization method of hob's wear resistance in gear manufacturing—application to the evaluation of various cutting edge preparations in high speed dry gear hobbing, Journal of Materials Processing Technology 209 (2009) 5152–5160.