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Linkage model and manufacturing process of shaping non-circular gears Fangyan Zheng a, Lin Hua a,⁎, Xinghui Han a, Bo Li b, Dingfang Chen b a b

School of Automotive Engineering, Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan University of Technology, Wuhan 430070, China School of Logistics Engineering, Wuhan University of Technology, Wuhan 430063, China

a r t i c l e

i n f o

Article history: Received 18 January 2015 Received in revised form 29 September 2015 Accepted 30 September 2015 Available online 11 November 2015 Editor: Philippe VELEX Keywords: Non-circular gear Gear shaping Manufacturing process

a b s t r a c t Current research of non-circular gear processing mainly focuses on gear hobbing. However, this method has many limitations including being unable to process internal gears or non-circular external gears with a concave pitch curve and is likely to undercut when processing noncircular gears of greater curvature with a smaller number of teeth. Gear shaping, in contrast, is a method that is able to overcome the limitations of gear hobbing. Relevant studies on gear shaping have focused on the theoretical bases rather than the concrete processes and tools. Through the combination of shaping theory and practice, this paper aimed to derive a linkage model for shaping non-circular gears, and to reﬁne the feeding strategy, and develop a cutter retraction and cutter setting method. Finally, with a pair of 3-order sinusoidal-gearratio non-circular gears, this paper demonstrated the entire process in a CNC gear-shaping machine, thus proving the accuracy of the mathematical model and providing a valid reference for shaping non-circular gears. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Due to their compact structure and accurate non-uniform transmissions, non-circular gears are widely used in function generators [1,2], gear pumps [3], and a variety of mechanical systems [4–6]. Until now, research on non-circular gears has been insufﬁcient for their various categories, complicated shapes, complex design calculations, and especially their challenging manufacturing processes. Many researchers have focused on the design of non-circular gears and their applications in mechanisms, but little attention has been paid to the processing aspects. Chen et al. [7] and Tan et al. [8] built a basic mathematical model for hobbing noncircular spur gears. Litvin, et al. [9] also built a model for hobbing non-circular helical gears. Liu et al. [10,11] and Xia et al. [12] constructed several hobbing systems and linkage models based on 4-axis and 5-axis hobbing machine respectively. However, there are some limitations to hobbing non-circular gears, including the inability to process internal gears or noncircular external gears with concave pitch curves, and they are apt to undercut when processing non-circular gears of greater curvature variation and smaller number of teeth; fortunately, a more universal method that does not involve the above-mentioned limitations is available: gear shaping. Cheng et al. [13] built a basic mathematical model for shaping non-circular spur gears; Cheng et al. [14] and Bair [15] proposed a computerized tooth proﬁle generation method and undercutting analysis method of noncircular gears manufactured with shaper cutter; Li et al. [16] proposed a numerical computing method of noncircular gear tooth proﬁles generated by shaper cutters; Tang et al. [17] discussed negative meshing in shaping non-circular internal gears;

⁎ Corresponding author. Tel./fax: +86 27 87168391. E-mail address: [email protected] (L. Hua).

http://dx.doi.org/10.1016/j.mechmachtheory.2015.09.010 0094-114X/© 2015 Elsevier Ltd. All rights reserved.

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Xiong et al. [18] researched the avoid method of cutting interference in non-circular gear shaping. Unfortunately, there are no processing programs available for these models. To solve these problems, this paper conducted the following exercises through a combination of shaping principles and manufacturing processes: a) Based on the normal vector of gear pitch curve, a simpliﬁed mathematical model of tooth proﬁlegenerating method with a shape cutter was established involving coordinate relationship, relative velocity, meshing equation, and b) in combination with the structure of the 3-axis linkage shaping machine and the process of shaping cylindrical gears, a mathematical model for shaping non-circular gears containing both a radial feed and rotary feed was developed. Furthermore, an equal-arc-length mathematical cutting model was proposed to avoid a varying cutting area in the model. c) A series of problems closely related to the process was discussed, including the feeding strategy, cutter-retracting approach, cutter preset method, and the design method of stock and ﬁxture. d) A pair of non-circular gears with a 3-order sinusoidal gear ratio was then used as an example; the stock and corresponding ﬁxture were designed, the processing parameters were set, the cutting process was simulated in the form of a computer graphic, and ﬁnally, the process was implemented with a 3-linkage CNC gear-shaping machine.

2. Mathematical model of generating noncircular gear 2.1. Generating method Fig. 1 shows the geometric relationship of tooth-proﬁle generating with a shape cutter. Its principle was to ensure the pure rotation between the pitch curve of a non-circular gear and the pitch circle of the shape cutter. Supposing that the shaped non-circular gear is ﬁxed on the ground, the coordinate system S0(O0 − x0y0) is rigidly connected to the gear. Its pitch curve is then deﬁned as r(φ). P represents the contact point between the pitch curve and pitch circle with a polar angle φ, thus:

xp ¼ r ðφÞ cosðφÞ : yp ¼ r ðφÞ sinðφÞ

ð1Þ

Assuming that t is the unit tangent vector of the pitch curve, and according to the basic knowledge of the planar curve [19], the tangent vector at P can be found by: 2

3 dðr ðφÞ cosðφÞÞ 0 6 7 dφ 7 ¼ r0 ðφÞ cosðφÞ−r ðφÞ sinðφÞ t0 ¼ 6 4 dðr ðφÞ sinðφÞÞ 5 r ðφÞ sinðφÞ þ r ðφÞ cosðφÞ dφ

where r 0 ðφÞ ¼ drðφÞ dφ

Fig. 1. Geometric relationships of tooth-proﬁle generating.

ð2Þ

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The module of the tangent vector is: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½r 0 ðφÞ cosðφÞ−r ðφÞ sinðφÞ2 þ ½r 0 ðφÞ sinðφÞ þ r ðφÞ cosðφÞ2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ½r 0 ðφÞ2 þ r ðφÞ2 :

jt0 j ¼

ð3Þ

Thus, the unit tangent vector is: 2

3 r 0 ðφ0 Þ cosðφÞ−rðφÞ sinðφÞ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ q 6 7 6 7 ½r 0 ðφÞ2 þ r ðφÞ2 t t 6 7 t¼ 0 ¼6 0 7 ¼ tx : jt0 j 6 r ðφÞ sinðφÞ þ r ðφÞ cosðφÞ 7 y 4 5 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½r 0 ðφÞ2 þ r ðφÞ2

ð4Þ

Assuming that the unit normal vector of a pitch curve is n at point P, as it is perpendicular to the unit tangent vector [20] then the following is true: 2

3 r 0 ðφÞ sinðφÞ þ r ðφÞ cosðφÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6 7 6 7 ½r0 ðφÞ2 þ r ðφÞ2 ty 6 7 n¼ ¼6 7: 0 −t x 6 −r ðφÞ cosðφÞ þ r ðφÞ sinðφÞ 7 4 5 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½r0 ðφÞ2 þ r ðφÞ2

ð5Þ

As shown in Fig. 1, a mobile coordinate system S1(O1 − x1y1) was set at the center of the shape cutter, its x1-axis and y1-axis parallel to the unit normal vector n and the unit tangent vector t, respectively; the distance between O1 and point P is the radius of the cutter's pitch circle, namely O1P = ro. The reference frame S2(O2 − x2y2) is ﬁxed on the shape cutter, and its angle relative to S1 is θ (the rotation angle of the shape cutter). The arc length of the pitch curve at point P is

SðφÞ ¼

Zφ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½r0 ðφÞ2 þ r ðφÞ2 dφ:

ð6Þ

0

With the pure rolling relationship [21], the rotating angle of the shape cutter can be represented as: Zφ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½r 0 ðφÞ2 þ r ðφÞ2 dφ θ¼

SðφÞ ¼ ro

0

ro

:

ð7Þ

The center of the shape cutter O2 is on a normal equidistant line of the pitch curve [22], namely:

xo2 ¼ rðφÞ cosðφÞ−r o t y : yo2 ¼ rðφÞ sinðφÞ þ r o t x

ð8Þ

Thus, the center distance between the cutter and non-circular gear can be found by: ⇀ E ¼ jO2 O0 j ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v u 2r o r ðφÞ u ﬃ: xo2 2 þ yo2 2 ¼ tr ðφÞ2 þ ro 2 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½r 0 ðφÞ2 þ rðφÞ2

ð9Þ

The polar angle of the cutter at point O2 is: " # y r ðφÞ sin φ þ r o t x γ ¼ a tan o2 ¼ a tan xo2 r ðφÞ cos φ−r o t y 8 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ9 > =

ð10Þ

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The angle between O0 O2 and PO2 is: 2

rðφÞ r o þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 3 0 ⇀ ⇀ x x −x ; y −y ð ; y Þ r ½ ð φ Þ2 þ r ðφÞ2 o2 p o2 p o2 o2 B PO2 O0 O2 C 4 5 ¼ a cos vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : α ¼ a cos@

⇀

⇀ A ¼ a cos u Er o 2r o r ðφÞ

PO2

O0 O2

ur ðφÞ2 þ r 2 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ t o 0 ½r ðφÞ2 þ r ðφÞ2 0

1

ð11Þ

Based on the geometric relations above, the coordinate transformation matrix between reference frames [23] S2 and S1 is: 2

M12

cosθ ¼ 4 − sinθ 0

3 sinθ 0 cosθ 0 5: 0 1

ð12Þ

The coordinate transformation matrix between reference frames S0 and S1 is easily obtainable through the use of the vector, and the coordinate basis vector of S0 is thus: 8 1 > > < i0 ¼ 0 : 0 > > : j0 ¼ 1

ð13Þ

Similarly, the coordinate basis vector of S1 is:

i1 ¼ n : j1 ¼ t

ð14Þ

Thus, the transformation matrix between reference frame S0 and S1 is: 2

M01

i0 i1 ¼ 4 j0 i1 0

i0 j1 j0 j1 0

3 2 ty xo2 yo2 5 ¼ 4 −t x 0 1

tx ty 0

3 xo2 yo2 5: 1

ð15Þ

The tooth proﬁles of the shape cutter are the same as those of spur gears, which are generated from rack cutters and can be deﬁned as r2 ðtÞ ¼ ½ x2 ðtÞ y2 ðtÞ 1 T in reference to Ref. [24]. Then the tooth proﬁle of the non-circular gear can be presented as the envelope curves of the tooth proﬁle of the shape cutter; that is: r ¼ M01 M12 r2 :

ð16Þ

2.2. Tooth proﬁle Section 2.1 explains the geometric relationship for shaping a non-circular gear, and presents the tooth proﬁle of a non-circular gear with envelope curves. Unfortunately, Eq. (16) signiﬁes the tooth proﬁle envelope curve, while the real tooth proﬁle is only the boundary of the envelope curve. In order to obtain the tooth proﬁle curve, Vasie et al. [25] proposed a numerical method that analyzed the features of each point on the envelope curve. Fortunately, there is a more effective and convenient way to do this, which is based on the meshing principle [26], namely nr vr ¼0

ð17Þ

where nr is the normal vector of the cutter's tooth proﬁle and vr is the relative velocity between the cutter and non-circular gear, as in this case: 2

3 dy2 ðt Þ − 6 dt 7 nr ¼ 4 5: dx2 ðt Þ dt

ð18Þ

Because the shaped non-circular gear (reference frame S0) is ﬁxed on the ground, the relative velocity vr is the velocity of the cutter (the velocity of the reference frame S2 relative to the ground reference frame S0), consisting of two velocity components: the movement along with the tangent of the pitch curve and the rotation together with its rotational axis.

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The tangential velocity along with the tangent of the pitch curve is: v t ¼ t0 :

ð19Þ

Supposing that the cutter's angular velocity along its rotational axis is ω2, then, the velocity at a point on the cutter's tooth proﬁle can be presented as [27]: 2

3 2 3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 3 0 x ðt Þ r ðφÞ2 þ ½r 0 ðφÞ2 −y2 ðt Þ 6 07 4 2 5 4 x2 ðt Þ 5: v2 ¼ ω2 r2 ¼ 4 dθ 5 y2 ðt Þ ¼ ro 0 0 dt

ð20Þ

The velocity of the cutter can be represented by: 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 0 2 2 6 r 0 ðφÞ cosðφÞ−r ðφÞ sinðφÞ−y ðt Þ ½r ðφÞ þ r ðφÞ 7 6 7 2 ro 6 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ7 v r ¼ vt þ v2 ¼ 6 7: 6 ½r 0 ðφÞ2 þ r ðφÞ2 7 4 0 5 r ðφÞ sinðφÞ þ r ðφÞ cosðφÞ þ x2 ðt Þ ro

ð21Þ

By substituting Eqs. (21) and (18) into the meshing equation (Eq. (17)), and then combining Eq. (17) and the envelope equation (Eq. (16)), the tooth proﬁle of the non-circular gear can be solved. 3. Linkage model of shaping a non-circular gear 3.1. Linkage model with equal-polar-angle cutting Fig. 2 shows the conﬁguration of the 3-linkage CNC shaping machine tool in which the A-axis is the revolving axis of the work piece (shaped gear); the B-axis is the revolving axis of the shape cutter; the X-axis is the linear axis of the center distance

Fig. 2. The conﬁguration of CNC shaping machine tool.

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between the cutter and the gear; Sp is the spindle of the machine tool, which implements the reciprocal cutting motion of the cutter. A-axis, B-axis, and X-axis are three linkage axes with interpolation. In terms of the mathematical model above, when the cutter and gear are perfectly meshed, the positional equation of each linkage axis is as follows: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 u > 2ro r ðφÞ u > > ﬃ X ð φ Þ ¼ E ¼ tr ðφÞ2 þ r o 2 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > > > r ð φ Þ2 þ ½r 0 ðφÞ2 > > 8 > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ9 > > > > > =

ð22Þ

According to the coordinate transformation, the transformation matrix between A-axis and X-axis is: 2

MXA

cos½AðφÞ ¼ 4 sin½AðφÞ 0

3 − sin½AðφÞ 0 cos½AðφÞ 0 5: 0 1

ð23Þ

The transformation matrix between the X-axis and the machine base can be expressed as: 2

MX0

1 1 ¼ 40 0 0 0

3 X ðφÞ 0 5: 1

ð24Þ

The transformation matrix between the machine base and the B-axis can be expressed as: 2

MB0

cos½BðφÞ ¼ 4 − sin½BðφÞ 0

3 sin½BðφÞ 0 cos½BðφÞ 0 5: 0 1

ð25Þ

Thus, the envelope equation of the cutter is: r¼MB0 MX0 MXA r2 :

ð26Þ

Although Eqs. (26) and (16) are based on different coordinates systems, in essence, they are equivalent. When shaping a gear, the feed of the cutter should perform motions in two directions instead of a single conjugate meshing of the gear. Similar to the process of a cylindrical gear, several process cycles are set in advance, the number of which is codetermined by the cutter parameters and the processing capability of the machine tool. An exceedingly small number of cycles will do harm to the cutter and the machine tool, while too many cycles will reduce process efﬁciency. Fig. 3 shows the geometric relationship in shaping a non-circular gear involving feed; and different from that shown in Fig. 1, the cutter pitch circle and gear's pitch curve are no longer in line with each other, namely O02 P ≠ r o . Supposing that O02 P 0 ¼ ho , in terms of Eq. (8), the position of O2 ' can be obtained with the following equation: (

0

xo2 ¼ rðφÞ cos φ−ho t y : 0 yo2 ¼ r ðφÞ sin φ þ ho t x

ð27Þ

Thus, the center distance is: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u ⇀ 2ho rðφÞ u 0 0 ﬃ: E ¼ jO2 O0j ¼ tr ðφÞ2 þ ho 2 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ðφÞ2 þ ½r 0 ðφÞ2

ð28Þ

198

F. Zheng et al. / Mechanism and Machine Theory 96 (2016) 192–212

Fig. 3. Geometrical relationship in shaping non-circular gear with feed.

And the polar angel of O2 ' can be expressed as: 8 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ9 > 0

ð29Þ

The angle between O0 O02 and P 0 O02 is obtained by: r ðφÞ2 ho þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ðφÞ2 þ ½r 0 ðφÞ2 0 α ¼ a cos vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : u 2ho r ðφÞ u tr ðφÞ2 þ ho 2 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ðφÞ2 þ ½r 0 ðφÞ2

ð30Þ

Therefore, the position of each linkage axis containing feed can be presented as follows: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 u > 2ho r ðφÞ u 0 0 > > ﬃ ð Þ ¼ E φ; h ¼ X tr ðφÞ2 þ ho 2 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > o > > > r ðφÞ2 þ ½r 0 ðφÞ2 > > 8 > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ9 > > > > > =

ð31Þ

The position of each axis contains two independent variables φ and ho. The process of shaping gears is always divided into several cycles, in each cycle; the ﬁrst feed is in the radial direction for a gear (ho increase while φ stays at zero), then the feed

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is in the tangential direction (φ increase while ho stays constant). The velocity of each axis in different feeding direction is as follows: 8 > ∂X 0 ðφ; ho Þ dφ dφ 0 φ > > ¼ X ðφ; ho Þ Vt X ðφ; ho Þ ¼ > > dt dt ∂φ > > > > > ∂A0 ðφ; ho Þ dφ dφ 0 φ > > ¼ A ðφ; ho Þ Vt A ðφ; ho Þ ¼ > > dt dt ∂φ > > > 0 > > ∂B ðφ; ho Þ dφ dφ 0 φ > > ¼ B ðφ; ho Þ < Vt B ðφ; ho Þ ¼ dt dt ∂φ 0 > > Vr 0 ðφ; h Þ ¼ ∂X ðφ; ho Þ dho ¼ X h ðφ; h Þ dho > > o o X > dt dt ∂ho > > > 0 > > ∂A ð φ; h Þ dh dho 0 h > o o > ¼ A ðφ; ho Þ Vr A ðφ; ho Þ ¼ > > dt dt ∂ho > > > 0 > > ∂B ðφ; ho Þ dho dho 0 h > > ¼ B ðφ; ho Þ : Vr B ðφ; ho Þ ¼ dt dt ∂ho

ð32Þ

where X φ(φ, ho), Aφ(φ, ho), and Bφ(φ, ho) are the corresponding differentials of functions X(φ, ho), A(φ, ho), and B(φ, ho) with respect to φ. Xh(φ, ho), Ah(φ, ho),and Bh(φ, ho) are the corresponding differentials of functions X(φ, ho), A(φ, ho), and B(φ, ho) with o o respect to ho. Vt is the velocity of each axis in tangential feed, while Vr is the velocity of each axis in the radial feed. dh and dφ are dt dt the feed rate of radial feed and that of the tangential feed, respectively. Generally, the feed rate remains constant in each process cycle [28], namely: 8 dφ > < ¼ fφ dt : dh > o : ¼ fh dt

ð33Þ

Thus, the velocity of each linkage axis is: 8 0 φ Vt X ðφ; ho Þ ¼ X ðφ; ho Þf φ > > > 0 φ > > Vt A ðφ; ho Þ ¼ A ðφ; ho Þ f φ > > > < Vt 0 ðφ; h Þ ¼ Bφ ðφ; h Þf B o o φ : 0 h > Vr ð φ; h Þ ¼ X ð φ; h > X o o Þf h > > > 0 h > Vr ðφ; ho Þ ¼ A ðφ; ho Þf h > > : A0 h Vr B ðφ; ho Þ ¼ B ðφ; ho Þf h

ð34Þ

3.2. Linkage model with equal-arc-length cutting According to the equal-polar-angle cutting linkage model shown in Section 3.1, the rotation of the gear (A-axis) is approximately uniform, meaning that rotating angle φ basically stays the same in each process step. The small inertial force of this uniform rotation helps to decrease kinematic errors of the machine tool, in turn improving its kinematic accuracy. However, along with its rotating angle, the radius of the non-circular gear's stock generally changes. Thus, the arc length of the pitch curve varies in different process steps, which means the cutting area changes from one process step to the next. The inconsistent cutting areas will result in varying degrees of roughness on the surfaces of the tooth proﬁles, and can cause severe damage to the cutter and spindle. To solve this problem, a linkage model with equal-arc-length cutting was proposed. Eq. (26) shows that the pitch-curve arc length of the non-circular gear and the rotating angle of the cutter constitute a linear relationship, which means if the rotation of the cutter is kept uniform, the arc length of the non-circular gear will stay the same in each process step. In this sense, to implement equal-arc-length cutting, the independent variable (the rotating angle of the noncircular gear φ) of the linkage model in Section 3.1 must be replaced with the rotor angle of cutter θ. In terms of Eq. (7): Zφ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ θr o ¼ r ðφÞ2 þ r 0 ðφÞ2 dφ:

ð35Þ

0

In term of Eq. (35), φ can be solved by the numerical method and its solution is an implicit function of θ, and can be presented as: φ ¼ φd ðθÞ:

ð36Þ

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Taking the derivative of Eq. (36), then: dθ ¼ dφ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ðφÞ2 þ r0 ðφÞ2 ro

¼ δðφÞ:

ð37Þ

By substituting φd(θ) into Eq. (31), the position of each linkage axis with equal-arc-length cutting can be presented as follows: 8 0 < X d ðθ; ho Þ ¼ X ½φd ðθÞ; ho 0 A ðθ; ho Þ ¼ A ½φd ðθÞ; ho : : d 0 Bd ðθ; ho Þ ¼ B ½φd ðθÞ; ho

ð38Þ

Because the feed rate in the equal-arc-length cutting method can be gained by taking the derivative of the angle of the cutter with respect to time, and with a constant feed rate in each process cycle, it can be assumed that: dθ ¼ f θ: dt

ð39Þ

Thus, the velocity of each linkage axis in feed with equal-arc-length cutting method is: 8 0 φ ∂X ðφd ðθÞ; ho Þ dφd ðθÞ dθ X ðφd ðθÞ; ho Þf θ > > > ¼ ð θ Þ ¼ Vdt > X > dθ dt δ½φd ðθÞ ∂φd ðθÞ > > > 0 φ > > ∂A ðφd ðθÞ; ho Þ dφd ðθÞ dθ A ðφd ðθÞ; ho Þf θ > > ¼ Vdt A ðθÞ ¼ > > dθ dt δ½φd ðθÞ ∂φd ðθÞ > < 0 ∂B ðφd ðθÞ; ho Þ dφd ðθÞ dθ Bφ ðφd ðθÞ; ho Þf θ : ¼ Vdt B ðθÞ ¼ > > dθ dt δ½φd ðθÞ ∂φd ðθÞ > > > > h > > VdrX ðθÞ ¼ X ðφd ðθÞ; ho Þf h > > > h > > VdrA ðθÞ ¼ A ðφd ðθÞ; ho Þf h > > : h VdrB ðθÞ ¼ B ðφd ðθÞ; ho Þf h

ð40Þ

3.3. Cutter retracting method When shaping a gear, the cutter should reciprocate [29]. To avoid scratches on the tooth ﬂank or damage to the cutter, a cutter retracting motion in the return stroke should be considered to keep the cutter away from the stock. The 3-linkage axis CNC shaping machine tool is conﬁgured with a relieving mechanism to implement the cutter retracting motion, and the motion strictly follows the center line between the cutter and the work piece. When processing cylindrical gears, as the normal direction of the gear pitch curve is consistent with the center line, cutter interference can be effectively avoided with this method. However, as far as non-circular gears are concerned, this method would be infeasible due to the normal direction of the gear's pitch curve being different from that of the center line. A practical method to avoid cutter interference while processing non-circular gears is to make the cutter center move in the direction of the gear pitch curve through the compound motion of the relieving mechanism and machine axes when the cutter approaches the bottom dead center. Fig. 4 shows the positional relationships of the machine axes during the compound relieving motion. Supposing that the relieving distance of the relieving mechanism is ΔE and the center distance between the gear and cutter is E ' ', then the position of the cutter center O2 ' ' is: (

00 00 0 0 xo2 φ ; ho ¼ xo2 ðφ; ho Þ þ ΔE cos γ 00 00 0 00 0: yo2 φ ; ho ¼ yo2 φ; ho þ ΔE sin γ

ð41Þ

Supposing that the position of the gear following the relieving motion is at P ' ' and its polar angle is φ ' ', then the coordinate value of P ' ' is: (

00

00 00 00 xp φ ¼ r φ cos φ

00 00 00 00 : yp φ ¼ r φ sin φ

ð42Þ

And the vector is: h iT 00 00 00 00 00 00 00 00 O2 P ¼ xp φ −xo2 ðφ; ho Þ yp φ −yo2 ðφ; ho Þ :

ð43Þ

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Fig. 4. The cutter retracting method of shaping a non-circular gear.

In terms of Eq. (2), the tangent vector at point P ' ' is: 00

t0 ¼

0 r ðφ}Þ cosðφ}Þ−r ðφ}Þ sinðφ}Þ : 0 r ðφ}Þ sinðφ}Þ þ r ðφ}Þ cosðφ}Þ

ð44Þ

By making the vector O2 ' ' P ' ' perpendicular to tangent vector t0 ' ', then the following is true: 00 00

00

O2 P t0 ¼0:

ð45Þ

The position of P ' ' can be solved with Eq. (45), and assuming that the solution is φ ' ' (θ), and by substituting it into Eq. (31), the position of each axis in the relieving motion is as follows: 8 0 < X r ðθ; ho Þ ¼ X ½φd }ðθÞ; ho 0 A ðθ; ho Þ ¼ A ½φd }ðθÞ; ho : : r 0 Br ðθ; ho Þ ¼ B ½φd }ðθÞ; ho

ð46Þ

Fig. 5. Presetting cutter datum for shaping non-circular gears.

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Table 1 Parameters of the gear. Gear ratio function i12(φ1)

Module m

Teeth number of drive gear z1

Pressure angle α

Addendum coefﬁcient ha

3.27 + 1.3 sin(φ1)

2.5

14

20°

1

Dedendum coefﬁcient h f

1.25

Therefore, the velocity of each linkage axis in the relieving motion can be found with the following: 8

X φ φ00d ðθÞ; ho f θ > > > 00 ð θ Þ ¼ Vrt > X > δ φd ðθÞ > > > φ 00 > > A φ > d ðθÞ; ho f θ > > ð Þ Vrt θ ¼ A > > δ φ00d ðθÞ > <

Bφ φ00d ðθÞ; ho f θ : Vrt ð θ Þ ¼ > B > > δ φ00d ðθÞ > > > h 00 > > Vrr ð θ Þ ¼ X φd ðθÞ; ho f h > X > > h 00 > > Vrr A ðθÞ ¼ A φd ðθÞ; ho f h > > > : h 00 Vrr B ðθÞ ¼ B φd ðθÞ; ho f h

ð47Þ

A cutter retracting method for shaping non-circular gears was developed in which the retracting motion was implemented by the compound motion of the relieving mechanism and machine axes. Unfortunately, the high-response control system and precise transmission in the machine tool reduced the processing efﬁciency. This created a new issue of how to improve the processing efﬁciency, which will be researched in subsequent papers. 3.4. Presetting cutter method The presetting cutter for shaping non-circular gears was complicated due to the irregular geometry stock. A series of rules apply to the process: a) Ensure that the stock and A-axis and the cutter and B-axis share the same respective center axis. b) Make sure the stock's starting point r(0) and the X-axis keep a certain rotation relation. And as is shown in Fig. 5, the relation is ascertained by modulating the angle between datum C of the stock and the X-axis. By substituting φ = 0, ho = h + ro into Eq. (10), this angle can be calculated by the following equation: 8 9 > > < = −hr0 ð0Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : γ st ¼ a tan > :h r ð0Þ þ rð0Þ r ð0Þ2 þ ½r 0 ð0Þ2 > ; o

ð48Þ

(a)

(b) Fig. 6. Proﬁle of the stock.

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Fig. 7. Stock and its corresponding ﬁxture.

c) Ensure a certain angle between the cutter's tooth proﬁle and the stock. As shown in Fig. 5, the angle is ascertained by modulating between datum D of the cutter and the X-axis. By substituting φ = 0, ho = h + ro into Eq. (11), this angle can be calculated as follows: 2

r ð0Þ h þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ð0Þ2 þ ½r 0 ð0Þ2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : θst ¼ a cos v u 2hrð0Þ u trð0Þ2 þ h2 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ð0Þ2 þ ½r0 ð0Þ2

ð49Þ

When processing the driven gear with the above cutter, the position angle should be as follows: 0

θst ¼ θst þ

π zo

ð50Þ

where zo is the tooth number of the shape cutter. d) Ensure a constant center distance of Est between the A-axis and B-axis by modulating the value of the X-axis. By substituting φ = 0, ho = h + ro into Eq. (28), the center distance can be obtained with the following equation: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u 2hrð0Þ u Est ¼ trð0Þ2 þ h2 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ: r ð0Þ2 þ ½r 0 ð0Þ2

ð51Þ

3.5. Manufacturing process of shaping non-circular gears The manufacturing process of shaping non-circular gears can be divided into Ni cycles, and just like the fabrication of cylindrical gears, each cycle consists of two independent feeds: the radial feed and the rotary feed. The feed rate and cutting depth are generally different between the different cycles. The radial feed, rotary feed, and the cutting depth are fhi, fθi, and Δhi, respectively, in each cycle.

Table 2 Parameters of the shape cutter. Module mo (mm)

Tooth number zo

Pressure angle αo (deg)

Addendum (mm) ha ' (mm)

Dedendum (mm) hf ' (mm)

2.5

11

20°

2.5

3.125

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Table 3 Cutter presetting position. Gear

A-axis γst (deg)

B-axis θst (deg)

X-axis Est (mm)

Drive gear Driven gear

9.212 4.5

7.722 28.798

35.251 71.924

With an increased cutting depth, the cutting area expands and the subsequent cutting force is also increased. Therefore, a decreasing feed strategy should be adopted, which means both the feed rate and cutting depth decrease along with the increasing number of cycles: 8 < f hiþ1 b f hi f θ b f θi ði ¼ 1⋯Ni −1Þ: : iþ1 Δhiþ1 b Δhi

ð49Þ

The feed rate and cutting depth should be set according to the parameters of the cutter, machine tool, materials, etc. In general cutting, a shaper cutter is used for both roughing and ﬁnishing, meaning that the precision and roughness of the gear ﬂanks are mainly determined in the last processing cycle; therefore, the feed rate and cutting depth of the last cycle should all be set at small amounts. The radial feed time in each cycle can be calculated as follows: thi ¼

Δhi ði ¼ 1⋯Ni Þ: f hi

ð50Þ

The rotary feed time in each cycle can also be ﬁgured by: tθi ¼

2πz ði ¼ 1⋯Ni Þ zo f θi

ð51Þ

where z is the tooth number of the shaped gear and zo is the tooth number of the cutter. The total process time can be calculated as follows:

tA ¼

Ni X ðthi þ tθi Þ:

ð52Þ

i¼1

Fig. 8. Presetting position of drive gear.

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angular error of cutter (deg)

0.14

equal-arc-length cutting method equal-polar-angle cutting method

0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

processing steps Fig. 9. Angular error of cutter for setting processing step.

4. Example and analysis 4.1. Parameters of shaped gears and shape cutter A pair of non-circular gears with 3-order sinusoidal gear ratio function was used as an example. And the parameters of the gear are shown in Table 1. The arc length of the drive gear pitch curve is calculated using Eq. (6), thus giving:

Sð2πÞ ¼

Z2π qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r 1 ðφÞ2 þ r01 ðφÞ 2 dφ ¼ mz1 π ¼ 109:956 mm: 0

And the pitch curve of the drive gear is as follows: r 1 ðφ1 Þ ¼

a a ¼ : 1 þ i12 ðφ1 Þ 4:27 þ 1:3 sinðφ1 Þ

Hence the center distance can be solved by a ¼ 69:428 mm:

positional error of cutter (mm)

0.0350

equal-polar-angle cutting method equal-arc-length cutting method

0.0325 0.0300 0.0275 0.0250 0.0225 0.0200 0.0175 0.0150 0

500

1000

1500

2000

2500

3000

3500

4000

processing steps Fig. 10. Positional error of cutter for setting processing step.

4500

5000

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Table 4 Process parameters setting.

Cutting depth Δhi (mm) Radial feed rate fhi (mm/s) Rotary feed rate θh (deg/s) Radial feed time thi (s) Rotary feed time of drive gear tθi (s) Rotary feed time of driven gear tθi (s) Process time of drive gear tA (min) Process time of driven gear tA (min)

Cycle 1

Cycle 2

Cycle 3

Cycle 4

Cycle 5

2 0.1 0.03 20 266.559 799.678 62.578 182.544

1.7 0.05 0.02 34 399.839 119.517

1.4 0.03 0.015 46.667 533.119 1599.36

0.5 0.01 0.01 50 799.678 2399.03

0.025 0.005 0.005 5 1599 4798.07

The pitch curve of the driven gear is: r 2 ðφ1 Þ ¼ a−r 1 ðφ1 Þ ¼ 69:428

3:27 þ 1:3 sinðφ1 Þ : 4:27 þ 1:3 sinðφ1 Þ

The rotating angle of the driven gear is: Zφ1 φ2 ðφ1 Þ ¼ 0

1 dφ: i12 ðφÞ

The tooth number of the driven gear is: z2 ¼

2πz1 ¼ 42 φ2 ð2πÞ

The tooth addendum is ha ¼ ha m ¼ 2:5, and by substituting ho = ha into Eq. (27), the stock proﬁle curve can be calculated as follows:

xa ¼ r ðφÞ cosðφÞ−ha t y : ya ¼ r ðφÞ sinðφÞ þ ha t x

Fig. 6 shows the stock proﬁle in which (a) is the stock proﬁle of the drive gear and (b) is the stock proﬁle of the driven gear. Both the stock and its corresponding ﬁxture were fabricated and assembled as exhibited in Fig. 7. To avoid undercutting and pointed teeth, the parameters of the shape cutter were set in term of Ref. [15]. Table 2 presents the parameters of the shape cutter. The presetting position of each axis was calculated using Eqs. (48), (49), (50), and (51) (as shown in Table 3). Fig. 8 illustrates the shape of the cutter and the presetting position of the drive gear.

Fig. 11. Envelope curve in Cycle 1.

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Fig. 12. Envelope curve in Cycle 2.

4.2. Error analysis of the processing Because of a linear interpolation method of the CNC system, theoretical errors depend on the processing step setting [30], regardless if the error was caused by the cutter, machine or material. Taking the drive gear as an example, the total processing steps were set as Ne ¼ 5000: Thus the step length of equal-polar angle cutting obtained for the maximal polar angle of drive gear is 2π. Δφ ¼ 2π=Ne ¼ 0:00126rad ¼ 0:072 deg While the step length with equal-polar angle cutting is obtained in term of Eq. (7) Δθ ¼ Sð2πÞ=ðr o Ne Þ ¼ 2z1 π=ðzo Ne Þ ¼ 0:0016rad ¼ 0:091 deg

Fig. 13. Envelope curve in Cycle 3.

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Fig. 14. Envelope curve in Cycle 5.

The value of the parameter in two cutting method is as follows in each cutting step ei

φi ¼ ei Δφ e ¼ 0; 1⋯Ne : θi ¼ ei Δθ i

By substituting φi and φd(θi) into Eq. (7), the angle of the cutter θi in the coordinate system of the gear can be obtained, and the angular error of the cutter can be obtained as Δθ = θi − θi − 1. Fig. 9 shows the angular error of the cutter in the setting processing step, the red curve is the angular error with equal-polar-angle cutting while the blue one is the angular error with equal-arc-length cutting. It shows that the angular error with equal-polar-angle cutting is change while that with equal-arclength cutting held steady; the maximum error with equal-polar-angle cutting was 0.1224° while that with equal-arc-length cutting was 0.0918°. By taking ho = 0, and substituting φi and θi into Eqs. (31) and (38), the position of each axes can be obtained as (Xφi, Aφi, Bφi) and (Xθi, Aθi, Bθi). By taking the center of cutter r2e ¼ ½ 0 0 1 , and substituting the position of each axis into Eq. (26). The position of the cutter center can be obtained as rφi and rθi in the coordinate system of the gear. Finally, the positional error of the cutter can be obtained as

Δeφi ¼ jr φi −r φði−1Þ j : Δeθi ¼ jr θi −rθði−1Þ j

Fig. 10 shows the positional error of cutter for the setting processing step, the red curve is the positional error with equalpolar-angle cutting, and the blue one is the positional error with equal-arc-length cutting. It is shown that the positional error with equal-polar-angle cutting changed while that with equal-arc-length cutting held steady; the maximum error with equalpolar-angle cutting was 0.0293 mm while that with equal-arc-length cutting was 0.0221 mm.

Fig. 15. Velocity of each machine axis in tangential feeding for shaping the drive gear.

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Fig. 16. Velocity of each machine axis in tangential feeding for shaping the driven gear.

Based on error analysis of the two cutting method, both the angular error and the positional error of the cutter with equal-arclength cutting were less than that with equal-polar-angle cutting. Thus the equal-arc-length cutting was adopted; the expected errors in processing are shown in Figs. 9 and 10 for drive gear. 4.3. Process setting and simulation analysis There were altogether ﬁve cycles for shaping the drive gear and the driven gear. The corresponding feed rate, cutting depth, and processing time are listed in Table 4. Firstly, the position of each machine axis was calculated using Eq. (38), next substituted into Eq. (26), and then the envelope curve of the cutter in each cycle was then obtained (except for cycle 4 due to its resemblance to the last cycle), as shown in Figs. 11–14. The velocity of each machine axis was calculated in terms of Eq. (40). Figs. 15 and 16(a), (b), and (c) illustrate the velocity of the A-axis, B-axis, and X-axis of each gear in the tangential feed, respectively. The tooth proﬁles of gears were obtained based on the mathematical model discussed in Section 2 and shown in Fig. 17. The geometrical shape of all teeth was different for drive gear, while that of all teeth for driven gear was third order symmetric. As it is shown in Fig. 17, shaped tooth proﬁles without points and undercutting. 4.4. Numerical control processing There are two machine tool control methods; one is the velocity control based on the NC platform of ARM & DSP & FPGA [31] with the linkage model of Eq. (40). The other control method is the position control based on the CNC system with the linkage model of Eq. (38). Here, the second method was taken: the cutting processes were implemented with the help of a numerical control gear shaping machine based on the SINUMERIK 840D system. Fig. 18 shows the processing of the drive gear (a) and the driven gear (b). Fig. 19 shows the ﬁnished drive gear (a) and driven gear (b), where the tooth proﬁle of each ﬁnished gear is the same as the corresponding tooth proﬁle shown in Fig. 17. The surface roughness of the ﬁnished gears seems ﬁne in Fig. 19, while the manufacturing precision still needs further investigation. Fig. 20 shows that the two gears are in a good contact state and they could roughly meet the design requirement. However, further research of an effective test method is needed.

Fig. 17. Tooth proﬁle geometry of shaped gear.

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Fig. 18. Processing of each gear.

5. Conclusions In combination with the principles of shaping non-circular gears, kinematic relation of the machine, and the manufacturing processes, this paper presented a 3-linkage model with an equal arc-length cutting method involving feed. Additionally, the feeding parameter settings, stock design method, retracting interpolation, and cutter datum presetting relating to the process were discussed in this paper. The ﬁndings of this research were as follows: 1) The process of shaping non-circular gears is in-depth and comprehensive. The feed, machine tool, cutting and retracting method, cutter datum, and other related issues should all be taken into consideration. Thus, a practical and feasible mathematical model should include each these factors. 2) The process of non-circular shaping discussed in this paper can be implemented using a cylindrical gear CNC gear-shaping machine, which has been widely adopted in manufacturing, with no need to develop a special machine tool. It is an ideal way to fabricate non-circular gears at low cost and with good ﬂexibility. 3) There are two cutting strategy shown in this paper, the equal-polar-angle cutting and equal-arc-length cutting. By using error analysis, the equal-arc-length cutting method is proved to produce less but more stable theoretical errors. 4) In order to target the presetting cutter position, a circumferential datum should be set on both the stock and ﬁxture due to the non-regular stock shape. Circumferential datum should also be set on the shape cutter, as the circumferential position of the cutter's tooth proﬁle may affect the tooth proﬁle of the non-circular gear being processed. However, doing so may create a new source of errors and in turn decrease the accuracy of the gear.

Fig. 19. Finished gears.

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Fig. 20. Contact state of the two gears.

Acknowledgments The authors would like to thank Innovative Research Team Development Program of Ministry of Education of China (No. IRT13087), and Science and Technology Support Program of Hubei Province (No. 2014BAA024) for the support given to this research.

Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.mechmachtheory.2015.09.010.

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