新型空間漸開線齒輪的解析研究與模擬

(1)

行政院國家科學委員會補助專題研究計畫行政院國家科學委員會補助專題研究計畫行政院國家科學委員會補助專題研究計畫

行政院國家科學委員會補助專題研究計畫 ■ ■ 成果報告 ■ ■ 成果報告成果報告成果報告

□

□ □

□期中進度報告期中進度報告期中進度報告期中進度報告

新型空間漸開線齒輪的解析研究與模擬新型空間漸開線齒輪的解析研究與模擬新型空間漸開線齒輪的解析研究與模擬新型空間漸開線齒輪的解析研究與模擬

計畫類別：個別型計畫 □整合型計畫計畫編號：NSC 96-2221-E-006-214

執行期間：96 年 8 月 1 日至 97 年 7 月 31 日

計畫主持人：黃金沺，國立成功大學機械工程學系教授共同主持人：

計畫參與人員：李乘龍、劉柏村、洪世杰、郭武彰

成果報告類型(依經費核定清單規定繳交)：■精簡報告 □完整報告

本成果報告包括以下應繳交之附件：

□赴國外出差或研習心得報告一份

□赴大陸地區出差或研習心得報告一份

出席國際學術會議心得報告及發表之論文各一份

□國際合作研究計畫國外研究報告書一份

處理方式：除產學合作研究計畫、提升產業技術及人才培育研究計畫、

列管計畫及下列情形者外，得立即公開查詢

□涉及專利或其他智慧財產權，□一年□二年後可公開查詢執行單位：國立成功大學機械工程學系

中華民國 97 年 7 月 31 日

(2)

行政院國家科學委員會專題研究計畫成果報告行政院國家科學委員會專題研究計畫成果報告行政院國家科學委員會專題研究計畫成果報告行政院國家科學委員會專題研究計畫成果報告

新型空間漸開線齒輪的解析研究與模擬新型空間漸開線齒輪的解析研究與模擬新型空間漸開線齒輪的解析研究與模擬 新型空間漸開線齒輪的解析研究與模擬

計畫編號：NSC 96-2221-E-006-214

執行期限：96 年 8 月 1 日至 97 年 7 月 31 日

主持人：黃金沺，國立成功大學機械工程學系教授共同主持人：

計畫參與人員：李乘龍、劉柏村、洪世杰、郭武彰

中文摘要中文摘要

中文摘要中文摘要— 最近才被提出的空間漸開線

齒輪為平面漸開線齒輪的一般化形式，使用此種空間齒面的主要優點為齒輪對之轉速比不受裝配誤差之影響。為了深入了解此一新的齒輪原理，本計劃進行了齒面接觸分析。本計劃採用解析方法並援用傳統的運動學法推導共軛滑移軌跡及共軛漸開線齒面之方程式。齒面接觸分析方程式用以分析一組標準齒輪對及另一組具有裝配誤差齒輪對，齒面接觸分析的結果證實了此新型空間漸開線齒輪對的轉速比的確不受裝配誤差之影響。

關鍵詞：空間漸開線齒面、滑移軌跡、齒面接觸分析

Abstract—The general spatial involute gear (GSIG), recently discovered by Jack Phillips, is the spatial generalization of the planar involute gear. The main advantage of using this new gear tooth surface is that assembly errors of a GSIG pair do not alter its speed ratio. In order to better understand this new develop- ment in gearing theory, this project conducts tooth contact analysis (TCA) of the GSIG.

This project takes an analytical approach and adopts the conventional kinematic method to derive parametric equations of conjugate slip tracks and involute helicoids. The equations for TCA are then derived to analyze a stan- dard gear pair and a gear pair with assembly errors. The result of TCA is consistent with the marvellous feature that the speed ratio of

a GSIG pair is insensitive to assembly errors.

Keyword: general spatial involute gear, slip track, involute helicoid, tooth contact analysis

I. Introduction

The involute curve is the most popular planar gear tooth profile. The most important feature of planar involute gearing is that the speed ratio is not altered by changing the distance between two parallel gear axes.

Many spatial gears had been designed based on involute curves. However, until recently the feature of insensitivity to assembly errors could not be found in spatial gearing. Then Phillips [1,2] discovered the theory of general spatial involute gearing (GSIG) and intro- duced the involute helicoid surface for spatial gearing. Gear flanks of involute helicoid surfaces, similar to planar involute tooth profiles, have the merit of insensitivity to assembly errors. An alternative algebraic proof of the GSIG and an investigation of geometric properties at the point of contact can be found in [3,4]. Jack Phillips’ discovery of GSIG was presented based on profound geometric reasoning [1]. This project, however, takes a different approach by adopting the conventional analytical method [5,6] to analyze the GSIG.

This project adopts the widely used kinematic method developed by Litivin [5,6], who utilized homogeneous coordinate transformation matrices in deriving parametric

(3)

equations of curves and surfaces and developed the kinematic method for determining conjugate curves and surfaces. The system- atic approach developed by Litvin [5,6], as oppose to classical differential geometry method, is concise and more convenient for engineering applications. In this project, we take advantage of this approach to derive parametric equations of slip tracks and involute helicoids [1]. In order to better understand the GSIG theory, one may think of the slip track as the spatial generalization of the involute curve, while the involute helicoid can be considered the (spatial) involute surface that corresponds to the involute curve in planar geariung. However, the notion that the slip track and the involute helicoid are the generalizations of the (planar) involute curve was not used in [1].

In addition to the derivation of parametric equations of slip tracks and involute helicoids, this project conducts tooth contact analysis (TCA) of GSIG by employing the numerical TCA technique [5,6]. We aim to demonstrate that the novel GSIG is insensitive to assembly errors. The authors also seek to help bridge the gap between the approaches of [1]

and [5,6].

II. Parametric Equations of Conjugate Slip Tracks

Involute helicoids are ruled surfaces discovered by Phillips [1] for the novel general spatial involute gears. An involute helicoid is constructed based on the concept of slip track [1], which can be thought of as the 3-D generalization of the planar involute curve. This section utilizes the coordinate transformation method [5,6] to derive the parametric equations of a slip track and its conjugate curve.

A planar involute curve is the locus of a point on a straight line that rolls over a circle.

As shown in Fig. 1, a planar involute curve is traced by point G₁ as line A G₁ ₁ rolls over the base circle, starting from θ₁=0. The parameter of the motion-generated curve is θ₁. Now consider line A B₁ ₁ on the tangent plane of the base circle. The angle that line A B₁ ₁ makes with line A G₁ ₁ is α₁. Point B₁ is the projection of point G₁ on line A B₁ ₁. The locus traced by

point B₁, as line A G₁ ₁ rolls over the base circle from θ₁=0, is the slip track [1]. When α₁=0, the slip track degenerates into the planar involute curve.

( )1

, 1

A z

B1

G1

x1

zp

Slip track 1

H

xp A₁ α¹ θ1

a1

Involute 1

Fig. 1. The slip track—the spatial involute curve

Using the coordinate transformation approach [5,6], the parametric equation of the slip track is derived as follows:

1 1

1 1 1 1

1 1 1 1 1

1

1 1 1 1

1

cos sin 0 cos 0

sin cos 0 sin cos

0 0 1 0 sin

0 0 0 1 1

cos cos sin

sin cos cos

sin 1

P P

a

a H

H

a H

H

θ θ θ

θ θ θ α

α

θ α θ

α

=

 −   

   

=   

   

   

 − 

 

 + 

= 

 

 

r M r

(1)

where r_P denotes the homogeneous coordinates of point B₁ in S_P , a reference frame rigidly attached to line A G₁ ₁ , and

1 1cos 1

H=aθ α is the length of line A B₁ ₁. M_1P is the homogeneous coordinate transformation matrix from S_P to S₁, a reference frame attached to the base circle.

Similar to involute curves in planar gearing, if a constant speed ratio of a spatial gear pairs is specified, the conjugate curve to a slip track is also a slip track. What follows is the derivation of the conjugate slip track of a given slip track. As shown in Fig. 2, A^{( )}¹ and

( )2

A denote the axes of gears 1 and 2, respectively. The angle and distance between axes

(4)

( )1

A and A^{( )}² are ^∑ and C, respectively. Two reference frames, S₁ and S₂, are rigidly attached to gears 1 and 2, respectively. L₁ and

L2 are the distances from the common normal of A^{( )}¹ and A^{( )}² to the base circles. A fixed reference frame S_f is also assigned, and S_a is an auxiliary reference frame. Let the speed ratio of the gear pair be k. The rotation angle of gear 1 is denoted by ϕ₁ and that of gear 2 by ϕ2(=kϕ1). According to the kinematic approach discussed in [5,6], the curve (in S₂) that is conjugate to r₁ (in S₁) can be obtained as follows:

( ) ( )

2 1 1 21 1 1

(12)

1 1

,

( , ) 0

f

θ ϕ θ

θ ϕ

 =

 = ⋅ =



r M r

N v (2) where r2(θ ϕ1, 1) is the parametric equation of the locus of slip track 1 in S₂, and the scalar constraint f( ,θ ϕ =₁ ₁) 0 is the equation of meshing, the necessary condition for the exis- tence of the envelope of r2(θ ϕ1, 1). The envelope of r2(θ ϕ1, 1) is the curve (in S₂) conjugate to r₁ (in S₁).

The parametric equation of r2(θ ϕ1, 1) is obtained, by using coordinate transformations:

2 = 2a af f1 1

r M M M r

The coordinates of r₂ are expanded as follows:

2 1 1

1 1 1 1 1 1

2

1 1 1 1 1 1 1

2

1 1 1 1 1 1

1 1 1 1 1

2 1 1 1

cos( ) sin

cos sin sin cos( ) sin( ) ( cos cos sin )( cos sin cos( ) cos sin( )) ( cos sin

cos )(cos cos( ) cos sin sin( )) cos( ) sin sin( )

x L k

a k C k

a k

k a

k k

y C k L k

a

ϕ

θ α α ϕ ϕ

θ α θ θ ϕ ϕ

ϕ ϕ θ α θ

θ ϕ ϕ ϕ ϕ

ϕ ϕ

= − Σ

− Σ −

+ − − Σ

+ +

+ Σ +

= − + Σ

+ _{1 1} ₁ ₁ ₁ ₁ ₁ ² ₁ ₁

1 1 1 1 1

2

1 1 1 1 1 1 1

1 1

2 2 1 1 1 1 1

2

1 1 1 1 1 1

cos sin sin sin( ) ( cos sin cos )(cos( ) sin cos cos sin( ))

( cos sin sin )(cos( ) cos cos sin sin( ))

cos ( cos sin )

cos cos sin sin sin

k a

k k

a k

k

z L L a

a a

θ α α ϕ θ α θ

θ ϕ ϕ ϕ ϕ

θ α θ θ ϕ ϕ

ϕ ϕ

θ α α

θ α ϕ θ

Σ +

+ − Σ

+ −

+ Σ

= − + Σ +

+ Σ + ₁ ₁

2

1 1 1 1 1 1

sin sin

cos sin (cos cos sin )

a

θ ϕ

θ ϕ θ α ϕ

Σ

+ Σ −

( )¹ , _f, 1

A z z U

ϕ2

x2

x1

ϕ1

( )² , _a, 2

A z z

T L1

O

C L2

∑

D

f, a

y y

Fig. 2. Coordinate systems of a crossed-axis gear pair

In order to derive the equation of meshing, we begin with the normal vector N₁ of slip track 1 at B₁. As shown in Fig. 1, it is obvi- ous that

1= 1− 1

N r a

where a₁ denotes the position vector of A₁ in

S1. We can then transform N₁ from S₁ to S₂:

2 = 2a af f1 1

N L L L N (3) The relative velocity v⁽¹²⁾ between gears 1 and 2 at a point can be obtained as [5,6]:

(12) 2 1

2

1

d dt

ϕ ϕ

= ∂

∂

v r (4)

where dϕ₁/dt is the angular velocity of gear 1, ω(1). Finally, take the inner product of Eqs. (3) and (4) to obtain the equation of meshing:

(12)

1 1 2 2

(1)

1 1 1 1

1 1 1 1 1

1 1 1

( , )

( sin sin ( sin( ))

cos ( cos cos sin

cos ( sin( ))

sin sin sin ) 0

f

k C a

a kL

k a C

kL θ ϕ

ω α θ ϕ

α θ ϕ

θ ϕ

= ⋅

= Σ + −

+ + Σ

− Σ + −

+ Σ

= N v

To simplify the above equation, let t=(θ₁−ϕ₁), and we obtain

1 1

( , ) sin cos 0

f θ ϕ =a t+b t+ =c (5) where

(5)

1 1 1

1 1

1 1 1 1 1

sin sin cos cos

cos sin

sin sin cos cos cos

a a k kC

b kL

c Ck a ka

α α

α

α α α

= Σ − Σ

= Σ

= Σ + − Σ

The solution to Eq. (5) is

1

1 1 tan x

θ =ϕ + ⁻ y (6) where

2 2 4 2 2

2 2

ac a b b b c x

a b c ax

y b

 − ± + −

 = +



 − −

 =

Note that there are two solutions to θ₁ in Eq.

(6).

The equation of the conjugate curve can then be obtained by substituting Eq. (6) into

( )

2 θ ϕ1, 1

r to obtain r2( )ϕ1 , which is the parametric curve (in S₂) that is conjugate to r₁ (in

S1). It can be shown that r2( )ϕ1 is also a slip track, whose axis is A^{( )}² .

A geometric construction of conjugate slip tracks was also discussed in [1]. As shown in Fig. 3, given the parameters of slip track 1: a₁,

α1, and L₁ , two contact normal vectors N

and N′ can be determined from the two solutions in Eq. (6). The parameters of the conjugate slip track, a₂,α₂,L₂ can be obtained accordingly. As shown in Fig. 4, the radius of base circle 2, a₂, is the distance between N

and A⁽²⁾. The common perpendicular of N and A⁽²⁾ intersects N and A⁽²⁾ at points A₂ and U , respectively. Angle α₂ is the angle that N makes with base circle 2. The other solution of slip track 2 can be found similarly.

( )¹, _f

A z A( )²

U

D L2

L′2

xf

O

A2

yf

A′2

N′

U ′ N

1 A L

T A′

Fig. 3. Two solutions of conjugate slip tracks

( )² A

U

yf

N

a2

L2

D A2

α2

Fig. 4. Geometric construction of slip track 2

. Parametric Equations of Conjugate

Ⅲ

Involute Helocoids

As discovered in [1], the ruled surface traced by line G B₁ ₁, as shown in Figs. 5 and 6, can be used as the gear tooth surface of GSIG.

The ruled surface is called an involute helicoid. As shown in Fig. 5, line G B₁ ₁ is tangent to the base cylinder at point W₁. Let the coordinates of B₁ in S₁ be ( x_B, y_B, z_B ) and those of W₁ be ( x_W, y_W, z_W ). Note that ( x_B, y_B, z_B )

can be obtained from r₁( )θ₁ in Eq. (1). The coordinates of the unit vector directed from W1 to B₁ is

1 1

, ,

B W B W B W

x x y y z z

W B

F F F

W B

− − −

 

= =  

 

q

where

2 2 2

( _B _W) ( _B _W) ( _B _W) F = x −x + y −y + z −z

The locus of W₁ is a helix on the base cylinder, as shown in Fig. 6. The pitch of the helix is a₁cotα₁. Therefore the coordinates of W₁ are identical to the parametric equations of the helix. Namely,

1 1

1 1 1

cos sin cot

W W W

x a

y a

z a θ

θ

θ α

=

= −

=

The parametric equation of involute helicoid 1 is

( ) ( )

1 θ1, m1 = 1 θ1 +m1

r r q (7)

(6)

where m₁ is a scalar parameter.

W1 Base cylinder

G1

a1 1θ a1

θ1

α1

B1

H

Hyperboloid

Base circle Slip track

A1

Fig. 5. Construction of the involute helicoid

Fig. 6. The involute helicoid—the GSIG tooth surface

The parametric equation of the involute helicoid (in S₂) that is conjugate to r1(θ1, m1) can be obtained by the following equations:

( ) ( )

2 1 1 1 21 1 1 1

(12)

1 1

, , ,

( , ) 0

m m

f

θ ϕ θ

θ ϕ

 =

 = ⋅ =



r M r

N v

where r2(θ1,m1,ϕ1) is the locus of r1(θ1, m1) in

S2.

.Tooth Contact Analysis of

Ⅳ

Ⅳ General Spa-

tial Involute Gears with Assembly Er- rors

In this section, we conduct the TCA of GSIG. Most importantly, we seek to verify that the speed ratio of a GSIG pair is insensi-

tive to assembly errors. Assembly errors may cause C, ^∑, L₁, and L₂ to deviate from their nominal values. As discussed in [5,6], the conditions for conjugate gear tooth surfaces to remain in contact are (1) the coincidence of contact points and (2) the coincidence of normal vectors at the coincident contact point, as shown in Fig. 7. To satisfy these conditions, the position vectors of the contact points of both gear flanks need to be transformed into S_f , from S₁ and S₂:

(1) 1 1

f = f

r M r

(2)

2 2

f = f

r M r

For two gear flanks to remain in contact at a point, we have

(1) (2)

f = f

r r (8) The second condition can be obtained similarly. We need to transform the contact normal vectors of both gear flanks into S_f . Since the locations of the normal vectors are imma- terial, only the rotation portion L of a coordinate transformation matrix M is concerned.

As shown in Fig. 7, the normal vector N₁ of involute helicoid 1 can be obtained as:

1 1

1

1 m1

θ

∂ ∂

= ×

∂ ∂

r r N

N1 can be normalized to a unit vector:

1 1

1

= N n N

It is then transformed into S_f :

(1) 1 1

f = f

n L n

where L_f₁ is the rotation portion of M_f₁, and

1 1

1 1 1

cos sin 0

= sin cos 0

0 0 1

f

ϕ ϕ

 − 

 

 

L

Similarly, the contact normal vector of involute helicoid 2, can be derived as follows:

2 2

2

2 m2

θ

∂ ∂

= ×

∂ ∂

r r N

2 2

2

= N

n N

(7)

(2)

2 2

f = f

n L n

where

2 2

2 2 2

2 2

cos cos cos sin sin

sin cos 0

cos sin sin sin cos

f

ϕ ϕ

Σ − Σ Σ

 

 

=  

− Σ Σ Σ

 

L

, 2

n n1

2

θ2

∂

∂ r

2

m2

∂

∂ r

m1

∂

∂ r1

θ1

∂

∂ r1

Involute helicoid 1

Slip track 1 Involute Slip track 2

helicoid 2

Fig. 7. Normal vectors at the contact point For the coincidence of normal vectors at the contact point, we have

(1) (2)

f = f

n n (9) The above equation gives only two independ- ent scalar equations because n⁽¹⁾_f and n⁽²⁾_f are unit vectors. The two scalar equations are:

(1) (2)

fx fx

fy fy

n n

n =n (10)

(1) (2)

fx fx

fz fz

n n

n =n (11) Eqs. (8)), (10), and (11) contain five scalar equations in six variables:

1, m1, 1, 2, m2, and 2

θ ϕ θ ϕ . For any given value of ϕ1 , we can, for example, use Newton’s method to solve for the other five variables.

The simulation result provides the informa- tion about the speed ratio and line of action of the meshing.

In what follows, we use the gear set given in example WkEx#1 of [1] to demonstrate the TCA of GSIG. As shown in Fig. 8, tooth flanks 1D and 1C form a tooth of gear 1, and tooth flanks 2D and 2C form a tooth of gear 2.

Tooth flank 1D meshes with tooth flank 2D, while tooth flank 1C meshes with tooth flank 1D in reverse motion.

This project discusses the TCA of tooth flanks 1D and 2D only. The TCA of tooth flanks 1C and 2C can be conducted similarly.

Given two crossed gear axes with C=80 and Σ =50°, the corresponding coordinate systems Sf , S_1D, S_2D are set up accordingly, as shown in Fig. 9. The parameters of both tooth flanks are listed in Table 1. Using the above discussed TCA method [5,6], the relation between the input and output rotation angles, ϕ₁ and ϕ₂, is obtained, as shown by the solid line in Fig. 10. Note that the speed ratio of the gear pair is the slope of the line, which is -0.6.

The line of action, which is the locus of contact points in the fixed frame, is also obtained, as shown by the solid line in Fig. 11.

Flanks 2D

Flanks 2C

Flanks 1C

Flanks 1D Axis 2

Axis 1 Gear 1

Gear 2

Fig. 8. Gear flanks of conjugate GSIG [1]

Table 1. Parameters of gear flanks 1D and 2D Flank

1D

1_D 4.83181

α = °, a₁_D =49.078057,

1_D 131.966605 L = , θ >0 Flank

2D

2_D 4.83181

α = °, a₂_D =81.796761,

2_D 116.962403 L = , θ >0

In order to verify that a GSIG pair is insensitive to assembly errors, we use the same involute helicoids but make the following changes in assembly parameters:

(8)

1

2

: 80 82 : 50 48

:131.966605 130 :116.962403 115

D D

C

L L

→

∑ →

→

We conduct the TCA again, and the results are shown by the dotted lines in Figs. 10 and 11.

As shown in Fig. 10, the dotted line is parallel to the solid line. This result indicates that the speed ratio remains unchanged with the assembly errors imposed. The dotted line shown in Fig. 11 indicates that the assembly errors result in a different line of action and that the new line of action is also a straight line.

z2

Axis 2 yf

Axis 1

Involute helicoid 2D

ϕ1

L2D

x2

ϕ2

L1D

x1

, 1

zf z

∑ C

Involute helicoid 1D

Fig. 9. Tooth contact analysis of flanks 1D and 2D

20 40 60 φ1Hdeg L

200 210 220 230 φ2Hdeg Lϕ₂(deg)

1(deg) ϕ

Fig. 10. TCA reault: speed ratios

V. Conclusion

This project utilizes conventional analytical methods to investigate the general spatial involute gears discovered by Jack Phillip.

The parametric equations of slip tracks and involute helicoids are derived using coordinate transformations, and conjugate slip tracks and involute helicoids are obtained using the kinematic method of envelope theory. Numerical tooth contact analysis is then conducted to investigate the meshing of general spatial involute gears.

The result of tooth contact analysis is consistent with the proclaimed feature, which states that general spatial involute gears, similar to planar involute gears, are insensitive to assembly errors. In addition to provid- ing some analytical equations for this elegant gearing theory, this project may also help to bridge the gap between conventional analytical methods for gearing and the geometric presentation of the novel theory of GSIS.

x y

z

Fig. 11. TCA result: lines of action

References

[1] Phillips, J., 2003, General Spatial Involute Gearing, Springer-Verlag, Berlin.

[2] Phillips, J., 2000, ‘‘From the Trailed Disc Plough with Ball to General Invo- lute Gearing,” Proceedings of the Ball Symposium, July 9-12, Cambridge, UK.

[3] Stachel, H., 2004, “On Jack Phillips spatial involute gearing,” Proceedings of the Eleventh International Confer- ence on Geometry and Graphics, pp.

43-48, Giangzhou, China, August 2004.

(9)

[4] Stachel, H. 2004, “On spatial involute gearing,” Proceedings of the Sixth in- ternational Conference on Applied In- formatics, Eger, Hungary, January 2004, to be found also at Geometry Preprints, No. 119, TU Vienna, 2004.

[5] Litvin, F. L., 1968, Theory of Gearing, 2^nd edition, Nauka, (in Russian).

[6] Litvin, F. L., 1994, Gear Geometry and Applied Theory, Prentice-Hall Inc., New Jersey.

計畫成果自評計畫成果自評計畫成果自評

計畫成果自評

本計劃已完成原預期完成之工作項目。本計劃的解析及模擬研究建立了共軛空間漸開線及共軛空間之漸開線齒面之運動模型，接著並以齒面接觸分析證實該新型齒面對裝配誤差不敏感的特性。本研究以傳統的運動學分法證明新型齒面的優異特性，其研究結果除了具有齒輪原理理論上的重要性外。並可做為未來進行齒面設計應用之重要基礎。