Comparison of different methods in gear curvature

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Comparison of different methods in gear curvature analysis using a new approach F Di Puccio , M Gabiccini, and M Guiggiani Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione, Universita` di Pisa, Pisa, Italy The manuscript was received on 8 February 2005 and was accepted after revision for publication on 21 July 2005. DOI: 10.1243/095440605X31986

Abstract: In the literature, some methods for curvature analysis of gears can be found, apparently very different from one another. This article presents a comparison of three approaches to stress their similarities or differences and field of application: the classic one by Litvin, its development by Chen and another one proposed by Wu and Luo and revisited by the present authors. All these methods are re-examined and expressed in a new form by means of a new approach to the theory of gearing that employs vectors and tensors. An extension of the relative motion is also considered, assuming translating axes of the gear pair and modified roll. Keywords: curvature analysis, theory of gearing, conjugate surfaces, equation of meshing

1

INTRODUCTION

Curvature analysis of a surface can be carried out quite easily by means of well-established results of classical differential geometry [1, 2]. However, their application to the investigation of the surfaces of a gear pair is still a difficult task, mainly because it requires solving the equation of meshing with respect to one of the involved parameters [3]. Another difficulty is the extreme complexity of the equations of the generated surface and of its derivatives. Therefore, alternative approaches have been sought by many researchers working on the theory of gearing. They can be represented mainly by three different methods. 1. The indirect meshing method proposed by Litvin [4]. 2. The indirect generating method proposed by Chen [5]. 3. The direct generating method proposed by Di Puccio et al. [6] and by Wu and Luo [7]. In the indirect meshing method [4], relationships between the curvatures of the gear pair are obtained

Corresponding author: Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione, Universita` di Pisa, via Diotisalvi 2, 56126 Pisa, Italy. email: [email protected]

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starting from the analysis of the motion of the contact point. That is why it is defined here as indirect. As the two gears are not necessarily obtained by means of an enveloping process, this method differs also for considering meshing and not generating conditions. The method has been proposed by Litvin employing kinematic relationships and scalar components of the involved vectors. An extension to a generalized motion has been proposed in reference [8]. In the indirect generating method [5], Chen follows Litvin’s approach for the case of an enveloping process between the gears. Some fundamental expressions, again in kinematic terms, for the case of fixed axes and constant gear ratio are obtained. The direct generating method has been recently proposed by the authors in references [6, 9] and is similar to Wu and Luo’s approach [7]. The equations for the normal curvature and geodesic torsion of the surface are obtained simply by applying their definitions, therefore in a direct way, for the case of the enveloping process. Among other curvature analyses, at least two are worth mentioning: one was proposed by Dooner [10] and the other by Ito and Takahashi [11]. In the first one, the author employs the screw algebra to derive the so-called ‘third law of gearing’ and formulates the limiting relationship between the radii of curvature, which is only valid, though, for Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

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F Di Puccio, M Gabiccini, and M Guiggiani

the reference pitch surfaces. In the second one, the curvatures in hypoid gears are investigated starting from a classical differential geometry point of view, but then kinematic relationships are introduced. In this article, all three main approaches are extended to the case of translating axes and variable gear ratio, and re-written using a purely geometric approach [9], which does not require reference systems. A coordinate-free approach, based on geometric algebra, has also been proposed by Miller [12] but it only deals with the generating process and the curvature analysis is not addressed. The new feature of this article is also the introduction of the tensor of curvature which plays a key role in the basic equations.

2

THEORETICAL BACKGROUND: CURVATURE OF A SURFACE 3

Let us consider, in the Euclidean space E , a regular surface S whose generic point will be denoted by P(j, u), where (j, u) are the parametric coordinates of the surface. Once chosen a fixed point O of E3 , the position vector p of P(j, u) is p(j, u) ¼ P(j, u) O

(1)

so that the unit tangent vector to c is given by t(s) ¼

dc ¼ p,j j,s þ p,u u,s ¼ p,s ds

(5)

and the derivative along c of the unit normal vector is dmu ¼ m,uj j,s þ m,uu u,s ¼ m,us ds

(6)

If s is not the arc length, the vector in equation (5) is still tangent to the curve but has not unit magnitude. The normal curvature kn and the geodesic torsion tg of S along t at a point P can be easily computed by means of the following equations kn ¼ m,us t tg ¼ mu m,us

t

(7)

(8)

and they are features of the surface. It is well known that the normal curvature assumes its extreme values k1 and k2 along two principal (orthogonal) directions of unit tangent vectors t1 and t2 , respectively, having also null geodesic torsion [1, p. 128]. The normal curvature and the geodesic torsion along any other direction at P are not independent and, according to Euler’s and Bertrand’s formulas, [1, pp. 132 and 159], are given by the following equations

By definition, the normal vector m to S is given by @p(j, u) @p(j, u) m(j, u) ¼ ¼ p,j p,u @j @u

(2)

and, due to regularity of the surface, it is also m = 0. The normal unit vector to the surface is

kn ¼ k1 cos2 b þ k2 sin2 b

(9)

tg ¼ (k2 k1 ) sin b cos b

(10)

the unit tangent vector being (11)

t ¼ cos b t1 þ sin b t2 m(j, u) mu (j, u) ¼ jm(j, u)j

2.1

(3)

Normal curvature and geodesic torsion of a surface

In differential geometry, the second-order properties of a surface, as curvature and torsion, are conveniently derived from their evaluation for curves belonging to the surface [1, p. 121]. Therefore, let us consider an arbitrary regular curve c on the surface S defined by the position vector c(s) ¼ p(j(s), u(s))

Therefore, to determine kn for any direction at P, the principal curvatures k1 and k2 of the surface and the principal directions are required. If the normal curvatures and geodesic torsions are given along two orthogonal but non-principal directions of unit tangent vectors ta and tb , the generalized Euler’s and Bertrand’s formulas can be employed kn ¼ ka cos2 b þ kb sin2 b þ ta sin 2b

(12)

tg ¼ ta cos 2b þ (kb ka ) sin b cos b

(13)

where in this case

(4) t ¼ cos b ta þ sin b tb

where j(s) and u(s) are analytic functions and s is a generic parameter. In the following equation, we will conveniently assume s to be the arc length of c, Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

(14)

In reference [5], a more general form for nonorthogonal direction is given in equation (45). C02105 # IMechE 2005

Comparison of different methods in gear curvature analysis

Let us observe that all unit tangent vectors lie in the tangent plane to the surface in P, therefore

equations (11) and (15), we obtain K t ¼ k1 cos b t1 þ k2 sin b t2

mu ¼ t1 t2 ¼ ta tb

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(20)

and when i ¼ ta (and j ¼ tb ), for equations (14) and (16), it becomes 2.2

Tensor of curvature

The curvature at a point P of the surface S can be simply described, using the tensor of curvature K [2, 13]. We denote with ½K the matrix form of the tensor, i.e. its elements, that expressed in principal directions is

k1 ½K ¼ 0

0 k2

(15)

ka ta

tb kb

¼

ka ta

ta kb

i i0 ½T ¼ j i0

i j0 j j0

cos w ¼ sin w

(16)

sin w cos w

(17)

where w is the angle between i and i0 and applying the tensor law of transformation ½KS0 ¼ ½TT ½KS ½T

(18)

In the following section let us assume to express K in an orthogonal reference frame, so that ½K will be a symmetric matrix.

2.3

The inner product between a tensor and a vector is the vector whose components, in an orthogonal reference frame S(P, i, j), are given by the product of K and t components (19)

For example, when i ¼ t1 (and j ¼ t2 ) employing C02105 # IMechE 2005

kn tg

tg kn?

1 0

K t ¼ kn t þ tg (mu t) (22) kn? being the normal curvature along j; this equation holds even for a non-unit vector v parallel to t K v ¼ kn v þ tg (mu v)

(23)

According to equation (22), Euler’s and Bertrand’s equations become kn ¼ (K t) t

(24) u

tg ¼ (K t) (m t)

(25)

which correspond to equations (9) and (10), when K is written in the principal directions as in equation (15), or alternatively to the generalized versions (12) and (13) if we employ equation (16). It may be interesting to note that equations (24) and (25) hold in any reference system, orthogonal or not. The previous relationships become very useful when applied in Rodrigues’ formula, which relates the derivative of the normal unit vector to the curvature. The most common version of this formula employs scalar components in a principal reference frame, as in equation (20) ½m,us

Rodrigues’ formula

½K t ¼ ½K ½t

½K t ¼

being tb ¼ ta . Therefore, in an orthogonal reference frame, ½K is symmetric whereas in a non-orthogonal one ½K is more complex, not symmetric and its elements do not correspond to normal curvature and torsion, as detailed in Appendix 2. Once K is expressed in an orthogonal reference frame S(i, j), either principal or not, its components in another orthogonal system S0 (i0 , j0 ) can be obtained by using the rotation matrix T between them

(21)

If i ¼ t (and j ¼ mu t), from equation (19), a general useful expression can be obtained

whereas in non-principal (but orthogonal) directions is ½K ¼

K t ¼ (ka cos b þ ta sin b)ta þ (ta cos b þ kb sin b)tb

k ¼ 1 0

0 ½t k2

(26)

but a more general expression can be used as well, exploiting equation (22) m,us ¼ K t ¼ kn t tg (mu t)

(27)

The previous equation holds even if s is not the arc Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

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length, with p,s instead of t m,us ¼ K p,s ¼ kn p,s tg (mu p,s )

3

As in equation (2), the normal vector me to Se is given by (28)

(31)

and me = 0 for the regularity of the surface; the unit normal vector mue is

ROTATION OPERATOR

In this study, three different techniques in curvature analysis are compared after being re-elaborated by means of a new approach presented in reference [9], which is based on the use of the rotation operator R to describe the kinematics of the gear pair. In this article, only the basic concepts of this new approach are given, referring to reference [9] for a deeper insight. To express the rotation of a point P around an axis a of an angle a, a point O is taken on the axis to define the position vector p ¼ P–O and then Euler’s relation is applied, corresponding to the compact notation R(p, a, a) p^ (a) ¼ P^ O ¼ R(p, a, a) ¼ (p a)a þ ½ p (p a)a cos (a) þ a p sin (a)

me (j, u) ¼ pe,j pe,u

mue (j, u) ¼

me (j, u) me (j, u) me (j, u) ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 jme (j, u)j D(j, u) EG F

where E, F, G are the coefficients of the first fundamental form of Se [1, p. 82], that is E ¼ pe,j pe,j G ¼ pe,u pe,u

F ¼ pe,j pe,u pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ EG F 2 (33)

It is useful for the development of the analysis to introduce a curve ce on the surface Se that according to equation (4) is defined as ce (se ) ¼ pe (je (se ), ue (se ))

(29)

(32)

(34)

where se is the arc length but a generic parameter could do as well. The unit tangent and normal vectors to the surface along the curve are respectively

where P^ is the image of P after the rotation and a is the unit vector that marks the axis direction. As equation (29) is defined on vectors, this approach has the main advantage of involving vectors as such and not their components, required, for example, by homogeneous transformations, frequently employed in gear generation. Some relevant properties of rotating vectors are described in Appendix 3.

Let us assume that the tensor of curvature K e at each point of Se is known, in particular, its principal curvatures are k1e and k2e and the corresponding directions are marked by unit vectors e1 and e2 .

4

4.2

SURFACES OF THE GEAR PAIR

The aim of the curvature analysis is the definition of a relationship between the tensors of curvature of a gear pair, which can be a pinion and a gear or a generating tool and the generated gear. In this section, the definitions of section 2 are applied to the surfaces of the pair.

te (se ) ¼ ce,se ¼ pe,j je,se þ pe,u ue,se

(35)

muce (se )

(36)

Surface of the first gear

The first gear, pinion or generating tool, is assumed to be a (fixed) regular surface Se in the Euclidean space E3e . Given a fixed point Oe conveniently taken on the axis of the pinion, it is possible to associate a position vector pe of R3 to each point Pe (j, u) of the surface pe (j, u) ¼ Pe (j, u) Oe

(30)

Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

ue (se ))

Surface of the second gear

In a similar way, another surface Gg in the Euclidean space E3g is introduced. Two different cases are considered for defining this surface: in the first one (meshing), Litvin assumes it to be a generic surface, whereas in the other case (generating) it is the envelope surface [5, 9]. 4.2.1

4.1

¼

mue (je (se ),

Meshing case

In Litvin’s approach, a second gear is considered whose curvature features are investigated, to make it mesh with the pinion Se under known relative motion. Since no equations are available, it is assumed that the gear surface is expressed using the parametric coordinates (x, n) and accordingly its position vector pl is pl (x, n) ¼ Pl (x, n) Og

(37)

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The normal and unit normal vectors to the surface are ml (x, n) and mul (x, n), evaluated according to equations (2) and (3), respectively. Also in this gear, we introduce a curve cl defined as in equation (4) cl (sl ) ¼ pl (x(sl ), n(sl ))

(38)

The normal unit vector along the curve is mucl (sl ) ¼ mul (x(sl ), n(sl )) 4.2.2

(39)

Generating case

In the second case, the generated surface is related to pe (j, u) by the enveloping process, involving also the relative motion between the pair. As better described in section 5.2, it is possible to define the position vector of Gg as pg ¼ pg (j, u, f),

f (j, u, f) ¼ 0

where f (j, u, f) ¼ 0 is the equation of meshing [4]. mg (j, u, f) and mug (j, u, f) are, respectively, the normal and unit normal vector to Gg , that is at points where the equation of meshing is satisfied. A curve cg on Gg is defined, as done for ce and cl , taking into account the special definition of points of the surface Gg

f (jg (sg ), ug (sg ), f(sg )) ¼ 0 (40) The unit tangent and normal vectors along the curve are tg (sg ) ¼ cg,sg ¼ pg,j jg,sg þ pg,u ug,sg þ pg,f fg,sg

(41)

mucg (sg )

(42)

¼

mug (jg (sg ),

ug (sg ), f(sg ))

place. The first gear (pinion) rotates of an angle c about the translating axis a identified by means of one of its (moving) points Oa and of a fixed unit vector a. In a similar way, the second generated gear rotates of an angle w about the translating axis b defined by Ob and b. Both the rotation angles c and w and the points Oa and Ob depend on the parameter of motion f. The surface S^ e (f) of the pinion, isomorphic to Se , rigidly rotates around the first axis a, while moving with the transfer motion of Oa . Denoting by ^ j, u, f) its generic point, the corresponding posP( ition vector p^ a with respect to Oa is ^ j, u, f) Oa (f) ¼ R(pe (j, u), a, c(f)) p^ a (j, u, f) ¼ P( (43) employing notation in equation (29). Similarly, with respect to the moving point Ob , the position vector ^ j, u, f) is p^ b (j, u, f) of P( ^ j, u, f) Ob (f) ¼ p^ a (j, u, f) d(f) p^ b (j, u, f) ¼ P( (44) where (Fig. 1) d(f) ¼ Ob (f) Oa (f)

cg (sg ) ¼ pg (jg (sg ), ug (sg ), f(sg ))

1283

(45)

Both position vectors p^ a (j, u, f) and p^ b (j, u, f) of R3 describe the same family of surfaces Ff of E3f . ^ ue to each regular surface The unit normal vector m ^Se (f) of the family (hence for fixed f) is by definition

again where the equation of meshing holds. Actually in both the meshing and generating cases, the explicit relation between position vector and parametric coordinates is considered to be too complex and not convenient for investigating the features of the surface, as described in section 2 [14]. Therefore, in all the approaches compared in this study, the aim is to determine the tensor of curvature K g of surface Gg , in terms of tensor of curvature of the pinion K e , for a given (known) motion of the gears. 5

GENERAL DEFINITION OF THE MESHING/GENERATING PROCESS

Let us consider a third fixed Euclidean space E3f where the enveloping or meshing process takes C02105 # IMechE 2005

Fig. 1 Mating surfaces in the fixed space E3f Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

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given by

advantage of the properties of the rotation operator, the last expression can be developed further

^ ue (j, u, f) ¼ R(mue (j, u), a, c(f)) = 0 m

(46)

where equations (41), (43), and (31) were employed. At this point, we have to distinguish between a meshing process considered by Litvin and a generating process as done in references [5, 9], but in both cases, the aim is the investigation of the relationship between the tensors of curvature of the gears, once their motion is given.

5.1

Meshing process

In Litvin’s approach, a second gear is considered in the fixed space, which is a surface G^ g (f) isomorphic to Gg introduced in section 2.1. Denoting by P^ l (x, n, f) the generic point of G^ g (f), its position vector p^ l (x, n, f) with respect to the moving point Ob is given by the following expression p^ l (x, n, f) ¼ P^ l (x, n, f) Ob (f) ¼ R(pl (x, n), b, w(f)) (47) ^ ul to each regular surface G^ g (f) The normal vector m (hence for fixed f) is given by ^ ul (x, n, f) ¼ R(mul (x, n), b, w(f)) m

(48)

In curvature analysis, we assume that G^ g (f) is in meshing contact with the surface S^ e (f), therefore three relationships are supposed to be satisfied: same position vector and same normal vector (for fixed f) and the equation of meshing (in kinematic terms v(1) n(1) ¼ v(2) n(2) [4]), respectively p^ b (j, u, f) ¼ p^ l (x, n, f) ^ ue (j, u, f) ¼ m ^ ul (x, n, f) m

(49) (50)

^ ue ¼ (w0 b p^ l (x, n, f)) m ^ ul (c0 a p^ a (j, u, f) d0 ) m (51) Strictly speaking equation (50) should be ^ ul (x, n, f), but since at the end the ^ ue (j, u, f) ¼ +m m sign does not affect results, we assume it to be positive. Taking into account equations (49) and (50), the equation of meshing (51) becomes ^ ue (c0 a p^ a (j, u, f) (w0 b p^ l (x, n, f)) d0 ) m 0 0 0 ^ ue ¼ ((c a w b) p^ a (j, u, f) þ w b d(f) d0 ) m

¼ (we (f) pe (j, u) þ qe (f)) mue (j, u) ¼ 0 f~ (j, u, f) ¼ he (j, u, f) mue (j, u) ¼ 0 (53) already obtained in reference [9], where three vectors have been introduced we (f) ¼ R(w(f), a, c(f))

(54)

qe (f) ¼ R(w0 b d(f) d0 , a, c(f)))

(55)

he (j, u, f) ¼ we (f) pe (j, u) þ qe (f)

(56)

It is interesting to note that he (j, u, f) corresponds to the rotated relative velocity between the gears, denoted by v(12) in reference [4]. As in reference [7], a different notation is used when the equation of meshing is written using the normal vector or the unit normal vector f (j, u, f) ¼ he (j, u, f) me (j, u) ¼ 0 f~ (j, u, f) ¼ he (j, u, f) mue (j, u) ¼ 0 It can be interesting to underline that in the meshing case the gears can be in point or in line contact, depending on the solutions of equations (49)-(50). The important point is that in curvature analysis, the interest is not actually in solving the three conditions stated earlier (as in a tooth contact analysis (TCA) case), but it can be assumed that they are satisfied (at least at one point). This means that although it is not known explicitly, because of the complexity of the solution of the TCA, there exists a relationship between (x, n) and (j, u) for each value of the parameter of motion f.

5.1.1

Conjugate curves in the meshing process

A curve ce of Se and a curve cl of Gg , defined, respectively, in equations (34) and (38), are said to be conjugate if, for each f, they are in contact at one point. This means that during meshing, their arc lengths vary with the parameter of motion, so that c^ e (se (f)) d(f) ¼ c^ l (sl (f))

^ ue ¼ 0 ¼ (w p^ a (j, u, f) þ w0 b d(f) d0 ) m (52) 0

R(w p^ a (j, u, f) þ w0 b d(f) d0 , a, c(f)) mue (j, u)

(57)

or simply

0

where w(f) ¼ c a w b is directed as the screw axis of the relative motion between the gears. Taking Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

c^ e (f) d(f) ¼ c^ l (f)

(58)

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where c^ e (f) ¼ R(ce (f), a, c(f)) ¼ R(pe (je (f), ue (f)), a, c(f)) ¼ C^ e (f) Oa (f) (59)

points of the family satisfying the equation of meshing, which in this case is also the triple product of the partial derivatives of the position vectors pg (j, u, f)

c^ l (f) ¼ R(cl (f), b, w(f)) ¼ R(pl (x(f), n(f)), b, w(f)) ¼ C^ l (f) Ob (f) (60) The unit normal vectors along conjugate curves, defined in equations (36) and (39), are related as follows

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pg,j

pg,u

pg,f ¼ mg pg,f ¼ f (j, u, f) ¼ 0

corresponding to equation (53) as detailed in reference [9]. As already mentioned in section 4.2.2, position vectors of points of the envelope surface are given by the system

^ uce (f) ¼ R(muce (f), a, c(f)) m ^ ucl (f) ¼ R(mucl (f), b, w(f)) ¼ m

It may be worth noting that in case of point contact between the gears, there is only one couple of conjugate curves, corresponding to the trajectories of the contact point over the two surfaces S^ e and G^ g . In case of line contact, there are infinitely many couples of conjugate curves, but each curve ce corresponds to only one curve cl .

5.2

pg ¼ pg (j, u, f),

(61)

R(pg (j, u, f), b, w(f)) ¼ p^ b (j, u, f) ¼ R(pe (j, u), a, c(f)) d(f), f (j, u, f) ¼ 0 (66) and R(mue (j, u), a, c(f)) ¼ R(mug (j,u, f), b, w(f))

When Gg is the generated gear, it is obtained from Se by an enveloping process and E3g is the moving space where the gear surface Gg will be obtained as a fixed one [9]. In this space, we first introduce a family of surfaces Fg whose envelope Gg we are interested in; each surface of Fg is isomorphic to Se . If Pg denotes the generic point of Fg , the corresponding position vector pg ¼ Pg –Og is obtained by rotating p^ b around b by an angle w(f) (minus sign to take the gear back in a fixed position)

(67)

which are automatically satisfied by the definitions of pg and mug . 5.2.1

Conjugate curves in the generating process

The same steps of the meshing case can be repeated exactly, for obtaining the condition for ce on Se and cg on Gg to be conjugate curves, i.e. for a given f,

pg (j, u, f) ¼ R(^pb (j, u, f), b, w(f))

c^ e (se (f)) d(f) ¼ c^ g (sg (f))

(68)

c^ e (f) d(f) ¼ c^ g (f)

(69)

or

¼ R(^pa (j, u, f) d(f), b, w(f)) ¼ R(R(pe (j, u), a, c(f)) d(f), b, w(f)) (62) According to equations (31) and (46), the normal vector mg to each regular surface Sg (f) of the family Fg (hence for fixed f) is given by ^ e (j, u, f), b, w(f)) mg (j, u, f) ¼ pg, j pg,u ¼ R(m ¼ R(R(me (j, u), a, c(f)), b, w(f)) (63) and similarly (64)

The generated gear is the envelope Gg of the family of surfaces Fg in the Euclidean space E3g given by the C02105 # IMechE 2005

(65)

In the generating case, equations (49) and (50) correspond to

Generating process

mug (j, u, f) ¼ R(R(mue (j, u), a, c(f)), b, w(f))

f (j, u, f) ¼ 0

where c^ e (f) has already been introduced in equation (59) and c^ g (f) ¼ R(cg (sg (f)), b, w(f)) ¼ R(pg (jg (f), ug (f), f), b, w(f)) ¼ C^ g (f) Ob (f)

(70)

In a similar way, for the unit normal vectors along conjugate curves, defined in equations (36) and (39) ^ uce (f) ¼ R(muce (f), a, c(f)) ¼ R(mucg (f), b, w(f)) m ^ ucg (f) ¼m

(71)


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It is interesting to note that in this case, conjugate curves share the same relations between the parametric coordinates and the arc lengths

je (se (f)) ¼ jg (sg (f)) ¼ j(f),

^ ucl,f ^ uce,f ¼ m m

LITVIN’S APPROACH TO CURVATURE: INDIRECT MESHING METHOD

The classical curvature analysis proposed by Litvin moves from the determination of the velocity of the contact points to relate the curvature features of the gear Gg meshing with a pinion Se . In reference [4], Litvin first considers a generic contact point P defined by r(1) ¼ r(2) þ d and n(1) ¼ n(2) , which in the present approach become equations (49) and (50) for a given set of parameters (x , n , j , u , f ) satisfying also the equation of meshing

Equations (75) and (76) correspond to Litvin’s conditions r_ (1) ¼ r_ (2) and n_ (1) ¼ n_ (2) . By employing the properties of the rotation operator and taking into account equations (59), (60), (73), and (74), the last two results can be further developed. In particular, equation (75) becomes R(ce (f), a, c(f)),f d0 ¼ R(cl (f), b, w(f)),f c0 a c^ e þ R(ce,f , a, c(f)) d0 ¼ w0 b c^ l þ R(cl,f , b, w(f)) w c^ e (f) þ w0 b d(f) d0 þ R(ce,f , a, c(f)) ¼ R(cl,f , b,w(f)) R(w c^ e (f) þ w0 b d(f) d0 , a, c) þ ce,f

(77)

l

f (j , u , f ) ¼ 0 (72) Then, Litvin investigates the motion of the contact point, requiring that there is still contact between the gears after an infinitesimal time increment. In this context, time does not play any role, so we express an infinitesimal time increment as an infinitesimal increment of the motion parameter f. This small rotation causes the contact point to move to in the neighbourhood of the an unknown point Q, previous P. The trajectories of the contact point over the two rotating surfaces S^ e (f) and G^ g (f) are the rotating conjugate curves c^ e and c^ l , for which equations (58) and (61) hold. Accordingly, the first two equations in equation (72) can also be written as c^ e (f ) d(f ) ¼ c^ l (f ) ^ u (f ) ^ u (f ) ¼ m m ce

(76)

¼ R(R(cl,f , b, w(f)), a, c) he þ ce,f ¼ cle,f

p^ b (j , u , f ) ¼ p^ l (x , n , f ) ^ u (x , n , f ) ^ u (j , u , f ) ¼ m m e

^ uce,f df ¼ m ^ ucl (f ) þ m ^ ucl,f df ^ uce (f ) þ m m or, for equation (74) simply

ue (se (f)) ¼ ug (sg (f)) ¼ u(f) f (j(f), u(f), f) ¼ 0

6

whereas the normal unit vector is

(73) (74)

cl

where he has been defined in equation (56). In a similar way, equation (76) becomes R(muce (f), a, c(f)),f ¼ R(mucl (f), b, w(f)),f ^ uce þ R(muce,f , a, c(f)) c0 a m ^ cl þ R(mucl,f , b, w(f)) ¼ w0 b m ^ uce þ R(muce,f , a, c(f)) ¼ R(mucl,f , b, w(f)) wm ^ uce (j, u, f), a, c(f)) þ muce,f R(w m ¼ R(R(mucl,f , b, w(f)), a, c(f))) we mue þ mce,f ¼ mcle,f (78) It is worth observing that the derivatives ce,f , cl,f , mce,f , and mucl,f , as well as cle,f and mcle,f are meaningful only along the curves and should be calculated as ce,f ¼ pe,j j,f þ pe,u u,f ¼ pe,f muce,f ¼ mue,j j,f þ mue,u u,f ¼ mue,f cl,f ¼ pl,x x,f þ p,ln n,f ¼ pl,f

and for the new contact point Q

mucl,f ¼ mul,x x,f þ mul,n n,f ¼ mul,f c^ e (f ) þ c^ e,f df d(f ) d df ¼ c^ l (f ) þ c^ l,f df 0

(79)

that, for equation (73), can be simplified as c^ e,f d0 ¼ c^ l,f

(75)


Therefore, to determine the motion of the contact point, the values at P of the four scalar functions j,f , u,f , x,f , and n,f are required. However, as C02105 # IMechE 2005


explained in section 5.1, such functions are very difficult to obtain in an explicit form and so they are practically unknown. Instead of trying to solve them, it is equivalent to assume as unknowns all the derivatives vectors pe,f , pl,f , mue,f , and mul,f , which correspond respectively (and rotated) to the (2) (1) (2) unknowns v(1) r , vr , nr , and nr in reference [4]. All these derivatives are related through Rodrigues’ formula (28) mue,f ¼ K e pe,f

(80)

mul,f

(81)

¼ K g pl,f

Applying rigid rotations to the second equation, surface features as K g do not change, therefore (R(mul,f ,b,w),a, c) ¼ R(R(K g pl,f , b,w),a, c) R(R(mul,f ,b, w), a, c) ¼ K g R(R(pl,f , b, w), a, c) mule,f ¼ K g ple,f (82) Substituting in equation (78), we find we mue K e pe,f ¼ K g ple,f we mue K e (ple,f he ) ¼ K g ple,f (K e K g ) ple,f ¼ K e he þ we mue K eg ple,f ¼ K e he þ we mue (83) K eg being the relative tensor of curvature. The unknown vectors must also satisfy the derivative of the equation of meshing df ¼ he,f mue þ he mue,f df ¼ (w0e pe þ we pe,f þ q0e ) mue þ he me,f ¼ 0 (84) that inserting equations (77) and (80) becomes df ¼ (w0e pe þ we pe,f þ q0e ) df

1287

and finally (K e he þ we mue ) ple,f ¼ (w0e pe þ q0e ) mue þ (K e he þ we mue ) he (85) Let us observe that the hypothesis of expressing K e in an orthogonal reference frame has been applied for commutating the dot product, i.e. (K e he ) pe,f ¼ (K e pe,f ) he . In the following equations, a new vector k will be used, which is defined as k ¼ K e he þ we mue

(86)

normal to the contact line, as shown in section 7, that will help to have more compact expressions. Employing k, the solving equations (83) and (85) in the unknown ple,f can be written as k ple,f ¼ (w0e pe þ q0e ) mue þ k he

(87)

K eg ple,f ¼ k

(88)

which represent a system of three scalar equations in two scalar unknowns, the components of ple,f in the ^ corresponding to equation tangent plane at P, (8.4.42) in reference [4]. Curvature features are determined requiring infinitely many solutions for the case of line contact between the gears. Litvin discusses this issue by writing the matrix of the linear system obtained after introducing a reference frame and expressing vectors and tensor in their scalar components. It can be interesting to follow another way, employing a different form for equation (87); from the dot product k ple,f ¼ (w0e pe þ q0e ) mue þ k he ¼ c

(89)

it also follows that the component of ple,f along k is determined, because c is a known scalar quantity; therefore, the vector ple,f has an unknown component y in the direction normal to k in the tangent plane. The solving equations become c k þ (mue k)y kk K eg ple,f ¼ k

ple,f ¼

(90) (91)

mue þ he (K e pe,f ) ¼ (w0e pe þ q0e ) mue þ (K e he þ mue we ) pe,f ¼ (w0e pe þ q0e ) mue (K e he þ we mue ) (ple,f he ) ¼ 0

C02105 # IMechE 2005

Having infinitely many solutions of ple,f means that the earlier mentioned equations must be satisfied for every value of y; so from y ¼ 0 c K eg k ¼ k kk

(92)


1288


and therefore, K eg (mue k) ¼ 0

(93)

When compared with equation (23), equation (93) means that both the normal curvature and the geodesic torsion are null (as expected) in the direction mue k, parallel to the contact line, which is a principal direction for the relative curvature. From equations (92) and (93) k1eg ¼ 0 along mue k kk along k k2eg ¼ c

(94) (95)

Therefore, the relative tensor of curvature is completely determined and to have the tensor of curvature K g , we only need to remind (96)

K g ¼ K e K eg

In scalar components in the reference system S(e1 , e2 ), the tensor of curvature is

k1e 0 cos s sin s T ½K g ¼ 0 k2e sin s cos s 0 0 cos s sin s 0

k2eg

sin s

cos s

(97)

where s is the angle between mue k and e1 , positive if a counterclockwise rotation of s around mue moves mue k over e1 . In case of point contact, K eg has both principal values different from zero and there is only one value of y and one ple,f ; we have inserting equation (90) in equation (91) c k þ (mue k)y ¼ k K eg kk

CHEN’S APPROACH TO CURVATURE: INDIRECT GENERATING METHOD [5]

All the previous procedure results, in part, simplified when the second gear is the envelope gear, generated according to equations in section 5.2. As already mentioned, equation (72) is automatically satisfied from the definition of pg and mug in equation (62) and (64). In the initial contact point P we have pe ¼ pe (j , u );

f (j , u , f ) ¼ 0;

mue ¼ mue (j , u )

As in section 6, we follow the contact point after an infinitesimal increment of the motion parameter f and its trajectories over the two rotating surfaces S^ e (f) and Gg (f), which again correspond to the rotating conjugate curves c^ e and c^ g . Requiring the contact leading to equations (75) and (76), in the in points Q, generating case already known properties of the rotation operator are obtained pe,j ¼ R(R(pg,j , b, w(f)), a, c(f)) pe,u ¼ R(R(pg,u , b, w(f)), a, c(f)) he ¼ R(R(pg,f , b, w(f)), a, c(f)) mue,j ¼ R(R(mug,j , b, w(f)), a, c(f)) mue,u ¼ R(R(mug,u , b, w(f)), a, c(f)) that directly give relationships equations (77) and (78)

equivalent

to

(99)

muce,f þ we mue ¼ mucge,f

c K eg k ¼ yK eg (mue k) kk

(100)

where cge,f ¼ R(R(cg,f , b, w(f)), a, c(f))

that corresponds to

mucge,f ¼ R(R(mug,f , b, w(f)), a, c(f))

c k K eg k (K eg (mue k)) ¼ 0 kk As the previous equation is a cross product of vectors in the tangent plane, it may be written in the following scalar equation c k K eg k (K eg (mue k)) mue ¼ 0 kk

7

ce,f þ he ¼ cge,f

from which k

that corresponds to the null determinant required by Litvin in the case of point contact, i.e. equation (8.4.52) in reference [4].

(98)


In this case, we need to take care of the notation of the derivatives of pg with respect to f, in particular @pg (j, u, f) @f dpg (j(f), u(f), f) ¼ df

pg,f ¼

(101)

cg,f

(102)

C02105 # IMechE 2005


The expressions of the derivative vectors are

The derivatives of the function f (j, u, f) are f~ ,j ¼ mue,j he þ mue he,j

ce,f ¼ pe,j j,f þ pe,u u,f ¼ pe,f

¼ mue,j he þ mue (ce (f) pe,j (j, u))

muce,f ¼ mue,j j,f þ mue,u u,f ¼ mue,f

¼ mue,u he þ mue (ce (f) pe,u (j, u))

mucge,f ¼ mug,j j,f þ mug,u u,f þ mug,f (103) where the unknowns are the values at P of the derivatives j,f and u,f , which, from the first equation, Chen writes as

u,f ¼ pe,j

pe,f pe,f

1 me 1 u me me mue

(104) (105)

with me ¼ mue me . In this way, instead of having two scalar unknowns j,f and u,f , we have the unknown vector pe,f that lies in the tangent plane. Rodrigues’ formula applied to Gg and rotated, as in equation (82), becomes mug,f ¼ K g cg,f R(R(mug,f ,b,w),a, c) ¼ R(R(K g cg,f ,b,w),a, c) mucge,f ¼ K g cge,f (106) that inserted in equation (100) with equations (80) and (99) gives we mue K e pe,f ¼ K g cge,f we mue þ K e he ¼ K eg cge,f

f~ ,f ¼ ¼

mue mue

he,f (c0e (f) pe (j, u) þ q0e (f))

(111)

which, although fairly long, are easily computable functions requiring the knowledge of the generating surface equations and the relative motion. Putting all together in equation (108), a scalar equation in the unknown quantities j,f and u,f is obtained. Substituting all together in equation (108), we have df 1 ¼ ( f,j pe,u þ f,u pe,j ) (pe,f mue ) þ f,f ¼ 0 df me df ¼ (me ce (mue,j he )pe,u þ (mue,u he )pe,j ) df (pe,f mue )

1 þ f, f ¼ 0 me

corresponding to equation (10) in reference [5]. Employing Rodrigues’ formula, some simplifications can be done leading to an expression already obtained in Litvin’s approach (cf. equation (87)) (me ce ((K e pe,j ) he )pe,u pe,f mue þ ((K e pe,u ) he )pe,j ) þ f ,f me ¼ (me ce ((K e he ) pe,j )pe,u pe,f mue þ ((K e he ) pe,u )pe,j ) þ f ,f me pe,f mue ¼ (me ce me (K e he ) mue ) þ f ,f me ¼ (ce (K e he ) mue ) (pe,f mue ) þ f ,f ¼ k pe,f þ (c0e (f) pe (j, u) þ q0e (f)) mue ¼ 0 (112)

K eg (pe,j j,f þ pe,u u,f þ he ) ¼ we mue þ K e he ¼ k (107) in the unknowns j,f and u,f . The scalar unknowns must also satisfy the derivative of the equation of meshing

C02105 # IMechE 2005

(110)

¼ k pe,f þ f ,f ¼ 0

It is also

df~ ¼ f ,j j ,f þ f ,u u ,f þ f ,f df

(109)

f~ ,u ¼ mue,u he þ mue he,u

cg,f ¼ pg,j j,f þ pg,u u,f þ pg,f

j,f ¼ pe,u

1289

(108)

Therefore, the solving equations are (107) and (112) K eg (pe,j j,f þ pe,u u,f þ he ) ¼ k (c0e (f) pe (j, u) þ q0e (f)) mue ¼ k (pe,j j,f þ pe,u u,f þ he ) which can be discussed as already done for the meshing case. At this point Chen, assuming again infinitely many solutions for pe,f and also assuming Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

1290


the line of contact to be a principal direction, chooses to express equations (107) and (112) along lines at constant j or u on the generating tool, and along the direction k. For example, on a curve at constant u pe,f ¼ pe,j j,f

j ¼ jg (sg ), u ¼ ug (sg ),

and from equation (107) the relative curvature in the direction of pe,j is kjeg ¼

k (he þ pe,j j,f )

(118)

and gj ¼ jg,sg (sg ),

gu ¼ ug,sg (sg ),

gf ¼ fg,sg (sg )

(119)

The parametric coordinates in equation (118) are not completely independent but must satisfy the equation of meshing f~ (j , u , f ) ¼ 0. In addition, the components (gj , gu , gf ) in equation (41) must be such that tg (sg ) tg (sg ) ¼ 1 and the total derivative of the equation of meshing vanishes

(c0e (f) pe (j, u) þ q0e (f)) mue he þ pe,j

In this approach, although not proven by Chen, it is possible to recognize that k is orthogonal to the contact line by deriving the equation of meshing with respect to the arc length of a curve ce df ¼ f ,j j,se þ f ,u u,se þ f ,s f,se ¼ 0 dse

df ¼ k pe,se þ f ,f f,se ¼ 0 ds

(114)

(115)

confirming that k is orthogonal to the contact line. One of the main points in Chen’s paper is the description of the curvature analysis in case of nonorthogonal reference system; in this comparison this is not taken into account, but can be developed employing the relations in Appendix 2.

NEW APPROACH TO CURVATURE: DIRECT METHOD

In the direct method, [6, 9] (see also reference [15] for a preliminary explanation), K g is determined moving directly from definition of curvature and torsion in equations (7) and (8) kng ¼ mug,sg tg tgg ¼ mug mug,sg

tg

(120) g

g

The direction is chosen for evaluating kn and tg on the tool surface Se by means of the unit vector te ¼ R(R(tg (j , u , f ), b, w(f )), a, c(f ))

If ce is a contact line, being f,se ¼ 0, Eq. (114) gives k pe,se ¼ k te ¼ 0

f~ ,j gj þ f~ ,u gu þ f~ ,f gf ¼ 0

(113)

It is rather easy to repeat the steps in equation (112) and to obtain

8

f ¼ fg (sg )

(he þ pe,j j,f )2

where, from equation (112)

j ,f ¼

Again, we want to evaluate the normal curvature corresponding to c g ¼ at a specific point P, pg (jg (sg ), ug (sg ), fg (sg )), and along a given direction tg ¼ tg (sg ). The point and the direction are selected by means of the numbers

(116) (117)


(121)

which, when the tool and the gear get in contact at (j , u , f ), marks the same direction of tg . It is important to stress that in this case, we do not use conjugate curves as in the other approaches, because they do not satisfy equation (121). The tangent vector te lies in the tangent plane therefore can also be written as linear combination of pe,j (j , u ) and pe,u (j , u ) te ¼ pe,j ej þ pe,u eu

(122)

with suitable scalars ej and eu , satisfying also (equation (33)) te te ¼ E ej2 þ 2F ej eu þ G eu2 ¼ 1

(123)

Although ej and eu have been introduced as numbers, they could also be seen as the values at (j , u ) of the derivatives of functions like in equation (35). Inserting equations (41) and (122) in equation (121), employing equations (99) and (103), we find pe,j ej þ pe,u eu ¼ pe,j gj þ pe,u gu þ he gf

(124)

The dot products of equation (124) with pe,j and pe,u together with the derivative of the equation of meshing with respect to sg give the following system of C02105 # IMechE 2005


three linear equations

1291

(86), (99), (103), (122), (124), and (129), then

8 < E gj þ F gu þ he pe,j gf ¼ E ej þ F eu F gj þ G gu þ he pe,u gf ¼ F ej þ G eu :~ f ,j gj þ f~ ,u gu þ f~ ,f gf ¼ 0

kneg ¼ (mue,j ej þ mue,u eu (mue,j gj þ mue,u gu ) we mue gf ) te

(125)

¼ (mue,j (ej gj ) þ mue,u (eu gu ) we mue gf ) te ¼ (mue,j ej þ mue,u eu ) (pe,j (ej gj )

which is solvable at non-singular points on Gg [9, equation (60)] since E 1 ~ j, u, f) ¼ 2 F g( D ~ f,

j

F G f~ ,u

he pe,j he pe ,u = 0 f~ ,

þ pe,u (eu gu )) þ te we mue gf ¼ ((mue,j ej þ mue,u eu ) he þ mue we te )gf ¼ (mue,se he þ mue we te )gf

(126)

¼ ((K e te ) he þ mue we te )gf

f

¼ ((K e he ) te þ mue we te )gf ¼ (k te )gf

For instance, gf is given by

that, taking into account equations (114) and (127), becomes

E F Eej þ Feu 1 G Fej þ Geu gf ¼ 2 F D g~ f~ ,j f~ ,u 0 E F 0 1 G 0 ¼ 2 F D g~ ~ f ,j f~ ,u f~ ,j ej þ f~ ,u eu f~ ,f 1 ¼ (f~ ,j ej þ f~ ,u eu ) ¼ f,se g~ g~

kneg ¼

(127)

The normal curvature of Gg defined in equation (116), owing to equations (99), (103), and (100), along with property (140) can be written as

1 (k te )2 g~

(132)

This result confirms that k is a principal direction and that the relative normal curvature along the contact line, orthogonal to k, is null. When compared to equations (94) and (95), equation (132) underlines ~ that c ¼ (k k)g. Without reporting all the steps that can be found in reference [6], the geodesic torsion of Gg along cg defined in equation (117) can be written as tgg ¼ mue

(mue,j gj þ mue,u gu þ we mue gf )

(128)

and for equation (121)

teg g ¼ k te

(130)

provided equation (121) holds. Employing equations C02105 # IMechE 2005

¼

(129)

This is a computable expression where everything is evaluated at (j , u , f ) and expressed in terms of the tool shape and the generating relative motion. A simpler equation can be obtained for the relative normal curvature kneg between surfaces Se and Gg at contact points [3, p. 286] along the same direction; by definition, it is kneg ¼ kne kng ¼ (mue,se te mug,sg tg )

while the relative geodesic torsion teg g at contact points of the surfaces Se and Gg is given by the following equation

kng ¼ (mue,j gj þ mue,u gu þ we mue gf ) (pe,j ej þ pe,u eu )

te

(133)

kng ¼ (mue,j gj þ mue,u gu þ we mue gf ) (pe,j gj þ pe,u gu þ he gf )

(131)

9

1 k mue g~

mue gf te (k te )

(134)

CONCLUDING REMARKS

Three different approaches in curvature analysis have been compared with a new method, which is based on geometric instead of kinematic relationships. The indirect approach obtains curvatures starting from the analysis of the motion of contact points from the following conditions. 1. Contact in a point P (a) corresponding position vectors of the gear surfaces in P in the fixed space; Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

1292


(b) equal normal unit vectors to the gear surfaces in P in the fixed space; (c) equation of meshing satisfied in P: 2. Contact in a point Q, after df (a) equal derivatives of position vectors of the gear surfaces with respect to f in P in the fixed space; (b) equal derivatives of the normal vectors to the gear surfaces in P with respect to f in the fixed space; (c) derivative of equation of meshing with respect to f satisfied in P. 3. Rodrigues’ formula. In Litvin’s approach, the two gear surfaces are generic and previous conditions are used to relate the curvature of one gear to the curvature of the other one knowing relative motion. As the contact point P is supposed to exist, the first three conditions are not really employed; the fundamental relations are obtained from the last four requirements. Chen considers a generating case where one gear is obtained from the other by an enveloping process; in this case, the first three conditions are automatically guaranteed, and also conditions on the derivatives of the position and normal vectors are satisfied thanks to properties of the rotation operator. The derivative of the equation of meshing and Rodrigues’ formula is simply applied. It does not seem to be a relevant advantage to restrict the analysis to the generating case instead of the general meshing study considered by Litvin. In both these indirect approaches, although not done in the original version, conjugate curves have been introduced as trajectories of the contact points over the two surfaces which should make some steps clearer. In the direct approach, conjugate curves are not involved but rotation operator properties are used to relate the generating tool characteristic vectors, as tangent and normal vectors, to those of the generated gear. In particular, curves sharing the same tangent vector in the contact point are considered during meshing. Employing a relationship between the derivatives of the unit normal vectors along such curves and the derivative of the equation of meshing, the very same results of the previous approaches are obtained but in a direct way. The present comparison employs the tensor of curvature K, not explicitly considered in other papers, which simplifies many steps of the analysis leading to very compact and simple expressions without requiring reference systems.

ACKNOWLEDGEMENT The support of Avio SpA is gratefully acknowledged. Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

REFERENCES 1 Kreyszig, E. Differential geometry, 1991 (Dover Publications, New York). 2 Do Carmo, M. P. Differential geometry of curves and surfaces, 1976 (Prentice-Hall, Englewood Cliffs). 3 Litvin, F. L. Theory of gearing, 1989, NASA Reference Publication 1212 (NASA, Washington). 4 Litvin, F. L. Gear geometry and applied theory, 2004 (PTR Prentice-Hall, Englewood Cliffs). 5 Chen, N. Curvatures and sliding ratios of conjugate surfaces. ASME J. Mech. Design, 1998, 120, 126 – 132. 6 Di Puccio, F., Gabiccini, M., and Guiggiani, M. Curvature analysis of general gear surfaces via a new approach. Mech. Mach. Theory, 2005 (in press). 7 Wu, D. R. and Luo, J. S. A geometric theory of conjugate tooth surfaces, 1992 (World Scientific, Singapore). 8 Chen, C. H., Chiou, S. T., Fong, Z. H., Lee, C. K., and Chen, C. H. Mathematical model of curvature analysis for conjugate surfaces with generalized motion in three dimensions. J. Mech. Eng. Sci., 2001, 215(4), 487–502. 9 Di Puccio, F., Gabiccini, M., and Guiggiani, M. Alternative formulation of the theory of gearing. Mech. Mach. Theory, 40(5), 2005, 613 – 637. 10 Dooner, D. B. On the three laws of gearing. ASME J. Mech. Design, 2002, 124, 733 – 744. 11 Ito, N. and Takahashi, K. Differential geometrical conditions of hypoid gears with conjugate tooth surfaces. ASME J. Mech. Design, 2000, 122, 323 – 330. 12 Miller, S. M. Kinematics of meshing surfaces using geometric algebra. In Proceedings of DETC03. ASME paper ID PTG-48086, 2003. 13 Taubin, G. Estimating the tensor of curvature of a surface from a polyhedral approximation. In Proceedings of the Fifth International Conference on Computer Vision (ICCV’95). IEEE, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA, 1995. 14 Vogel, O., Grienwak, A., and Ba¨r, G. Direct gear tooth contact analysis for hypoid bevel gears. Comp. Method. Appl. Mech. Eng., 2002, 191, 3965 – 3982. 15 Gabiccini, M., Di Puccio, F., and Guiggiani, M. New investigation on the geometry of the contact in gear generation. In Proceedings of DETC03. ASME paper ID PTG-48087, 2003.

APPENDIX 1 Notation a b d ¼ Ob Oa kn K m mu

unit vector parallel to the pinion axis unit vector parallel to the gear axis vector connecting the pinion and gear axes normal curvature tensor of curvature normal vector to the surface unit normal vector to the surface C02105 # IMechE 2005


Oa Ob p

point on the pinion axis point on the gear axis position vector of points of the surface arc length of a curve on the surface unit tangent vector to a line of the surface vector parallel to the screw axis

s t w

angle of the pinion rotation around its axis a angle of the gear rotation around its axis b parameter of motion geodesic torsion

c w f tg Subscripts e g l

features of the pinion features of the generated gear features of the meshing gear

1293

generic S0 (i0 , j0 ) ½KS0 ¼ ½T1 2

k1

0

0

k2

½T 3

k1 cos a sin b

6 k cos b sin a k1 k2 7 2 7 6 sin (2b) 7 6 2 sin (b a) 7 6 sin (b a) 7 6 ¼6 7 k2 cos a sin b 7 6 7 6 5 4 k1 k2 k1 cos b sin a sin (2a) 2 sin (b a) sin (b a) As a general rule, writing ½K in a reference system S(i, j), that can be orthogonal or not, as k11 k12 ½K ¼ k21 k22 the normal curvature and geodesic torsion along the axes directions i and j can be obtained from the elements of the matrix as kn(i) ¼ (K i) i ¼ k11 þ k21 i j u u t(i) g ¼ (K i) (me i) ¼ k21 (me i) j

kn(j) ¼ (K j) j ¼ k22 þ k12 i j

APPENDIX 2

u u t(j) g ¼ (K j) (me j) ¼ k12 (me j) i

Tensor of curvature in a non-orthogonal reference frame The tensor of curvature at a point P of the surface can be expressed in a non-orthogonal reference frame S(i, j) but its elements do not correspond to the curvature and torsions along the axes as in equation (16). To obtain a more general form for ½K it is convenient, first of all, to extend the rotation in equation (17) to a more general transformation case, from S(i, j) to S0 (i0 , j0 ) both with generic directions ½T ¼

0

ii j i0

0

ij j j0

¼

cos a sin a

cos b sin b

½KS0 ¼ ½T ½KS ½T

(135)

(136)

To give an idea of a possible form of ½K in a nonorthogonal reference frame, the previous equations are applied to transform ½K expressed in principal directions S(P, i ¼ t1 , j ¼ t2 ) as in equation (15), to a C02105 # IMechE 2005

Properties of rotating vectors The rotation operator has many important and useful properties that are employed in this approach. Given the rotated vectors u^ ¼ R(u, a, a),

v^ ¼ R(v, a, a),

w^ ¼ R(w, a, a) (137)

where a and b are the angles between i0 and j0 and i, respectively; instead of equation (18) in this case we obtain

1

APPENDIX 3

we immediately have the following algebraic properties u ¼ R(R(u, a, a), a, a) u^ þ v^ ¼ R(u þ v, a, a) u^ v^ ¼ u v v^ w^ ¼ R(v w, a, a) ^ ¼ u (v w) ¼ u v u^ (^v w)

(138) (139) (140) w

(141) (142)

As far as differential properties are concerned, we have p^ ,j (j, u, f) ¼ R(p,j (j, u), a, a(f)) p^ ,u (j, u, f) ¼ R(p,u (j, u), a, a(f)) (143)


1294


whereas the derivative with respect to f that controls the rigid rotation is

like in p^ (j, u, f) ¼ R(p(j, u,f), a, a(f))

p^ ,f (j, u, f) ¼ R(p(j,u), a, a(f)),f ¼ a0 a R(p(j, u), a, a(f)) ¼ a0 a p^ (j, u, f)

(145)

a combination of the previous results is required (144)

where a0 (f) ¼ da(f)=df. In a more general case,


p^ ,f (j,u, f) ¼ R(p,f (j, u, f), a, a(f)) þ a0 a p^ (j, u, f)

(146)

C02105 # IMechE 2005