A Spatial Version of Octoidal Gears Via the Generalized Camus Theorem

7/25/2019 A Spatial Version of Octoidal Gears Via the Generalized Camus Theorem

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Giorgio Figliolini1

Mem. ASME

Professor

Department of Civil and Mechanical Engineering,

University of Cassino and Southern Lazio,

Via G. Di Biasio 43,

Cassino, FR 03043, Italy

e-mail: [email protected]

Hellmuth StachelProfessor Emeritus

Institute of Discrete Mathematics and Geometry,

Vienna University of Technology,

Wiedner Hauptstrasse 8-10/104,

Wien A-1040, Austria

e-mail: [email protected]

Jorge AngelesFellow ASME

Professor

Department of Mechanical Engineering and CIM,

McGill University,

817 Sherbrooke Street W,

Montreal, QC H3A 03C, Canadae-mail: [email protected]

A Spatial Version of OctoidalGears Via the GeneralizedCamus TheoremUnderstanding the geometry of gears with skew axes is a highly demanding task, whichcan be eased by invoking Studys Principle of Transference. By means of this principle,spherical geometry can be readily ported into its spatial counterpart using dual algebra.This paper is based on Martin Distelis work and on the authors previous results, whereCamus auxiliary curve is extended to the case of skew gears. We focus on the spatialanalog of one particular case of cycloid bevel gears: When the auxiliary curve is speci-fied as a pole tangent, we obtain pathologic spherical involute gears; the profiles arealways interpenetrating at the meshing point because of G2-contact. The spatial analogof the pole tangent, a skew orthogonal helicoid, leads to G2-contact at a single point com-bined with an interpenetration of the flanks. However, when instead of a line a plane isattached to the right helicoid, the envelopes of this plane under the roll-sliding of the aux-iliary surface (AS) along the axodes are developable ruled surfaces. These serve as con-jugate tooth flanks with a permanent line contact. Our results show that these flanks aregeometrically sound, which should lead to a generalization of octoidal bevel gears, oreven of bevel gears carrying teeth designed with the exact spherical involute, to thespatial case, i.e., for gears with skew axes. [DOI: 10.1115/1.4031679]

Keywords: gears with skew axes, involute gearing, octoidal gears, spatial Camustheorem, curvature theory

1 Introduction

Let the motions of two wheels, R2; R3, the pinion and the gear,respectively, with respect to the machine frame R1 be given, i.e.,the rotations R2=R1; R3=R1 about fixed skew axes I21 and I31with angular velocities x21, x31, respectively. Throughout thepaper, dual algebra is used extensively, with a dual quantity,whether a scalar, a vector, or a matrix is represented with a hat on top, its primal and dual parts with the same unhatted vari-

able, its dual part carrying the subscript 0 [1,2]. In this context,lines are represented with a dual unit vector (DUV), i.e., a dualvector whose primal part is of unit magnitude.

We use a Cartesian coordinate frameF O;x1;x2;x3 with thex1- and x2-axes represented by the DUV e1 and e2. Axis x1 isdefined as thedual bisectrix of lines I21 and I31, namely, the linethat (a) passes through the midpoint of the segment of the com-mon normal to the two lines defined by the two perpendicular feetand (b) is parallel to the bisectrix of the angle between the twolines. Moreover, x3 is defined as the common normal to the twolines, whilex2completes a right-handed coordinate frame with x1andx3. Using thedual anglea a ea0 between the x1-axis andI21 (see Fig.1), the DUVs p21 and p31 representing the axes I21andI31 are

p21 cos ae1 sin ae2; p31 cos ae1 sin ae2 (1)

We limit ourselves to the case of skew axes, while assuming

0< a < p=2 and a0 6 0 (2)

though most of the arguments hold also in the spherical case, witha0 0, and in the planar case witha 0 and parallel axes.

In addition, let u denote the dual angle between e1 and theinstant screw axis (ISA) I32, which is represented by p32. Then,we obtain

p32 cos ue1 sinue2n (3a)

x32p32 x31p31 x21 p21 (3b)

Fig. 1 Skew axesp21; p31 of the gear wheels, the ISA p32 andthe axis p41 of the AS P4 R4 in the particular caseb5u1p=2.The Frenet frame f1; f2; f3 of the axodes remains fixed to themachine frame R1.

1Corresponding author.Manuscript received March 17, 2015; final manuscript received August 28, 2015;

published online November 24, 2015. Assoc. Editor: David Dooner.

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Comparison of coefficients and recalling a pertinent relation2 leadto

tan ux31 x21x31 x21

tan a (4a)

u0 R sin 2u with R a0

sin2a (4b)

The vector product of both sides of Eq. (3b) with p21 and p31results in

x21

sin u a

x31

sin u a

x32

sin 2a (5)

which sometimes is called thedual sine-theorem, as applied to thedual trianglex21 p21; x31 p31 and x32 p32. This implies

x32 x21sin 2a

sin u a (6)

and, consequently, the pitch h32 of the relative motion R3=R2 isreadily derived [3]

h32x320

x32R cos2a cos2u 2R cos2a cos2u

(7)

The axodes of the relative motion R3=R2 are one-sheet hyper-boloids P3 R3 and P2 R2, swept by the relative axis I32under the inverse rotations R1=R2 and R1=R3 about I21 and I31,respectively.

In Sec. 2, we recall Distelis spatial Camus theorem togetherwith some of the basic formulas. Section 3 offers an account ofthe differential geometry of ruled surfaces, and Sec.4 provides adeeper insight intoG1- and G2-contact along lines between ruledsurfaces. In Sec. 5, we study the differential geometry of toothflanks obtained by means of the spatial Camus theorem. Finally,in Sec.6, we apply all this to spatial analogs of spherical involute

and octoidal gears.

2 The Spatial Camus Theorem

The result below is due to Disteli [4].LEMMA1. For given wheels R2; R3 there exists a frame R4 such

that the screws ofR4=R2; R4=R3, andR3=R2 are equal at everyinstant if and only if the instant axis I41ofR4=R1 is located on thePlucker conoid3 W, but different from I32.

Letbbe the dual angle between thex1-axis and p41, as depictedin Fig.1. Then,

p41 cos be1 sin be2 (8)

If we specifyI41 W different fromI21;I31;I32, thenu 6 6a;b.From Eq.(4b), which defines the Plucker conoid, we obtain

b0 R sin2b (9)

The dual sine-theorem, as applied to the trianglex21p21; x41p41,and x42 p42 x42 p32, yields

x21

sin u b x41

sin u a

x42

sin a b (10)

Likewise, forx31 p31; x41p41, and x43 p43 x43p32

x31

sin u b x41

sin u a

x43

sin b a (11)

The instant pitchh41 x410=x41 is defined as

h41x410

x41R cos 2a cos2b (12)

Let P4 be the ruled helical surface4 traced by the relative axis

I32 under the helical motion R1=R4 about I41 with pitch h41. Wecall P4 the AS. Further details are provided by the authors else-where [5]. P4 forms together with P2 and P3, the axodes of therelative motions ofR4 with respect to R2 and R3, i.e., the motionsR4=R2 and R4=R3 are defined by the rolling and sliding of P4along the hyperboloidsP2 and P3, respectively. The roll-slidingbetween the axodes implies the mutual rolling of their sphericalimages, and hence

x21sinu a x31sinu a x41sinu b (13)

which equals the primal part of Eq.(5).The importance of the AS P4 R4 lies in the result below [6]:THEOREM 1. (Spatial Camus Theorem) For any line G attached

to R4, the surfaces U2; U3 traced by G under the relative motionsR4=R2 and R4=R3, respectively, are conjugate tooth flanks ofR3=R2. At any instant, the meshing points for these flanks arelocated on a straight line.

With respect to the machine frame R1, the locus of the meshinglines, i.e., themeshing surface orsurface of action, is traced byGunder R4=R1 with the fixed twist q41 x41 p41. Consequently,the said surface is helical, with axis p41. By virtue of Lemma 1

p42 p43 p32 f1 and Eqs. (10) and (11), the twists ofR4=R2and R4=R3 are

q42 x42 p42 x41p41 x21 p21

x42f1 x21 sin a b

sin u b f1

q43 x43 p43 x41p41 x31 p31

x43f1 x31 sin b a

sin u b

f1

(14)

while, on the other hand, the relative motion R3=R2 of the gearsundergoes the twist

q32 x32 p32 x31 p31 x21 p21 x32f1

x21 sin2a

sin u a f1

x31 sin2a

sin u a f1

We can confirm that, according to Lemma 1, the twists q42; q43,and q

32 differ by real factors only. This leads, by virtue of

Eqs.(5)and(7), to

x42 x21sin b a

sin u b 1 e h32

x43 x31sin b a

sin u b 1 e h32

x32 x21sin 2a

sin u a 1 e h32 x42 x43

(15)

2Equation (7) in Ref. [3].3Sometimes referred to as thecylindroid.

4In this paper, the term ruled surface stands for a twice continuouslydifferentiable one-parameter set of oriented lines.

021015-2 / Vol. 8, APRIL 2016 Transactions of the ASME



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Therefore,

x42 : x43 : x32

sinu asinb a : sinu asinb a : sin2a sinu b

At every instant, the three axodes P2; P3, and P4 are in contact

at all points of the common generator I32. Therefore, theyshare the striction point S, i.e., the point whose tangent plane isorthogonal to the asymptotic plane, the limit of the tangentplane when the contact points tend to infinity. The striction pointS 0;0; u0is the point of intersection between the ISA I32 andthe common normal ofI21 and I31, as seen in Fig.1.

3 The Disteli Axes of a Ruled Surface

Along each nontorsal generatorG of a ruled surface, a Frenetframe, also known as aSannia frame[7], can be defined, consist-ing of:Gitself; thecentral normalN, which is the surface normalat the striction point; and the central tangentT [1,8]. This tripletof mutually orthogonal axes meets at the striction point S ofG,defined on thestriction curve, as per Fig.2. The central tangent isorthogonal to the asymptotic plane and tangent to the surface atthe striction point. The motion of the Frenet frame along the ruledsurface is calledFrenet motion.

Let, in dual-vector notation,5 the ruled surface be given by thetwice-differentiable dual-vector function gt for t in the openinterval J R. Then, the derivatives of the DUVs of the Frenetframe, g; n, and t, satisfy the Frenet equationsEq. (9.11) ofRef. [1]namely,

_g kn q g

_n kg l t q n

_t ln q t

with q lg k t xg; g g 1 (16)

Hence, ^x

2

^

k

2

^l

2. Such as the Darboux vector of a smoothspatial curve represents a vector proportional to the angular veloc-ity vector of the curve Frenet frame [9], the dual Darboux vectorq xg represents the twist of the dual Frenet motion. TheDUV g

is the instantDisteli axis[10] oraxis of curvature [1] ofthe ruled surface, providedk60.6

Remarks

(a) Even when the generators of the ruled surface are oriented,the central normal can be oriented in two different ways.One could obtain uniqueness by demandingk > 0.

(b) Conversely, when the dual-vector functions gt; nt;tt; t2 J, represent a moving orthonormal frame satisfy-ing the Frenet equations(16), this frame defines a ruled sur-face with generators gt, with central normals nt andcentral tangents tt. Here, fork 0 also cylindric genera-tors are included. By the same token, also the directed linestt define a ruled surface with the same given movingframe and the same Disteli axis; only g and t change theirroles as well as the coefficientskand l.

The Frenet equations (16) contain two dual coefficients,

k k ek0 and l l el0. Various formulas expressing invar-iants of the ruled surface in terms ofk,k0,l, and l0are availablein Ref. [8].7 Here, we adopt a different approach.

The dual representationgt gt e g0t; t2 J, of the ruledsurface gives rise to a real parametrization, namely,

xt;u gt g0t u gt; t; u 2 JR (17)

Here, we recall that g g0 is the position vector of the point ofgclosest to the origin of the underlying coordinate frame. Thederivatives ofg are readily obtained:

d

dtg _g _g e _g0

kn kn e k0n kn0

d2

dt2 ^g

gg e

g0

^

k

2^g

_

k^n

^

kl^t

k2g _kn klt e2kk0g k2g0

_k0n _kn0 k0lt kl0t klt0 (18)

which determine the partial derivatives of the parametrizationxt; u

@x

@t _g g0 g _g0 u _g;

@x

@ug

and

@2x

@t2 g g0 2 _g _g0 g g0 ug

@2x

@t@u _g kn; @2x

@u2 0

We study the derivatives at the points of a single generator, say, at

t 0. To this end, we use the triplet g0; n0; t0as the newcoordinate frame; now the striction point s0ofg0is the originof the frame in question. We can then set

g0

1

0

0

264

375; n0

0

1

0

264

375; t0

0

0

1

264

375

g00 n00 t00 0

which yields

_g0

0

k

0

264

375 e

0

k0

0

264

375

g0

k2

_k

kl

2664

3775 e

2kk0_k0

k0l kl0

264

375

Fig. 2 Frenet frame g; n; t and striction curve of a ruledsurface

5We identify an oriented lineG with its DUV g in a given coordinate frame. Inthis sense, we speak of the line g.

6By virtue of the last condition, we exclude stationary (singular) generators.

7For example, the dual part q 0 of the twist q equals the velocity vector of theorigin s. Consequently, for the striction point S of Fig. 3, we obtain tan r k=l,where angle r between g and the striction curve is called the striction angle or,simply, thestriction.

Journal of Mechanisms and Robotics APRIL 2016, Vol. 8 / 021015-3



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Therefore,

@x

@t0; u

0

ku

k0

2

4

3

5; @x

@u0; u

1

0

0

2

4

3

5 (19)

and

@2x

@t2 0; u

k2u

k0l kl0 _ku

_k0 klu

26664

37775

@2x

@t@u0; u

0

k

0

2664

3775; @

2x

@u2 0;u

0

0

0

2664

3775

(20)

The vector product b @x=@t @x=@u is a normal vector ofthe ruled surface, provided the surface point is regular, which

meansb 60. The coordinates

b0;u 0

k0ku

24

35 (21)

reveal that at generators with kk06 0, the angle w between thecentral normal vectorb0; 0 knand the normal vectorb0; u,depicted in Fig.3, satisfies the equation

tan wku

k0

u

d with d

k0

k (22)

The ratio d is called the distribution parameter. This is a geo-metric invariant, i.e., indifferent to surface parametrization.

Moreover, Generators with k0 0 and hence d 0 are torsal. Here, all

points with u60 have the same tangent plane, which isorthogonal to T. The striction point (u 0) is singularbecauseb0; 0 0.

Cylindrical generators are characterized by _g 0 or k 0.Here, all points are possible striction points. These generatorsoccur whend ! 1.

When a parametrization of the Frenet frame is given, the originis automatically a striction pointalso in the case of cylindricalgenerators. In the case of an isolated cylindrical generator, i.e., alocal zero of the function kt, the origin is the limit of the stric-tion point when neighboring generatorsgt tend to g0.

4 Two Ruled Surfaces With Line Contact

For our study on cycloid gearing, we need some results con-cerning the Disteli axes g

of a ruled surface. According to

Eq.(16), q xg is the twist and thereforeg the instant screwaxis of the moving Frenet frame. From Eqs.(16)and(18), the fol-lowing relation is obtained:

_g g kn k2

g _kn kl t k

2xg

(23)

The dual angle c c ec0 between the generator g and thecorresponding Disteli axisg satisfies

cot cl

k) cot c

l

k and c0

kl0 k0l

k2 l2(24)

a consequence of the two standard products

g g cos cl

x and g g sin cn

k

xn

and of the dual extension of an analytic real function f(t), namely,ft ft et0 ft et0_ft, which yields

cotc cotc ec01 cot2c

The dual angle between the moving^gt and the fixed

^g

0 is sta-tionary of second-order8 att 0[8]. Due to the spherical analogy,cotc can be called the dual (geodesic) curvature of the ruled sur-face. We have, further,

LEMMA2. If two ruled surfaces are in contact at all points of acommon generator and if they share the corresponding Frenetframe and the Disteli axis, then their dual coefficients in the Fre-net equations differ at the corresponding parameter values onlyby a real factor c 60.

Proof.Let gtfort2 Jand ~g~tfor~t2 ~Jbe the dual paramet-rizations of the two surfaces, such that for t ~t 0, the corre-sponding Frenet frames coincide; hence, also their strictionpoints coincide. The contact at all points of the commongenerator implies, by virtue of Eq.(22), equal distribution param-

eters d0 ~d0. Consequently, there is a real constant c60

with ~k ck and ~k0 ck0 at t ~t 0 and hence^~k0 c k0.

On the other hand, the coincidence of the Disteli axes implies

cot ~c cotc ; hence, by virtue of Eq. (24), ~l=~k l=kand, there-fore,~l0 c l0.

Without loss of generality, we can set c 1 because we canapply the parameter transformation~t! t c ~tto the second sur-

face. This yields the new dual parametrizationgt ~gt=c, andhence

d

d tg

1

c

d

d ~t~g ) k

1

c~k and l

1

c~l

By Lemma 2, this implies k0 k0 ; likewise, l0 l0.In analogy with Eq.(17), the dual parametrization g t yields

the parametrization xt; u ~xt=c; u of the second ruled sur-

face, such that the tangent vectors

a x t0; u b xu0; u and a xt0; u b xu0; u (25)

of the two corresponding ruled surfaces are identical for alla; b 2 R2 andu 2 R.

THEOREM2. Letgt and ~g~t be two twice-differentiable ruledsurfaces, which, at t ~t 0, share the same Frenet frame, thedistribution parameter d0 ~d0, and the Disteli axis. Then,the surfaces have a G2-contact at the striction point of the com-

mon generator. Moreover, if ~k0 c k0 and ~l0 c l0,

Fig. 3 The tangent plane Tx at the point x of the generator g isdefined by the distribution parameterd via tanw52u=d

8That is, up to its second t-derivative.




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then by Lemma 2 there is a G2-contact at all points of the common

generatorg0 ~g 0 if and only if _~d0 c _d0.Proof. The dual-vector function gt gt e g0t deter-

mines the real parametrizationxt; u of the ruled surface, as givenin Eq.(17). The partial derivatives at t 0, as given in Eq.(19),define the coefficients of the first fundamental formas [11]

E 0; u @x

@t@x

@tk2u2 k20

F 0; u

@x

@t

@x

@u 0; G 0; u

@x

@u

@x

@u 1

(26)

For the second surface, parametrized by t, we obtain the analogcoefficients E; F; G with E0; u E0; u; F0; u F0; u,and G0; u G0; u.

The second-order partial derivatives at t 0 are displayed inEq.(20). Together with the normal vectorb of Eq.(21), the coeffi-cients of the second fundamental form [11] at the points of thegeneratort 0 are defined as

L 1

jjbjjb

@2x

@t2

1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik20 k

2u2q k0 k0l kl0 _kk0 k _k0

u k2l u2

h i

M 1

jjbjjb

@2x

@t@u

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik20 k

2u2q kk0

N 1

jjbjjb

@2x

@u2 0 (27)

The analog coefficients L; M; N of the parametrization xt; uof the second surface satisfy M0; u M0; u and N0; uN0; u. However, L0; u L0; u holds if and only if eitheru 0 or

d k

d tk0 k

d k0

d t

t0

dk

dt k0 k

dk0

dt

t0

which is equivalent to

d

d td 0

1

k2

k d k0

d t

d k

d tk0

t0

1

k2 k

dk0

dt

dk

dtk0

t0

d

dtd 0

Because ofdt ~dt=c, we obtain _~d c _d.According to a standard formula from differential geometry, for

the first surface the normal curvature of the tangent vectora@x=@t b@x=@u60, i.e., the curvature of the orthogonal sec-tion through this tangent is given by the quotient of the two funda-mental forms [11], namely,

jnII a; b

I a; b La2 2Mab Nb2

Ea2 2Fab Gb2

Therefore, for our two given ruled surfaces the two statementsbelow are equivalent:

(i) The two second fundamental forms have equal coefficientsat the pointx0;u x0; u.

(ii) For all tangent vectors at this point, the normal curvaturesare equal.

Statement (ii) characterizes the G2-contactof the two surfacesat this point.

For our particular case of cycloid gearing, we use the converseof Lemma 2 and Theorem 2:

THEOREM3. Letgt and ~g~t be two twice-differentiable ruledsurfaces, which share the same Frenet frame and the distributionparameterd0 ~d0< 1 at t ~t 0.

If, for t ~t 0, the spherical images are osculating and thesurfaces have a G2-contact at the striction point, then the surfacesshare the Disteli axis, too.

Proof.Let gt fort2 Jand ~g~t for~t2 ~Jbe the dual paramet-rizations of the two surfaces such that for t ~t 0 the corre-sponding Frenet frames coincide. Hence, also their striction pointscoincide. According to the proof of Lemma 2, there is a real con-

stant c60 with ^~k0 c k0. The osculation of the spherical

images implies

l 0

k 0

~l 0

~k 0 ; ) ~l 0 c l 0

provided k0 6 0. By virtue of Eqs. (26) and (27), the coeffi-cients of the fundamental forms at the striction point (u 0)follow:

E0; 0 k20; F0; 0 0; G0; 0 1L0; 0 k0l kl0; M0; 0 k; N0; 0 0

(28)

By a suitable parameter substitution, as in the proof of Theorem 2,we can conclude that G2-contact at the striction point implies

~L0; 0 c2

L0; 0, and hence

~k0~l ~k~l0 j~t0 c2k0l kl0jt0

From k0 6 0 follows ~l00 c l00. But ~k0 c k0 and

~l0 c l0imply that the two surfaces share the Disteli axis att ~t 0.

5 The Curvature of the Ruled Tooth Flanks

In the realm of gearing, we need two different Frenet frames,

the frame f1; f2; f3 R1 for the axodes with the ISA f1 (seeFig.1) and the frameg1; g2; g3 for conjugate tooth flanks U2;U3with g1 as the meshing line (see Fig.4). Let R5denote the moving

Fig. 4 The triplet g1; g2; g3 is the Frenet frame for the conju-gate tooth flanks U2 and U3. The corresponding Disteli axes g

are defined by the spatial EulerSavary equation(36).




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space attached to the foregoing frame. Then, R5=R2 and R5=R3are the Frenet motions along U2and U3, respectively.

5.1 The Frenet Frame of the Axodes. Upon gear meshing,theFrenet frame f1; f2; f3 of the axodes with f1 p32 remainsfixed to the machine frame R1. The second axis f2 equals thespear9 e3 along the common perpendicular of the gear axes p21and p31. In terms of the basis e1; e2; e3, we obtain, fromEq.(3a)and Fig.1,

f1f2 f3 e1e2e3cosu 0 sinusinu 0 cosu

0 1 0

24 35 (29)

or, conversely,

e1e2e3 f1f2f3cosu sinu 0

0 0 1

sinu cosu 0

24

35

by simple transposition.The origin of the Frenet frame is the striction point

S 0;0; u0of the axodes, the point of intersection between theISA p32 and the common normal of p21 and p31. The motionof this frame along the axode P2 R2 is the rotation R1=R2

about the axis p21 with the angular velocity x21. Therefore,p21 cosu af1 sinu af3 is the permanent Disteli axisof P2. From Eq. (1), the corresponding Frenet equations (note

e3 f2) begin with

_f1 x21p21 f1 x21sinu a f2

x21sinu a eu0 a0cosu af2

which implies for the axode P2the distribution parameter10

d2 u0 a0cotu a

and the coefficients

k2 x21sinu a; l2 x21cosu a

The last equation follows from the third Frenet equation (16),_f3 x21p21 f3, which confirms, for the dual angle c2between the generator p32 f1 and the Disteli axis p21, from

Eq. (24), that c2 u a with cotc2 l2=k2 as the dual curva-ture ofP2, provided thatk2 60.

The Frenet motion along the other axode, the one-sheethyperboloid P3 R3, is the rotation R1=R3 about p31cosu af1 sinu af3, with the velocity x31. Here,c3 u a is the dual angle between the generatorp32 f1 andthe Disteli axisp31. We obtain forP3the distribution parameter

d3 u0 a0cotu a

and the coefficients

k3 x31sinu a; l3 x31cosu a

The conditiond2 d3, which can also be concluded from Eq.(5),guarantees the line contact between P2 and P3, i.e., at least G

1-contact at all points of the ISAp32[12].

In the Frenet equations of the AS P4 R4 with axis p41cosu bf1 sinu bf3 and dual velocity x41, weobtain the coefficients

k4 x41sinu b; l4 x41cosu b (30)

As a consequence, P4 has, by virtue of Eq.(22), the distributionparameter

d4 h41 u0 b0cotu b

The equations d4 d3 d2 can be verified via Eqs. (4),(9), and(12). The axis ofP4 makes, with all generators P4, the dual anglec4 u b.

5.2 The Frenet Frame of the Tooth Flanks.According toTheorem 1, any line g attached to the AS P4 traces conjugatetooth flanks U2 and U3 under the respective relative motionsR4=R2 and R4=R3 with the AS P4 roll-sliding on the axodes P2and P3, respectively. The motion R4=R2 is the composition ofR4=R1with the Frenet motion R1=R2along P2[13,14].

We can set up the moving lineg as

g cos gf1 sin g cos nf2 sin n f3 (31)

This follows because the common perpendicular11 k between g

and the ISA f1 can be written as k sin nf2 cos nf3, as perFig.4. The dual angles n and g can be seen as dual geographicallongitude and colatitude, respectively.

The common perpendiculark is already the central normal n ofthe tooth flanks. This follows because, for the trajectory of gunderR4=R2, we obtain

_g x42f1 g x42sin g cos n f3 sin n f2 x42sin gk

Therefore, the Frenet frame g1 g; g2 n k; g3 tfor theconjugate tooth flanks U2 and U3 has the initial pose

g1g2g3 f1 f2 f3cos g 0 sin g

sin g cos n sin n cos g cos nsin g sin n cos n cos g sin n

2

4

3

5(32)

or, conversely

f1 f2f3 g1g2g3cos g sin g cos n sin g sin n

0 sin n cos nsin g cos g cos n cos g sin n

264

375

(33)

Upon differentiation of _g x42f1 g and considering thatx42 const, the following relation is obtained:

g x42_f1 g f1 _g

During the motion R4=R2, the ISA f1 traces P2 with angularvelocity x21. Therefore,

_f1 x21 p21 f1 x21sinu a f2

and hence

g x42x21sinu af2 g f1 x42f1 g

x42x21sinu af2 gx42f1 gf1 f1 f1 g9Spearis a line with a direction, usually represented with an arrow indicating thepositive direction.

10Leta,b denote the sem-iaxes of a one-sheet hyperboloid of revolution, where b

is measured along the axis of rotation. Then, b equals the absolute value of thedistribution parameter of the generators, i.e.,jdj b.

11Under sin g 0, the moved line g is parallel to the ISA f1 and a cylindricalgenerator of its trajectory. In this case, the common perpendiculark is not unique.


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By virtue of Eq.(33), we can express the first and second deriva-tives ofg in the Frenet frame g1; g2; g3 as

_g _g1 x42f1 g x42 sin gg2g x42x42sin

2gg1 x21 sinu acos ncos gg2

x21sinu asin n x42sin gcos gg3

which, upon comparison with Eq. (18), yields the instantaneousinvariants of the tooth flank U2 underg 60, i.e., g not parallel to

the ISA^f1, as

kU2 x42 sin g

lU2 x21sin u a sin n

sin g x42cos g

(34)

The conjugate tooth flank U3 is the trajectory of line g underthe composition R4=R3 of the helical motion R4=R1 of the AS P4and the Frenet motion R1=R3along P3. We obtain,

_g x43f1 g

and

g x43_f1 g f1 _g

where

_f1 x31 p31 f1 x31sinu af2

In terms of the Frenet frameg1; g2; g3, we obtain, further

_g x43 sin gg2g x43x43sin

2gg1 x31 sinu acos ncos gg2

x31sinu asin n x43sin gcos gg3

The instantaneous invariants ofU3are

kU3 x43 sin g

lU3 x31sin u a sin n

sin g x43cos g

(35)

providedg 60.By virtue of Eq. (15), the invariants kU2 and

kU3 differ only bya real factor. This is, of course, a consequence of theG1-contactalong g. The distribution parameter along the instant meshingline is

dUkU2 0

kU2

kU30

kU3

x320sin g x32g0cos g

x32sin g

This confirms that all linesg that underR3=R2trace ruled surfaceswith the same instant distribution parameterd constitute a quad-

ratic line complex, i.e., the six (normalized) real coordinates ofg g1; g2; g3 eg10; g20; g30 in Eq. (31)satisfy a homogene-ous quadratic equation

x320 d x32g22 g

23 x32g2g20 g3g30 0

which includes, ford 0, the complex of tangent lines.12

We can set up the instant Disteli axis gU2

ofU2 in analogy toEq.(31), in light of Fig.4, as

gU2

cos g U2

f1 sin gU2

cos n f2 sin gU2

sin n f3

On the other hand, from Eq. (16), a dual multiple of gU2

equalsthe twist

q52 lU2 g1 kU2 g3

of the Frenet motion R5=R2 along U2. After expressing this dualvector in the basis f1; f2; f3, the comparison of the coefficientsoff1and f3results, from Eq.(6), in

cot gU2 cot g

lU2 cot g kU2

lU2 kU2 cot g cot g

x42

x21sin u a sin n

sin a b

sin u a sin u b

sin n

provided that sinn sin g60. Thus, we can verify the spatialEulerSavary equation (see the Appendix)

cotg U2

cotg sin n cotc2 cotc4

cotu a cotu b(36)

for the motion R4=R2, which generates the tooth flank U2.In the same way, we can confirm that, for the conjugate

tooth flank U3, the Frenet motion R5=R3 undergoes the twistq53 lU3 g1 kU3 g3and the Disteli axis becomes

gU3

cos g U3

f1 sin gU3

cos n f2 sin gU3

sin n f3

which satisfies

cot g U3

cot g sin n cot c3 cot c4

cotu a cotu b(37)

Upon subtraction of the two EulerSavary equations (36) and(37), we obtain

cot g U2

cot g U3

sin n cotc2 cotc3

thereby proving the spatial version of a result which is well knownin planar and spherical kinematics, namely,

THEOREM4. LetU2 andU3 be conjugate ruled gear-tooth flankswith permanent line contact. Then, the Disteli axes g

U2

andgU3

ofthe instant meshing line satisfy the EulerSavary equation for therelative motion R3=R2between the two gears.

6 A Spatial Analog of Involute Gearing

In planar cycloid gearing there are two auxiliary curves,namely, two circles, which usually are laid out in a symmetric rel-ative position with respect to the pole tangent. The same is true on

the sphere. However, when the auxiliary circles are specified asgreat circles they become identical, coinciding with the sphericalpole tangent t. The axis p41 of the great circle t is orthogonal tothe ISAp32. The corresponding profiles are involutes of the polo-des; in the mean pose, they have the pressure anglea 0 deg.

This is the particular case of involute gearing, where the pitchcircles coincide with the base circles. These profiles arenotgeo-metrically feasible because of one reason (see Fig. 5): At themeshing point M on the instant pole tangent t, the profiles haveeither

aG2-contact with mutual penetration, ora cusp, and, if gears are external, the curves open toward oppo-site sides.12See Theorem 56 in Ref. [15].




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We obtain the corresponding spatial version when we specifythe axis p41 orthogonal to the ISA p32 on the Plucker conoid (seeFig.1). This is the case analyzed below in detail, following a pre-liminary study [16].

By virtue of Eqs. (4)(9), the representation p41 sin be1 cos be2implies

13

b u p

2 ; b0 u0; u b

p

2 2eu0 (38)

Therefore,

sinu b 1 and cosu b 2eu0 (39)

From Eqs.(5),(10), and(12)follow, for our particular choice

x41 x21sinu a x31sinu a

h41 Rcos 2a cos2u (40)

The AS P4 is a skew orthogonal helicoid with axis p41 andpitchh41, the ISA p32 being its initial generator. The invariants of

P4, by virtue of Eq.(30), are

k4 x41; l4 2eu0 x41 (41)

The dual angle between the generators ofP4and its axis is

c4 u b p

2 2eu0 with cot c4 l4=k4 2eu0

From Eq.(4a), the distance c40between axis and generators van-ishes if and only ifu 0, i.e., ifx21 x31.

By Eq. (14), the generating motions R4=R2 and R4=R3 of the

tooth flanks U2 and U3 have the twists q42 x42f1 and

q43 x43f1, respectively; in our particular case, we have

x42 x21cosu a eu0 a0sinu ax43 x31cosu a eu0 a0sinu a

(42)

Hence,

x43 : x42 tanu a : tanu a

u0 a0 :u0 a0 (43)

6.1 The ISA as a Line of Regression.Analog to the planarand spherical cases, in spatial cycloid gearing the ISAp32 is a sin-gular generator of the two tooth flanks U2 and U3. All the genera-tor points are uniplanar, the tangent planes along p32 beingequally distributed along a regular generator with distribution pa-rameterd R cos2a. As revealed in Fig.6, the ISA I32 does notlook singular at all; it is the border line of the two components,originating from two symmetrically placed ASs. However, in ourparticular case the two ASs coincide with the skew helicoid P4.The ISA is, in fact, a line of regression for both tooth flanks. Inexternal gears, as depicted in Fig. 6, the two flanks open towardopposite sides. Hence, when the ISA becomes the meshing line,no transmission of forces can take place. Figure6 shows the con-

jugate tooth flanks as wire-frames, the depicted thick lines beingthe intersections of the flanks with planes perpendicular to theISA.

6.2 G2-Contact at the Striction Point.Now a questionarises: What corresponds in skew gears to the osculation of toothprofiles when the pole tangentserves as auxiliary curve?

Figure7 shows an example14 where the meshing line g differsfrom the ISA. But g is parallel to the ISA and intersects the centraltangent of the axodes at right angles. This central tangent passes

Fig. 5 Involute bevel gearing with the pole tangent as auxiliarycurvea case which is geometrically unfeasible because theconjugate profiles c2and c3always penetrate each other at themeshing point Mi

Fig. 6 When the ISA coincides with the meshing lineg, the sin-gular lines of the two flanks U2; U3 come together sharing the

tangent plane at each point ofg, but the flanks open toward op-posite sides

13One could also set b u p=2. However, this has no effect on the AS. It onlyreverses the orientation ofp41and changes the signs ofx41and x410.

14Data: 2a60:0 deg; 2a0 70:0 mm,x31 :x21 2 :3, and distance betweenthe ISA and the initial meshing line g : SSg 35:0 mm.


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through the striction point S of the axodes and is parallel to theaxisp41 of the AS P4note

f3in Fig.1.The spatial EulerSavary equation (36) for the motion R4=R2

recalled below

cot g cot g sin n x

kcot c2 cot c4

holds only under sin n60, but we can replace it by Eq. (A2) inthe Appendix, namely,

k sin ncos g sin g sin g cos g x sin g sin g 0

Under the relation sin n 0, i.e., k f3 in Fig. 4, it is apparentthat sin g 60 implies sin g 0. In other words, when g 6 p32intersects the striction tangent f3 of the axodes at right angles, theDisteli axis g

coincides with the ISA. The same holds for themotion R4=R3, which means that under this condition the twotooth flanks share the instant Disteli axis. According to Theorem2, U2 and U3 are underG

2-contact at the common striction pointSg. In Fig.7, the thick lines, which are in contact at marked pointson the meshing lineg, are level lines of the two flanks, i.e., inter-sections with planes orthogonal to the ISA. The mean sectionshows the G2-contact at the striction point Sg, which causes theinterpenetration.

The case of osculating cylindrical or spherical tooth flanks is

misleading. In the true spatial version, there is noG

2

-contact at allother points ofg for one reason: According to Theorem 2, in this

case the condition _~d0 c _d0 must be satisfied. However,

because of the permanent line contact the flanks have the same

distribution parameter ~dt dt for each t2 J. This implies_~d0 _d0, but by Eqs. (34), (35) and (43), the constant c in

kU3 c kU2 is

c tanu a= tanu a u0 a0 :u0 a0 61

The different postures depicted in Fig.8reveal that there is alsoa mutual penetration of the conjugate tooth flanks U2 and U3 atthe other postures. Since the surfaces share this curve of

intersection as well as the tangent planes at all points of the mesh-ing line, there must be aG2-contact at the point where the curveof intersection meets the meshing line. This point is close to thestriction point; however, it must be different fromSg for the fol-lowing reason.

If a G2-contact takes place at the striction point, then byTheorem 3 the two gear-flanks must share the Disteli axis g

.

However, when we plug the pairg; gof lines into the left-handside of the EulerSavary equation, it cannot happen that under

sin n6 0 we obtain two different results on the right-hand side,either cot c2 cot c4 for R4=R2, Eq. (36), or cot c3 cot c4 forR4=R3.

This can also be confirmed by studying the coefficients of thefundamental forms for U2 and U3 at the striction point of themeshing lines g. With respect to the machine frame R1, the

Fig. 7 Two conjugate flanksU2 and U3 with G2-contact at the

common striction point Sg. The meshing line g is here parallelto the ISA and a cylindric generator of U2 and U3.

Fig. 8 Snapshots of the penetrating tooth flanks with theirstriction curves upon meshing




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meshing lines trace the helicoid P4. In the frame f1; f2; f3, wecan set up

g

cos s

sin s024 35

e

h41s sin s

h41s cos s2u01 cos s24 35

The corresponding dual anglesn and g according to Fig.4are

n p 2eu0tan s

2 and g s eh41s

As in the proof of Theorem 3, we have to compare the valueslisted in Eq.(28)forU2with those ofU3. Equations(34)and(35),

together with Eq.(43), show that kU2 and kU3 differ by the same

real factorc 61, as mentioned before. We find the same factorcin the second terms oflU2 and lU3 ; however, the first terms are

equal. So, the necessary condition LU3 0; 0 c2LU2 0; 0 for a

G2-contact at the striction point is not satisfied, except in thecase sin n0 0, which is only possible for u0 0. In this case,the helicoid P4 is not skew, and its axis p41 equals the striction

tangent f3 of the axodes. Then, all meshing lines g intersect thestriction tangent of the axodes at right angles.

6.3 A Spatial Analog of Octoidal Gears. In the plane aswell as on the sphere, thegeneralized Camus Theorem states that,for any curvec4attached to the auxiliary curvep4 R4, the enve-lopesc2and c3under motions R4=R2and R4=R3, respectively, areconjugate tooth profiles, both being conjugate to a virtual gearwith pitch curvep4and the tooth profile c4.

In the particular planar case depicted in Fig. 9, the auxiliarycurvep4is the pole tangenttand the attached curvec4is a line. Inall its postures, the linec4shows the same inclination with respectto the gear frame R1. At each pose, the enveloping point Mofc4

is the pedal point with respect to the pole I32. The right triangleenclosed by c4, p4, and the line I32M shows that the angle15 ac

between the meshing normal and the polodep4 is constant, whichleads to the case of involute gearing.

The foregoing statement does not hold in spherical geometrysince for spherical triangles, the sum of the interior angles is notconstant. This sum is always greater thanp, the amount by whichthe sum exceeds p being proportional to the area of the triangle.Therefore, we cannot conclude for the analog specification inbevel gears that the angleacbetween the meshing normal and thepolodes remains constant. Quite the contrary, the sum ofac andthe constant anglebcbetween the great circlesc4and p4is always

greater thanp=2 and increases with the spherical distance betweenI32and c4, as shown in Fig.10. We obtain what is known asoctoi-dal gears, as reported in Refs. [18] and [19].

Figure10shows octoidal bevel gears, the envelopes c2 and c3of the great circle c4 under the respective motions R4=R2 andR4=R3 are displayed at different postures c

12;; c

42 and c

13;; c

43.

The corresponding meshing points M1;;M4 trace the meshingline m. Under the motion R4=R1, which is the rotation about theaxis p41, the great circle c4 envelopes a pair of antipodal circleswith axis p41. The complete meshing linem is thepedal curve ofthese circles with respect to the relative pole I32. It consists of twoantipodal curves of octoid shape [18].

The spatial analog of the generalized Camus Theorem [6] leadsto: For any surface U4 attached to the AS P4, the envelopes U2and U3 under the respective relative motions R4=R2 and R4=R3are conjugate tooth flanks. We choose again P4 as the skeworthogonal helicoid and specify U4as a plane.

The envelope of a plane under any spatial motion is in general

a torse, i.e., a developable ruled surface. In each pose, the movingplane contacts its envelope along a line. Let the one-parametricset of planes be represented by the linear equation nt x dtwith twice-differentiable functions nt and d(t) for all t in a realintervalJ. Then, the corresponding line of contact with the enve-lope satisfies simultaneously the two equations below [11,20,21]

nt x dt_nt x _dt

(44)

The cuspidal point of this generator satisfies also the third equa-tionnt x dt.

Suppose the instant pose of the plane U4 R4 with respect tothe frame R2 satisfies the equationn2 x2 d2. Then, the motion

Fig. 10 Octoidal bevel gears: The conjugate profilesc2and c3are the envelopes of the great circle c4under the motions R4=R2and R4=R3

Fig. 9 Planar version of the generalized Camus theorem in theparticular case leading to involute gears

15Notation after [17], p. 274.




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R4=R2 with the twist q42 q42 e q420 x42f1, as given inEq.(14), yields the derivatives

_n2 q42 n2 and _d2 q420 n2 (45)

Hence, the instant generator of the envelope U4 is represented, byvirtue of Eq.(44), as the intersection of two perpendicular planes.Analog results hold for the envelope ofU4underR4=R3.

In this way, we obtain a pair of conjugate torses U2;U3 withpermanent line contact. In Fig. 11, one example is depicted thatindicates that these flanks should work correctly. Contrary tothe general case of Phillips involute gearing [22], contact isnot punctual, but along a line. The thick curves in Fig. 11 arethe intersections of the flanks with planes perpendicular to theinstant meshing line, which is depicted as magenta double line.

Figure 12 shows snapshots of the conjugate torses uponmeshing.16

7 Conclusions

Based on the Camus Theorem and on Martin Distelis work, weshowed in this paper that the flanks of spatial cycloid gears can besynthesized by means of an AS. In the first part, we studied indual-vector notation the curvature of the tooth flanks. Uponchoosing the skew orthogonal helicoid as AS, the tooth flanks ofthe spatial equivalent of octoidal bevel gears are obtained. Thefinal example with torses as tooth flanks looks promising but stillneeds a detailed analysis: Interference avoidance, singularity anal-ysis, pressure angle, and sliding velocity are still areas warrantingfurther research.

Acknowledgment

The third author acknowledges the support received from theNSERC, Canadas Natural Sciences and Engineering ResearchCouncil, through Discovery Grant No. RGPIN/4532-2008 andfrom the McGill University through a James McGillProfessorship.

Nomenclature

Rj=Rk motion of frame Rjwith respect to frame RkR1 fixed frame (machine frame)

R2; R3 frames attached to the two wheels

Spur and Bevel Gears

c2,c3 conjugate tooth profiles ofR3=R2

Iij instant center of rotation ofRi=RjI21,I31 instant centers of rotation of pinion and gear

with respect to R1p2,p3 pitch circles of the gears

p4 auxiliary curveR4 frame attached to the auxiliary curve

Spatial Case (Gears With Skew Axes)

e1; e2 DUVs representing thedirected bisectrixofI21and I31and its intersecting perpendicular,as indicated in Fig.1

g;n; t Frenet frame of a ruled surface Ug

instant Disteli axis ofU

Fig. 11 Skew gears with torses as conjugate tooth flanksU2;U3and permanent line contact

Fig. 12 Snapshots of the conjugate torses U2 and U3 uponmeshing x31 : x21522: 1

16Data: 2a 56deg; 2a0 50 mm, x31 : x21 2 : 3. The initial pose of theenveloping plane U4 R4 is defined with respect to the frame f1; f2; f3 in thefollowing way: Its normal vector has the geographical longitude 81:8 deg and thelatitude 11:9 deg, and U4 passes through the point0; 0; 47:3.




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hij xij0=xij instant pitch ofRi=RjIij ISA ofRi=Rj

I21; I31 ISAs of pinion and gear with respect to R1I41 ISA ofP4with respect to the machine frame

R1

pij unit vector parallel to the instant axis ofRi=Rj

pij0 moment of the axisIijwith respect to theorigin

pij pij epij0 DUV representingIij

qij xijpij twist ofRi=Rja a ea0 dual angle between p21and e1

b dual angle betweene1and p41c dual angle between the generatorg ofU and

the corresponding Disteli axisg cot c dual

curvature of the generatorg ofUd instant distribution parameter ofU

P2;P3 axodes of the relative motion R3=R2P4 AS, axode ofR4=R2 and R4=R3R4 frame attached to the AS P4u dual angle betweene1and the ISA p32

U2;U3 conjugate tooth flanks ofR3=R2xij signed instant angular velocity ofRi=Rj

xij0 signed instant translation velocity ofRi=Rjxij xij e xij0 instant dual velocity ofRi=Rj

x21,x31 angular velocities of pinion and gear,respectively2a angle between gear axesI21and I31

2a0 distance between gear axesI21and I31

Appendix: The Spatial EulerSavary Equation

The spatial version of the EulerSavary equation dates back toDisteli [10] and has since been discussed in the literature [8,23].The discussion goes along the lines of Ref. [8], where this equa-tion was first derived as the exact dual extension of the sphericalversion in Ref. [24].

First, we compute the Disteli axis g of the ruled surface U,

which is traced by the directed line g under the spatial motionR3=R2 with the twist q32 x32p32. To this end, we recall that by

virtue of the Frenet equations(16)

_g g kn _

kn kkg l t k2k t lg k

2xg

(A1)

Hence, the vector product of the first and the second derivative ofg is a dual multiple of the Disteli axis g

.Let P2 and P3 denote the fixed and the moving axode of the

relative motion R3=R2. At each instant, the Frenet frames of theaxodes P2 and P3 for the instant axis p32 are coincident,

thus defining a framef1; f2; f3 R1 with f1 p32, as shown inFig. 4. Therefore, the motion R3=R2 arises when the Frenetmotion R1=R2 along the fixed axode is superimposed with theinverse R3=R1 of the Frenet motion R1=R3 along the movingaxode U3. The twist ofR3=R2is obtained as

q32 q12 q13 x32f1; where

q1i li f1 kif3 for i 2; 3

Hence, q32 l2 l3 f1 and k2 k3, where we have recalledthat the dual curvature of the axode Piis given by cotci li=ki.

We follow the notation explained in Fig. 4 and use the dual

anglesn n en0 and g g eg0 for defining the relative posi-tion of the moving line g with respect to the Frenet frame

f1; f2; f3 of the axodes. Further, from Eq.(31)

g cos gf1 sin gh with h cos n f2 sin n f3

the derivative ofg for the motion R3=R2is derived below

_g q32 g x32 f1 g x32sin g f1 h

x32sin g cos n f3 sin n f2

which proves that the central normal ofU coincides with the com-mon perpendiculark to g and the ISAf1. Also, the Disteli axis g

intersects k at right angles, as shown in Fig. 4. Now the secondderivative follows:

g _x32 f1 g x32 _f1 g x32f1 _g

where the motion of f1 with respect to R2 along the fixed axodeP2 leads to

_f1 k2f2

Hence,

g _x32 f1 g x32k2 cos g f3 sin n sin gf1

x232sin g cos nf2 sin nf3

Therefore,

_g g x232sin g cos nf3 sin n f2

k2cos gf3 sin nsin g f1

x32sin g cos nf2 sin n f3

x232sin gk2sin ncos g x32sin g f1k2sin nsin gh

By virtue of Eq. (A1), this product equals a dual multiple of theDisteli axisg

, which can be expressed as (see Fig.4)

g cos gf1 sin g

h

Hence, if x32sin g60, we obtain, upon comparison ofcoefficients

k2sin ncos gsin g sin gcos g x32sin gsin g

0 (A2)

Under the additional condition k2sin g 6 0, we may divide the

equation above by k2sin gsin g. Moreover, we can replace

x32= k2 by l2 l3= k2 and express this, by virtue of k2 k3,in terms of the dual curvature of the axodes, thereby obtaining thedesired result

cot g cot g sin nx32

k2cot c 2 cot c 3

which is the spatial EulerSavary equation.

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http://dx.doi.org/10.1007/978-3-319-01851-5http://dx.doi.org/10.1007/978-3-319-01851-5http://dx.doi.org/10.1016/S0094-114X(99)00077-4http://dx.doi.org/10.1115/1.4006654http://dx.doi.org/10.1115/1.4006654http://dx.doi.org/10.1115/DETC2013-13401http://dx.doi.org/10.1115/1.4029287http://dx.doi.org/10.1017/CBO9780511547126http://dx.doi.org/10.1115/1.1900147http://dx.doi.org/10.1002/9781118255056http://dx.doi.org/10.1002/9781118255056http://dx.doi.org/10.1115/1.3267394http://dx.doi.org/10.1115/1.3267394http://dx.doi.org/10.1115/1.4029185http://dx.doi.org/10.1115/1.4029185http://dx.doi.org/10.1007/978-3-662-05302-7http://dx.doi.org/10.1115/1.2735339http://dx.doi.org/10.1115/1.2735339http://dx.doi.org/10.1115/1.2735339http://dx.doi.org/10.1007/978-3-662-05302-7http://dx.doi.org/10.1115/1.4029185http://dx.doi.org/10.1115/1.4029185http://dx.doi.org/10.1115/1.3267394http://dx.doi.org/10.1002/9781118255056http://dx.doi.org/10.1002/9781118255056http://dx.doi.org/10.1115/1.1900147http://dx.doi.org/10.1017/CBO9780511547126http://dx.doi.org/10.1115/1.4029287http://dx.doi.org/10.1115/DETC2013-13401http://dx.doi.org/10.1115/1.4006654http://dx.doi.org/10.1115/1.4006654http://dx.doi.org/10.1016/S0094-114X(99)00077-4http://dx.doi.org/10.1007/978-3-319-01851-5http://dx.doi.org/10.1007/978-3-319-01851-5

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