the computational fundamentals of spatial cycloidal gearing

The Computational Fundamentals

of Spatial Cycloidal Gearing

Giorgio Figliolini, Universita degli Studi di CassinoHellmuth Stachel, Vienna University of Technology

Jorge Angeles, McGill University, Montreal

Computational Kinematics 2009, May 6–9, Duisburg/Germany

Table of contents

1. Introduction

2. Reuleaux’s principle of gearing in the plane

3. Reuleaux’s principle of gearing in 3-space

4. Consequences for skew gearing

5. Conclusion

Computational Kinematics 2009, May 6–9, Duisburg/Germany 1

1. Introduction

Wanted: A gear set in order to transmit the rotary motion from wheel Σ2 aboutaxis a21 to the second wheel Σ3 with axis a31 such that the ratio of angular

velocities ω31/ω21 is constant.

spur gears bevel gearsskew gears

(e.g., worm gears)

a21, a31 parallel a21, a31 crossing a21, a31 skew



I41

I∗41

I31

I21

C

I32

P3

P2

C3C2

E

c2

p2

p4

p3

t

n

c3

c∗2

c∗3

p∗3

Σ2

Σ3

Σ4

ω21

ω31

ω41

Principle of gearing by? . . . Ch.E.L. Camus (1733),F. Reuleaux (1875)

Let a smooth auxiliary curvep4 roll on the polodes p2, p3.Then any point C attachedto p4 traces conjugate toothprofiles c2, c3.

At cycloidal gears (see left) p4

is a circle and C ∈ p4.



I41

I∗41

I31

I21

C

I32

P3

P2

C3C2

E

c2

p2

p4

p3

t

n

c3

c∗2

c∗3

p∗3

Σ2

Σ3

Σ4

ω21

ω31

ω41

Adaption for 3D:

The auxiliary curve p4 definesa system Σ4 which can movesuch that for the relative polesholds: I42 = I43 = I23.


Comments on Reuleaux’s principle in the plane

Almost all conjugate tooth profiles can be generated by Reuleaux’s principle.E.g., for involute gearing the auxiliary curve is p4 ⊂ Σ4 is a logarithmic spiral.

Generalization:

Instead of a point C ∈ Σ4 a smooth curve c4 ⊂ Σ4 can be given. Then theenvelopes c2, c3 of c4 under Σ4/Σ2 and Σ4/Σ3, resp., give conjugate tooth flanks.

Spherical kinematics:

Reuleaux’s principle is also valid in the spherical case, i.e., for bevel gears.



Given: Rotations Σ2/Σ1, Σ3/Σ1 about fixed skew axes a21, a31 with angularvelocites ω21, ω31.

Question: Is there a frame Σ4 moving such that the instantaneous screw motionsof Σ4/Σ2, Σ4/Σ3 and Σ3/Σ2 are equal, i.e., equal axis and screw parameter, butdifferent velocities.

We find the answer by means of dual vectors.


Dual vectors in spatial kinematics

bijection: Spear g 7→ dual unit vector g = g + εg0

under ε2 = 0,

g · g = g · g + 2g · g0 = 1 + ε 0 = 1.



bijection: Spear g 7→ dual unit vector g = g + εg0

under ε2 = 0,

g · g = g · g + 2g · g0 = 1 + ε 0 = 1.

ϕ = ϕ + εϕ0 is called dual angle.

cos ϕ = g · h = g ·h + ε(g0·h + g ·h0),

sin ϕ n = g × h =

= g×h + ε[(g0×h) + (g×h0)].g

h

x

y

n

ϕ

ϕ0



Bijection: Instantaneous motion of Σi/Σj 7→ twist qij = ωij pij

with pij as spear of the instantaneous axis and ωij = ωij + εωij0 with

ωij and ωij0 as instantaneous angular- and translatory velocity, respectively.

Aronhold-Kennedy Theorem:

Given: Σ1,Σ2, Σ3 and instantaneous twists q21, q31 of Σ2/Σ1, Σ3/Σ1, resp.,

=⇒ q32 = q31 − q21 is the twist of the relative motion Σ3/Σ2.


Plucker’s conoid

Let Σ2 and Σ3 rotate about fixed skew axes p21, p31 with variable angularvelocities ω21, ω31, respectively.

p21p21p21p21p21p21p21p21p21

p21p

21p21p21p21p

21p21p21

p31p31p31p31p31p31p31p31p31

p31p

31p31p31p31p

31p31p31

ΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨ

Then in any case the twist

ω32 p32 = ω31 p31 − ω21 p21

is a real linear combination ofp21 and p31 =⇒

p32 is located on Plucker’sconoid Ψ (= cylindroid).

The position of p32 on Ψdefines the pitch h32 =ω320/ω32 uniquely.



Given: Rotations Σ2/Σ1, Σ3/Σ1 about fixed skew axes p21, p31 with angularvelocites ω21, ω31.

Question: Is there a frame Σ4 moving such that the instantaneous screw motionsof Σ4/Σ2, Σ4/Σ3 and Σ3/Σ2 are equal, i.e., equal axis and screw parameter, butdifferent velocities.

q42 = λ2q32, q43 = λ3q32, λ1, λ2 ∈ R, =⇒ q41 − qi1 = λi(q31 − q21),

q41 = λ1q31 + (1 − λ1)q21 = (1 + λ2)q31 − λ2q21

Necessary and sufficient: The instantaneous axes p41 of Σ4/Σ1 are located onPlucker’s conoid Ψ and the pitch h41 is corresponding to this position.


A tribute to Martin Disteli

M. Disteli: Uber die Verzahnung der

Hyperboloidrader mit geradlinigem Eingriff.

Z. Math. Phys. 59, 244–298 (1911)

Martin Disteli, (1862–1923)Dresden, Karlsruhe



In analogy to cycloid gearing in the plane, we specify the axis p41

of Σ4/Σ1 onPlucker’s conoid and keep it fixed in the machine frame Σ1.

Then we move simultaneously Σ2 with the twist q21, Σ3 with twist q31 and Σ4

with twist q41 (axis p41 and pitch h41) such that the relative axes p42, p43 andp32 coincide.

Main Theorem: For each line g attached to Σ4 the ruled surfaces Φ2, Φ3 tracedby g under Σ4/Σ2 and Σ4/Σ3, resp., are conjugate tooth flanks.

The contact takes place at of points of g.



Proof: The distribution of tangent planes of Φ2 and Φ3 along the line g is definedby the derivatives ˙g, i.e., by

Σ4/Σ2 : ˙g = g × q42, Σ4/Σ3 : ˙g = g × q43.

Since q42 and q43 are real multiples of q32, the same holds for ˙g. They have thesame distribution of tangent planes.

The axodes of Σ3/Σ2 are one-sheet hyperboloids of revolution Π2, Π3.Under Σ4/Σ2 and Σ4/Σ3 a ruled helical surface Π4 (auxiliary surface) is slidingand rolling (German: schroten) along the hyperboloids.

In analogy to planar are spherical cycloidal gearing we start with the generatorg = p32 = ISA.



αα

α0

α0

ϕ

β

ϕ0

β0

O

x1

x2

x3

bp32

bp21

bp31

bp41

There are 4 axes involved — all locatedon Plucker’s conoid

• Axes p21, p31 of the wheels,

• the ISA p32, and

• the axis p41 of the auxiliary systemΣ4.



bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31




Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2


Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3 planar case









spherical case









Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3


skew case



g

p21

p31

p41

p32

ω21

ω31

Φ2

Φ3

g

p21

p31

p41

p32

ω21

ω31

Φ2

Φ3

conjugatetooth flanks


g

p21

p31

p32

ω21

ω31

Φ2

Φ3

These are conjugate cycloidal tooth flanksΦ2, Φ3.

The displayed red and blue curvesare the intersections with planesorthogonal to the line g of contact.

In this particular case g is the ISA andtherefore a singular line of the ruled surface— like a cusp in the plane.

Each tooth flank is composed from twoparts glued together along g. In can beproved that such pairs of surfaces have acommon tangent plane at each point of g.



Theorem: For each surface Φ4 attached to Σ4 the envelopes Φ2,Φ3 of underΣ4/Σ2 and Σ4/Σ3, resp., are conjugate tooth flanks.

Proof: For any pose of Φ4, a point C is a point of contact between Φ4 and itsenvelope Φi under Σ4/Σi, i = 2, 3, if and only if the surface normal n of Φ4 atC belongs to the linear line complex

q4i0·n + q

4i·n0 = 0.

Twists differing by a real factor define the same linear line complex.


5. Conclusions

The main goal of this paper is to shed light on the geometry of the tooth flanksof gears with skew axes.

To this end, the authors follow and extend results reported by Martin Disteli,thereby deriving ruled surfaces as conjugate tooth flanks in contact along a line.

A detailed analysis of the tooth flanks obtained for constant and for variable axisp41 ∈ Ψ is left for future research.

Conjecture: Any pair of conjugate ruled tooth flanks with contact along lines canbe obtained by this principle.


References

• J. Angeles: The Application of Dual Algebra to Kinematic Analysis.In J. Angeles, E. Zakhariev (eds.): Computational Methods in

Mechanical Systems, Springer-Verlag, Heidelberg 1998, Vol. 161, pp. 3-31.

• G. Figliolini, H. Stachel, J. Angeles: A new look at the Ball-Disteli

diagram and its relevance to spatial gearing. Mech. Mach. Theory 42/10,1362–1375 (2007).

• C. Huang, P.-C. Liu, S.-C. Hung: Tooth contact analysis of the

general spatial involute gearing. Proceedings 12th IFToMM World Congress,Besancon/France, 2007, paper no. 836.

• M. Husty, A. Karger, H. Sachs, W. Steinhilper:Kinematik und Robotik. Springer- Verlag, Berlin Heidelberg 1997.


• F.L. Litvin, A. Fuentes: Gear Geometry and Applied Theory. 2nd ed.,Cambridge University Press 2004.

• J. Phillips: General Spatial Involute Gearing. Springer Verlag, New York2003.

• H. Pottmann, J. Wallner: Computational Line Geometry. SpringerVerlag, Berlin, Heidelberg 2001.

• H. Stachel: Instantaneous spatial kinematics and the invariants of the

axodes. Proc. Ball 2000 Symposium, Cambridge 2000, no. 23.

• H. Stachel: On Jack Phillips’ Spatial Involute Gearing. Proc. 11th ICGG,Guangzhou/P.R.China, 2004, pp. 43–48.

• H. Stachel: Teaching Spatial Kinematics for Mechanical Engineering

Students. Proc. 5th Aplimat, Bratislava 2006, Part I, pp. 201–209.


• G.R. Veldkamp: On the Use of Dual Numbers, Vectors, and matrices in

Instantaneous Spatial Kinematics. Mech. and Mach. Theory 11, 141–156(1976).

• W. Wunderlich: Ebene Kinematik. Bibliographisches Institut, Mannheim1970.


the computational fundamentals of spatial cycloidal gearing

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