the computational fundamentals of spatial cycloidal gearing
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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
Computational Kinematics 2009, May 6–9, Duisburg/Germany 2
2. Reuleaux’s principle of gearing in the plane
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 3
2. Reuleaux’s principle of gearing in the plane
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 4
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 5
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 5
3. Reuleaux’s principle of gearing in 3-space
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 6
3. Reuleaux’s principle of gearing in 3-space
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 6
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 6
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.
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
Computational Kinematics 2009, May 6–9, Duisburg/Germany 7
Dual vectors in spatial kinematics
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 8
Dual vectors in spatial kinematics
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 8
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 9
3. Reuleaux’s principle of gearing in 3-space
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 10
3. Reuleaux’s principle of gearing in 3-space
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 10
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
Computational Kinematics 2009, May 6–9, Duisburg/Germany 11
Computational Kinematics 2009, May 6–9, Duisburg/Germany 12
Computational Kinematics 2009, May 6–9, Duisburg/Germany 13
Computational Kinematics 2009, May 6–9, Duisburg/Germany 14
4. Consequences for skew gearing
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 15
4. Consequences for skew gearing
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 15
4. Consequences for skew gearing
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 16
4. Consequences for skew gearing
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 16
4. Consequences for skew gearing
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 17
4. Consequences for skew gearing
bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31
bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21
bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32
ΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨ
Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2
Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3
Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3 planar case
Computational Kinematics 2009, May 6–9, Duisburg/Germany 18
4. Consequences for skew gearing
bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31
bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21
bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32
ΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨ
Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2
Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3
spherical case
Computational Kinematics 2009, May 6–9, Duisburg/Germany 19
4. Consequences for skew gearing
bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31bp31
bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21bp21
bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32bp32
ΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨ
Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2Π2
Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3Π3
Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3Φ3
Π4Π4Π4Π4Π4Π4Π4Π4Π4Π4Π4Π4Π4Π4Π4Π4Π4
skew case
Computational Kinematics 2009, May 6–9, Duisburg/Germany 20
4. Consequences for skew gearing
g
p21
p31
p41
p32
ω21
ω31
Φ2
Φ3
g
p21
p31
p41
p32
ω21
ω31
Φ2
Φ3
conjugatetooth flanks
Computational Kinematics 2009, May 6–9, Duisburg/Germany 21
4. Consequences for skew gearing
Computational Kinematics 2009, May 6–9, Duisburg/Germany 22
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 23
4. Consequences for skew gearing
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 24
4. Consequences for skew gearing
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 24
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 25
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.
Computational Kinematics 2009, May 6–9, Duisburg/Germany 25
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diagram and its relevance to spatial gearing. Mech. Mach. Theory 42/10,1362–1375 (2007).
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general spatial involute gearing. Proceedings 12th IFToMM World Congress,Besancon/France, 2007, paper no. 836.
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