A STUDY ON SPATIAL CYCLOID GEARING

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2 16 TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS 214 ISGG 4 8 AUGUST, 214, INNSBRUCK, AUSTRIA A STUDY ON SPATIAL CYCLOID GEARING Giorgio FIGLIOLINI 1, Hellmuth STACHEL 2, and Jorge ANGELES 3 1 University of Cassino & Southern Lazio, Italy 2 Vienna University of Technology, Austria 3 McGill University, Montréal, Canada ABSTRACT: Understanding the geometry of gears with skew axes is a complex task, hard to grasp and to visualize. However, due to Study s Principle of Transference, the geometric treatment based on dual vectors can be readily derived from that of the spherical case. This paper is based on Martin Disteli s work and on the authors previous results where Camus concept of an auxiliary curve is extended to the case of skew gears. We focus on the spatial analogue of the following case of cycloid bevel gears: When the auxiliary curve is specified as a pole tangent, we obtain pathologic spherical involute gears with vanishing pressure angle. The profiles are always penetrating at the meshing point because of G 2 -contact. In view of the Camus Theorem, the spatial analogue of the pole tangent is a skew orthogonal helicoid Π 4 as auxiliary surface. Its axis lies on the cylindroid and is normal to the instant screw axis (ISA). Under the roll-sliding of Π 4 along the axodes Π 2 and Π 3 of the gears, any generator g of Π 4 traces a pair of conjugate flanks Φ 2,Φ 3 with permanent line contact. Again, these flanks are not realizable because of the reasons below: (1) When g coincides with the ISA, the singular lines of the two flanks come together. At each point of g the two flanks share the tangent plane, but in the case of external gears the surfaces open toward opposite sides. (2) We face the spatial analogue of a spherical G 2 -contact, which surprisingly does not mean a G 2 - contact at all points of g but only at a single point combined with a mutual penetration of the flanks Φ 2 and Φ 3. However, when instead of a line g aplaneφ 4 is attached to the right helicoid Π 4, the envelopes of Φ 4 under the roll-sliding of Π 4 along Π 2 and Π 3 are torses that serve as conjugate tooth flanks Φ 2,Φ 3 with a permanent line contact. So far, it seems that these flanks, Φ 2 and Φ 3, are geometrically feasible. This is a possible spatial generalization of octoidal gears or even of planar involute gears. Keywords: Gears with skew axes, Cycloidal gearing, Involute gearing, Cylindroid, Camus Theorem 1. INTRODUCTION Let the motions of two gears Σ 2, Σ 3, against the gear box Σ 1 be given, i.e., the rotations Σ 2 /Σ 1, Σ 3 /Σ 1 about fixed skew axes p 21 and p 31 with angular velocities, ω 31, respectively. The dual unit vectors representing the axes p 21 and p 31 are denoted by p 21 and p 31, respectively. We use a Cartesian coordinate frame F (O;x 1,x 2,x 3 ) with ê 1,ê 2 denoting the dual unit vectors of the x 1 -andx 2 -axis. The given axes p 21 and p 31 of the wheels are assumed to be symmetrically placed with respect to the x 1 -axis such that the x 3 -axis is the common normal of the gear axes. Using the dual angle α = α + εα between the x 1 -axis and p 21, we can set (see Fig. 1) p 21 = cos α ê 1 sin α ê 2, p 31 = cos α ê 1 + sin α ê 2. We limit ourselves to the skew case and assume (1) < α < π/2 andα (2) Paper #

3 though most of the arguments hold also in the spherical case, α =, and in the planar case, α =, with parallel axes. α α S O ê 3 = f 2 β α α ϕ ê 2 f3 f1 = p 32 p 41 p 21 ϕ β = ϕ ω 31 ê 1 p 31 ISA Figure 1: Skew axes p 21, p 31 of the gear wheels, the ISA p 32 and the axis p 41 of the auxiliary surface Π 4 Σ 4 in the particular case β = ϕ +π/2. The oriented lines f 1, f 2, f 3 form the Frenet frame of the axodes. This frame remains fixed in the gear frame Σ 1. In addition, let ϕ denote the dual angle between ê 1 and the ISA, i.e., the relative axis p 32. Then we obtain p 32 = cos ϕ ê 1 + sin ϕ ê 2 and (3) ω 32 p 32 = ω 31 p 31 p 21. The comparison of coefficients and [4, Eq. (7)] lead to tanϕ = ω 31 + tanα and ω 31 ϕ = Rsin2ϕ with R = α (4) sin2α. The vector product of both sides of the last equation in (3) with p 21 and p 31 (compare [6, Eq. (12)]) results in sin( ϕ α) = ω 31 sin( ϕ + α) = ω 32 sin2 α, (5) which sometimes is called the dual Sine- Theorem applied to the dual triangle p 21 and ω 31 p 31 and ω 32 p 32. This implies ω 32 = sin2 α (6) sin( ϕ α) and, consequently, for the pitch of the relative motion Σ 3 /Σ 2 like [4, Eq. (15)] h 32 = ω 32 = R(cos2α cos2ϕ) ω 32 (7) = 2R(cos 2 α cos 2 ϕ). The axodes of the relative motion Σ 3 /Σ 2 are one-sheet hyperboloids Π 3 Σ 3 and Π 2 Σ 2, swept by the relative axis p 32 under the inverse rotations Σ 1 /Σ 2 and Σ 1 /Σ 3 about p 21 and p 31, respectively. 2. THE SPATIAL CAMUS THEOREM The following lemma was first published by Disteli (see [6, Theorem 2] and the references therein). Lemma 1. For given wheels Σ 2, Σ 3 there exists a frame Σ 4 such that the screws of Σ 4 /Σ 2, Σ 4 /Σ 3 and Σ 3 /Σ 2 are equal at every instant if and only if the instant axis p 41 of Σ 4 /Σ 1 is located on the Plücker conoid Ψ, but different from p 32. Let β be the dual angle between between the x 1 -axis and p 41 (Fig. 1). Then we can write p 41 = cos β ê 1 + sin β ê 2. (8) If we specify p 41 Ψ different from p 21, p 31, p 32, then ϕ ±α,β. From the equation [6, Eq. (8)], which defines the Plücker conoid, we obtain β = Rsin2β. (9) The dual Sine-Theorem applied to the triangle p 21, ω 41 p 41 and ω 42 p 42 = ω 42 p 32 gives (compare [6, Eq. (12)]) sin( ϕ β) = ω 41 sin( ϕ + α) = ω 42 sin( α + β) (1)

4 and, analogously, for ω 31 p 31, ω 41 p 41 and ω 43 p 43 = ω 43 p 32 ω 31 sin( ϕ β) = ω 41 sin( ϕ α) = ω 43 sin( β α). (11) The instant pitch h 41 = ω 41 /ω 41 is defined by [6, Eq. (9)] as h 41 = ω 41 = R(cos2α cos2β). (12) ω 41 Let Π 4 be the ruled helical surface 1 traced by the relative axis p 32 under the helical motion Σ 1 /Σ 4 about p 41 with pitch h 41. We call Π 4 the auxiliary surface (for further details see [5]). It forms together with Π 2 and Π 3 the axodes of the relative motions of Σ 4 against Σ 2 and Σ 3, i.e., the motions Σ 4 /Σ 2 and Σ 4 /Σ 3 are defined by the rolling and sliding of Π 4 along the hyperboloids Π 2 and Π 3, respectively. The importance of the auxiliary surface Π 4 Σ 4 lies in [6, Theorem 3] which we recall as below: Theorem 2. [Spatial Camus Theorem] For any line g attached to Σ 4, the surfaces Φ 2, Φ 3 traced by g under the relative motions Σ 4 /Σ 2 and Σ 4 /Σ 3, respectively, are conjugate tooth flanks of Σ 3 /Σ 2. At any instant, the meshing points for these flanks are located on a straight line. With respect to the gear frame Σ 1,thelocus of the meshing lines, i.e., the meshing surface or surface of action, is traced by g under Σ 4 /Σ 1 with the fixed twist q 41 = ω 41 p 41. Consequently, it is a helical surface with axis p THE DISTELI AXES OF A RULED SUR- FACE Along each non-torsal generator g of a ruled surface afrenet frame can be defined, consisting of: g itself; the central normal n, which is the surface normal at the striction point; and the central tangent t (see, e.g., [1, 2]). This triplet of mutually orthogonal axes meets at the striction point S of g, defined on the striction curve (Fig. 2). The central tangent is orthogonal to the asymptotic plane and tangent to the surface at the striction point. ĝ n S t Figure 2: Frenet frame (ĝ, n, t) and striction curve of a ruled surface. Let, in dual-vector notation 2, the ruled surface be given by the twice-differentiable dual vector function ĝ(t), t I. Then, the derivatives of the Frenet frame (ĝ, n, t) satisfy the Frenet equations Eq. (1) of [1] namely, ĝ = λ n = q ĝ n = λ ĝ + μ t = q n t = μ n. = q t with q = μ ĝ + λ t = ω ĝ, (13) ĝ with ĝ ĝ = 1beingtheDisteli axis and ω 2 = λ 2 + μ 2, provided λ. By the last condition we exclude stationary (= singular) generators. The Frenet equations (13) contain two dual coefficients, λ = λ + ελ and μ = μ + εμ. Various formulas expressing invariants of the ruled 1 In this paper the term ruled surface stands for a twice continuously differentiable one-parameter set of oriented lines From now on we identify oriented lines with their dual unit vector with respect to any well-defined coordinate frame. In this sense we speak of the line ĝ.

5 surface in terms of λ, λ, μ,andμ can be found in [1, Theorems 1 3]. 3 Here we adopt a different approach. The dual representation ĝ(t) =g(t)+εg (t), t I, of the ruled surface gives rise to a real parametrization, namely x(t,u)=[g(t) g (t)] + ug(t), (14) (t,u) I R. Here we recall that g g is the position vector of the pedal point of the generator ĝ with respect to the origin of the underlying coordinate frame. The derivatives d dt ĝ = ĝ = ġ + εg = λ n = λn + ε (λ n + λn ), d 2 dt 2 ĝ = ĝ = g + εg = λ 2 ĝ + λ n + λ μ t t = λ 2 g + λn + λμt + ε ( 2λλ g λ 2 g + λ n + λn ) x(t, u) ĝ + λ μt + λμ t + λμt (15) determine the partial derivatives of the parametrization x(t,u): S n x t =(ġ g )+(g g )+uġ, x u = g and x tt =( g g )+2(ġ g )+(g g )+u g, x tu = ġ = λn, x uu =. Figure 3: The distribution parameter δ defines We study the derivatives at the points of a single the tangent planes T x along the generator ĝ by generator, say, at t =. For this purpose we use the triplet ( ĝ(), n(), t() ) tanψ = u/δ. The angle σ between ĝ and the as the new coordinate frame; now the striction point s() of ĝ() striction. striction curve is called the striction angle or the is the origin of the frame in question. Thus we may set The vector product b = x t x u is a normal g()= 1, n()= 1, t()= vector of the ruled surface, provided the surface, point is regular, which means b. The coordinates 1 g ()=n ()=t ()=. b(,u)= λ (18) This yields λu ĝ()= λ + ε λ, 3 For example: The dual part q of the twist q equals the instant velocity vector of the origin s. Consequently, for the striction σ (see Fig. 3) we get tanσ = λ/μ ĝ()= λ 2 λ λμ + ε 2λλ λ λ μ + λμ and therefore 1 x t (,u)= λu, x u (,u)=, (16) λ λ 2 u x tt (,u)= λ μ λμ + λu, λ + λμu (17) x tu (,u)= λ, x uu (,u)=. T x ψ u reveal that at generators with λλ the angle ψ between the central normal vector b(,) = λn and the normal vector b(,u) (see Fig. 3) satisfies the equation σ tanψ = λu λ = u δ with δ = λ λ. (19) T s

6 The quotient δ is called the distribution parameter. This is a geometric invariant, i.e., invariant against parameter transformations. Generators with λ = and hence δ = are called torsal : Here all points with u have the same tangent plane; the striction point (u = ) is singular because of b(, )=. Cylindrical generators are defined by ġ = or λ =. Here, all points are possible striction points, for which we set δ :=. 4. TWO RULED SURFACES WITH LINE CONTACT For our study on cycloid gearing we need some results concerning the Disteli axes ĝ of a ruled surface. According to (13), q = ω ĝ is the twist and therefore ĝ the instant screw axis of the moving Frenet frame. From Eqs. (15) and (13) follows the relation below: ĝ ĝ = λ n ( λ 2 ĝ + λ n + λ μ t) = λ (2) 2 ω ĝ. Due to [1, Theorem 3, 3], the dual angle γ = γ + εγ between the generator ĝ and the corresponding Disteli axis ĝ satisfies cot γ = μ λ, hence cotγ = μ λ and γ = λμ (21) λ μ λ 2 + μ. 2 This is a consequence of the two standard products ĝ ĝ = cos γ = μ ω, ĝ ĝ = sin γ n = λ ω n, and of the rule that the dual extension of an analytic real function f (t) is defined as f ( t )= f (t + εt )= f (t)+εt ḟ (t), which yields cot γ = cotγ + εγ (1 + cot 2 γ). The dual angle between the moving ĝ(t) and the fixed ĝ () is stationary of order 2 at t = (see [1, Theorem 3, 4]). Due to the spherical analogy, cot γ can be called the dual (geodesic) curvature of the ruled surface. 5 Lemma 3. If two ruled surfaces are in contact at all points of a common generator and if they share the corresponding Frenet frame and the Disteli axis, then their dual coefficients in the Frenet equations differ at the corresponding parameter values only by a real factor c. The proof is straightforward and left for the reader. Theorem 4. Let ĝ(t) and ĝ( t) be two twicedifferentiable ruled surfaces which at t = t = share the Frenet frame, the distribution parameter δ() = δ() and the Disteli axis. Then, the surfaces have a G 2 -contact at the striction point of the common generator. If by Lemma 3 λ()=c λ() and μ()=c μ(), then there is a G 2 -contact at all points of ĝ()= ĝ() if and only if δ()=c δ(). Proof: The dual vector function ĝ(t) determines the real parametrization x(t, u) of the ruled surface as presented in (14). The partial derivatives at t =, as given in (16), define the coefficients of the first fundamental form as E(,u)=x t x t = λ 2 u 2 + λ 2, F(,u)=x t x u =, (22) G(,u)=x u x u = 1. For the coefficients of the second fundamental form we obtain L = 1 b b x 1 [ tt = λ 2 + λ 2 u 2 λ (λ μ + λμ )+( λλ λλ )u λ 2 μ u ], 2 M = 1 b b x 1 tu = λλ, λ 2 + λ 2 u N = 1 b b x uu =. (23) For the sake of brevity we skip the detailed analysis, which reveals that at any point x(,u) on the common generator t = the equations Ẽ = c 2 E, L = c 2 L,and M = cm characterize the G 2 - contact between the two surfaces.

7 5. THE CURVATURE OF THE RULED TOOTH FLANKS In the realm of gearing, we need two different Frenet frames, the frame ( f 1, f 2, f 3 ) for the axodes with the ISA f 1 (see Fig. 1) and the frame (ĝ 1,ĝ 2,ĝ 3 ) for conjuate tooth flanks with ĝ 1 as the meshing line (see Fig. 4). 5.1 The Frenet Frame of the Axodes Upon gear meshing, the Frenet frame ( f 1, f 2, f 3 ) of the axodes with f 1 = p 32 remains fixed in the gear frame Σ 1. The second axis f 2 equals the spear ê 3 along the common perpendicular of the gear axes p 21 and p 31. In terms of the basis (ê 1,ê 2,ê 3 ) we obtain from (3) (see Fig. 1) f1 cos ϕ sin ϕ f2 = f3 1 sin ϕ cos ϕ ê 1 ê 2 ê 3. (24) The origin of this Frenet frame is the striction point S =(,,ϕ ) of the axodes, the point of intersection between the ISA p 32 and the common normal of p 21 and p 31. The movement of this frame along the axode Π 2 Σ 2 is the rotation Σ 1 /Σ 2 about the axis p 21 with the angular velocity. Therefore p 21 = cos( ϕ + α) f 1 + sin( ϕ + α) f 3 is the permanent Disteli axis of Π 2. Due to (1), the corresponding Frenet equations (note ê 3 = f2 ) begin with f 1 = p 21 f 1 = sin( ϕ + α) f 2 = [sin(ϕ + α)+ε(ϕ + α )cos(ϕ + α)] f 2, which implies for the axode Π 2 the distribution parameter 4 δ2 =(ϕ + α)cot(ϕ + α) and the coefficients λ2 = sin( ϕ + α), μ 2 = cos( ϕ + α). 4 For the generators of a one-sheet hyperboloid of revolution with semiaxes a,b the absolute value of the distribution parameter equals the secondary semiaxis, i.e., δ = b The last equation follows from the third Frenet equation f3 = p 21 f 3 in (13), and it confirms for the dual angle γ 2 between the generator p 32 = f 1 and the Disteli axis p 21 by (21) γ 2 = ϕ + α with cot γ 2 = μ 2 / λ 2 as dual curvature of Π 2 according to [1, Theorem 3]. In a similar way we obtain for Π 3 the distribution parameter and the coefficients δ 3 =(ϕ α )cot(ϕ α) λ3 = ω 31 sin( ϕ α), μ 3 = ω 31 cos( ϕ α). The equation δ 2 = δ 3, which can also be concluded from (5), guarantees the contact between Π 2 and Π 3 at all points of the ISA p 32. In the Frenet equations of the auxiliary surface Π 4 Σ 4 with axis p 41 = cos( ϕ β) f 1 + sin( ϕ β) f 3 and dual velocity ω 41 we obtain the coefficients λ4 = ω 41 sin( ϕ β), μ 4 = ω 41 cos( ϕ β). (25) As a consequence, Π 4 has, by virtue of (19), the distribution parameter δ 4 = h 41 +(ϕ β )cot(ϕ β). The equation δ 4 = δ 3 = δ 2 can be verified using Eqs. (4), (9), and (12). The axis of Π 4 makes, with all generators Π 4, the dual angle γ 4 = ϕ β. 5.2 The Frenet Frame of the Tooth Flanks According to Theorem 2, any line ĝ attached to the auxiliary surface Π 4 traces conjugate tooth flanks Φ 2 and Φ 3 under the respective relative motions Σ 4 /Σ 2 and Σ 4 /Σ 3 with the auxiliary surface Π 4 roll-sliding on the axodes Π 2 and Π 3, respectively. The motion Σ 4 /Σ 2 is the composition of Σ 4 /Σ 1 with the Frenet motion Σ 1 /Σ 2 along Π 2.

8 We can set up the moving line ĝ by ĝ = cos η f 1 + sin η cos ξ f 2 + sin η sin ξ f 3. (26) This follows because the common perpendicular k between ĝ and the ISA f 1 (see Fig. 4) can be written as k = sin ξ f 2 + cos ξ f 3. The dual angles ξ and π/2 η can be seen as dual geographical longitude and latitude, respectively. f1 = p 32 ĝ 1 ĝ ĝ Φ 3 has the inital pose ĝ 1 f1 ĝ 2 = M f2 with ĝ 3 f3 cos η sin η cos ξ sin η sin ξ M = sin ξ cos ξ. sin η cos η cos ξ cos η sin ξ (27) From ĝ = ω 42 f1 ĝ follows by differentiation because of ω 42 = const. the relation below: ĝ = ω 42 [( f1 ĝ)+( f 1 ĝ) ]. ĝ 3 η η During the motion Σ 4 /Σ 2 the ISA f 1 traces Π 2 with angular velocity. Therefore η S g k=ĝ 2 f 1 = p 21 f 1 = sin( ϕ + α) f 2 and hence, ξ η ĥ ĝ = ω 42 [ ω21 sin( ϕ + α)( f 2 ĝ) + f 1 ω 42 ( f 1 ĝ) ] S ξ ξ f2 f3 [ = ω 42 ω21 sin( ϕ + α)( f 2 ĝ) ( + ω 42 ( f1 ĝ) f 1 ( f 1 f 1 )ĝ )]. By (27), we can express the first and second derivatives of ĝ in the Frenet frame (ĝ 1,ĝ 2,ĝ 3 ) as ĝ = ĝ 1 = ω 42 f1 ĝ = ω 42 sin η ĝ 2, Figure 4: The triplet (ĝ 1,ĝ 2,ĝ 3 ) is the Frenet frame for the conjugate tooth flanks Φ 2 and Φ 3. The corresponding Disteli axes ĝ are defined by the spatial Euler-Savary equation. The common perpendicular k is already the central normal n of the tooth flanks. This follows because, for the trajectory of ĝ under Σ 4 /Σ 2,we obtain ĝ = ω 42 f1 ĝ = ω 42 sin η (cos ξ f 3 sin ξ f 2 ) = ω 42 sin η k. Therefore, the Frenet frame (ĝ 1 = ĝ, ĝ 2 = n = k, ĝ 3 = t) for the conjugate tooth flanks Φ 2 and ĝ = ω 42 [ ω42 sin 2 η ĝ 1 + sin( ϕ + α)cos ξ cos η ĝ 2 + ( sin( ϕ + α)sin ξ + ω 42 sin η cos η ) ĝ 3 ], which, upon comparison with (15), yields the instantanous invariants of the tooth flank Φ 2 under η, i.e., ĝ not parallel to the ISA f 1,as λφ2 = ω 42 sin η, sin( ϕ + α)sin μ ξ Φ2 = + ω 42 cos η. sin η (28) For the conjugate tooth flank Φ 3 we obtain likewise λφ3 = ω 43 sin η, μ Φ3 sin( ϕ α)sin ξ = ω 31 + ω 43 cos η. sin η (29)

9 When the dual angle η Φ i characterizes the instant Disteli axis of Φ i we can verify the spatial Euler-Savary equation (see [1]) (cot η Φ 2 cot η )sin ξ = cot γ 2 cot γ 4 = cot( ϕ + α) cot( ϕ β) (3) for the motion Σ 4 /Σ 2, which generates Φ 2. In the same way we can confirm that the Disteli axis ĝ Φ 3 of Φ 3 satisfies (cot η Φ 3 cot η )sin ξ = cot γ 3 cot γ 4 = cot( ϕ α) cot( ϕ β). Upon subtraction of the two Euler-Savary equations we obtain (cot η Φ 2 cot η Φ 3 )sin ξ = cot γ 2 cot γ 3, thereby proving the spatial version of a result which is well known in planar and spherical kinematics, namely Theorem 5. Let Φ 2 and Φ 3 be conjugate ruled tooth flanks with permanent line contact. Then the Disteli axes ĝ Φ 2 and ĝ Φ 3 of the instant meshing line satisfy the Euler-Savary equation for the relative motion Σ 3 /Σ 2 between the two gears. 6. A SPATIAL ANALOGUE OF INVOLUTE GEARING In planar cycloid gearing there are two auxiliary curves, namely two circles, which usually are laid out in a symmetric relative position with respect to the pole tangent. The same is true on the sphere. However, when the auxiliary circles are specified as great circles they become identical, coinciding with the spherical pole tangent t. Theaxis p 41 of the great circle t is orthogonal to the ISA p 32. The corresponding profiles are involutes of the polodes; they are characterized by the constant pressure angle α =. This is the particular case of involute gearing where the pitch circles coincide with the base circles. These profiles are not geometrically feasable because of one reason: At the meshing point P on the instant pole tangent t the profiles have either ag 2 -contact with mutual penetration, or a cusp, and at external gears the curves open towards opposite sides. We obtain the corresponding spatial version when we specify the axis p 41 orthogonal to the ISA p 32 on the Plücker conoid (see Fig. 1). This is the case we analyze below. Due to Eqs. (4) (9), the representation p 41 = sin β ê 1 + cos β ê 2 implies Therefore, β = ϕ + π 2, 5 β = ϕ, ϕ β = π 2 + 2εϕ. (31) sin( ϕ β)= 1, cos( ϕ β)=2εϕ. (32) From Eqs. (1), (5), and (12) follows for our particular choice ω 41 = sin( ϕ + α) (33) h 41 = R(cos2α + cos2ϕ). The auxiliary surface Π 4 is a skew orthogonal helicoid with axis p 41 and pitch h 41,theISA p 32 being its initial generator. The invariants of Π 4 are, by virtue of (25), λ4 = ω 41, μ 4 = 2εϕ ω 41. (34) The dual angle between the generators of Π 4 and its axis is γ 4 = ϕ β = π 2 + 2εϕ with cot γ 4 = μ 4 / λ 4 = 2εϕ. From (4), the distance γ 4 between axis and generators vanishes if and only if ϕ =, i.e., = ω 31. The generating motions Σ 4 /Σ 2 and Σ 4 /Σ 3 of the tooth flanks Φ 2 and Φ 3 have the twists q 42 = 5 One could also set β = ϕ π/2. However, this has no effect on the auxiliary surface. It only reverses the orientation of p 41 and changes therefore the sign of ω 41 and ω 41.

10 ω 42 f1 and q 43 = ω 43 f1, respectively; in our particular case we have ω 42 = [cos(ϕ + α)+ε(ϕ α )sin(ϕ + α)], ω 43 = ω 31 [cos(ϕ α)+ε(ϕ + α )sin(ϕ α)]. (35) Hence, ω 43 : ω 42 = tan(ϕ + α) :tan(ϕ α) =(ϕ + α ) : (ϕ α ). 6.1 The ISA as a Line of Regression p 31 ω 31 ISA p 32 Φ 2 Φ 3 (36) p 21 distributed just as along a regular generator with distribution parameter δ = Rcos2α. Figure 8 in [6] reveals that the ISA doesn t look singular at all; it is the common border line of the two components, orginating from two symmetrically placed auxiliary surfaces. However, in our particular case the two auxiliary surfaces coincide with the skew helicoid Π 4. The ISA is, in fact, a line of regression for both tooth flanks. In external gears, as depicted in Fig. 5, the two flanks open toward opposite sides. Hence, when the ISA becomes the meshing line, no transmission of forces can take place. Figure 5 shows the conjugate tooth flanks as wire-frames; the depicted fat red and blue lines being the intersections of the flanks with planes perpendicular to the ISA. 6.2 There is a G 2 -contact at the Striction Point What corresponds in skew gears to the osculation of tooth profiles when the pole tangent serves as auxiliary curve? Figure 6 shows an example 6 where the initial meshing line ĝ differs from the ISA. But ĝ is parallel to the ISA and intersects the central tangent of the axodes at right angles. This central tangent passes through the striction point S of the axodes and is parallel to the axis p 41 of the auxiliary surface Π 4 (note f 3 in Fig. 1). The spatial Euler-Savary equation (3) (see [1, Theorem 6]) for the motion Σ 4 /Σ 2 Figure 5: When the ISA coincides with the meshing line ĝ, the singular lines of the two flanks Φ 2, Φ 3 come together sharing the tangent plane at each point of ĝ; but the flanks open toward opposite sides. The fat red and blue lines indicate sections orthogonal to the ISA. Analogue to the planar and spherical cases, in spatial cycloid gearing the ISA p 32 is a singular generator of the two tooth flanks Φ 2 and Φ 3. As pointed out in [6, Theorem 5], all its points are uniplanar, the tangent planes along p 32 being 9 12 (cot η cot η)sin ξ = ω λ = cot γ 2 cot γ 4, holds only under sinξ, but we can replace it by the equation [1, page 13] λ sin ξ (cos η sin η sin η cos η ) + ω sin η sin η =. Under the relation sin ξ = (i.e., k = f 3 in Fig. 4) it is apparent that sin η implies sin η =. In other words, when ĝ p 32 intersects the striction tangent f 3 of the axodes at right angles, the 6 Data: 2α = 6.,2α = 7.mm, ω 31 : = 2: 3, and distance between the ISA and the initial meshing line ĝ: SS g = 35.mm.

11 p 31 Φ 3 ω 31 ĝ ISA= p 32 p 21 S g Φ 2 S p 41 Figure 6: Two conjugate flanks Φ 2 and Φ 3 with G 2 -contact at the common striction point S g. The meshing line ĝ is parallel to the ISA and a cylindric generator of Φ 2 and Φ 3. Disteli axis ĝ coincides with the ISA. The same holds for the motion Σ 4 /Σ 3, which means that under this condition the two tooth flanks share the instant Disteli axis. According to Theorem 4, Φ 2 and Φ 3 have a G 2 -contact at the common striction point S g. In Fig. 6, the fat blue and read curves, which are in contact at marked points on the meshing line ĝ, are level lines of the two flanks, i.e., intersections with planes orthogonal to the ISA. The mean section shows the G 2 -contact at the striction point S g, which causes the penetration. The case of osculating cylindrical or spherical tooth flanks is misleading. In the true spatial version there is no G 2 -contact at all other points of ĝ for one reason: According to Theorem 4, in this case the condition δ()=c δ() must be satisfied. However, because of the permanent line contact the flanks have the same distribution parameter δ(t) =δ(t) for each t I. This implies δ()= δ(), but by Eqs. (28), (29) and (36), the constant c with λ Φ3 = c λ Φ2 is c = tan(ϕ + α)/tan(ϕ α) =(ϕ + α ) : (ϕ α ) 1. The different poses depicted in Fig. 7 reveal that there is also a mutual penetration of the conjugate tooth flanks Φ 2 and Φ 3 at the other poses. Since the surfaces share this curve of intersection as well as the tangent planes at all points of the meshing line, there must be a G 2 -contact at the point where the curve of intersection meets the meshing line. This point is close to the mar

12 ϕ 21 = ϕ 21 = ϕ 21 = ϕ 21 = 12.8 ϕ 21 =. ϕ 21 = 12.8 ϕ 21 = ϕ 21 = ϕ 21 = Figure 7: Snapshots of the penetrating tooth flanks with their striction curves upon meshing

13 ked striction point; however it can be proved that the point of G 2 -contact must be different from the striction point S g up to the symmetric case ω 31 = ; hence ϕ =. ISA= p 32 p A Spatial Analogue of Octoidal Gears In the plane as well as on the sphere, the generalized Camus Theorem states that for any curve c 4 attached to the auxiliary curve p 4 Σ 4 the envelopes c 2 and c 3 under motions Σ 4 /Σ 2 and Σ 4 /Σ 3, respectively, are conjugate tooth profiles. c 2 I 21 p 31 p 41 c 4 ω 31 Φ 2 p 2 I 32 C α Σ 4 p 4 Φ 3 p 3 b 3 ω 31 }{{} I 31 c 3 Figure 9: Skew gears with torses as conjugate tooth flanks Φ 2,Φ 3 and permanent line contact. The fat red and blue lines indicate sections orthogonal to the meshing line. Figure 8: Planar version of the generalized Camus Theorem in the particular case leading to involute gears. In the particular planar case, depicted in Fig. 8, the auxiliary curve p 4 is the pole tangent t and the attached curve c 4 is a line. In all its poses, the line c 4 shows the same inclination with respect to the gear frame Σ 1. At each pose, the enveloping point C of c 4 is the pedal point with respect to the instant pole I 32. The right-angled triangle enclosed by c 4, p 4 and the line I 32 C shows that the pressure angle α remains constant, which leads to the case of involute gearing. The foregoing statement does not hold in spherical geometry since for spherical triangles the sum of the interior angles is not constant. This sum is always greater than 18, the amount by which the sum exceeds 18 being proportional to the area of the triangle. Therefore, we cannot conclude for the analog specification in bevel gears that the pressure angle α is constant; it increases with the distance between I 32 and c 4. We obtain what is known as octoidal gears, as reported in [3]. The validity of the spatial analogue of the generalized Camus Theorem was proved in [6, Theorem 6]: For any surface Φ 4 attached to the auxiliary surface Π 4 the envelopes Φ 2 and Φ 3 under the respective relative motions Σ 4 /Σ 2 and Σ 4 /Σ 3 are conjugate tooth flanks. We choose again Π 4 as the skew orthogonal helicoid and specify Φ 4 as a plane. Then we obtain a pair of conjugate torses Φ 2,Φ 3 with permanent line contact. In Fig. 9 one example is depicted which indicates

14 ϕ 21 = 44.2 ϕ 21 = ϕ 21 = p 21 ISA= p 32 Φ 3 p 31 ω 31 Φ 2 p 41 ϕ 21 = ϕ 21 = ϕ 21 = 44.2 Figure 1: Snapshots of the conjugate torses Φ 2 and Φ 3 upon meshing (ω 31 : = 2:1). that these flanks should work correctly. Contrary to the general case of J. Phillips involute gearing [7], contact is not punctual, but along a line. ThefatredanbluecurvesinFig.9aretheintersections of the flanks with planes perpendicular to the instant meshing line, which is depicted as magenta double line. Figure 1 shows snapshots of the conjugate torses upon meshing CONCLUSIONS Based on the Camus Theorem and on Martin Disteli s work, we showed in this paper that the flanks of spatial cycloid gears can be synthesized by means of an auxiliary surface. Upon choos- 13 ing the skew orthogonal helicoid as auxiliary surface, the tooth flanks of the spatial equivalent of octoidal gears are obtained. The final example with torses as conjugate tooth flanks looks promising but still needs a detailed analysis. REFERENCES [1] Stachel, H. (2). Instantaneous spatial kinematics and the invariants of the axodes, Proc. Ball 2 Symposium, Cambridge (article no. 23). [2] Blaschke, W. (196). Kinematik und Quaternionen, VEB Deutscher Verlag der Wissenschaften, Berlin.