ON JACK PHILLIP'S SPATIAL INVOLUTE GEARING

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1 The th International Conference on Geometry and Graphics, -5 August, 004, Guangzhou, China ON JACK PHILLIP'S SPATIAL INVOLUTE GEARING Hellmuth STACHEL Vienna University of Technology, Vienna, AUSTRIA ABSTRACT This is a geometric approach to spatial involute gearing which has recently been developed by Jack Phillips (003). After recalling Phillips' fundamental theorems and other properties, some geometric questions around this interesting type of gearing are discussed. Keywords: Spatial gearing, involute gearing, helical involute PRELIMINARIES In the series of International Conferences on Geometry and Graphics there had been several papers dealing with spatial gearing, e.g. Podkorutov et al. (998 and 00), Andrei et al. (00) or Brailov (998). The function of a gear set is to transmit a rotary motion of the input wheel Σ about the axis p 0 with angular velocity ω 0 to the output wheel Σ rotating about p 0 with ω0 in a uniform way, i.e., with a constant transmission ratio i : = ω ω = const. () 0 0 According to the relative position of the gear axes p 0 and p we distinguish the following types: 0 a) Planar gearing (spur gears) for parallel axes p0, p 0, b) spherical gearing (bevel gears) for intersecting axes p, and p0, 0 c) spatial gearing (hyperboloidal gears) for skew axes p0, p0, in particular worm gears for orthogonal p. p0, 0. Planar gearing In the case of parallel axes p0, p 0 we confine us to a perpendicular plane where two systems Σ, Σ are rotating against Σ 0 about centers 0, 0 with velocities ω0, ω 0, respectively. Two curves c Σ and c Σ are conjugate profiles when they are in permanent contact during the transmission, i.e., ( c, c ) is a pair of enveloping curves under the relative motion Σ Σ. Due to a standard theorem from plane kinematics (see, e.g., Wunderlich (970) or Husty et al. (997)) the common normal at the point E of contact must pass through the pole of this relative motion. Due to the planar Three-Pole-Theorem this point divides the segment 0 0 at the constant ratio i and is therefore fixed in the frame link Σ 0. We summarize: Theorem (Fundamental law of planar gearing): The profiles c Σ and c Σ are conjugate if and only if the common normal e (= meshing normal) at the point E of contact (= meshing point) passes through the relative pole. Due to L. Euler (765) planar involute gearing (cf. e.g. Wunderlich (970)) is characterized by the condition that with respect to Σ 0 all meshing normals e are coincident. This implies (i) The profiles are involutes of the base circles. (ii) For constant driving velocity ω 0 the point of contact E runs relative to Σ 0 with constant velocity along e. (iii) The transmitting force has a fixed line of action. (iv) The transmission ratio i depends only on the dimension of the curves c, c and not on their relative position. Therefore this planar gearing remains independent of errors upon assembly.. Basics of spatial kinematics There is a tight connection between spatial kinematics and 3 the geometry of lines in the Euclidean 3-space IE. Therefore we start with recalling the use of appropriate line coordinates: Any oriented line (spear) g = a + IR g can be uniquely represented by the pair of vectors ( g, gˆ ), the direction vector g and the momentum vector g ˆ, with g g = and gˆ : = a g. It is convenient to combine this pair to a dual vector g : = g + ε g, ˆ where the dual unit ε obeys the rule ε = 0. We extend the usual dot product of vectors to dual vectors and notice g g = g. gˆ + g. g ˆ = + ε 0 =. Hence we call g a dual unit vector. The dot product of dual vectors is defined by g h = g. h + ε ( gˆ. h + g. h ˆ). In the case of dual unit vectors this product has the following geometric meaning: g h = cosϕ = cosϕ εϕˆ sinϕ () where ϕ = ϕ + ε ˆ ϕ is the dual angle (see Fig. ) between the corresponding lines g und h, i.e., with ϕ = < ) gh and ˆϕ as shortest distance (cf. Pottmann, Wallner (00), p. 55 ff).

2 Figure : Two spears g, h enclosing the dual angle ϕ = ϕ + εϕˆ. The components ϕ and ˆϕ of ϕ are signed according to any chosen orientation of the common perpendicular n. When the orientation of n is reversed then ϕ and ˆϕ change their sign. When the orientation either of g or of h is reversed then ϕ has to be replaced by ϕ + π (mod π ). There is also a geometric interpretation for non-normalized dual vectors: At any moment a spatial motion Σi Σ j assigns to the point X Σ i (coordinate vector x) the velocity vector v = qˆ + ( q x) (3) X relative to Σ j. We combine the pair (q, q ˆ ) again to a dual vector q. This vector can always be expressed as a multiple of a unit vector, i.e., q : = q + ε qˆ = = ( ω + ε ˆ ω )( p + ε p ˆ ) = ω p with p p =. It turns out that X v coincides with the velocity vector of X under a helical motion (= instantaneous screw motion) about the instantaneous axis (ISA) with dual unit vector p. The dual factor ω is a compound of the angular velocity ω and the translatory velocity ˆ ω of this helical motion. q is called the instantaneous screw. In this sense, spears are the screws of rotations with angular velocity. For each instantaneous motion (screw q ) the path normals n constitute a linear line complex, the (= complex of normals) as the dual unit vector n = n + ε n ˆ of any normal n obeys the equation qˆ. n + q. n ˆ = 0 ( q n IR). (5) n a X n = This results from v. 0 and nˆ = x n. By () it is equivalent to ˆ ω ( ω + εωˆ )cosα IR i.e. ω = ˆ α tan α. (6) with α denoting the dual angle between orientation of n. p and any The spatial Three-Pole-Theorem states (cf. Husty et al. (997) or Stachel (000)): If for three given systems Σ0, Σ, Σ the dual vectors 0, q are the instantaneous screws of q 0 ˆϕ ϕ h g Σ Σ 0, Σ Σ 0, resp., then q = q q (7) p is the instantaneous screw of the relative motion Σ Σ. As a consequence, if a line n which intersects the ISAs p 0 of Σ Σ 0 and p 0 of Σ Σ 0 orthogonally, then it does the same with the axis p of Σ Σ, provided ω 0. αˆ0 For given skew axes p0, p 0, but arbitrary dual velocities ω0, ω 0 the axes p of the relative motions Σ Σ constitute a cylindroid or Plücker conoid (see, e.g., Pottmann, Wallner (00), p. 8). Fig. gives an impression of the cylindroid by showing some generators 'between' p 0 and p 0. α 0 Figure : Generators of the cylindroid spanned by the screws p 0 and p 0. p 0 For each ratio ω0 ω0 0, the polodes Π, Π of the relative motion Σ Σ are one-sheet hyperboloids of revolution which are in contact along p and have axes p0, p0, respectively. The common normal lines of Π and Π at the points of p intersect both, p 0 and p 0, and constitute an orthogonal hyperbolic paraboloid. We summarize: Lemma The generators p of the cylindroid are characterized as those straight lines in space where a line contact can take place between one-sheet hyperboloids of revolution, Π with mutually skew axes p0, p 0. Π These hyperboloids are the spatial equivalents for root and addendum circles from planar gearing. They were used at the gearing displayed in Fig Basics of spatial gearing Let the systems Σ, Σ rotate against Σ 0 about the fixed axes p0, p 0 with constant angular velocities ω0, ω 0, respectively. Then the instantaneous screw of the relative motion Σ Σ is constant in Σ 0, too.

3 It reads q = ω p ω p for ω0, ω0 IR. (8) When two surfaces Φ Σ and Φ Σ are conjugate tooth flanks for a uniform transmission, then Φ contacts Φ permanently under the relative motion Σ Σ. In analogy to the planar case we obtain Theorem (Fundamental law of spatial gearing) The tooth flanks Φ Σ and Φ Σ are conjugate if and only if at each point E of contact the contact normal e is included in the complex of normals of the relative motion Σ Σ. Due to (5) the dual unit vector e of any meshing normal e obeys the equation of the linear line complex q e = ω ( p e) ω ( p e ) IR Hence () implies for the dual angles α, α between e and p 0 and p 0, resp., (see Fig. 3, compare Phillips (003), p. 46, Fig..0) ω ˆ α sinα ω ˆ α sinα = and therefore ω0 ˆ α sinα i = ω =. (9) ˆ α sinα 0 p 0 ω 0 ˆ α α e Figure 3: Spatial involute gearing: The point E of contact traces a fixed contact normal e. SPATIAL INVOLUTE GEARING ˆα 0 Spatial involute gearing is characterized in analogy to the planar case as follows (cf. Phillips (003)): All contact normals e are coincident in Σ 0 and skew to p 0 and p 0. We exclude also perpendicularity between e and one of the axes. According to (9) a constant contact normal e implies already a constant transmission ratio i. Note that in general this gearing offers single point contact only. n n θ p ˆα ω 0 E ε α 0 p 0 α. Slip tracks First we focus on the paths of the meshing point E relative to the wheels Σ, Σ. These paths are called slip tracks c, c : Σ Σ 0 is a rotation about p 0, and with respect to Σ 0 point E is placed on the fixed line e. Therefore conversely the slip track c is located on the one-sheet hyperboloid Ψ of revolution through e with axis p0. On the other hand, the slip track c is located on the tooth flank Φ, and for each posture of Φ line e Σ 0 is orthogonal to the tangent plane ε at the instantaneous point of contact (see Fig. 3). Therefore the line tangent to the slip track c is orthogonal to e. This gives the result: Lemma The path c of E relative to Σ is an orthogonal trajectory of the e-regulus on the one-sheet hyperboloid Ψ through e with axis p 0. We use a cartesian coordinate system 3 with the z-axis at p 0 and the x-axis along the common perpendicular n between p 0 and e (see Fig. 3). Then the plane z = 0 contains the throat circle of the hyperboloid Ψ. Ψ ( u, v) = A( u) e( v ) with cos u sin u 0 ˆ α A( u) = sin u cos u 0, e( v) = v sin αα 0 0 cos v is a parametrization of Ψ in matrix form and based on the parametrization e( v ) of the line e. The curve c ( t) = Ψ[ u( t), v ( t)] intersects the v -lines orthogonally if d A d e de e( v) u + A( u) v A( u) = 0. du d v d v This is equivalent to the differential equation ˆ α sinα u + v = 0. T Hence the slip track c which starts in the plane z = 0 on the x-axis can be parametrized as or c ( u ) = A ( u ) e ( ˆ α sin α u ) (0) c ( u) = ˆ α sin u + ˆ αu sin α cos u. 0 cotα i cos u sin u () We obtain all other slip tracks on Ψ by rotation about p 0. In order to describe the absolute motion of the point E of contact, i.e., its motion with respect to Σ 0, we superimpose the rotation of the hyperboloid Ψ about p 0 with angular velocity ω 0 with the movement of E along c by setting u = ω0 t. Thus we obtain the path e( ω0 ˆ α sin αt). As line e is different from the relative axis p, this one-sheet hyperboloid of revolution Ψ differs from the polodes Π mentioned in Lemma. 3 A more geometric approach for this fact can be found in Stachel (004).

4 p 0 ω 0 Ev 0 E c g c Ψ Φ Figure 4: Slip tracks c as orthogonal trajectories on the one-sheet hyperboloid Ψ ( ˆ α sinα < 0). This proves that E traces line e with the constant velocity vector 0 Ev 0 = ω0 ˆ α sinα sin α. () cos α. The tooth flanks The simplest tooth flank for single point contact is the envelope Φ of the plane ε of contact in Σ. By () this plane ε through E and perpendicular to e is translated along e and simultaneously rotated about p 0. Therefore Φ it is a helical involute (see Fig. 5), the developable swept by tangent lines of a helix. The parametrization in () can be rewritten as cost sin t c ( u) = ˆ α sin t ˆ αt sin α cos t t tanα tanα thus showing c as a curve on Φ. (3) The first term on the right hand side parametrizes the edge of regression of Φ. The second term has the direction of tangent lines, the generators g Φ. Lemma 3 The slip track c is located on a helical involute Φ with the pitch ˆ α tan α. At each point E c there is an orthogonal intersection between Φ and the one-sheet hyperboloid Ψ of Lemma. Different slip tracks on Φ arise from each other by helical motions about p 0 with pitch ˆ α tan α. Figure 5: The tooth flank Ψ (helical involute) with the slip track c of point E g ( ˆ α sinα > 0). We summarize: Theorem 3 (Phillips' st Fundamental Theorem) The helical involutes Ψ, Ψ are conjugate tooth flanks with point contact for a spatial gearing where all meshing normals coincide with a line e fixed in Σ 0..3 Two helical involutes in contact The following theorem completes the confirmation that the advantages (ii) (iv) of planar involute gearing as listed above are still true for spatial involute gearing: Theorem 4 (Phillips' nd Fundamental Theorem) If two given helical involutes Φ, Φ are placed in mutual contact at point E and if their axes are kept fixed in this position, then Φ and Φ serve as tooth flanks for uniform transmission whether the axes are parallel, intersecting or skew. According to (9) the transmission ratio i depends only on Φ and Φ and not on their relative position. Therefore this spatial gearing remains independent of errors upon assembly. Proof: When the two flanks Φ, Φ rotate with constant angular velocities ω0, ω 0 about their axes p 0, p 0 and point E runs relatively along the slip tracks with appropriate velocities, then with respect to Σ 0 point E traces e with the velocities ˆ sin ˆ sin, ω0α α and ω0α α respectively, due to (). By (9) these velocities are equal at any moment. Hence the initial contact between Φ and Φ at E is preserved under simultaneous rotations with transmission ratio i.

5 p 0 p 0 ω 0 ω 0 p α 0 Figure 6: Spatial involute gearing together with the effective slip tracks on the flanks. List of dimensions: Transmission ratio i = 3, numbers of teeth: z = 8, z = 7 ; contact ratio.095; dual angle between p 0 and p 0 : α 0 =.35, ˆ α 0 = 7.0; dual angles between p i0 and the fixed contact normal e: α = 60.0, ˆ α = 45.0, α = 76.98, ˆ α = 60.0, and (compare Fig. 3) angle θ = 4.0. ω ω 0 0 e ϕ 0 = 0 ϕ 0 = 3.6 ϕ 0 = 7. ϕ 0 = 0.8 ϕ 0 = 4.4 ϕ 0 = 8.0 Figure 7: Different postures of meshing involute teeth for inspecting the backlash. e is the line of contact. Interval of the input angle ϕ 0 = 3.6, interval of the output angle ϕ 0 =.4. ω 0 ω 0 Figure 8: Different postures of meshing involute gear flanks together with the effective slip tracks, seen in direction of the contact normal e. The second wheel is displayed as a wireframe.

6 .4 Involute gear flanks with line contact Suppose there is any pair of flanks Φ, Φ with permanent line contact while each point E of contact traces a straight line e in Σ 0. Then the line g of contact generates in Σ 0 a ruled surface Γ while g remains an orthogonal trajectory of the generators. Due to a standard theorem of Gauss any two orthogonal trajectories intersect all generators of Γ in segments of equal length. Therefore all points E g Σ i must have the same absolute velocity E v 0 by (), i.e., the dual angles ν i between surface normals n of Φ i along g and the axis p i0 obey ˆ ν sinν = i ˆ ν sin ˆ ν = c = const. 0. (4) This defines a (nonlinear) congruence of lines. Conversely, with the arguments used in the proof of Theorem 4 we can confirm that these equations are sufficient for continuing line contact between Φ and Φ. Theorem 5 Φ, Φ are flanks for involute gears with permanent contact along a curve g if and only if g is an orthogonal trajectory of a ruled surface Γ with generators included in the line congruence (4). The flanks Φ i are swept by slip tracks passing through the points E g. Particular examples of such flanks are the helical involutes mentioned in Theorem 3. Their generators are the lines of contact; the ruled surface Γ is a plane parallel to both axes p 0 and p 0 (cf. Stachel (004)). This particular example arises when the congruence (4) contains a line n which is located in a plane Γ parallel to p 0 and p0. Then all lines parallel to n and included in Γ obey (4), too. Orthogonal trajectories of this pencil of parallel lines are generators of helical involutes Φ and Φ as mentioned in Lemma 3. In Fig. 5 a generator g Φ is depicted together with equal velocity vectors Ev 0 at different points E g. Podkorutov, A.N., Malcev, D.V., 998, The geometry modelling of conjugate curved surfaces, excluding interference, scientific basic, 8th ICECGDG Austin, vol., Podkorutov, A.N., Pavlyshko, A.V., Podkorytov, V.A., 00, Design of high-efficiency fair manyways wormy mills. Proc. 0th ICGG, vol., 3. Pottmann, H., Wallner, J., 00, Computational Line Geometry, Springer Verlag, Berlin, Heidelberg. Stachel, H., 000, Instantaneous spatial kinematics and the invariants of the axodes. Proc. Ball 000 Symposium, Cambridge, no. 3. Stachel, H., 004, On Spatial Involute Gearing.TU Wien, Geometry Preprint No. 9. Wunderlich, W., 970, Ebene Kinematik, Bibliographisches Institut, Mannheim. ABOUT THE AUTHORS Hellmuth Stachel, Ph.D, is Professor of Geometry at the Institute of Discrete Mathematics and Geometry, Vienna University of Technology, and editor in chief of the Journal for Geometry and Graphics. His research interests are in Higher Geometry, Kinematics and Computer Graphics. He can be reached by stachel@dmg.tuwien.ac.at, by fax: (+43) , or through the postal address: Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wiedner Hauptstr A 040 Wien Austria, Europe. 3 REFERENCES Andrei, L., Andrei, G., Mereuta E., 00, Simulation of the curved face width spur gear generation and mesh using the solid modeling method, Proc. 0th ICGG, vol., Brailov, A.Yu., 998, The Exclusion Method of Interference in Conjugated Helicoids, Proc. 8th ICECGDG Austin, vol., Husty, M., Karger, A., Sachs, H., Steinhilper, W., 997, Kinematik und Robotik, Springer-Verlag, BerlinHeidelberg. Phillips, J., 003, General Spatial Involute Gearing, Springer Verlag, New York.