WO2006018665A1

WO2006018665A1 - Spherical 3-dimensional gears transmission of motion along any of the three dimensions at any moment.

Info

Publication number: WO2006018665A1
Application number: PCT/GR2005/000017
Authority: WO
Inventors: Michael Mytilineos
Original assignee: Michael Mytilineos
Priority date: 2004-08-19
Filing date: 2005-05-16
Publication date: 2006-02-23
Also published as: GR1004886B

Abstract

Spherical 3-dimensional gears, which are characterized by polyhedral (or conical) teeth - or polyhedral (or even conical) holes respectively - homogeneously distributed across a spherical surface and which can transmit motion to all three dimensions. They are subdivided into 'positives' (P) (Figure 4), 'negatives' (N) (Figure 6) and 'concave positives' (Cp3) (Figure 17) or 'concave negatives' (CN2) (Figure 14). The 'conjugate' gears (P-N) (Figure 7), those that fit with each other, so that they transmit the motion from one to the other, must be positive with negative, and with corresponding geometry (shape, size) and number of teeth-holes, in order to block each other. Based on this idea we establish and describe the following three methods of mounting such gears within the space with the corresponding ways of motion transmission from one to another: Σ1. Mounting by ring (Figures 9, 10, 11, 12) Σ 2. Articulated (simple) mounting (Figures 13, 14) Σ 3. Articulated-planet mounting (Figures 15, 16, 17, 18) The bases of teeth (or the openings of holes) (Figures 21, 22, 23, 24) constitute in the ideal case faces of regular polyhedra, however in practice they form usually faces of pseudoregular polyhedra. Our idea, after choosing triangular bases of trihedral teeth or holes, does not exclude polyhedra of other form of their faces (i.e. beyond triangles only) or even mixed forms that produce the respective pyramid teeth or holes. In the case of faces with a great number of sides, which are circles essentially (circular bases), the teeth and the holes will become cones.

Description

SPHERICAL 3-DIMENSIONAL GEARS

Transmission of motion along any of the three dimensions at any moment.

DESCRIPTION

This invention refers to a new kind of gears that can transmit motion along all three dimensions at any moment.

In the traditional gears (like those in Figure 1), the transmission of motion (M) takes effect always along one dimension per gear. Each gear-tooth forms no "plain" angle (1) it is shown in Figure 1, but this angle is instead dihedral (2) - Figure 2.

In fact, how a gear might be so that it could transmit motion along all three dimensions, more precisely along any dimension at any moment? The problem translates to how each gear- tooth should be, so as all of them to be homogeneously distributed on the surface of the gear and transmit the motion towards all directions. A reasonable picture that we can imagine is that of "trihedral" angle (3) like the one shown in Figure 3, that is, an angle with one more face than the angles of an ordinary tooth. Based on that thought, we can conceive the picture of the whole gear, which is supposed to transmit motion to all directions, as one that has a spherical distribution of its teeth. A spherical distribution of trihedral teeth (3) will have a form like the one shown in Figure 4.

Considering traditional gears, we can see something else too. If we assume that the 1^st gear (p) on the left of Figure 1 generates the motion and the 2^nd gear (n) transmits it, name moreover (p) "positive" and (n) "negative" and, for the sake of simplicity assume also that (p) consists of a circle of radius Ri with teeth (1) on its circumference while (n) consists of a circle of radius R₂ with holes (4) on its circumference, as it is shown in Figure 5, then we see that the negative gear (n) with radius R2 (circle R₂ with holes) is actually nothing more than a positive gear (p) with radius Ri (circle Ri with teeth). In other words, in traditional (2- dimensional) gears the positives have the same shape as the negatives.

As concerns spherical-3 dimensional gears, however, this is not the case. Here it is obvious that two gears having the form shown in Figure 4 do not "fit" (block) each other. Therefore we adopt a configuration involving sphere Ri with teeth (3) - positive gear (P) (Figure 4) and sphere R₂ with holes (5) - negative gear (N) (Figure 6). A "conjugation" (P - N) of a positive- negative gear will have then the form shown in Figure 7, where the teeth (3) of the positive (P) penetrate and block the holes (5) of the negative (N) and hence they transmit motion. Figure 8, on the other hand, shows that in order to reduce friction, bearing balls (8) could be placed upon the apexes of the faces of both positive (P) and negative (N) gears, as well as the edges (9) of their teeth and holes could be smoothed out. Figures 4, 6 and 7 illustrate our idea schematically.

Next we are describing three methods of mounting such gears, each one corresponding to a different type of motion transmission. These methods are:

∑l . Mounting by ring:

There is an embedded (6) ring (7) (not part of a cylindrical surface but of a spherical one) that covers the gear, being either positive (Pi) or negative (Ni) but nevertheless doing it in such a way that leaves a substantial part of the gear's surface uncovered to both its circular (symmetrical) openings (Di), as Figures 9, 10 and 1 1 show. The transmission of motion (by blocking teeth-holes) is resumed in Figure 1 1. In case that gears have arms (B ι) attached upon them, therefore the motion of arms within the space is associated with theirs, we can get a picture like the one shown in Figure 12. The degrees of freedom of those arms depend on the gears' dimensions (i.e. radii, teeth, holes) as well as on the dimensions of rings.

Σ2. Articulated (simple) mounting:

In (simple) articulated mounting, there is a gear (either positive (P₂) or negative (N₂)) "nested" within a "concave" one which would be negative (C_N2) or positive (Cp₂) respectively, with a circular opening (D₂) on its surface, as we can see in the respective 2- dimensional model of the configuration shown in Figure 13. We shall name "concaves" spherical 3-dimensional gears with a corresponding surface distribution of teeth or holes like those presented earlier, in which however their teeth or holes are located in the interior of their surface (instead of the exterior). If the nested gear is positive (P₂), the concave one is negative (C_N2) in the sense that its internal surface has holes, as it is shown in the 3-dimensional aspect of the configuration of Figure 14. If the nested gear is however negative, the concave one is positive i.e. there are teeth in its internal surface.

There are arms (B₂) (Figures 13, 14) attached (embedded) in the two gears, and thus the motion of arms affects the motion of gears and vice versa. The degrees of freedom of arms depend on the gears' dimensions (i.e. radii, teeth, holes) as well as on the opening of the concave gear.

Σ3. Articulated-planet mounting:

There are in principle four gears, namely "planets" (PL), which support a "kernel" gear (K) as it is shown in Figure 15. The centres of the planets define the apexes of a regular tetrahedron. The central gear (kernel) (K) is affected to the same extent to which it affects itself the motion of the planets (PL). This system (kernel - 4 planets) is covered by a concave gear (its teeth or holes are in its interior), which has a circular opening (D₃) in its surface (Figures 16, 17). If we name the surrounding concave gear "crust", we can say that our purpose is to make the kernel transmit the motion to the crust and vice versa.

If the kernel is positive (Kp₃), as it is the case in Figures 16 and 17, then the planets will be negative (PL_N3) and the crust concave-positive (Cp₃). If the kernel is negative though, the planets will be positive and the crust concave-negative. A 2-dimensional respective model of the crust-kernel configuration is shown in Figure 16, while a 3-dimensional model is shown in Figure 17. From the opening (D₃) of the crust in Figure 17, we can see the kernel (K_P3), the planets (PL_N3) and the internal teeth of the crust (Cp₃). If we attach (embed) then arms (B₃) to the kernel and the crust, it will result in the configuration of Figure 18. The motion and the location of each arm affect the motion and the location of the other arm. The degrees of freedom of the arms depend also on the dimensions of the gears (i.e. radii, teeth, holes) as well as upon the opening of the concave gear.

We deem especially useful to point out here the following observation that concerns the geometry of spherical-3 dimensional gears: The bases of the teeth (or the openings of the holes) may form in the perfect case regular polyhedra, however in practice they usually form pseudo-regular polyhedra. Our idea does not also exclude polyhedra of other form of their faces (i.e. beyond triangles only) or even mixed forms that produce the respective pyramid-like teeth or holes. The faces-bases can be for example only pentagons, or mixed pentagons and hexagons, or even circles (that will produce conical teeth and holes). More specifically, we preferred a polyhedral distribution of teeth and holes with triangle faces-bases because:

To the best of our knowledge, existing algorithms of Computational Geometry can construct up to regular 34-hedra_; which does not means though that all of their precedents (i.e. 33-, 32- , 31-, ..., 5- and 4-hedra) are also constructible. By taking however a constructible polyhedron, e.g. dodecahedron (Figure 19), we can "triangulate" each face (Figure 20) and then project (radically) the apexes of the resultant triangles upon a circumscribed spherical surface so that the sides of the new triangles, defined by these apexes, will be strings of that sphere and the triangles themselves be transformed into faces (Figures 21, 22). Obviously, the resultant teeth or the holes respectively will have differences between them (precisely as their faces-bases), yet these differences will be homogeneously (and periodically) distributed in such a way that, whenever they are found in a positive gear, the expected ones (i.e the expected same differences) be found also in its negative conjugate. In Figure 23 for example we can see a group (10) of equal faces which create equal teeth or equal holes respectively. If in our example (12-hedron with 80 triangles on each face) we group all the equal faces, we result in Figure 24.

Our present observational approach concerns the possibility of not being able to construct 3- dimensional gears based on regular polyhedra of a great number (and of other shape) of faces. The symbols we used (except the numbering 1, 2, ... of the figures) consist of: i) letters that denote the type or the role of a gear, that is: p, n for positive and negative 2-dimensional gears, respectively. P, N for positive and negative 3-dimensional gears, respectively. C for concave gears and D for diameters of circular openings. PL, K for planets and kernel, respectively. ii) indexes 1, 2, 3 which refer to the three methods of mounting ∑l, Σ2 and Σ3, respectively, iii) indexes P and N, which denote whether the gear, that the preceding capital letters refer to, is positive or negative.

For example, Pi means positive gear in the ∑l method of mounting and C_N2 means concave negative gear in the Σ2 method of mounting.

A succinct description of the figures is as follows:

Figure 1. 2 dimensional gears - single dimensional transmission of motion.

Figure 2. Dihedral (2-face) teeth in 2 dimensional gears. Figure 3. Trihedral (3 -face) tooth.

Figure 4. Spherical distribution of trihedral teeth - "positive" spherical 3-dimensional gear.

Figure 5. "Positive" and "negative" 2-dimensional gears.

Figure 6. Spherical distribution of trihedral holes - "negative" spherical 3-dimensional gear. Figure 7. Positive - negative gear "conjugation".

Figure 8. Friction damping with bearing balls.

Figure 9. Front view of the "ring mounting" configuration.

Figure 10. Side view of the "ring mounting" configuration.

Figure 1 1. Transmission of motion by "ring mounting". Figure 12. Attachment of arms - their associated motion in the "ring mounting".

Figure 13. 2-dimensional equivalent model of the configuration of "articulated (simple) mounting" .

Figure 14. 3-dimensional view of the configuration of "articulated (simple) mounting".

Figure 15. "Kernel" - "planets". Figure 16. 2-dimensional equivalent model of "crust-kernel". Figure 17. 3-dimensional aspect of "crust-kernel" - configuration of "articulated planet — mounting" . Figure 18. Attachment of arms - their associated motion in the "articulated planet — mounting" . Figure 19. Example of constructive regular polyhedron - Dodecahedron. Figure 20. "Triangulation" of each face of a polyhedron. Figure 21. Transformation of triangles into faces. Figure 22. Pseudo-regular polyhedron with triangular faces. Figure 23. Grouping of equal triangles-faces. Figure 24. Mapping of groups of equal faces.

Claims

1. Spherical 3-dimensional gears, which are characterized from polyhedral (or even conical) teeth (3) (Figure 4) - or polyhedral (or even conical) holes (5) (Figure 6) respectively - homogeneously distributed across a spherical surface, which moreover can transmit motion along all the three dimensions. They are subdivided into "positives" (P), "negatives" (N) and "concave" (C)-positives or negatives. Specifically:

Positives (P): Gears with spherical distribution of polyhedral teeth (3) (Figure 4). Negatives (N): Gears with spherical distribution of polyhedral holes (5) (Figure 6). Concaves (C): Concave spherical gears i.e. those that have in the interior of their spherical surface homogeneously distributed polyhedral teeth-"concave positives" (C_P3) (Figure 17) or polyhedral holes-"concave negatives" (C_N2) (Figure 14).

2. Spherical 3-dimensional gears, according to the claim 1, in which the "conjugates" (P-N) (Figure 7), those that "fit" together so that they transmit the motion between them, must be positive with negative so as to block each other. Obviously the conjugate gears have corresponding geometry (shape, size) and number of teeth-holes.

3. Spherical 3-dimensional gears, according to the claims 1 and 2, in which the imaginary faces-bases of polyhedral teeth or holes, respectively, need not be all equal, but they must be distributed into "groups" (10) (Figure 23) of equals the same way, both in positive (P) gears and their negative (N) conjugates. Thus the teeth of a specific group of a positive gear will fit into the holes of the corresponding group of its conjugate negative gear.

4. Spherical 3-dimensional gears, according to the claims 1 and 2 in which, there are bearing balls (8) on the apexes of faces (of a positive or negative gear), and the edges (9) of the teeth and holes have been smoothed out, in order to reduce the friction (Figure 8).

5. Spherical 3-dimensional gears, according to the claims 1, 2 and 3, which can be mounted and transmit motion to each other (by penetration of teeth (3) of the positive gear into the holes (5) of the negative one (Figures 7, 1 1)) via the following three methods: ∑l. Mounting by ring: Each gear is covered by an embedded (6) ring (7) (Figures 9, 10) (not part of a cylindrical surface but of a spherical one), in such a way that leaves a satisfactory part of the surface (of the gear) uncovered to both its symmetrical circular openings (Di). The uncovered parts of gears (Pi, Ni) (Figures 11, 12) transmit the motion (to each other) by blocking each other, and they can have also arms (Bi) attached upon them (Figure 12), hence their motion is associated with the motion of their arms.

Σ2. Articulated (simple) mounting: There is a gear, positive (P₂) or negative (N₂) (Figures 13, 14), which is "nested" within a concave gear, negative (C_N2) or positive (Cp₂) respectively, with a circular opening (D₂) on its surface. The arms (B₂) (Figures 13, 14) are attached to (embedded into) the two gears, and thus the motion of the arms affects the motion of the gears and vice versa.

Σ3. Articulated-planet mounting: There are four gears "planets" (PL), which support a "kernel" gear (K) which is affected to the extent to which it affects itself the motion of planets (PL) (Figure 15). The centres of the planets (PL) define the apexes of a regular tetrahedron. The system (kernel - 4 planets) is covered by a concave gear-the "crust", which has a circular opening (D₃) (Figures 16, 17) on its surface. In this way the kernel transmits the motion to the crust and vice versa. If the kernel is positive (Kp₃), then the planets will be negative (PL_N3) and the crust concave-positive (C_P3) (Figures 16, 17), but if the kernel is negative, then the planets will be positive and the crust concave-negative. The arms (B₃) (Figure 18) are embedded into the kernel and the crust, and the motion of each arm affects the motion of the other arm.

6. Spherical 3-dimensional gears, according to the claims 1, 2, 3 and 5, which are' characterized by trihedral teeth (3) - or trihedral holes (5) (Figures 4, 6).