(1) (2) be the position vector for a generic point. If this point belongs to body 2 (with helical motion) its velocity can be expressed as follows:

Transcription

1 The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: /IFToMM.14TH.WC.OS6.025 A Rolling-Joint Higher-Kinematic Pair for Rotary-Helical Motion Transformation J. Meneses 1, J.C. García-Prada 2, C. Castejón 3, H. Rubio 4, E. Corral 5 University Carlos III de Madrid, Spain. Abstract: Higher kinematic pair that converts rotary motion into helical motion is presented as an alternative to the screw joint which is a lower kinematic pair. First, the existence of a rolling (and hence desmodromic) transmission pair for a rotary to helical motion conversion is proven. Then, the corresponding pair of rolling surfaces and the position relative to each other is defined, and depends on the kinematic parameters of the transmission. Finally, a MATLAB code that calculates these surfaces for a given set of transmission parameters, and some examples are presented. Keywords: Higher kinematic pair, Gear transmission, Rotary-helical motion transmission Without loss of generality, we will consider the helical motion of body 2 around the z-axis. The motion is then characterized by the following linear and angular velocities, respectively: Meanwhile, body 1 rotates around the E-axis, which lies in a plane defined by, and is parallel to the y-z plane (see Fig. 1), so that its angular velocity can be expressed as follows: (1) (2) I. Introduction Gear mechanisms are commonly used to transform rotary motion into either rotary or linear motion. In rotary to rotary motion transmissions, where both axes are parallel or intersecting, it is easy to obtain the corresponding pair of rolling contact surfaces, being their axodes. These are two cylinders and two cones, respectively, having either diameter ratio or cone base diameter ratio equal to the transmission ratio. But no set of rolling surfaces exists for transmission between two skew axes (non-parallel nonintersecting) since their axodes do not roll without sliding. However the operating pitch surfaces are defined for gears with crossed axes [1],[2]. Based on these pitch surfaces (cylinders or cones), helical or spiral gears can be used for any two crossed axes at any angle, by a suitable choice of helix angle, in addition to worm and hypoid gears for crossed axes at right angle. This article addresses the existence of a pair of rolling surfaces for transmitting rotary motion into helical motion between crossed axes at any angle. E d II. A Rolling Transmission Kinematics Pair to Convert Rotary into Helical Motion In this section two rigid solids are considered: one performing a pure rotation (body 1), and the other performing a helical motion (body 2). It is shown that, for any values of their angular velocities, for any value of the helical linear velocity of body 2, and for any position of the corresponding axes relative to each other, there is a line of points having the same velocity as belonging to each body in motion. Then, a pair of ruled surfaces (each one attached to each body) exists such that each rolls without slipping over the other. 1 meneses@ing.uc3m.es 2 jcgprada@ing.uc3m.es 3 castejon@ing.uc3m.es 4 hrubio@ing.uc3m.es 5 ecorral@ing.uc3m.es Fig. 1 Kinematic parameters of the rotary motion into helical motion transmission Let be the position vector for a generic point. If this point belongs to body 2 (with helical motion) its velocity can be expressed as follows: This is the velocity field of body 2. As regards to the velocity field of body 1, it can be written as follows: (4) Equating both velocities, the subspace of points having the same velocity in both solids is obtained. These points can be referred to as rolling points : (3)

2 Or in matrix form: (5) (6) Now we will discuss the existence of solutions of the system (6) according to the Rouché-Capelli theorem [3]. The matrix of the system and the extended matrix will be referred to as (A) and (A B) respectively: If the translation velocity v is zero, then equations (8) are fulfilled only if, ie, no transmission exists. However, if body 1 performs a strictly helical motion, with and v 0, then a straight line of points having the same velocity on both solids exists so long as condition (9) is met. This line would be the contact line between the rolling surfaces in the rotation motion into helical motion transmission. In fact the parameters that characterize the transmission, v,,, and d, are not independent, but interrelated by (9). For instance, we can express d in terms of the other four parameters. By replacing components and by and respectively (see Fig. 1), (9) yields: (9) (10) The contact rolling line is then obtained by solving the system (5): (11a,b) (7) Since the determinant of the matrix (A) is always zero, if a solution exists it will be not unique. Actually, the system will be consistent (undetermined) if and only if rank (A) = rank (A B), and its set of solutions will be, a plane or a line if the value of the rank is 0, 1 or 2, respectively. The system will be inconsistent if rank (A) rank (A B). The solutions are discussed by case: i) If and, then rank (A) = 0. Rank (A B) is also equal to zero only if and. This is a trivial solution: no relative motion between both bodies exists. ii) rank (A) 1, as (A) T = (A): if any element of is not zero, neither is it its transpose element, and then it exists a nonzero second order determinant. iii) If or both, then rank (A) = 2. Rank (A B) is also equal to 2 (the solution points form a straight line) only if the following equations are simultaneously fulfilled: Equation (11b) indicates that the rolling line is also parallel to the yz plane, and that it is located at a distance from the yz plane, which is dependent on the direction angle between axes, as well as on the ratio. In equation (11a) angle defines the direction of the rolling line. This also depends on the angle between axes, and on the ratio 2 / 1. Therefore, both rolling surfaces are ruled surfaces tangent to each other along the common generatrix line defined by equation (11). The surface corresponding to the pure rotating motion is the hyperboloid generated by rotating the generatrix about the E-axis, whereas that corresponding to the helical motion is a ruled z-axis helicoid, which is generated by the generatrix line describing that helical motion. The case where implies that and corresponds to the usual rotation transmission between parallel axes: both rolling surfaces are cylinders. Fig. 2 shows the rolling surfaces corresponding to a rotary-helical transmission, where:,,, and (all in S.I. units), as obtained with a MATLAB code. These values determine the distance between axes, d = 31 mm. In this case, the hyperboloid throat diameter is 30 mm, and that of the helicoid is 32 mm. (8a) (8b) Note that the determinant of the remaining 3 rd order submatrix in (A B) is always equal to zero. Equations (8) are fulfilled simultaneously for if and only if:

3 Fig. 2. Rotary-helical transmission rolling surfaces as obtained with MatLab code, in which (S.I. units):,,, and, III. Examples of Rolling Surfaces for Rotary into Helical Transmission Depending on the kinematic properties of the transmission, different geometries of rolling surfaces are found, some of which would be impossible to implement. To begin with, there are two singularities: one occurs when both axes are parallel to each other, that is, when ; and the other occurs when the generatrix line is horizontal (, in this case the ruled helicoid degenerates, additionally d = 0. Other cases that are impossible to implement occur when both surfaces intersect, even as being tangent to each other along their common generatrix line. In Fig. 3 some sets of rolling surfaces are shown for, and, and different values of. As can be seen, for the case when, the rolling surfaces intersect; when, then d < 0 and hence the hyperboloid is behind the helicoid; and for the case, d > 0 and hence the hyperboloid is before the helicoid. In Fig. 4, the rolling surfaces are calculated for, and, with the values of previously stated. Fig. 4 shows that when, the hyperboloid throat diameter is greater than that of the helicoid. Actually, when, contact takes place at the inner face of the hyperboloid. Finally, in Fig. 5 the rolling surfaces are calculated for, and, for the same values of. Now contact takes place at the inner face of the helicoid when (in order to facilitate visualization of this, the first graph in Fig. 5 has been rotated).

4 Fig. 3. Rolling surfaces for,, and Fig. 4. Rolling surfaces for,, and

5 Fig. 5. Rolling surfaces for,, and IV. Conclusion and Future Work In this article, a rolling-joint higher kinematic pair that converts between rotary and helical motions has been presented and investigated. An algorithm that calculates and plots the rolling surfaces for any set of transmission parameters has been developed, and this has been demonstrated by various examples. In order to practically implement this kinematic pair, a ruled involute gear profile can be formed on the hyperboloid, as shown in Fig. 6, whereas the corresponding gear profile on the helicoid will be obtained via interference analysis using a CAD program. Both gears are being made on a 3D printer. References [1] Litvin F.L. and Fuentes A. Gear Geometry and Applied Theory. Cambridge University Press, [2] Radzevich S.P. Theory of Gearing. Kinematics, Geometry and Synthesis. CRC Press, [3] Giaquinta M. and Modica G. Mathematical Analysis. Linear and Metric Structures and Continuity Fig. 6. Ruled involute gear profile on a hyperboloid Acknowledgment The authors would like to thank David K. Anthony for the English language review.