Possibilities of Exact Grinding of Conical and Globoid Worms

Transcription

1 International Review of Mechanical Engineering (I.RE.M.E.), Vol. xx, n. x Possibilities of Exact Grinding of Conical and Globoid Worms László Dudás Abstract There is a demand in the machine industry and especially in the production of gearing elements for high precision and exact geometry. These requirements may be achieved using grinding in the finishing process. The grinding of these complicated 3D surfaces needs a special grinding machine and unique technology. This article introduces a special patented worm grinder machine construction and related technology that resolves the problem of geometrically exact grinding of conical and globoid worms. The new method proposed here applies a special grinding wheel having the same number of threads as the worm has. The novelty of this machine lies in the special grinding wheel that is not surface of revolution form, because the working surface of the wheel is generated as a conjugate surface pair of the worm. Keywords: Conical worms, globoid worms, grinding, grinding machine construction I. The Problem of Theoretically Exact Grinding of Non-helical Worms Worm drives with a non-helical worm, especially globoid worm drives and conical worm gearings, have a special advantage since having several connecting teeth at once results in large transmittable torque [1]. These gearings have been analysed in great detail to reveal the characteristics of connection and areas of application. The conical worm gearing was introduced by Bohle, who first applied the registered spiroid name to it [2]. Nelson analysed the Archimedean spiroid [3] and Saari accomplished a mathematical analysis of the kinematical relations [4]. However, researchers have spent less time on dealing with manufacturing problems; the grinding of these types of gearings has long been seen as problematic. The problem of the grinding of non-helical worms, which originates from the changing curvature values of tooth surface and/or changing diameter along the axis, was discovered a long time ago, see [5]. Globoid worms - for example Cone-types - and spiroid worms are characterised by threads having different diameters. This property leads to changing curvature values along the threads of the worm, so application of the classical grinding wheel with surface of revolution form is impossible because the wheel is not able to change its shape during grinding. However, this ability is required for the grinding of globoid or spiroid worms because the contact line between the machined worm and grinding wheel changes its shape in time. If we try to use a traditional surface of revolution grinding wheel, the resulting surface will have geometrical errors. Some researchers have suggested applying grinding wheels of small diameter. This method will give reduced errors and a fairly precise worm surface but the working cycle time between the two dressings of the wheel will be short and this procedure is not suitable for mass production of worms [6]. Two possibilities for reducing geometrical error arising from the manufacturing of spiroid worms have been proposed by I. Dudás [7]. One of the suggested methods determines the optimal position along the thread of the worm for generating a surface of revolution shaped grinding wheel resulting in smaller errors of the same size at both ends of worm. The second, patented technology applies a CNC grinding machine and CNC wheel dresser combination that continuously dresses the wheel during grinding. The dresser dresses the wheel and produces the appropriate wheel-profile at every moment. The accuracy of this method increases with increased frequency of dressing. Actually, the error may be less than the tolerance given for the worm surface. This gives an approximately accurate worm surface, but will not be theoretically exact because of feed of the dressing tool. Clearly this method uses up the wheel quickly and thus is not suitable for mass production. It is possible to eliminate geometrical errors and to produce theoretically exact spiroid worms using an involute surface. These types of worms were developed and introduced by Gansin [8] and Tajnafői [9]. For conical worm drives, the problem of exact manufacturing can be solved by applying a roundabout process for producing the gearing elements. This new method was introduced by Litvin et al. [10]. The method applies a tilted head-cutter or grinding tool for generation of the gear surface and the worm surface alike which results in a localised bearing pattern between gear and worm. The advantage of this method Manuscript received January 2007, revised January 2007

2 is the applicability of a simple surface of revolution form grinding tool for finishing of the worm and the gear. The disadvantage is the discrete head-cutter or grinding tool positioning required for each tooth of the face gear, as opposed to the continuous generating motion of the conical hob. There is a special generating method to produce globoid worms having possibility for exact grinding. In this case, the generating surface for globoid worm and for the gear is a surface of revolution [11]. Despite the studies mentioned above, we can declare that theoretically exact grinding of general spiroid worms and general globoid worms is not yet a solved task, especially in ways suitable for mass production. Here, the author proposes special grindable Archimedean and convolute spiroid worm constructions featuring a cylindrical foot surface. The motion of the grinding wheel is parallel to the worm axis but the pitches are different for the left and right side of the thread. This possibility for grinding results from the fact that the line as a geometrical entity can be translated along itself. We have grinded an Archimedean spiroid worm shown in Fig. 1 using this technology. If we would like to grind general spiroid or globoid worms in a theoretically exact manner, we can apply a new method that will be introduced in the next section. Fig. 1. The grinded Archimedean spiroid worm with cylindrical foot surface Fig. 2. The exact profile of the wheel has to change along the threads II. New Procedure and Grinding Machine Construction It is well-known that most worms with threads of changing diameters are not grindable using wheels of surface of revolution form. A wheel that is set in the perfect position and has the proper profile for that position will cause a local undercut on the other part of the worm thread. This occurrence theoretically results in an edge on the surface of grinding wheel if it goes along the worm threads without turning. The theoretical background is the same for the enveloped edge on the gear surface of Cone-type globoid gearings. If the grinding wheel is turning, this means that the wheel must have different profiles at the different parts of the thread in order to produce a geometrically exact worm surface, as shown in Fig. 2. If the wheel profile cannot change its shape - for example it has an exact profile calculated at the middle of the worm and this profile will work on the two ends of the worm - it will leave an allowance on the thread surfaces with larger diameters and will cause geometrical error by removing a little material under the needed worm surface at the smaller diameters. Consequently, the resulted worm surface will have errors. It is clear that exact machining needs a special grinding wheel that can change its profile while grinding, because of the changing contact line resulting from changing curvature parameters caused by different thread-diameters along the worm. A mathematical expression of the contact line between the wheel and the worm that depends on motion parameter is given in [12]. If we would like to grind general spiroid or globoid worms in a theoretically exact manner, we may apply the technology and grinding machine construction invented and patented by the author [13]. The reason for the problem when we use a surface of revolution grinding wheel is that we would like to grind the threads using the same wheel but they need different wheel profiles. One way to resolve this problem is if the different surface-parts of the worm can connect with different parts of the wheel, such as a special grinding wheel that has a working surface generated theoretically by the worm surface. The new grinding machine construction includes this wheel, see Fig. 3 on the next page.

3 The surface of the wheel and the surface of the worm surface. This idea, the use of non-helical worm surface, appeared later in the invention of Tan, whose proposed method for dressing a grinding worm is documented in [14]. In his method the grinding wheel surface is enveloped by an inner segment of the theoretical surface of a conical pinion with involute teeth, not by the face gear surface that will be grinded with this wheel. The result of the grinding wheel dressing is a toroid-like grinding wheel, also not a surface of revolution form. The gearing itself is similar to a bevel gearing. An analysis of this type of gearing can be found in [15]. Fig. 3. Scheme of the new grinding machine. Notation: 1. Bedplate 2. Z axis, horizontal slide 3. X axis, column 4. Y axis, vertical slide 5. A rotary axis, carrier 6. B rotary axis, workpiece holder 7. C rotary axis, workpiece 8. C' rotary axis, grinding wheel 9. Numerical controller 10. C / C' angle speed ratio adjuster. are conjugate surfaces. We apply the same angle speed quotient between the wheel and the worm for calculation of wheel surface in the manufacturing process. This angle speed quotient is generally equal to 1. In this situation the worm surface and the wheel surface have full contact, that is, every surface point on the worm has a pair on the wheel surface. Grinding under these conditions may seem impossible. However, the large sliding velocity between the worm and the wheel surfaces makes it possible. This sliding is caused by the different diameters of the worm and the wheel, resulting in different speeds at the contact points in spite of the fact that the angle speeds are the same. Using a higher angle speed causes a higher sliding speed. In the above-mentioned example we presumed identical tangential speed directions (opposite angle speed vectors), but opposite tangential speed directions (identical angle speed vectors) can also be applied. In this way, the appropriate sliding or relative speed for material removal can be produced. Auxiliary motions can also be applied in the machining process, such as a continuous feed with an end at the finishing position. The form of the new type of grinding wheel may remind one of a Reishauer-type wheel or can take another form, like a toroid gearingelement. However, there is an important difference between the Reishauer-type wheel and this: the proposed wheel almost never has a helical working II.1. The grinding process using the proposed grinding machine Before grinding, the worm is fixed in the workpiece holder and some of the CNC controlled axes are positioned and held in their given positions. When the machining begins, the CNC equipment controls the relative motion of the worm and the grinding wheel, thus producing the needed continuous conjugate relations between the workpiece and the tool. Auxiliary moving components to produce feed can be used in this integrated relative motion. II.2. Advantages of the new grinding machine It makes theoretically exact grinding of globoid and spiroid worms possible. It can use grinding wheels coated with superhard abrasive grain layer that need no dressing. Before the grinding process dressing of the wheel can be applied for as long as desired, and as accurately as needed, using a dressable grinding wheel. It results in high output and prolonged accuracy because of the large working surface of the grinding wheel. It makes the grinding of worms modificated along the threads possible. This machine is applicable as a standard cylindrical or thread grinder, using a wheel of surface of revolution or as a Reishauer spur-gear grinder using a Reishauer-type grinding wheel. This type of grinding machine is advantageous in the mass production of worm surfaces, as opposed to continuous dressing during grinding, which is useful for production of tool surfaces and smaller series of gearing elements. II.3. Limits of applicability of the new grinding machine This type of grinding machine would be suitable only for mass production because of its expensive and complex wheel.

4 The machine is expensive and complicated because of the seven simultaneously controlled NC axes, although CNC grinding machines with similar complexity do exist, e.g. the Gleason BPG Blade Profile Grinding Machine with seven NC axes and the Koepfer & Sohne Model KX120 CNC Gear Grinding Machine, with eight axes and a multithread grinding worm delivering a continuous grinding process. III. Generating the Grinding Wheel Surface The first step in the above-mentioned technology is the generation of the needed grinding wheel surface. We can use the Surface Constructor software developed by the author for this task, among others. The original theory and the features of the software system have been discussed in [16] and [17]. The capabilities of the software include an analysis of undercut situations. III.1. The structure of the kinematical modelling system IMPLICIT FORM of IMPLICIT FORM of F2 F2i ZKV ZKVi u,v VG R, T VGKV ff Ri, Ti Φ Φi F2glob ZKV R, T F2lok NUMERICAL F2 F2al Φ R - Φ u,v E T T VG VGKV ff NUMERICAL F2i v_a PT ZKVi Ri, Ti The system sketched in Fig. 4 has three main representation levels: the symbolic level, which uses symbolic algebraic representation of the objects in the kinematical model, the numerical level, which stores the given and computed objects using numerical form, and the visualization level, which allows the analysis of views and motion of the objects. The selection and visualisation options are as follows: F2glob: global computational method and appropriate result F2lok: local computational method and appropriate result F2al: computation and visualisation of the occurrences of local undercuts Φ : computation and visualisation of moving path of selected points R-Φ : computation and visualisation of R-Φ functions as a special ability of this software v_a: computation and visualisation of the space of relative speed and acceleration PT : computation and visualisation of axoids. Most of the mentioned notions are discussed in detailed form in [17]. The next section will introduce the determination of the surface of the grinding wheel for finishing an Archimedean conical worm, a type of spiroid worm. The kinematical relations of the grinding machine and the worm surface can be entered into the software program using the motion (i.e. time) parameter. The software uses symbolic algebraic representation so we have to give the two-parametric function of the worm surface or one space curve function using one parameter and a generating kinematical relation between the coordinate system of the curve and the co-ordinate system of the surface. This kinematical relation is given by the second parameter. The generating line of the spiroid worm tooth surface is given in Fig. 5. Φ Φi z70 K70 F2al I T2 ZKV Φ P1 b P x70= P1*cos(BETA) y70= RT z70= P1*sin(BETA) BETA F2lok x70 F2glob PT Fig. 5. Defining the generator line in the K70 co-ordinate system R - Φ v_a Fig. 4. Structure of the gear investigation system The general point P of the generator line b is given by P1 parameter in the K70 co-ordinate system. BETA is the profile angle. The RT throat radius is zero for Archimedean conical worms. The relation between the K70 and K100 co-ordinate systems, where K100 is

5 fixed to the worm, defines the generating conical spiral motion, see Fig. 6. z100 z70 K7 b P2*PA y70 x70 x100 K100 P2*PR x100 K100 Fig. 6. The surface generating motion K70 moves on a spiral path while b sweeps the conical worm surface. P2 is the sweeping parameter, PA is the axial, and PR is the radial screw parameter. After entering the expressions of rotation and two translations, symbolic normal and inverse transformation matrices are created automatically. The kinematical relation between worm and grinding wheel can be entered as a chain of co-ordinate system pairs, sketched in Fig. 7. All of the symbolic algebraic matrices are created and handled automatically. The different constants correspond to the size of individual machine parts while X, Y, Z, A, B, C, C are numerically controlled axes. After data entry, the generating computation can begin to produce the numerical representation of the wheel surface. It is possible to visualise each component of the kinematical chain and study the motions on the screen. Fig. 8 on the next page shows a K7 P2 b x70 y100 y70 sample screen-shot of the analysis of grinding a spiroid worm by the new method. Different display windows can be opened for visualisation of the working surface of the Archimedean spiroid worm, the calculated conjugate wheel surface and the co-ordinate systems of the grinding machine parts. IV. Conclusion The exact finishing of worms is important in order to achieve the designed quality of mesh, to realize a long lifecycle for worm gearings and to guarantee the maximum effectiveness of torque transmissions. This paper has presented a new grinding machine construction and method for grinding non-helical worm surfaces. The accomplished modelling of the grinding of an Archimedean spiroid worm proves the suitability of the proposal. Though the method solves the theoretically exact grinding of conical and globoid worms, the production of this special grinding wheel brings new technical challenges. Since the generation of this type of wheel by dressing seems to be difficult, the method of producing coated wheels having superhard abrasive grain layer seems more feasible. In any case, this type of wheel would be suitable for mass production only. Modelling of the grinding of globoid worms was analysed using the previous version of Surface Constructor. This successful work can be continued using the novel capabilities of the new program to discover the full range of applicability of the proposed grinding method. A y202 y201 y100 y230 y203 Y z202 K202 x202 z201 XCONST2 B K201 x201 z100 XCONST1 C C K100 x100 K230 x230 Z ZCONST z230 YCONST X z203 K203 x203 Fig.7. The kinematical schema of the grinding machine

6 Fig. 8. The Surface Constructor gear investigation software in the midst of spiroid worm grinding analysis Acknowledgements This work was supported by the National Scientific Research Foundation, Grant No. T The author gratefully acknowledges the valuable help of Professor Piotr Skawiński. References [1] E. Eitel, Power Transmission: Even the worm will turn Motion System Design, Penton Media Inc, Ohio, USA, January, pp , [2] F. Bohle, Spiroid Gears and Their Characteristics, Machinery, Vol. 88, n. 6, pp , [3] W. D. Nelson, Spiroid gearing. Part 1, 2, 3, Machine Design, Vol. 33, n. 4, pp , n. 5, pp , n. 6, pp , [4] O. Saari, The Mathematical Background of Spiroid Gears, Industrial Mathematical Series, No. 7, Detroit, pp , [5] E. Boecker, G. Rochel, Messprobleme bei der Fertigung von snecken-getrieben, Werkstatt und Betrieb, No.2, pp , [6] Z. Huan, The Study on a Grinding Method of Straight Outline Hourglass Worm, Machine Design and Research, No.2, pp.71-76, [7] I. Dudás, Grinding possibilities of the conical worm of spiroid driving pairs, Proc. microcad'96 International Computer Science Conf., Miskolc, Hungary, sect. F, 1996, pp [8] V. A. Gansin, Grinding Worm of Involute Spiroid Gearings, Machine Tools and Tools, No.5., pp , [9] J. Tajnafői, Theory and a Few Application of Motion Describing Capabilities of Machine Tools, Ph.D. Thesis, Dept. of Machine Tools, Univ. of Heavy Industry, Miskolc, (in Hungarian) [10] F. L. Litvin, A. Nava, Q. Fan, A. Fuentes, New geometry of face worm gear drives with conical and cylindrical worms: generation, simulation of meshing, and stress analysis, Computer Methods in Applied Mechanics and Engineering, Vol. 191, n. 27, pp , [11] I. Siposs, Globoidschneckengetriebe mit Kreisbogenprofil, Publications of the University of Miskolc, Series C, Mechanical Engineering, Vol. 44, no. 1, pp. 7-19, [12] I. Dudás: Theory and Practice of Worm Gear Drives (Penton Press, London, 2000, p.332) [13] L. Dudás, Grinding machine for grinding free form surfaces located on surfaces of revolution, especially for spiroid and globoid worms, Invention No.OTH19089/ Publishing No. H/3615, Hungarian Patent Office [14] J. Tan, Method for forming a grinding worm for forming a conical face gear that meshes with a conical involute pinion, US Patent No , Assigned to The Boeing Co., Chicago, [15] G. F. Heath, R. R. Filler, J. Tan, Development of Face Gear Technology for Industrial and Aerospace Power Transmission, NASA Contractor Report CR , May [16] L. Dudás, A Consistent Model for Generating Conjugate Surfaces and Determining All the Types of Local Undercuts

7 and Global Cut, Proc. of UMTIK'96 International Machine Design and Production Conf., Ankara, Sept pp [17] L. Dudás, Surface Constructor - a Tool for Investigation of Gear Surface Connection, Proc. of CIM 2003, Wisla, Poland, 2003, pp Author information László Dudás is Associate Professor and Vice Head of the Department of Information Engineering of the University of Miskolc, Hungary. He received an MSc degree in manufacturing engineering, specialisation in system organization from the University of Miskolc in 1980, and a PhD degree in Engineering Science as a scholar of Hungarian Scientific Academy in For his scientific and educational work he has received the Signum Aureum Facultatis medal of University of Miskolc. He is the owner of two inventions and has published more than 50 articles in scientific journals and conference proceedings. His research interests include symbolic algebraic computation, pattern matching, computer graphics and modelling of connection of conjugate surfaces. Dr. Dudás is a member of the editorial board of the Publications of University of Miskolc, Mechanical Engineering Proceedings.