Microsoft Word - B_03_Bodzas_R.doc HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPRÉM Vol. 39(2) pp. 173-176 (2011) CONNECTION THEORY OF CONICAL WORM GEAR DRIVES S. BODZÁS1, I. DUDÁS2 1,2College of Nyíregyháza, Department of Technical Preparatory and Production Engineering H-4400 Nyíregyháza, Sóstói u. 9-11., HUNGARY 1,2University of Miskolc, Department of Production Engineering, H-3515 Miskolc, Egyetemváros, HUNGARY E-mail: bodzassandor@nyf.hu E-mail: illes.dudas@uni-miskolc.hu We worked out a mathematical modell for the production geometry and mathematical analysis of spiroid worm gear drives. This modell is adapted for analysis of spiroid worm gear drives with random profile. Using this modell it could be possible for defining of the equations of cog surfaces, the surface normal vector, contact curves and connection surface in one concrete case. Keywords: spiroid, transformation matrix, normal vector Introduction In technical practice conical worm surfaces, which can be used in many ways, are most widely applied as a function surface of conical worms. The conical worm – crown wheel pairs spiroid drive, can be used for example as jointless drives of robots and tool machines [1]. The jointless drives are attained by simply shifting (setting) the worm in an axial direction. The cog surface of the conical worm of the spiroid drives (Fig. 1) can be attained the same way as that of the cylindrical worm, but besides the axial shift of the hob, a tangential shift must be done depending on the conicity of the worm. Different – evolvent, Archimedean and convolute – helical surfaces can be defined in case of spiroid worm surface similar to the line surface cylindrical worm. Figure 1: Spiroid worm gear drive The dentation of crown wheel is produced with hob which tiler surface is similar to conical worm surface. This is called direct motion mapping. With these modern drive pairs, which are characterized by favourable hidrodinamic conditions, great strength and high efficiency, the energy loss in the gear can be reduced significantly [1]. In power dissipation it is important to apply those cog geometrical characteristics which result in good connection terms. Defining of the spatial coordination systems ϕ1 ϕ1 ϕ2 ϕ2 Figure 2: Evading rotation axis coordinate system for defining of cog surfaces 174 We worked out this model based on the General mathematical model of Dr. Illés Dudás [1, 2]. Defining of minimum four coordinate systems are needed for analysing of motion transmission between evading axis and defining of cog surface describing spatial coordinates: two fixed rotation coordinate systems for the first part K1F (x1F, y1F, z1F) and the second part K2F (x2F, y2F, z2F) and two standing coordinate systems for the first part K1 (x1, y1, z1) and the second part K2 (x2, y2, z2), where the positions of the rotating coordinate systems can be defined (Fig. 2). The rotation axis of the elements are z1 and z2, the turning direction is positive watching from the directions of the axis (opposite for the clock working), the turning angles, the motion parameters are φ1 and φ2. The transformation matrixes between the rotation coordinate system K1F (x1F, y1F, z1F) for the first part and the standing coordinate system K1 (x1, y1, z1) for the first part are: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = 1000 0100 00cossin 00sincos 11 11 1,1 ϕϕ ϕϕ FM (1) ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = 1000 0100 00cossin 00sincos 11 11 1,1 ϕϕ ϕϕ FM (2) The transformation matrixes between the standing coordinate system K1 (x1, y1, z1) for the first part and the standing coordinate system K2 (x2, y2, z2) for the second part are: Figure 3: Connection between the K1 (x1, y1, z1) and the K2 (x2, y2, z2) standing coordinate systems ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ −− = 1000 010 100 001 1,2 c b a M (3) ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − −− = 1000 010 100 001 2,1 b c a M (4) The transformation matrixes between the standing coordinate system K2 (x2, y2, z2) for the second part and the rotation coordinate system K2F (x2F, y2F, z2F) for the second part are: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = 1000 0100 00cossin 00sincos 22 22 2,2 ϕϕ ϕϕ FM (5) ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = 1000 0100 00cossin 00sincos 22 22 2,2 ϕϕ ϕϕ FM (6) The transformation matrixes between the rotation coordinate system K1F (x1F, y1F, z1F) for the first part and the rotation coordinate system K2F (x2F, y2F, z2F) for the second part are: =⋅⋅= FFFF MMMM 1,11,22,21,2 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅+⋅⋅−⋅ ⋅+⋅−⋅⋅− = 1000 0cossin cossincossinsincossin sincossinsincoscoscos 11 2222112 2221212 c ba ba ϕϕ ϕϕϕϕϕϕϕ ϕϕϕϕϕϕϕ (7) =⋅⋅= FFFF MMMM 2,22,11,12,1 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ⋅−⋅⋅−⋅ ⋅−⋅−⋅⋅− = 1000 0cossin cossincossinsinsincos sincossinsincoscoscos 22 1111212 1112112 b ca ca ϕϕ ϕϕϕϕϕϕϕ ϕϕϕϕϕϕϕ (8) The equations of conical thread surface The gr r leading curve is given in the K0 ( , , ) tool coordinate system and its equation of the η coordinate function. That is: )(ηgg rr rr = (9) Since we consider the coordinate an independent variable, the equation of the leading curve is: kjirg rrrr ⋅+⋅+⋅= )()( ηζηηξ (10) Carrying out a pa axial and pr radial helical motion of the K0 ( , , ) coordinate system – which includes the gr r leading curve – along the z axis and the y axis alternatively includes, the leading curve touches a conical helical surface in the K1F (x1F, y1F, z1F) an independent position and equals K0 coordinate system before the helical motion (Fig. 4). 175 ϕ1 ϕ1 ϕ ζ η ξ υ υ Figure 4: Touched thread surface by leading curve The helical surface touched by gr r curve in the K1F (x1F, y1F, z1F) coordinate system is: gFF rMr rr ⋅= 0,11 (11) The transformation matrix between the two coordinate systems is: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ ⋅ − = 1000 100 0cossin 00sincos 0,1 ϑ ϑϑϑ ϑϑ a r F p p M (12) That is: ϑηϑηξ sincos)(1 ⋅−⋅=Fx ϑϑηϑηξ ⋅+⋅+⋅= rF py cossin)(1 (13) ϑηζ ⋅+= aF pz )(1 We gave the equations of the helical surface in the K1F (x1F, y1F, z1F) rotation coordinate system (13). The given describing thread surface ),(11 ϑηFF rr rr = two parameters vector – scalar function can be transformed from the K1F coordinate system to the K2F coordinate system: FFFF rMr 11,22 rr ⋅= (14) 1212 ϕϕ ⋅= i Direct case The ),(11 ϑηFF rr rr = is given, the two parametric vector- scalar function in the coordinate system K1F (x1F, y1F, z1F) for the surface to be generated. In the difference geometry for the independence of the η and θ parameters are essential condition: 011 ≠ ∂ ∂ × ∂ ∂ ϑη FF rr rr (15) The η and θ parameters are the curve line coordinates of the surface. υ=υ3 υ=υ2 υ=υ1 η=η1 η=η2 η=η3 υ η Figure 5: Definition of the surface normal vector We suppose the surface is continuous on the working parts of cogs, it is continuous function of η and θ parameters; two coordinate lines have to be crossed throught on every M point of the surface: a) η = const, b) θ = const the tangents of this lines do not coincide in this point. The working parts of the cog surfaces could contain only general points. The normal vector belong to the surface and the tangent plane will be decided only in the general point. The plane defined by the tangents of the η∂ ∂ Fr1 r and ϑ∂ ∂ Fr1 r parameter lines is the tangent plane of the surface in the given point. The surface normal vector Fn1 v is perpendicular for the tangent plane and it can be defined: ϑη ∂ ∂ × ∂ ∂ = FFF rr n 111 rr v (16) The normal vector in the K1F coordinate system is: ϑϑϑ ηηηϑη ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ × ∂ ∂ = FFF FFFFF F zyx zyx kji rr n 111 11111 1 rrr rr v (17) The relative velocity of the two surfaces can be determined in coordinate system K2F using the transformation between K1F (x1F, y1F, z1F) coordinate system for worm and the K2F (x2F, y2F, z2F) coordinate system for worm gear: ( ) FFFFF rMdt d r dt d v 11,22 )12( 2 rrr ⋅=⋅= (18) The vector )12(2 Fv r should be transformed into coordinate system K1F (x1F, y1F, z1F) to determine the necessary connection surface, so: ( ) FFFFFFFFFF rPrMdt d MvMv 1111,22,1 )12( 22,1 )12( 1 rrrr ⋅=⋅⋅=⋅= (19) where: ( )FFFF Mdt d MP 1,22,11 ⋅= (20) the matrix for kinematic generation. 176 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅⋅−⋅ ⋅⋅⋅ ⋅⋅−⋅−− = 0000 0sincos sinsin01 coscos10 11 11 11 1 iaii ibi ibi P ϕϕ ϕϕ ϕϕ (21) On the connecting cog surfaces of elements, as on tiler each other surfaces the contact curve can be defined by the concomitant solving of expression of Connection I. statement 0 )12()12( 22 )12( 11 =⋅=⋅=⋅ vnvnvn FFFF rrrvrv (22) contact equation and describing cog surfaces vector – scalar function. The connection equation expresses the correlation between the η and θ surface parameters and the φ1 motion parameter, that is ( ) 0,, 11 =ϕϑηFF (23) Defining of the contact curves of the ∑1 and ∑2 cog surfaces in the K1F coordinate system is: ( ) 0,, 11 =ϕϑηFF ),(11 ϑηFF rr rr = (24) The tiler surfaces of contact curves forming the equations of cog surface of the second part are in the K2F coordinate system: ( ) 0,, 11 =ϕϑηFF ),(11 ϑηFF rr rr = (25) FFFF rMr 11,22 rr ⋅= The connection equation is in the K1F coordination system: 0 )12( 11 =⋅ FF vn rv (26) The relative velocity is: =⋅= FF rPv 11 )12( 1 rr ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅+⋅⋅−⋅⋅ ⋅⋅+⋅⋅+ ⋅⋅−⋅⋅−− = 0 sincos sinsin coscos 1111 1111 1111 iaiyix ibizx ibizy FF FF FF ϕϕ ϕϕ ϕϕ (27) with which the connection equation in the K1F coordinate system is: ( ) ( ) ( ) 0sincos sinsin coscos 11111 11111 11111 =⋅+⋅⋅−⋅⋅⋅ +⋅⋅+⋅⋅+⋅ +⋅⋅−⋅⋅−−⋅ iaiyixn ibizxn ibizyn FFFz FFFy FFFx ϕϕ ϕϕ ϕϕ (28) The application of the model We designed a conical worm of which we carried out the virtual model and using the mathematical model (Fig. 6) we carried out the virtual model of the connecting worm gear (Fig. 7). ϕ ϕ ϕ ϕ Figure 6: Defining of the coordinate systems Figure 7: Our designed worm gear drive model Summary We worked out a mathematical model for production geometry and mathematical analysis of spiroid worm. This modell is appropriate for every spiroid worm gear drives with random profile. We designed a spiroid worm gear drive and using this model we carried out the virtual model of this drive pair. REFERENCES 1. I. DUDÁS: The Theory and Practice of Worm Gear Drives. Penton Press, London, 2000. (ISBN 1 8571 8027 5) 2. I. DUDÁS: Gépgyártástechnológia III., Miskolci Egyetemi Kiadó 2005. (ISBN 963 661 572 1) 3. H. JÓZSEF: Untersuchugen zur Anwendung von Spiroidgetrieben. Diss. A. TU. 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