A FORM GRINDING METHOD FOR MANUFACTURING THE VARIABLE TOOTH THICKNESS HOBON CNC GEAR GRINDING MACHINE

Transcription

1 A FORM GRINDING MEHOD FOR MANUFACURING HE VARIABLE OOH HICKNESS HOBON CNC GEAR GRINDING MACHINE Van-he ran Department of Mechanical Engineering, Hung Yen University of echnology and Education, Hung Yen City, Vietnam ABSRAC he variable tooth thickness (V) hob is usually applied for longitudinal crowning work gear surface with twist free tooth flank. It is an important cutting tool for manufacturing the high precision helical gear. However, the manufacturing process for a V hob has not investigated yet. herefore, in this study, we proposed a method for generating profile of V hob by using form grinding wheel on CNC grinding machine. A mathematical model for the tooth profile of V hob is established by setting the center distance between the grinding wheel and hob as a second order function of hob s longitudinal feed movement. A numeral example is presented to illustrate and verify the merits of the proposed form grinding method. Keywords: variable tooth thickness hob, grinding wheel, CNC grinding machine. INRODUCION he grinding process is the most popular finishing process because of its high accuracy, efficiency, and flexibility in tooth flank modification. Actually, two major grinding methods with high productivity for grinding cylindrical workpieces are usually used. One is a single-indexing process called form grinding method applying a form wheel(developed by Höfler, Niles, and Gleason-Pfauter) and other one is a continuous-indexing process called generating grinding method applying a worm wheel(developed by Reishauer and Gleason). he form grinding is the most suitable process for achieving flexible profile and lead modifications on the finished tooth flanks. he newly developed gear grinding machines are all Cartesian-type structures with movable axes controlled by computer numerical control (CNC). Because of state-of-the-art CNC technology, these machines can offer precise simultaneous five-axis movement that enables free-form flank modification. he details the geometric design, tooth surface envelope theory and tooth contact analysis of various gears is published by Litvin and Fuentes [1]. Practical applications of various gears, including gear processing methods, tool design and the strength analysis based on the finite element method. In recently, Hsu and Fong [2] and Hsu and Su [3] proposed a hobbing method for antitwist tooth flank of work gear by using variable tooth thickness (V). Besides, the topologies, contact ellipses and transmission errors of the double-crowned work gear pairs generation by modified hob in gear-hobbing process are investigated. Some related literatures on form grinding process are published. Yoshino et al. [4] proposed a flank correction method for form grinding through compensation of the wheel profile and the position between the wheel and the work gear. Subsequently, in extensive research, Nishida et al. [5] derived the wheel profile using an analytical method. And Kobayashi et al. [6] proposed an optimum contact-line shape by modulating the setting angle of the wheel for tooth trace modification. hen the accuracy of the gear tooth profile is estimated corresponding to the wheel setting errors [7]. You et al. [8] proposed a model that profiles the CBN shape of the grinding wheels using the geometric properties of the contact points. Radzevich [9] proposed a methodology for calculating the parameters of the plunge shaving cutter and the respective form grinding wheel from measured discrete points of the modified gear. Nishimura et al. [10] offer the ZE series for gear grinding machine that feature higher efficiency, higher precision, lower running cost, and easier operation. In 2009, Chiang et al. [11] proposed a simplified two-dimensional numerical simulation method for form grinding the thread on cylindrical workpieces without solving the simultaneous system equations that produce numerically unstable solutions in the presence of undercutting, interference, or double enveloping. And Chiang and Fong [12] proposed a tilt form grinding method and a methodology to determine the minimum tilt angle for avoiding undercutting and secondary enveloping. Profile and lead modifications are more applied in the past years to fulfill high torque load demands and low running noise [13]. After that, Wei et al. [14] established a mathematical model for calculating the processing error of the rotor tooth profile caused by the form grinding wheel profile. Lately, the free-form flank topographic correction method on a five-axis computer numerical control gear profile grinding machine proposed by Shih and Chen [15, 16]. In this paper, a method for manufacturing the V hob is proposed by using form grinding wheel on CNC grinding machine. A mathematical model for the tooth profile of the V hob is established. Numeral results show that the proposed V hob surface generated by form grinding wheel and theoretical V hob surface are close with each other. he proposed V hob surface is much smoother than the theoretical V hob surface. 3501

2 MAHEMAICAL MODEL OF HE GRINDING WHEEL Basically, the theoretical axis profile of a grinding wheel is a projection of the involute curve on traverse plane, as shown in Figure-1. Accordingly, the position vector and unit normal vector of the grinding wheel s right-hand side profile can be expressed in coordinate system Sg ( xg, yg, zg) as follows: r [ x ( ), y ( ), z ( ),1], (1) g g g g and n [ n ( ), n ( ), n ( )], (2) where g xg yg zg x r (cos cos sin( )), (3) g og p p y 0, (4) g z r (sin cos cos( )), (5) g og p p n sin( ), (6) xg p n 0, (7) y g And n cos( ). (8) zh p where is the profile parameters, r og is the radius of the pitch circle of grinding wheel, and p is the profile angle in transverse section, as shown in Figure-1. p r og 1 2 w Figure-1. Surface parameters of the grinding wheel. Since the grinding wheel surface is a surface of revolution, therefore, the grinding wheel profile is used to generate the grinding wheel surface by rotating the profile defined in Eq. (1) around its axis-symmetric z g. By applying the homogeneous coordinate transformation matrix equation from Sg to S w, the locus and unit normal vector of the grinding wheel surface can be represented in coordinate system S as follows: w 3502

3 r (, ) M ( ) r ( ), (9) w w wg w g and n (, ) L ( ) n ( ), (10) w w wg w g he matrix M ( ) describes the coordinate wg transformation from S g to w S w and is represented as: cosw sinw 0 0 sinw cos 0 0 w M wg ( w). (11) and the transformation matrix L wg ( w) is the sub-matrix of Mwg ( w) by deleting the last column and row. After some mathematical operations, the locus of grinding wheel surface can be simplified as follows: r (, ) [ x (, ), y (, ), z (, ),1], (12) w w w w w w w w and n (, ) [ n (, ), n (, ), n (, ),1], (13) where w w xw w yw w zw w x r (cos cos sin( )) cos, (14) w og p p w y r ( cos cos sin( )) sin, (15) w og p p w z r (sin cos cos( )), (16) w og p p n sin( )cos, (17) xw p w n sin ( )sin, (18) y w p w and n cos( ). (19) zw p By using Eqs. (12) and (13), the 3D model of the grinding wheel surface is shown in Fig. 2. Figure-2. 3D model of the grinding wheel surface. MAHEMAICAL MODEL OF HE V HOB a. Mathematical model of the theoretical V hob he profile of the generating surface of the standard rack cutter is the same as that of the standard helical gear. And the theoretical V rack cutter profile is generated by modifying the rack cutter tooth thickness of the standard rack cutter as a second order function of the 2 lead parameter, st bv. t, as shown in Figure-3(a). Accordingly, the position vector and unit normal vector of the right-hand side of the theoretical V rack cutter profile can be expressed in coordinate system St( xt, yt, z t) as follows: 2 t [ xt, yt, zt,1] [ ut cos on, ut sin on bvt, vt,1] r., (20) t xt yt zt on on t on and n [ n, n, n ] [sin,cos, bv. cos ],(21) where u 1 and v 1 are the surface parameters, on is the pressure angle of the standard rack cutter and b is the V coefficient of rack cutter, as shown in Figure-3(a). According to Figure-3(b), by applying the homogenous coordinate transformation matrix equation, the locus and unit normal vectors of V rack cutter surface represented in coordinate system Sht ( xht, yht, z ht ) are attained by cost cos o1sint sin o1sin t rog(cos t t sin t) sint cos o1cost sin o1cos t rog(sint t cos t) rht ( ut, vt, t ) Mht ( t ) rt ( ut, vt ) t( ut, vt), 0 sino1 coso1 0 r (22)

4 and n ( u, v, ) L ( ) n ( u, v ), (23) ht t t t ht t t t t where L ht ( t ) is the upper-left (3 3) sub-matrix of the (4 4) homogeneous coordinate transformation matrix M ( ). ht t According to the theory of gearing, the equation of meshing between the right-hand side rack cutter surface and left-hand side hob surface can be obtained by [ xht ( ut, vt, t ), yht ( ut, vt, t ), zht ( ut, vt, t )] fht ( ut, vt, t ) n ht ( ut, vt, t ) 0 (24) t x t z t x 3,t z3 zt x 1,2 z 2 s bv. t 2 t o1 y 3 y 2 on u t v t O 3,t y t l r. t og t x ht O 2 z ht,1 r og Ot yt O ht,1 y ht t y 1 (a) (b) Figure-3. Surface parameter of the V rack cutter and coordinate systems for the schematic generation of the theoretical V hob. b. Mathematical model of the proposed V hob surface generated by form grinding wheel For generating the profile of V hob, a 3D model of CNC form profile grinding machine is presented in Figure-4. his model has six-axis movements that need for grinding profile of V hob: rotational axis A 1 is applied for setting the crossed angle between the grinding wheel and hob axes, rotational axes B 1 and B2 are applied for controlling rotation angles of the grinding wheel and hob, respectively, horizontal axis X 1 is applied for controlling the radial feed along the center distance between the grinding wheel and hob, vertical axis Y 1 is applied for setting the position of the grinding wheel and hob, axis Z 2 is applied for controlling the longitudinal feed along hob axis. he coordinate systems for grinding process of hob are shown in Figure-5, wherein coordinate systems Sw( xw, yw, z w) and h( h, h, h) connected to the grinding wheel and hob, respectively, S x y z are rigidly while the coordinate system (,, ) c c c c S x y z is rigidly connected to the frame of grinding machine, and Sa( xa, ya, z a), Sb( xb, yb, z b) and Sd( xd, yd, zd) are auxiliary coordinate systems for simplification of coordinate transformation. he crossed angle hw of the hob and work gear axes is usually a machine-tool setting. o generate the profile of V hob, this study proposes a grinding method by changing the center distance between the grinding wheel and hob as a second order function of hob s longitudinal feed movement in grinding process. he center distance between the grinding wheel and hob is proposed as follows: 2 t o c h. E E a l (25) where Eo is the original center distance between grinding wheel and hob, ac is the center distance variation coefficient and lh is the hob s longitudinal feed movement. 3504

5 A 1 Y 1 B 1 X 1 B 2 Z 2 Figure-4. 3D model of CNC form profile grinding machine. x cba,, y w W h O baw,, hw z b y a y z b aw, x w l h E t E t x h h x d O c W g yc yd O dh, z hd, y h Figure-5. Coordinate systems for the form profile grinding machine. he position vector r w(, w) (see Eq. (12)) and unit normal n w(, w) (see Eq. (13)) present locus of grinding wheel surface, respectively. By applying the homogeneous coordinate transformation matrix equation from S w to S h, the locus and unit normal vector of grinding wheel surface can be represented in coordinate system S h as follows: r (,,, l ) M (, l ) r (, ), (26) h w h h hw h h w w and n (,,, l ) L (, l ) n (, ), (27) h w h h hw h h w w where M cosh cos hwsinh sin hwsinh Et cosh sinh coshwcosh sin hwcosh Et sin h (, l ). 0 sinhw coshw lh hw h h (28) and the transformation matrix L hw( h, l h ) is the submatrix of M hw ( h, l h ) by deleting the last column and row. After some mathematical operations, the locus of grinding wheel surface can be simplified as follows: 3505

6 r (,,, l ) [ x (,,, l ), y (,,, l ), z (,,, l ),1], (29) h w h h h w h h h w h h h w h h and n (,,, l ) [ n (,,, l ), n (,,, l, n (,,, l ),1], (30) h w h h xh w h h yh w h h zh w h h where xh ( Et xw) cosh ( zw sin hw yw cos hw )sin h, (31) yh ( yw cos hw zw sin hw) cos h ( Et xw) sin h, (32) zh lh zwcoshw ywsin hw, (33) nx nx cos h ( nz sin hw ny cos hw) sin h, (34) h w w w n ( n sin n cos ) cos n sin, (35) hw hw h h y z y x h w w w and n n cos n sin. (36) hw zh zw yw In the gear grinding process, there are two independent kinematic parameters l h and h that need for finish tooth flank of work gear. herefore, there are two equations of meshing between the hob and the grinding wheel as follows: hw xh(, w, h, lh), yh(, w, h, lh), zh(, w, h, lh) f1(, w, h, lh) n h 0, (37) h xh(, w, h, lh), yh(, w, h, lh), zh(, w, h, lh) and f 2 (, w, h, lh) n h 0, (38) l h he tooth surface of the hob can be defined by solving Equations (29), (30), (37) and (38), simultaneously. NUMERICAL EXAMPLE AND DISCUSSIONS he purpose of this example is to verify the merits of proposed form grinding method. he basic data of the hob and the form grinding wheel are given in able- 1. he grinding machine setup data are calculated according to the basic meshing conditions as illustrated in Ref. [1]. able-1. Basic data of the hob and the form grinding wheel. Items Value (Unit) Number teeth of hob ( N 1) 1 Normal module ( m ) 2 pn Normal circular-tooth thickness of hob ( s pn1 ) mm Normal pressure angle ( ) 20.0 Helix angle of hob ( p1 ) 87.8 R.H. pn Hob pitch radius( r p1 ) mm Hob form radius( r f 1) mm Hob outer radius( r a1 ) mm Hob length ( F w1 ) 135 mm Grinding wheel radius( r og ) 41.0 mm Center distance between grinding wheel and hob ( E ) o mm Crossed angle between grinding wheel and hob( hw ) 87.8 Center distance variation coefficient ( a ) V coefficient ( b ) he opography of the theoretical V hob surface and proposed V hob surface generated by form grinding wheel are shown in Figures 6-7, respectively. he maximum deviation of the tooth flank of the proposed V hob surface, M d 10.23m, is approximately equal to that of the theoretical V hob surface, M d 10.0m. And the ratio of deviation of the proposed V hob surface, Rd 0.66, is also approximately equal to that of the theoretical V hob surface, Rd It reveals that the proposed V hob surface generated by form grinding wheel fits well with the theoretical V hob surface. In addition, the proposed V hob surface is much smoother than the theoretical V hob surface, as shown Figure

7 Gear root Gear root op land unit: m Figure-6. opography of the theoretical V hob surface (maximum deviation of the tooth flank M = 10.0 m and ratio of deviation R 0.67 ). d d Gear root Gear root op land unit: m Figure-7. opography of the proposed V hob surface generated by form grinding wheel (maximum deviation of the tooth flank M 10.23m and ratio of deviation R 0.66 ). d d CONCLUSIONS In this study, a new form grinding method for manufacturing the V hob is proposed by setting the center distance between the grinding wheel and hob as a second order function of hob s feed movement (Eq. (25)). Mathematical models for the tooth profile of the theoretical V hob and proposed V hob are established. he compared results show that the proposed V hob surface generated by form grinding wheel fits well with the theoretical V hob surface. he proposed V hob surface is much smoother than the theoretical V hob surface. REFERENCES [1] F. L. Litvin, A. Fuentes Gear Geometry and Applied heory. 2 nd ed., Cambridge University Press, Cambridge, UK. [2] R.H. Hsu, Z.H. Fong Novel Variable-tooththickness Hob for Longitudinal Crowning in the Gearhobbing Process. hirteenth World Congress in Mechanism and Machine Science, Gearing and transmissions, Guanajuato, Mexico. 3507

8 [3] R.H. Hsu, H.H. Su ooth Contact Analysis for Helical Gear Pairs Generated by a Modified Hob with Variable ooth hickness. Mechanism and Machine heory. (71): [4] H. Yoshino, K. Ikeno Error Compensation for Form Grinding of Gears. ransactions of the Japan Society of Mechanical Engineers Part C. 57(543): [5] N. Nishida, Y. Kobayashi, Y. Ougiya, N. sukamoto Profile Calculation Method for Form Grinding Wheels (1st report). Grinding Wheels for Helical Gears and Avoidance of Interferences. Journal of the Japan Society for Precision Engineering. 58(4): [6] Y. Kobayashi, N. Nishida, Y. Ougiya and H. Nagata ooth race Modification Processing of Helix Gear by Form Grinding Method. rans. Jpn. Soc. Mech. Eng., Ser. C. 61(590): [13] A. ürich Producing profile and lead modifications in threaded wheel and profile grinding. Gear echnology 27 (1): [14] J. Wei, Q. Zhang, Z.Z. Xu and S. K. Lyu Study on Precision Grinding of Screw Rotors Using CBN Wheel. Int. J. Precis. Eng. Manuf. 11(5): [15] Y.P. Shih, S.D. Chen A flank correction methodology for a five-axis CNC gear profile. Mechanism and Machine heory. (47): [16] Y.P. Shih, S.D. Chen Free-Form Flank Correction in Helical Gear Grinding Using a Five- Axis Computer Numerical Control Gear Profile Grinding Machine. ASME J. Manuf. Sci. Eng. 134(4): [7] Y. Kobayashi, N. Nishida, Y. Ougiya and H. Nagata Estimation of Grinding Wheel Setting Error in Helical Gear Processing by Form Grinding. rans. Jpn. Soc. Mech. Eng., Ser. C. 63(612): [8] H.Y. You, P.Q. Ye, J.S. Wang, X.Y. Deng Design and application of CBN shape grinding wheel for gears. International Journal of Machine ools and Manufacture. (43): [9] S.P. Radzevich Computation of parameters of a form grinding wheel for grinding of shaving cutter for plunge shaving of topologically modified involute pinion. Journal of Manufacturing Science and Engineering, ransactions of the ASME. 127(4): [10] Y. Nishimura, Y. Ashizawa,. Katsuma, Y. Yanase, K. Masuo Gear Grinding Processing Developed for High-Precision Gear Manufacturing. Mitsubishi Heavy Industries echnical Review. 45(3): [11] C.J. Chiang, Z.H. Fong, J.. seng Computerized simulation of thread form grinding process. Mech Mach heory. 44(4): [12] C.J. Chiang, Z.H. Fong Undercutting and interference for thread form grinding with a tilt angle. Mech Mach heory. 44(11):