2016 International Conference on Mechanics Design, Manufacturing and Automation (MDM 2016) ISBN: 978-1-60595-354-0

Research on NC Grinding Tooth Surface Morphology of Spiral Bevel Gear Rui MING1,a, Hong-Zhi YAN2,b, Xing-Zu MING3,c,*, Jin-Hua LIU3,d 1School

of Information and Electrical Engineering, Hunan University of Science and Technology, Xiangtan, 411201, China

2College

3School

of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, China

of Mechanical Engineering, Hunan Industry University, Zhuzhou 412007, China

a844153340qq.com, [email protected], [email protected], [email protected]

*Corresponding author Keywords: Surface morphology, Simulation, Experiment, Spiral bevel gear, Grinding.

Abstract. Based on the analysis of expanding cup wheel topography, grinding wheel model, whose grain diameter and space position on the wheel surface are distributed randomly, is established and used to simulate grinding tooth surface topography. Based on the analysis of the formation mechanism of grinding tooth surface topography, the relative movement relationships between grinding wheel and tooth surface, and the interference and superimposition principle of grain tracks, the modeling and simulation of grinding tooth surface morphology of spiral bevel gear are made. The simulation morphology of grinding tooth surface of spiral bevel gear is obtained by Matlab software. By the comparative study on experiment, The results show that the simulation morphology and experimental observing one is basically identical. These indicate that the topography simulation model of grinding tooth surface of spiral bevel gear established in this paper is more accurate and reasonable. Introduction At present, in order to assure the quality and precision of gear grinding and improve the transmission performance, spiral bevel gears are basically grinded on multi-axis NC machine tool. If grinding surface morphology was researched in a traditional method, a lot of repetitive experiments needed to be done. With the development of computer technology, the research of grinding was greatly simplified. Now the two kinds of most common methods are used by Matlab software[1] and Visual C++ programming software by means of 3D graphics software standard interface OpenGL[2], by which the computer simulation study on the grinding process are carried on. Matlab software was used for modeling and analysis in this paper. Basing on the profile formation mechanism of grinding tooth surface of spiral bevel gear, the research on modeling of geometrical morphology of grinding tooth surface was made in this paper. Firstly, the grinding wheel topography was analyzed. The sphere structure was used as the basic shape of grain, and considering the randomicity of the size and position distribution of grain, the simulation model of grinding wheel was constructed as possible as close to the truth. Next, based on the grinding method, parameters and the no clearance meshing movement relationships between cutting tool and gear, the theoretical model of tooth surface of spiral

bevel gear was obtained. Afterwards, according to the movement interference of grains of grinding wheel model and tooth surface, and formation mechanism of tooth surface contour, the simulation model of grinding tooth surface topography of spiral bevel gear was established. By using Matlab software to compile a computer program, simulation research on the morphology of grinding tooth surface was done; the simulation morphology and experimental one of tooth surface were compared and analyzed. Building Mathematical Model of Grinding Wheel Topography Analysis of Grinding Wheel When NC grinding, surface topography of grinding wheel had a direct impact on grinding surface topography, so the rationality and accuracy of surface topography modeling of grinding wheel would directly determine the accuracy of modeling simulation of grinding surface topography. Grinding wheel was made of grits of porous body bonded by binder. In the various influence factors, the grain granularity and organization size of grinding wheel had a great influence on surface topography of grinding wheel, and the surface topography and grinding parameters had a considerable influence on surface morphology of grinding tooth. The size, shape, direction of the grain which distributing in the grinding wheel surface were not identical, protrusion height of abrasive bumps changed randomly, but after the grinding wheel was appropriately modified by grinding, the protrusion height of grain was basically obeyed normal distribution[3]. The grain shape of grinding wheel had great influence on the surface topography and accuracy of grinding tooth, this article assumed that the grinding grain shape was spherical and its diameter was distributed randomly. When analyzing the surface morphology of grinding tooth, surface topography of grinding wheel played a decisive role. Only understanding the topography of grinding wheel in detail, the surface morphology of grinding tooth could be studied more accurately. If the grinding wheel granularity M was known, the grain maximum size dgmax and average size dgavg could be approximately calculated by the following formula[4] dg max (mm) = 15.2M -1

(1)

dgavg (mm) = 68M -1.4

(2)

Experimental investigation[5] showed that the dgmax and dgavg which were calculated by the formula (1), (2) were very close to maximum protrusion height hmax and average protrusion height havg of grinding grain, and grain protrusion height Hi (also the grain diameter) was obeyed (μ,σ2 ) Gaussian distribution, its averageμ was davg, the standard deviation wasσ, that was m = dg avg

(3)

s = (dgmax - dgavg ) / 3

(4)

The distance between the grinding grain was related to the organization size S of grinding wheel. According to S, the percentage density Vg of grinding wheel was gained approximately as follow

Vg = 2(32 - S )

(5)

Assumed that the grinding grain distribution space Δwas uniform, then  6

dg 3 avg  Vg / 100   3

(6)

Thus, Δ(mm) was calculated as follow D = 137.9M -1.4 3

p 32 - S

(7)

As shown in Fig. 1, Np was the grinding grain number intersected randomly by per unit area AB inside grinding wheel, namely it was the grinding grain number of 1×dgavg volume in grinding wheel surface, then Np ⋅

p 3 dg avg = Vg /100 ⋅ (1´ dg avg ) 6

Np =

(8)

6Vg 100p dg 2 avg

(9)

Figure 1. Grains Distribution inside Grinding Wheel. Therefore, the average plane grain interval ω(mm) of grinding wheel surface was w=

1 Np

= 196.3M -1.4

p 32 - S

(10)

In general, the average plane grain interval ω was much bigger than grinding grain distribution space Δ. Firstly, the reason for this was that the grinding wheel dressing made the grinding grain of wheel surface fall off, lead to own loss, and the ω was bigger than the grain interval inside wheel grinding [6]. Secondly, the smaller the grinding wheel granularity and the lower the hardness was, the greater the differential value betweenω and Δ was. Modeling of Grinding Wheel When grinding the spiral bevel gear, grinding wheel of microcrystalline ceramic alumina (SG) was often used. Here grinding of spiral bevel big gear with expanding cup wheel SG60-JV [7] was taken as an example for modeling analysis. Through analysis of the grinding wheel topography, the grain diameter of grinding wheel was obeyed normal distribution(μ,σ2) and grains was distributed randomly in the area, then the wheel model was established for grinding surface topography simulation. Because actual grinding wheel was larger and it had lots of

grains, it was very difficult to establish a complete wheel model for simulation computing, and grinding grains of a circle along bus direction of conical surface of grinding wheel was used to establish the model. Here grains of a whole circle of grinding wheel surface within 5dgavg width in vertical grinding direction and along 2dgavg in the radial thickness direction was used to model, grains distribution of whole circle of grinding wheel was shown in Fig. 2. Assuming that the grinding grains on the surface space of grinding wheel was distributed evenly, local grains distribution topography of grinding wheel was shown in Fig. 3. If the equivalent mean radius from the cone circle of grinding wheel modeling was rs, the grinding grains number N of in modeling area VT of grinding wheel surface was VT = 2prs ⋅ 5dgavg ⋅ 2dgavg N=

VT

(11)

D3

(12)

Figure 2. Grains Distribution of Whole Circle of Grinding Wheel.

Figure 3. Local Grains Distribution of Grinding Wheel.

In simulation computing, the corresponding spatial locations of grain could be calibrated by the grain sets of grinding wheel surface with its center coordinates, and grain radius was generated randomly. If the center coordinate and radius were formulated by (x, y, z) and r, the mathematical model of grinding wheel topography was  x1 x GN ( x, y, z, r )   2    xN

y1

z1

y2

z2





yN

zN

r1  r2    rN 

Modeling and Simulation of Grinding Tooth Surface Morphology of Spiral Bevel Gear Theoretical Modeling of Tooth Surface Motion Coordinate System of Tooth Surface Grinding of Spiral Bevel Gear. In order to study on the morphology of tooth surface grinding of spiral bevel gear, theoretical modeling and analysis of tooth surface should be done. Here big spiral bevel gear was grinded by double-sided method in six axes five linkages CNC grinding machine with expanding cup wheel, and this was shown in Fig. 4. Basic parameters of big spiral bevel gear were listed in table 1, and their machine adjusted parameters were listed in table 2.

Figure 4. Structure of CNC Grinding Machine. Table 1. Basic Parameters of Big Gear. Item Number of teeth Module m(mm) Pitch circle diameter (mm) Spiral Angle β(°) Pressure angle α(°) External cone distance Re(mm) Pitch cone distance R (mm) Pitch cone angle(°) Root cone angle (°) Body cone angleδa1(°) Tooth face width b2(mm) Tooth dedendum(mm) Tooth addendum (mm)

Parameters z2=46, left spin 8.22 d2=378.12 35 20 198.857 8 170.282 8 δ2=71.939 5 δf2=68.658 5 b2=57.15 hef2=11.4 hea2=4.12

Table 2. Machine Adjusted Parameters of Big Gear. Item Nominal cutter diameter r 0(mm) Cutter tooth profile angle αp(°) Tooth blank setting angle(°) Radial cutter spacing(mm) Angular cutter spacing(°) Bed position(mm) Perpendicular gear position(mm) Axial gear position(mm) Roll ratio io1

Adjusted parameters 152.4 22 γ2=68.658 5 Sr2=149.83 q2=56.42 XB2=0 E02=0 Xg2=0 io2=0.95

In order to model tooth surface, the cone of expanding cup wheel was considered as the one of cutting edge of cutter head. Based on the kinematic relationships between grinding wheel and gear, and no clearance meshing principle, the theoretical model of tooth surface was established. Grinding motion coordinate system [8] on the NC machine tool was set up, and it was shown in Fig. 5. In gear grinding machine fixed coordinate system Sm2(Om2xm2ym2zm2), its origin Om2 was located in the center of the machine tool, and the plane Om2(xm2ym2zm2) was located in machine tool and in a plane perpendicular to the spindle of imaginary shape wheel. Coordinate system Sc2(Oc2xc2yc2zc2) was fixed on imaginary shape wheel production, and Sc2 rotated on axis zm2 of Sm2 when generating process. At the initial position, Sc2 overlapped with Sm2, and φ was the current angle for Sc2. Coordinate system Sp2(Op2xp2yp2zp2) was fixed on Sc2, it was connected in

the plane installed for wheel, and the origin Op2 was located in the center of expanding cup wheel. Among these parameters, q2 was angular cutter spacing, Sr2 was radial cutter spacing. Sa2(Oa2xa2ya2za2) was auxiliary coordinate system which was used to describe the installation position of the big gear on the machine, the coordinate system was connected to Sm2, and the angle between xa2 and xm2 was γ2 (root cone angle of big gear). Coordinate system S2(O2x2y2z2) was connected to the machining big gear, the origin O2 was the pitch cone vertex of the machining big gear. S2 rotated on axis xa2 of Sa2 when generating process. At the initial position, S2 overlapped with Sa2, and ф was the current angle of S2, XB2 was bed position, Xg2 was axial gear position, ω(g) was the angular velocity vector of rotating table, ω(2) was the angular velocity vector of machining big gear rotating.

Figure 5. Coordinate System of Big Gear Grinding. Inside and outside grinding cone of double side expanding cup wheel was shown in Fig. 6. According to the installation location of wheel in NC machine tool and the coordinate system of tooth surface grinding above, the wheel work cone of big gear grinding was expressed in the coordinate system Sp2 as follow. é( R p + s p sin a p ) cos b ù ê ú ê ( Rp + s p sin a p )sin b ú ê ú rp 2 ( s p , b ) = ê ú s cos a p p ê ú ê ú 1 ëê ûú

(13)

At the point of grinding contact between wheel and gear in the coordinate system Sp2, the unit vector tp2 along the bus direction of wheel grinding edge and unit normal vector np2 of wheel work cone could be represented respectively as follow

ésinα p cosb ù ê ú t p2 = êê sinα p sinb úú ê cosα ú p êë úû

(14)

écosα p cosb ù ê ú n p2 = êê cosα p sinb úú ê -sinα ú p ëê ûú

(15)

Figure 6. Two Cone of Grinding Wheel. In above formula, αp was the cutter tooth profile angle, choosing negative value for the outside cutter tooth profile angle αw (concave grinding of big gear) and positive value for the inside cutter tooth profile angle αn (convex grinding of big gear). Rp was the tool tip radius for processing big gear, and Rp=r0±W2/2 (choosing ‘+’ while outside edge processing, and choosing ‘-’ while inside edge processing in the formula). Among them, sp was the distance from tool tip on wheel work cone to grinding contact point along the bus direction, and β was grinding wheel angle. Expression of Grinding Wheel Working Face and Coordinate Transformation. In the coordinate system S2,grinding wheel surface r2, unit vector t2 along the bus direction and unit normal vector n2 were represented respectively as follow r2 ( s p , b , j ) = M 2 a 2 M a 2 m 2 M m 2 c 2 M c 2 p 2 rp 2 ( s p , b )

(16)

t2 ( , )  L2a2 La2m2 Lm2c2 Lc2p2tp2

(17)

n2 ( , )  L2a2 La2m2 Lm2c2 Lc2p2np2

(18)

Among them, M2a2, Ma2m2, Mm2c2, Mc2p2 were transformation matrix between the coordinates, and matrix L2a21, La2m2, Lm2c2 and Lc2p2 were gotten by removing their last line and last column of matrix M respectively. Matrix M were 1 0 M 2a2 =  0  0

0 cos 

0 -sin 

sin  0

cos  0

 cos  2  0 M a2m2 =    sin  2   0  cosφ  -sinφ M m2c2 =   0   0

0 0  0  1 ,

0 1

sin  2 0

0 0

cos  2 0

sinφ cosφ 0 0

0 0 1 0

0 0  0  1

,

 X B 2 sin  2  X g 2   E 02    X B 2 cos  2  1 ,

1 0 M c2p2 =  0  0

0 1 0 0

0 0 1 0

S r2 cosq 2  S r2 sinq 2   0  1 

In the generating process of big gear, roll ratio io2 was constant, the relationship between the angle φ of big gear and the angle Ф of rotating table was

f = i02 ⋅ j Establishment of the meshing equation of grinding movement. When processing big gear, the meshing equation was established in static coordinate system Sm2 of the machine tool as follows (19) In above formula, nm2 was the normal of wheel grinding surface, and vm2(g2) was relative speed between big gear and grinding points in the coordinate system Sm2. Assumed that the rotating speed of imaginary flat shape wheel、the rotating speed of gear, wheel cone and the mesh point of gear in the coordinate system Sm2 were respectively wm 2( g ) , wm2(2) , rm 2 and Rm2 , made wm 2(2)  1 , then wm 2 ( g )

rm 2 =M m2c2 M c2p2 rp2 (s p , )

é ù ê ú ê 0 ú ê ú = êê 0 úú ê 1ú ê- ú ê i ú ë o2 û ,

wm 2

(2)

é- cos g 2 ù ê ú =ê 0 ú ê ú ê - sin g ú 2û ë ,

, Rm 2

It was easily known that

rm 2

  X g 2 cos  2     E02    X B 2  X g 2 sin  2   

was gained by removing the last line of

rm¢2 ,

thus

nm2 ( , )  Lm2c2 Lc2p2 np2

(20)

vm2(g2)  m2(g)  rm2 -wm2(2)  (rm2  Rm2 )

(21)

Mathematical Model of Grinding Tooth Surface. After the tooth surface of spiral bevel gear was rotated and projected on a axial section, its rotating projection could be gotten. Now origin O2 of the coordinate system S2 was taken as the origin, rotating projection coordinate system xuOuyu was established by regarding the positive direction of x2 axis as the positive direction of x, axis yu was radial rotation projection of tooth surface, and its rotation projection and grid nodes were shown in Fig. 7. then ì ï ï xu = x2 í 2 2 ï ï î yu = y 2 + z 2

(22)

A grid node Pij on the rotating projection plane of tooth surface was chosen randomly, its corresponding point on tooth surface was M0, then the tooth surface model within the neighborhood M0 could be gotten basing on the above analysis.

Figure 7. Rotation Projection and Grid Nodes of Tooth Surface. By substituting Eq.(20), (21) in Eq.(19), x2, y2, z2 in Eq.(16) in Eq.(22), and combining Eq.(19), (22), a three element equations about (Sp, β, φ) could be gotten. Then, substituting the discrete coordinates (xu, yu) in rotating projection plane in the above three equations and solving, (Sp, β, φ) could be inferred. Finally, by substituting (Sp, β, φ) in Eq.(13), the grinding point coordinates of wheel could be gotten correspondingly, then the equivalent radius of wheel at the grinding point could be calculated. By substituting the radius in Eq.(16), the tooth surface coordinates of big gear could be gotten, and the tooth surface model was established. Modeling, Simulation Analysis of Morphology of Grinding Tooth Surface The morphology formation of grinding tooth surface was a dynamic and highly nonlinear complex process. In order to make the tooth surface modeling and simulating feasible, some assumptions and simplifications were made as fellow: (1)Assuming that the grinding grains of wheel were spherical, and evenly distributed randomly in the wheel surface space. (2)Topography of grinding tooth surface was formed by pure cutting completely. (3) Ignoring the grain wear of grinding wheel and vibration of machine tool. (4)Assuming grinding wheel was rigid perfectly, inelastic concession and so on. Basing on the above assumptions, under the certain grinding parameters, movement tracks of many grains randomly distributed in grinding wheel surface were interposed and superimposed, and micro geometry morphology of tooth surface of spiral bevel gear was formed eventually. Here modeling and analysis were made to chose the point M0 of tooth surface within the neighborhood 4dgavg×5dgavg[6]. According to the equivalent average radius rs of wheel at corresponding point M0, the wheel topography model within the grinding area was established. Based on the relative motion relationships between grinding wheel and gear in the grinding process and motion simulation, the tooth surface morphology corresponding to different grinding parameter could be gotten. In the simulation calculation of movement interference, the calculation started from the time while the lowest point of wheel model began to enter the research area of tooth surface to the time while the lowest point left the research area, and the calculation on wheel grain was made once while every rotating half a grain size. Namely wheel grains which might participate in the research area of grinding tooth surface at every position of wheel were kept, and every grains were superimposed (the grains superposition was the union solution of grains set kept every calculating). The grids within the research area of tooth morphology were discretized, and according to the sphere equation of grain, the lowest point which grains could be cut at each discrete point was calculated. Namely the point was one

forming on the grinding tooth surface[8]. Lastly, M files for simulation calculation were programmed by the software Matlab. Here the grinding wheel granularity M was 60, and the modeling and simulation of grinding tooth surface of spiral bevel big gear were carried out under different grinding parameters (the wheel grinding speed vs, grinding depth a, generating speed v2). When a and vs were constant, the simulation morphology of big gear convex under different v2 were shown in Fig.8.

(a) v2=1.5m/min

(b) v2=2.0m/mi

(c) v2=5.6m/min

Figure 8. Simulation Morphology of Big Gear Convex under Different v2 (a=0.02mm, vs=25.6m/s). Experiment Study of Grinding Tooth Surface Morphology Experiment testing of grinding tooth surface morphology Experimental gear material was selected 20CrMnTi, and spiral bevel big gear was grinded by formation method in six axes five linkages CNC grinding machine installed expanding cup SG wheel with a 30 degree cone angle. Grinding wheel diameter D was 300mm, basic parameters and machine adjusted ones of big gear were respectively shown in table 1, table 2. The grinding way was cut-out inverse grinding, the requirement of hardening layer depth was 0.80 to 1.20mm, tooth surface hardness was 58 to 64 HRC, and heart hardness was 35 to 42HRC after grinding. Experimental research on tooth surface morphology of grinding specimens of spiral bevel big gear under different processing parameters was made by using 3D digital electron microscope HIROX KH-7700. Experimental testing was gotten. When a and vs were constant, the morphology of big gear convex under different v2 were shown in Fig. 9.

(a)v2=1.5m/min

(b)v2=2.0m/min

(c)v2=5.6m/min

Figure 9. Morphology of Big Gear convex Under Different v2(a=0.02mm,vs=25.6m/s,×400). Comparative Analysis of Experiment and Simulation Morphology Grinding tooth surface morphology had a direct relationships with grinding wheel topography, grinding wheel velocity vs, grinding depth a, and generating speed v2, and its parameters would directly affect the surface quality. According to contrastive analysis between simulation and experiment of tooth surface morphology under different grinding parameters,

the results were gotten. The tooth surface microstructure texture of experimental observing and one of simulation were basically identical, and the law was the same with the influence of grinding parameters. When grinding depth and generating speed were certain, the faster the speed of grinding wheel, the more the flat and smooth of surface morphology. This suggested that the grinding surface morphology quality was improved as increasing the grinding wheel speed, and the surface morphology became coarse as enlarging the speed of the generating motion. Thus, in order to obtain better surface morphology quality, the generating speed must be reduced and the grinding wheel speed enlarged under ensuring the production efficiency. The influence of grinding depth on surface morphology was mainly due to the causing of grinding grain passivation, and the plastic bulge on the tooth surface was intensified. When grinding depth was appropriate small, it had a little effect on grinding surface topography. These indicated that the modeling and simulation had certain accuracy. Conclusions In this paper, the study on modeling and simulation of grinding tooth surface morphology of spiral bevel gear was carried out after appropriate assumptions and simplifications were made for extremely complex grinding. Therefore, the wheel morphology which directly impacted on tooth surface morphology was analyzed, and grinding wheel model was established. According to the relative positions and kinematic relationships between grinding wheel and gear in grinding process, topography modeling and simulation analysis were made within a small area of tooth surface. Finally, the morphology simulation calculations of tooth surface were carried out, and experimental study on grinding tooth surface morphology of spiral bevel big gear was done. Through comparison analysis of simulation and experimental morphology of tooth surface, it was showed that simulation model of grinding tooth surface morphology in this paper was more accurate and effective. Acknowledgement This research was financially supported by National Natural Science Foundation of China (Grant No. 51375161, 51575533, 51375160), and Hunan Provincial Natural Science Foundation of China (2015JJ5018). References [1] Tan P.L. Tooth surface geometrical microstructure modeling and analysis of spiral bevel gear under the gear grinding process [D]. Changsha: Changsha University of Science and Technology, 2009: 47-53. (in Chinese) [2] Su C.Y., Zhao H.H., et al. The development of virtual grinding wheel and analysis of grinding performance [J]. Journal of Computer Aided Design and Graphics. 2008, 20(5): 560-564. (in Chinese) [3] Liang F. Some research of virtual grinding wheel [D]. Shenyang: Northeastern University, 2005: 22-84. (in Chinese) [4] Zhou X., XI F. Modeling and predicting surface roughness of the grinding process [J]. International Journal of Machine Tools & Manufacture, 2002, 42 (7): 969–977.

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