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On the snagging operation
Precision Engineering 28 (2004) 261–269
On the snagging operation Part I: Modeling and simulation of wheel wear characteristics S.D.G.S.P. Gunawardane, Hirotaka Yokouchi∗ Department of Mechanical Systems Engineering, Muroran Institute of Technology, Muroran-shi, Mizumoto-cho 27-1, Hokkaido 050-8585, Japan Received 16 June 2003; received in revised form 29 September 2003; accepted 2 October 2003
Abstract A complete mathematical model for snag-grinding operation is introduced for both machine dynamics and the wheel wears. Machine dynamics is identified as machining with impact and is treated as a quasi-bilinear problem. The grit breaking phenomena is treated as a stochastic process of breaking brittle materials. The cutting stiffness (K2 ) is estimated by the method described by Yokouchi [J. Jpn. Soc. Prec. Eng. 47 (7) (1983) 90] and machine parameters were found by experimental methods. The equations of motion are solved by numerical computations, and wheel wear patterns are simulated and compared with the experimental results. Similar results are obtained in the case of auto balancer, but the qualitative results of the fixed balancer were deviated to some extent. Possible explanations for the deviation are discussed. The wheel engaging positions with the work are identified and hence the cause of uniformly distributed wear behavior around the wheel is discovered. © 2004 Published by Elsevier Inc. Keywords: Snagging operation; Wheel rattling; Quasi-bilinear; Stochastic process; Auto balancer; Fixed balancer; Wheel wear
1. Introduction The first part of this two-paper series is devoted to building up a complete mathematical model for the snagging operation to simulate the wear phenomenon of the grinding wheel under different operating conditions. The second part is devoted to discussing the detailed dynamical behavior of the snagging machine and its influence on the wheel wear behavior. Since early times, grinding has been the major contributor to machining, which constitute 25% of the total machining cost [2]. Grinding usually refers to the final touching of the work in modern manufacturing operations. On the other hand, snagging can be categorized as either grinding or stock removing operation that is widely used in steel industries for conditioning billets or slabs before performing next operations such as rolling. As a stock removing operation, snagging is probably the simplest form of all grinding operations and is relatively less important in terms of the machined surface quality. Even though it could not be neglected by only considering the quality since it spends a large amount of production cost. Very limited literatures on snagging can be found and most of them dealt only with experimental results and dis∗ Corresponding
author. Tel.: +81-143-46-5325; fax: +81-143-46-5341. E-mail addresses: [email protected] (S.D.G.S.P. Gunawardane), [email protected] (H. Yokouchi). 0141-6359/$ – see front matter © 2004 Published by Elsevier Inc. doi:10.1016/j.precisioneng.2003.10.006
cussions concerning wheel wear performance of the various types of wheels under different operating conditions [3,4].
2. Objectives Most of grinding processes that have been analyzed so far have been based on the assumption of continuous contact of the wheel with the work. The first of those analyses was for the chatter dynamics of the cylindrical plunge grinding reported by Hahn [5]. Later Thompson explained many details of his extensive work with time domain analysis [6]. Similar types of studies can be found in the work of Soneys and Brown [7]. Most of these analyses were based on the steady state with no unbalance except lobes since the operations are carried out under trued and balanced wheel conditions. Similar types of theoretical modeling could not be applied to snagging process because the wheel rotates at a higher frequency than the natural frequency of the machine, which leads to rattling because of relatively large unbalance, a lobe due to the wheel eccentricity and high gyroscopic moments. Therefore, the main objective of this paper is to introduce the extensive theoretical model based on the work done by Yokouchi et al. [1], and remodel it using numerical simulations to discuss the wheel wear behavior in order to fill the gap of the analysis between continuous grinding and grinding with impact.
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Nomenclature Cy , Cz d
dc dj dc0 , dj 0 Idy , Idz Ip Ky , Kz l me R r V v ε0 φy , φz θ
damping factors along Y, Z axis wheel depth co-ordinate; wheel surface position measured in Y direction from the work surface d at engaging d when wheel detaches the work to jump dc , dj of the first cut second moment of inertia at vibration center around Y, Z axis, respectively polar moment of inertia of the wheel system around X axis stiffness along Y, Z axis distance from the vibration center to the wheel unbalance mass moment mean radius of the wheel curvature radius of the wheel profile grinding speed (wheel surface speed) feed speed lobe height or eccentricity angular displacement of the wheel axis center around the Y and Z axis, respectively lobe position angular co-ordinate
of motion can be written as follows. Idz 0 φ¨ z Cz 0 φ˙ z + ¨ 0 Idy 0 Cy φy φ˙ y Gz 0 φz lmeω2 sin(ωt) + = 0 Gy φy −lmeω2 cos(ωt)
(1)
where Cz = Cy l2 , Cy = Cz l2 , Gz = Ky l2 , Gy = Kz l2 . The gyroscopic moments acting on the system around the Z and Y axis can be written as follows [9]. MGz = −(Idz φ¨ z + ωIp φ˙ y ),
MGy = −(Idy φ¨ y − ωIp φ˙ z )
Therefore, from Eq. (1), the equation of motion with the gyroscopic motion becomes as, 2Idz φ¨ z Cz 0 ωIp φ˙ z + 0 2Idy −ωIp Cy φ¨ y φ˙ y φz Gz 0 lmeω2 sin(ωt) + = (2) 0 Gy φy −lmeω2 cos(ωt) Fig. 2 shows the self-generated force system acting on the wheel in cutting. The total exciting force acting on the wheel in Y direction Fcy , due to the lobe interaction and cutting stiffness K2 can be written as follows. Fcy = −[K2 ε0 sin(ωt + η) + K2 lφz ]
3. Theory of snagging 3.1. Equation of motion According to observations by means of a high-speed camera, M.C. Shaw reported that only 30–50% of the wheel circumference grinds during a single rotation [8]. On the basis of these facts, the snagging operation could be considered as a quasi-bilinear impact problem since the wheel rattles on the work. Therefore, equations of motion can be obtained in two parts, jumping and cutting. On the basis of the co-ordinate system illustrated in Fig. 1, in the case of jumping (referred to as air cutting), the equation
(3)
According to the general grinding principle, the counterpart of the cutting force acting in the Z direction Fcz is regarded as about a half of the force acting in the Y direction,Fcz = (1/2)Fcy . Therefore, from Eqs. (2) and (3), the equation of motion in cutting action is φ¨ z 0 2Idz 0
+ =
2Idy Cz
φ¨ y Gz φ˙ z ωIp + 1 2 Cy φ˙ y 2 K2 l z
0
φz
−ωIp φy Gy 2 lmeω sin(ωt) − K2 lε0 sin(ωt + η) −lmeω2 cos(ωt) − 21 K2 lε0 sin(ωt + η)
(4)
where Gz = Gz + K2 l2 .
WHEEL
DRIVE
w
X Y
Z
WORK
Fig. 1. General schematic of the snagging operation.
Fig. 2. Schematics of the self-generated force system applied on the wheel.
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269
To solve Eqs. (2) and (4) for φy and φz , all the parameters except the cutting stiffness K2 , can be obtained from the snagging machine physical construction and by simple experimental procedures. The contacting and separating conditions of the cutting and jumping during the wheel rattling on the work will be discussed in Section 3.3.
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As shown in Fig. 3, the instantaneously generated surfaces on the cut are defined as follows. The surface generated by the previous cut (x + wj )2 z2 + + y = dj0 2r 2R
(6)
The surface generated at the end of the cut 3.2. Estimation of the effective cutting stiffness K2 3.2.1. Geometrical analysis First of all, the basic phenomenons underlying the cutting conditions have to be analyzed in detail. Detailed descriptions of the estimation of chip volume formed in one cutting cycle, the depth of cut and the geometrical analyses have been reported by Yokouchi et al. [1], and some of the important details will be discussed again for the proper understanding of the flow of this paper. Fig. 3 shows a cutting path, depth of cut and the consequences for one pitch feed of wj at impact velocity vc in the direction of Y in the steady state condition when dj = dj0 . In this section of the analysis, the following major assumptions were made. 1. A partial circle can be replaced by a parabola in the cross-section of the torus surface of the wheel. 2. Cutting occurs only in the Y direction with the down feed velocity of vf (d), and the pitch feed wj is used instead of the continuous feed speed v. 3. The down motion of the wheel surface is considered to be a harmonic motion with ωt. Associating with the assumption 3, the deceleration of the wheel down motion during cutting can be considered as a constant [1]. Then the wheel inserting velocity in to the work vf can be written as a function of the depth d as follows. dj − d vf (d) = vc (5) dj − d c where vf (dc ) = vc , vf (dj ) = 0.
z2 x2 + + y = dj 2r 2R
(7)
The surface of the wheel at the engagement x2 2r
+
z2 + y = dc 2R
(8)
The instantaneous surface generated during the cutting operation assuming dj r x2 z2 + +y =d 2r 2R
(9)
The effective cutting length (2AA in Fig. 3a), lc (d, x) is defined as the length of the line generated by the intersection of plane and the Y = 0 plane. Hence, from Eq. (9) 1/2 R lc (d, x) = 2 (10) (2rd − x2 ) r The effective contact area Say is nearly equal to the projected area Sa on the Y = 0 plane, since d r. This approximation and the assumption 1 are made to overcome complex integration difficulties. Therefore, from Eq. (10), Say can be obtained as follows. xB Say (d) ≈ Sa (d) = lc (d, x) δx (11) xA
Solving Eqs. (6) and (9), the length OA = xA , can be found as, xA = {2r(dj0 − d) − w2j }/2wj and similarly from Eq. (9). √ The length OB becomes xB = 2rd. √ Substituting x/ 2rd = sin θ into Eq. (11) √ Sa (d) = d rR[π − 2θA (d) − sin 2θA (d)] (12) where, θA (d) = sin
−1
2r(dj0 − d) − w2j √ 2wj 2rd
(13)
Using Eqs. (5) and (11), M(d), the instantaneous stock removal rate can be obtained as xB xB M(d) = M(d, x) δx = l(d, x)vf (d) δx xA
= Sa (d)vf (d)
Fig. 3. The geometry of the machined surface (section at z = 0 plane and plane view at y = 0 plane) (a) and (b) is the enlarged top view of (a).
xA
(14)
3.2.2. Tangential cutting force Ft (d) and the radial cutting force FR (d) Equating the instantaneous energy balance of the cutting operation Ft (d)V = Ks M(d)
(15)
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where Ks is the specific grinding energy and it can be defined as having a well-known relation with the instantaneous undeformed chip sectional area am (d) given by Wheel
Ks = K0 {am (d)}−ε
where K0 and ε are constants and they can be found experimentally by the other independent grinding test [1]. increment of instantaneous volume of metal removal am (d) = effective number of cutting edges × incremental cutting length δ(Vol(d)) Sa (d)δd δd/δt 1 vf (d) = = = = Ce Sa (d)δl Ce Sa (d)δl Ce δl/δt Ce V (17) where Ce is the effective number of cutting edges per unit area. Therefore, from Eqs. (12) to (17), tangential cutting force, Ft (d) can be derived as follows.
1−ε
vf (d) Ft (d) = K0 Ceε V √ × d rR[π − 2θA (d) − sin 2θA (d)]
(18)
The radial cutting force can be written as FR (d) = 2Ft (d) (as described in Section 3.1). Therefore, from Eq. (18), radial cutting force FR (d) can be obtained as follows. vf (d) 1−ε FR (d) = 2K0 Ceε V √ × d rR[π − 2θA (d) − sin 2θA (d)]
(19)
It is obvious that the gradient of the graph shown in Fig. 4 for FR (d) with d obtained by Eq. (19) does show small values at the start of the cut and soon it reaches a fairly constant value until just before the jump. Considering the simplicity of the model, the mean gradient of the left part of the curve (FR ) is regarded as the cutting stiffness K2 for a specific cut.
Radial grinding force(FR ) (N)
5000 4000 3000 2000 1000 dc
0 1.66
1.68
dj 1.70
Wheel
(16)
1.72 1.74 1.76 1.78 1.80 Wheel penitration depth(d) (mm)
1.82
Y
Y
r
2f
Vf =Vc
Vf =0 A
r
2f
f
v B
B
(a)
Work
(b)
Fig. 5. (a) The first contact of the wheel with the work and (b) the wheel position about to separate from the work.
3.3. Wheel–work interaction and geometrical compatibilities The judgment of the wheel contacting and separating can be made by dynamical and geometrical conditions of the wheel and the work. The first engagement should satisfy both the velocity and the geometrical compatibilities (Fig. 5a), y˙ ≥ −v tan φ and d ≥ dc , respectively. Where, median angle of the slope φ = cos−1 {(r − d)/2r}. The wheel separation occurs when the vertical velocity of the wheel exceeds the leading velocity of the machined surface in the Y direction y˙ < −v tan φ (Fig. 5b). From the geometrical compatibility, dc can be approximated as follows. wj 2 dc = (20) dj0 − √ 2r The pitch feed wj , is equal to v(tc + tj ), where tc and tj are cutting and jumping time of the previous cycle, respectively. 3.4. Wheel wear model The wheel wear phenomenon is basically developed on the theory of fracture of brittle materials [10]. The grit breaking behavior of the wheel is assumed to be brittle, considering that the resinoid bonds become brittle at high temperatures, which eventually occurs due to burning. The wheel wear or the grit breaking phenomenon mainly depends on the stress acting on a grit σ and the time duration t before breaking [10]. On the basis of stochastic process of breaking of brittle materials, the grit breaking probability Pf can be derived by the following concepts.
1.84
Fig. 4. The variation of the radial cutting force with the instantaneous depth of cut for known experimental conditions [1] (R = 160 mm, r = 82.2 mm, V = 4000 mm/s, Vc = 142 mm/s, dc = 1.689 mm, dj = dj0 = 1.845 mm, wj = 0.752 mm/s, Ce = 0.233 mm−2 , K0 = 275 J/mm3 , ε = 0.482).
P(t):probability that a grit does not break until time t. µ(t):probability density of breaking at time t. Then, the probability that a grit does not break until time t but does in the time increment δt can be written as P(t)µ(t) dt.
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269
Reversely, it can also become as follows.
From Eq. (16), the average cutting force acting on a grit can be derived as (29) f¯ t = Ks a¯ m = K0 (¯am )1−ε
d [1 − P(t)] = −dP dt Therefore, P(t)µ(t) dt = −dP
(21)
Then, µ(t) dt can be derived from Eq. (21) as follows. dP(t) µ(t) dt = − = −d(log P) (22) P(t) According to Hirata’s series of experiments, the variations of the logarithmic value of the breaking probability of glass with time shows a constant gradient line for constant load [10]. Therefore, µ(t) does not depend on time t but exponentially varies with the load applied. On the basis of the analysis µ(t) can be assumed to be µ(t) = µ = A exp(Bσ)
(23)
where A and B are constant and σ is replaced by the cutting force ft applied on a grit with another constant B . From Eqs. (22) and (23), probability of a grit not breaking until time t can be written as, P(t) = exp[−At exp(B ft )]
(24)
Therefore, using Eq. (24), the grit breaking probability Pf becomes Pf = 1 − exp[−At exp(B ft )]
(25)
B
where A and are constants and could be inherent parameters of a particular wheel type. They should be identified independently, but in the present state these were determined from the G values obtained by the snag grinding performed in several load and speed conditions. Pf for individual grit should be identified for more precise wear volume calculations. But for simplicity of the modeling, the average probability P¯ f is used by replacing t and ft by averages of them ¯t and f¯ t , respectively. The average time duration of cut for grits ¯t can be obtained as follows. ¯l g¯ ¯t = c = vol (26) V¯ a¯ m V¯ where g¯ vol is the mean chip volume removed per grit at a single cut and can be obtained as follows. Svol g¯ vol = (27) Ce Swe where Swe is the total swept area of the wheel surface for a single cut derived from geometrical analysis and Ce Swe is the total cutting edges contributed to a single cut [1]. The total stock removed volume per cut Svol can be derived from geometrical methods described in Section 3.2. dj 4 (28) Svol = Sa (d) δd = Wj dj 2Rdj 3 dc From Eqs. (26) to (28) ¯t can be obtained.
265
where the average of am (d) is dj 2 vc d am (d) δd a¯ m = c d = j 3 Ce V δd dc
The grit breaking probability Pf can be calculated from (25) substituting the values for ¯t and f¯ t from Eqs. (26) and (29), respectively. Then the total wear volume of the wheel per single cut Wvol can be obtained as follows. −(3/2)
Wvol = Swe Ce C0
Pf
(30)
where C0 is the theoretical number of grits per unit area and −(3/2) C0 is the equivalent wheel volume occupied by a grit. 4. Numerical experiments Due to the complexity of the equation of motion, a simple analytical solution to the current model could not be expected. In order to simulate the cutting process for a number of consecutive cutting cycles, the mathematical model was solved numerically. The fourth order Runge–Kutta–Gill method was employed for the numerical integrations. Mathematical model was coded by FORTRAN and the flow of the simulation is briefly shown in Fig. 6. The program consists of two main sections called SPS and M3. SPS is the main body of the simulation and it calculates the wheel jumping and cutting instances and their limitations, which are estimated and controlled by geometrical and dynamic compatibilities of each cutting cycle. M3 is the tool to calculate the previous depth of cut dj 0 , the previous engaging depth dc0 , and K2 grad , the gradient function of dc0 , dj 0 , wj and vf that is used to calculate the cutting stiffness K2 . In real practice, at the beginning of the operation, wheel is allowed to rotate freely, and then is forced to touch the work at a comparatively low load. The load is then increased to achieve the required depth of cut while maintaining the stability of the operation. Modeling the real situation, we are not aware of the contacting radial velocity vc and the engaging position of the wheel surface (due to wheel vibration), and there are no good arguments for predicting them since the lobe height is not much less than the wheel vibrating amplitude, and the work surface is expected to be uneven. Therefore, it is assumed that the rms value of the velocity of the wheel surface in the Y direction is used as the first contacting velocity in the subprogram M3. The flag MCVCR is used to run the program through M3 only at the beginning. To determine the geometry of the cut in the beginning, quasi-steady state solution to the problem was obtained by repeating Runge–Kutta solution until the solution approached
266
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Fig. 7. The experimental snagging machine.
Fig. 6. The simplified flow chart of the wear simulation.
3% tolerance level in 10 consecutive cycles. Then the program was run up to required number of cutting cycles (ITERA). For a set of Runge–Kutta solution per single cutting cycle (air cut + cut), the ending conditions of jumping (air cut) was used as the initial condition for subsequent cutting (cut) and vise versa. At the end of each cut, the wear volume was calculated by the method defined in Section 3.4 and removed from the wheel at the cutting portion of the wheel. Then the new unbalanced mass vector |me|, ∠η and the new wheel inertia was calculated and fed back to the program. The time step of the Runge–Kutta integration was chosen at 26.539 s, which corresponds to 0.5◦ of wheel rotation at the speed of 3140 rpm. Therefore, the resolution of the contacting angular position was maintained within a difference of 0.5◦ . The wheel was partitioned into 360 divisions around the circumference for identifying the location of engagement with the work. Employing single lobe model, wheel surface was smoothened by Fourier methods after each 3000 revolutions in order to avoid unexpected sharp lobes, which are not present in practice. The smoothing was not required for the auto balancer since the cutting engagement locations flow rapidly and smoothly around the circumference. The evidence of this will be discussed in Section 6.
5. Experimental and theoretical comparison The results obtained from the simulation work were compared with the experimental results. The experiments were
conducted using an experimental grinding machine built up on the bed of a turret lathe. Two hydraulic rams were employed to load the wheel and to feed the work on the carrier (Fig. 7). The loading ram with a hydraulic actuator was designed to maintain a constant load throughout the operation using compensating reaction principle. A cylindrical work was used instead of a sheet work and the feeding was provided by a spline connector by means of the turret gearbox. A belt and pulley system was used to drive the wheel. CrMo alloy steel (SCM4, JIS) was used as the work material and A20MB wheels were used. The experiments were carried out with both a mercury auto balancer and a fixed balancer. However, once the fixed balancer was adjusted initially, then it was not adjusted during the whole operation. The operating conditions and the machine parameters are listed in Table 1.
Table 1 Parameters of machine, wheel, work and cutting conditions Wheel Work Wheel speed (ω) Feed rate (v) Work diameter Wheel edge radius (r) Machine damping (Cy , Cz ) Machine stiffness (Ky , Kz ) Load applied on the wheel (W) A B Ce C0 Idy Idz Ip l K0 ε
A20MB SCM4(JIS) 3140 rpm 31.7 mm/s 172 mm 86 mm 0.532, 2.02 N s/mm 1.38, 19.6 kN/mm 470 N 1.89 s−1 0.32 N−1 0.349 mm−2 1.047 mm−2 164.5 kg mm2 133.5 kg mm2 1.65 kg mm 110 mm 275 J/mm3 0.482
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269
267
183 181
Wheel radius (mm)
179 177 175 173 171 169 167 165 0
50
100
150
200
250
300
350
Wheel surface position (deg) Rev.10500(Exp.) Rev.29000(Exp.)
Initial surface profile(Exp. and Simu.) Rev.20500(Exp.) Rev.36000(Exp.)
Fig. 8. The simulated and experimented wheel wear progress of A20MB wheel with SCM4(JIS) as the work at 3140 rpm with the auto balancer. (The solid lines represent the simulated results in the same descending order of the revolutions.)
6. Results and discussions As shown in Fig. 8, it is clear that the simulation results in the case of the auto balancer are similar to the experimental results in both qualitatively and quantitatively. Fig. 9 shows one of the best comparative results, achieved for the fixed balancer in numerical experiments starting with initial small mass moment vector |me|, ∠η (20 g mm, ∠0◦ ), which sensi-
tively influenced the qualitative aspects of the wear progress. The magnitude of the mass moment vector |me| is selected as 20 g mm since it was nearly the minimum measurable limit with the experimental machine and the phase η might not be accurate. In real experiments, there were several interruptions, which were necessary for taking measurements but in numerical experiments there were no interruptions. This also might have some effects on the wear progress.
202 200
Wheel radius (mm)
198 196 194 192 190 188 186 184 182 0
50
100
150
200
250
300
350
Wheel surface position (deg) Initial surface profile(Exp. and Simu.)
Rev.10500(Exp.)
Rev.24000(Exp.)
Rev.28000(Exp.)
Rev.36000(Exp.)
Rev.46000(Exp.)
Fig. 9. The simulated and experimented wheel wear progress of A20MB wheel with SCM4(JIS) as the work at 3140 rpm with the fixed balancer. (The solid lines represent the simulated results in the same descending order of the revolutions.)
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269 Wheel contacting angular pos (deg)
268 350 300 250 200 150 100 50 0 0
500
1000
1500
2000 2500 3000 Number of revolutions
3500
4000
4500
5000
350
Wheel contacting angula position (deg)
Wheel contacting angular pos ( deg)
Fig. 10. The distribution of the wheel contacting angular position up to 5000 revolutions for auto balancer.
300 250 200 150 100 50
130 120 110 100 90 80 12000
12010
12020
12030
12040
12050
Number of revolutions
0 4040
4050
4060
4070
4080
4090
4100
4110
Fig. 13. An example of neighborhood distribution of the wheel contacting position from 12,000 to 12,050 revolutions for fixed balancer.
Number of revolutions Fig. 11. An example of neighborhood distribution of the wheel contacting position from 4050 to 4100 revolutions for auto balancer.
Normally, in other fine grinding processes, the wheel continuously contacts with the work that resulted in fairly uniform wear around the wheel circumference. On contrary to that, in the snagging operation, the wheel rattles on the work. Assuming snagging operation as a linear system with forced vibration, the next contact position of the wheel with the work should be almost the same as the previous contact position for smooth work surface. Therefore, it is reasonable to expect the wheel to have a tendency to be worn in a limited area. However, in the experiments as well as in practice, it was not observed. The wheel always maintains its circular shape. Moreover, in both cases of auto balancer and fixed balancer,
Wheel contacting angu position (deg)
140
the lobe height ε0 and its position θ (measured from reference point P on the wheel, Fig. 2) are attracted to a nearly constant condition that resulted in fairly uniform wear around the wheel (Figs. 8 and 9). Fig. 10 shows the variation of the wheel angular co-ordinate of the engaging position on consecutive cuts obtained by numerical experiments for the auto balancer up to 5000 revolutions. The engaging pattern is repeated smoothly over a wide range with similar behavior even beyond 5000 revolutions as well. Fig. 11 shows the neighborhood variations of the engaging positions. The unclear pattern could be the results of non-linearity in the dynamical system, which can lead to chaotic or random behavior. The counterpart of the fixed balancer shows an un-predictable flow of engaging
350 300 250 200 150 100 50 0 0
2000
4000
6000 8000 Number of revolutions
10000
12000
Fig. 12. The distribution of the wheel contacting angular position up to 14,000 revolutions for fixed balancer.
14000
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269
pattern nearly up to 12,000 revolutions (in a period of developing me due to wheel wear) and then the flow settles into a steady pattern (Fig. 12). As shown in Fig. 13, the neighborhood variations are very close. That could be the result of the presence of a comparatively high unbalance mass established in the wheel, which dominates the jumping behavior.
7. Conclusions 1. A complete mathematical model was introduced for the wheel wear behavior of the snag-grinding operation. 2. The mathematical modeling of the wheel wear characteristics produced close qualitative and quantitative results with the experiments. 3. A small variation in the initial unbalance mass moment vector |me|, ∠η largely influences to the wheel wear behavior. 4. The repetitive wheel-engaging pattern that describes the phenomenon of the uniform wear progress even with wheel rattling was discovered.
269
References [1] Yokouchi H, Onouchi Y, Kikuchi K. Study on the snagging-theoretical analysis of grinding mechanisms when wheel rattles on the work. J Jpn Soc Prec Eng 1983;47(7):90–8 [in Japanese]. [2] Francis C. Moon. Dynamics and chaos in manufacturing processes. Wiley Series in nonlinear science; 1998. [3] Matsuo T, Ueda N, Sonoda S, Oshima E. Evaluation of zirconiaalumina wheels by laboratory snag grinding and abrasive cutting-off tests. In: Proceedings of the Fourth International Conference on Production Engineering Tokyo; 1980. p. 667–72. [4] Matsuo T, Takagi T. Basic research on snagging. J Jpn Soc Prec Eng 1977;43(7):56–61 [in Japanese]. [5] Hahn RS. Worcester, mass. On the theory of regenerative chatter in precision-grinding operations. Trans ASME 1954;76(1):593–7. [6] Thompson RA. On the doubly regenerative stability of a grinder: the effect of contact stiffness and wave filtering. ASME J Eng Ind 1992;114:53–60. [7] Soneys R, Brown D. Dominating parameters in grinding wheel and workpiece regenerative chatter. In: Proceedings of 10th International M.T.D.R. Conference 1969. University of Birmingham, Birmingham, England. p. 325–48. [8] Shaw MC. Principles of abrasive processing. Oxford Science Publications; 1996. p. 189. [9] Hahn RS. Gyroscopically induced vibrations in grinding spindles. Ann CIRP 1966;XIII:381–8. [10] Hirata S. Statistical phenomenon in engineering sciences [5]. Sci Machines 1949;1(5):234– [in Japanese].
On the snagging operation Part I: Modeling and simulation of wheel wear characteristics S.D.G.S.P. Gunawardane, Hirotaka Yokouchi∗ Department of Mechanical Systems Engineering, Muroran Institute of Technology, Muroran-shi, Mizumoto-cho 27-1, Hokkaido 050-8585, Japan Received 16 June 2003; received in revised form 29 September 2003; accepted 2 October 2003
Abstract A complete mathematical model for snag-grinding operation is introduced for both machine dynamics and the wheel wears. Machine dynamics is identified as machining with impact and is treated as a quasi-bilinear problem. The grit breaking phenomena is treated as a stochastic process of breaking brittle materials. The cutting stiffness (K2 ) is estimated by the method described by Yokouchi [J. Jpn. Soc. Prec. Eng. 47 (7) (1983) 90] and machine parameters were found by experimental methods. The equations of motion are solved by numerical computations, and wheel wear patterns are simulated and compared with the experimental results. Similar results are obtained in the case of auto balancer, but the qualitative results of the fixed balancer were deviated to some extent. Possible explanations for the deviation are discussed. The wheel engaging positions with the work are identified and hence the cause of uniformly distributed wear behavior around the wheel is discovered. © 2004 Published by Elsevier Inc. Keywords: Snagging operation; Wheel rattling; Quasi-bilinear; Stochastic process; Auto balancer; Fixed balancer; Wheel wear
1. Introduction The first part of this two-paper series is devoted to building up a complete mathematical model for the snagging operation to simulate the wear phenomenon of the grinding wheel under different operating conditions. The second part is devoted to discussing the detailed dynamical behavior of the snagging machine and its influence on the wheel wear behavior. Since early times, grinding has been the major contributor to machining, which constitute 25% of the total machining cost [2]. Grinding usually refers to the final touching of the work in modern manufacturing operations. On the other hand, snagging can be categorized as either grinding or stock removing operation that is widely used in steel industries for conditioning billets or slabs before performing next operations such as rolling. As a stock removing operation, snagging is probably the simplest form of all grinding operations and is relatively less important in terms of the machined surface quality. Even though it could not be neglected by only considering the quality since it spends a large amount of production cost. Very limited literatures on snagging can be found and most of them dealt only with experimental results and dis∗ Corresponding
author. Tel.: +81-143-46-5325; fax: +81-143-46-5341. E-mail addresses: [email protected] (S.D.G.S.P. Gunawardane), [email protected] (H. Yokouchi). 0141-6359/$ – see front matter © 2004 Published by Elsevier Inc. doi:10.1016/j.precisioneng.2003.10.006
cussions concerning wheel wear performance of the various types of wheels under different operating conditions [3,4].
2. Objectives Most of grinding processes that have been analyzed so far have been based on the assumption of continuous contact of the wheel with the work. The first of those analyses was for the chatter dynamics of the cylindrical plunge grinding reported by Hahn [5]. Later Thompson explained many details of his extensive work with time domain analysis [6]. Similar types of studies can be found in the work of Soneys and Brown [7]. Most of these analyses were based on the steady state with no unbalance except lobes since the operations are carried out under trued and balanced wheel conditions. Similar types of theoretical modeling could not be applied to snagging process because the wheel rotates at a higher frequency than the natural frequency of the machine, which leads to rattling because of relatively large unbalance, a lobe due to the wheel eccentricity and high gyroscopic moments. Therefore, the main objective of this paper is to introduce the extensive theoretical model based on the work done by Yokouchi et al. [1], and remodel it using numerical simulations to discuss the wheel wear behavior in order to fill the gap of the analysis between continuous grinding and grinding with impact.
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Nomenclature Cy , Cz d
dc dj dc0 , dj 0 Idy , Idz Ip Ky , Kz l me R r V v ε0 φy , φz θ
damping factors along Y, Z axis wheel depth co-ordinate; wheel surface position measured in Y direction from the work surface d at engaging d when wheel detaches the work to jump dc , dj of the first cut second moment of inertia at vibration center around Y, Z axis, respectively polar moment of inertia of the wheel system around X axis stiffness along Y, Z axis distance from the vibration center to the wheel unbalance mass moment mean radius of the wheel curvature radius of the wheel profile grinding speed (wheel surface speed) feed speed lobe height or eccentricity angular displacement of the wheel axis center around the Y and Z axis, respectively lobe position angular co-ordinate
of motion can be written as follows. Idz 0 φ¨ z Cz 0 φ˙ z + ¨ 0 Idy 0 Cy φy φ˙ y Gz 0 φz lmeω2 sin(ωt) + = 0 Gy φy −lmeω2 cos(ωt)
(1)
where Cz = Cy l2 , Cy = Cz l2 , Gz = Ky l2 , Gy = Kz l2 . The gyroscopic moments acting on the system around the Z and Y axis can be written as follows [9]. MGz = −(Idz φ¨ z + ωIp φ˙ y ),
MGy = −(Idy φ¨ y − ωIp φ˙ z )
Therefore, from Eq. (1), the equation of motion with the gyroscopic motion becomes as, 2Idz φ¨ z Cz 0 ωIp φ˙ z + 0 2Idy −ωIp Cy φ¨ y φ˙ y φz Gz 0 lmeω2 sin(ωt) + = (2) 0 Gy φy −lmeω2 cos(ωt) Fig. 2 shows the self-generated force system acting on the wheel in cutting. The total exciting force acting on the wheel in Y direction Fcy , due to the lobe interaction and cutting stiffness K2 can be written as follows. Fcy = −[K2 ε0 sin(ωt + η) + K2 lφz ]
3. Theory of snagging 3.1. Equation of motion According to observations by means of a high-speed camera, M.C. Shaw reported that only 30–50% of the wheel circumference grinds during a single rotation [8]. On the basis of these facts, the snagging operation could be considered as a quasi-bilinear impact problem since the wheel rattles on the work. Therefore, equations of motion can be obtained in two parts, jumping and cutting. On the basis of the co-ordinate system illustrated in Fig. 1, in the case of jumping (referred to as air cutting), the equation
(3)
According to the general grinding principle, the counterpart of the cutting force acting in the Z direction Fcz is regarded as about a half of the force acting in the Y direction,Fcz = (1/2)Fcy . Therefore, from Eqs. (2) and (3), the equation of motion in cutting action is φ¨ z 0 2Idz 0
+ =
2Idy Cz
φ¨ y Gz φ˙ z ωIp + 1 2 Cy φ˙ y 2 K2 l z
0
φz
−ωIp φy Gy 2 lmeω sin(ωt) − K2 lε0 sin(ωt + η) −lmeω2 cos(ωt) − 21 K2 lε0 sin(ωt + η)
(4)
where Gz = Gz + K2 l2 .
WHEEL
DRIVE
w
X Y
Z
WORK
Fig. 1. General schematic of the snagging operation.
Fig. 2. Schematics of the self-generated force system applied on the wheel.
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269
To solve Eqs. (2) and (4) for φy and φz , all the parameters except the cutting stiffness K2 , can be obtained from the snagging machine physical construction and by simple experimental procedures. The contacting and separating conditions of the cutting and jumping during the wheel rattling on the work will be discussed in Section 3.3.
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As shown in Fig. 3, the instantaneously generated surfaces on the cut are defined as follows. The surface generated by the previous cut (x + wj )2 z2 + + y = dj0 2r 2R
(6)
The surface generated at the end of the cut 3.2. Estimation of the effective cutting stiffness K2 3.2.1. Geometrical analysis First of all, the basic phenomenons underlying the cutting conditions have to be analyzed in detail. Detailed descriptions of the estimation of chip volume formed in one cutting cycle, the depth of cut and the geometrical analyses have been reported by Yokouchi et al. [1], and some of the important details will be discussed again for the proper understanding of the flow of this paper. Fig. 3 shows a cutting path, depth of cut and the consequences for one pitch feed of wj at impact velocity vc in the direction of Y in the steady state condition when dj = dj0 . In this section of the analysis, the following major assumptions were made. 1. A partial circle can be replaced by a parabola in the cross-section of the torus surface of the wheel. 2. Cutting occurs only in the Y direction with the down feed velocity of vf (d), and the pitch feed wj is used instead of the continuous feed speed v. 3. The down motion of the wheel surface is considered to be a harmonic motion with ωt. Associating with the assumption 3, the deceleration of the wheel down motion during cutting can be considered as a constant [1]. Then the wheel inserting velocity in to the work vf can be written as a function of the depth d as follows. dj − d vf (d) = vc (5) dj − d c where vf (dc ) = vc , vf (dj ) = 0.
z2 x2 + + y = dj 2r 2R
(7)
The surface of the wheel at the engagement x2 2r
+
z2 + y = dc 2R
(8)
The instantaneous surface generated during the cutting operation assuming dj r x2 z2 + +y =d 2r 2R
(9)
The effective cutting length (2AA in Fig. 3a), lc (d, x) is defined as the length of the line generated by the intersection of plane and the Y = 0 plane. Hence, from Eq. (9) 1/2 R lc (d, x) = 2 (10) (2rd − x2 ) r The effective contact area Say is nearly equal to the projected area Sa on the Y = 0 plane, since d r. This approximation and the assumption 1 are made to overcome complex integration difficulties. Therefore, from Eq. (10), Say can be obtained as follows. xB Say (d) ≈ Sa (d) = lc (d, x) δx (11) xA
Solving Eqs. (6) and (9), the length OA = xA , can be found as, xA = {2r(dj0 − d) − w2j }/2wj and similarly from Eq. (9). √ The length OB becomes xB = 2rd. √ Substituting x/ 2rd = sin θ into Eq. (11) √ Sa (d) = d rR[π − 2θA (d) − sin 2θA (d)] (12) where, θA (d) = sin
−1
2r(dj0 − d) − w2j √ 2wj 2rd
(13)
Using Eqs. (5) and (11), M(d), the instantaneous stock removal rate can be obtained as xB xB M(d) = M(d, x) δx = l(d, x)vf (d) δx xA
= Sa (d)vf (d)
Fig. 3. The geometry of the machined surface (section at z = 0 plane and plane view at y = 0 plane) (a) and (b) is the enlarged top view of (a).
xA
(14)
3.2.2. Tangential cutting force Ft (d) and the radial cutting force FR (d) Equating the instantaneous energy balance of the cutting operation Ft (d)V = Ks M(d)
(15)
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where Ks is the specific grinding energy and it can be defined as having a well-known relation with the instantaneous undeformed chip sectional area am (d) given by Wheel
Ks = K0 {am (d)}−ε
where K0 and ε are constants and they can be found experimentally by the other independent grinding test [1]. increment of instantaneous volume of metal removal am (d) = effective number of cutting edges × incremental cutting length δ(Vol(d)) Sa (d)δd δd/δt 1 vf (d) = = = = Ce Sa (d)δl Ce Sa (d)δl Ce δl/δt Ce V (17) where Ce is the effective number of cutting edges per unit area. Therefore, from Eqs. (12) to (17), tangential cutting force, Ft (d) can be derived as follows.
1−ε
vf (d) Ft (d) = K0 Ceε V √ × d rR[π − 2θA (d) − sin 2θA (d)]
(18)
The radial cutting force can be written as FR (d) = 2Ft (d) (as described in Section 3.1). Therefore, from Eq. (18), radial cutting force FR (d) can be obtained as follows. vf (d) 1−ε FR (d) = 2K0 Ceε V √ × d rR[π − 2θA (d) − sin 2θA (d)]
(19)
It is obvious that the gradient of the graph shown in Fig. 4 for FR (d) with d obtained by Eq. (19) does show small values at the start of the cut and soon it reaches a fairly constant value until just before the jump. Considering the simplicity of the model, the mean gradient of the left part of the curve (FR ) is regarded as the cutting stiffness K2 for a specific cut.
Radial grinding force(FR ) (N)
5000 4000 3000 2000 1000 dc
0 1.66
1.68
dj 1.70
Wheel
(16)
1.72 1.74 1.76 1.78 1.80 Wheel penitration depth(d) (mm)
1.82
Y
Y
r
2f
Vf =Vc
Vf =0 A
r
2f
f
v B
B
(a)
Work
(b)
Fig. 5. (a) The first contact of the wheel with the work and (b) the wheel position about to separate from the work.
3.3. Wheel–work interaction and geometrical compatibilities The judgment of the wheel contacting and separating can be made by dynamical and geometrical conditions of the wheel and the work. The first engagement should satisfy both the velocity and the geometrical compatibilities (Fig. 5a), y˙ ≥ −v tan φ and d ≥ dc , respectively. Where, median angle of the slope φ = cos−1 {(r − d)/2r}. The wheel separation occurs when the vertical velocity of the wheel exceeds the leading velocity of the machined surface in the Y direction y˙ < −v tan φ (Fig. 5b). From the geometrical compatibility, dc can be approximated as follows. wj 2 dc = (20) dj0 − √ 2r The pitch feed wj , is equal to v(tc + tj ), where tc and tj are cutting and jumping time of the previous cycle, respectively. 3.4. Wheel wear model The wheel wear phenomenon is basically developed on the theory of fracture of brittle materials [10]. The grit breaking behavior of the wheel is assumed to be brittle, considering that the resinoid bonds become brittle at high temperatures, which eventually occurs due to burning. The wheel wear or the grit breaking phenomenon mainly depends on the stress acting on a grit σ and the time duration t before breaking [10]. On the basis of stochastic process of breaking of brittle materials, the grit breaking probability Pf can be derived by the following concepts.
1.84
Fig. 4. The variation of the radial cutting force with the instantaneous depth of cut for known experimental conditions [1] (R = 160 mm, r = 82.2 mm, V = 4000 mm/s, Vc = 142 mm/s, dc = 1.689 mm, dj = dj0 = 1.845 mm, wj = 0.752 mm/s, Ce = 0.233 mm−2 , K0 = 275 J/mm3 , ε = 0.482).
P(t):probability that a grit does not break until time t. µ(t):probability density of breaking at time t. Then, the probability that a grit does not break until time t but does in the time increment δt can be written as P(t)µ(t) dt.
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269
Reversely, it can also become as follows.
From Eq. (16), the average cutting force acting on a grit can be derived as (29) f¯ t = Ks a¯ m = K0 (¯am )1−ε
d [1 − P(t)] = −dP dt Therefore, P(t)µ(t) dt = −dP
(21)
Then, µ(t) dt can be derived from Eq. (21) as follows. dP(t) µ(t) dt = − = −d(log P) (22) P(t) According to Hirata’s series of experiments, the variations of the logarithmic value of the breaking probability of glass with time shows a constant gradient line for constant load [10]. Therefore, µ(t) does not depend on time t but exponentially varies with the load applied. On the basis of the analysis µ(t) can be assumed to be µ(t) = µ = A exp(Bσ)
(23)
where A and B are constant and σ is replaced by the cutting force ft applied on a grit with another constant B . From Eqs. (22) and (23), probability of a grit not breaking until time t can be written as, P(t) = exp[−At exp(B ft )]
(24)
Therefore, using Eq. (24), the grit breaking probability Pf becomes Pf = 1 − exp[−At exp(B ft )]
(25)
B
where A and are constants and could be inherent parameters of a particular wheel type. They should be identified independently, but in the present state these were determined from the G values obtained by the snag grinding performed in several load and speed conditions. Pf for individual grit should be identified for more precise wear volume calculations. But for simplicity of the modeling, the average probability P¯ f is used by replacing t and ft by averages of them ¯t and f¯ t , respectively. The average time duration of cut for grits ¯t can be obtained as follows. ¯l g¯ ¯t = c = vol (26) V¯ a¯ m V¯ where g¯ vol is the mean chip volume removed per grit at a single cut and can be obtained as follows. Svol g¯ vol = (27) Ce Swe where Swe is the total swept area of the wheel surface for a single cut derived from geometrical analysis and Ce Swe is the total cutting edges contributed to a single cut [1]. The total stock removed volume per cut Svol can be derived from geometrical methods described in Section 3.2. dj 4 (28) Svol = Sa (d) δd = Wj dj 2Rdj 3 dc From Eqs. (26) to (28) ¯t can be obtained.
265
where the average of am (d) is dj 2 vc d am (d) δd a¯ m = c d = j 3 Ce V δd dc
The grit breaking probability Pf can be calculated from (25) substituting the values for ¯t and f¯ t from Eqs. (26) and (29), respectively. Then the total wear volume of the wheel per single cut Wvol can be obtained as follows. −(3/2)
Wvol = Swe Ce C0
Pf
(30)
where C0 is the theoretical number of grits per unit area and −(3/2) C0 is the equivalent wheel volume occupied by a grit. 4. Numerical experiments Due to the complexity of the equation of motion, a simple analytical solution to the current model could not be expected. In order to simulate the cutting process for a number of consecutive cutting cycles, the mathematical model was solved numerically. The fourth order Runge–Kutta–Gill method was employed for the numerical integrations. Mathematical model was coded by FORTRAN and the flow of the simulation is briefly shown in Fig. 6. The program consists of two main sections called SPS and M3. SPS is the main body of the simulation and it calculates the wheel jumping and cutting instances and their limitations, which are estimated and controlled by geometrical and dynamic compatibilities of each cutting cycle. M3 is the tool to calculate the previous depth of cut dj 0 , the previous engaging depth dc0 , and K2 grad , the gradient function of dc0 , dj 0 , wj and vf that is used to calculate the cutting stiffness K2 . In real practice, at the beginning of the operation, wheel is allowed to rotate freely, and then is forced to touch the work at a comparatively low load. The load is then increased to achieve the required depth of cut while maintaining the stability of the operation. Modeling the real situation, we are not aware of the contacting radial velocity vc and the engaging position of the wheel surface (due to wheel vibration), and there are no good arguments for predicting them since the lobe height is not much less than the wheel vibrating amplitude, and the work surface is expected to be uneven. Therefore, it is assumed that the rms value of the velocity of the wheel surface in the Y direction is used as the first contacting velocity in the subprogram M3. The flag MCVCR is used to run the program through M3 only at the beginning. To determine the geometry of the cut in the beginning, quasi-steady state solution to the problem was obtained by repeating Runge–Kutta solution until the solution approached
266
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Fig. 7. The experimental snagging machine.
Fig. 6. The simplified flow chart of the wear simulation.
3% tolerance level in 10 consecutive cycles. Then the program was run up to required number of cutting cycles (ITERA). For a set of Runge–Kutta solution per single cutting cycle (air cut + cut), the ending conditions of jumping (air cut) was used as the initial condition for subsequent cutting (cut) and vise versa. At the end of each cut, the wear volume was calculated by the method defined in Section 3.4 and removed from the wheel at the cutting portion of the wheel. Then the new unbalanced mass vector |me|, ∠η and the new wheel inertia was calculated and fed back to the program. The time step of the Runge–Kutta integration was chosen at 26.539 s, which corresponds to 0.5◦ of wheel rotation at the speed of 3140 rpm. Therefore, the resolution of the contacting angular position was maintained within a difference of 0.5◦ . The wheel was partitioned into 360 divisions around the circumference for identifying the location of engagement with the work. Employing single lobe model, wheel surface was smoothened by Fourier methods after each 3000 revolutions in order to avoid unexpected sharp lobes, which are not present in practice. The smoothing was not required for the auto balancer since the cutting engagement locations flow rapidly and smoothly around the circumference. The evidence of this will be discussed in Section 6.
5. Experimental and theoretical comparison The results obtained from the simulation work were compared with the experimental results. The experiments were
conducted using an experimental grinding machine built up on the bed of a turret lathe. Two hydraulic rams were employed to load the wheel and to feed the work on the carrier (Fig. 7). The loading ram with a hydraulic actuator was designed to maintain a constant load throughout the operation using compensating reaction principle. A cylindrical work was used instead of a sheet work and the feeding was provided by a spline connector by means of the turret gearbox. A belt and pulley system was used to drive the wheel. CrMo alloy steel (SCM4, JIS) was used as the work material and A20MB wheels were used. The experiments were carried out with both a mercury auto balancer and a fixed balancer. However, once the fixed balancer was adjusted initially, then it was not adjusted during the whole operation. The operating conditions and the machine parameters are listed in Table 1.
Table 1 Parameters of machine, wheel, work and cutting conditions Wheel Work Wheel speed (ω) Feed rate (v) Work diameter Wheel edge radius (r) Machine damping (Cy , Cz ) Machine stiffness (Ky , Kz ) Load applied on the wheel (W) A B Ce C0 Idy Idz Ip l K0 ε
A20MB SCM4(JIS) 3140 rpm 31.7 mm/s 172 mm 86 mm 0.532, 2.02 N s/mm 1.38, 19.6 kN/mm 470 N 1.89 s−1 0.32 N−1 0.349 mm−2 1.047 mm−2 164.5 kg mm2 133.5 kg mm2 1.65 kg mm 110 mm 275 J/mm3 0.482
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269
267
183 181
Wheel radius (mm)
179 177 175 173 171 169 167 165 0
50
100
150
200
250
300
350
Wheel surface position (deg) Rev.10500(Exp.) Rev.29000(Exp.)
Initial surface profile(Exp. and Simu.) Rev.20500(Exp.) Rev.36000(Exp.)
Fig. 8. The simulated and experimented wheel wear progress of A20MB wheel with SCM4(JIS) as the work at 3140 rpm with the auto balancer. (The solid lines represent the simulated results in the same descending order of the revolutions.)
6. Results and discussions As shown in Fig. 8, it is clear that the simulation results in the case of the auto balancer are similar to the experimental results in both qualitatively and quantitatively. Fig. 9 shows one of the best comparative results, achieved for the fixed balancer in numerical experiments starting with initial small mass moment vector |me|, ∠η (20 g mm, ∠0◦ ), which sensi-
tively influenced the qualitative aspects of the wear progress. The magnitude of the mass moment vector |me| is selected as 20 g mm since it was nearly the minimum measurable limit with the experimental machine and the phase η might not be accurate. In real experiments, there were several interruptions, which were necessary for taking measurements but in numerical experiments there were no interruptions. This also might have some effects on the wear progress.
202 200
Wheel radius (mm)
198 196 194 192 190 188 186 184 182 0
50
100
150
200
250
300
350
Wheel surface position (deg) Initial surface profile(Exp. and Simu.)
Rev.10500(Exp.)
Rev.24000(Exp.)
Rev.28000(Exp.)
Rev.36000(Exp.)
Rev.46000(Exp.)
Fig. 9. The simulated and experimented wheel wear progress of A20MB wheel with SCM4(JIS) as the work at 3140 rpm with the fixed balancer. (The solid lines represent the simulated results in the same descending order of the revolutions.)
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269 Wheel contacting angular pos (deg)
268 350 300 250 200 150 100 50 0 0
500
1000
1500
2000 2500 3000 Number of revolutions
3500
4000
4500
5000
350
Wheel contacting angula position (deg)
Wheel contacting angular pos ( deg)
Fig. 10. The distribution of the wheel contacting angular position up to 5000 revolutions for auto balancer.
300 250 200 150 100 50
130 120 110 100 90 80 12000
12010
12020
12030
12040
12050
Number of revolutions
0 4040
4050
4060
4070
4080
4090
4100
4110
Fig. 13. An example of neighborhood distribution of the wheel contacting position from 12,000 to 12,050 revolutions for fixed balancer.
Number of revolutions Fig. 11. An example of neighborhood distribution of the wheel contacting position from 4050 to 4100 revolutions for auto balancer.
Normally, in other fine grinding processes, the wheel continuously contacts with the work that resulted in fairly uniform wear around the wheel circumference. On contrary to that, in the snagging operation, the wheel rattles on the work. Assuming snagging operation as a linear system with forced vibration, the next contact position of the wheel with the work should be almost the same as the previous contact position for smooth work surface. Therefore, it is reasonable to expect the wheel to have a tendency to be worn in a limited area. However, in the experiments as well as in practice, it was not observed. The wheel always maintains its circular shape. Moreover, in both cases of auto balancer and fixed balancer,
Wheel contacting angu position (deg)
140
the lobe height ε0 and its position θ (measured from reference point P on the wheel, Fig. 2) are attracted to a nearly constant condition that resulted in fairly uniform wear around the wheel (Figs. 8 and 9). Fig. 10 shows the variation of the wheel angular co-ordinate of the engaging position on consecutive cuts obtained by numerical experiments for the auto balancer up to 5000 revolutions. The engaging pattern is repeated smoothly over a wide range with similar behavior even beyond 5000 revolutions as well. Fig. 11 shows the neighborhood variations of the engaging positions. The unclear pattern could be the results of non-linearity in the dynamical system, which can lead to chaotic or random behavior. The counterpart of the fixed balancer shows an un-predictable flow of engaging
350 300 250 200 150 100 50 0 0
2000
4000
6000 8000 Number of revolutions
10000
12000
Fig. 12. The distribution of the wheel contacting angular position up to 14,000 revolutions for fixed balancer.
14000
S.D.G.S.P. Gunawardane, H. Yokouchi / Precision Engineering 28 (2004) 261–269
pattern nearly up to 12,000 revolutions (in a period of developing me due to wheel wear) and then the flow settles into a steady pattern (Fig. 12). As shown in Fig. 13, the neighborhood variations are very close. That could be the result of the presence of a comparatively high unbalance mass established in the wheel, which dominates the jumping behavior.
7. Conclusions 1. A complete mathematical model was introduced for the wheel wear behavior of the snag-grinding operation. 2. The mathematical modeling of the wheel wear characteristics produced close qualitative and quantitative results with the experiments. 3. A small variation in the initial unbalance mass moment vector |me|, ∠η largely influences to the wheel wear behavior. 4. The repetitive wheel-engaging pattern that describes the phenomenon of the uniform wear progress even with wheel rattling was discovered.
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