ISSN 1052-6188, Journal of Machinery Manufacture and Reliability, 2017, Vol. 46, No. 4, pp. 394–403. © Allerton Press, Inc., 2017. Original Russian Text © S.A. Voronov, Ma Veidun, 2017, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2017, No. 4, pp. 85–94.
RELIABILITY, STRENGTH, AND WEAR RESISTANCE OF MACHINES AND STRUCTURES
Mathematical Modeling of the Cylindrical Grinding Process S. A. Voronov* and Ma Veidun Bauman Moscow State Technical University, Moscow, 105005 Russia *e-mail:
[email protected] Received November 21, 2016
Abstract—This paper proposes a dynamic model of cylindrical grinding with a tool owning specified distribution of abrasive grains. Cutting forces have been calculated, the surface geometry formed after a grinding wheel pass has been determined, the influence of the process dynamics on cutting forces and machined surface geometry has been taken into account, and the effects of cutting condit on vibrations being generated in the process of grinding have been investigated. DOI: 10.3103/S1052618817030177
INTRODUCTION Grinding is a complex process of removing material with a large number of grains affecting each other through the surface being processed. This process makes it possible to produce parts with high precision and surface quality. However, unlike a lathe tool, the cutting edges of the grinding wheel grain have stochastic geometry and are randomly distributed on the surface layer of the tool. This feature makes the process of grinding difficult to analyze and can become an obstacle to predict the results of processing and optimize the process regimes and parameters [1–3]. A grinding wheel contains a lot of abrasive grains distributed over the surface and fixed with a binding substance [4–6]. Every abrasive grain can be treated as a single element with its cutting edge involved in the grinding process. The complexity of the process is that, during the interaction, each grain in the contact area of the wheel and the work piece performs several microscopic regimes, including sliding, scratching, and cutting [6–8]. To explore this process, we have to find a correlation between the forces of interaction between the grain and the material being processed (cutting force), as well as the grain parameters and processing regimes. GRINDING WITH A SINGLE GRAIN: MODELING RESULTS Let us simulate the interaction of a single grain and the material of the work piece in order to understand how technological parameters affect the process. In the simulation, we assume that the cutting thickness hcu and the cross-sectional area of the grain and material contact Ac are the key variables that influence the cutting forces and all other parameters, such as geometric and physical parameters of the grain and the material of the work piece, are given. The simulation of a single-grain grinding process will be analyzed with Abaqus software as follows: The grain is embedded in the material of the work piece at the speed of the cutting (by setting V = 5 m/s) at different thicknesses of the chipped layer hcu = 1, 2, 3, 4, 5 μm. During this translational motion, the grain penetrates the material at the depth of hcu and passes 150 μm. The grain is assumed to be shaped as a cone with the vertex angle of 30°, the fillet radius of 10 μm, and a height of 80 μm (see Fig. 1a). At the center of the grain implementation route, we select a cross section where we determine the cutting forces and area of the cross section of the contact between the grain and material Ac. Figure 1d shows the relationship of the tangent and normal component of the cutting forces on the area of the cross section of the grain and material contact area, Ac. You can see that the dependence of cutting forces on Ac is nearly linear. Based on an analysis of the modeling results for a single grain, we can write down an expression for the cutting force of the jth grain on Ac as follows:
⎛A ⎞ Ft, j = K tc ⎜ c, j + k te ⎟ , A ⎝ 0 ⎠
⎛A ⎞ Fr , j = K rc ⎜ c, j + k re ⎟ , A ⎝ 0 ⎠ 394
MATHEMATICAL MODELING
O''
O''
O''
B−B grain(j)
Z
B s
s
h0
Ac
Ft, j
B
Fr, j
hcuj(t) (a)
FX, Z, N 1.0
s
A0
Y X
B−B
ϕj(t)
A0
O
395
A0 Ac
≈ hcuj(t)
θ
(b)
Ac
hcuj(t)
(с)
0.9 0.8 0.7 0.6 0.5
FZ
0.4 0.3 0.2 FX
0.1 0
20
40
60
80
100
120
140
160 Ac, μm2
Fig. 1.
where Ft, j, Fr, j are tangent and normal components of the cutting forces acting on the jth grain; Ac,j is the cross-sectional area of the spot of the jth grain contact with the material; Ktc = 5.866 N and Krс = 29.791 N are proportionality factors; kre = 0.00100; kre = 0.00077; and A0 = 4600 μm2 is a certain value with an area dimension. In our model, the grains are assumed to be shaped like cones with rounded vertices with radii of 10 μm. We will now define the cross section of the contact spot of the grain and material Ac,j at a known thickness of the chipped layer hcu, j. Figure 1a shows the force and geometric schemes of the grain in polar coordinates. The projection of the grain-material contact area on the B–B plane is shown in Fig. 1b. It can be approximately calculated as an area of the longitudinal section of the cone shell with its spherical vertex (see Fig. 1c). The cross-sectional area Ac, j can be calculated according to as follows:
Ac, j
⎧r 2 arccos((r − hcu, j )/ r ) − (r − hcu, j ) ⎪ 2 ⎪× −hcu, j + 2hcu, j r, ⎪if 0 ≤ h ≤ r − r sin θ, cu, j ⎪⎪ 2 = ⎨r (π/2 − θ − sin 2θ/2) ⎪+ [(hcu, j − r + r sin θ) tan θ + r cos θ] ⎪ ⎪× (hcu, j − r + r sin θ), ⎪if r − r sin θ < hcu, j ≤ h0. ⎪⎩
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OZ
KZ Z'
O' grain(j − 1) ϕr(t)
KX X' CX
rs
Vs grain( − 1) grain(j grain(j) grain(j + 1)
ae FX FZ
VW
work piece
Fig. 2.
MATHEMATICAL MODELING OF CYLINDRICAL GRINDING PROCESS Cutting force leads to the relative displacement of the tool and the work piece, and the thickness of the chipped layer changes causing waves on the surface due to vibrations. The dynamics of the system during the cutting is described by the delayed differential equations. Depending on the thickness of the chipped layer, the behavior of the dynamic system may be unstable because of exciting increasing vibrations, which results in an increase in the chip thickness and, thus, of the forces of cutting and vibration, which may possibly lead to grain wear and fracture [1, 4]. We investigate this problem by simulating a dynamic system during grinding. Mathematical modeling and stability analysis enable the design of the cutting tool to be optimized and dynamic system parameters to be calculated, as well as the most productive cutting conditions during the grinding process to be predicted. The grinding process was simulated under the following assumptions: a simplified instrument model with a one-dimensional grain is used in cases with and without vibrations and a tool model with randomly distributed grains of random sizes also with and without vibrations. Simplified Cylindrical Grinding Model A kinematic diagram of the flat grinding model is presented in Fig. 2. The main parameters that characterize the cutting process include the following processing regimes: grinding wheel rotation velocity Vs, work piece velocity Vw (feed rate), and the wheel cutting depth ae. Now, we introduce the following assumptions: the grinding wheel moves in plane as a solid body on elastic supports with specified stiffness and damping; the distribution of grains on the surface of the grinding wheel in one row is uniform; all grains have the same geometric dimensions, as shown in Fig. 2. The original data is provided in Table 1. The grinding process is simulated as a flat system with two orthogonal degrees of freedom, as shown in Fig. 2. The cutting forces that occur when cutting into the material of the work piece will be represented as components along the feed direction (X) and normally to it (Z), which cause dynamic offsets. It is assumed that the simulation does not address angular oscillations of the grinding wheel, i.e., the angular velocity ω is constant and does not change due to vibrations. JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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Table 1. Angular velocity of wheel ω
200 rad/s
Work piece feed velocity Vw
5 m/s
Grinding wheel radius rs
100 mm
Cutting depth ae
1 mm
Number of grains at the grinding wheel N
5000
Number of grains in the contact area n
arccos((rs − ae )/ rs ) N ≈ 112 2π
Table 2.
Coordinate system
Description
OXYZ
Bounded to the work piece, stationary
O'X'Y 'Z'
Located at the theoretical center of the grinding wheel, moves with velocity of the original coordinate location ( X o, Z o ), at t = 0, along direction –Z, at the feed velocity V w , f as feed of the tool for one grain, f as = V w (T / N )
Polar coordinate system (ρ, θ)
Point O'', the center of the coordinate system, is located in the actual center of the grinding wheel, with displacement X o' '' (t ), Z o' '' (t ) , and depends on the position of the theoretical center O; coordinates of O'' point in the OX'Y 'Z' system: ( X o' '' (t ), Z o' '' (t ))
Geometric Ratios for Calculating the Thickness of the Chipped Layer Taking into Consideration the Vibrations In order to analyze how the cutting of the previous grain affects the formation of surface by the current grain, we select grains j and (j – 1) that are located at the area of contact of the wheel and the work piece at moments t and (t – T/N) (see Fig. 3а). Geometric ratios that describe the cutting process with grains (j – 1) and j, as well as coordinate systems, are provided in Table 2. Formulas for converting from the OXYZ to the O'X'Y 'Z' coordinate system: X ' = X − X o ' + V wt, Z ' = Z − Z o '. The polar coordinate system (ρ, θ) and the OXYZ system are related as follows:
ρ = [ X − ( X o ' − V wt + X o' ''(t ))] + [Z − (Z o ' + Z o' '' (t ))] , 2
2
⎛ X − ( X − V t + X ' (t )) ⎞ o' w o '' ⎟. θ = − π − arctan ⎜ ⎜ Z − (Z + Z ' (t )) ⎟ 2 ⎝ ⎠ o' o '' For the jth grain the polar angle equals θ j (t ) = ϕ j (t ) − π/2 . Figure 3b shows the geometric relations for the formation of a new surface. Let us express the dynamic motions in the radial direction for a rotating jth grain in the polar coordinate system as follows: v j = Δ x sin ϕ j + Δ z cos ϕ j , where Δ x, Δ z are projections of dynamic displacement on the X and Z axis and ϕ j is the instantaneous angular immersion of the jth grain measured clockwise from the normal axis Z. The immersion angle changes over time as ϕ j (t ) = − 2π (n − j ) − ω t when the wheel rotates at angular velocity ω . N Let us write the equation system that describes the formation of new surfaces as follows: D j (t ) = [L(t − T / N ) + V w T sin(ϕ j (t )) − Δ x sin(ϕ j (t )) − Δ z cos(ϕ j (t ))]g(ϕ j (t )), N hcu, j (t ) = max(0, D j (t )), Π(t ) = Trend[Π j (t − T / N ) − hcu, j (t )], L(t ) = 0,
t ≤ 0,
⎧1, g(ϕ j (t )) = ⎨ ⎩0,
(2)
if 0 ≤ ϕ j (t ) ≤ ϕ ex , otherwise,
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Z'
Z'
O'' Y'
O'' Y '
X'
X' |Vw|
s
(a)
ϕj(t) s
ϕj − 1(tt − T/N) T
t grain( grain(j)
grain( − 1) grain(j
t − T/N
grain(j)
Z Y X
L(t − T/N)
Π(t − T/N)
t
t − T/N (b)
Dj(t)
grain(j)
ΔX ΔY
grain(j − 1)
fas T/N sin(ϕj(t)) fas = VW T/N Fig. 3. Kinematic notch scheme: (a) coordinate system and position of the (j – 1) and jth grains at cutting; (b) determination of thickness of chipped layer for jth grain.
where T/N is the grain passing period; L(t – T/N) is the deflection of the work piece surface at the polar coordinate system at t – T/N (we create an information bank in Matlab software to capture the surface configuration Π (t)); Dj(t) is the distance from the cutting edges to the raw surface, which consists of the static part L(t − T / N ) + V w T sin(ϕ j (t )) and the dynamic part v j = Δ x sin ϕ j + Δ z cos ϕ j . hcu, j(t) is the N thickness of the chipped layer of the jth grain at time t, g(ϕ j (t )) is a singular function that determines whether the jth grain is in the contact area; and the ϕ ex is the angle of grain emergence from immersion in the material. For interpolation, we select the Trend method, which is commonly used in texture modeling in geography. This method selects a mathematical function, a polynomial of the specified order that fits all of the source points; it is based on the least-squares method and creates surfaces with the minimal deviation from the original values. The trend describes the most general features of a surface, or the general trend in the behavior of the value being analyzed [9–11]. The drawback of this interpolation method is its inability JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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Z, mm 54 1
2
(a) jth grain
3
399
grinding wheel 110
52
I
111
112
II
50 185
190
195
Z, mm 50.60
200
50.08 (b)
50.58 50.56
X, mm
(c)
50.07
grain(j − 1)
50.06 ten passes
grain(j) grain(j + 1)
50.05
50.54 50.04 50.52
50.03
50.50
50.02
50.48 50.01 189.0 189.2 189.4 189.6 189.8 190.0 190.2 196.0
196.5
197.0
197.5
198.0 X, mm
Fig. 4.
to describe the following surface shapes: a surface with sharp angles (noncontinuous derivative) at separate points, as well as functions with ambiguous coordinates. KINEMATICS IGNORING DISPLACEMENTS CAUSED BY VIBRATIONS We introduce the following assumptions: at t = 0, coordinates of the center of the grinding wheel in the OXYZ coordinate system, ( X o ' (0), Z o '(0)) = (200,150) [mm], L(0) = 0. In the case of the high rigidity of the part–tool elastic system, one can disregard displacements caused by vibrations. In this case, Δ x = 0 , Δ z = 0 and system of equations (1) can be rewritten as follows:
D 'j (t ) = [L'(t − T / N ) + V w T sin(ϕ j (t ))]g(ϕ j (t )), N ' , j (t )], Π '(t ) = Trend[Π 'j (t − T / N ) − hcu
⎧1, g(ϕ j (t )) = ⎨ ⎩0,
' , j (t ) = max(0, D 'j (t )), hcu L'(t ) = 0,
t ≤ 0,
if 0 ≤ ϕ j (t ) ≤ ϕ ex , otherwise.
Figure 4 shows the work piece surface after a pass of grains in the time range of t = 0 to t = 10T/N without taking into account the vibration displacement; ten passes of the cutting grains. Figures 4b and 4c show enlarged areas I and II respectively, surface formation after ten grain passes, which show that there are sev' ,j = eral zones (marked with circles) where adjacent surfaces intersect. This means that, in these cases, hcu 0, i.e., the current grain does not penetrate the material of the work piece. DYNAMICS TAKING INTO ACCOUNT VIBRATION DISPLACEMENTS Let us convert the tangent and normal part of the cutting forces for the individual grain Ft, j, Fr, j in OXYZ coordinate systems to forces Fx, j and Fz, j (projections of cut forces to X and Z axes) using their relations as follows: F x, j = Ft, j cos ϕ j + Fr , j sin ϕ j , F z, j = −Ft, j sin ϕ j + Fr , j cos ϕ j . JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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hcu(t)
hcu(t) (а)
1.6
j=1
(b)
30
1.2
j = 40
20
0.8 10
0.4 0
0.5
1.0
1.5
2.0 j = 80
(c)
16
2.5 0
0.5
1.0
1.5
2.0 j = 100
(d)
12
2.5
12 8 8 4
4 0
0.5
1.0
1.5 t, s
2.0
2.5 0 ×10−4
0.5
1.0
1.5 t, s
2.0
2.5 ×10−4
Fig. 5.
By summing up the cutting forces acting on all grains in the area of the wheel and the work piece contact, we obtain forces that influence the grinding wheel as follows: F x =
∑
n j =1
F x, j , F z =
∑
n j =1
F z, j .
The system of differential equations that describe the movement of the grinding wheel is as follows:
mx + C x x + K x x = F x ,
mz + C z z + K z z = Fz ,
(3)
where m is the grinding wheel mass; Cx, Cz are the factors in the damping of the tool mountings in the X and Z axes direction; and Kx, Kz are the values of the stiffness of the tool mountings along the X and Z axes. The following values were used in calculations: m = 2 kg, Cx = 632.4 kg/s, Cz = 309.8 kg/s, Kx = 20 kN/mm, Kz = 30 kN/mm. We set initial conditions: at t = 0, x(0) = 0, x(0) = 0 , z(0) = 0, z(0) = 0 . The results of the test case simulation with the original data shown in Table 1 are presented in Figs. 5– 6 below. The maximum thickness of the chipped layer reaches 54 μm at j = 26 at time instant t = 2.15 × 10–6 s. Figure show the time dependence of the chipped layer cut by an individual jth (j = 1, 40, 80, 100) grain for every corresponding grain. Figure 5a shows that at t ≥ T/N, hcu, 1(t) = 0, i.e., the first grain emerges from the work piece after the first period of grain pass. Figure 5b shows the thickness of a material chip cut with the 40th grain in the case of the present vibrations. Figures 5c, 5d show the thickness of the material chip cut with the 80th and 100th grains; the influence of vibrations is also apparent here. The pictures show that vibrations sometimes break the contact of the grain and the work piece, i.e., hcu(t) = 0. We can separate all of the grains in contact zone into three groups as follows: during cutting, grains 1–20 emerge smoothly from the work piece, scratching with chip formation with small shear thickness; grains 20–80 cut the main material of the work piece in the presence of vibrations with chip formation and scratching with large shear thickness; grains 80–120 partially cut the material of the work piece that implements scratching with sliding. JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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Z, mm 51.2 (a) 50.8 I 50.4 II 50.0 185
190
195
Z, mm 50.40
200 X, mm
50.12
grain(j − 1)
(b) 50.38
ten passes
(с)
grain(j) grain(j + 1)
50.10
50.36 50.08 50.34 50.06
50.32 50.30
191.1
191.3
191.5
191.7
191.9
50.04 195.0
195.2
195.4
195.6
196.0 X, mm
Fig. 6.
Figure 6 shows a comparison of the work piece surface configurations Π(t) (solid lines) taking into account vibrations and Π'(t) (dashed lines) disregarding vibrations. Figures 6b and 6c show enlarged formed surfaces after ten consecutive grain passes in areas I and II. Figure 6c shows that there are also sev' , j = 0, i.e., grains do not cut off eral areas (encircled) where adjacent surfaces intersect; i.e., in this case hcu the material of the work piece. Grinding forces are significantly distorted due to the influence of the dynamics taking into account vibration displacements. The change in the cutting forces occurs due to the discontinuous notching of individual grains that move along an uneven contact surface between the tool and the work piece, as well as additional offsets caused by tool vibrations. The mean value of the grinding forces: Fz = 10.93 N, Fx = 3.73 N; mean square deviation: σFz = 2.59 N, σFx = 0.79 N. After the spectral decomposition, the main frequency of the grinding forces with maximum amplitude is 4665 Hz. Simulation does not take into account the friction between the link material, work piece, and chips. The assumption of equal distribution and same geometric sizes of grains ensures an almost stable amount of chipped material at a given feed, so the grinding forces are also stable. ANALYSIS SOLUTIONS BASED ON VIBRATIONAL DISPLACEMENT Let us introduce the following dimensionless variables and symbols:
ξ= x, h0
η= z, h0
t = T τ,
Kx , fx = T 2π m
Kz , fz = T 2π m
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ξ, η 0.9
0.04 ξ
0.8 0.03
0.7 0.6
0.02
0.5
ξ
0.4
0.01
0.3
0
0.2
−0.01 0.0145
0.1
0.0150
0.0155
0.0160
0.0165
0.0170
0.0175
η
0 −0.1
η
0
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.0180 t, s
0.040 t, s
Fig. 7.
ζz =
Cz , 2 K zm
Fx =
Fx , mh0
Fz =
Fz . mh0
The tool motion equations (2) in dimensionless form will be as follows:
ξ + 4πζ x f x ξ + 4π 2 f x2ξ = F x ,
+ 4πζ z f z η + 4π 2 f z2η = F z . η
(4)
Kx m Kz m = 503.5 Hz, f nz = = 616.7 Hz; the dimension2π 2π less frequencies are fx = 11.18, fz = 13.69; and we will use the following damping factors: ζ x = 0.05, ζ z = 0.02 . The natural tool frequencies are f nx =
The initial conditions are at t = 0, ξ(0) = 0 , ξ(0) = 0 , η(0) = 0, η(0) = 0. Differential equations (3) are numerically integrated in the time interval (0, 0.04 s). The simulation results are presented in Fig. 7. Figure 7 shows how the dimensionless tool shifts ξ, η change over time. You can see that, after each period of grain pass, the tool offsets take a small leap as the next grain penetrates the work piece. In order to integrate the equations to determine the offsets at the (j + 1)th period of the grain pass, it is necessary to consider the surface position formed at the jth grain pass period. During this time frame, the amplitude of vibrations decreases and dimensionless shifts of the tool gradually stabilize. In this case, the dimensionless amplitude of vibrations is less than 1, i.e., the amplitude of vibrations is less than the grain height h0 = 80 μm. It is important to note that this work does not consider interactions between the wheel bond material and the material of the work piece that occurs at large vibration amplitudes. Let us analyze the spectrum of the instrument vibrations in a stabilized regime, i.e., at t ≥ 0.015 s, where the tool vibrations are almost periodic. In this area, the main frequency of the dimensionless offsets of the tool with maximum amplitude is 4231 Hz. Table 3 shows that the main displacement frequency is close to the main frequency of the grinding forces, which suggests that there is a dominance of forced vibrations (influence of the cut forces on the wheel) that play an important role in this dynamic system. The main frequencies of the tool offsets and changes in the cutting forces do not coincide because the cutting force is a result of all the cutting forces from individual grains and the effect of the change in the contact surface between the wheel and the work piece. JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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Table 3. Grain frequency Natural tool frequencies Main frequency of changes in cutting forces Main frequency of tool vibrations
fgrain = 1/(T/N) = 159 235.6 Hz fnx = 503.5 Hz, fnz = 616.7 Hz 4665 Hz 4231 Hz
CONCLUSIONS The developed models enable the evaluation of the cutting forces and vibrations, as well as an analysis of their effects on the surface to be formed when grinding in specified processing regimes during the design phase of the technological process. REFERENCES 1. Brakhage, K.-H. and Makowski, M., Grinding wheel modeling, development of a mathematical model, Proc. MASCOT11-IMACS/ISGG Workshop IAC-CNR, Rome, 2013, pp. 31–40. 2. Abdalslam Darafon, Measuring and modeling of grinding wheel topography, Dissertation, 2013, pp. 6–15. 3. De Pellegrin, D.V. and Torrance, A.A., Characterization of abrasive particles and surfaces in grinding, Diamond at Work Conf., Barcelona, 2006. 4. Xuekun Li, Modeling and simulation of grinding processes based on a virtual wheel model and microscopic interaction analysis, Dissertation, 2010, pp. 14–19. 5. Malkin, S. and Guo, C., Grinding Technology – Theory and Applications of Machining with Abrasives, New York: Industrial Press, 2008. 6. Voronov, S.A., Kiselev, I.A., Ma, V., and Shirshov, A.A., Imitation dynamical model of figurine-shaped detection grinding process. Simulation processes development, Nauka Obraz., 2015, no. 5, pp. 40–57. 7. Chen, X. and Rowe, W.B., Analysis and simulation of the grinding process. Part I: generation of the grinding wheel surface, Int. J. Mach. Tools Manuf., 2011, vol. 36, pp. 871–882. 8. Brinksmeier, E. and Aurich, J.C., Advances in modeling and simulation of grinding processes, Ann. CIRP, 2006, vol. 55, pp. 667–696. 9. Laikin, V.I., Lectures on Geoinformational Systems and Surface Simulation, Moscow, 2011. http://studopedia.ru/3_84782_interpolyatsii.html. 10. Dem’yanov, V.V. and Savel’eva, E.A., Geostatistika teoriya praktika (Geostatistics: Theory and Practice), Moscow: Russian Academy of Sciences, 2010, pp. 99–103. 11. Savel’ev, A.A. and Mukharamova, S.S., Geostatisticheskii analiz dannykh v ekologii i prirodopol’zovanii (Geostatistical Data Analysis in Ecology and Environmental Management), Kazan: Kazan Univ., 2012, pp. 45–50.
Translated by I. Kashukov
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