Predictive modeling of surface roughness in grinding

International Journal of Machine Tools & Manufacture 43 (2003) 755–761

Predictive modeling of surface roughness in grinding Rogelio L. Hecker a, Steven Y. Liang b,∗ b

a Facultad the Ingenieria, Universidad Nacional de La Pampa, General Pico, LP, 6360, Argentina George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA

Received 13 September 2002; received in revised form 4 February 2003; accepted 14 February 2003

Abstract The surface roughness is a variable often used to describe the quality of ground surfaces as well as to evaluate the competitiveness of the overall grinding system. This paper presents the prediction of the arithmetic mean surface roughness based on a probabilistic undeformed chip thickness model. The model expresses the ground finish as a function of the wheel microstructure, the process kinematic conditions, and the material properties. The analysis includes a geometrical analysis of the grooves left on the surface by ideal conic grains. The material properties and the wheel microstructure are considered in the surface roughness prediction through the chip thickness model. A simple expression that relates the surface roughness with the chip thickness was found, which was verified using experimental data from cylindrical grinding.  2003 Elsevier Science Ltd. All rights reserved. Keywords: Grinding; Surface roughness; Grinding chip

1. Introduction The quality of the surface generated by grinding determines many workpiece characteristics such as the minimum tolerances, the lubrication effectiveness and the component life, among others. A typical surface is characterized by clean cutting paths and plowed material to the sideway of some grooves. However, many other marks can be found such as cracks produced by thermal impact, back transferred material, and craters produced by grain fracture [1]. Other factors that characterize a ground surface are the traverse and longitudinal waviness produced by the random nature of the grinding process and by machine vibration. When considering all these factors, a complete prediction of the surface topography is a complicated problem. A typical parameter that has been used to quantify the quality of a surface topography is the surface roughness, which is represented by the arithmetic mean value, Ra, the rootmean-square-average, Rq, and the maximum roughness height, Rt. In general, the longitudinal surface roughness

Corresponding author. Tel.: +1-404-894-8164; fax: +1-404-8949342. E-mail address: [email protected] (S.Y. Liang). ∗

has a lower value than the traverse surface roughness, therefore the latter is more frequently used in industry. Analytical models for surface roughness were based on the microstructure of the grinding wheel in both one and two dimensions. The wheel microstructure was described using simplification factors such as constant distance between cutting edges and uniform height of the cutting edges. Similar assumptions were used to describe the surface roughness based on chip thickness models, i.e. [2]. A general equation derived from analytical models was presented by [3], from where it can be concluded that besides the analytical effort, some parameters such as wheel topography and material properties were often represented by general empirical constants. Empirical surface roughness models are a function of kinematic conditions, such as the one presented by Malkin [4], and have had more success in the industry because they do not need the effort of material and surface wheel characterizations. However, the empirical constants involved must be adjusted for every workpiece material, lubricant, and type of wheel. The surface roughness was also described as a function of the equivalent chip thickness. These models can be considered a subclass of empirical models due to the empirical constants that need to be adjusted. Neither of the above mentioned models was based on

0890-6955/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0890-6955(03)00055-5

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the stochastic nature of the grinding process, governed mainly by the random geometry and the random distribution of the cutting edges on the wheel surface. To account for this, the ground surface was generated by simulation of the interaction of each grain with the workpiece material, where the relative cutting edge positions were either randomly generated [5–11], or deterministically given by measurement of the wheel topography [12,13]. A closed-loop simulation was performed, where thermo-mechanical equilibrium was established during grinding and an explicit description of the workpiece material was used [13]. Simulations can closely reproduce the ground surface using probabilistic analysis, but they are time consuming and they have limitations for online prediction. An analytical model given by simple equations and based in a probabilistic analysis is desired to represent the stochastic nature of the grinding process. Basuray et al. [14] made probably the only attempt to develop a simple mathematical expression for the surface roughness based on the probabilistic analysis of their simulations; however, many parameters and the material properties were lumped into empirical constants. This paper presents a model to predict the arithmetic mean surface roughness directly from the estimated chip thickness probability density function developed in [15]. The expected close relation between the surface roughness and the chip thickness is analytically justified based on a probabilistic analysis, where the main random variable is the chip thickness. The simple relation found is based on the stochastic nature of the grinding process that allows for a time efficient solution.

2. The surface roughness and the chip thickness

再

2/2s2

(h / s2) e⫺h

hⱖ0

0

h⬍0

Chip thickness probability density function.

critical depth of cut, hcr. However, the probability density function (p.d.f.) must be used to describe the surface roughness because all the engaged grains, including those that only plow material, leave a groove of depth h in the surface. The parameter s, that completely defines this p.d.f., was calculated as a function of the material properties (hardness), grinding wheel microstructure (grain shape, static cutting edge distribution), dynamics effects (local grain deflection and wheel-workpiece contact zone deflection), and grinding kinematic conditions (depth of cut, wheel and workpiece tangential velocity) [15]. This undeformed chip thickness was used to predict the normal and tangential force, and now a model to calculate the arithmetic mean surface roughness is presented.

3. Surface roughness model. Theoretical analysis

The workpiece profile in grinding is generated by the grooves left by grain paths when the material is either plowed or removed as a chip. In both cases, the depth of these marks is equal to the undeformed chip thickness. This undeformed chip thickness (called here the chip thickness, h) can be described by the Rayleigh probability function [15], as f(h) ⫽

Fig. 1.

(1)

with an expected value and a variance expressed as E(h) ⫽ 冑p / 2 s

(2a)

sd(h) ⫽ 冑0.429 s

(2b)

given a spectrum of possible undeformed chip thickness or depth of grain engagement as Fig. 1 shows, and where it can be observed that the material will be removed as a chip for those depths of engagement greater than the

The surface roughness can be generally described by the arithmetic mean value, Ra, defined as Ra ⫽

1 L

冕

L

|y⫺yCL| dl

(3)

0

where yCL is the position of the center-line so that the areas above and below the line are equal. The quantity Ra represents the summation of the areas above and below the line divided by the total length of the profile. The surface roughness, Ra, can be directly calculated from the grooves generated during the grain engagements using the probability density function described in Eq. (1). Since a complete description of the surface topography generated is of a complex matter in grinding, the following assumptions have been established: (1) The grooves generated are of the same characteristics either when the grain is plowing or cutting. In both cases, the grooves are characterized by the depth of engagement or undeformed chip thickness, h. (2) There

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is no groove overlapping. (3) Each groove has a triangular shape that comes from the projection of the assumed conical shape for the active grains. This shape is described by the internal angle (2q) of the cone as shown in Fig. 2. (4) There is no back transferred material or marks due to grain fracture. 3.1. Center-line calculation The surface roughness can be directly calculated from the summation of the areas above and below the centerline (Eq. (3)); therefore, the position of this line must be calculated first. Among all the grooves generated, two types can be distinguished when their depths, h, are either less or greater than yCL as shown in Fig. 2. The grooves with a depth (h⬘) less than yCL contribute to Ra with an expected area value of E(A⬘) ⫽ tanq(2yE(h⬘)⫺E(h⬘ )) 2

E(A⬙top) ⫽ 1 / 2 tanq y2

(5)

and E(A⬙bottom) ⫽ tanq (E(h⬙2)⫺2yE(h⬙) ⫹ y2cl)

(6)

where E(A⬙bottom) and E(A⬙top) are the areas below and above the center line as Fig. 2 shows. By definition, the areas below and above the center line must be equal, or p⬘E(A⬘) ⫹ p⬙2E(A⬙top) ⫽ p⬙E(A⬙bottom)

(7)

where p⬘ and p⬙ are the probabilities of a groove depth to be less and greater than yCL respectively. These probabilities can be mathematically defined using the chip thickness probability density function, f(h), as p⬘ ⫽

冕

f(h)dh

(8)

冕

f(h)dh.

(9)

yCL

0

and p⬙ ⫽

⬁ yCL

Eq. (7) can be rewritten using Eqs (4)–(6) to be 2yCL(p⬘E(h⬘) ⫹ p⬙E(h⬙)) ⫽ p⬘E(h⬘2) ⫹ p⬙E(h⬙2).

(10)

The calculation of the expected values above requires the definition of the probability density functions for those cases where the chip thickness is smaller (h⬘) and greater (h⬙) than the center-line yCL. Therefore, two new probability density functions must be defined in each region as f⬘(h) ⫽

冕

f(h)

0ⱕh ⬍ yCL

yCL

(11)

f(h)dh

0

for those chips smaller than yCL, and f⬙(h) ⫽

(4)

and the areas that belong to the grooves with a depth (h⬙) greater than yCL can be calculated as

757

冕

f(h)

yCLⱕh ⬍ ⬁

⬁

(12)

f(h)dh

yCL

for the rest of the chip thickness. The redefined probability functions, Eq. (11) and (12), and the probabilities expressed in Eqs. (8) and (9), can be substituted into Eq. (10) to find, after mathematical simplifications, an expression for the centre-line as; yCL ⫽

1E(h2) 2 E(h)

(13)

or by replacing the expected values by their explicit expressions, Eq. (2a) and (A2), yCL is yCL ⫽ 冑2 / p s.

(14)

3.2. Surface roughness calculation Two groups of chips thickness or surface marks have been defined to calculate the position of the center-line. These two groups of chips, with h either smaller or greater than yCL, contributes differently to the surface roughness calculation. Therefore, the total expected value for the surface roughness can be calculated as a weighted contribution as E(Ra) ⫽ p⬘ E(R⬘a) ⫹ p⬙ E(R⬙a).

(15)

By definition, the surface roughness can be calculated by adding the area between the profile and the centerline and dividing it by the total profile length. Therefore, the expected value of the surface roughness contribution of those chips thickness less than yCL, E(R⬘a), can be calculated by

冉

E(R⬘a) ⫽ E

Fig. 2.

Theoretical profile generated by grain grooves.

A⬘ 2 h⬘ tanq

冊

(16)

and the contribution of those chips greater than yCL, E(R⬙a), is calculated as

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R.L. Hecker, S.Y. Liang / International Journal of Machine Tools & Manufacture 43 (2003) 755–761

冉

E(R⬙a) ⫽ E

冊

2A⬙top ⫹ A⬙bottom . 2 h⬙ tanq

ened steel 52100 with 62 HRC. The tool is an aluminum oxide, 32A80KVBE, which was dressed before every experiment with a sharp (cone angle of 75°) diamond tool with 100 mm of dressing lead and 20 mm of dressing depth. The amount of material removed was the same for each experiment to maintain a consistent state of the wheel surface. The chip thickness expected value was calculated for each experiment from the model developed in [15]. Fig. 3 shows the simulated and the experimental surface roughness for cylindrical grinding process, where a ( ±) standard deviation was plotted with the experimental data. The calibrated experimental factor in Eq. (22) was Rfactor = 0.87, which is close to 1 as is desired to justify the analytical analysis. It can be observed in Fig. 3 that some predicted values of surface roughness are inside one standard deviation from the experimental mean value (exp. 4, 8, and 13), which is considered here a reasonable prediction. Some other experiments need two standard deviations to fit the predicted data with the experimental data. However, this standard deviation was calculated from a set of measurements of the surface roughness taken in different positions of the workpiece for a single experiment; therefore, it does not represent the scattering observed in grinding from one experiment to another under the same conditions.

(17)

Eq. (16) and (17) can be rewritten using the expression for the areas A⬘, A⬙bottom, and A⬙top in Eqs (4)– (6) respectively, to have E(R⬘a) ⫽ yCL⫺0.5 E(h⬘)

(18)

and E(R⬙a) ⫽ y2CLE(1 / h⬙) ⫹ 0.5 E(h⬙)⫺yCL

(19)

where the center-line value, yCL, is given by Eq. (14) and the expected values E(h⬘), E(h⬙), and E(1 / h⬙) are presented respectively in Eqs. (A8), (A11), and (A15). Based on (15), (18), and (19), the expected surface roughness can be mathematically shown to be: E(Ra) ⫽ 0.46 s

(20)

or it can be expressed as a function of the chip thickness expected value, E(h), as E(Ra) ⫽ 0.37 E(h).

(21)

4. Surface roughness model: Empirical calibration A proportional relation between the surface roughness and the chip thickness expected value was derived from the analytical analysis in the previous section. However, a correction factor, Rfactor, is necessary to adjust the empirical values to the analytical expression obtained in Eq. (21), resulting in: E(Ra) ⫽ Rfactor 0.37 E(h).

(22)

This empirical factor accounts for the non-modeled phenomena mentioned in the introduction section. Those phenomena can include imperfect grain conical shape, plowing, back transferred material, and groove overlapping. A series of experiments were performed in cylindrical grinding configuration to calibrate and subsequently validate the model. The main kinematic parameters for each experiment are the depth of cut, a, and the speed ratio, q, as Table 1 shows. The material used was hard-

Fig. 3.

Ra simulated and experimental data from cylindrical grinding.

Table 1 Experimental conditions Exp. number

a (um)

1 2 3 4 5 6 7

2 22 5 1 1 4 4

q (Vs / Vw) 363 356 90 192 92 188 131

Exp. number

a (um)

q

8 9 10 11 12 13 14

7 2 4 6 1 3 8

179 118 118 118 356 324 280

(Vs / Vw)

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759

Another useful parameter to evaluate a model is the absolute value of the relative error between the predicted and the measured values. This parameter is shown in Fig. 4, where it can be observed that the maximum error is 20% and the average error is approximately 10%. This is considered a good prediction of the surface roughness of ground surfaces, where a typical scattering is present.

5. Surface roughness predictions The surface roughness model described above is based in a chip thickness model that includes many parameters such as: the wheel microstructure, the kinematic conditions, and the material properties. Therefore, the model can be used to predict the surface roughness under different conditions of these parameters. The depth of cut and the speed ratio are the two most common kinematic variables set on the machine to obtain the desired grinding outputs. Fig. 5(a) shows the surface roughness versus the depth of cut for three different speed ratios. It can be observed that variations on the depth of cut, in its lower range, produce significant changes on the value of the surface roughness. The model also predicts that at higher speed ratios the surface produced is smoother. This is because at higher speed ratios, that is at higher wheel velocity and/or lower workpiece velocity, more grains participate in removing a given volume of material, hence the depth of engagements is lower, producing smooth surfaces. The equivalent diameter is defined as de = ds / (1 ± ds / dw), where ds is the wheel diameter, dw is the workpiece diameter, the minus/plus sign is for internal/external grinding. The equivalent diameter is a measurement of the conformity between the wheel and the workpiece surfaces. A greater value of de implies a better conformity that is traduced in a bigger area of

Fig. 4. Absolute relative error between the measured and the predicted Ra values.

Fig. 5. Surface roughness vs depth of cut at (a) 3 speed ratios and (b) 3 equivalent diameters.

contact between the wheel and the workpiece. This means that a higher number of grains can become active and consequently, there is a chip thickness reduction and smoother surfaces. This conclusion can be observed in the predicted data shown in Fig. 5(b), where the surface roughness is plotted versus the depth of cut for three different equivalent diameters. The wheel microstructure plays a major role in the quality of the ground surfaces. The wheel and dressing conditions used for the model calibration and validation were the same for each experiment. This wheel condition was specified to the model by the static wheel parameters such as the static grain density given by Cs(z) = A zk and the grain tip diameter, D. These parameters were directly measured by [16], given the values A = 0.96, k = 1.2, and D = 60 mm, and they represent the ‘wheel 1’ condition in Fig. 6. This figure also shows a second wheel condition, ‘wheel 2’, defined by A = 0.7, k = 1, and D = 100 mm. Wheel 2, with a lower static grain density and a larger grain diameter, represents a rougher wheel surface that produces a lower dynamic grain density hence, a rougher ground surface, as Fig. 6 shows. A finer wheel produces better surface finish but it will

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square of the undeformed chip thickness. This expected value can be expressed as E(h2) ⫽

冕

⬁

冕

⬁ 3

h ⫺ h2 e 2s2dh 2 0s

h2 f(h)dh ⫽

0

(A1)

that gives the following solution E(h ) ⫽ e

⫺

2

h2 2s2

|

⬁

(⫺2s ⫺h ) ⫽ 2s2 2

2

(A2)

0

where s is the parameter that defines the probability density function in Eq. (1).

Fig. 6. Surface roughness and dynamic grain density vs depth of cut for two wheel conditions.

cause higher forces and higher power due to the higher specific energy governed by a smaller expected value of the chip thickness.

6. Conclusions An analytical model for surface roughness prediction of ground parts has been presented. The model is based on the analysis of the grooves left by the grains that interact with the workpiece, which is characterized by the undeformed chip thickness. A geometric analysis based on a probabilistic approach was used to describe the arithmetic mean value, Ra; where the random variable was defined by the probability density function of the undeformed chip thickness. A simple relationship between both variables of surface roughness and undeformed chip thickness was found. The wheel microstructure, the kinematic and the dynamic grinding conditions, and the material properties were included in the model through the prediction of the undeformed chip thickness. The predicted surface roughness shows a good agreement with experimental data obtained from different kinematic conditions in cylindrical grinding. The effects on the surface roughness due to grinding parameters changes was simulated and discussed. These grinding parameters include changes in the depth of cut, speed ratio, equivalent diameter, and wheel microstructure.

Appendix A. Mathematical calculation of expected values

Surface roughness calculation (Eqs (15), (18), and (19)) The probability that an undeformed chip thickness value (h) is smaller than the center-line position, yCL, can be calculated using p⬘ ⫽

冕

yCL

f(h)dh ⫽ 1⫺e

⫺

2s2

(A3)

0

therefore, by definition of a probability density function, the probability of a chip thickness to be grater than yCL is p⬙ ⫽ 1⫺p⬘ ⫽ e

⫺

y2 CL 2s2

.

(A4)

It is also useful to solve the following integration

冕

L2

冋 冉冑 冊册

h2

h f(h) dh ⫽ ⫺h e⫺2s2

L1

⫹

冑2p 2

s Erf

(A5)

h

L2

2s2

L1

which helps to solve the following expected values. The expected value of those chip thicknesses with a value lower than yCL, can be calculated using the redefined probability density function in Eq. (11) that is equal to f⬘(h) = f(h) / p⬘, resulting in the expected value of this region: E(h⬘) ⫽

1 p⬘

冕

yCL

h f(h)dh.

(A6)

0

Now, by using the general solution of Eq. (A5), the above integration is solved as 1

E(h⬘) ⫽

1⫺e⫺

y2 CL

冉

⫺yCL e

2s2

Center–line calculation (Eq. (13)) The calculation of the center-line position, Eq. (13), requires the calculation of the expected value of the

y2 CL

⫹

冑2p 2

冉冑 冊冊

sErf

yCL

2s2

⫺

y2 CL 2s2

(A7)

R.L. Hecker, S.Y. Liang / International Journal of Machine Tools & Manufacture 43 (2003) 755–761

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or

or E(h⬘) ⫽ 0.51 s.

(A8)

In a similar way, the expected value of those chips in the other region, where h ⬎ yCL, can be calculated as E(h⬙) ⫽

冕

⬁

1 p⬙

h f(h)dh.

(A9)

yCL

Noting also that Eq. (12) was rewritten as f⬙(h) = f(h) / p⬙. It can be also solved using the general solution of Eq. (A5) resulting in E(h⬙) ⫽

e⫺ ⫺

冑2p 2

冉

冑2p

s y2 CL

⫹e

2

y2 CL

⫺ 2 2s

yCL

(A10)

2s2

冉冑 冊冊 yCL

Erf

2s2

or E(h⬙) ⫽ 1.53 s.

(A11)

Another expected value needed is the following E(1 / h⬙) ⫽ ⫽

冕

1 p⬙

冕

⬁

1 / h f⬙(h)dh

(A12)

yCL

⬁

1 / h f(h)dh

yCL

for which a general solution is as follows E(1 / h⬙) ⫽

冑2p ⫺

2se

y2 CL 2s2

冉 冉冑 冊冊| Erf

⬁

h

2s2

(A13)

yCL

and after the limit evaluation can be expressed as E(1 / h⬙) ⫽

冑2p 2s e⫺

y2 CL 2s2

冉 冉冑 冊冊 1⫺Erf

yCL

2s2

(A14)

E(1 / h⬙) ⫽

0.73 . s

(A15)

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