A probabilistic approach to predict surface roughness in ceramic grinding

A probabilistic approach to predict surface roughness in ceramic grinding

Sanjay Agarwal, P. Venkateswara Rao*

Department of Mechanical Engineering, Indian Institute of Technology, New Delhi 110 016, India

Received 29 July 2004; accepted 13 October 2004

Available online 23 November 2004

Abstract

The quality of the surface produced during ceramic grinding is important as it influences the performance of the finished part to great

extent. Hence, the estimation of surface roughness can cater to the requirements of performance evaluation. But, the surface finish is

governed by many factors and its experimental determination is laborious and time consuming. So the establishment of a model for the

reliable prediction of surface roughness is still a key issue for ceramic grinding. In this study, a new analytical surface roughness model is

developed on the basis of stochastic nature of the grinding process, governed mainly by the random geometry and the random distribution of

cutting edges. This model has been validated by the experimental results of silicon carbide grinding. The theoretical analysis yielded values

which agree reasonably well with the experimental results.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Analytical model; Ceramic grinding; Surface roughness; Chip thickness

1. Introduction

Advanced ceramics have been widely used in the

manufacturing of the workpieces with complex demands

in which materials of high performance are required due to

the unique combination of properties [1]. The advantage of

ceramics over other materials includes high hardness and

strength at elevated temperatures, chemical stability,

attractive high temperature wear resistance and low density

[2]. Structural ceramics such as silicon nitride, silicon

carbide are now being increasingly used in valves, packing

(sealing) elements, bearings, pistons, rotors and other

applications where a close dimensional tolerances, high

accuracy and good surface finish is required. However, the

benefits mentioned above are accompanied by difficulties

associated with machining in general and with grinding in

particular because of the high values of hardness and

stiffness of the ceramics and very low fracture toughness as

compared to metallic materials and alloys. Precision

ceramic components require strict adherence to close

tolerances and surface finish as the performance and

0890-6955/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijmachtools.2004.10.005

* Corresponding author. Tel.: C91 11 2659 1443; fax: C91 11 2658

2053.

E-mail address: [email protected] (P.V. Rao).

reliability of these components are greatly affected by the

accuracy and surface finish produced during the grinding

process. The grinding process, the most important step of

the advanced ceramic machining process, is highly complex

and involves the contact between a large number of abrasive

particles with the surface of the workpiece. The grinding

process allows a precise dimensional control and a good

superficial finishing but requires good decision making and

ability from the machine operator. Furthermore, the

grinding characteristics of advanced ceramics are very

much different from the ones for metals [3] and therefore, it

is necessary to perform further studies to achieve a more

comprehensive understanding and a better control of the

parameters in order to achieve good surface finish. The

finish produced on the workpiece surface is one of the most

important parameters in finish grinding. Despite various

research efforts in ceramic grinding over last two decades,

much needs to be established to standardize the theoretical

models for the prediction of surface roughness for

improving product quality and increasing productivity to

reduce machining cost. The effective use of ceramics in

industrial applications demands the machining of ceramics

with good surface finish and low surface damage. In order to

assess the effectiveness of ceramic grinding, experimental

or theoretical evaluation of surface roughness is essential.

International Journal of Machine Tools & Manufacture 45 (2005) 609–616

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Nomenclature

t undeformed chip thickness, mm

b parameter of probability density function

E expected value

ycl center line distance, mm

p probability

A shaded area, mm2

ae wheel depth of cut, mm

vw table feed rate, m/min

vs wheel speed, m/s

Ra surface roughness, mm

erf error function

S. Agarwal, P.V. Rao / International Journal of Machine Tools & Manufacture 45 (2005) 609–616610

But, the experimental methods of surface finish evaluation

are costly and time consuming. So, an attempt has been

made to develop a theoretical model for the prediction of

surface roughness for the grinding of silicon carbide with

diamond abrasive.

A ground surface is produced by the action of large

number of cutting edges on the surface of the grinding

wheel. The groove produced on the workpiece surface by an

individual grain closely reflects the geometry of the grain tip

with little side flow of the work material. Thus, it is possible

to evaluate the surface roughness from the considerations of

the grain tip geometry. Since the size of these cutting edges

on the wheel surface is random in nature, the surface

roughness produced during grinding cannot be predicted in

a deterministic manner. Because of this randomness, a

probabilistic approach for the evaluation of surface rough-

ness is more appropriate and hence any attempt to estimate

surface roughness should be probabilistic in nature.

2. Literature review

The development of ceramic materials with enhanced

properties is leading to their widespread consideration for

structural applications. However, the actual utilization of

advanced ceramics has been quite limited mainly because of

the high cost of the machining these materials by grinding

while ensuring the workpiece quality. A technological basis

to achieve more efficient utilization of the grinding process

for ceramics requires an understanding of the interaction

between the abrasive and the workpiece which has direct

bearing on the surface finish produced. Extensive research

has been carried out to predict the surface roughness of the

workpiece manufactured by grinding. On the basis of

information available in the literature, theoretical methods

of surface roughness evaluation can be classified into

empirical and analytical methods. In the empirical method,

surface roughness models are normally developed as a

function of kinematic conditions [4]. The empirical model

developed by Suto and Sata [5] relates surface finish to the

number of active cutting edges using the experimental data

and it has been found to be having a logarithmic

relationship. Although empirical models have the advantage

that they require minimum efforts to develop and are used in

all fields of grinding technology but the inherent problem

associated with this method is that the model developed

under one grinding condition, cannot be used for surface

roughness prediction at other conditions, i.e. it can be used

for accurate description of the process within the limited

range of chosen parameters only. Hence the scope is limited.

To overcome the above-mentioned problem, analytical

models for surface roughness were tried out to predict the

surface roughness at different grinding conditions. The

analytical models are always preferred to empirical models

as these models are based on the fundamental laws which

use mathematical formulations of qualitative models.

Hence, these results can be made applicable to a wide

range of process conditions. The analytical models for

surface roughness have always been characterized by the

description of the microstructure of the grinding wheel in

one dimensional form taking the grain distance, the width of

cutting edge and the grain diameter into account and in two

dimensional forms by considering the grain count and the

ratio of width of cut to depth of cut. Lal and Shaw [6] used

similar approach to describe the surface roughness based on

chip thickness model. But this model does not require the

microstructure of the grinding wheel, thus it is more

successful in industry as it does not need the effort of wheel

characterization. Tonshoff et al. [7] have described the state

of art in the modeling and simulation of grinding processes

comparing different approaches to modeling. Furthermore,

the benefits as well as the limitations of the model

applications and simulation were discussed. This work

identified one simple basic model where all the parameters

such as wheel topography, material properties, etc. are

lumped into the empirical constant. Models developed for

the grinding process in the surface roughness analysis

[8–10] have assumed an orderly arrangement of the abrasive

grains on the grinding wheel. Zhou and Xi [11] have used a

conventional method to determine the surface roughness

based on the model using the mean value of the grain

protrusion heights. However, the predicted value of the

surface roughness based on traditional method is found to be

less than the measured value. To overcome this problem,

proposed method takes into consideration the random

distribution of the grain protrusion heights.

Several random models have been proposed [12–15] to

simulate the surface profile generated during grinding based

on the stochastic nature of the grinding process. In these

models, the abrasive grains on the grinding wheel have been

Fig. 1. Schematic view of the workpiece in Cartesian coordinate system.

Fig. 2. Sectional view showing the shape of groove generated.

S. Agarwal, P.V. Rao / International Journal of Machine Tools & Manufacture 45 (2005) 609–616 611

thought of as a number of small cutting points distributed

randomly over the wheel surface. Assuming a particular

probability distribution of these random cutting points,

output surface profiles have been generated for known input

surface profile and input grinding conditions. To simulate

the relative cutting path of grains, Steffens [16] has

performed a closed loop simulation, presupposing that

thermo-mechanical equilibrium has been established during

the grinding process. The input for this simulation program

will be the quantities like grinding wheel topography,

physical quantities of the system, set-up parameters of the

machine tool, and the temperature dependent material

properties, etc. Simulations can closely reproduce the

ground surface using probabilistic analysis; however,

the applicability of this programme is limited since the

simulation programme is based on the measurement of

microstructure of grinding wheel. This method is time-

consuming. Although many analytical models have been

developed based on the stochastic nature of the grinding

process but Basuray et al. [17] have proposed a simple

model for evaluating surface roughness in fine grinding

based on probabilistic approach. The concept of radial

distribution parameter and an effective profile depth

associated with the stochastic model have been used to

obtain the distribution of the grains on the wheel surface.

Results of the approximate analysis yield values that agree

reasonably well with the experimental results. However

many parameters and properties of materials were merged

into the empirical constants in this analysis. Hecker and

Liang [18] have also developed an analytical model for the

prediction of the arithmetic mean surface roughness based

on the probabilistic undeformed chip thickness model. This

model uses ground finish as a function of the wheel

microstructure, the process kinematic conditions and the

material properties. The material properties and the wheel

microstructure are considered in the surface roughness

through the chip thickness model. A simple expression that

relates the surface roughness with chip thickness was found,

which was verified using experimental data from cylindrical

grinding; however, the geometrical analysis of the grooves

left on the surface has been carried out considering the ideal

conic shape of grains which may not be true.

In most of the models developed so far, the transverse

shape of the grooves produced has been assumed to be

triangular. A simple abrasive grain on the wheel surface

generally has many tiny cutting points on its surface.

Experiments conducted by Lal and Shaw [19] with single

abrasive grain under fine grinding conditions indicates that

the grain tip could be better approximated by circular arc.

Therefore, it is evident that the groove produced by an

individual grain can be better approximated by an arc of a

circle. Based on the above analysis, it can be observed that

there is a need to develop an analytical model to predict the

surface roughness based on probabilistic approach to

represent the stochastic nature of the grinding process

considering the grooves to be a part of circular arc.

In this paper, an analytical model has been developed to

evaluate the arithmetic mean surface roughness from the

chip thickness probability density function and the relation-

ship between surface roughness and the chip thickness has

been established with the chip thickness as random variable.

3. Mathematical model for estimation

of surface roughness

A schematic diagram showing the interaction of the grain

tip to the workpiece is given in Fig. 1. At any transverse

section m–m, the profile of groove generated by any grain is

as shown in Fig. 2. Since an individual grain has many tiny

cutting points on its surface and the speed ratio is high, the

groove produced by an individual grain can be assumed to

be an arc of a circle.

It can be further assumed that the material is either

plowed with little side pile-up or removed in the form of

chips whenever grain-workpiece interference occurs. Since

the occurrence of grains on the wheel surface is random, a

probability density function is required to describe the

surface roughness for all the grains engaged. Thus, the

undeformed chip thickness t can be better described by

Rayleigh’s probability density function f(t) [18] as

f ðtÞ ¼t

b2

� �exp �

1

2

t

b

� �2� �for tR0

0 for t!0

8<: (1)


where b is a parameter that completely defines the

probability density function and it depends upon the cutting

conditions, the direction of grinding wheel, the properties of

workpiece material, etc. The expected value of the above

function is given as

EðtÞ Z

ffiffiffiffip

2

rb

The surface roughness, Ra, is defined as the arithmetic

average of the absolute values of the deviations of the

surface profile height from the mean line within the

sampling length l. Therefore, the surface roughness Ra can

be expressed as

Ra Z1

l

ðl

0jy Kycljdl (2)

where ycl denotes the distance of the center line, drawn in

such a way that the areas above and below it are equal (Fig. 3).

It can also be expressed statistically as

Ra Z1

l

ðymax

ymin

jy KycljpðyÞdy

where ymax and ymin are the lowest and highest peak height of

the surface profile and p(y) is the probability that height of

grain has a particular value y and may be expressed as

pðyÞ Z limDy/0

probabilityfy K ð1=2ÞDy%y%y C ð1=2ÞDyg

Dy

The surface roughness, Ra, can be calculated using

probability density function defined in Eq. (1). The complete

description of surface generated is very difficult due to the

complex behavior of different grains producing grooves

because of the random grain-work interaction. Thus, certain

assumptions have to be made while predicting the surface

roughness. The assumptions are given below:

(1)
An individual grain has many tiny cutting points in its
surface, therefore, for simplicity, the grain tips are

approximated as hemispheres of diameter dg (Z2t),

randomly distributed throughout the wheel volume.

Fig. 3. Profile of groo

(2)

ves g

The profile of the grooves generated is same and

completely defined by the depth of engagement or

undeformed chip thickness t

(3)
There is no groove overlapping
(4)
On an average, the expected area of interference of grain
tip and workpiece surface is about half of the area of a

circle.

Under these assumptions, the profile generated by the

grain is as shown in Fig. 3.

As per definition of surface roughness, the area above and

below the centre line must be equal. Hence the total expected

area can be written as

EfAðtÞg Z 0 (3)

The above expression can be represented in terms of the

probability density function f(t) asðN

0AðtÞf ðtÞdt Z 0 (4)

During the grain–work interaction in the grinding, two

types of grooves are generated depending upon their depth of

engagement is either less or greater than centre line ycl. For

the case when the depth of engagement is less than ycl, the

expected value of area can be expressed as

EfAðt1Þg Z

ðycl

0Aðt1Þf ðtÞdt Z

ðycl

0A1f ðtÞdt (4a)

Similarly, for the groove with depth of engagement

greater than ycl, it can be expressed as

EfAðt2Þg Z

ðN

ycl

Aðt2Þf ðtÞdt Z

ðN

ycl

ðAupper2 KAlower

2 Þf ðtÞdt

(4b)

Substituting the values from Eqs. (4a) and (4b), in Eq. (3),

equation becomesðycl

0A1f ðtÞdt C

ðN

ycl

ðAupper2 KAlower

2 Þf ðtÞdt Z 0 (5)

or,

p1EðA1ÞCp2fEðAupper2 ÞKEðAlower

2 Þg Z 0 (6)

enerated.


where p1 and p2 are the probabilities defined in terms of the

chip thickness probability density function f(t) and are given

by

p1 Z

ðycl

0f ðtÞdt for t!ycl (7)

p2 Z

ðN

ycl

f ðtÞdt for tOycl (8)

The expected area value, for the groove with depth less

than centre line contributing to surface roughness Ra, can be

calculated as

EðA1Þ Z 2yclEðt1ÞKp

2Eðt2

1Þ (9)

where A1 is the intercepted area between grain and the centre

line contributing to surface roughness (Ra) as shown in Fig. 3.

Similarly, the expected area value for the groove with

depth greater than the centre line contributing to surface

roughness (Ra) will be given as

EðAupper2 Þ Z 2yclEðt2ÞK

p

2Eðt2

2ÞCE t22 sinK1 k

t2

� ��

KE k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt2

2 Kk2Þ

q� �ð10Þ

EðAlower2 Þ Z E t2

2 sinK1 k

t2

� �� KE k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt2

2 Kk2Þ

q� �(11)

where 2k is the intercepted length on the center line by the

grain having depth of engagement greater than center line

distance and, Alower2 and A

upper2 are the areas below and above

the centre line as shown in Fig. 3. Rewriting the Eq. (6) after

substituting the expected values from Eqs. (9)–(11) as

2yclfp1Eðt1ÞCp2Eðt2Þg Zp

2fp1Eðt2

1ÞCp2Eðt22Þg (12)

To calculate the expected values in the above equation

requires another probability density function for the cases

where the chip thickness is smaller and greater than the centre

line ycl. Therefore, for the grains having depth of engagement

lying between 0 and ycl, the probability density function of

the chip thickness will be given by the conditional probability

density function f1(t) as

f1ðtÞ Z f1ðtj0% t!yclÞ Zf ðtÞÐ ycl

0 f ðtÞdt(13)

and for rest of the chip thickness, i.e. for the grains lying

above ycl, the conditional probability density function f(t2)

will be given by

f2ðtÞ Z f2ðtjycl% t!NÞ Zf ðtÞÐN

yclf ðtÞdt

(14)

Substituting the Eqs. (7), (8), (13) and (14) in Eq. (12) to find

ycl. After simplification the expression for centre line can be

expressed as

ycl Zp

4

� Eðt2Þ

EðtÞ

� �(15)

substituting the expected values from Eqs. (A2) and (A4), and

after mathematical simplification, the value of the centre line

will be

ycl Z

ffiffiffiffip

2

rb (16)

For the calculation of surface roughness, two types of

grooves are considered. Since the contribution of the two

types of grooves considered is different, thus, the total

expected value of surface roughness can be calculated as

EðRaÞ Z p1EðRa1ÞCp2EðRa2Þ (17)

where E(Ra1) and E(Ra2) are the expected values of the

surface roughness for depth of engagement smaller or greater

than ycl and these values can be calculated by the definition of

the surface roughness. As per this, the surface roughness can

be calculated by adding the area between the profile and the

centre line and divide it by the total profile length. Hence

from Fig. 3, the values can be written as

EðRa1Þ Z EA1

2t1

� �(18)

EðRa2Þ Z EA

upper2 CAlower

2

2t2

� �(19)

Rewriting the Eqs. (18) and (19) after substituting the

expressions of A1, Aupper2 and Alower

2 from Eqs. (9)–(11) as

EðRa1Þ Z ycl Kp

4

�Eðt1Þ (20)

EðRa2Þ Z ycl Kp

4

�Eðt2ÞCE t2 sinK1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 K

y2cl

t22

s !

KyclE


y2cl

t22

s !(21)

Substituting the expected values of E(t1), E(t2),

E t2 sinK1


y2cl

t22

s !and E


y2cl

t22

s !

from Eqs. (A8), (A10), (A13) and (A16) in Eqs. (20) and (21)

and then from Eq. (17), the expected value of surface

roughness can be expressed as

EðRaÞ Z 0:532b (22)

It can also be expressed in terms of the expected value of

chip thickness E(t) by replacing b in terms of E(t) from

Eq. (A2) as given below:

b Z 0:795EðtÞ

Table 1

Experimental values of surface roughness at different values of kinematic parameters

Experiment no. 1 2 3 4 5 6 7 8 9

ae (mm) 5 5 5 10 10 10 15 15 15

vs/vw 440 220 146 440 220 146 440 220 146

Ra,experimental (mm) 0.184 0.271 0.329 0.206 0.297 0.367 0.220 0.331 0.394


substituting this value in Eq. (22), expression becomes

EðRaÞ Z 0:423EðtÞ (23)

Eq. (23) shows a proportional relationship between the

surface roughness and the chip thickness expected value

under the assumption that the profile of grove generated by an

individual grain to be a semi-circular in shape and the

phenomena like back transferring of material, grain overlap-

ping, etc. are absent.

4. Validation of the surface roughness model

The grinding process has been the object of technological

research for last few decades. Grinding processes, from a

very early period, have been described by means of the

models. These models have their own domain of application

and hence the selection of the required model depends on the

need. In order to reduce the difficulty associated with the

experimental analysis, grinding processes are modeled

theoretically. Models contribute significantly to the compre-

hension of the process itself, and form the basis for the

simulation of the grinding process. They, thus create a

precondition for increased efficiency while ensuring a high

quality. The present work focuses on the prediction of surface

roughness based on the chip thickness model. The chip

thickness models play a major role in predicting the surface

quality. The chip thickness is a variable often used to describe

the quality of ground surfaces as well as to evaluate the

competitiveness of the overall grinding system. A new chip

thickness model has been developed for the performance

assessment of SiC grinding by incorporating the modulus

Fig. 4. Comparison of calculated and experimental va

elasticity of the grinding wheel and workpiece system into

account for elastic deformation in addition to other factors

such as speed ratio, depth of cut, the equivalent diameter of

the wheel, etc. [20]. This model is based on the deflections of

the wheel caused by grinding forces.

A model has to be evaluated to determine its validity. The

validity of the model is assessed through a comparison

between the predicted value and measured value of the

surface roughness within the predefined range of parameters.

The analytical surface roughness model has to be validated

by comparing the results obtained at different kinematic

conditions with the experimental results of the earlier study

[20]. The work material taken was silicon carbide with

hardness 2700 Hv and modulus of elasticity 410 GPa and the

tool was diamond grinding wheel (ASD240R100 B2) with

modulus of elasticity 70 GPa and aluminium as core material.

The other conditions taken for the experimentation are as

follows: wheel speed 36.6 m/s, wheel diameter 250 mm,

wheel width 19 mm. The kinematic parameters considered

for each experiment are depth of cut (ae) and the speed ratio

(vs/vw) where vw is the feed rate and vs is the wheel speed,

along with the experimental value of surface roughness as

shown in Table 1.

By making use of model developed in [20], the expected

value of chip thickness was calculated for each experiment.

The deviation of the surface roughness calculated with the

new model from the experimental values, for various values

of feeds and depth of cut is shown in Fig. 4. The center-line

average value of surface roughness (Ra) has been

compared with results obtained from theoretical model. It

can be found from Fig. 4 that there is a good agreement

between the two.

lues of surface roughness in ceramic grinding.


5. Conclusions

In this paper, a simple analytical model is proposed for

estimating the surface roughness by considering the random

distribution of the grain protrusion heights and by assuming

the profile of groove generated by an individual grain to be an

arc of a circle. The values obtained with the proposed model

yields results that are consistent with the experimental values.

This model can be used for the performance evaluation of the

ceramic grinding process without conducting laborious

experimentation.

Appendix A

While developing the mathematical model for the

estimation of surface roughness, various expected values

are required to be calculated. These expected values are

required for the calculation of center line position and

establishment of surface roughness equation. These expected

values are substituted at various stages of mathematical

analysis. Calculations for these expected values are given

below.

A.1. Calculation of center line position, ycl (Eq. (15)

The calculation of the center line position, ycl, Eq. (15)

requires the calculation of the value of the undeformed chip

thickness t and square of the undeformed chip thickness t2.

These expected values can be calculated as

EðtÞ Z

ðN

0tf ðtÞdt Z Kt eKt2=2b2

C

ffiffiffiffip

2

rb erf

tffiffiffiffiffiffiffi2b2

p !" #N

0

(A1)

that gives the value of E(t) as

EðtÞ Z

ffiffiffiffip

2

rb Z 1:257b (A2)

Similarly,

Eðt2Þ Z

ðN

0t2f ðtÞdt Z

ðN

0

t3

b2eKt2=2b2

dt (A3)

it will give the value as

Eðt2Þ Z ½eKt2=2b2

ðK2b2 K t2ÞN0 Z 2b2 (A4)

where b is the parameter that completely defines the

probability density function as in Eq. (1).

A.2. Calculation of surface roughness, Ra (Eqs. (17), (20),

and (21))

The probability that an undeformed chip thickness value t

is smaller than the center line position, ycl, can be calculated

as

p1 Z

ðycl

0f ðtÞdt Z 1 KeKy2

cl=2b2

(A5)

Thus, as per definition of probability density function, the

probability of a undeformed chip thickness to be greater than

center line position, ycl, will be

p2 Z 1 Kp1 Z eKy2cl=2b2

(A6)

The expected value of chip thickness smaller than ycl can

be calculated by using the conditional probability density

function (Eq. (13)), giving the expected value in this region

as:

Eðt1Þ Z1

p1

ðycl

0tf ðtÞdt (A7)

after solving the above integration and limit evaluation, it can

be expressed as

Eðt1Þ ¼1

1 KeKy2cl=2b2 Ky cl eKy2

cl=2b2

þ

ffiffiffiffip

2

rb erf

yclffiffiffiffiffiffiffi2b2

p !" #

After simplification, it can be written as

Eðt1Þ Z 0:76b (A8)

In the same way, the expected value of chip thickness

greater than center line position, ycl, can be calculated as

Eðt2Þ Z1

p2

ðN

ycl

tf ðtÞdt (A9)

Using the Eq. (14), the above integration after limit

evaluation can be expressed as

Eðt2Þ Z1

eKy2cl=2b2

ffiffiffiffip

2

rCeKy2

cl=2b2

ycl K

ffiffiffiffip

2

rerf

yclffiffiffiffiffiffiffi2b2

p !" #

or,

Eðt2Þ Z 1:85b (A10)

Two more expected values are required to be calculated to

compute the surface roughness for the chips whose chip

thickness value is more than ycl, as given by Eq. (21).

Expected value,

E


y2cl

t22

s !;

can be calculated as

EfFðt2Þg Z1

p2

ðN

ycl

FðtÞf ðtÞdt (A11)

where

Fðt2Þ Z


y2cl

t22

s


After transformation of Eq. (A11) in terms of z, which is a

function of t, the equation becomes

EfFðt2Þg Z

ðN

0eKzf ðzÞdz (A12)

where

z Zt2

2b2and f ðzÞ Z eK

ffiffiffiffipz

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 C

ffiffiffiffip

z

rs

Solving the above integration gives the value asðN

0eKzf ðzÞdz Z 0:534

Therefore, the expected value is

E


y2cl

t22

s !Z 0:534 (A13)

In the same way, the expected value,

E t2 sinK1


y2cl

t22

s !

can be written as

EfGðt2Þg Z1

p2

ðN

ycl

GðtÞf ðtÞdt (A14)

where

Gðt2Þ Z t2 sinK1


y2cl

t22

s

It can be expressed, in terms of z, which is a function of t,

as

EfGðt2Þg Z b

ðN

0eKzf ðzÞdz (A15)

where

z Zt2

2b2and f ðzÞ Z eK

ffiffiffiffipz

pffiffiffiffiffi2z

pC

ffiffiffip2

p� �2ffiffiffiffiffi2z

p

!sinK1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 K

1

2ffiffiffizp

pC1

�2

vuutAfter solving the above integration, the expected value

can be written as

E t2 sinK1


y2cl

t22

s !Z 1:23b (A16)

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Documents

A probabilistic approach to predict surface roughness in ceramic grinding