A Comparative Analysis of Structure of Machine Tool Component using Fuzzy Logic

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 06 | June -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 2673
A Comparative Analysis of Structure of Machine Tool Component Using
Fuzzy Logic
Lomit Kumar Sao1, Dr. Harsh Pandey2
1M-Tech Scholar, Dept. of Production Engg., Dr. C. V. Raman University, Kota- Bilaspur(C.G.), India.
2HOD, Dept. of Mechanical Engg., Dr. C. V. Raman University, Kota- Bilaspur(C.G.), India.
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Abstract –Design of structures for various machine toolsis
very complex and conventional methods suggestedbyauthors
of text books present different methodologies. Trial and error
is adopted to arrive at optimum values. Utilization of FEM,
Multi value optimization and tools like ANSYS is reported in
literature. In this work a fuzzy basedapproachissuggested for
selection of optimum structure for bed. A comparison of I-
section and box section is presented. Result clearly shows that
I-section gives lower stress values compared to box section.
Advantages of fuzzy approach are presented.
Key Words: Fuzzy Logic, Machine Tool Design and Lathe
Bed.
1. INTRODUCTION
Machine tool industry of the world produces a large number
of machine tools differing in purpose,processingcapabilities
and size. Every machine tool producedcontainsa supporting
frame, bed or column or knee. In machine tools this
structures account for 70-90% of the total weight of the
machine. Hence design and development of beds is one of
the important part of the production engineering
management. Structures are designed for strength and
rigidity. Different materials are used for design ofbeds.Gray
cast iron and carbon steel are among the widely used
material.
Design of bed structures byauthorsoftextbooks onmachine
tools like Mehta [1], Acherkan [2], Sen and Bhattacharya [3]
present trial and error based methodologies to arrive at
optimum values. A great deal about general engineering
design is presented in Dieter [4]. Klir and Yuan[5]presented
an application of fuzzy set theory, fuzzy logic and fuzzy
measure theory in the field of engineering. Antonsson and
Otto [6] presented a method on turning parameters in
engineering design. Otto et al. [7] presented method of
imprecision, a formal method, based on mathematics of
fuzzy sets, for representing and manipulatingimprecision in
engineering design. Antonsson et al. [8; 9; 10] suggested a
technique to perform design calculations on imprecise
representations of parameters. The level of imprecision in
the description of design elements is typically high in the
preliminary phase of engineering design.Thisimprecision is
represented using the fuzzy calculus. Zadeh [11] explained
about fuzzy sets that a fuzzy set is a class of objects with a
continuum of grades of membership. Such a set is
characterized by a membership function which assigns to
each object a grade of membership ranging between zero
and one. The technology of fuzzy set theory and its
application to systems was presented by Ross [12].
2. COMPARATIVE DESIGN OF BED
Design of bed using a conventional method [1] is presented
with the help of an example.
2.1 Design by Conventional Method
A part of diameter 200 mm andlength1500mmissupported
between centres with a tightening load of 5000 N and
machined on a lathe. The cutting force components were
recorded with a dynamometer as Px = 2500 N, Py = 5000 N
and Pz = 10000 N.
Fig -1: Composite I-Section of the Bed
Design for I-Section:
Considering an I-section as shown in Fig. 1 as given below
with assumed dimensions. The dimensions of the bed
sections are as follows b1 = 110 mm, b2 = 20 mm, b3 = 115
mm, d1 = 30 mm, d2 = 315 mm, d3 = 25 mm, A = 365 mm, B =
145mm, C = 275 mm, D = 235 mm, E = 370 mm, F = 140 mm.
The data related to the machine tool is as follows: Height of
centres h = 300 mm, distance of the feed pinion from thebed
surface hfp = 50 mm. Ignoring the weight of the work-piece.
Check the strength of the bed if the allowable stress of the
bed materials is 12 N/mm2.

The first step is to determine the position of the neutral axis
of the bed section. Taking the moment of the section about
the lower most edge, we find
Hence, ẑ= z1 = 181.05 mm and z2= d1+d2+d3 - z1 = 188.95 mm
a1 = z1 - d1 = 151.05 mm and a2 = z2 - d3 = 163.95 mm
As the two I-sections are symmetrical about the Z-axis
ẏ = 0
Now, the moment of inertia about Y-Y and Z-Z-axis are
Iyy = 466508061.3 mm4,
Izz = 419005416.7 mm4 and
It = 885513478 mm4
We now determine the maximum normal and shearstresses
in the bed section
For the composite bed section, zmax = max [z1, z2] = max
[181.05, 188.95] Hence, zmax = 188.95 mm. Ignoring the
weight of the work-piece, we find σzmax = 2.58 N/mm2
For the composite bed section, ymax = max [A/2,E/2] = max
[365/2, 370/2] Hence, ymax = 185.00 mm. Substituting the
values of the various parameters, wefindσymax =0.83N/mm2
For the given composite bed section, max[ymax, zmax] = max
[185.00, 188.95] Hence, max[ymax, zmax] = 188.95 mm.
Substituting the values of the various parameters, we find
τmax = 0.53 N/mm2
Knowing σymax; σzmax and τmax we determine the principal
stress from the expression,
On substituting the values, we find σpmax = 2.73 N/mm2
As σpmax is less than the allowable stress it may be concluded
that the bed design is safe in terms of strength.
Design for Box Section:
Using similar dimensions for box section as shown in Fig. 2.
The first step is to determine the position of the neutral axis
of the box section. Taking the moment of the section about
the lower most edge, we find
ẑ = 185 mm
Hence, ẑ= z1 = 185 mm and z2 = 185 mm
a1 = z1 - d1 = 155 mm and a2 = z2 - d3 = 160 mm
As the box sections are symmetrical about the Z-axis
ẏ = 0
Now, the moment of inertia about Y-Y and Z-Z-axis are
Iyy = 372696875 mm4,
Izz = 79062500 mm4 and
It = 451759375 mm4
We now determine the maximum normal and shearstresses
in the bed section
For the composite bed section, zmax = max [z1, z2] = max
[185, 185] Hence, zmax = 185 mm. Ignoring the weight of the
work-piece, we find σzmax = 3.1644349 N/mm2
For the composite bed section, ymax = max [A/2,E/2] = max
[365/2, 370/2] Hence, ymax = 185.00 mm. Substituting the
values of the various parameters, we find σymax = 4.3873518
N/mm2
For the given composite bed section, max[ymax, zmax] = max
[185.00, 188.95] Hence, max[ymax, zmax] = 185 mm.
Substituting the values of the various parameters, we find
τmax = 1.0237751 N/mm2
Knowing σymax; σzmax and τmax we determine the principal
stress from the expression,
On substituting the values,wefindσpmax =4.9683684N/mm2
As σpmax is less than the allowable stress it may be concluded
that the bed design is safe in terms of strength.
Fig -2: Composite Box Section of the Bed
2.2 Design by Fuzzy Logic Method
The most important parameter in the design of bed is the
cross section geometry and corresponding sectionmodulus.
Since the geometry of I-section and boxsectionaresimple,it
is generally considered as first choice for analysis. The
parameters which govern the overall stress calculations are
the section modulus Iyy, Izz and It. The preference regarding
values of the above parameters can be estimated roughly
based on approximation and range values can be tested to
find optimum dimensions.

The section modulus for I-section and box sectionabout
Y- and Z-axis are represented by the following fuzzy sets
which are triangular fuzzy numbers showninTable1below;
Table - 1: Fuzzy Set for I-section and Box section
I-Section Iyy = {186603225, 466508061, 793063704}
Izz = {167602167, 419005417, 712309208}
It = {354205391, 885513478, 1505372913}
Box Section Iyy = {149078750, 372696875, 633584688}
Izz = {31625000, 79062500, 134406250}
It = {180703750, 451759375, 767990938}
where the middle value represent the term with highest
preference and the two extremes are lower and upperlimits
respectively.
The discrete α-cuts of the three variables from the Chart 1, 2
and 3 are shown in Table 2 and 3.
Table -2: Section Modulus of I-Section for Various Values
of α-cuts
α-Cuts Iyy (108) mm4 Izz (108) mm4 It (108) mm4
0.2 {2.38, 7.23} {2.11, 6.51} {4.15, 13.8}
0.4 {2.95, 6.57} {2.65, 5.90} {5.61, 12.3}
0.6 {3.55, 5.91} {3.15, 5.33} {6.72, 11.2}
0.8 {4.08, 5.23} {3.65, 4.69} {7.74, 10.0}
1.0 {4.69} {4.12} {8.85}
Table -3: Section Modulus of Box Section for Various
Values of α-cuts
α-Cuts Iyy (108) mm4 Izz (107) mm4 It (108) mm4
0.2 {1.91, 5.80} {4.08, 12.2} {2.30, 6.94}
0.4 {2.35, 5.26} {5.05, 11.1} {2.81, 6.38}
0.6 {2.85, 4.76} {5.97, 10.0} {3.42, 5.77}
0.8 {3.28, 4.26} {6.94, 8.93} {3.98, 5.15}
1.0 {3.72} {7.86} {4.54}
Based on these discrete α-cuts of the input parameters,
applying the functional relationship valid for the problem,
we get the following discrete α-cuts of the performance
parameter, σ, as from the Chart 4.
0.2σ = {9.09, 18.73}
0.4σ = {10.42, 17.48}
0.6σ = {11.52, 16.30}
0.8σ = {12.77, 15.20}
1.0σ = {13.95}
Chart -1: Fuzzy Set for Section Modulus Iyy
Chart -2: Fuzzy Set for Section Modulus Izz
Chart -3: Fuzzy Set for Section Modulus It

From the values of discrete α-cuts of the inducedsetG onthe
performance parameter, σ, the membership function of the
set G is drawn on Chart 4.
Chart -4: Fuzzy Set for Superposition of σ, G and NEp
Now, the performance parameter, σ i.e. the allowable stress
should not be excessive. For a given material, let the rangeof
values be 12 to 20 N/mm2. The fuzzy set NE defined as
Comparing the value of σ for whichthemembershipfunction
has its maximum with NE as shown in Chart 4, we find that
this value is 18.73, hence it does satisfy therequirementthat
the allowable stress should not be excessive. Consequently
the values of the input parameters with the highest
preference could not be used. To obtained the optimal value,
σ* of the performance parameter, we find a value of σ that
lies on the intersection of the sets G and NE with the highest
membership grade. That is, σ* is the value of σ for which
is obtained. This results in σ* = 13.11 N/mm2 Corresponding
to this value of σ*, we have the membership grade as α =
G(σ*) = 0.86.
Corresponding to this value of α, the α-cuts of the input
parameters are
Table - 4: α-Cuts of the Input Parameters
I-Section
0.86Iyy = {4.29×108, 5.03×108}
0.86Izz = {3.85×108,4.57×108}
0.86It = {8.06×108,9.49×108}
Box Section
0.86Iyy = {3.48×108,4.12×108}
0.86Izz = {7.27×107,8.91×107}
0.86It = {4.24×108,4.85×108}
Then, the corresponding optimum values of the input
parameters Iyy
*, Izz
*, and It
* for the I-section and box section
are
Table - 5: Optimum Parameters
I-Section
Iyy
* = 5.03×108 mm4
Izz
* = 4.57×108 mm4
It
* = 9.49×108 mm4
Volume = 18712500 mm3
Box Section
Iyy
* = 4.12×108 mm4
Izz
* = 8.91×107 mm4
It
* = 4.85×108 mm4
Volume = 36000000 mm3
3. RESULT
Considering the volume of the sections it is found that the I-
section is having lower values compared to the box section
and corresponding stress is lesser as well hence
recommended for design of bed.
4. CONCLUSIONS
Conventional methodology requires trial and error in order
to reach conclusion and depends on designer’s intuitionand
knowledge. Fuzzy approach uses range of values to arrive at
optimum values. Initial guessisrequiredandtheimprecision
in initial phase is taken care of by the method. Further work
is being done to incorporate many other structural
characteristic as well.
REFERENCES
[1] Mehta, N. K. (2012), Machine Tool Design and Numerical
Control. Tata McGraw Hill Education Private Limited, New
Delhi.
[2] Acherkan, N.;Push,V.;Ignatyev,N.andKudinov,V.(1969),
Machine Tool Design Vol. 3. Mir Publication, Moscow.
[2] Antonsson, E. K., and Wood, K. L. (1990), Modeling
Imprecision and Uncertainty in Preliminary Engineering
Design. Mechanism and Machine Theory, 25(3):305-324.
[3] Sen, G. C. and Bhattacharya, A. (2009), Principals of
Machine Tools. New Central Book Agency (P) Ltd, Kolkata.
[4] Dieter, G. E. (2000), Engineering Design (Third Ed.).
Singapore: Mcgraw Hill.
[5] Klir, G. J., and Yuan, B. (2000), Fuzzy Sets and Fuzzy
Logic: Theory andApplications.Prentice-Hall ofIndia Private
Limited, New Delhi.
[6] Otto, K. N., and Antonsson, E. K. (1991), Research of
Imprecision Methods for Quality Engineering Design at
Caltech. California Institute of Technology, Mechanical
Engineering, California.
[7] Wood, K. L., Otto, K. N., and Antonsson, E. K. (1992),
Engineering Design Calculations with Fuzzy Parameters.
Fuzzy Sets and Systems, 52(1):1-20.
[8] Wood, K. L., and Antonsson, E. K.(1989) , Computations
with Imprecise Parameters in Engineering

Design:Background and Theory. ASME Journal of
Mechanisms, Transmissions and Automation in Design,
111(4):616-625.
[9] Wood, K. L., Antonsson, E. K., and Beck, J. L. (1990),
Representing ImprecisioninEngineeringDesign:Comparing
Fuzzy and Probability Calculus. Research in Engineering
Design, 1(3-4):187-203.
[10] Wood, K. L., Otto, K. N., and Antonsson, E. K. (1990), A
Formal Method for Representing Uncertainties in
Engineering design. Workshop on Formal Methods in
Engineering Design Fort Collins, Colorado, 202-246.
[11] Zadeh, L. A. (1965), Fuzzy sets. In Information and
Control, pp.338-353.
[12] Ross, T. J. (2010), Fuzzy Logic with Engineering
Applications. A John Wiley and Sons Ltd., Publication,
University of New Mexico, USA.

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