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Copyright © 2012 by Modern Scientific Press Company, Florida, USA
International Journal of Modern Mathematical Sciences, 2013, 6(1): 9-27
International Journal of Modern Mathematical Sciences
Journal homepage: www.ModernScientificPress.com/Journals/ijmms.aspx
ISSN: 2166-286X
Florida, USA
Article
Fuzzy Finite Element Method for Vibration Analysis of
Imprecisely Defined Bar
Nisha Rani Mahato 1
, Diptiranjan Behera 1, S. Chakraverty
1,*
1Department of Mathematics, National Institute of Technology, Rourkela, Odisha – 769 008, India
* Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.:
+91661-2462713; Fax: +91661-2462713-2701.
Article history: Received 18 December 2012, Received in revised form 3 February 2013, Accepted 28
February 2013, Published 11 March 2013.
Abstract: This paper investigates the vibration of a bar for computing its frequency
parameter with interval and fuzzy material properties using fuzzy finite element method.
First crisp values of material properties are considered. Then the problem has been
undertaken taking the properties as interval and fuzzy. Initially, Young’s modulus and
density as material properties have been considered as interval in two different cases, one
for homogenous and other for non-homogenous material properties. Then the problem has
been analyzed using Young’s modulus and density properties as fuzzy. The finite element
method with fuzzy material properties (in term of fuzzy number that is triangular and
trapezoidal fuzzy number) is solved using -cut to obtain corresponding intervals. Using
interval computation frequency parameters are obtained and the fuzzy results are depicted
in term of plots.
Keywords Finite Element Method (FEM); Interval; Fuzzy Set; Fuzzy Number; Triangular
Fuzzy Number (TFN); Trapezoidal Fuzzy Number; Vibration; Natural Frequency.
1. Introduction
Finite Element Method is being extensively used to find approximate results of complicated
structures of which exact solutions cannot be found. The finite element method for the vibration
problem is a method of finding approximate solutions of the governing partial differential equations by
transforming it into an eigenvalue problem.
For various scientific and engineering problems, it is an important issue how to deal with
variables and parameters of uncertain value. Generally, the parameters are taken as constant for
simplifying the problem. But, actually there are incomplete information about the variables being a
result of errors in measurements, observations, applying different operating conditions or it may be
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
10
maintenance induced error, etc. Rather than the particular value of the material properties we may have
only the bounds of the values. Recently investigations are carried out by various researchers
throughout the globe by using the uncertainty or the fuzziness of the material properties.
Various generalized model of uncertainty have been applied to finite element analysis to solve
the vibration and static problems by using interval or fuzzy parameters. Although FEM in vibration
problem is well known and there exist large number of papers related to this. As such few papers that
are related to FEM and fuzzy FEM are discussed here. Elishakoff et al. [1] investigated a method of
successive iterations to yield closed-form solutions for vibration of inhomogeneous bars, where the
author studied the method of successive approximations so as to obtain closed form solutions for
vibrating inhomogeneous bars. Panigrahi et al. [2] presented a discussion about the vibration based
damage detection in an uniform strength beam using genetic algorithm. Dimarogonas [3] studied the
interval analysis of vibrating systems, where the author presented the theory for vibrating system
taking interval rotor dynamics. Naidoo [4] studied the application of intelligent technology on a multi-
variable dynamical system, where a fuzzy logic control algorithm was implemented to test the
performance in temperature control. The generalized fuzzy eigen value problem is investigated by
Chiao [5] who used the Zadeh’s extension principle to find the solution. Fuzzy finite element analysis
for imprecisely defined system is presented by Rao and Sawyer [6]. Recently Gersem et al. [7]
presented a discussion about the non-probabilistic fuzzy finite element method for the dynamic
behavior of structures using uncertain parameter. Verhaeghe et al. [8] discussed the static analysis of
structures using fuzzy finite analysis technique based on interval field. Very recently Mahato et al. [9]
studied the fixed free bar with fuzzy finite element method for computing its natural frequency. Moens
and Hanss [10] gave a brief overview of recent research activities on non-probabilistic finite element
analysis and its application for the representation of parametric uncertainty in applied mechanics. The
overview focuses on interval as well as fuzzy uncertainty treatment in finite element analysis. The
paper concentrates on implementation strategies for the application of the interval and fuzzy finite
element method to large finite element problems. Short Transformation Method (STM) with uncertain
parameters for dynamic analysis of structures is introduced by Donders et al. [11]. Chen et al. [12]
developed a method by using matrix perturbation and interval extension principle for interval
eigenvalue problem. In this paper cantilever beam and auto mobile frame are taken into consideration
with interval parameters. Qui et al. [13] studied the eigenvalue bounds of structures with interval
parameters. They compared the eigenvalue results obtained by different method. Also Qui et al. [14]
gave the exact bound of static response of the structures with interval parameters. Chen and Rao [15]
discussed the fuzzy finite element approach and developed algorithm to solve uncertain eigenvalue
problem for imprecisely defined structural systems. Fuzzy finite element analysis of smart structures is
investigated by Akpan et al. [16]. In this paper, vertex fuzzy analysis technique is used to find
frequency response. Rama Rao and Reddy [17] used the fuzzy finite element approach for analysis of a
cable-stayed bridge with multiple uncertainties present in the material properties. Interval based
approach is used for uncertain mechanics problems by Muhanna and Mullen [18]. Neumaier and
Pownuk [19] proposed method for linear system with large uncertainties. The uncertain-bounds of the
response have been found for truss structure in that paper.
In the present paper Fuzzy Finite Element method (FFEM) has been discussed to study vibration
of imprecisely defined bar. As already mentioned, generally, the values of variables or properties are
taken as crisp but in actual case the accurate crisp values cannot be obtained. To overcome the
vagueness we use interval and fuzzy numbers in place of crisp values. As such simulation with various
numbers of elements with crisp, interval and fuzzy material properties in the vibration of a bar has
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
11
been investigated here. The main aim in this paper is to solve the interval/fuzzy eigenvalue problem on
interval/fuzzy system matrices as obtained from FFEM [10, 12-14].
2. Preliminaries
In the computation of the said models we need to use interval and fuzzy computation. These
computations follow the interval arithmetic for interval and the fuzzy alpha cut which is actually a
form of interval computation for fuzzy numbers. As such, in this section, some definitions related to
the present work are given.
Definition 1 Interval
An interval A is a subset of R such that },,|{],[ 212121 R aaatataaA .
Definition 2 Interval arithmetic
If ],[ 21 aaA and ],[ 21 bbB are two intervals, then the arithmetic operations are defined as
i. ],[ 2211 babaBA
ii. ],[ 1221 babaBA
iii. }],,,max{},,,,[min{ 2212211122122111 babababababababaBA
iv. 0, where,]/1,/1[],[/ 211221 bbbbaaBA
Definition 3 Fuzzy set
A fuzzy set A of the real line R can be defined as the set of ordered pairs such
that ]}1,0[)(,/))(,{( xxxxA AA R , where )(xA is called the membership function or grade of
membership of x .
Definition 4 Fuzzy number
A fuzzy number A is convex normalised fuzzy set A of the real line R such that
},]1,0[:))({ RR xxA
where, )(xA is called the membership function of the fuzzy set and it is piecewise continuous.
One may find there are different types of fuzzy numbers present as described in [23, 25- 26].
Here only triangular and trapezoidal fuzzy numbers are taken into consideration for the present
investigation. These two fuzzy numbers are defined as follows.
Definition 5 Triangular fuzzy number
A triangular fuzzy number (TFN), A = (a, b, c) is shown in Fig.1 and its membership function is given
by )(xA as
cx
cbxbc
xc
baxab
axax
xA
, 0
],[,
],[,
, 0
)(
Definition 6 Trapezoidal fuzzy number
Also, a trapezoidal fuzzy number, A = (a, b, c, d) is depicted in Fig.1 and its membership function
)(xA is defined as
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
12
dx
dcxcd
xd
cbx
baxab
ax
ax
xA
, 0
],[,
,,1
],[,
, 0
)(
As mentioned in the Introduction, a fixed free bar is considered for the detail methodology and
analysis. In the following section the classical finite element method is used to find the expressions for
eigenvalue problem to obtain the frequency parameters. Here the material properties are taken first as
crisp.
Fig.1: Triangular and Trapezoidal Fuzzy numbers
3. Structural Finite Element Model for a Bar
The structure is considered to be an one-dimensional fixed free bar as shown in Fig. 2 with the
governing equation
2
2
2
2
t
UA
x
UAE
(1)
Here E , A , and U are Young’s modulus of elasticity, cross sectional area, density and displacement
at any point respectively.
1u 2u
Fig.2: A fixed free bar element
We develop first the necessary equation for single element (Zienkiewicz [20], Klaus [21],
Seshu [22]) using shape functions-l
xx 1)(1 and
l
xx )(2 .
l
1 2
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
13
The displacement at any point is given by )()(1),( 21 tul
xtu
l
xtxU
Accordingly the kinetic energy T and potential energy V are given respectively by
)()()()(6
1),(
2
121
22
21
2
0tutututumltxUmdxT
l
(2)
)()(2)()(22
121
22
21
0
2
tutututul
EAdx
x
UEAV
l
(3)
Using Lagrange’s Equation 0
q
L
q
L
t with Lagrange’s operator VTL and generalized
coordinator, )(),( 21 tutuq , we obtain the equation of motion as
0)()( tUKtUM (4)
In matrix form (Zienkiewicz [20], Klaus [21], Seshu [22]), it can be written as
0
0
11
11
21
12
6 2
1
2
1
u
u
l
EA
u
uml
For free vibration, taking tiWeU , where, W is the vector of the nodal displacements and is the
natural frequencies, we have
WMWK 2 (5)
This is a typical eigenvalue problem and is solved numerically as a generalized eigenvalue
problem.
The above is for single element case. In the next section, we will introduce the bar model with
more number of elements having crisp, interval and fuzzy material properties respectively.
4. Homogenous Fixed Free Bar with Crisp Values of Material Properties
Let us consider a fixed free bar having young’s modulus, density, area of cross section and
length viz. LAE and,,, respectively as crisp values of material properties for determining the
frequency parameter. The bar is simulated numerically with finite element models taking one, two,
three and four element discretization. For each model mass and stiffness matrix with the corresponding
eigenvalue equations are obtained by satisfying the boundary condition. Then frequency parameters
are obtained for all models after getting the global mass and stiffness matrices through assembling the
corresponding stiffness and mass matrix for each element. Although this section includes the well-
known expressions obtained by FEM but for clear understanding these have been included here. The
eigenvalue equations for various models satisfying boundary condition may easily be written with the
reduced stiffness and mass matrix as follows.
22
26
2 uAL
uL
EA (P. Seshu [12]) (6)
3
22
3
2
21
14
1211
122
u
uAL
u
u
L
EA (7)
4
3
2
2
4
3
2
210
141
014
18110
121
012
3
u
u
uAL
u
u
u
L
EA
(8)
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
14
5
4
3
2
2
5
4
3
2
2100
1410
0141
0014
24
1100
1210
0121
0012
4
u
u
u
u
AL
u
u
u
u
L
EA (9)
As discussed above the variables and parameters taken for the investigation may contain
uncertainty or vagueness. So, one may model these uncertainties in terms of interval or fuzzy number
(triangular or trapezoidal). In the subsequent sections imprecisely defined bar viz. taking the material
properties in term of interval and fuzzy numbers for homogenous and non-homogenous cases are
discussed. First homogeneous and next, non-homogeneous bar is simulated with uncertain Young’s
modulus and density for different cases.
5. Interval Material Properties for Homogenous Bar
In this section a homogeneous fixed free bar is considered for vibration analysis with interval
values of material properties. Finite element method is used to obtain the bounds of the frequency
parameters. The bar is simulated numerically with interval finite element models taking different
number of element discretization. For each model, interval mass and stiffness matrices with the
corresponding interval eigenvalue equations are obtained by satisfying the boundary condition.
Young’s modulus and density properties of bar are considered as interval. First single parameter
uncertainty that is only Young’s modulus or the density are considered. Next multiple uncertainty viz.
taking both Young’s modulus and density as uncertain in term of interval is considered. The
expressions obtained for different number of discretization for interval material properties using finite
element method are discussed in the following sub sections.
5.1. Homogenous Fixed Free Bar with Young’s Modulus ( E ) as an Interval
Now, we consider the fixed free bar having density, area of cross section and length viz. LA and,, respectively as crisp. Young’s modulus has been taken as interval that is ],[ EEE (Hanss
[23]). Frequency parameters are obtained for second, third and fourth models after getting the global
interval mass and stiffness matrices. Interval eigenvalue equations for various models satisfying
boundary condition may easily be written with the reduced stiffness and mass matrices as follows.
3
2 AL
L
AE and
3
2 AL
L
AE (10)
3
22
3
2
21
14
12
22
u
uAL
u
u
EE
EE
L
A and
3
22
3
2
21
14
12
22
u
uAL
u
u
EE
EE
L
A (11)
4
3
2
2
4
3
2
210
141
014
180
2
023
u
u
uAL
u
u
u
EE
EEE
EE
L
A
and
4
3
22
4
3
2
210
141
014
1810
2
023
u
u
uAL
u
u
u
E
EEE
EE
L
A (12)
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
15
5
4
3
2
2
5
4
3
2
2100
1410
0141
0014
24
00
20
02
002
4
u
u
u
u
AL
u
u
u
u
EE
EEE
EEE
EE
L
A and
5
4
3
2
2
5
4
3
2
2100
1410
0141
0014
24
00
20
02
002
4
u
u
u
u
AL
u
u
u
u
EE
EEE
EEE
EE
L
A (13)
5.2. Homogenous Fixed Free Bar with Density )( as an Interval
In this case, the homogenous fixed free bar is taken into consideration as discussed in the sub
section 5.1 but now with Young’s modulus E as crisp and density as interval viz. ],[ .
Governing interval eigenvalue equations satisfying the boundary condition for various models are
obtained. Although results with various number of finite elements are computed but here only the case
of four element discretization is given in Eq. (14).
5
4
3
2
2
5
4
3
2
200
40
04
004
24
1100
1210
0121
0012
4
u
u
u
u
AL
u
u
u
u
L
EA
and
5
4
3
2
2
5
4
3
2
200
40
04
004
24
1100
1210
0121
0012
4
u
u
u
u
AL
u
u
u
u
L
EA
(14)
5.3. Homogenous bar with Density ( ) and Young’s Modulus ( E ) both as Intervals
Next, the above homogenous fixed free bar is considered with both Young’s modulus and
density as interval (Jaulin et al. [24]) viz. ],[ EEE and ],[ . Governing interval eigenvalue
equations satisfying the boundary condition for various models may again be obtained and matrix
equation for four element discretization is written as
5
4
3
2
2
5
4
3
2
200
0
041
004
24
00
20
02
002
4
u
u
u
u
AL
u
u
u
u
EE
EEE
EEE
EE
L
A
and
5
4
3
2
2
5
4
3
2
200
40
04
004
24
00
20
02
002
4
u
u
u
u
AL
u
u
u
u
EE
EEE
EEE
EE
L
A
(15)
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
16
6. Homogenous Bar with Material Properties as Fuzzy
Here two types of fuzzy uncertainty viz. triangular and trapezoidal has been considered.
Accordingly first, a homogeneous fixed free bar is considered for vibration analysis with material
properties as triangular fuzzy number having length L . Finite element method is used to obtain the
fuzzy frequency parameters. The bar is simulated numerically with fuzzy finite element models taking
one, two, three and four element discretization. For each model fuzzy mass and stiffness matrix with
the corresponding fuzzy eigenvalue equations are obtained by satisfying the boundary condition.
Single parameter uncertainty in term of triangular fuzzy number with Young’s modulus ( E ) and
density )( individually are considered first. Then multiple uncertainties in the formulation of the
model have been considered by taking both Young’s modulus and density as triangular fuzzy number.
Through cut approach triangular fuzzy number material properties viz. ),,( cbaE and
),,( fed can be expressed as )(,)(],[ bccaabEEE and
)(,)(],[ effdde . So for these the respective equations of motion may be written
as in Eqs. (10) to (15).
Similarly, one may model the uncertainties in term of trapezoidal fuzzy number for different
cases as discussed above. For example let us consider a fixed free bar having Young’s modulus as
),,,( 1111 dcbaE . Then its corresponding -cut forms is )(,)(],[ 111111 cddaabEEE .
7. Non-homogeneous Fixed Free Bar with Crisp Values of Material Properties
A non-homogeneous fixed free bar is considered having length L . The bar is simulated with
four models for determining the frequency parameters using finite element method. For first, second,
third and fourth model the material is discretized into one, two, three and four number of elements with
equal length respectively.
Area of cross section for each element is uniformly distributed as A . As discussed above
Young’s modulus and density are different for every element. So Young’s modulus and density are
considered as iiE , for each model. For first, second, third, and fourth model 1i ; 2,1i ; 3,2,1i and
4,3,2,1i respectively.
Then eigenvalues are obtained for all models after getting the global mass and stiffness
matrices through assembling the corresponding stiffness and mass matrix for each element. Although,
this section includes the well-known expressions obtained by FEM but for clear understanding these
have been included here. The Young’s modulus and density varies for different elements along the bar.
The eigenvalue equations for first, second, third and fourth model satisfying the boundary condition
may easily be written again with the reduced stiffness and mass matrix as
21
21
3u
ALu
L
AE (16)
3
2
22
221
3
2
22
221
2
)(2
12
2
u
uAL
u
u
EE
EEE
L
A
(17)
4
3
2
33
3322
221
2
4
3
2
33
3322
221
20
)(2
0)(2
180
03
u
u
uAL
u
u
u
EE
EEEE
EEE
L
A
(18)
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
17
200
)(20
0)(2
00)(2
24
00
0
0
00
4
5
4
3
2
44
4433
3322
221
2
5
4
3
2
44
4433
3322
221
u
u
u
u
AL
u
u
u
u
EE
EEEE
EEEE
EEE
L
A
(19)
8. Non-homogenous Fixed Free Bar with Young’s Modulus ( E ) and Density ( )
both as Intervals
Consider the same non-homogeneous bar for fourth model (four element discretization) having
material properties as area of cross section and length for each element are A and 4/ L respectively.
Now let us take Young’s modulus and density material properties as interval for first, second, third and
fourth element respectively as
, 111 EEE , , , 222 EEE , 333 EEE , , 444 EEE and 111 , , 222 , , 333 , an
d 444 , .
Now with these material properties, using finite element method the interval eigenvalue
problem for four element discretization is obtained. After satisfying the boundary condition it is
expressed as two generalized eigenvalue problem as below
5
4
3
2
44
4433
3322
221
2
5
4
3
2
44
4433
3322
221
200
)(20
0)(2
00)(2
24
00
0
0
00
4
u
u
u
u
AL
u
u
u
u
EE
EEEE
EEEE
EEE
L
A
and
00
)(20
0)(2
00)(2
24
00
0
0
00
4
5
4
3
2
44
4433
3322
221
2
5
4
3
2
44
4433
3322
221
u
u
u
u
AL
u
u
u
u
EE
EEEE
EEEE
EEE
L
A
(20)
9. Uniform Non-homogenous Fixed Free Bar with Young’s Modulus ( E ) and
Density ( ) both as Triangular Fuzzy Number
Let us take Young’s modulus and density material properties as triangular fuzzy number (Ross
[11]) for first, second, third and fourth element respectively as
),,,( 1111 cbaE ),,,( 2222 cbaE ),,,( 3333 cbaE ),,( 4444 cbaE
and ),,,( 1111 fed ),,,( 2222 fed ),,( 3333 fed , ),,( 4444 fed
As mentioned earlier, these material properties for different elements may be written in -cut
form respectively as
, )(,)( 111111 bccaab , )(,)( 222222 bccaab , )(,)( 333333 bccaab
)(,)( 444444 bccaab and
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
18
, )(,)( 111111 effdde )(,)( 222222 effdde , )(,)( 333333 effdde ,
)(,)( 444444 effdde where,
1111 )( acbE , )( 1111 bccE , 2222 )( acbE , )( 2222 bccE ,
, )( 3333 acbE )( 3333 bccE 2444 )( acbE )( 4444 bccE ,)( 1111dde
),( 1111 eff ,)( 2222dde ,)( 2222
dde ,)( 3333dde ),( 3333 eff
)( 4444 eff and )( 4444 eff
Using these bounds of material properties one may obtain the fuzzy eigenvalue problem by
fuzzy finite element method as discussed in previous sections.
10. Numerical Results and Discussions
Two examples are used to demonstrate the procedures of this paper. The first is a homogeneous
fixed free bar. Next is a uniform non-homogeneous fixed free bar, where the Young’s modulus and
density are assumed to vary independently. In both the examples various cases have been studied with
single and multiple uncertainties in the material properties.
10.1. Homogeneous Bar
To verify the discussed procedures for homogeneous fixed free bar with crisp and uncertain
material properties, here an example problem is considered with four models. The cross section area
and length of the bar respectively are taken as 2-6 m1030 A and m1L for the homogeneous bar.
Here different cases for fixed free bar with crisp and uncertain material properties are studied to obtain
the frequency parameters using the equations as discussed above. In the following paragraphs
examples for heads 4 to 6 are given in term of cases 1 to 7. In cases 8 to 10 the frequency parameters
are computed with material properties as trapezoidal fuzzy number.
Case 1: In this case, the Young’s modulus and the density of the models are crisp parameters given by 3211 kg/m 7800,N/m102 E .
The frequency parameters are obtained by solving the crisp generalized eigenvalue problem for
one, two, three and four numbers of discretization by using the expressions from Eqs. (6) to (9). The
results are given in Table 1.
Case 2: Here, the density of the models is crisp parameter given by 3kg/m 7800 . Young’s modulus
of the bar is uncertain and its interval value is 21111 N/m10002.2,101.998, EEE .
The interval frequency parameters are obtained by solving interval eigenvalue problem for one,
two, three and four numbers of discretization by using the expressions from Eqs. (10) to (13) and are
given in Table 2.
Table 1. Crisp value of frequencies with LAE ,,, as crisp
Number of elements
Modes
1 2 3 4
1 8770.6 8160.724 8045 7791
2 28505 26312 25596
3 47733 45059
4 71475
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
19
Table 2. Interval values of frequencies with Young’s modulus as interval value
Number of elements
Modes
1 2 3 4
1 [8766.2,8775] [8136, 8183 ] [7989, 8101] [7692, 7889 ]
2 [28503, 28508] [26299, 26325] [25567, 25624]
3 [47133, 48679 ] [45049, 45069]
4 [71473, 71476 ]
Case 3: Now, the Young’s modulus of the models is considered as crisp given by 211 N/m102 E and
density of the bar is uncertain and its interval value is 3kg/m 8000,7500],[ .
The bounds of the frequency parameters are computed for different number of elements and the
results are given in Table 3.
Table 3. Interval values of frequencies with density as interval
Number of elements
Modes
1 2 3 4
1 [8660.3, 8944.3] [8057, 8321 ] [7944, 8205] [7693, 7946]
2 [ 28147, 29070] [25981, 26833] [ 25274, 26103]
3 [47133, 48679 ] [44493, 45952]
4 [70575, 72890 ]
Case 4: Both, Young’s modulus and the density of the models are considered as interval parameters
given by 321111 kg/m 8000,7500],[ ,N/m10002.2,101.998, EEE .
The interval eigenvalues are computed for one, two, three and four numbers of discretization
by using interval finite element method are incorporated in Table 4.
Table 4. Interval values of frequencies with E, as intervals
Number of elements
Modes
1 2 3 4
1 [ 8655.9, 8948.7] [8034, 8346] [7888, 8261] [7596, 8046]
2 [28144, 29072] [25968, 26846] [25245, 26132 ]
3 [47131, 48680 ] [44482, 45962]
4 [70574, 72891 ]
Case 5: Consider, the density as crisp parameter given by 3kg/m 7800 and Young’s modulus of the
bar in term of triangular fuzzy number is 2111111 N/m )10002.2 ,102 ,101.998 ( E
The uncertain frequency parameters are obtained for different number of elements by using the
expressions from Eqs. (10) to (13). In this case corresponding fuzzy eigenvalue problem is solved.
Only the results for model four are depicted (triangular fuzzy number) in Fig. 3.
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
20
(a ) (b)
(c ) (d )
Fig. 3: First(a), second(b), third(c) and fourth(d) frequency for 4 element discretization of case 5
Case 6: The Young’s modulus and density of the models are considered as crisp and triangular fuzzy
number respectively as 211 N/m102 E , 3kg/m )8000 ,7800,7500( .
The frequency parameters in term of triangular fuzzy numbers are computed for different
number of elements by solving the fuzzy eigenvalue problem. For the 4 element discretization the
computed fuzzy frequency parameters and the results are depicted in Fig. 4.
Case 7: In this case, both the Young’s modulus and the density of the models are in term of triangular
fuzzy number which are considered as 2111111 N/m )10002.2 ,102 ,101.998 ( E and 3kg/m )8000 ,7800,7500(
Corresponding frequency parameters are computed which are fuzzy number (TFN) for different
element discretization and for four element results are shown in Fig. 5.
Case 8: Consider the density as crisp parameter given by 3kg/m 7800 and Young’s modulus of the
bar in term of trapezoidal fuzzy number as 211111111 N/m )10002.2 ,10001.2,10999.1 ,101.998 ( E
Corresponding fuzzy eigenvalues are computed which are trapezoidal fuzzy number. The
results for four element discretization are cited in Fig. 6.
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
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21
(a ) (b)
(c ) (d)
Fig. 4: First(a), second(b), third(c) and fourth(d) frequency for 4 element discretization of case 6
(a ) (b )
(c ) (d)
Fig. 5: First(a), second(b), third(c) and fourth(d) frequency for 4 element discretization of case 7
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
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22
(a) (b)
(c) (d)
Fig. 6: First(a), second(b), third(c) and fourth(d) frequency for 4 element discretization of case 8
Case 9: In this example, Young’s modulus and density of the models are considered as crisp and
trapezoidal fuzzy number given by 211 N/m102 E and 3kg/m )8000 ,7900,7700,7500( respectively
The fuzzy frequency parameters are computed for 4 element discretization and the results in
term of trapezoidal fuzzy number are depicted by Fig. 7.
Case 10: For this case, both the Young’s modulus and the density of the models are trapezoidal fuzzy
number as 211111111 N/m )10002.2,102.001 ,101.999 ,101.998 ( E and 3kg/m )8000 ,7900,7700,7500(
Corresponding frequency parameters are computed and those are obtained as trapezoidal fuzzy
number. Results from four element discretization are shown by Fig. 8.
10.2. Non-homogeneous Bar
Again, to verify the discussed procedures for non-homogeneous fixed free bar with crisp and
uncertain material properties four models depending upon different number of elements are
considered. For nonhomogeneous bar the cross section area and length of the bar are taken as same as
of homogeneous case that is 2-6 m1030 A and m1L respectively.
In the following paragraphs examples for heads 8 to10 are given in term of case 1 to 3. In the
first case material properties are taken as crisp. Next for second and third case Young’s modulus and
density properties both are considered as interval and triangular fuzzy number respectively.
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
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23
(a) (b)
(c) (d)
Fig. 7: First(a), second(b), third(c) and fourth(d) frequency for 4 element discretization of case 9
(a) (b)
(c) (d)
Fig. 8: First(a), second(b), third(c) and fourth(d) frequency for 4 element discretization of case 10
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24
Case 1: In this case, the Young’s modulus and the density of the models are taken as crisp parameters
given by 211
3211
2211
1 N/m104,N/m103N/m102 EE,E , 2114 N/m101E
, 31 kg/m7800 3
43
33
2 kg/m 8500,kg/m 7500,kg/m 8200
The eigenvalues are obtained for one, two, three and four numbers of discretization by using
the expressions from Eqs. (16) to (19) and are given respectively in Table 5.
Table 5: Crisp values of frequency parameters for non-homogenous bar
Number of elements
Modes
1 2 3 4
1 8770.6 8195 8636 1034.7
2 33533.565 3158.1 2397
3 6436.2 3883.6
4 7305.1
Case 2: Here, the Young’s modulus and the density of the models are interval parameters given by
,N/m 10002.3,102.998,N/m 10002.2,101.998 211112
211111 EE
,N/m 10002.4 ,103.998 211113 E
21111
4 N/m 10002.1,100.998 E and
, 31 kg/m8000,7500 ,kg/m 8500,8000 3
2 ,kg/m 7700 ,7200 33 3
4 kg/m 8700 ,8200 The interval frequency parameters are computed for different numbers of discretization by
using interval finite element method. The results are incorporated in Table 6.
Case 3: Both the Young’s modulus and the density of the models are considered in term of triangular
fuzzy number given by
,N/m )10002.2 ,102 ,101.998 ( 21111111 E ,N/m )10002.3 ,103 ,102.998 ( 2111111
2 E
,N/m)10002.4 ,104 ,103.998 ( 21111113 E
21111114 N/m )10002.1 ,101 ,100.998 ( E
and
,kg/m )8000,7800,7500( 31 ,kg/m )7700 ,7500,7200(,kg/m )8500 ,8200,8100( 3
33
2 3
4 kg/m )8700 ,8500,8200( Table 6: Interval values of frequencies for E, as intervals
Number of elements
Modes
1 2 3 4
1 [8655.9, 8948.7] [8036,8290 ] [ 8456, 8818 ] [1009.6 , 1061.8]
2 [3299.9, 3390.7] [3111.7, 3209.2] [2366.2, 2441.4 ]
3 [ 6342, 6537] [3827 , 3945]
4 [7199.8,7415.9]
Corresponding fuzzy frequency parameters are computed which are triangular fuzzy number
(TFN) for different element discretization. Results for four element discretization are shown in Fig. 9.
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
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25
(a) ( b)
(c) (d)
Fig. 9: First(a), second(b), third(c) and fourth(d) frequency of fourth model of case 3
It may be seen from the above numerical results that the frequency parameters gradually
decrease with increase in number of elements as it should be. In crisp values of frequency for
homogenous and non-homogenous bar, the first frequency parameter got reduced to 8006.248 from
8770.32. Similar trend of reduction may also be seen for interval and fuzzy cases. Moreover, in Table
3 the interval width for frequency parameter also reduces with increase in elements (first frequency
parameter reduces to (7693, 7946) from (8660.3, 8944.3). This is true for only in the density case.
However in case of Young’s modulus (as interval) it is increasing. In the case of multiple uncertainty
that is when Young’s modulus and density both are as interval, the width again increases as we
increase the number of elements. It is interesting to note also that the addition of the computed
frequency widths for the cases of homogeneous bar viz. interval/fuzzy andE (such as Tables 2 and
3) gives the interval width of frequency parameter in Table 4. That is to say if we add the widths of
frequency interval in Table 2 with the widths of Table 3 for each of the frequency parameters, one may
get the width of the frequency from Table 4. So, the frequency width for multiple uncertainty model
may be obtained (approximately) by adding the frequency width of each of the single uncertainty
model. This is also true for fuzzy case. Here, one may also note a special case related to the eigenvalue
Int. J. Modern Math. Sci. 2013, 6(1): 9-27
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26
equations that result for crisp case can be obtained simply by putting 1 in fuzzy case. Moreover by
substituting 0 in fuzzy case we can obtain the results for the interval case.
11. Conclusions
The investigation presents here the fuzzy FEM in the vibration of a fixed free bar. Simulation
with different number of elements in the model has been studied in detail. Related generalized
eigenvalue problem with respect to the interval and fuzzy material properties are solved to obtain the
uncertain bounds of frequency parameters depending upon the number of elements taken in the
discretization. Single and multiple uncertainties are also investigated in detail. Two types of fuzzy
numbers viz. triangular and trapezoidal have been considered in the analysis. The case of non-
homogeneous material properties is also considered. Investigation presented here may find real
application where the material properties may not be obtained in term of crisp values but a vague value
in term of either interval or fuzzy is known. From the results of interval/fuzzy uncertainty, it is worth
mentioning that frequency width may approximately be obtained for multiple uncertainties by adding
the frequency widths of each of single uncertainty models.
Acknowledgements
This work is financially supported by Board of Research in Nuclear Sciences (Department of Atomic
Energy), Government of India. We would like to thank the reviewers for their valuable comments and
suggestions to improve the paper.
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