19
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, * 1 Department 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

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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

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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.

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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

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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.

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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.

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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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