modal analysis of p-fgm rectangular...

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International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 8, August 2017, pp. 1202–1211, Article ID: IJMET_08_08_120

Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=8

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

MODAL ANALYSIS OF P-FGM RECTANGULAR

PLATE MADE OF ORTHOTROPIC MATERIALS

UNDER SIMPLY SUPPORTED AND CLAMPED

BOUNDARY CONDITIONS IN BOTH

SYMMETRIC AND ANTISYMMETRIC MODES

OF VIBRATION

S. Karthikeyan

Department of Mathematics, Government Arts College, Salem, Tamilnadu

S. Rama

Department of Mathematics, Mount Carmel College, Bangalore, Karnataka

ABSTRACT

The functionally graded materials have a wide range of applications over medical

science and aerospace engineering. In general the functional gradation has been

exposed only on the two different nature materials. It is necessary to determine the

resonant frequencies of these FGM used in biosensors, actuators and memory devices.

With the help of this resonant frequencies the smart structures capability has been

increased. This paper deals with the determination of the resonant frequencies of

FGM with thickness gradation made by pyroelectric-piezo magnetic materials. The

gradation function has been considered as power-law function of thickness variable.

The initial disturbance has been made by the wave propagation in both symmetric and

antisymmetric modes. For the two types of boundary conditions viz all the sides of the

plate are simply supported and opposite sides are clamped, the non-dimensionalized

frequencies for different exponent and wave numbers have been determined along

with the damping effect. The dispersion curves are drawn. For numerical study, the

FGM plate graded with Barium Titanate (pyroelectric material) and Copper Ferrous

Sulphate (piezo-magnetic material) has been considered.

Keywords: Functionally graded materials, Piezo magnetic, pyroelectric, resonant

frequencies.

Cite this Article: Karthikeyan and S.Rama, Modal Analysis of P-FGM Rectangular

Plate Made of Orthotropic Materials under Simply Supported and Clamped Boundary

Conditions in both Symmetric and Anti-symmetric Modes of Vibration, International

Journal of Mechanical Engineering and Technology 8(8), 2017, pp. 1194–1211.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=8

S. Karthikeyan and S.Rama


1. INTRODUCTION:

Piezo magnetic materials are mainly used as nonvolatile memory units which have electrically

written data and would be read magnetically. Pyroelectric materials majorly used as bio-

sensors which have more applications in medical field and also in aerospace engineering. The

composite structures of pyroelectric-piezo magnetic materials have wide range of

applications.

Cho and Tingley (2000) analyzed the thermal stress characteristics of FGM introduced in

layered composite materials and optimize the FGM properties to control the thermal stress.

Efrain (2011) derived an accurate formula for natural frequencies of FGM plate and the

isotropic plate made of materials with different poison’s ratio. Nakamura, Wang and Sampath

(2000) used inverse analysis process to determine the displacement at several load magnitudes

and to make best estimates for the unknown parameters. Ramu and Mohanty (2014) carried

out the modal analysis of FGM isotropic plate using FEM and FGM plate program has been

coded in MATLAB software. Weon (2016) investigated the vibration and buckling analysis

of FGM plate using 8 noded shell elements. Liux and Lam (2001) developed a hybrid

numerical method to compute the wave field in an FGM plate for a given material properties

and the gradation has been considered in the thickness direction. Sung, Lomboy and Kim

(2007) investigated the natural frequencies and buckling loads of FGM plates and shells using

4 noded quasi conforming shell elements with the assumptions that the poison ratios of the

materials are constant. Liu, Wang and Chen (2010) analyzed the free vibration of FGM

isotropic rectangular plate by considered the in-plane inhomogeneity of the material

properties. Saritha and Vinayak (2016) performed the thermal analysis and structural analysis

of disc brake by varying the materials like cast Iron, Carbon fiber and used FGM as interface

zone.

In this chapter the functionally graded material with thickness gradation has been

considered and the resonant frequencies for a thin rectangular plate have been determined.

The gradation function has been taken as the power-law function in terms of thickness

variable. Hence the FGM plate is referred as p-FGM plate. The governing and constitutive

equations for the coupled field given in [3]

have been considered and using variational

formulation technique the frequency equations are derived and numerically solved for a set of

boundary conditions viz simply supported boundary conditions and clamped boundary

conditions. The geometry has been exerted to the harmonic wave propagation in different

modes of vibration. The resonant frequencies along with the damping frequencies are

determined and dispersion curves are drawn.

2. P-FGM PLATE THICKNESS GRADATION AND MATERIAL

PROPERTIES:

The geometry of the FGM plate considered has been given in Fig1. The bottom surface with

Pyroelectric material (BaTiO3-Barium Titanate) and the top surface with piezo magnetic

material (CoFe2O4-Copper Ferrous Sulphate) with the dimension (Lx,Ly,h).

Figure 1 Geometry of the p-FGM plate

Modal Analysis of P-FGM Rectangular Plate Made of Orthotropic Materials under Simply Supported

and Clamped Boundary Conditions in both Symmetric and Anti-symmetric Modes of Vibration


The initial assumptions made are the surfaces of the plate are traction free, coated with

electrodes and insulated both thermally and magnetically. The gradation of the plate has been

made along thickness direction with the gradation function [10]

( )

p

h

hz

zg

+

= 2

(1)

With p a non-negative real number as exponent. Also the volume fractions obey the above

said power-law such that

Vc(z)=g(z) and Vb(z)+Vc(z)=1 (2)

Where Vc(z) volume fraction of piezo-magnetic material and Vb(z) volume fraction of

pyroelectric material. Clearly Vc(z), Vb(z) ( )hC ,0∂∈ . If p approaches one then the

pyroelectric layer extended and the plate act as a pyroelectric plate whereas p approaches zero

the piezo-magnetic layer extended and the plate act as a piezo-magnetic plate. The material

properties are depending on the thickness variable and volume fractions

( ) bcFGM zgzg γγγ )(1)( −+= (3)

Where cγ -material property of piezo-magnetic material and bγ - material property of

pyroelectric material. By considering the power-law function the material properties of FGM

vary uniformly over the plate in the thickness direction.

3. FREQUENCY EQUATION:

This problem has been considered as multifield problem involving the four fields viz elastic,

electric, thermal and magnetic fields. The constitutive equations and the governing equations

of the plate are considered from [3]

.

3.1 Variational formulation:

In the multifield problem the quantities determine the system are ),,,( ψθφU where

U=(u,v,w)-mechanical displacement, φ -electric potential, θ -change in temperature with

reference to the initial temperature 0θ and ψ -magnetic potential. By variational method

divergence equations are multiplied with the virtual quantities ),,,( δψδθδφδU respectively

and integrated over the domain V. Using the initial approximations in section 1, the integral

weak form of divergence equations

( ) ( )

( )

( ) ( )

( )

0][

][

0

=

⋅∇

+⋅∇

⋅∇

+⋅∇

∫

∫

∫

∫

V

V

V

V

BdV

dVh

DdV

dVUUU

δψ

ηθδθδθ

δφ

ρδσδ

&

&&

(4)

These integral equations are solved by finite element technique. Dividing the model into

finite number of elements and the unknowns are determined at the nodes by Rayleigh-Ritz



method. Initially the unknowns are approximated as linear combination of the nodal values

and shape functions at each element.

[ ]( ){ } )16(616][ ××× = n

e

n

eeXNX

(5)

Where n-number of nodes per element. Plugging these approximations into the

constitutive equations and has been evaluated at each element. Substituting this

approximation and the transformed constitutive equations into the integral equation (5) yield

the system of homogeneous differential equations at each element referred as element

stiffness equations. The stiffness matrices are given in appendix A.

0}]{[}]{[}]{[ =++ ee

III

ee

II

ee

I XKXKXK &&& (6)

3.2 Approximations and frequency Equation:

Consider the coordinate system (x,y,z) represent the FGM geometrically. The shape

functions are approximated as functions of (x,y,z,t) by separating the in plane variables and

the unknowns vary harmonically in the time domain. Let the system vibrate with an angular

frequency ω which is referred as the resonant frequency of the system. Hence the shape

functions are approximated

[ ] [ ]tie ezhyxfN ω−= )(),( (7)

Where the functions f and h are appropriately chosen to satisfy the mechanical boundary

conditions of the FGM plate. By plugging eqn. (7) in eqn.(6) the element stiffness equation

transformed into frequency equation at each element and assembling the element matrices

yield the global frequency equation

[ ] 0}{][][][ 2 =−− e

IIIIII XKKiK ωω (8)

4. NUMERICAL APPROXIMATIONS AND SOLUTIONS:

For numerical study the material properties for barium titanate and copper ferrous

sulphate materials are considered. At each node there are six degrees of freedom

( )ψθφ ,,,,, wvu . In symmetric mode of vibration, the shape functions for each unknown are

approximated so as to satisfy the mechanical boundary conditions (simply supported and

clamped)

[ ] ( ) ( ) tiju exxzN ω−−= 21

1 qsinqcos

[ ] ( ) ( ) tijv exxinzN ω−−= 21

1 qcosqs

[ ] ( ) ( ) tijw exxinzNNNN ωψθφ −−==== 21

1 qsinqs][][][ (9)

Where j=1:n, xL

lkq

π11 = ,

yL

lkq

π12 = , αcos=l , αsin=m and α is the direction of

harmonic wave propagation and k1 is the wave number. For antisymmetric mode of vibration

sine would be replaced by cosine and cosine would be replaced by sine. Solving the frequency

equation in eqn.(8) for both symmetric and antisymmetric mode of wave propagation the

natural frequencies along with the damping frequencies are obtained and their corresponding

non-dimensionalized values are determined [2]




( )b

byx

FGMFGMch

LL

1,1

ρωω

=

(10)

Where bρ and c(1,1)b are density and elastic coefficients of pyroelectric material

respectively.

4.1 Simply supported boundary condition:

The material constants are taken for barium titanate and copper ferrous sulphate and the angle

of wave propagation is considered in the direction 30=α . The geometry of the plate is taken

as (0.01, 0.01, 0.03). While solving the frequency equation, the imaginary eigenvalues are

obtained. The real part of the eigenvalues proportional to the natural frequency whereas the

imaginary part corresponds to the damping frequency. The maximum damping frequencies in

symmetric mode of vibration for different exponent and wave numbers are given in figure.2-

figure.5

Figure 2 Exponent=0.125

Figure 3 Exponent=0.5



Figure 4 Exponent=1

Figure 5 Exponent=4

In the antisymmetric mode of vibration the dispersion curves corresponding to the

damping frequencies are drawn for different exponent (Figure 6)

Figure 6 Dispersion curves for antisymmetric mode of vibration

The results are validated with the literature [4]

for the exponent p=1




4.2 Clamped boundary conditions:

For symmetric mode the maximum natural frequencies and damping frequencies are

tabulated and are given in table.1

Exponenet=0.125 0.5 1 1.5 2 2.5 3 3.5

Max frequency

0.04377+

i9.145e-11

0.113+

i0.0009476

0.1351+

i0.003426

0.1286+

i0.000122

0.1313+

i0.004796

0.1504+

i0.000267

0.1554+

i0.0001715

Max Damping

4.319e-8+

i0.01433

1.854e-06+

i7.182

1.44e-

07+i0.793

0.04343+

i0.03238

0.006046+

i0.07623

4.307e-8+

i0.2846

5.66e-08+

i0.9703

Exponenet=0.5 0.5 1 1.5 2 2.5 3 3.5

Max frequency

0.04536+

i2.247e-12

0.1206+

i0.001015

0.1441+

i0.003118

0.1367+

i0.0001208

0.1323+

i0.002187

0.1606+

i0.0002475

0.1659+

i0.000157

Max Damping

2.247e-08+

i0.01858

1.055e-6+

i8.161

1.223e-07+

i0.8072

0.04197+

i0.03529

0.0661+

i0.01597

1.254e-07+

i0.313

5.86e-08+

i1.059

exponent=1 0.5 1 1.5 2 2.5 3 3.5

Max frequency

0.04705+

i1.061e-11

0.1285+

i0.001034

0.1534+

i0.002805

0.1451+

i9.187e-05

0.1372+

i6.97e-09

0.1712+

i0.000226

0.1768+

i0.000142

Max Damping

2.564ee-

09+i0.02532

1.143e-06+

i9.224

3.519e-

07+i0.9134

0.03873+

i0.03981

7.6397e-5+

i0.04754

0.01816+

i0.08013

2.214e-07+

i1.198

exponent=4 0.5 1 1.5 2 2.5 3 3.5

Max frequency

0.05149+

i3.419e-11

0.1495+

i0.0009483

0.177+

i0.002032

0.1671+

i9.513e-05

0.1554+

i3.052e-07

0.1981+

i0.0001683

0.2045+

i0.0001039

Max Damping

9.365e-09+

i0.07289

1.67e-08+

i23.2

3.588e-07+

i2.307

0.02526+

i0.8237

0.0023+

i0.08075

3.928e-07+

i6.194

6.124e-08+

i3.026

Table 1 Natural frequencies and damping frequencies for symmetric mode of vibration

For antisymmetric mode of vibration the natural frequencies, damping frequencies and the

damping ratio for different exponent and different wave numbers are tabulated and are given

in Table 2-5

Wave

number

Maximum Imaginary part Maximum Real part

Damping

frequency Real part Damping ratio

Natural

frequency

Damping

frequency Real part

Damping

ratio

Natural

frequency

0.01 0.0005566 0.00848 0.997851831 0.008496251 9.50E-07 0.02924 1 0.02924

0.5 0.0994 1.15E-07 1.15191E-06 0.0994 2.04E-09 0.07486 1 0.07486

1 0.01234 5.05E-07 4.09076E-05 0.01234 7.54E-05 0.08135 1 0.08135003

5

1.5 0.04138 0.0269 0.54503108 0.049354984 3.36E-09 0.1323 1 0.1323

2 7.722 4.03E-06 5.21756E-07 7.722 1.42E-04 0.1126 1 0.11260009

2.5 0.5869 4.23E-08 7.20736E-08 0.5869 4.19E-05 0.1335 1 0.13350000

7

3 0.0595 1.30E-07 2.18992E-06 0.0595 2.37E-04 0.1435 1 0.14350019

5

3.5 0.00201 5.06E-11 2.51642E-08 0.00201 8.27E-05 0.148 1 0.14800002

3

Table 2 Exponent p=0.125



Wave

number

Maximum imaginary part Maximum real part

Damping


Natural

frequency

Damping

frequency Real part

Damping

ratio

Natural

frequency

0.01 0.001574 0.01609 0.995249231 0.016166805 1.43E-05 0.03039 1 0.030390003

0.5 0.344 2.45E-08 7.13372E-08 0.344 1.03E-08 0.0792 1 0.0792

1 0.01219 2.95E-09 2.42248E-07 0.01219 1.39E-12 0.08806 1 0.08806

1.5 0.02151 0.04511 0.902634778 0.049975916 1.86E-09 0.1392 1 0.1392

2 8.422 2.03E-06 2.41035E-07 8.422 1.56E-04 0.1201 1 0.120100101

2.5 0.6407 1.36E-08 2.118E-08 0.6407 4.16E-05 0.142 1 0.142000006

3 0.0003749 1.53E-01 0.999996998 0.153000459 3.75E-04 0.153 1 0.153000459

3.5 0.08597 1.15E-01 0.800434869 0.143422038 7.75E-05 1.58E-01 1 0.158000019

Table 3 exponent p=0.5

Wave

number


Damping


Natural

frequency

Damping

frequency Real part

Damping

ratio

Natural

frequency

0.01 0.0002498 0.00427 0.998293186 0.004277301 1.25E-05 0.00844 1 0.008440009

0.5 0.07765 1.35E-08 1.73986E-07 0.07765 1.11E-06 0.0837 1 0.0837

1 0.01925 3.81E-03 0.194253836 0.019623808 4.63E-09 0.09266 1 0.09266

1.5 0.0563 3.22E-07 5.71048E-06 0.0563 8.03E-10 0.1463 1 0.1463

2 9.52 1.30E-06 1.3645E-07 9.52 1.65E-04 0.128 1 0.128000106

2.5 0.725 8.38E-08 1.15601E-07 0.725 4.03E-05 0.1508 1 0.150800005

3 1.20E+43 9.08E+44 0.999912429 9.0828E+44 1.20E+43 9.08E+44 0.99991 9.0828E+44

3.5 0.006095 6.38E-02 0.995467731 0.064090475 7.15E-05 1.68E-01 1 0.168300015

Table 4 exponent p=1

Wave

number


Damping

frequency

Real

part

Damping

ratio

Natural

frequency

Damping

frequency Real part

Damping

ratio

Natural

frequency

0.01 0.01234 4.55E-11 3.68963E-09 0.01234 3.67E-05 0.01636 1 0.016360041

0.5 0.3172 1.57E-07 4.95902E-07 0.3172 3.35E-10 0.09539 1 0.09539

1 0.05091 7.36E-05 0.001445294 0.050910053 1.09E-11 0.1045 1 0.1045

1.5 0.245 4.81E-08 1.96163E-07 0.245 5.10E-10 0.1649 1 0.1649

2 23.95 1.03E-06 4.28392E-08 23.95 1.45E-04 0.1488 1 0.148800071

2.5 1.832 9.19E-02 0.050084444 1.834302074 3.31E-05 0.1733 1 0.173300003

3 1.22E-01 1.17E-05 9.57377E-05 0.122000001 4.57E-03 1.66E-01 0.99962 0.165663046

3.5 0.125 3.60E-11 2.8824E-10 0.125 5.40E-05 1.95E-01 1 0.194600007

Table 5 exponent p=4




5. CONCLUSIONS:

The natural and damping frequencies of the p-FGM plate made of pyroelectric-piezo

magnetic material have been determined. The two different types of boundary conditions are

considered namely simply supported boundary conditions and clamped boundary conditions.

As exponent approaches 1 the plate behaved as a pyroelectric plate. The following assertions

are obtained from the tabulated results

i. As the wavenumber is very close to zero the damping frequencies are increasing.

ii. The increase of damping frequencies results in a decrease of natural frequency by the

amount (1- 2ζ ). ( −ζ damping ratio)

iii. In case of simply supported boundary conditions as the exponent increase results an

increase in both natural and damping frequencies corresponds to both the mode of

vibrations.

iv. In case of clamped boundary conditions, the symmetric mode of vibration yields the

assertions that the wave numbers increase there is a fluctuation in the damping

frequencies and in antisymmetric mode the maximum real part of the eigenvalues

corresponds to the critical damping and the maximum imaginary part of the

eigenvalues corresponds to the under damping situation for the given dynamical

system. By considering different power laws for the gradation function, the same

mathematical model would be applied to obtain the resonant frequencies of the given

dynamical system.

REFERENCES:

[1] J.R.Cho and J.Tingley orden (2000), Functionally graded material a parametric study on

thermal stress characteristics using the crank-nicolson-galerkin scheme, Computer

methods in applied mechanical and engineering, 188, 17-38.

[2] Elia Efraim (2011), Accurate formula for determination of natural frequencies of FGM

plates basig on frequencies of isotrpic plates”, Engineering procedia, 10,242-247.

[3] S.Karthikeyan and Rama S (2013).Wave Propagation in Magneto Pyroelectric Plates,

Mathematical Sciences International Research Journal, 2(2), 600 - 605.

[4] S.Karthikeyan and Rama S (2016). Free Vibration Analysis of Orthotropic Rectangular

Plate Made of Pyroelectric Material, International Journal of Pure and Applied

Mathematical sciences, 9(2), 273-282.

[5] D.Y.Liu, C.Y.Wangand W.Q.Chen (2010), “ Free vibration of FGM plates with in-plane

material inhomogeneity”, Journal of composite structures, 92(5),1047-1051.

[6] G.R.Liux and Han K.Y.Lam (2001), Material characterization of FGM plate using elastic

waves on an inverse procedure, Journal of composite structures, 35(11), 954-971.

[7] T.Nakamura, T.Wang and S.Sampath (2000), Determination of properties of graded

materials by inverse analysis and instrumented indentation, Acta materialia, 48, 4293-

4306.

[8] I.Ramu and S.C.Mohanty (2014), Modal analysis of FGM plates using finite element

method, Procedia materials science, 6,460-67.

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[14] D. Rajesh, V. Balaji, A. Devaraj and D. Yogaraj, An investigation on Effects of Fatigue

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Journal of Mechanical Engineering and Technology 8(7), 2017, pp. 85–91

APPENDIX A

The Stiffness matrices are

[ ] [ ] [ ][ ]dVBcBK U

V

T

UUU ∫=

[ ] [ ] [ ] [ ] [ ]dVBNLLV

U

TTT

UU ∫== λθ θθθ 0

[ ] [ ] [ ][ ]dVBBKV

T

φφφφ ξ∫=

[ ] [ ] [ ] [ ] [ ]dVBpNLLV

TTT

∫== φθφθθφ θ0

[ ] [ ] [ ][ ]dVBkBKV

T

θθθθ ∫=

[ ] [ ] [ ][ ]dVBBKV

T

ψψψψ µ∫=

[ ] [ ] [ ]dVNNM U

V

T

UUU ∫= ρ

[ ] [ ] [ ] [ ]dVBdBKV

TT

UU φφ ∫=

[ ] [ ] [ ]dVNNMV

T

θθθθ ζθ ∫= 0

[ ] [ ] [ ] [ ]dVBqBKV

TT

UU ψψ ∫=

[ ] [ ] [ ] [ ]dVBdBKV

TT

ψφφψ ∫=

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ][ ] [ ] [ ] [ ]

−−

−

−

==

ψψθψφψψ

θθ

φψφθφφφ

ψθφ

KLKK

K

KLKK

KLKK

K

TTT

U

T

U

UUUUU

I 000 [ ] [ ] [ ] [ ]

−==

0000

0000

0000

θψθθφθθ LMLLK TT

U

II

[ ]

==

0000

0000

0000

000UU

III

M

K

[ ] [ ] [ ] [ ] [ ]dVBNLLV

U

TTT

∫== µθ θψθθψ 0

modal analysis of p-fgm rectangular...

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