Computer Based Free Vibration Analysis of Isotropic Thin Rectangular Flat CCCC Plate

The International Journal Of Engineering And Science (IJES)
|| Volume || 5 || Issue || 7 || Pages || PP -38-41 || 2016 ||
ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805
www.theijes.com The IJES Page 38
Computer Based Free Vibration Analysis of Isotropic Thin
Rectangular Flat CCCC Plate
O. M.Ibearugbulem1
, E. I.Adah2
,D. O.Onwuka3
, C.P.Anyadiegwu4
.
1,2,3
Departmentof Civil Engineering, Federal University Of Technology, Owerri, Nigeria
4
Department of Civil Engineering, Federal Polytechnic Nekede, Owerri, Nigeria
--------------------------------------------------------ABSTRACT-----------------------------------------------------------
The paper is aim at the development of a computer program for free vibration analysis of an all-round clamped
(i.eCCCC) rectangular thin isotropic plate. Polynomial-based shape function for a CCCC plate, was used in
Ritz energy equation to formulate an equation in term of a non-dimensional parameter 'k' for calculating the
fundamental natural frequency of the plate. Computer program based on the formulated equation was develop
in Matlab language for calculating fundamental natural frequency of a CCCC rectangular thin isotropic plate
subjected to free vibration. To validate the values of the fundamental natural frequency obtained from the
program, comparison was made between the values obtained and those available in relevant existing
literatures. And the average percentage difference was 0.079% which is acceptable. Hence, it can be deduce
that the computer program can be adopted as a better and faster alternative to the former rigorous approaches
for obtaining the fundamental natural frequency of CCCC plate.
Keywords: Computer Program, Free Vibration, Fundamental Frequency, MATLAB, Thin Rectangular
Isotropic Plate.
-------------------------------------------------------------------------------------------------------------------------------------
Date of Submission: 17 May 2016 Date of Accepted: 05 July 2016
I. INTRODUCTION
What is important in the analysis of free vibration is the fundamental natural frequency which is the value of an
externally induced frequency ofa vibrating plate that causes it to resonate. Structural elements, such as plates
frequently undergo vibration which is usually undesirable. Several scholars have carried out free vibration
analysis of CCCC rectangular plates using classical and approximate methods based on Fourier or trigonometric
series. Among them areChakraverty (2009); Lee (2004); Misra(2012);Ventsel& Krauthammer (2001);
Werfalli&Karaid(2005); Sakata et al (1996).
Due to the rigorousness of the classical and approximate methods using fourier series, recent researchers such as
Ibearugbulem et al.(2013); Njoku et al.(2013); Ebirim et al.(2014); Ezeh et al.(2014);Ibearugbulem et al.
(2014), have gone ahead to use Taylor Maclaurin's series for the analysis of vibration ofrectangular plates.
Theapproach they adopted is less rigorous but still full of lengthy and time consuming computations.
None of the scholars have to the knowledge of the writers; develop computer program(s)based on this new
approach that uses polynomial series. Thus, thiswork concerned with designing a computer program for
analyzing CCCC thin rectangular isotropic flat platethat uses polynomial series.
II. THEORETICAL BACKGROUND
Consider a plate clamped all round (i.e CCCC) shown in Fig. 1. The total potential energy functional for free
vibration of rectangular isotropic plate as developedby Ibearugbulem(2013) using Ritz method, is given in (1) as
Figure 1: Schematic Representationof CCCC Rectangular Plate.
where a and b are plate dimensions, s is aspect ratio, while X and Y are axis

Computer Based Free Vibration Analysis Of Isotropic Thin Rectangular Flat CCCC Plate
Π =
D
2𝑏2 ∫∫ [
1
𝑃3 (w''R
)2
+
2
P
(w''RQ
)2
+ p(w''Q
)2
] ∂R∂Q-pρhω2
b2
∫∫ w2
∂R∂Q (1)
where p = a/b is the aspect ratio, D is flexural rigidity, w is plate deflection, R = X/a and Q =Y/b are non-
dimensional parameters, and w''R
=
∂2w
∂𝑅2 ; w''Q
=
∂2w
∂𝑄2 ; w''RQ
=
∂2w
∂R ∂Q
, ρ is the specific gravity of the plate
material use (steel), h is plate thickness and ω is the fundamental natural frequency. The double integrals are
within 0≤R≤1 and 0≤Q≤1
Assuming w = Ak, then total potential energy functional, Π, becomes
Π =
D𝐴2
2𝑏2 ∫∫ [
1
𝑃3 (k''R
)2
+
2
P
(k''RQ
)2
+ p(k''Q
)2
] ∂R∂Q-pρhω2
b2
A2
∫∫ k2
∂R∂Q (2)
where A is amplitude of deflection, and k is the R and Q components of the shape function.
Minimizing (2) and making ω2
the subject of the formula, gives (3)
ω2
=
∫ ∫ [
1
𝑃4 (k′′R )2+
2
𝑃2 (k′′RQ )2+ (k′′Q )2] ∂R ∂Q
∫ ∫ 𝑘2 ∂R ∂Q
*
D
ρh𝑏4 (3)
For aspect ratio s = b/a = 1/p, the (3) becomes (4)
ω2
=
∫ ∫ [ (k′′R )2+
2
𝑆2(k′′RQ )2+
1
𝑆4 (k′′R )2 ∂R ∂Q
∫ ∫ 𝑘2 ∂R ∂Q
*
D
ρh𝑎4 (4)
Let fs
2
=
∫ ∫ [ (k′′R )2+
2
𝑆2(k′′RQ )2+
1
𝑆4 (k′′R )2 ∂R ∂Q
∫ ∫ 𝑘2 ∂R ∂Q
(5)
Therefore, fs is given by (6)
fs= [
∫ ∫ [ (k′′R )2+
2
𝑆2(k′′RQ )2+
1
𝑆4 (k′′R )2 ∂R ∂Q
∫ ∫ 𝑘2 ∂R ∂Q
]1/2
(6)
From (4), the square of the fundamental natural frequency can be obtained as:
ω2
= fs
2
*
D
ρh𝑎4 (7)
Hence, ω =
ƒ 𝑠
𝑎2
D
ρh
(8)
where fs (or ƒ 𝑠
) is the coefficient of fundamental natural frequency.
But for CCCC plate, Adah(2016)gave k value asin (9)
k = (R2
-2R3
+R4
)(Q2
-2Q3
+Q4
) = UV (9)
(9) can be substituted into (7),and the resulting equation is used to develop a Matlabprogram for easy and quick
analysis.
III. DEVELOPMENT OF COMPUTER PROGRAM
To ease and quicken the analysis, a program was developed to aid free vibration analysis of CCCC plate using
Matlab. According to Chapra (2012), Matlab offers better advantage as a result of its ability to handle complex
mathematical manipulations and have a user friendly environment. Matlab codes were applied to the derived
expression systematically to develop the program for the analysis of free vibration of CCCC rectangular
isotropic plate. Fig. 2 shows the flow chart of the develop program.
Figure 2: Flow Chart

The program is presented in Appendix and the values of the coefficients of the fundamental natural frequency 'fs'
obtained from the program are presented on Table 1.
IV. RESULTS AND DISCUSSIONS
When the developed program was executed by aspect ratio s = b/a, the values of the coefficient of the
fundamental natural frequency, fs,given in Table 1 were obtained.
Table 1:Coefficients 'fs
'
of Fundamental Natural frequency
Aspect Ratio
s= b/a
CCCC
ω =
ƒ 𝑠
𝑎2
D
ρh
ƒS
ω =
ƒ1𝑠
𝑎2
D
ρh
ƒ1S =fs/π2
1.0 36.000 3.648
1.2 30.774 3.118
1.5 27.047 2.740
1.6 26.333 2.668
2.0 24.648 2.497
4.1 Comparison of Coefficients of Fundamental Natural Frequency, Ƒ.
The values of the fundamental natural frequency 'f' obtained from the executed program were validated by
carrying out a comparative analysis in Table 2, with those of Njoku et al.(2013) and Ibearugbulem et al.(2014).
A close look at Table 2, shows that the percentage differences are all less than 0.1% when compared with values
obtained by Njoku et al. (2013). The implication of this is that the values of,‘f’, obtained from this program, are
very close to those values obtained by Njoku et al. (2013). Also, comparing with Ibearugbulem et al.(2014), the
percentage differences ranges from the lowest value of 0.040% to the highest value of 0.110%, which are all
negligible. And so, the computer results closely agree with the results obtained by Ibearugbulem et al. (2014).
V. CONCLUSION
Since, the values of the results from the computer program, are very close, the computer program written in
MATLAB, can predict accurately fundamental natural frequency, ω. Thus, the conclusion is that, the use of the
develop computer program, is an easier and quicker approach for accurate analysis of free vibration of a thin
isotropic rectangular plate clamped all round.
Table 2:Comparison of computer values of coefficients of fundamental natural frequency 'fs'with those of Njoku
et al. (2013) and Ibearugbulem et al. (2014)
Aspect
Ratio
S = b/a
S
Computer
Values of
Coeff.
ƒ1
Njoku et al
(2013)
ƒ2
%difference
100(f1-f2)/f2
Computer
Values of
Coeff.
ƒ3
Ibearugbulem et
al (2014)
ƒ4
%difference
100(f3-f4)/f4
1.0 36.000 35.9709 0.081 3.648 3.644 0.110
1.2 30.774 30.7508 0.075 3.118 3.115 0.096
1.5 27.047 27.0305 0.061 2.740 2.738 0.073
1.6 26.333 26.3175 0.059 2.668 2.666 0.075
2.0 24.648 24.6377 0.041 2.497 2.496 0.040
Aver.
%diff.
0.063 0.079
REFERENCE
[1]. S. Chakraverty, Vibration of Plates, (Taylor & Francis Group,London, 2009).
[2]. S. J. Lee, Free vibration analysis of plates by using a four-node finite element formulation with assumed natural transverse shear
strain.Journal of sound and vibration, 278, 2004, 657-684.
[3]. R. K. Misra, Static and Dynamic analysis of rectangular isotropic plates using multiquadric radial basis function. International
journal of management, Tech. & Engineering, 2(8), 2012, 166-178.
[4]. E. Ventsel&T. Krauthamer, Thin Plates and Shells: Theory, Analysis and Applications, (New York, Marcel Dekker, 2001).
[5]. N. M. Werfalli&A. A. Karaid, Free vibration analysis of rectangular plates using Galerkin-based infinite element method,
International Journal of Mechanical Engineering, 2(2), 2005, 59-67.

[6]. T. Sakata, K. Takahashi, &R. B. Bhat, Natural frequencies of Orthotropic Rectangular plates obtained by iterative reduction of
partial differential equation. Journal of sound vibration, 189, 1996, 89-101.
[7]. O. M. Ibearugbulem, S. I. Ebirim&J. C. Ezeh,Vibration Analysis of CSSF and SSFC Panel. International Journal of Engineering
Research and Applications,3(6), Nov.-Dec. 2013, 703-707.
[8]. K. O. Njoku, O. M. Ibearugbulem, J. C. Ezeh, L. O. Ettu,&L. Anyaogu, Free Vibration Analysis of Thin Rectangular Isotropic
CCCC Plate using Taylor Series Formulated Shape Function in Galerkin's Method,Academic Research International Part-I:
Natural and Applied Science,4(4), July 2013, 126-132.
[9]. S. I. Ebirim, J. C. Ezeh, O. M. Ibearugbulem, Free Vibration analysis of Isotropic Rectangular Plate with one edge free of
support (CSCF and SCFC plate), International Journal of Engineering &Technology, 3(1), 2014, 30-36.
[10]. J. C. Ezeh, O. M. Ibearugbulem,&S. I. Ehirim,Vibration Analysis of Plate with one Free Edge Using Energy Method (CCCF
Plate), International Journal of Engineering and Technology, 4(1), January 2014, 85-92.
[11]. O. M. Ibearugbulem, J. C. Ezeh,&L. O. Ettu, Energy Methods in Theory of Rectangular Plates: Use of Polynomial Shape
Functions,(Liu House of Excellence Ventures, Owerri, 2014).
[12]. E. I. Adah, Development of Computer Programs for Analysis of Single Panel and Continuous Rectangular Plates, M. Eng.
Thesis, Federal University of Technology, Owerri, 2016.
[13]. S. C. Chapra, Applied Numerical Methods with Matlab for Engineers and Scientists,(3rd ed), (McGraw-Hill, New York,2012).
APPENDIX
clc
%PROGRAM FOR ANALYZING FUNDAMENTAL NATURAL FREQUENCY OF CCCC PLATE
v = input('Enter value of poission ratio v:');
a = input('Enter plate dimension along x-axis -length- a(m):');
b = input('Enter plate dimension along y-axis -width- b(m):');
h = input('Enter the thickness h(m):');
E = input('Enter the value of young modulus E:');
p = input('Enter the value of specific density p:');
echo on
s = b/a
echo off
%The flexural Rigidity of plate D is
D = E*h^3/(12*(1-v^2));
syms r q
U = r^2-2*r^3+r^4;
V = q^2-2*q^3+q^4;
diff(U,2);
(diff(U,2))^2;
y1 = int((diff(U,2))^2,r,0,1);
z1 = int(V^2,q,0,1);
Y1 = y1*z1;
diff(V,2);
(diff(V,2))^2;
y2 = int(U^2,r,0,1);
z2 = int((diff(V,2))^2,q,0,1);
Y2 = y2*z2;
diff(U,1);
diff(V,1);
y3 = int((diff(U,1))^2,r,0,1);
z3 = int((diff(V,1))^2,q,0,1);
Y3 = y3*z3;
y4 = int(U,r,0,1);
z4 = int(V,q,0,1);
Y4 = y4*z4;
Y6 = y2*z1;
%Fundamental Natural frequency
f = vpa(sqrt((Y1+(2*Y3/s^2)+(Y2/s^4))/Y6),7)
f1 = f/pi^2
F = vpa((f/a^2)*sqrt(D/p*h),7)

Computer Based Free Vibration Analysis of Isotropic Thin Rectangular Flat CCCC Plate

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Computer Based Free Vibration Analysis of Isotropic Thin Rectangular Flat CCCC Plate