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IJSRD - International Journal for Scientific Research & Development| Vol. 3, Issue 03, 2015 | ISSN (online): 2321-0613
All rights reserved by www.ijsrd.com 797
Vibration Analysis of Laminated Composite Plate using Finite Element
Analysis
Subrat Sarkar1 Mr. Ankur Malviya
2
1M. Tech Student
2Assistant Professor
1,2Department of Mechanical Engineering
1,2CCET, Bhilai
Abstract— This work is concerned with vibration (Modal)
analysis of a laminated composite plate structure with
different Angle orientation of ply. The finite Element
method has been used computational by means of ANSYS
14.5. a main reason for adopting ANSYS 14.5 is that there is
no such analytical model has been develop for Laminate
composite plate structure in presence of singularities i.e.
Cutout. Moreover parametric analysis has also been carried
out for detailed convergence study plate vibration in
presence of cutout. Following analysis have been carried out
for different boundary condition i.e. CFFF and SSSS to
allocate optimum location of cutout for optimum natural
frequency in order to make laminate composite structure
statically and dynamically balanced The obtained FEM
results are validated with experimental and analytical result
of available literature and it result shows good agreement.
Key words: Ansys, Laminate, SSSS and CFFF.
I. INTRODUCTION
Today, composite laminates have many applications as
advanced engineering materials, primarily as components in
power plants, aircrafts, ships, civil engineering structures,
cars, robots, rail vehicles, sports equipment, prosthetic
devices, etc. The major attribute of composite material is
capability of the controllability of fiber alignment. By
arranging layers and fiber direction, laminated material with
required stiffness and strength properties to specific design
conditions, can possibly be achieved.
Vibration damping is the ability of a material to
dissipate energy during vibration. Damping of composites is
higher than that of conventional metals such as steel and
aluminum. Damping of composites depends on fiber volume
fraction, orientation, constituent properties, and stacking
sequence. Damping in composites is measured by
calculating the ratio of energy dissipated to the energy
stored. The primary reasons for using composites are that
they improve the torsional rigidity of the racquet and reduce
risk of elbow injury due to vibration damping which results
in significant advantages such as higher specific stiffness
and specific strength, better resistance to impact and fatigue
damages and to corrosion, as well as greater autonomy for
multifunctional design whereby the design serves many
purposes at once. These advantages account for the
extensive use of composites in the, aerospace, marine and
consumer goods areas. In modern industry, net and near-net
production schemes are becoming more essential in
increasing effort is made to avoid waste. This has made
nondestructive testing methods to gain more importance.
Vibration test methods are among the simplest and most
indicative available.
II. LITERATURE REVIEW
Kenneth et. al 1985 Present a finite element formulation for
examining the large amplitude, steady-state, forced vibration
response of arbitrarily laminated composite rectangular thin
plates. [1] To analysis nonlinear forced vibration laminated
composite rectangular plate, the nonlinear stiffness and
harmonic force matrices has been developed and parametric
variation in form of laminate angles and number of plies, in-
plane boundary conditions has also been presented. The FE
results are compared with available approximate continuum
solutions.
A.Mukherjee and M.Mukhopadhayay 1988
introduced an isoparametric quadratic stiffened plate
bending element and the effect of several parameters of the
stiffener- eccentricity, shape, torsional stiffness etc.-on the
natural frequencies of the stiffened plates are studies.[2] The
studies shows that the natural frequencies agree very well up
to the first 15 modes.
Kehdir and L.Librescu 1988 applied the higher
order plate theory developed in the first paper of them and
theory and making use of powerful technique based on the
state space concept, the state of stress and displacement of
cross-ply laminates was analyzed in an unitary manner. [3]
A variety of boundary conditions is considered for the free
vibration and buckling problem of such rectangular
laminated plates were studied.
Mottram demonstrates that high-order modelling
predicts more precisely the effect of shear deformation than
the FEM. [4] A specific high-order laminated plate with
central patch load has been developed and found that by
providing S is greater than 35, then the maximum
contribution to deflections and stresses from shear
deformation for multi-layered plates is 10% in comparison
with CPT
Zamanov 1999 calculate natural frequency of thin
rectangular plate by applying periodically bent structures in
a thin rectangular plate made of a composite material. The
effect bending parameters are analyzed by using variation
principle by FEM.[6]
Nayak and Bandyopadhay 2002 presented a finite
element analysis for free vibration behavior of doubly
curved stiffened shallow shell. [8] The shell type are
examined are the elliptic and hyperbolic paraboloids, the
hyper and the conoidal shell. The accuracy of the
formulation was established by comparing FEA results of
specific problem.
Rastgaar et. al 2006 exploits order shear
deformation theory of plates (TSDT) for evaluating natural
frequencies of square laminated composite plates for
different supports at edges. [10]A novel set of linear
equations of motion for square multi-layered composite
Vibration Analysis of Laminated Composite Plate using Finite Element Analysis
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plates has been derived. The FEM result of natural
frequency for different combination of layers and supports
are postulated and the compared result shows good
agreement with equivalent single layer theories.
Ferreira et. al 2007 develop an innovative
numerical scheme, collocation by radial basis functions and
pseudo-spectral methods are combined to produce highly
accurate results for symmetric composite plates [11].
Yongshenga and Shuangshuang 2007 numerically
investigate the Large Amplitude Flexural Vibration of Shape
Memory Alloy Fibers embed in composite plate. The
obtained result shows the significance of temperature on
forced reaction behavior during phase alteration from
Martensite to Austenite.[12]
Ngo-cong et. al 2011present a new radial basis
function (RBF) collocation technique for analyzing free
vibration analysis of laminated composite plates. [13] First
order shear deformation theory has been used. A Cartesian
grid is employed for discretizing rectangular or non-
rectangular plates. Results obtained are compared with the
exact solutions and numerical results by other techniques in
the literature to investigate the performance of the proposed
method.
Zeki et. al 2013 experimentally and numerically
examine the effect of interfacial root crack on the lateral
buckling and free vibration responses of a sandwich
composite beam[14]. Lateral buckling loads and natural
frequencies in a thin sandwich composite cantilever beam
with root crack are resolved.
Süleyman et. al 2014 investigates the nonlinear
dynamic response of a hybrid laminated composite plate
composed of composite material such as basalt,
Kevlar/epoxy and Eglass/ epoxy under the blast load with
damping effects. [15] von Kármán type of geometric
nonlinearities are considered and parameters such as
damping ratios, aspect ratios and different peak pressure
values are analyzed
Sarmila Sahoo [17] 2014 applied finite element
method to analyze free vibration problems of laminated
composite stiffened shallow spherical shell panels with
cutouts employing the eight-noded curved quadratic
isoparametric element for shell with a three noded beam
element for stiffener formulation.
Xin and Hu 2015 applied new develop hybrid
approach for analyzing free vibration of simply supported
and multilayered magneto-electro-elastic plates.[18] The
Approach include state space approach (SSA) and the
discrete singular convolution (DSC) algorithm which is
based on three-dimensional elasticity theory.
III. MATHEMATICAL MODEL
A mathematically exact stress analysis of a thin laminate
plate—subjected to loads acting normal to its surface—
requires solution of the differential equations of three-
dimensional elasticity [1.1.1].Let considered an rectangular
plate with side a in X direction and b in Y direction and
transverse deflections w(x, y) for which the governing
differential equation is of fourth order, requiring only two
boundary conditions to be satisfied at each edge.
Fig. 1: Shows Laterally Loaded Rectangular Laminate Plate
Fig. 2: Shows External and Internal Forces on the Element
of the Middle Surface
Vibration Analysis of Laminated Composite Plate using Finite Element Analysis
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A. Plate Equation in Cartesian coordinate system:
Let us express, for instance, that the sum of the moments
of all forces around the Y axis is zero. The governing
differential equation for thin plates, we employ, for
pedagogical reasons, a similar method as used in elementary
beam theory.
Equilibrium of Plate Element:
0xM , 0yM ’ 0zp (1)
Figure 3 Shows Stresses on a lam
x x yx yx x x(m ) dy – m dy (m dy) dx – m dx – (q dx) dy q dy 02 2
yxx xmm q dx dx
dxx x x
(2)
After simplification, we neglect the term containing
since it is a small quantity of higher-order. Thus Eq. (2)
becomes ½(∂qx/∂x)(dx)2dy since it is a small quantity of
higher-order. Thus Eq. (2) becomes.
xdxdy dydx – q dxdy 0yxx
mm
x y
(3)
and, after division by dx dy, we obtain
yxxx
mmq
x y
(4)
In a similar manner the sum of the moments around
the X axis gives
xyxy
mmq
y x
(5)
The summation of all forces in the Z direction
yields the third equilibrium equation:
zdxdy dxdy p dxdy 0,
yxqq
x y
(6)
Which, after division by dx dy, becomes
yxz
qqp
x y
(7)
Substituting Eqs. (4) and (5) into (7)and observing that mxy
= myx , we obtain
2 22
2 22 ( , ).
xy yx
z
m mmp x y
x yx y
(8)
The bending and twisting moments in Eq. (8)
depend on the strains, and the strains are functions of the
displacement components (u, v, w).
Relation between stress, strain and displacement
x x yE (9)
y y xE (10)
Substituting (10) into (9), we obtain. 21
x x y
E
(11)
And similar manner 21
y y x
E
(12)
The torsional moments mxy and myx produce in-plane shear
stresses τxy and τyx (Fig.1), which are again related to the
shear strain γ by the pertinent Hookean relationship, giving
2(1 )xy xy xy yx
EG
(13)
Fig. 4: Shows Stresses on a Plate Element
Fig. 5: Shows Section Before and After Deflection The curvature changes of the deflected middle surface are
defined by 2
2,x
w
x
2
2y
w
y
and 2w
x y
(14)
Internal Forces Expressed in Terms of w ( /2)
( /2)
h
x x
h
m zdz
,
( /2)
( /2)
h
y y
h
m zdz
(15)
( /2)
( /2)
h
xy xy
h
m zdz
,
( /2)
( /2)
h
yx yx
h
m zdz
(16)
3 2 2
2 2 212(1 )x
Eh w wm
x y
,
3 2 2
2 2 212(1 )y
Eh w wm
y x
(17) 3
212(1 )
EhD
(18) Represent the bending of the plate. Similarly the
expression of twisting moment in terms of the lateral
deflections is obtained: ( /2) ( /2) 2
2
( /2) ( /2)
2
h h
xy yx
h h
wm m zdz G z dz
x y
2
(1 ) (1 ) .w
D Dx y
(19)
Thus the governing differential equation of the
plate subjected to distributed lateral loads obtained by
substitution equations: (19),(18)and(17) into eqs.(8).
Vibration Analysis of Laminated Composite Plate using Finite Element Analysis
(IJSRD/Vol. 3/Issue 03/2015/195)
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4 4 4
4 2 2 4
( , )2
w w w Pz x y
x x y y D
(20)
Material Properties Ref. [10,13]
Orthotropic (Composite) Value
Density, ρ 1600 kg/m3
Young's Modulus
Ex 280GPa
Ey 7GPa
Ez 7GPa
Poisson’s Ratio, νx=νy=νz 0.25
Shear Stress, Gx 4.2GPa
Shear Stress, Gy 3.5GPa
Shear Stress, Gz 4.2GPa
Table 1: Material Properties
IV. RESULT AND DISCUSSION
The governing equations of the problem were solved,
numerically, using a Element method, and finite element
Analysis (FEA) used in order to calculate the Vibration
characteristics of a Composite Laminated plate. As a result
of a grid independence study, a grid size of 100x100 was
found to model accurately the Natural Frequency of a
Composite Laminated plate described in the corresponding
results.
The accuracy of the computer model was verified
by comparing results from the present study with those
obtained by Rastgaar [10], Ngo-Cong [13] Experimental,
Analytical and FEA results.
Non dimensional Natural frequencies of a SSSS square
laminated composite cross-ply (0/90/90/0) with different
E1/E2 ratios and a/h = 5
E1
/E
2
TSDT
Ref.
[10]
HSDT
Ref.
[3,10]
FSDT
Ref.
[3,10]
CLPT
Ref.
[3,13]
Present
FEA
(ANSYS)
10 8.2741 8.294 8.2982 10.65 8.2512
20 9.5312 9.5439 9.5671 13.948 9.3309
30 10.265
1 10.284 10.326 16.605 10.2591
40 10.791
2 10.794 10.854 18.891 10.5619
Table 2: Validation of Non Dimensional Natural Frequency
with Varying Elastic Ratio
Natural Frequency of laminated plate in CFFF a=b= 0.2032
m, t=0.0013716 m
Mo
de
Mukherjee
Ref. [2]
Nayak
Ref.[8]
Sahoo
Ref. [17]
Present
FEA
(ANSYS)
1 711.8 725.2 733 721.2
2 768.2 745.3 748.25 746.7
3 1016.5 987.6 991.23 988.1
Table 3: Validation Natural Frequency of Laminated Plate
in CFFF
Fig. 6: Validation of Non Dimensional Natural Frequency
with Varying Elastic Ratio in SSSS
Fig. 7: Validation Natural Frequency of Laminated Plate in
CFFF
In figure 6-7 shows the validation of FEM result
obtained from the ANSYS tool. It has been seen that the
obtained result of Laminated composite plate with diffent
boundary condition shows good agreement with the
analalytical, Experimental and FEM of avalilable literature.
The small variation is results is due to variation in grid
sizing, operating condition, material properites, etc. but the
obtained result shows the same trend so that the results are
suitably verified.
Figure 8 shows the mode shapes of Laminated
composite plate for the two boundary condition i.e. Simply
supported (SSSS) and Clamped (CFFF). The pattern of
mode shapes are also verifed with the available literature i.e.
D. Ngo-Cong [13]. Since other researchers perform
analytical investigation and plot there mode shapes by
means of MATLAB and the pattern of their mode shapes are
still been same in comparison with present FEM results i.e.
D. Ngo-Cong [13], Ferreira [11].
Vibration Analysis of Laminated Composite Plate using Finite Element Analysis
(IJSRD/Vol. 3/Issue 03/2015/195)
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Fig. 8: Mode Shapes in SSSS and CFFF Boundary
Condition
Fig. 9: Variation of Natural Frequency of Laminated Plate
with Different Ply Orientation in SSSS
Figure 9 illustrates the Variation of Natural
Frequency of Laminated Plate with Different Ply orientation
in SSSS. It has observed that on varying ply orientation the
natural frequency significantly effects. For simply supported
condition the optimal natural frequency has been seen at
0/45/45/0 orientation. From Laminate design point of
minimum natural frequency is selected and it can also be
conclude that for simply supported boundary condition
0/45/45/0 orientation is recommended due to have better
interlock capability.
Vibration Analysis of Laminated Composite Plate using Finite Element Analysis
(IJSRD/Vol. 3/Issue 03/2015/195)
All rights reserved by www.ijsrd.com 802
Fig. 10: Variation of Natural Frequency of Laminated Plate
with Different Ply Orientation in CFFF
Figure 10 illustrates the Variation of Natural
Frequency of Laminated Plate with Different Ply orientation
in CFFF. It has observed that on varying ply orientation the
natural frequency extensively effected but in low range of
frequency as compared to SSSS. For Clamped condition the
optimal natural frequency has been seen at 0/30/60/0
orientation. From Laminate design point of minimum
natural frequency is selected and it can also be conclude that
for Clamped boundary condition 0/30/60/0 orientation is
recommended.
Fig. 11: Variation of Natural Frequency of Laminated Plate
with Cutout and Different Ply Orientation in SSSS
Figure 11 shows the Variation of Natural
Frequency of Laminated Plate with cutout and Different Ply
orientation in SSSS. It has observed that on varying ply
orientation the natural frequency significantly affects. For
simply supported condition the optimal natural frequency
has been seen at 0/45/45/0 orientation in presence of cutout.
From Laminate design point of minimum natural frequency
is selected and it can also be conclude that for simply
supported boundary condition 0/45/45/0 orientation is
recommended due to have better interlock capability.
Fig. 12: Variation of Natural Frequency of Laminated Plate
with Cutout and Different Ply Orientation in CFFF
Figure 12 shows the Variation of Natural
Frequency of Laminated Plate with cutout and Different Ply
orientation in CFFF. It has observed that on varying ply
orientation the natural frequency extensively effected but in
low range of frequency as compared to SSSS. For Clamped
condition the optimal natural frequency has been seen at
0/30/60/0 orientation with cutout. From Laminate design
point of minimum natural frequency is selected and it can
also be conclude that for Clamped boundary condition
0/30/60/0 orientation is recommended with presence of
cutout.
Fig. 13: Variation of Natural Frequency of Laminated Plate
with Varying Cutout Diameter in SSSS
Figure 13 shows the Variation of Natural
Frequency of Laminated Plate with varying cutout Diameter
in SSSS. It has been seen that on increasing cutout diameter
the natural frequency first decreases till d/a 0.2 and then
starts increasing drastically. This means that the laminated
plate become more stable as stiffness and mass changes.
The variation of trend is similar for higher mode.
Vibration Analysis of Laminated Composite Plate using Finite Element Analysis
(IJSRD/Vol. 3/Issue 03/2015/195)
All rights reserved by www.ijsrd.com 803
Fig. 14: Variation of Natural Frequency of Laminated Plate
with Varying Cutout Diameter in CFFF
Figure 14 shows the Variation of Natural
Frequency of Laminated Plate with varying cutout Diameter
in CFFF. It has been observed that on increasing cutout
diameter natural frequency gradual goes on decreases. This
is due to abrupt change in mass and stiffness of laminated
plate but the behavior varies according to subjected
boundary condition. At higher modes the trend is not
similar.
Fig. 15: Variation of Natural Frequency of Laminated Plate
with Same Area and Varying Cutout Location in SSSS and
CFFF
Figure 15 shows the Variation of Natural
Frequency of Laminated Plate with same area and varying
cutout Location in SSSS and CFFF. It has observed that on
change cutout location the remarkable change in natural
frequency has been seen for both shapes of cutout of having
same area i.e. circle and square. At centre of laminated
plate the optimal natural frequency has been seen in case of
simply supported boundary condition (SSSS) and for
clamped the optimal frequency is at left edge side of the
laminated plate.
V. CONCLUSIONS
[1] In the view of parametric Vibration analysis of
Laminated Composite plate structure following
conclusions have been drawn
[2] Change ply orientation the natural frequency
remarkably changes
[3] The presence of cutout affects the frequency
differently in different boundary condition
[4] In simply support (SSSS) boundary condition the
frequency is 6-7 times higher in comparison with
clamped (CFFF) boundary condition.
[5] On introducing cutout at centrally in simply
supported (SSSS) Laminated plate the natural
frequency decreases significantly. But in CFFF
boundary condition the decreases in frequency is
3% less in comparison with simply support (SSSS)
boundary condition.
[6] Increasing the thickness by adding a core in the
middle increases resistance.
[7] The natural frequency of circular cutout and square
cutout with same area are not same. The natural
frequency of circular is less in comparison with
Square one.
[8] The change in natural frequency is different for
isotropic material and orthotropic material. This is
due to variation is elastic properties of plate
structure.
[9] On changing the cutout location the natural
frequency significantly affect differently for
different angle oriented ply.
[10] For laminated composite plate the natural
frequency is minimum at centre of the plate in
simply supported condition. Similarly, in clamped
condition the frequency is least at the edge from the
clamped side.
[11] On reducing mass in laminated composite plate the
natural frequency significantly decreases in CFFF
and the trend is unlike in higher mode. Whereas in
SSSS the natural frequency first decreases upto 0.2
d/a and then starts increasing and the trend is
similar even at higher mode.
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