Geometrically Non-Linear Vibrations of Fully Clamped … · 2015-08-25 · Shear Deformation Theory...
Transcript of Geometrically Non-Linear Vibrations of Fully Clamped … · 2015-08-25 · Shear Deformation Theory...
Applied Mathematical Sciences, Vol. 9, 2015, no. 109, 5401 - 5415
HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.56427
Geometrically Non-Linear Vibrations of
Fully Clamped Asymmetrically
Laminated Composite Skew Plates
Moulay Abdelali Hanane
Université Mohammed V Agdal-
Ecole Mohammadia D’ingénieurs
Avenue Ibn Sina, B.P.765
Agdal, Rabat, Morocco
Benamar Rhali
Université Mohammed V Agdal-
Ecole Mohammadia D’ingénieurs
Avenue Ibn Sina, B.P.765
Agdal, Rabat, Morocco
Copyright © 2015 Moulay Abdelali Hanane and Benamar Rhali. This article is distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, a theoretical model based on Hamilton’s principle and spectral
analysis is used to obtain a solution for geometrically non-linear vibration of
symmetrically and asymmetrically cross ply laminated skew plates. The
Analysis was considered regarding the influence of different parameters such as
fiber orientation, ply properties, plate skew angle and aspect ratio. Comparison
between the numerical and analytical results and those available in previous
studies showed a good agreement. The associated non-linear bending stresses
were calculated and compared showing a higher increase with increasing the skew
angle for asymmetrically laminated composite skew plates.
Keywords: Non-linear vibrations, Composite laminated skew plate, asymmetric,
bending stress
5402 Moulay Abdelali Hanane and Benamar Rhali
1 Introduction
In fields like Aerospace, mechanical and civil engineering, thin laminated
composite skew plates are commonly used. Very often, such structures are
working in severe environmental conditions and subjected to high vibration
amplitudes. This induces non-linear effects in the lamina of the composite
material. Analytical methods are important for understanding the influence of
different parameters on the response of the structure, and complete and validate
the numerical methods. Numerous studies have been devoted to analytical and
numerical methods [1-2] for investigating plate large vibration amplitudes.
The lack of symmetry about the mid-plane of asymmetric laminated composite
plates leads to bending- stretching coupling inducing errors if linear theories are
applied [3]. Whitney and Leissa [4] considered the effect of this coupling in a
non-linear dynamic plate theory. The unbalanced residual stresses created after
curing an asymmetric plate transforms the original plane design of the plate [5].
However, use of asymmetric laminated plates may be interesting if the laminate is
subjected to a non-symmetrical thermal field. Asymmetric composite laminates
have been the subject of different technique and design approaches. Few years
ago, asymmetric laminates have been studied using the classical lamination theory
[6, 7]. Asymmetric composite beams subjected to large vibration amplitudes were
analyzed using the finite element method by Singh et al. [8]. They presented the
bending-extension coupling by two second order, ordinary, differential non-linear
equations. The non-linear free vibration response of simply supported angle-ply
plates has been studied by Bannet [9]. Chandra and Raju examined the large
vibration amplitudes of cross-ply and angle-ply plates using a two-term
perturbation solution for non-symmetric laminates [l0-12]. Ugo Icardi [13-14]
analysed a non-linear asymmetric cross-ply laminate using a higher order layer
theory and applying a third -order zig-zag approach. He concluded that the use of
the equilibrium equations of elasticity provides a good accuracy for the value of
the transverse shear stress. Hegaze [15] has been interested by the dynamic
analysis of unsymmetric stiffened and unstiffened composite laminated plates
with a higher-order finite element. Kharazi and Ovesy [16] used the first-order
Shear Deformation Theory (FSDT) and the Rayleigh–Ritz method to study the
large deflections of unsymmetric delaminated composite plates. Yazdani and
Ribeiro [17] presented a geometrically non-linear static analysis of unsymmetric
composite plates with curvilinear fibers using the p-version layer approach. They
studied the effect of unsymmetric stacking sequences and fiber orientations.
Dhurvey and Mittal [18] presented a review on various studies on composite
laminates with ply drop-off. Pirbodaghi, Fesanghary and Ahmadian [19] analyzed
the non-linear vibrations of laminated composite plates resting on non-linear
elastic foundations using the homotopy analysis method (HAM). This study
showed that only a first-order approximation of the HAM leads to highly accurate
solutions for this type of non-linear problems.
Geometrically non-linear vibrations 5403
The aim of this paper was the extension of the theoretical model developed by
(Benamar Bennouna and White 1993) [23] using a technique of homogenization
to analyze the geometrical non-linear free dynamic response of asymmetrically
laminated skew plates in order to investigate the effect of non-linearity on the
non-linear resonance frequencies and the non-linear fundamental mode shape at
large vibration amplitudes. The general formulation of the model for non-linear
vibration of laminated plates at large vibration amplitudes is presented. Fully
clamped boundaries have been considered here and periodic displacements were
assumed. The results obtained have been compared to the previously published
ones. The first non-linear mode shape and the associated bending stresses have
been investigated. The amplitude dependence of the non-linear resonance
frequency ratio nl/l on the vibration amplitude associated to the first non-linear
mode shape, for various values of the plate skew angle, the number of layers, the
aspect ratio and the fiber orientation are discussed.
The results showed a hardening type of nonlinearity with a higher increase in the
nonlinear frequency ratio with increasing the skew angle and the amplitude of
vibration for the case of symmetrically cross ply laminated composite skew plates,
compared to the asymmetric case. The bending stress also increases with
increasing the skew angle and reaches its maximum value for the skew angle =45°
in the case of an asymmetrically laminated skew plate. The maximum is obtained
near to the plate centre.
2 Geometrically Non-Linear Vibrations of Fully Clamped
Laminated Composite Skew Plates
2.1 Constitutive Equation of a Laminated Skew Plate at Large Deflections
Consider the skew plate with a skew angle shown in Fig.1. For the large
vibration amplitudes formulation developed here, it is assumed that the material of
the plate is elastic, isotropic and homogeneous. The thickness of the plate is
considered to be sufficiently small so as to avoid the effects of shear deformation.
The skew plate has the following characteristics: a, b, S: length, width and area
of the plate; x-y: plate co-ordinates in the length and the width directions; -η, H:
Skew plate co-ordinates and plate thickness; E, : Young’s modulus and
Poisson’s ratio; D, : plate bending stiffness and mass per unit volume.
5404 Moulay Abdelali Hanane and Benamar Rhali
Fig.1 Composite Skew plate in x-y and -η co-ordinate system
For the classical plate laminated theory, the strain-displacement relationship for
large deflections are given by:
{𝜀} = {𝜀0} + 𝑧 {𝑘} + {0} (1)
in which {𝜀0}, {𝑘} and {0} are given by:
{ε0} =
[
∂U
∂x∂V
∂y
∂U
∂y+
∂V
∂x]
; {k} = [
kx
ky
kxy
] =
[
−∂2W
∂x2
−∂2W
∂Y2
−2 ∂2W
∂xy ]
; {0} = [
x0
y0
xy0
] =
[ 1
2(∂W
∂x)2
1
2(∂W
∂y)2
∂W
∂x ∂W
∂y ]
(2)
U, V and W are displacements of the plate mid-plane, in the x, y and z directions
respectively. For the laminated plate having n layers, the stress in the Kth layer can
be expressed in terms of the laminated middle surface strains and curvatures as:
{σk} = [Q̅]k{ε} (3)
in which {𝜎}𝑘𝑇 = [𝜎𝑥𝜎𝑦𝜎𝑥𝑦 ] and terms of the matrix [�̅�] can be obtained by the
relationships given in reference (Timoshenko 1975) [20]. The in-plane forces and
bending moments in a plate are given by:
Geometrically non-linear vibrations 5405
[NM
] = [ A B B D
] [{ε0} + {0}
{k}] (4)
A, B and D are the symmetric matrices given by the following Equation 5. [𝐵] = 0
for symmetrically laminated plates (Mallick 1993) [21].
(Aij, Bij, Dij) = ∫ Qij(k)(1, z, z2)
H2⁄
−H2⁄
dz (5)
Here the 𝑄𝑖𝑗(𝑘)
are the reduced stiffness coefficients of the kth layer in the plate
co-ordinates. The transverse displacement function W may be written as in
reference (Ribeiro and Petyt 1999) [22] in the form of a double series:
W = {Ak}T{W} sinkωt (6)
Where {Ak}T = {a1
k, a2k, … , an
k} is the matrix of coefficients corresponding to the
kth harmonic, {W}T = {w1, w2, … , wn} is the basic spatial functions matrix, k is
the number of harmonics taken in to account, and the usual summation convention
on the repeated index k is used. As in reference (Benamar, Bennouna and White
1993) [23], only the term corresponding to k=1 has been taken into account, which
has led to the displacement function series reduced, to only one harmonic: i.e.
W = ai wi(x, y)sinωt (7)
Here the usual summation convention for the repeated indexes i is used. i is
summed over the range 1 to n, with n being the number of basic functions
considered. The expression for the bending strain energy Vb, axial strain energy Va
and kinetic energy T are given in reference (Harras 2001) [24] in the rectangular
co-ordinate (x,y). The skew co-ordinates are related to the rectangular co-ordinate
(,) by: =x-y tan ; =y/cos. So, the strain energy due to bending Vb, axial
strain energy Va and kinetic energy T are given in the -η co-ordinate system. In
the above expressions, the assumption of neglecting the in-plane displacements U
and V in the energy expressions has been made as for the fully clamped rectangular
isotropic plates analysis considered in reference (Benamar, Bennouna and White
1993) [23]. Discretization of the strain and kinetic energy expressions can be
carried out leading to:
Vb =1
2sin2(ωt)aiajkij ; Va =
1
2sin4(ωt)aiajakalbijkl ; T =
1
2ω2cos2(ωt)aiajmij (8)
in which mij, kij and bijkl are the mass tensor, the rigidity tensor and the geometrical
non-linearity tensor respectively. Non-dimensional formulation of the non-linear
vibration problem has been carried out as follows.
5406 Moulay Abdelali Hanane and Benamar Rhali
𝐰𝐢(, 𝛈) = 𝐇𝐰𝐢∗ (
𝐚,𝛈
𝐛) = 𝐇𝐰𝐢
∗(∗, 𝛈∗) (9)
where ∗and 𝛈∗are non-dimensional co-ordinates ∗ =
𝐚 𝐚𝐧𝐝 𝛈∗ =
𝛈
𝐛 one then
obtains:
kij =aH5E
b3 kij∗ ; bijkl =
aH5E
b3 bijkl∗ ; mij = ρH3abmij
∗ (10)
where the non-dimensional tensors m*ij, k*ij and b*ijkl are given in terms of
integrals of the non-dimensional basic function wi*, non-dimensional extensional
and bending stiffness coefficient A*ij and D*
ij , skew angle and aspect ratio α.
Upon neglecting energy dissipation, the equation of motion derived from
Hamilton’s principle is:
δ∫ (V − T)2π
0= 0 (11)
where V= Vb +Vm+Vc. The expression for the bending strain energy Vb, the
membrane strain energy Vm, the coupling strain energy Vc and the kinetic energy T.
The skew co-ordinates are related to the rectangular co-ordinate (,) by: =x-y
tan ; =y/cos. So, the strain energy due to bending Vb, the membrane strain
energy Vm, the coupling strain energy Vc and the kinetic energy T are given below.
In order to simplify the expressions and reduce the problem to that of a
homogeneous skew plate, the change of variable z1=z-c is made, with c =Bij
Aij.
Consequently, the coupling term Vc vanishes and the bending strain and the
membrane strain energy expressions are written in the -η co-ordinate system.
Insertion of Equations 8 into Equation 11, and derivation with respect to the
unknown constants ai, leads to the following set of non-linear algebraic equations:
𝟐𝐚𝐢𝐤𝐢𝐫∗ + 𝟑𝐚𝐢𝐚𝐣𝐚𝐤𝐛𝐢𝐣𝐤𝐫
∗ − 𝟐𝛚∗𝐚𝐢𝐦𝐢𝐫∗ = 𝟎, r=1…n. (12)
These have to be solved numerically. To complete the formulation, the procedure
developed in (Harras 2001) [24] is adopted to obtain the first non-linear mode. As
no dissipation is considered here, a supplementary equation can be obtained by
applying the principle of conservation of energy, which leads to the equation:
𝛚∗𝟐 =𝐚𝐢𝐚𝐣𝐤𝐢𝐣
∗ +(𝟑/𝟐)𝐚𝐢𝐚𝐣𝐚𝐤𝐚𝐥𝐛𝐢𝐣𝐤𝐥∗
𝐚𝐢𝐚𝐣𝐦𝐢𝐣∗ (13)
This expression for ω*2 is substituted into Equation 12 to obtain a system of n
non-linear algebraic equations leading to the contribution coefficients ai, i=1 to n. ω
and ω* are the non-linear frequency and non-dimensional non-linear frequency
parameters related by:
Geometrically non-linear vibrations 5407
ω2 =D
b4cos4θω∗2
(14)
To obtain the first non-linear mode shape of the skew plate considered, the
contribution of the first basic function is first fixed and the other basic functions
contributions are calculated via the numerical solutions of the remaining (n-1)
non-linear algebraic equations.
2.1. Bending stress analysis
Many analytical and numerical methods used to model the geometrically non-linear
vibration of plates and beams considered transverse displacement only [25]. It is
well known that the estimation of the total non-linear stress must include the
in-plane displacement U and V. But, the scope of the present work permitted only
consideration of bending stress, which allows only a qualitative understanding of
the non-linearity effects. The bending stress associated to the fundamental
non-linear mode shape was examined, which allows a quantitative understanding
of the non-linearity effects. The maximum bending strains b and b obtained for
z=H/2 are given by:
εb =H
2(∂2W
∂2). (15)
εb =H
2(∂2W
∂2
1
cosθ−
2tanθ
cosθ
∂2W
∂∂+
∂2W
∂2tan2θ). (16)
For z=H/2 which corresponds to a layer with =90°, one has Q16=Q26=0. For the
skew plate, the associated stresses b and b can be obtained by applying the
plane stress Hooke’s law as:
b =HQ11
2(∂2W
∂2) +
HQ12
2(∂2W
∂2
1
cosθ−
2tanθ
cosθ
∂2W
∂∂+
∂2W
∂2tan2θ). (17)
b =HQ12
2(∂2W
∂2) +
HQ22
2(
∂2W
∂2
1
cosθ−
2tanθ
cosθ
∂2W
∂∂+
∂2W
∂2tan2θ). (18)
In terms of the non-dimensional parameters defined in the previous section, the
non- dimensional stresses *b and *b are then given by:
∗b = (2 ∂2W∗
∂∗2 + (∂2W∗
∂∗2
1
cosθ−
2tanθ
cosθ
∂2W∗
∂∗ ∂∗ + 2 ∂2W∗
∂∗2 tan2θ)). (19)
∗b = (γ
∂2W∗
∂∗2
1
cosθ− γ
2tanθ
cosθ
∂2W∗
∂∗ ∂∗ + γ2 ∂2W∗
∂∗2 tan2θ + 2 ∂2W∗
∂∗2 ). (20)
where =b/a, =Q11/Q12 and =Q22/Q12.
5408 Moulay Abdelali Hanane and Benamar Rhali
The relationships between the dimensional and non-dimensional stresses are:
=Q12H2
2b2 ∗. (21)
3. Results and Discussion
The aim of this section is to apply the theoretical model presented above to
analyze the geometrical non-linear free dynamic response of skew fully clamped
symmetrically and asymmetrically laminated plates in order to investigate the
effect of non-linearity on the non-linear resonance frequencies and non-linear
fundamental mode shape at large vibration amplitudes. Convergence studies are
carried out, and the results are compared with those available from the literature
through a few examples of laminated composite skew thin clamped plates with
different fiber orientation and aspect ratio. The material properties, used in the
present analysis are: (1) Plate 1, 3 and 6-layers symmetrically cross-ply
APC2/AS4 Peek/carbon laminate square plate [0°/90°/0°] and [90°/0°/90°]sym ET
= 131 GPa; EL = 5 GPa; GLT=2.855 Gpa LT = 0.25. (2) Plate 2, square
antisymmetrically cross-ply [0°/90°] fiberglass reinforced plastic clamped plate;
EL=206.84 Gpa; ET=5.171 Gpa; GLT=2.855 Gpa; LT = 0.25. (3) Plate 3, 4-layers
cross-ply asymmetrically laminated plate [0°/90°/0°/90°]; with the same
proprieties as plate 2. Plate 4, 4-layers cross-ply symmetrically laminated plate
[0°/90°/90°/0°] with the same proprieties as plate 2, where E, G and are Young’s
modulus, shear modulus and Poisson’s ratio. Subscripts L and T represent the
longitudinal and transverse directions respectively with respect to the fibers. All
the layers are of equal thickness. Calculation was made by using 18 functions
corresponding to three symmetric beam functions in the direction and three
symmetric beam functions in the η direction, and three anti-symmetric beam
functions in the direction and three anti-symmetric beam functions in the η
direction. Table 1 shows nonlinear to linear frequency ratios (nl/l) for square
symmetrically cross-ply APC2/AS4 laminated plate 1 obtained using only 18
well-chosen plate functions. It can be seen that there is a good convergence with
results presented in reference (Harras 2001) [24]. The variation of
non-dimensional nonlinear frequency ratio nl/l with respect to non-dimensional
maximum amplitude wmax/h is evaluated for antisymmetrically cross-ply [0°/90°]
fiberglass reinforced plastic clamped plate 2 for an aspect ratio =1 and is shown
in Tables 2, along with published results. It is observed that present results are in
close agreement with those of assumed direct numerical integration (five-point
Gauss rule) presented in reference [26]. Next, the nonlinear free vibration
behaviors of fully clamped cross-ply asymmetrically and symmetrically thin
square and skew plates 3 and 4 are examined in Table 3. It is observed that the
results obtained here closely match with increasing skew angle for the case of
asymmetrically cross ply laminated composite skew plates 3, however the
nonlinear frequency ratio increases with increasing skew angle and amplitude of
vibration for the case of symmetrically cross ply laminated composite skew plates
Geometrically non-linear vibrations 5409
4. It is indicating hardening type of nonlinear behavior. Furthermore, for the
chosen amplitude, it can be noted that, with the increase in skew angle, the degree
of nonlinearity is high.
Table 1 Nonlinear to linear frequency ratios (nl/l) for square (a/b=1) symmetrically
cross-ply APC2/AS4 laminated plate 1.
W/h [0°/90°/0°] [90°/0°/90°/90°/0°/90°]
nll
Present
results
Ref
[24]
Error
%
Present
results
Ref
[24]
Error
%
0,2 1,0074 - - 1,008 1,0074 0,06
0,4 1,0288 - - 1,0315 1,0295 0,19
0,5 1,0464 1,0455 0,09 1,0507 - -
0,6 1,0665 - - 1,0684 1,0645 0,36
0,8 1,1125 - - 1,1163 1,1105 0,52
1 1,1693 1,1671 0,18 1,1751 1,1657 0,8
Table 2 Nonlinear to linear frequency ratios (nl/l) for square antisymmetrically
cross-ply [0°/90°] fiberglass reinforced plastic clamped plate (a/h=150).
W/h [0°/90°]
nll
Present results Ref [26] Relative error %
0,3 1,0507 1,048 0,3
0,6 1,1798 1,1827 0,2
0,9 1,3604 1,3762 1,2
1,2 1,5451 1,6069 4
1,5 1,8038 1,861 3,2
1,8 2,0104 2,1304 6
2,1 2,253 2,4099 7
5410 Moulay Abdelali Hanane and Benamar Rhali
Table 3 Nonlinear to linear frequency ratios (nl/l) for skew cross-ply asymmetrically
and symmetrically clamped plate.
W/h [0°/90°/0°/90°] [0°/90°/90°/0°]
nl/l
Skew angle ° Skew angle °
0° 30° 45° 0° 30° 45°
0,2 1,0086 1,0087 1,0087 1,0076 1,0097 1,0134
0,4 1,036 1,0354 1,0344 1,0309 1,039 1,0518
0,6 1,0797 1,0764 1,0736 1,0653 1,0818 1,1113
0,8 1,1393 1,1318 1,1247 1,1122 1,1393 1,187
1 1,2126 1,2053 1,1926 1,1716 1,211 1,2795
1,2 1,2756 1,2657 1,2589 1,2327 1,2838 1,3717
Figure 2 and Figure 3 shows respectively the non-linear stresses along the
section *=0.5 at W*max=2 for the rectangular case =0° and the skew cases
=30° and =45° for the asymmetrically and symmetrically composite skew
plates 3 and 4. It can be noticed that the stress was increasing with increasing
skew angle and had maximum for the skew angles =45° in the asymmetrically
laminated skew plate near to the centre of the plate.
Fig. 2 Comparison between the non-dimensional bending stress distribution associated
with the fundamental non-linear mode of a fully clamped asymmetrically composite plate
3 for =0(°), =30(°) and =45(°) with W* max =2 and =1 along the section *=0.5.
=30°
=0°
=45°
𝜎∗
∗
Geometrically non-linear vibrations 5411
Fig. 3 Comparison between the non-dimensional bending stress distribution associated
with the fundamental non-linear mode of a fully clamped symmetrically composite plate
4 for =0(°), =30(°) and =45(°) with W* max =2 and =1 along the section *=0.5.
Conclusion
The theoretical model established and applied to beams, plates and shells
(Benamar, Bennouna and White 1993) [23] has been successfully applied to
calculate the first non-linear mode shape of fully clamped skew symmetrically
and asymmetrically laminated plates. A technique of homogenization was used
for asymmetrically laminated plates for various values of the skew angle, the
number of layers, the plate thickness ratio and different fiber orientation. The
present formulation has been verified with the available results in the literature.
Limited numerical studies are conducted to examine the effect of the skew angle,
the fiber orientation, and the aspect ratio on the large-amplitude frequency of
composite skew plates. As has been shown in this work, the fundamental
non-linear mode shape of the fully clamped skew asymmetrically and
symmetrically laminated plate can be obtained with a sufficient accuracy by using
18 basic plate functions. This provides a more accurate description of the
non-linear mode shape, compared with previous results based on the use of nine
symmetric basic plate functions since it allows the non-symmetry induced by the
fiber orientation to be taken into account. The present study reveals a hardening type of nonlinearity for the asymmetrically and symmetrically laminated skew plate
=30°
=0°
=45°
𝜎∗
∗
5412 Moulay Abdelali Hanane and Benamar Rhali
and the nonlinear frequency ratio in general increases with the increase in the
skew angle. The asymmetrically laminated plate presented higher values of
bending stresses compared with the symmetrically laminated plate in the
direction.
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Received: June 24, 2015; Published: August 25, 2015