VIBRATIONAL ANALYSIS OF SANDWICH BEAMS
VIBRATIONAL ANALYSIS OF SANDWICH BEAMS
A PROJECT REPORT SUBMITTING IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE IN
Bachelor of Technology
In
Mechanical Engineering
By
ALANKAR SHARAN
&
ATUL KUMAR
Under the Guidance of
Prof. R.K. BEHERA
Department of Mechanical Engineering
National Institute of Technology
Rourkela
2008
1
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA
CERTIFICATE
This is to certify that the thesis entitled, “VIBRATIONAL ANALYSIS OF SANDWICH
BEAMS” submitted by Sri Alankar Sharan and Sri. Atul Kumar in partial fulfillment of the requirements for the award of Bachelor of Technology Degree in Mechanical Engineering at the NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA (Deemed University) is an authentic work carried out by them under my supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been submitted to any other University / Institute for the award of any Degree or Diploma .
Date : 12.05.2007 Prof. R . K .Behera
Dept. of Mechanical Engineering
National Institute Of Technology
Rourkela – 769008 .
2
ACKNOWLEDGEMENT
We deem it a privilege to have been given such a project work as a part of our undergraduate course at this point of our student life. For this we are sincerely thankful to our
Institution, NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA AND OUR
MECHANICAL ENGINEERING DEPARTMENT.
We would like to sincerely thank our project guide Prof. R.K. Behera who has been a real source of inspiration, motivation and who at every point in the due course of our project guided us, helped us with his precious suggestions, and was ever willing to clear our doubts.
We would also like to sincerely thank our Head of the Department, Dr. B.K. Nanda
& Dr. K.P. Maity who with their valuable comments and suggestions during the vivavoice helped us immensely. We would like to thank them because they were the ones who constantly evaluated us, corrected us and had been the guiding light for us.
Alankar Sharan
Roll No. 10403028
Atul Kumar
Roll No. 10403030
B.Tech. 4 th
Yr.
Mechanical Engg.
NIT Rourkela
3
CONTENTS
Sl No
.
1.
2.
3.
4.
5.
6.
7.
8.
Topic Name
Abstract
Introduction
Higher Order Sandwich Panel Theory
Mathematical Assumptions
Validation
Numerical Study and discussion
Conclusion
References
Page No.
5
6  9
10  11
12  18
19 23
24 – 28
29
30
4
ABSTRACT
The project report deals first with the brief introduction on composites and their types and uses. Then it explains about the vibration of sandwich beams. The consistent higherorder dynamic formulation for foamtype (soft) core sandwich beams was extended to the case of composite sandwich plates. Eight dynamic governing equations and the corresponding boundary conditions were derived through the application of Hamilton’s principle. The extended formulation was applied to the free vibration analysis of softcore and honeycombcore sandwich plates with antisymmetric and symmetric layups.
The vibration results for the thin and thick composite sandwich plates obtained using the extended formulation were consistent with the predictions of the higher order mixed layer wise theory for laminated and sandwich plates. To simplify the formulation for the case of symmetric sandwich plates, the general dynamic formulation was decoupled into two systems of equations representing symmetric and antisymmetric vibrations. The numerical study demonstrates the importance of the present formulation for the prediction of higher mode vibration response of composite sandwich plates.
5
INTRODUCTION
1.1 History
Composite Materials, which made a new revolution in materials industry, are known to mankind since ancient time. Composite have been used by humans in past. The most primitive composite material that was made by man was a brick for building construction. It comprised straw and mud, wherein straw acted like fibers and mud as matrix. Mankind reinvented the benefits of composite materials in 1960s. Since then, Composite materials have created an impact on the development of the aerospace structures because of their superior fatigue characteristics, damage tolerance and strength to weight ratio compared to that of metals. The use of composite is not only restricted to Aerospace domain and now it is being used extensively in Automotive, Industrial applications and Civil structures domain.
Another important reason for the popularity of composite materials is due to the significant operating cost reduction due to its use. Considerable weight reduction is achieved using composite materials.
The composite plates are used in various applications, ranging from Aerospace to marine industry. As composites are having better properties than metals, they can be exposed severe loading conditions and environment.
1.2 What are composites?
In their broadest form, composites are a material which consists of two or more constituents. The constituents are combined in such a way that they keep their individual physical phases and they are neither are not soluble in each other nor to form a new chemical compound. One constituent is called reinforcing phase and the one in which the reinforcing phase is embedded is called matrix. Historical or natural examples of composites are abundant: brick made of clay reinforced with straw, mud wall with bamboo shoots, concrete, concrete reinforced with steel rebar, granite consisting of quartz, mica and feldspar, wood
(cellulose fibers in lignin matrix), etc.
6
1.3 What are hybrid composites?
They are composites hybridized with other materials (either other composites or unreinforced materials) or composites using multiple reinforcements and matrices. For example, carbon fibers and glass fibers are both included in the epoxy matrix; a compressed natural gas tank consists of aluminum liner and carbon fiber/epoxy composite overwrapping.
1.4 Advantages of the Composites
• High specific stiffness and high specific strength .weight reduction for the same strength applications.
• High corrosion resistance. Acid, alkali resistance of polymers, chemical and marine applications, infrastructure applications.
• High impact resistance .High internal damping of Kevla fiber/epoxy composites, ballistics protection.
• High wear resistance .Ceramic particle reinforced metal matrix composites, ceramic matrix composites.
• Resistance to high temperature and extreme mechanical, environmental conditions:
• Tailorable properties. Design both materials and structures.
Choose appropriate combination of reinforcements and matrices.
Choose optional fiber orientation and layup sequences.
1.5 Disadvantages of composite materials
• Higher cost
• Complexity in mechanical characterization and difficulty in analysis
• Weak in transverse direction and low toughness
• Difficulty in attaching (Joining)
• Environmental degradation (Polymer matrix absorb moisture)
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1.6 Classification of the composite
1. According to the types of fiber reinforcement
a. Fiberreinforcement composite (Fibrous composite)
b. Particlereinforcement composite (Particulate composite)
2. According to the matrix material a. Metal matrix composite (MMC) b. Polymer matrix composite (PMC)
3. According to the particulate orientation a. Random orientation composite b. Preferred orientation composite
4. According to the number of fiber constituent a. Simple composite b. Hybrid composite
5. According to the fiber type a. Continuous fiber composite b. Discontinuous fiber composite
6. According to the direction of the fiber a. Unidirectional reinforcement composite b. Bidirectional reinforcement composite
1.7 Application of the Composite materials
1. Aerospace industries
2. Civil engineering
3. Space industries
4. Marine applications
5. Defense industries
6. Automobile industries
7. Electrical applications
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1.8 Terminology used in composite materials
1. Isotropic materialsProperties of the materials are same in all the direction. (Short, discontinuous fiber reinforcement composite)
2. Anisotropic materialsProperties are different in different direction (Long continuous fiber reinforcement composite)
3. Reinforcementcomposite consists of one or more continuous/discontinuous phase embedded in the continuous phase. The discontinuous phase, generally stronger than the continuous phase is called reinforcement or reinforcing material whereas continuous phase is called as matrix.
4. Controlled anisotropymeans the desired ratio of the property values in different direction can easily obtained. Ex. Unidirectional composite in which ration of longitudinal to transverse modulus ratio can easily altered by changing the fiber volume fraction.
5. Volume fractionFractional volume of the any constituent (fiber/matrix material) in a composite is referred as a volume fraction.
6. Weight fraction Fractional weight of the any constituent (fiber/matrix material) in a composite is referred as weight fraction of the individual constituent.
7. Aspect ratioThe ratio of diameter of the fiber to length of the fiber
8. LaminaA single layer of the composite material is generally referred as a ply or lamina. (Thickness=013mm)
9. Laminate Several laminas are bonded together to form the laminates. Here each lamina should have different properties.
10. Homogeneous anisotropy Material properties are the function of the direction but do not change from point to point
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2. Higher Order Sandwich Panel Theory
The free vibrations of composite sandwich plates have been extensively studied using classical analytical formulations, finite element analyses based on first and higherorder shear deformation theories. In the majority of these studies, the sandwich core is assumed to be incompressible in the vertical direction. This assumption is practically accurate only for the vibration analysis of sandwich plates with a honeycomb core. However, in the case of a flexible sandwich core (for example, a foam core), this assumption will preclude modeling of the symmetric vibration modes where the two face sheets move outofphase. Furthermore, modeling of a flexible sandwich core with the aid of generalpurpose commercial finite element software requires the use of 3D solid elements, which consumes significant computational resources (memory and cpu time).
The higherorder sandwich panel theory was derived to model the behavior of sandwich plates with a flexible core. This model is based on the nonlinear throughthethickness displacement field in the core in both longitudinal and vertical directions. However, the corresponding acceleration field in the core is assumed to vary linearly with height, which introduces inconsistency in the formulation. For sandwich beams, this inconsistency has been overcome in the recently developed formulation that accounts for a nonlinear acceleration field in the core.
In the present work, the onedimensional beam formulation described is generalized to the twodimensional plate analysis. The main difference between the present approach and the higher order mixed formulation in Ref. [3] is that in the latter case, the assumed throughthethickness displacement field is used, whereas in the former, nonlinear displacement field in the core is derived based on welldefined physical assumptions. In what follows, a consistent free vibration formulation for composite sandwich plates is developed in terms of the system of governing partial differential equations and the corresponding boundary conditions.
The derived formulation is applied to the vibration analysis of softcore and honeycombcore sandwich plates with antisymmetric and symmetric layups. Excellent agreement between the calculated vibration response for the thin and thick composite sandwich plates and the results of the higher order mixed layer wise theory is demonstrated.
10
For the case of symmetric sandwich plates, simplification is achieved through the decoupling of the general formulation into two independent systems of equations representing symmetric and antisymmetric motions. The numerical study of the free vibration response of composite sandwich plates based on the derived formulation follows next. Differences between the vibration responses of the softcore sandwich plates with antisymmetric and symmetric composite layups are studied and discussed. The effect of honeycomb core modulus on the symmetric vibration response of sandwich plates with isotropic face sheets is investigated. Finally, the importance of the present formulation for the prediction of higherfrequency vibration response is discussed and conclusions are drawn.
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3. Mathematical Assumptions
3.1. Assumptions
Sandwich Plate conventions: (a) Geometry, coordinate systems and displacement functions of the sandwich plate. (b) Stress field in the core
The present formulation is concerned with the linear vibration analysis of sandwich plates (see Fig. 1). The face sheets of a sandwich plate are assumed to behave as Kirchhoff thin plates with negligible shear deformations. The vertically compressible core layer is considered as an antiplane, threedimensional elastic medium with orthotropic outofplane shear properties. Here, an antiplane assumption implies that the stresses in the core in planes parallel to the face sheets are neglected, and only the out of plane shear and normal stresses are accounted for the analysis.
This assumption is nearly exact for a honeycomb core, and is an appropriate approximation for an isotropic foam core, where the core material modulus is significantly lower than the modulus of the face sheets (see Section 1). Note that as a result of the compressibility of the core, the core height may change under loading and the cross section
12
may not remain planar. The interface layers between the face sheets and the core are assumed to provide perfect continuity of the deformations at the interfaces. The acceleration fields of the face sheets are assumed to vary linearly with height, whereas the acceleration field of the compressible core varies nonlinearly with height in the plane and outofplane directions.
Thus, the dynamic fields in the face sheets and core are consistent with the corresponding static patterns as normally assumed in dynamic analyses
3.2. Equations of motion
The governing differential equations and the corresponding boundary conditions are derived here through the use of Hamilton’s variational principle, namely, where T is the kinetic energy and V is the strain energy of a sandwich plate; t is the time coordinate; and t1 and t2 are the values of the time coordinate at the beginning and end of the motion, respectively. The strain energy of the composite face sheets is given by
V
F
= [1/2A i11 u
0i,x
2
+1/2A i22 v
0i,y
2
+1/2A i66
(u
0i,y
+v
0i,x
)
2
+A i 12 u
0i,x v
0i,y
+
A i16
(u
0i,y
+v
0i,x
)u
0i,x
+ A i26
(u
0i,y
+ v
0i,x
)v
0i,y
–
B i11 u
0i,x w i,xx
– B i22 v
0i,y w i,yy
– 2B i66
(u
0i,y
+ v
0i,x
)w t,xy
–
B i12
(v
0i,y w i,xx
+ u
0i,y w i,yy
) – B i16
(u
0i,y
+ v
0i,x
)w i,xx
– B i26
(u
0i,y
+ v
0i,x
)w i,yy
– 2B i16 u
0i,x w i,xy
– 2B i26 v
0i,y w i,xy
+ ½ D i11 w i,xx
2
+
1/2D i22 w i,yy
2
+ 2D i66 w i,xy
2
+D i12 w i,xx
+ 2D i16 w i,xx w i,xy
+
2D i26 w i,yy w i,xy
]dx i dy i
(2) where a and b are the inplane dimensions of the sandwich plate; u
0i
(x i
, y i
, t), v
0i
(x i
, y i
, t), and w i
(x i
, y i,
t) are the longitudinal, transverse and vertical unknown displacement functions of
13
the centroid line of the face sheets, respectively (i = t, b); t and b refer to quantities affiliated to top and bottom face sheets, respectively; A ijk
, B ijk
, and D ijk
(i = t, b; j, k = 1, 2, 6) are the coefficients of the stiffness matrices A, B, D in the force–strain relations for the face sheets; and the comma stands for differentiation with respect to the spatial coordinate. The kinetic energy of the face sheets reads
0t
2
+ w t
2
) + 1/12ρ t d t
3
(w t,y
2
+w t,x
2
)]dx t dy t
+ T
F
=
½ [ρ b d b
(u
0b
2
+ v
0b
2
+ w b
2
) + 1/12ρ b d b
3
(w b,y
2
+ w b,x
2
)]dx b dy b where d t
and d b
are the thicknesses of the upper and lower face sheets, respectively; ρ t
and ρ b are the densities of the upper and lower face sheets, respectively; and the dot represents time differentiation.
As described in Section 3.1, the transversely flexible core is treated as a threedimensional elastic medium with small deformations, where the core height may change under loading, and the core crosssection does not remain planar (nonlinear displacement field in the core). The inplane stresses in the core are neglected because of the negligible inplane strength compared to the face sheets, which implies that
σ
xxc
= σ
xxy
= τ
xyc where
σ
xxc
;
σ
xxy
and
τ
xyc
are the inplane normal and shear stresses in the core. For this case, the partial differential equations describing the equilibrium in the core can be uncoupled and solved analytically for the longitudinal, transverse and vertical displacements, and the resultant expressions are generalized here to include the time parameter, as shown below u c
(x c y c
,z c,t
) = [{z c
2
(2z c
c)}/12](τ xx,xx
(x c
,y c
,t)+τ yx,xy
(x c
,y c
,t))+[{z c
2
}/2c](w t,x
(x c
,y c
,t) w bx,x
(x c
,y c
,t)+z c
τ xzc
(x c
,y c
,t)z c w t,x
(x c
,y c
,t)+u
0t
(x c
,y c
,t)1\2d t w t,x
(x c
,y c
,t) (5)
14
v c
(x c
,y c
,z c
,t) = [{z c
2
(2z c
3c)}/12E c
](τ xx,xy
(x c
,y c
,t)+τ yzc,yy
(x c
,y c
,t))+ [{z c
2
}/2c](w t,y
(x c
,y c
,t)
– w b,y
(x c
,y c
,t)) + [z d t w t,y
(x c
,y c
,t) c
/G cy
] τ yx
(x c
,y c
,t) – z c w t,y
(x c
,y c
,t) + v
0t
(x c
,y c
,t) – ½
(6) w c
(x c
,y c
,z c
,t) = [{z c
(z c
– c)}/2E c
] (τ xzc,x
(x c
,y c
,t) + τ yzc,y
(x c
,y c
,t)) – {w t
(x c
,y c
,t) +
w b
(x c
,y c
,t)}/c]z c
+ w t
(x c
,y c
,t) (7)
Here, E c
is the vertical Young’s modulus of the core, G cx
and G cy
are the vertical shear moduli of the core, τ xzc
and τ yzc
are the shear stresses in the core, and c is the thickness of the core. The strain energy and kinetic energy of the transversely flexible core can be calculated from the displacements in Eqs. (5)–(7). The strain energy of the core reads
Vc = [c
3
/24Ec(τ xzc,x
+ τ yzc,y
)
2
+E c
/2c(w t
– w b
)
2
+ c/2(τ xzc
2
/G cx
+ τ yzc
2
/G cy
)]dx c dy c
(8)
The expression for the kinetic energy of the core reads
T c
= 1\2 ρ c
(u c
2
+v c
2
+w c
2
)dx c dyz c dz c
(9) where ρ c
is the density of the core.
Eqs. (5)–(7) account only for the compatibility between the top face sheet and the core. The compatibility conditions at the lower interface are accounted for by u c
(x c
,y c
,c,t) = u
0b
(x c
,y c
,t)+1\2d b w b,x
(x c
,y c
,t) v c
(x c
,y c
,c,t) = v
0b
(x c
,y c
,t)+1\2d b w b,y
(x c
,y c
,t)
(10)
(11)
The governing equations are then obtained from Eq. (1) using the Lagrange multiplier method with the auxiliary equations given in Eqs. (10) and (11). The Lagrange multipliers can be shown to be the shear stresses in the core, τ xzc
and τ yzc
[15].
15
3.3. Symmetric sandwich plate
The majority of sandwich plates used in practical applications have symmetric crosssections. Therefore, it is worthwhile to simplify the general formulation derived in Section
3.2 for the symmetric case. For this purpose, the general horizontal and vertical motions of the face sheets are represented as the superposition of the symmetric and antisymmetric displacements (see Fig. 2). u oi
= u’ ± u’ v oi
= v’ ± v’
w i
= w’ ± w’
(12)
(13)
(14) where i = t, b stands for the top and bottom face sheets, respectively, the plus and minus signs are used for the upper and lower face sheets, respectively, and u’, v’, w’ denote the symmetric and antisymmetric displacements of the face sheets, respectively.
In the following, the displacement field in the core flexible in the vertical direction, corresponding to symmetric and antisymmetric vibrations, is derived. Substitution of the relations u
0t
= u
0b
= u’ and v
0t
= v
0b
= v’, which characterize the symmetric response, into
Eqs. (5)–(7) yields
c
3
\12E c
(τ xzc,xx
+τ yzc,xy
)+c\G c xτ xyc
= 0
c
3
\12E c
(τ xzc,xy
+τ yzc,yy
)+c\G cy
τ yzc
= 0
(15)
(16)
These equations are satisfied for τ xzc
= τ yzc
= 0. This is a mathematical manifestation of the fact that the symmetric motion of the sandwich plate is characterized only by the expansion and compression of the core material, which are caused by the outofphase vertical displacements of the face sheets, in the absence of shear deformation. Substitution of the relations u
0t
= u
0b
= u’, v
0t
= v
0b
= v’ and τ xzc
= τ yzc
= 0 into Eqs. (5)–(7) produces the following displacement field in the core for the case of symmetric motion
16
u’ c
= z c
2 w’
,x
(z c
+d\2)w’
,x
+ ū
v c
’ = z c
2
\2w’
,y
(z c
+d\2)w’
,y
+v’
(17)
(18)
w’ c
= (12z c
\c)w’ (19)
Eqs. (17)–(19) show that the tangential displacements in the core for symmetric motion are given by polynomials of the second order in the vertical coordinate z c
, whereas the vertical displacements depend linearly on z c
. Note particularly that neglecting the nonlinear terms in Eqs. (17) and (18) will result in a tangential displacement field for ordinary plates consistent with the Kirchhoff assumptions. From Eq. (19), the vertical displacements of the core are seen to be the linear interpolation of the vertical displacements of the face sheets. Similarly, for the antisymmetric motion, substitution of the relations u
0t
= u
0b
= u’, v
0t
= v
0b
= v’ and w
0t
= w
0b
= w’ into Eqs. (5)–(7) yields the following displacement field in the core:
17
u’ c
= [{2z c
3
3cz c
2
\12E c
}](τ xzc,xx
+τ yzc,xy
)+{[z c
\G cx
}]τ xzc
(z c
+d\2)w’, x
+ū (20) v’ c
= [{2z c
3
3cz c
2
}]12E c
(τ xzc,xy
+τ yzc,yy
)+[{z c
\G cy
}]τ yc
_(z c
+d\2)w’
,y
+v’ (21)
w’ c
= [{z c
2
+cz c
}]\2E c
(τ xzc, x+τ yzc,y
)+w’ (22)
Eqs. (20)–(22) reveal that the nonlinear behavior of the core depends on the secondorder derivatives of the shear stress. Therefore, for harmonic motion, the influence of the nonlinear terms on the response of the core will increase with increasing frequency of the vibration modes (see Section 4). Finally, using the auxiliary Eqs. (10), (11) and Eqs. (5), (6), the tangential displacements in the core can be expressed in the form u c
’=[{z
2
(2z3c)\c
3
}](2u’+c\G cx
τ xyc
(c+d)w’, x
)+z\G cx
τ xzc
+u’(z+d\2)w’, x
(23) v c
’=[{z
2
(2z3c)\c
3
}](2v’+c\G cx
τ yzc
(c+d)w’, y
+z\G cy
τ yzc
+v’(z+d\2)w’, y
(24) where the secondorder derivatives of the shear stress in the core have been eliminated. Thus, the general formulation for the free vibrations of symmetric softcore sandwich plates with composite laminated face sheets has been decoupled into two formulations representing symmetric and antisymmetric motions. The symmetric motion is represented by the three governing equations, nine boundary conditions, and three equations describing the displacements in the core, Eqs. (17)–(19), in terms of u’, v’ and w’. The antisymmetric vibrations are represented by five governing equations, 17 boundary conditions and three equations for the displacements of the core, Eqs. (20)–(22), in terms of u’ , v’, w’, τ xzc
and
τ yzc
.
18
4. Validation
For the free vibrations of a simply supported sandwich plate, the boundary conditions are identically satisfied by the following harmonic functions
u
0t
=U t cos(m \a x)sin(n \b y) (25) u
0b
=U b cos \a x)si \b y) (26) v
0t
=V t sin(m \a x)cos(n \b y) v
0b
=V b sin(m \a x)cos(n \b y)
(27)
(28) w t
=W t sin(m \a x)sin(n \b y) (29) w b
=W b sin(m \a x)sin(n \b y)
τ xzc
=Г x cos(m \a x)sin(n \b y)
τ yzc
=Г y cos(m sin(n
(30)
(31)
(32)
Substitution of Eqs. (25)–(32) into the general governing differential equations, produces the generalized algebraic Eigen value problem from which the natural frequencies and corresponding vibration modes of a sandwich plate can be calculated
4.1. Sandwich plate with composite face sheets and a soft core
The free vibration response of a fivelayer (0/90/core/0/90), simply supported sandwich plate with planar dimensions a x b is analyzed here. The computed results are subsequently
19
compared with the predictions of the higher order mixed layer wise theory for laminated and sandwich plates. The material constants for graphiteepoxy T300/934 composite face sheets and the isotropic core used in the calculations are given in Table 1. Note that the elastic moduli of the face sheets are given with respect to the material coordinates.
A comparison between the normalized natural frequencies, Ω, calculated using the present formulation and the results of Rao and Desai is presented in Table 2. The normalized frequencies are given by the equation Ω = ωb
2
(ρ t
/E
2
)
1/2
/H, where Ω is the natural frequency and H is the total thickness of the sandwich plate. Table 2 demonstrates excellent agreement between the predictions based on the present formulation and those in Ref. [3] Note particularly that the close agreement between the two theories holds in a broad range of lengthtothickness ratios. This means that the present formulation can be successfully used for the vibration analysis of both thin and thick soft core sandwich plates.
Note also that for each pair (m, n) (see Eqs. (25)–(32)), there are eight natural frequencies with eight corresponding vibration modes. Physically, this represents various combinations of the displacement patterns of the face sheets and the core corresponding to the same pair of wave numbers. Variations of the inplane and vertical displacements through the thickness of the sandwich plate under consideration vibrating at the fundamental frequency (m = 1, n = 1), as obtained by the present analysis and that Ref. [3] , appear in Fig.
3.
The variations of the longitudinal displacements (along the xaxis) are given at the middle of the left edge of the sandwich plate (x = 0, y = b=2), Fig. 3a. Similarly, the throughthethickness variation of the transverse displacements (along the yaxis) are given at the middle of the front edge of the plate (x = a/2, y = 0), Fig. 3b.
Table 1 : Material parameters of the antisymmetric sandwich plates
Component Elastic modulus(GPa)
Poisson’s Ratio
Face Sheets E
1
= 131
E
2
= 10.34
Core v
12 v
13
= 0.22
= 0.22
E
3
= 10.34
E c
= 6.89 x 10
3 v
23
= 0.49 v c
≈ 0
Shear modulus(GPa)
Mass density(kg/m3)
G
12
= 6.985
G
13
= 6.985
ρ
ρ b t
= 1627
= 1627
G
23
= 6.205
G c
= 3.45 x 10
3
ρ c
= 97
20
Table 2 : Comparison of non dimensional frequencies Ω of (0/90/core/0/90) sandwich plate with a/b = 1 and c/d t
= 10
M a/H = 100
1
1
1
2
2
3 a/H = 10
1
1
1
2
2
3 n
1
2
3
2
3
3
1
2
3
2
3
3
Rao and Desai [3]
11.9401
23.4017
36.1434
30.9432
41.4475
49.7622
1.8480
3.2196
5.2234
4.2894
6.0942
7.6762
Present Analysis
11.8593
23.3419
36.1150
30.8647
41.3906
49.7091
1.8470
3.2182
5.2286
4.2882
6.0901
7.6721
Fig. 3a Fig. 3b
Fig. 3c
21
Finally, the distribution of the vertical displacements through the height of the plate are calculated in the center of the plate (x = a/2, y = b/2), as shown in Fig. 3c.
Fig. 3 shows that the distributions of the inplane displacements from the present analysis practically coincide with those of Rao and Desai. The displacements in Fig. 3 were normalized to the longitudinal (u surface
), transverse (v surface
), and vertical (w surface
) displacement components at the outer surface of the lower skin (see Fig. 1). The same normalization is used in the following figures. A negligible discrepancy between the two theories (~0.3%) is detected for the vertical displacements of the core, which is explained by the fact that in the present formulation, the horizontal stresses in the core are neglected.
4.2 Symmetric sandwich plate with isotropic face sheets and a honeycomb core
A simply supported sandwich plate (1.83 m x 1.22 m) with symmetric crosssection, aluminum face sheets, and an aluminum honeycomb core is considered next. The material properties of the plate constituents are given in Table 3. In Table 4, the natural frequencies of the honey comb core sandwich plate computed using the present analysis are compared with the analytical results of Ref. [1]. and the experimental and analytical results of Ref. [4].. As shown in Table 4, the frequencies, which are obtained using the present formulation, are consistent with both previous experimental and numerical predictions. Thus, the present formulation yields accurate results for both soft core and honeycombcore sandwich plates.
Table 3 : Material Parameters from Ref [3].
Component Thickness
Face Sheets 0.4064
Core 6.35
(mm)
Elastic modulus
(GPa)
68.984
0.1379
Poisson’s ratio
0.3
0
Shear modulus(GPa)
Mass density
(kg/m
3
)
2768 25.924
G
xy
= 0
G
xz
= 0.13445
G
yz
= 0.05171
121.8
22
Table 4: Natural frequencies of sandwich plate with aluminum face sheets and aluminum
honeycomb core
Natural Ref. [4]
Frequency
Experimental Analytical
FEM SFPM Present
Analysis f
1 f
2 f
3 f
4 f
5 f
6

45
69
78
92
125
FEM : Finite Element Method
SFPM : Spline Finite Point Method
23
44
71
80
91
126
23
44
70
80
90
125
23.29
44.47
71.15
78.78
91.57
125.10
23.04
44.16
69.76
79.17
90.24
124.27
23
5. Numerical study and discussion
5.1. Vibration response of softcore sandwich plates with symmetric and antisymmetric layups of composite laminated face sheets
A comparison of the vibration response of softcore sandwich plates with antisymmetric (0/90/core/0/90) and symmetric (0/90/core/90/0) layups is considered. The properties of the composite laminates and the core are given in Table 1. The first six normalized natural frequencies Ω for both antisymmetric and symmetric sandwich plates are presented in Table 5. Note that the vibration frequencies of both thin (a/H = 100) and thick
(a/H = 10) softcore sandwich plates are presented. As evident from Table 5, the differences between the natural frequencies of the antisymmetric and symmetric sandwich plates are negligible. Note that this equivalence of values for antisymmetric and symmetric cases was tracked up to the pair of wave numbers m = 101, n = 101.
Table 5 : Natural frequencies of antisymmetric and symmetric sandwich plates
M N
2
2
1
1 a/H = 100
1 1
3 a/H = 10
1
1
1
2
2
3
2
3
2
3
3
1
2
3
2
3
3
Antisymmetric
(0/90/core/0/90)
11.8593
23.3419
36.1150
30.8647
41.3906
49.7091
1.8470
3.2182
5.2286
4.2882
6.0901
7.6721
Symmetric
(0/90/core/0/90)
11.8674
22.7200
34.9339
30.8859
40.7379
49.7455
1.8483
3.1645
5.1399
4.2845
6.0441
7.6753
24
However, the corresponding vibration modes exhibit quite different throughthethickness, inplane displacement patterns, as illustrated in Fig. 4. As expected, for the symmetric sandwich plate, the inplane displacements of the face sheets corresponding to the fundamental frequency are identical in magnitude and opposite in sign, as shown in Fig. 4a and b.
Fig(a):longitudinal displacement along xaxis Fig(b):transverse displacement along yaxis
Fig(c) : vertical displacement along z axis
Fig 4 : Comparison of the throughthethickness variations of the normalized displacements corresponding to the fundamental frequency (m = 1, n = 1) of the antisymmetric and symmetric sandwich plates with a/H = 10
25
5.2. Effect of the modulus of a honeycomb core on the symmetric vibration response of sandwich plates with isotropic face sheets
The vibration response of a honeycombcore sandwich plate is antisymmetric up to very high frequencies because a honeycomb core is extremely stiff in the vertical direction. Therefore, symmetric vibration modes of a honeycomb core sandwich plate are observed only in the much higher frequency range. For example, in the frequency range below 200 Hz, the vibration response of the sandwich plate with aluminum face sheets and honeycomb core, which was considered in Section 3.2, is characterized by antisymmetric vibration modes.
On the other hand, the first symmetric vibration mode for this plate is numerically predicted at a frequency above 600 Hz, even for a core with a very low value of elastic modulus in the vertical direction. To illustrate this point, the effect of the vertical stiffness of the honeycomb core Ec on the magnitude of the first symmetric natural frequency f
1 s
of the sandwich plate is presented in Fig. 5. As shown in Fig. 5, the magnitude of f
1 s
strongly depends on the vertical stiffness of the core up to Ec ≈ 0.3 MPa.
Beyond this value, the magnitude of the first symmetric frequency is not affected by the increase in E c
. This behavior is explained by the fact that there are three natural frequencies and three corresponding vibration modes for each pair (m, n) in the case of the symmetric vibration response (compare with Section 3.1). For the low value of E c
the magnitude of the vertical displacements W is significantly larger than that of the inplane displacements (see Fig. 6a).
On the other hand, if E c
is increased by several orders of magnitude, the reverse picture is observed. Namely, the inplane displacements become significantly larger than the vertical displacements (see Fig. 6b). Therefore, for the low value of E c
, the pronounced vertical motion of the sandwich plate occurs, whereas for the significantly increased value of
E c
, the longitudinal motion dominates the symmetric vibration response. This inplane vibration response happens to be independent of the value of E c
(see the plateau in Fig. 5)
26
Fig 5 : Dependence of the first symmetric natural frequency on the Young’s modulus of the core (m = 1, n = 1)
Fig 6 : The horizontal (U s
) and vertical (W s
) displacement amplitudes at x = a/2 corresponding to the first symmetric frequency (f
1 s
) of the honeycombcore sandwich plate with isotropic face sheets for: (a) Ec ≈ 0:07 MPa, {U s
= 
0.5271e  4, V s
=  0.7908e  4, W s
= 1.0000}; and (b) Ec ≈ 70:0 MPa, {U s
= 
.8320, V s
= 0.5547, W s
= 0.1336e  9} (  , U; —, W).
27
5.3 Importance of the present formulation for the prediction of the higherfrequency vibration response
As shown in Fig. 7, the variation of the normalized inplane displacements corresponding to the antisymmetric vibration modes, which are characterized by the dominant vertical displacements, changes significantly with increasing mode pair (m,n).
Namely, the throughthethickness variation of the inplane displacements is linear for m = 1, n = 1, whereas it becomes increasingly nonlinear for higher values of m and n. This is because the higher order terms in Eqs. (20) and (21) involve the second derivatives of the shear stress in the core, which increase with the mode numbers m and n in the harmonic assumptions of Eqs. (31) and (32). Therefore, the linear assumptions for the acceleration field in the core [9] are suitable only in the case of low vibrations modes. The present analysis, however, can be applied equally well for all frequency ranges of a sandwich plate.
Fig 7 : Distribution of the antisymmetric longitudinal displacements u’ at x = 0; y = b/2
[—, (m = 1, n = 1),   , (m = 5, n = 5),· · ·, ( m = 7, n = 7)].
28
6. Conclusions
The consistent higherorder free vibration equations and the corresponding boundary conditions for sandwich plates have been derived, taking into consideration the nonlinear acceleration field in the core. In addition, the general formulation was decoupled into two systems of equations representing symmetric and antisymmetric vibrations. The present formulation has been validated by comparisons with results of the higher order mixed layer wise theory for laminated and sandwich plates and the results of experimental studies.
A numerical study of the free vibration analysis of soft and honeycombcore thin and thick sandwich plates with antisymmetric and symmetric layups was undertaken. The study showed that the vibration modes of the softcore sandwich plates with antisymmetric and symmetric layups exhibit quite different throughthethickness inplane displacement patterns in spite of the almost identical natural frequencies. In the case of honeycomb sandwich plates, it was shown that the change in the modulus of the core not only brings about the shift in the first symmetric frequency, but also the qualitative change in the corresponding vibration pattern.
The importance of the present formulation for the prediction of the higherfrequency response of sandwich plates was demonstrated. The main difference between the present approach and the higher order mixed formulation is that unlike the latter, which is based on the assumed throughthethickness displacement field in the core, the former uses the nonlinear displacement field that is derived based on the welldefined physical assumptions. The present twodimensional formulation constitutes an efficient alternative to the application of commercial finite element software, which requires threedimensional modeling to achieve comparable results.
Furthermore, in the case of symmetric sandwich plates, the decoupled formulations, which represent symmetric and antisymmetric motions, can be easily coded and applied to the accurate and efficient assessment of the free vibration response of simply supported sandwich plates with either foam or honeycomb cores.
29
7. References
SL NO. REFERENCE
NO.
1. 1.
2.
3.
4.
5.
2.
3.
4.
5.
NAME OF THE BOOK AND AUTHOR
Monforton GR, Schmit LA Jr. Finite element analysis of sandwich plates and cylindrical shells with laminate faces.
Rao MK, Desai YM. Analytical solutions for vibrations of laminated and sandwich plates using mixed theory
Zhou HB, Li GY. Free vibration analysis of sandwich plates with laminated faces using spline finite point method
Raville ME, Veng ES. Determination of natural frequencies of vibration of a sandwich plate
TonganWang,Vladimir Sokolinsky,Shankar Rajaram,Steven
R Nutt.Consistent higherorder free vibration analysis of composite sandwich plates.
30
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