Dynamic Stability of a Sandwich Plate Under Parametric Excitation

Dynamic Stability of a Sandwich Plate Under Parametric Excitation
Dynamic Stability of a Sandwich Plate
Under Parametric Excitation
Thesis submitted in partial fulfillment of the requirements
for the degree of
Master of Technology
In
MECHANICAL ENGINEERING
(Specialization: Machine Design and Analysis)
By
RAGHUPATHI AVULA
Roll No. 207ME102
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-769008
MAY, 2009
Dynamic Stability of a Sandwich Plate
Under Parametric Excitation
Thesis submitted in partial fulfillment of the requirements
for the degree of
Master of Technology
In
MECHANICAL ENGINEERING
(Specialization: Machine Design and Analysis)
By
RAGHUPATHI AVULA
Roll No. 207ME102
Under the guidance of
Dr. S. C. Mohanty
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-769008
MAY, 2009
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled “DYNAMIC STABILITY OF A
SANDWICH PLATE UNDER PARAMETRIC EXITATION” submitted by Raghupathi
Avula in partial fulfillment of the requirements for the award of Master of Technology
Degree in Mechanical Engineering at the National Institute of Technology, Rourkela
(Deemed University) is an authentic work carried out by him 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
Dr. S. C. Mohanty
Assistant Professor
Department of Mechanical Engineering
National Institute of Technology
Rourkela
I ACKNOWLEDGEMENT
I avail this opportunity to express my hereby indebtedness , deep gratitude and
sincere thanks to my guide, Dr. S.C. Mohanty, Assistant Professor, Mechanical Engineering
Department for his in depth supervision and guidance, constant encouragement and cooperative attitude for bringing out this thesis work.
I extend my sincere thanks to Dr. R.K.Sahoo, Professor and Head of the
Department, Mechanical Engineering Department, N.I.T. Rourkela for his valuable
suggestions for bringing out this edition in time.
Finally I extend my sincere thanks to all those who have helped me
during my dissertation work and have been involved directly or indirectly in my endeavor.
Raghupathi Avula
Roll No-207ME102
Department of Mechanical Engg
National Institute of Technology
Rourkela
II CONTENTS
Description
Page No.
Certificate………………………..…………………………………………..…………………I
Acknowledgement …………………..……………………………………………………..…II
Contents ……………………………………...………………………………………...……III
Abstract……………………………………………………………………….……………….V
List of figures……………………………………………………………………………...…VI
List of Tables………………………………...………………...………………………...…VIII
Nomenclature…………………………….……………………….....…………………….....IX
Chapter 1 Introduction
1.1 Introduction……………………………………………………………………1
1.2 Types of sandwich plates………………………...……………………………2
1.3 Sandwich construction…………………………………………………..…….2
1.4 Properties of materials used in sandwich construction……………….……….6
1.5 Current applications…………………………………………………….…......6
Chapter 2 Literature Review
2.1 Introduction…………………………………………………………..………..9
2.2 Methods of stability analysis of parametrically excited system………...…..…9
2.3. Dynamic stability of plate……………………………………….……..…... 10
2.3.1 Isotropic Plates…………..…………………………………………….10
2.3.2Composite Plates………………………………………………………..13
Chapter 3 Finite Element Modeling
3.1 Introduction..…………………………………………………………….……17
3.2 Basic concept of finite element method………….……………………...……18
3.3 Finite element approaches………………….…………………………………19
3.3.1 Forced method………………………………………………….………19
3.3.2 Displacement method…………………………………………………..20
3.4 Advantages of fem…………………………………………………………….21
III 3.5 Limitations of fem……………....…...………………………………………..21
Chapter 4 Problem Formulation
4.1 Problem formulation…………….….…………………………………………22
4.2 Strain displacement relationships………..……………………………………23
4.3 Finite element formulation………………..…………………………………..26
4.3.1 Element shape function……………………….………………………..27
4.4 Equation of motion……………………………………………………………31
4.5 Regions of instability……………………………………………..…………...34
Chapter 5 Results and Discussions…………………………………...……………………36
Chapter 6 Conclusions and scope for the future work…………...………………………44
Chapter 7 References……………………………………………………………………….45
IV ABSTRACT
Vibration control of machines and structures incorporating viscoelastic
materials in suitable arrangement is an important aspect of investigation. The use of
viscoelastic layers constrained between elastic layers is known to be effective for damping of
flexural vibrations of structures over a wide range of frequencies. The energy dissipated in
these arrangements is due to shear deformation in the viscoelastic layers, which occurs due to
flexural vibration of the structures. Sandwich plate like structures can be used in aircrafts and
other applications such as robot arms for effective vibration control. These members may
experience parametric instability when subjected to time dependant forces. The theory of
dynamic stability of elastic systems deals with the study of vibrations induced by pulsating
loads that are parametric with respect to certain forms of deformation
The purpose of the present work is to investigate the dynamic stability of a
three layered symmetric sandwich plate subjected to an end periodic axial force. Equations of
motion are derived using finite element method. The regions of instability for simple and
combination resonances are established using modified Hsu’s method proposed by Saito and
Otomi[74].
It is observed that for plate with simply supported boundary conditions the
fundamental frequency, fundamental buckling load increase with increase in core thickness
parameter for higher values of core thickness parameter. The fundamental frequency,
fundamental buckling load decrease with increase in core thickness parameter for lower
values of core thickness parameter. The system fundamental loss factor increases with
increase in thickness ratio. The fundamental buckling load and fundamental frequency
increase with increase in shear parameter. The fundamental system loss factor has an
increasing tendency with increase in shear parameter. The increase in core thickness ratio and
shear parameter has stabilizing effect. Whereas increase in static load factor has a
destabilizing effect.
Keywords: Parametric excitation, Dynamic stability, Boundary condition, Sandwich plate.
V LIST OF FIGURES
Sl. No.
1.
TOPIC
Sandwich panel with (a) continuous corrugated-core (b) to-hat
PAGE NO
3
core (c) zed-core (d) channel-core and (e) truss core
2.
Laminate composite and sandwich composite
4
3.
Typical sandwich constructions
5
4.
Application of sandwich structure in aircraft
7
5.
Application of sandwich structure in helicopter
8
6.
The sandwich plate with an viscoelastic core and a constraining
22
layer
7.
Undeformed and deformed configurations of a sandwich plate
23
8.
Compatibility relation of systems.
25
9.
Sandwich plate element with four end nodes and seven DOF per
27
node
10.
A plate subjected to an in-plane dynamic load.
VI 32
11 Effect of core thickness parameter on buckling load parameter. 39 12 Effect of core thickness parameter on fundamental frequency 39 parameter. 13 Effect of core thickness parameter on fundamental loss factor 40 14 Effect of shear parameter on fundamental buckling load 40 parameter at h2/h1=1. 15 Effect of shear parameter on fundamental frequency at h2/h1=1. 41 16 Effect of shear parameter on fundamental loss factor at h2/h1=1. 41 17 Effect of core thickness parameter(h2/h1) on stability regions at 42 h2/h1=1, h2/h1=2. 18 Effect of static load factor(α) on stability regions at α =0, α=0.5. 42 19 Effect of shear parameter on stability regions at h2/h1=1, α=0, 43 g=5, g=10. VII LIST OF TABLES
SI.NO.
Title
Page No.
Table 5.1
1.
Comparison of Resonant frequency parameters and Modal loss
factors
calculated from present analysis with those of
reference[52]
36
Table 5.2
2.
Comparison of Resonant frequency parameters and Modal loss
factors
calculated from present analysis with those of
reference[47]
VIII 37
NOMENCLATURE
a
Length of of the sandwich plate.
b
Width of the sandwich plate..
D1
E1h3/[12(1-µ21)] flexural rigidity of the host plate.
Ei
Young’s moduli of face layers, i =1,3.
G
Shear modulus of the viscoelastic material.
hi
Thickness of the layer i, i=1,2,3.
h2/h1
Core thickness parameter.
i
Index for layer, i = 1, 2, 3.
K0
-P0h1b2/D1, non-dimensional static component of the external load.
Kt
Pth1b2/D1, non-dimensional dynamic component of the external load.
Nr
Number of elements in the r-direction.
Nz
Number of elements in the z-direction.
P0
Static component of the external load.
Pt
Dynamic component of the external load.
Ώ
Frequency of the external load..
{Г}
Set of generalized coordinates
ηv
Loss factor of the viscoelastic material.
µi
Poisson’s ratio of the layer i.
ρi
Mass density of the layer i, i =1,2,3.
ω
Natural frequency of the composite plate.
V
Total strain energy of the sandwich plate.
T
Total kinetic energy of the sandwich plate.
W
Work done by the periodic load.
[Ke]
Stiffness matrix of the sandwich plate.
[K]
Global Stiffness matrix of the sandwich plate.
[Me]
Element mass matrix of sandwich plate.
[M]
Global mass matrix of sandwich plate.
IX [Keg]
Geometric stiffness matrix of plate element.
[Kg]
Global stiffness matrix of sandwich plate.
P*
Critical buckling load of the equivalent plate.
{Δ}
Global nodal displacement matrix of the plate.
[Φ]
Normalized modal matrix.
S
Stable region
U
Unstable region
X CHAPTER-1
INTRODUCTION
1.1 INTRODUCTION
Sandwich plates have been successfully used for many years in the aviation
and aerospace industries, as well as in marine, and mechanical and civil engineering
applications. This is due to the attendant high stiffness and high strength to weight ratios of
sandwich systems. The theory of dynamic stability of elastic systems deals with the study of
vibrations induced by pulsating loads that are parametric with respect to certain forms of
deformation. A system is said to be parametrically excited if the excitation which is an
explicit function of time appears as one of the co-efficients of the homogeneous differential
equation describing the system, unlike external excitation which leads to an inhomogeneous
differential equation. A well known form of equation describing a parametric system is Hill’s
equation
x + ω 2α + ε xf (t ) = 0 ………………………..(1)
In the above equation when f(t) = cosΏt, Equation (1) is known as Mathieu’s equation .
Equation (1) governs the response of many physical systems to a sinusoidal parametric
equation.
In practice parametric excitation can occur in structural systems subjected
to vertical ground motion, aircraft structures subjected to turbulent flow, and in machine
components and mechanisms. Other examples are longitudinal excitation of rocket tanks and
their liquid propellant by the combustion chambers during powered flight, helicopter blades
in forward flight in a free-stream that varies periodically and spinning satellites in elliptic
orbits passing through a periodically varying gravitational field. In industrial machines and
mechanisms, their components and instruments are frequently subjected to periodic or
random excitation transmitted through elastic coupling elements, example includes those
associated with electromagnetic aeronautical instruments and vibratory conveyers saw blades
and belt drives. In parametric instability the rate of increase in amplitude is generally
exponential and thus potentially dangerous while in typical resonance the rate of increase is
linear. Moreover damping reduces the severity of typical resonance, but may only reduce the
rate of increase during parametric resonance. Moreover parametric instability occurs over a
region of parameter space and not at discrete points. The system can experience parametric
instability (resonance), when the excitation frequency or any integer multiple of it is twice the
natural frequency that is to say mΩ=2ω, m=1, 2, 3, 4.
1 The case Ω=2ω is known to be the most important in application and is
called main parametric resonance. A vital step in the analysis of parametric dynamic systems
is thus establishment of the regions in the parameter space in which the system becomes
unstable, these regions are known as regions of dynamic instability or zones of parametric
resonance. The unstable regions are separated from the stable ones by the so called stability
boundaries and a plot of these boundaries on the parameter space is called a stability diagram.
Vibration control of machines and structures incorporating viscoelastic materials in suitable
arrangement is an important aspect of investigation. The use of viscoelastic layers
constrained between elastic layers is known to be effective for damping of flexural vibrations
of structures over a wide range of frequencies. The energy dissipated in these arrangements is
due to shear deformation in the viscoelastic layers, which occurs due to flexural vibration of
the structures. Multilayered cantilever sandwich beam like structures can be used in aircrafts
and other applications such as robot arms for effective vibration control. These members may
experience parametric instability when subjected to time dependant forces. The conventional sandwich construction comprises a relatively thick core
of low-density material which separates top and bottom faceplates (or faces or facings) which
are relatively thin but stiff. The materials that have been used in sandwich construction have
been many and varied but in quite recent times interest in sandwich construction has
increased with the introduction of new materials for use in the facings (e.g. fiber- reinforced
composite laminated material) and in the core (e.g. solid foams) [96].
1.2 TYPES OF SANDWICH PANELS
Detailed treatment of the behavior of honeycombed and other types of
sandwich panels can be found in monographs by Plantema [32] and Allen [37]. These
structures are characterized by a common feature of two flat facing sheets, but the core takes
many generic forms; continuous corrugated sheet or a number of discrete but aligned
longitudinal top-hat, zed or channel sections (see Figures 1.1(a)-(e)). The core and facing
plates are joined by spot-welds, rivets or self-tapping screws [57].
1.3 SANDWICH CONSTRUCTION
Sandwich construction is a special kind of laminate consisting of a thick
core of weak, lightweight material sandwiched between two thin layers (called "face sheets")
of strong material figure (1.2). This is done to improve structural strength without a 2 corresponding increase in weight. The choice of face sheet and core materials depends
heavily on the performance of the materials in the intended operational environment.
Fig 1.1 Sandwich panel with (a) Continuous corrugated (b) top hat core
(c) Zed core (d) Channel core and (e) truss core
3 Because of the separation of the core, face sheets can develop very high
bending stresses. The core stabilizes the face sheets and develops the required shear strength.
Like the web of a beam, the core carries shear stresses. Unlike the web, however, the core
maintains continuous support for the face sheets. The core must be rigid enough
perpendicularly to the face sheets to prevent crushing and its shear rigidity must be sufficient
to prevent appreciable shearing deformations. Although a sandwich composite never has a
shearing rigidity as great as that of a solid piece of face-sheet material, very stiff and light
structures can be made from properly designed sandwich composites.
Fig 1.2 Laminate composite and sandwich composite
4 Fig 1.3 Typical sandwich constructions A useful classification of sandwich composites according to their core
properties by respective direction is shown in fig.1.3. To see the core effect upon sandwich
strength, let us consider the honeycomb-core and the truss-core sandwich composite.
The honeycomb sandwich has a ratio of shear rigidities in the xz and yz
planes of approximately 2.5 to 1. The face sheets carry in-plane compressive and tensile
loads, whereas the core stabilizes the sheets and builds up the sandwich section.
The truss-core sandwich has a shear rigidity ratio of approximately 20 to 1.
It can carry axial loads in the direction of the core orientation as well as perform its primary
function of stabilizing the face sheets and building up the sandwich section [76].
5 1.4 PROPERTIES OF MATERIALS USED IN SANDWICH COSNTRUCTION
No single known material or construction can meet all the performance
requirements of modern structures. Selection of the optimum structural type and material
requires systematic evaluation of several possibilities. The primary objective often is to select
the most efficient material and configuration for minimum-weight design [76]
Face Materials
Almost any structural material which is available in the form of thin sheet
may be used to form the faces of a sandwich panel. Panels for high-efficiency aircraft
structures utilize steel, aluminum or other metals, although reinforced plastics are sometimes
adopted in special circumstances. In any efficient sandwich the faces act principally in direct
tension and compression. It is therefore appropriate to determine the modulus of elasticity,
ultimate strength and yield or proof stress of the face material in a simple tension test. When
the material is thick and it is to be used with a weak core it may be desirable to determine its
flexural rigidity [37].
Core Materials
A core material is required to perform two essential tasks; it must keep the
faces the correct distance apart and it must not allow one face to slide over the other. It must
3
be of low density. Most of the cores have densities in the range 7 to 9.5 lb/ft . Balsa wood is
one of the original core materials. It is usually used with the grain perpendicular to the faces
of the sandwich. The density is rather variable but the transverse strength and stiffness are
good and the shear stiffness moderate. Modern expanded plastics are approximately isotropic
and their strengths and stiff nesses are very roughly proportional to density. In case of
aluminum honeycomb core, all the properties increase progressively with increases in
thickness of the foil from which the honeycomb is made [37].
1.5 CURRENT APPLICATIONS
Damped Structures
An increasing number of vibration problems must be controlled by
damping resonant response. By using a symmetric sandwich panel with a viscoelastic core,
various degrees of damping can be achieved, depending on the core material properties, core
thickness, and wavelength of the vibration mode [76] 6 Aerospace Field In Aerospace industry various structural designs are accomplished to fulfill
the required mission of the aircraft. Since a continually growing list of sandwich applications
in aircraft/helicopter (example-Jaguar, Light Combat Aircraft, Advanced Light Helicopter)
includes fuselages, wings, ailerons, floor panels and storage and pressure tanks as shown in
fig (1.4). Honeycomb sandwich structures have been widely used for load-bearing purposes
in the aerospace due to their lightweight, high specific bending stiffness and strength under
distributed loads in addition to their good energy-absorbing capacity [8]. In a new spaceformed system called "Sunflower," the reflector is of honeycomb construction, having a thin
coating of pure aluminum protected by a thin coating of silicon oxide to give the very high
reflectivity needed for solar-energy collection. Thirty panels fold together into a nose-cone
package in the launch vehicle.
Fig 1.4(a) Application of sandwich structure in aircraft. 7 Fig 1.4(b) Application of sandwich structure in helicopter
(1. Rotor Blades, 2. Main and Cargo Doors, 3. Fuselage Panels, 4. Fuselage, 5. Boom and
Tail section)
Building Construction
Architects use sandwich construction made of a variety of materials for
walls, ceilings, floor panels, and roofing. Cores for building materials include urethane foam
(slab or foam-in-place), polystyrene foam (board or mold), phenolic foam, phenolicimpregnated paper honeycomb, woven fabrics (glass, nylon, silk, metal, etc.), balsa wood,
plywood, metal honeycomb, aluminum and ethylene copolymer foam. Facing sheets can be
made from rigid vinyl sheeting (fiat or corrugated) ; glass-reinforced, acrylic-modified
polyester; acrylic sheeting; plywood; hardwood; sheet metal (aluminum or steel); glass
reinforced epoxy; decorative laminate; gypsum; asbestos; and poured concrete [76].
8 CHAPTER-2
LITERATURE REVIE
2.1 INTRODUCTION
Discovery of parametric resonance dates back to 1831. The phenomenon of
parametric excitation was first observed by Faraday[33], when he noticed that when a fluid
filled container vibrates vertically, fluid surface oscillates at half the frequency of the
container. Parametric resonance in the case of lateral vibration of a string was reported by
Melde[59]. Beliaev [6] was first to provide a theoretical analysis of parametric resonance
while dealing with the stability of prismatic rods. These are a few early works. Several review articles on parametric resonance have also been published.
Evan-Iwanowski [31], Ibrahim and coworkers [40-46], Ariarathnam [1] and Simitses [80]
gave exhaustive account of literature on vibration and stability of parametrically excited
systems. Review article of Habip [38] gives an account of developments in the analysis of
sandwich structures. Articles of Nakra [63-65] have extensively treated the aspect of
vibration control with viscoelastic materials. Books by Bolotin [10], Schmidt [77] and
Nayfeh and Mook [66] deals extensively on the basic theory of dynamic stability of systems
under parametric excitations. In this chapter further developments in subsequent years in the
field of parametric excitation of system with specific resonance to ordinary and sandwich
beams is reported. Reference cited in the above mentioned review works are not repeated
except at a few places for the sake of continuity. The reported literature mainly deals with the
methods of stability analysis, types of resonance, study of different system parameters on the
parametric instability of the system and experimental verification of the theoretical findings.
2.2 METHODES OF SABILITY ANALYSIS OF PARAMETRICALLY EXITED
SYSTEMS
There is no exact solution to the governing equations for parametrically
excited systems of second order differential equations with periodic coefficients. The
researchers for a long time have been interested to explore different solution methods to this
class of problem. The two main objectives of this class of researchers are to establish the
existence of periodic solutions and their stability. When the governing equation of motion for
the system is of Mathieu-Hill type, a few well known methods commonly used are method
proposed by Bolotin based on Floquet’s theory, perturbation and iteration techniques, the
Galerkin’s method, the Lyapunov second method and asymptotic technique by Krylov,
Bogoliubov and Mitroploskii. 9 Bolotin’s[10] method based on Floquet’s theory can be used to get
satisfactory results for simple resonance only. Steven [87] later modified the Bolotin’s
method for system with complex differentials equation of motion. Hsu proposed an
approximate method of stability analysis of systems having small parameter excitations .
Hsu’s method can be used to obtain instability zones of main, combination and difference
types. Later Saito and Otomi [74] modified Hsu’s method to suit systems with complex
differential equation of motion. Takahashi [88] proposed a method free from the limitations
of small parameter assumption. This method establishes both the simple and combination
type instability zones. Zajaczkowski and Lipinski [97] and Zajaczkowski [98] based on
Bolotin’s method derived formulae to establish the regions of instability and to calculate the
steady state response of systems described by a set of linear differential equations with time
dependent parameters represented by a trigonometric series. Lau et al. proposed a variable
parameter incrementation method, which is free from limitations of small excitation
parameters. It has the advantage of treating non-linear systems. Many investigators to study
the dynamic stability of elastic systems have also applied finite element method.
2.3. DYNAMIC STABILITY OF PLATES
General theories involving dynamic stability presented in section 2 can be
appropriately recast to study the instability behavior of both isotropic and composite plates.
2.3.1 ISOTROPIC PLATES
The dynamic stability of plates under periodic in-plane loads was
considered first by Einaudi [30] in 1936. A comprehensive review of early developments in
the parametric instability of structural elements including plates was presented in the review
articles. Simons and Leissa [78] explained the stability behavior of homogeneous plates
subjected to in-plane acceleration loads. Yamaki and Nagai [94] treated rectangular plates
under in-plane periodic compression. The dynamic stability of clamped annular plates is
studied theoretically by Tani and Nakamura [91] using the Galerkin procedure. It was found
that principal resonance was of most practical importance, but that of combination resonance
cannot be neglected when the static compressive force was applied. Dixon and Wright [29]
studied experimentally the parametric instability behavior of flat plates by normal or shear periodic in-plane forces. Oscillating tensile in-plane load at the far end causing parametric
instability effects around the free edge of the cutout is an interesting phenomenon in
10 structural instability. Carlson [11] conducted experiments on the parametric response
characteristics of a tensioned sheet with a crack like opening. Cutouts, cracks and other kinds
of discontinuities are inevitable in structures due to practical considerations. Datta [23]
investigated experimentally the buckling behavior and parametric resonance behavior of
tensioned plates with circular and elliptical openings. Datta [24] later studied the parametric
instability of tensioned panels with central openings and edge slot. The parametric resonance
experiments for different opening parameters indicate that the dynamic instability effects are
more significant for narrow openings than for wider openings. The studies on the dynamic
stability of plates by Ostiguy et al. [67] showed good agreement between theory and
experiment. The emergence of digital computers caused the evolution of various numerical
methods besides analytical and experimental procedures. Hutt and Salam [39] used the finite
element method for the dynamic stability analysis of homogeneous plates using a thin plate 4noded finite element model. Extensive results were presented on dynamic stability of
rectangular plates subjected to various types of uniform loads with/without consideration of
damping. Prabhakara and Datta [69] explained the parametric instability characteristics of
rectangular plates subjected to in-plane periodic load using finite element method,
considering shear deformation. Plates and shells are seldom subjected to uniform loading at
the edges. Cases of practical interest arise when the in-plane stresses are caused by localized
or any arbitrary in-plane forces. Deolasi and Datta [25] studied the parametric instability
characteristics of rectangular plates subjected to localized tension and compression edge
loading using Bolotin’s approach. The effect of damping on dynamic stability of plates
subjected to non-uniform in-plane loads was investigated by Deolasi and Datta [26] using the
Method of Multiple Scales (MMS). They further extended the work [27] to explain the
combination resonance characteristics of rectangular plates subjected to non-uniform loading
with damping. It was observed that under localized edge loading, combination resonance
zones were important as simple resonance zones and the effects of damping on the
combination resonances may be destabilizing under certain conditions. Deolasi and Datta
[28] verified experimentally the parametric response of plates under tensile loading.
Floquet’s theory was used by most of the investigators [25,39,69] to study
the dynamic stability of plates. The regions of dynamic instability regions were determined
by Bolotin’s method. Aboudi et al. [2] studied the instability of viscoelastic plates subjected
to periodic loads on the basis of Lyapunov exponents. The viscoelastic behavior of the plate
was given in terms of the Boltzmann superposition principle, allowing any viscoelastic
character. Square and rectangular plates were the subject of research for many investigators
11 [2,27,39,69, 79]. Shen and Mote [81] discussed the parametric excitation of circular plates
subjected to a rotating spring. The analytical works on dynamic stability analysis of annular
plates got new direction with the use of finite element method. Chen et al. [18] investigated
the parametric excitation of thick annular plates subjected to periodic uniform radial loading
along the outer edge, using higher order plate theory and axi-symmetric finite element. The
dynamic stability of annular plates of variable thickness was studied by Mermertas and Belek
[60]. The Mindlin plate finite element model was used to handle both the thin and thick
annular plates. Young et al. [95] presented results on the dynamic stability of skew plates
acted upon simultaneously by an aerodynamic force in a chordwise direction and a random
in-plane force in spanwise direction. The dynamic instability of simply supported thick
polygonal plates was analyzed by Baldinger et al. [5] and the corresponding stability regions
of the first and second order are calculated, considering shear and rotatory inertia. Structures
consisting of plates are often attached with stiffening ribs for achieving greater strength with
relatively less material. Srivastava et al. [82] investigated the parametric instability of
stiffened plates using the 9-node isoparametric plate element and stiffener element. The
results showed that location, size and number of stiffeners have a significant effect on the
location of the boundaries of the principal instability regions. As far as loading is concerned,
many studies involved dynamic stability of plates subjected to uniform [39, 79] in-plane
periodic loading. The dynamic stability of plates subjected to partial edge loading and
concentrated in-plane compressive edge loading was considered by Deolasi and Datta [2527]. Srivastava et al. [83-84] investigated the dynamic stability of stiffened plates subjected
to non-uniform in-plane periodic loading. Takahashi and Konishi [89] analyzed the
parametric resonance as well as combination resonance of rectangular plates subjected to inplane dynamic force. Takahashi and Konishi [90] further investigated the dynamic stability of
rectangular plates subjected to in-plane moments. Langley [53] examined the response of
two-dimensional periodic structures to point harmonic loading. The study has extensive
application to all types of two-dimensional periodic structures including stiffened plates and
shells and it raises the possibility of designing a periodic structure to act as a spatial filter to
isolate sensitive equipment from a localized excitation source. Young et al. [95] studied the
parametric excitation of plates subjected to aerodynamic and random in-plane forces. The
numerical studies involving dynamic stability behavior of plates with openings are relatively
complex due to non-uniform in-plane load distribution and are relatively new. Prabhakara
and Datta [70] investigated the parametric instability behavior of plates with centrally located
cutouts subjected to tension or compression in-plane edge loading. Srivastava et al. [85]
12 analyzed the dynamic stability of stiffened plates with cutouts subjected to uniform in-plane
periodic loading.
The study considered stiffened plates with holes possessing different
boundary conditions, cutout parameters, aspect ratios neglecting the in-plane displacements.
The interaction of forced and parametric resonance of imperfect rectangular plates was
explained by Sassi et al. [75]. In this study, the temporal response and the phase diagram
were used besides the frequency response and FFT curves to study the transition zones. The
effect of one particular spatial mode of imperfection on a different mode of vibration was
investigated for the first time. Cederbaum [12] through a finite element formulation studied
the effect of in-plane inertia on the dynamic stability of non-linear plates. Ganapathi et al.[34]
investigated the non-linear instability behavior of isotropic as well as composite plates,
subjected to periodic in-plane load through a finite element formulation. The analysis brought
out the existence of beats, their dependency on the forcing frequency, the influence of initial
conditions, load amplitude and the typical character of vibrations in different regions. Touati
and Cederbaum [92] analyzed the dynamic stability of non-linear visco-elastic plates. 2.3.2 COMPOSITE PLATES
The increasing use of fibre-reinforced composite materials in automotive,
marine and especially aerospace structures, has resulted in interest in problems involving
dynamic instability of these structures. The effects of number of layers, ply lay-up,
orientation and different types of materials introduce material couplings such as stretchingbending and twisting-bending couplings etc. All these factors interact in a complicated
manner on the vibration frequency spectrum of the laminates affecting the dynamic instability
region. The stability behavior of laminates was essential for assessment of the structural
failures and optimal design. As per Evan-Iwanowski , the earliest works on dynamic stability
of anisotropic plates were done by Ambartsumian and Khachaturian [3] in 1960.
Considerable progress has been made since the survey in this subject. There is a renewed
interest on the subject after Birman [9] studied analytically the dynamic stability of
rectangular laminated plates, neglecting transverse shear deformation and rotary inertia. The
effect of unsymmetrical lamination on the distribution of the instability regions was investigated in the above study. Mond and Cederbaum [61] analyzed the dynamic stability of
antisymmetric angle ply and cross ply laminated plates within the classical lamination theory,
13 using the method of multiple scales. It was observed that besides the principal instability
regions, other cases could be of importance in some cases. Srinivasan and Chellapandi [86]
analyzed thin laminated rectangular plates under uniaxial loading by the finite strip method.
The transverse shear deformation and in-plane inertia as well as rotatory inertia were
neglected and the region of parametric instability was derived using Bolotin’s procedure. Bert
and Birman [7] investigated the effect of shear deformation on dynamic stability of simply
supported anti-symmetric angle-ply rectangular plates neglecting in-plane and rotary inertia.
The parametric studies on the effects of the number of layers, aspect ratio and thickness-toedge length ratio were investigated. The dynamic instability of composite plates subjected to
in-plane loads was investigated by Cederbaum [13] within the shear deformable lamination
theory, using the method of multiple scales. Chen and Yang [19] investigated on the dynamic
stability of thick anti-symmetric angle-ply laminated composite plates subjected to uniform
compressive stress and/or bending stress using Galerkin's finite element. The thick plate
model included the effects of transverse shear deformation and rotary inertia. The effects of
number of layers, lamination angle, static load factor and boundary conditions were
investigated. Moorthy et al. [62] considered the dynamic stability of uniformly uniaxially
loaded laminated plates without static component of load and the instability regions were
obtained using finite element method. Extensive results were presented on the effects of
different parameters on dynamic stability of angle-ply plates. Kwon [51] studied the dynamic
instability of layered composite plates subjected to biaxial loading using a high order bending
theory. Chattopadhyay and Radu [17] used the higher order shear deformation theory to
investigate the dynamic instability of composite plates by using the finite element approach.
The first two instability regions were determined for various loading conditions using both
first and second order approximations. Pavlovic [68] investigated the dynamic stability of
anti-symmetrically laminated angle-ply rectangular plates subjected to random excitation
using Lyapunov direct method. Tylikowski [93] studied the dynamic stability of non-linear
anti-symmetric cross-ply rectangular plates. The parametric results on biaxial loading were
compared with those obtained by classical theory. Cederbaum [14] has investigated on the
dynamic stability of laminated plates with physical non-linearity. Librescu and Thangjitham
[55] analyzed the dynamic stability of simply supported shear deformable composite plates
along with a higher order geometrically non-linear theory for symmetrical laminated plates.
Gilai and Aboudi [36] obtained results on the dynamic stability of non-linearly elastic
composite plates using Lyapunov exponents. The non-linear elastic behavior of the resin
matrix was modeled by the generalized Ramberg-Osgood representation. The instability of
14 laminated composite plates considering geometric non-linearity was also reported using C
0
shear flexible QUAD-9 element by Balamurugan et al. [4]. The non-linear governing
equations were solved using the direct iteration technique. The effect of a large amplitude on
the dynamic instability was studied for a simply supported laminated composite plate. The
1
non linear dynamic stability was also carried out using C eight-nodded element by Ganapathi
et al.[35]. Numerical results were presented to study the influences of ply angle and lay-up of
laminate. The parametric resonance characteristics of composite plates for different
lamination schemes were also studied. Certain fiber reinforced materials, especially those
with soft matrices exhibit quite different elastic behavior depending upon whether the fiber
direction strain is tensile or compressive. The dynamic stability of thick annular plates with
such materials, called the bimodulus materials was studied by Chen and Chen [20]. The
annular element with Lagrangian polynomials and trigonometric functions as shape function
was developed. The non-axisymmetric modes were shown to have significant effects in the
annular bimodulus plates. The dynamic stability of thick plates with such bimodulus
materials were examined by Jzeng et al.[49]. The finite element method was used to
investigate the stability of bimodulus rectangular plates subjected to periodic in-plane loads.
The effects of shear deformation and rotatory inertia were considered using first order shear
deformation theory. The dynamic stability of bimodulus thick circular and annular plates was
analyzed by Chen and Juang [21]. Chen and Hwang [22] studied the axisymmetric dynamic
stability of orthotropic thick circular plates. Cederbaum [15] investigated on the dynamic
stability of viscoelastic orthotropic plates. The stability boundaries were determined
analytically by using the multiple scale method. Time dependent instability regions and
minimum load parameter were derived together with an expression for the critical time at
which the system, with a given load amplitude, would turn unstable. Cederbaum et al. [16]
studied the dynamic instability of shear deformable viscoelastic laminated plates by
Lyapunov exponents. Librescu and Chandiramani [56] analyzed the dynamic stability of
transversely isotropic viscoelastic plates subjected to in-plane biaxial edge load system. The
effects of transverse shear deformation, transverse normal stress and rotatory inertia effects
are considered in this study. Sahu and Datta [48] have investigated the dynamic stability of
composite plates subjected to non-uniform loads including patch and concentrated loads
using finite element method. The dynamic stability of laminated composite stiffened plates
15 or shells due to periodic in-plane forces at boundaries was discussed by Liao and Cheng [54].
The 3-D degenerated shell element and 3-D degenerated curved beam element were used to
model plates/shells and stiffeners respectively. The method of multiple scales was used to
analyze the dynamic instability regions.
16 CHAPTER-3
FINITE ELEMENT MODELING
FINITE ELEMENT MODELING
3.1 INTRODUCTION
The Finite Element Method is essentially a product of electronic digital
computer age. Though the approach shares many features common to the numerical
approximations, it possesses some advantages with the special facilities offered by the high
speed computers. In particular, the method can be systematically programmed to
accommodate such complex and difficult problems as non homogeneous materials, non linear
stress-strain behavior and complicated boundary conditions. It is difficult to accommodate
these difficulties in the least square method or Ritz method and etc. an advantage of Finite
Element Method is the variety of levels at which we may develop an understanding of
technique. The Finite Element Method is applicable to wide range of boundary value
problems in engineering. In a boundary value problem, a solution is sought in the region of
body, while the boundaries (or edges) of the region the values of the dependant variables (or
their derivatives) are prescribed. Basic ideas of the Finite Element Method were originated from advances
in aircraft structural analysis. In 1941 Hrenikoff introduced the so called frame work method,
in which a plane elastic medium was represented as collection of bars and beams. The use of
piecewise-continuous functions defined over a sub domain to approximate an unknown
function can be found in the work of Courant (1943), who used an assemblage of triangular
elements and the principle of minimum total potential energy to study the Saint Venant
torsion problem. Although certain key features of the Finite Element Method can be found in
the work of Hrenikoff (1941) and Courant (1943), its formal presentation was attributed to
Argyris and Kelsey (1960) and Turner, Clough, Martin and Topp (1956). The term “Finite
Element method” was first used by Clough in 1960. In early 1960’s, engineers used the method for approximate solution of
problems in stress analysis, fluid flow, heat transfer and other areas. A textbook by Argyris in
1955 on Energy Theorems and matrix methods laid a foundation laid a foundation for the
development in Finite Element studies. The first book on Finite Element methods by
Zienkiewicz and Chung was published in 1967. In the late 1960’s and early 1970’s, Finite
Element Analysis (FEA) was applied to non-linear problems and large deformations. Oden’s
book on non-linear continua appeared in 1972. Mathematical foundations were laid in the
1973.
17 3.2 BASIC CONCEPT OF FINITE ELEMENT METHOD
The most distinctive feature of the finite element method that separate it
from others is the division of a given domain into a set of simple sub domains, called ‘Finite
Elements’. Any geometric shape that allows the computation of the solution or its
approximation, or provides necessary relations among the values of the solution at selected
points called nodes of the sub domain, qualifies as a finite element. Other features of the
method include, seeking continuous often polynomial approximations of the solution over
each element in terms of solution and balance of inter element forces.
Exact method provides exact solution to the problem, but the limitation of
this method is that all practical problems cannot be solved and even if they can be solved,
they may have complex solution.
Approximate Analytical Methods are alternative to the exact methods, in
which certain functions are assumed to satisfy the geometric boundary conditions, but not
necessary the governing equilibrium equation. These assumed functions, which are simpler,
are then solved by any conventional method available. The solutions obtained from these
methods have limited range of values of variables for which the approximate solution is
nearer to the exact solution. Nodes of the sub domain, qualifies as a finite element. Other features of
the method include, seeking continuous often polynomial approximations of the solution over
each element in terms of solution and balance of inter element forces.
Exact method provides exact solution to the problem, but the limitation of
this method is that all practical problems cannot be solved and even if they can be solved,
they may have complex solution.
Approximate Analytical Methods are alternative to the exact methods, in
which certain functions are assumed to satisfy the geometric boundary conditions, but not
necessary the governing equilibrium equation. These assumed functions, which are simpler,
are then solved by any conventional method available. The solutions obtained from these
methods have limited range of values of variables for which the approximate solution is
nearer to the exact solution. Finite difference method, the differential equations are approximated by
finite difference equation. Thus the given governing equation is converted to a set of
18 algebraic equation. These simultaneous equations can be solved by any simple method such
as gauss Elimination, Gauss-Seidel iteration method, Crout’s method etc. the method of finite
difference yield fairly good results and are relatively easy to program. Hence, they are
popular in solving heat transfer and fluid flow problems. However, it is not suitable for
problems with awkward irregular geometry and suitable for problems of rapidly changing
variables such as stress concentration problems.
Finite element method (F.E.M) has emerged out to be powerful method
for all kinds of practical problems. In this method the solution region is considered to be built
up of many small-interconnected sub regions, called finite elements. These elements are
applied with in interpolation model, which is simplified version of substitute to the governing
equation of the material continuum property. The stiffness matrices obtained for these
elements are assembled together and boundary conditions of the actual problems are satisfied
to obtain the solution all over the body or region FEM is well-suited computer programming.
Boundary element method (B.E.M) like finite element, method is being used
in all engineering fields. In this approach, the governing differential equations are
transformed into integral identities applicable over the surface or boundary. These integral
identities are integrated over the boundary, which is divided into small boundary segments, as
infinite element method provided that the boundary conditions are satisfied, set of linear
algebraic equations emerges, for which a unique solution is obtained 3.3 FINITE ELEMENT APPROACHES
There are two differential Finite element approaches to analyze structures, namely
Force method
Displacement method
3.3.1 FORCE METHOD
The number of forces (shear forces, axial forces & bending moments) is the
basic unknown in the system of equations
19 3.3.2 DISPLACEMENT METHOD
The nodal displacement is the basic unknown in the system of equations. The
analysis of lever arm plate fixture has been using the concept of finite element method
(F.E.M). The fundamental concept of finite element method is that is that a discrete model
can approximate any continuous quantity such as temperature, pressure and displacement.
There are many problems where analytical solutions are difficult or impossible to obtain. In
such cases finite element method provides an approximate and a relatively easy solutions.
Finite element method becomes more powerful when combined rapid processing capabilities
of computers.
The basic idea of finite element method is to discretized the entire structure
into small element. Nodes or grids define each element and the nodes serve as a link between
the two elements. Then the continuous quantity is approximated over each element by a
polynomial equation. This gives a system of equations, which is solved by using matrix
techniques to get the values of the desired quantities.
The basic equation for the Static analysis is:
[K] [Q] = [F]
Where [k] = Structural stiffness matrix
[F] = Loads applied
[Q] = Nodal displacement vector
The global stiffness matrix is assembled from the element stiffness matrices.
Using these equations the model displacements, the element stresses and strains can be
determined. 20 3.4 ADVANTAGES OF FEM
The advantages of finite element method are listed below:
1. Finite element method is applicable to any field problem: heat transfer, stress
analysis, magnetic field and etc.
2. In finite element method there is no geometric restriction. The body or region
analyzed may have any shape.
3. Boundary conditions and loading are not restricted. For example, in a stress
analysis any portion of the body may be supported, while distributed or concentrated
forces may be applied to any other portion.
4. Material properties are not restricted to isotropy and may change from one element
to another or even within an element.
5. Components that have different behavior and different mathematical description
can be combined together. Thus single finite element model might contain bar, beam,
plate, cable and friction elements.
6. A finite element model closely resembles the actual body or region to be analyzed.
7. The approximation is easily improved by grading the mesh so that more elements
appear where field gradients are high and more resolution is required.
3.5 LIMITATIONS OF FEM
The limitations of finite element method are as given below:
1. To some problems the approximations used do not provide accurate results.
2. For vibration and stability problems the cost of analysis by FEA is prohibitive.
3. Stress values may vary from fine mesh to average mesh.
21 CHAPTER-4
PROBLEM FORMULATION
4.1. PROBLEM FORMULATION
The structure of a sandwich plate with a constraining layer and an
viscoelastic core is demonstrated in Fig. 4.1. Layer 3 is a pure elastic, isotropic and
homogeneous constraining layer. Layer 2 is an viscoelastic material. The base plate is
assumed to be undamped, isotropic and homogeneous and is designated as the layer 1. Before
the derivation procedures, the other assumptions used in this study must be mentioned:
1. No slipping between the elastic and viscoelastic layers is assumed.
2. The transverse displacements, w, of all points on any cross-section of the sandwich
plate are considered to be equal.
3. There exists no normal stress in the viscoelastic layer, and there exists no shear strain
in the elastic layer either. Fig.4. 1. The sandwich plate with viscoelastic core and a constraining layer
22 Fig.4. 2. Undeformed and deformed configurations of a sandwich plate
4.2 STRAIN DISPLACEMENT RELATIONS
By referring to Fig. 4.2, the strain–displacement relation of the elastic layer
can be expressed as: 2
∂ w
, ε =
−z
xi
i 2
∂x
∂x
∂ui
(2) 2
∂vi
∂ w
−z
,
ε =
yi
i 2
∂y
∂y
i = 1,3 Where εxi and εyi are the bending strains, ui and vi are the axial displacements of the midplane of layer i at the x and y directions, respectively, and zi is the distance of the mid-height
of layer i.
23 Considering the strain–displacement relation of the viscoelstic layer, the shear deformation
can be further expressed as
γ x2 =
γ y2 =
∂w ∂u2
+
,
∂x ∂z ∂w ∂v2
,
+
∂y ∂z (3)
(4) where u2 and v2 are the axial displacements in the x and y directions of the viscoelastic layer,
respectively. By referring to the geometric relationship between u1, u3, v1, v3 and ðw/ðz of the
face-plate (as shown in Fig. 4.2), it can be obtained that
h +h
u −u
∂u
2 = 1 3 ∂w + 1 3 ,
2h ∂x
∂z
h
2
2
h +h
v −v
∂v
2 = 1 3 ∂w + 1 3 ,
2h ∂y
∂z
h
2
2 (5) (6)
where h1, h2; and h3 are the thickness of layers 1, 2, and 3, respectively. Imposing the
displacement compatibility (as shown in Fig. 4.3) through the thickness, the following shear
strain in the mid-plane can be rewritten as
γ x2 =
γ y2 =
d ∂w u1 − u3
,
+
h ∂x
h
2
2
d ∂w v1 − v3
,
+
h ∂y
h
2
2 Where
h
h
d = 1 + h + 3 . 2 2
2
24 (7)
(8)
Fig. 4.3. Compatibility relation of systems.
The strain energy associated with the normal strain in the elastic layer can be obtained: 1
2 + ε 2 ) dv,
Vi = ∫ Di (ε xi
yi
2V
i = 1, 3,
(9)
Where Di is the differential operator matrix and listed in the following discussion in detail.
Then the strain energy of viscoelastic layer is obtained as follows: V = ∫ G (γ 2 + γ 2 ) dv,
2 V 2 x2
y2
Where G2 denotes the shear modulus of the viscoelastic fluid layer. 25 (10)
Let V be the total strain energy of the sandwich plate: then
V = V +V +V .
1 2 3 (11) The kinetic energy of the sandwich plate has the following three parts
1.The kinetic energy associated with the axial displacement
1
T = ∫∫ A [ ρ h (u 2 + v 2 ) + ρ h (u 2 + v 2 )]dxdy.
1
11 1 1
3 3 3 3
2
(12)
2.The kinetic energy associated with transverse displacement
1
T = ∫∫ A ( ρ h + ρ h + ρ h ) w 2 dxdy. 2 2
11
2 2
3 3
(13)
3.The kinetic energy associated with the rotation of the viscoelastic layer: 1
T = ∫∫ A I (γ 2 + γ 2 )dxdy, , (14) 3 2
2 x2
y2
Where I2 is the mass moment of inertia of the viscoelastic layer
Let T be the total kinetic energy of the sandwich plate, then
T = T +T +T .
1 2 3 (15) 4.3 FINITE ELEMENT APPROACH
The plate elements used in this study are two-dimensional element bounded
by four nodal points. The plate element is shown in Fig.4. 4. Each node has seven degrees of
freedom to describe the longitudinal displacements, transverse displacements, and slopes of
the sandwich plate. The transverse displacement, longitudinal displacement can be expressed
in terms of a nodal displacement vector and a shape function vector:
w( x, y, t ) = N w ( x, y ){q (t )},
(16)
u ( x, y, t ) = Nui ( x, y ){q(t )},
i
(17)
vi ( x, y, t ) = Nvi ( x, y ){q(t )}, i = 1,3,
(18)
T
where q (t ) = [u , v , u , v , wi , wxi , w yi ] , i = 1, 2,3, 4. 1i 1i 3i 3i
26 Fig.4. 4. A sandwich plate element with four end nodes and seven DOF per node.
N w ( x, y ), Nui ( x, y ), N ( x, y ), N ( x, y ), N ( x, y ) are the shape functions of the plate
u3
v1
v3
element.
4.3.1 ELEMENT SHAPE FUNCTIONS
N ( x, y ) = [ w1, w2, w3, w4],
w
(19)
x
y
x x2 y y2 1
x
y
y2 1
x
y
x2
1
w1 = [0,0,0, 0, (1 − )(1 − )(2 − −
− − ), b(1 − )(1 − )(1 − ), a(1 − )(1 − )(1 − )],
8
a
b
a a 2 b b2 8
a
b
a
b
b2 8
a2
1
x
y
x x2 y y 2 1
x
y
y2 1
x
y
x2
w2 = [0,0, 0,0, (1 + )(1 − )(2 + −
− − ), b(1 + )(1 − )(1 − ), a(1 + )(1 − )(1 − )],
8
a
b
a a 2 b b2 8
a
b
a
b
b2 8
a2
1
x
y
x x2 y y 2 1
x
y
y2 1
x
y
x2
w3 = [0,0, 0, 0, (1 + )(1 + )(2 + −
+ − ), b(1 + )(1 + )(1 − ), a(1 + )(1 + )(1 − )],
8
a
b
a a 2 b b2 8
a
b
a
b
b2 8
a2
1
x
y
x x2 y y 2 1
x
y
y2 1
x
y
x2
w4 = [0,0, 0,0, (1 − )(1 + )(2 − −
+ − ), b(1 − )(1 + )(1 − ), a(1 − )(1 + )(1 − )],
8
a
b
a a 2 b b2 8
a
b
a
b
b2 8
a2
N ( x, y ) = [u11, u12, u13, u14],
(20)
u1
27 N ( x, y ) = [u31, u32, u33, u34],
u3
(21)
Where
1
x
y
u11 = [ (1 − )(1 − ), 0, 0, 0, 0, 0, 0], 4
a
b
1
x
y
u 31 = [0, 0, (1 − )(1 − ), 0, 0, 0, 0], 4
a
b
1
x
y
u12 = [ (1 + )(1 − ), 0, 0, 0, 0, 0, 0], 4
a
b
1
x
y
u 32 = [0, 0, (1 + )(1 − ), 0, 0, 0, 0], 4
a
b
1
x
y
u13 = [ (1 + )(1 + ), 0, 0, 0, 0, 0, 0], 4
a
b
1
x
y
u 33 = [0, 0, (1 + )(1 + ), 0, 0, 0, 0], 4
a
b
1
x
y
u14 = [ (1 − )(1 + ), 0, 0, 0, 0, 0, 0], 4
a
b
1
x
y
u 34 = [0, 0, (1 − )(1 + ), 0, 0, 0, 0],
4
a
b
N ( x, y ) = [v11, v12, v13, v14],
v1
(22)
N ( x, y ) = [v31, v32, v33, v34],
v3
(23)
Where
1
x
y
v11 = [0, (1 − )(1 − ), 0, 0, 0, 0, 0], 4
a
b
x
y
1
v31 = [0, 0, 0, (1 − )(1 − ), 0, 0, 0], a
b
4
1
x
y
v12 = [0, (1 + )(1 − ), 0, 0, 0, 0, 0], 4
a
b
1
x
y
v32 = [0, 0, 0, (1 + )(1 − ), 0, 0, 0], 4
a
b
1
x
y
v13 = [0, (1 + )(1 + ), 0, 0, 0, 0, 0], 4
a
b
1
x
y
v33 = [0, 0, 0, (1 + )(1 + ), 0, 0, 0], 4
a
b
1
x
y
v14 = [0, (1 − )(1 + ), 0, 0, 0, 0, 0], 4
a
b
1
x
y
v34 = [0, 0, 0, (1 − )(1 + ), 0, 0, 0] . 4
a
b
28 The strain energy and kinetic energy derived in the above section can be rewritten in terms of
nodal displacement variables as follows:
1
V = {q (t )}T ([ K ] + [ K ] + [ K ] + [ K ] + [ K5 ]){q (t )},
1
2
3
4
2
(24)
Where
T
[ K ] = h ∫∫ [ N ] [ D ][ N ]dA,
1
1A 1
1p 1
(25)
T
[ K ] = ∫∫ [ N ] [ D ][ N ]dA,
2
b
1b
b
A
(26)
T
[ K ] = h ∫∫ [ N ] [ D ][ N ]dA,
3
3A 3
3p
3
(27)
T
[ K ] = ∫∫ [ N ] [ D ][ N ]dA,
4
b
b
3b
A
(28)
T
[ K5 ] = G h ∫∫ [ N g ] [ N g ]dA,
2 2A
(29)
Where
⎡
⎤
Nui, x
⎢
⎥
⎢
⎥, [ Ni ] =
N vi, y
⎢
⎥
⎢N
⎥
⎢⎣ ui, y + Nvi, x ⎥⎦
⎡ E
xi
⎢
⎢1 − μ xi μ yi
⎢
⎢ μ yi E xi
[ Dip ] = ⎢
⎢1 − μ xi μ yi
⎢
⎢
0
⎢
⎢⎣
μ xi E yi
1 − μ xi μ yi
E yi
1 − μ xi μ yi
0
⎤
⎥
⎥
⎥
⎥
⎥, 0
⎥
⎥
(1 − μ yi ) E xi ⎥
⎥
2(1 − μ xi μ yi ) ⎥
⎦
⎡ N w, xx ⎤
⎢
⎥
⎡ N ⎤ = ⎢ N w, yy ⎥ , ⎣ b⎦ ⎢
⎥
⎢⎣ 2 N w, xy ⎥⎦
29 0
⎡ E
xi
⎢
⎢1 − μ xi μ yi
⎢
⎢ μ yi E xi
[ D ] = Ii ⎢
ib
⎢1 − μ xi μ yi
⎢
⎢
0
⎢
⎢⎣
μ xi E yi
1 − μ xi μ yi
E yi
1 − μ xi μ yi
0
⎤
⎥
0
⎥
⎥
⎥
⎥,
0
⎥
⎥
(1 − μ yi ) E xi ⎥
⎥
2(1 − μ xi μ yi ) ⎥
⎦
i = 1,3
⎡ Nu1 − Nu3
⎤
+ N w, x ⎥
⎢
d
⎥ ⎡Ng ⎤ = d ⎢
⎣
⎦ h ⎢N −N
⎥
v3 + N
2 ⎢ v1
w, y ⎥
d
⎣
⎦
Where Ei, vi, Ii denote Young’s modulus, Poisson’s ratio, and the area moment of inertia of
the ith layer.
In addition, the kinetic energy of the sandwich plate is
1
T = {q (t )}T ([ M ] + [ M ] + [ M ] + [ M ]){q (t )},
1
2
3
4
2
(30)
Where
[ M ] = ∫∫ ( ρ h + ρ h + ρ h )[ N w ]T [ N w ]dxdy ,
1 A 11
2 2
3 3
(31)
[ M ] = ∫∫ ρ h ([ N ]T [ N ] + [ N ]T [ N ])dxdy,
u1
v1
v1
2 A 1 1 u1
(32)
[ M ] = ∫∫ ρ h ([ N ]T [ N ] + [ N ]T [ N ]) dxdy,
u3
v3
v3
3 A 3 3 u3
(33) [ M ] = ∫∫ I [ N g ]T [ N g ]) dxdy.
4 A 2
(34)
Considering the situation of a sandwich plate element with a periodic load. The work done by
the periodic load can be expressed as
W=
2
1
⎛ ∂w ⎞
P
(
t
)
∫∫
⎜ ⎟ dxdy.
2 A
⎝ ∂x ⎠
(35)
30 Substitute the interpolation function into the above equation, and we can obtain that
1
W = {q (t )T P (t )[ K ge ]q (t )},
2
(36)
Where
T
⎡ ∂N ⎤ ⎡ ∂N ⎤
[ K ge ] = ∫∫ A ⎢ w ⎥ ⎢ w ⎥ dxdy.
⎣ ∂x ⎦ ⎣ ∂x ⎦
4.4 EQUATION OF MOTION
According to the Hamilton’s principle, we have
δ t 2 (T − V + W )dt = 0.
∫t1
(37) By substituting the strain energy, kinetic energy, and the work done by the load force into the
Hamilton’s principle, the governing equation for the sandwich plate element is obtained as
Follows
⎡⎣ M e ⎤⎦ {q(t )} + ([ K e ] − P(t )[ K ge ]){q (t )} = 0,
(38)
Where
[ K e ] = [ K ] + [ K ] + [ K ] + [ K ] + [ K5 ],
1
2
3
4
(39) And
[ M e ] = [ M ] + [ M ] + [ M ] + [ M ].
1
2
3
4 (40) Assembling mass, elastic stiffness and geometric stiffness matrices of individual element, the
equation of motion for the sandwich plate is written as
[M ]{Δ}+ [K ]{Δ}− P (t ) [K g ]{Δ}=
(41)
0
Where {Δ} is the global displacement matrix.
31 The static component Ps and dynamic component Pt of the load P(t), can be represented in
terms of P* as Ps = α P* and Pt=β P* and, where P*=D/b/a2 and D = ∑ E ( 2i −1) I ( 2i −1) /(1 − υ i ) .
2
i =1,3
Hence substituting P(t ) = α P * + β P * cos Ω t , where α and β are static and dynamic load
factors respectively.
Fig. 4.5. A plate subjected to an in-plane dynamic load. Substituting P(t), eq.(41) becomes
[M ]{Δ }+ [K ]{Δ}− ( P
s
+ Pt cos Ω t) [K g ]{Δ }= 0
[M ]{Δ}+ ( [K ]− P [K ] ){Δ}− P
s
g S
t
cos Ωt [K g ]t {Δ}= 0
(42)
(43)
Where the matrices [K g ]s and [K g ]t reflect the influence of Ps and Pt respectively. If the static
and time dependent component of loads are applied in the same manner, then
[K g ]s = [K g ]t = [K g ] .
[M ]{Δ }+ [K ]{Δ }− β
P ∗ cos Ω t) [K g ]{Δ }= 0
32 (44)
[ ]
[ ]
Where K = [K ] − Ps K g
(45)
The global displacement matrix {Δ} can be assumed as
{Δ} = [Φ ]{Γ}
(46)
Where [Φ ] is the normalized modal matrix corresponding to
[M ]{Δ}+ [K ]{Δ} = {0}
(47)
and {Γ} is a new set of generalized coordinates .
Substituting eq.46 in eq.44, eq.44 is transformed to the following set of Nc coupled Mathieu
equations.
( )
Nc
m + ωm2 Γm + β P∗ cos Ω t ∑bmnΓn = 0
Γ
n=1
m =1,2,..........Nc ,
[ ]
(48)
Where (ω m2 ) are the distinct eigen values of [M ]−1 K and bmn are the elements of the
complex matrix [B ] = − [Φ ] −1 [M ] −1 [K g ] [Φ ] and
ω m = ω m. R + i ω
m. I
,
bmn = bmn. R + i bmn. I and i = - 1
33 4.5 REGIONS OF INSTABILITY
The boundaries of the regions of instability for simple and combination resonance are
obtained by applying the following conditions[78] to the eq.48.
Case (A): Simple resonance
The boundaries of the instability regions are given by
1/ 2
⎤
Ω
1 ⎡ β 2 (b2 μμ.R +b2 μμ.I )
2
− ωμ.R 〈 ⎢
−
16
ω
μ
.I ⎥
2ω0
4⎣
ω 2 μ.R
⎦
Where ω 0 = D ma 4 ,
μ =1, 2.........Nc
(49)
ωμ.R =ωμ.R / ω 0 , ωμ.I =ωμ.I / ω 0 , m is mass per unit length of the
sandwich plate.
When damping is neglected, the regions of instability are given by
1 ⎡ β (bμμ . R ) ⎤
Ω
− ω μ .R 〈 ⎢
⎥
2ω 0
4 ⎣⎢ ω μ .R ⎦⎥
μ = 1,2,......N c
(50)
Case (B): Combination resonance of sum type
__
Ω 1 __
1 (ω μ.I + ω υ.I )
− (ω μ.R + ωυ.R 〈
T 2ω0 2
8
(ω μ.I ω υ.I )1 / 2
μ ≠ν , μ,ν = 1,2,...Nc .
⎡ β 2 (bμυ.R bυμ.R + bμυ.I bυμ.I )
⎤
−16ωμ.I ωυ.I ⎥
⎢
ωμ.R ωυ.R
⎢⎣
⎥⎦
1/ 2
(51)
When damping is neglected
2
__
1 __
1 ⎡ β ( b μυ . R bυμ . R ) ⎤
Ω
− (ω μ . R + ω υ . R 〈 ⎢
⎥
2ω 0 2
4 ⎢⎣
ω μ .R ω υ .R
⎥⎦
1/ 2
μ ≠ ν , μ ,ν = 1, 2 ,... N c
(52)
Case (C): Combination resonance of difference type
The boundaries of the regions of instability of difference type are given by
2
⎤
Ω 1 __ __
1 (ω μ.I −ωυ.I ) ⎡ β (bμυ.I bυμ.I −bμυ.Rbυμ.R )
− (ωμ.R −ωυ.R) 〈
−16ωμ.Iωυ.I ⎥,
⎢
2ω0 2
8
ωμ.R ωυ.R
⎦⎥
(ω μ.I ωυ.I )1/ 2 ⎣⎢
ν 〉 μ , μ ,ν = 1,2,...N c
34 (53)
When damping is neglected, the unstable regions are
1/ 2
2
__
Ω 1 __
1 ⎡ β (bμυ.R bυμ.R ) ⎤
− (ω μ.R − ωυ.R ) 〈 ⎢ −
⎥
2ω0 2
4 ⎢⎣
ωμ.R ωυ.R ⎥⎦
ν 〉μ , μ,ν =1,2,...Nc .
35 (54)
CHAPTER-5
RESULTS & DISCUSSION
The dynamic stability problems of a sandwich plate with viscoelastic core and
constrained layer are studied by finite element method. To validate the proposed algorithm
and calculations, comparisons between the present results and the results of existing models
are made first. The solutions of natural frequencies and loss factors of a simply supported
sandwich plate with a viscoelastic layer are obtained. The numerical results are compared
with those obtained by Lall et al. [52] and Jia-Yi Yeh, Lien-Wen Chen [47] in Tables 1 and 2,
respectively. The solutions solved by present model are shown to have a good accuracy. A
good agreement can be observed in the above results with different geometry. The geometrical and physical parameters of the sandwich plate are as follows: a=0.3048m;
b=0.3480 m;
h1=h3=0.762mm;
E1=E3=6.89x1010 N/m2;
h2=0.254mm;
G2 =0.896x106 N/m2;
ρ2=999kg/m3;
ρ1=ρ3=2740 kg/m3;
µ=0.3;
η=0.5
Table 1
Comparison of Resonant frequency parameters and Modal loss factors calculated from
present analysis with those of reference [52] at h2/h1=0.33, η=0.5
Ref. [52]
Mode
natural
Frequency(rad/sec)
present
loss factor
Natural
Frequency(rad/sec)
loss factor
1.
59.05
0.206
58.69
0.201
2.
113.67
0.213
113.75
0.211
3.
128.89
0.207
129.16
0.208
4.
175.76
0.188
175.46
0.189
5.
196.67
0.179
193.79
0.183
36 Table 2
Comparison of Resonant frequency parameters and Modal loss factors calculated from
present analysis with those of reference [47] at h2/h1=10, η=0.5
Ref. [47]
Mode natural
Frequency(rad/sec)
1. 2. 3. 4. present
loss factor
natural
Frequency(rad/sec)
loss factor
975.17
0.044
972.89
0.044
2350.79
0.019
2346.45
0.019
2350.79
0.019
2346.45
0.019
3725.33
0.012
3711.90
0.012
The variation of the fundamental buckling load parameter (Pb), defined as the
ratio of fundamental buckling load to P*, with core thickness parameter (h2/h1) is shown in
figure-11. It is seen from the figure that for core thickness parameter 0.5 to 5.0 there is a
linear increase in fundamental buckling load parameter with increase in core thickness
parameter.
Figure-12 shows the effect of core thickness parameter on fundamental
frequency parameter (f). The fundamental frequency parameter is defined as the ratio of
fundamental frequency of the sandwich plate to ωo. The variation of fundamental frequency
parameter with core thickness parameter shows the similar trend as those for fundamental
buckling load.
The variation of fundamental system loss factor (η) with core thickness
parameter is shown in figure-13. The fundamental loss factor increases with increase in core
thickness parameter. It is revealed from the figure that the rate of increase of fundamental
loss factor with core thickness parameter is very high for low values (0.01 to 0.5) and for
higher values of h2/h1 though the η increases the rate of increase is comparatively less 37 Figure-14 shows the effect of shear parameter (g) on fundamental buckling
2
⎛a⎞
⎜ ⎟
(h2 / h1) ⎜⎝ h1 ⎟⎠
load. The shear parameter is defined as, g = G′
⎛2
⎜
⎜E
⎝
⎞
⎟ . It can be seen that with increase
⎟
⎠
in shear parameter the fundamental buckling load increases almost linearly for pinned-pinned
end condition.
Figure-15 shows the effect of shear parameter on fundamental frequency
parameter (f). With increase in g the fundamental frequency parameter increases.
Figure-16 shows the effect of shear parameter on system fundamental loss
factor (η). The system loss factor increases with increase in shear parameter. But for higher
values of g the effect becomes less dominant.
Figure-17 shows the effect of core thickness parameter on the first two
instability regions of simple resonance of the plate. It is seen that increase in core thickness
parameter shifts the occurrence of instability regions to higher excitation frequency and their
areas also decreases with increase in thickness ratio. So the increase in thickness ratio
improves the stability behaviour of the plate.
In figures-18 the effect of static load factor (α) on the stability behavior of the
plate is shown. The figure show the first two instability regions of simple resonance for α= 0.0
and 0.5. It is seen that increase in static load factor has destabilizing effect, because the
instability regions move to lower frequency of excitation and their areas also increase with
increase in static load factor.
The effect of shear parameter on the stability behavior of the plate has been
shown in figure-19. It can be seen that in addition to two instability regions of simple
resonance, the instability region of combination resonance (ω1 + ω2) type also exist. It can also
be seen from the figure that with increase in shear parameter with constant thickness ratio
improves the stability of the plate by shifting them to higher frequency of excitation and there
is also marginal reduction in their areas.
38 .
.
.
39 40 41 42 . 43 CHAPTER-6
CONCLUSION & SCOPE FOR
FUTURE WORK
6.1 CONCLUSION:
The present work investigates the dynamic stability of a sandwich plate with
simply supported, end condition. It is found that for plate with simply supported boundary
conditions the fundamental frequency, fundamental buckling load increase with increase in
core thickness parameter (h2/h1) for h2/h1 ≥ 0.5. For h2/h1 < 0.5, the fundamental frequency,
fundamental buckling load decrease with increase in core thickness parameter. The system
fundamental loss factor increases with increase in thickness ratio for 5.0 ≥ h2/h1 ≥0.01. The
fundamental buckling load and fundamental frequency increase with increase in shear
parameter. The fundamental system loss factor has an increasing tendency with increase in
shear parameter. The increase in core thickness ratio and shear parameter has stabilizing
effect. Whereas increase in static load factor has a destabilizing effect.
.
6.2 SCOPE FOR FUTURE WORK
The following works may be carried out as an extension of the present work.
1. Stability of sandwich plates with different boundary conditions.
2. Stability of sandwich plates of different cross sections like I section, trapezoidal
section etc.
3. Stability of sandwich plates of different cores like continuous corrugated-core, zedcore, channel-core and truss core
4. Stability of multilayered sandwich plates.
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