Dynamic Stability Of Layered Beams With Bolted Joints Under Parametric Excitation

Dynamic Stability Of Layered Beams With Bolted Joints Under Parametric Excitation
Dynamic Stability Of Layered Beams With Bolted Joints
Under Parametric Excitation
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Mechanical Engineering
By
Shivakumar Satyanarayan & Amarendra Kumar Patel
Department of Mechanical Engineering
National Institute of Technology
Rourkela
2007
1
Dynamic Stability Of Layered Beams With Bolted Joints
Under Parametric Excitation
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Mechanical Engineering
By
Shivakumar Satyanarayan & Amarendra Kumar Patel
Under the guidance of:
Prof S.C.Mohanty
Department of Mechanical Engineering
National Institute of Technology
Rourkela
2007
2
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled “Dynamic Stability of Layered Beams With Bolted Joints
Under Parametric Excitation.’’ submitted by Shivakumar Satyanarayan, Roll No: 10303013
and Amarendra Kumar Patel, Roll No: 10303038 in the partial fulfillment of the requirement for
the degree of Bachelor of Technology in Mechanical Engineering, National Institute of
Technology, Rourkela, is being carried out under my supervision.
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
Prof S.C.Mohanty
Department of Mechanical Engineering
National Institute of Technology
Rourkela-769008
3
ACKNOWLEDGEMENT
We avail this opportunity to extend our hereby indebtedness and sincere thanks to our guide Prof.
S. C. Mohanty, Department of Mechanical Engineering, for his valuable guidance, constant
encouragement and kind help at different stages for the execution of this dissertation work.
We would also like to thank Mr. Sadik Shaik for his assistance during the entire project.
We also express our sincere gratitude to Dr. B. K. Nanda, Head of the Department, Mechanical
Engineering, for providing valuable departmental facilities.
Submitted by:
Shivakumar Satyanarayan
Amarendra Kumar Patel
Roll No: 10303013
Mechanical Engineering
National Institute of Technology
Rourkela
Roll No: 10303038
Mechanical Engineering
National Institute of Technology
Rourkela
4
ABSTRACT
Time dependent excitation appearing as a variable co-efficient in the governing equation of motion
is called parametric excitation. Dynamic stability of elastic systems deal with the study of
vibrations, induced by pulsating loads that are parametric with respect to certain forms of
deformations. An important
aspect in the analysis of parametric dynamic system is the
establishment of the regions in the parametric space in which the system becomes unstable; these
regions are known as regions of dynamic instability. The boundary separating the stable zones
from unstable zones are called stability boundaries and the plot of these boundaries on the
parametric space is called the stability diagram. Beams are the simplest structural members in any
machine, mechanism or large structure. Layered structures are gaining importance in the present
day because of their good damping capacity characteristics. The objective of the proposed research
work are to carry out experimental investigations to study the dynamic stability of layered
structured beams with bolted joints with different configurations of constraining layers. It is to
investigate the following aspects of the dynamic stability of beams.
Experimental has been carried out to validate the theoretical findings. The results are in complete
match with the theoretical one.
5
NOMENCLATURE
Although all the principal symbols used in this thesis are defined in the text as they occur, a list of
them is presented below for easy reference. On some occasions, a single symbol is used for
different meanings depending on the context and thus uniqueness is lost. The contextual
explanation of the symbol at its appropriate place of use is hoped to eliminate the confusion.
English symbols
A
b
E
h
I
[K ]
Cross-sectional area of the uniform beam.
Width of the beam.
Young’s modulus of the beam material.
Height of the beam.
The second moment of inertia.
Global elastic stiffness matrix.
[K ]
Global geometric stiffness matrix.
l
L
Length of an element.
Length of the beam.
Global mass matrix.
g
[M ]
[M]
Element mass matrix.
P(t)
Ps
Pt
T
t
Axial periodic load.
Static component of the periodic load.
Time dependent component of the periodic load.
Kinetic energy of the beam.
Time coordinate.
Elemental kinetic energy.
(e)
T
U
(e)
U
v
w
x
R
[C]
Total strain energy of the beam.
Elemental potential energy.
Transverse displacement of the beam.
Transverse displacement of the beam.
Axial coordinate.
Modal co-ordinate
Damping Matrix
Greek symbols
α
β
γ
ζ
ρ(2k-1)
Static load factor.
Dynamic load factor.
Shear strain.
= x/l
Mass density of the (2k-1)th constraining elastic layer.
6
Ω
Excitation frequency of the dynamic load component.
Superscripts
(e)
Element
7
CONTENTS
a) Abstract
i
b) Nomenclature
ii
b) List of Figures
v
1 Introduction
1-2
2
3-4
Literature Review and Present Work:
3 Theoretical analysis
3.1 Formulation of the problem
3.2 FEM Analysis.
5-7
7-10
4. Experimental work
4.1 Introduction
11
4.2 Description of the experimental set up
11-13
4.3 Preparation of specimen
14-15
4.4 Experimental procedure
16
5. Results and Conclusions
6. References
17-19
20-21
8
FIGURE INDEX
SL NO.
FIGURE NO.
FIGURE
PAGE NO
1.
4.1
Schematic diagram of test
set up.
12
2.
4.2
photograph of set up
13
3.
4.3
attachments for clamped end
13
4.
4.4
photograph of prepared specimen
15
5.
5.1
plot of frequency ratio vs. dynamic
load factor for ξ=0.1
17
6.
5.2
plot of frequency ratio vs. dynamic
load factor for ξ=0.3
18
9
CHAPTER
1
INTRODUCTION
10
INTRODUCTION
Mechanical joints can have a significant effect on the dynamics of structures that contain
them. Bolted or riveted joints cause local stiffness and damping changes and are often the
primary source of energy dissipation and damping in assembled structures. In many such
structures, damping due to relative interfacial joint motion can account for as much as 90% of the
total . Thus, accurate prediction of dynamic response of assembled structures to external excitation
often hinges on efficacious modeling of the effect of the joint on structural behavior. Considerable
effort has being expended attempting to characterize the non-linear behavior of structures
containing joints. Liu and Ewins presented a ‘‘more general and more practical’’ method to extract
the ‘‘effective’’ mass, stiffness and damping parameters of a joint element from measured
frequency response functions. Note that a common characteristic of all of the above is that no
parametric model of the joint is required.
The aim is to develop predictive dynamic parametric models of mechanical joints for reliable
structural response analysis. The successful modeling of joints depends on understanding and
reproducing the basic physics associated with a jointed interface. Various studies have identified
micro- and macro-slip occurring along the interface as the source of change of interface stiffness
and energy dissipation, which constitutes the hysteresis mechanism of joints. Typically, the normal
interface pressure across a dynamically loaded joint is not uniformly distributed, and micro-slip
first occurs in regions where the contact pressure is insufficient to prevent it. The interface is, thus,
divided into zones of ‘‘stick’’ and ‘‘slip’’. As the magnitude of the transmitted load increases, slip
zones enlarge and coalesce, resulting in macroslip and the familiar hysteretic force–displacement
joint characteristic. This has been demonstrated experimentally for shear lap joints, loaded axially
and torsionally, by Gaul and Lenz .The notion that the non-linear hysteresis behavior of a joint is
the result of micro- and macro slip occurring along its interface was motivation for developing
detailed finite element joint models, which requires solving a contact problem at the interface
using an extremely fine mesh.
While it is possible to realistically model the behavior of the joint in this manner, the resulting
joint model is impractical for dynamic analysis of an arbitrary structure containing the joint. In
fact, it is computationally prohibitive and likely to remain so for some time. Lee et al. proposed a
technique to reduce the number of degrees of freedom of a detailed finite element contact model.
Chen and Deng noted that finite element analysis can provide a flexible and reliable tool for
11
understanding and characterizing the non-linear damping behavior of structural joints and
proposed to use the finite element method to generate numerical data for a typical slip joint. A
lumped-parameter model of small dimension to simulate the non-linear dynamic behavior of a
joint and, particularly, its effect upon the surrounding structure has long been deemed desirable. It
has been common to represent the friction occurring at contact interfaces by a Coulomb friction
model. However, a single Coulomb friction element is only capable of describing either the full
slip or full stick situation. Menq et al. developed a one-dimensional, physically motivated microslip model that allows partial slip on the friction interface; however, this model is not suited to
response analysis of complex structures. At present, non-linear, reduced order, full-joint models
that can be effectively used in structural dynamic response analysis do not exist.
12
CHAPTER
2
LITERATURE REVIEW AND PRESENT WORK
13
2. LITERATURE REVIEW AND PRESENT WORK:
The environmental interaction with the deformable continuum is usually represented by means
of body forces and surface tractions. When the body deforms, dead loads acting on the deformable
bodies retain their magnitude as well as their initial direction. In general the forces acting on the
body may not always be dead loads. The environmental mechanical action on a body may be due
to forces, which are motion and/or time dependent. Such forces are instationary in nature. When
these instationary external excitations are parametric with respect to certain form of deformation of
the body, they appear as one of the coefficients in the homogeneous governing differential
equation of motion of the system. Such systems are said to be parametrically excited and the
associated instability of the system is called parametric resonance. Whereas in case of forced
vibration of the systems, the equation of motion of the system is inhomogeneous and the
disturbing forces appear as in homogeneity. In parametric instability the rate of increase in
amplitude is generally exponential and thus potentially dangerous, while in typical resonance due
to external excitation the rate of increase in response is linear. More over damping reduces the
severity of typical resonance, but may only reduce the rate of increase during parametric
resonance. Parametric instability occurs over a region of parameter space and not at discrete
points. It may occur due to excitation at frequencies remote from the natural frequencies.
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 freestream 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. A few examples include those associated with electromagnetic and
aeronautical instruments, vibratory conveyers, saw blades, belt drives and robot manipulators etc.
The system can experience parametric instability, when the excitation frequency or any integer
multiple of it is twice the natural frequency that is to say
m nω2==Ω
m = 4,3,2,1,2ω
14
The case nω2=Ω is known to be the most important in application and is called main or principal
parametric resonance.
One of the main objectives of the analysis of parametrically excited systems is to establish the
regions in the parameter space in which the system becomes unstable. These regions are known as
regions of dynamic instability. The boundary separating a stable region from an unstable one is
called a stability boundary. Plot of these boundaries on the parameter space is called a stability
diagram.
Many machines and structural members can be modeled, as beams with different geometries, like
beams of uniform cross-section, tapered beams and twisted beams. These elements may have
different boundary conditions depending on their applications. Advances in material science have
contributed many alloys and composite materials having high strength to weight ratio. However
during manufacturing of these materials, inclusion of flaws affects their structural strength. These
flaws can be modeled as localized damage. The modulus of elasticity of the material is greatly
affected by the temperature. In high-speed atmospheric flights, nuclear engineering applications,
drilling operations and steam and gas turbines, the mechanical and structural parts are subjected to
very high temperature. Most of the engineering materials are found to have a linear relationship
between the Young’s modulus and temperature [44,114]. Geometry of the beam, boundary
conditions, localized damage and thermal conditions have greater effect on the dynamic behavior
of the beams and hence need to be studied in depth.
15
CHAPTER
3
THEORETICAL ANALYSIS
16
3. THEORETICAL ANANLYSIS
3.1
Formulation of the problem:
Beams with end conditions such as Fixed-Fixed
The matrix equation for free vibration of axially loaded discretised system is
[ M ]  q  + [ Ke]{ q} − P[ S ]{ q} =
••
 
0
---------------------(1)
Where, { q} = Assemblage nodal displacement vector [vi ,θ i ,ν j ,θ
The dynamic load P(t) is periodic and
j
]T.
can be expressed in the form
P = P0 + Pt cos Ω t
Where Ω is the disturbing frequency, Po the static and Pt the amplitude of time dependent
component of the load, can be represented as the fraction of the fundamental static buckling load
P* . Hence substituting
P = α P* + β P *cos Ω t , with α and β as static and dynamic load factors respectively.
The equation (1) becomes
[ M ]  q  +  [ K e ] − α
••
 

P * [ S S ] − β P * cos Ω t [ S t ]  { q} = 0 ----------------(2)

Where the matrices [S S] and [St] reflect the influence of Po and Pt respectively. If the static and
time dependent component of loads are applied in the same manner, then [SS] = [St] = [ S ]
Equation (2) represents a system of second order differential equations with periodic
coefficients of the Mathieu-Hill type. The development of region of instability arises from
Floquet’s theory which establishes the existence of the periodic solutions of period T and 2T,
17
where T =
2π
. The boundaries of the primary instability region with period 2T are of practical
Ω
importance (Bolotin–5) and the solution can be achieved in the form of trigonometric series.
q (t ) =
Kθ t
Kθ t 

∑  { a } sin 2 + {b } sin 2 
∞
K=1
k
k
--------------------------(3)
Putting this in equation (2) and if only first term of the series is considered, equating
coefficients of sin
θ t
θ t
and cos
, the equation becomes
2
2
2


[ K e ] − (α ± β / 2) P * [ S ] − Ω [ M ]  { q} = 0
4


-------------------------------(4)
Equation (4) represents an eigenvalue problem for unknown values of α , β and P* . This
equation gives two sets of eigenvalues ( Ω ) bounding the regions of instability due to presence of
plus and minus sign.
Also this equation (4) represents the solution to a number of related problems.
For free vibration: α = 0, β = 0 & λ =
(i)
Ω
2
Equation (4) becomes
([ K ] − λ [ M ] ){ q} = 0
2
e
(ii)
---------------------(5)
For vibration with static axial load:
β = 0, α ≠ 0, λ =
Ω
2
Equation (4) becomes
 [ K ] − α P * [ S ] − λ 2 [ M ]  { q} = 0
 e

--------------------------(6)
18
(iii)
For static stability: α = 1, β = 0, λ =
Ω
2
Equation 4 becomes
([ K e ] −
(iv)
P * [ S ] ){ q} = 0
---------------------------------------(7)
For dynamic stability, when all terms are present
 Ω 

Let Ω = 
ω1
ω
1
Where ω1 is the fundamental natural frequency as obtained from the solution of equation
(5). Equation (4) then becomes


β 

 [ K e ] −  α ±  P * [ S ] 
2



 Ω 
Where, θ =  
ω1
{ q} = θ
ω
2
1
4
[ M ] { q} ------------------------------(8)
2
The fundamental natural frequency ω1 and critical static buckling load P* can be solved
using the equations (5) and (7) respectively. The regions of dynamic instability can be determined
from equation (8).
3.2 FEM Analysis
The input file, stiffmat1.m, bolts.m, and boundary.m are executed in succession within Matlab to
form the model mass and stiffness matrices. These matrices can then be used to describe an
undamped eigenproblem [16,23] that will give the free vibration characteristics of the structure:
([K]−ω [M][x])= 0
2
(1)
where [K] is the stiffness matrix, [M] is the mass matrix, and ω 2 is an eigenvalue of the
problem. The eigenvalues give the undamped natural frequencies of the structure, while the
eigenvectors describe the mode shapes. For lightly damped structures, the undamped natural
19
frequencies are very close to the actual (damped) values.
FEMs can also be used to generate analytical FRFs. This can be done a number of ways. One
way is to use the model to generate time responses to a given excitation. The time responses can
then be used to generate FRFs. The process is as follows:
The mass and stiffness matrices described by the FEM represent a system of differential equation
of the form:
[ M] { } + [ K ] { x } = [ F ]
where
(2)
{ o} is a vector of nodal accelerations, {x} is a vector of nodal displacements, and {F}
is
a vector of forces applied at each node. This problem is accompanied by a set of initial positions
{x} and initial velocities {vo}. In general, the system will be coupled, which makes a solution
difficult. The modal analysis procedure is a coordinate transformation where all physical
coordinates {x} are transformed to modal coordinates {r}. This results in a system similar to (2):
(3)
where {r} is the transformed version of {x} and [Λ ] is a diagonal matrix containing the values
ω n2
The transformation to modal coordinates is accomplished through transformation matrices
defined by [K] and [M]. First, the matrix [M]−1 / 2 is calculated. This matrix is then used to
transform the stiffness matrix:
(4)
An eigensolution is then performed on K and the eigenvectors are arranged in order as
columns in a new matrix [P]:
[P] = [v1v2v3.....vn ]
(5)
The transformation matrices can then be calculated as
[S] = [M]-1/2[P]
20
[S] −1 = [P] T [ M] − 1
(7)
The transformation to modal coordinates can then be accomplished as
{r} = [S] {x}
(8)
−1
The transformation described by (8) gives the system in (3), in which the differential equations
are not coupled. This means that they can each be solved independently with a solution of the
form
{r} = [S] {x}
(8)
−1
The transformation described by (8) gives the system in (3), in which the differential equations
are not coupled. This means that they can each be solved independently with a solution of the
form
16
rn( t ) = An sin(ω nt +φn )
(9)
which is an undamped oscillatory motion (this solution assumes that all elements in {f } are zero,
giving homogeneous equations). The constants An and φn depend on the initial conditions of the
problem (each vector of initial conditions must also be transformed into modal coordinates).
Once the solution vector {r} is known, it can be transformed back to physical coordinates with
the transformation
{x} = [S]{r}
(10)
This solution vector {x} gives the time responses of each DOF due to the initial conditions. Note
that as each time response is an undamped motion, it will continue to oscillate and never decay to
zero. This is obviously unrealistic. Some form of damping must be incorporated into the
equations to give a more realistic description:
[ M] {
} + [C ]{
} + [K ]{ x } = { F }
(11)
where [C] is the damping matrix.
The finite element formulation gives the mass and stiffness matrices, but the damping matrix
must be described in another way. In the modal analysis process outlined above, the damping
matrix will often not decouple when the transformation is applied, preventing a solution.
Proportional damping is a type of damping where the damping matrix is a linear combination of
21
the mass and stiffness matrices:
[ C ] = α [ M] + β [ K ]
(12)
where α and β are constants. This type of damping is relatively easy to compute and lends
itself to an easy solution by modal analysis. When the damping matrix fits this formula, equation
(11) will decouple into the form
{ }+ 2[ς n ][ω n ]{ }+ [λ ]{r} = {f }
(13)
where [ς n] is a diagonal matrix containing the modal damping values and [ω n] is a diagonal
matrix containing the values ω n . Each equation in this system will have a solution of the
general form
rn ( t ) = An -ς n ω n t e sin(ωdnt + φ )
where again An and φn are constants that depend on the initial conditions (this solution is only
true for the unforced system). This solution form is an exponentially decaying oscillatory
motion, which could be a close approximation for some structures.
The constants α and β in equation (12) can be chosen to provide specific damping values, or
calculated from known damping values according to the equation
(15)
22
CHAPTER
4
EXPERIMENTAL WORK
23
4.0 EXPERIMENTAL WORK
4.1 Introduction
The aim of the experimental work is to establish experimentally the stability diagrams for multilayered structural beams with bolted joints under axial loading. For multi-layered beams, the
stability diagrams have been experimentally established for 2, 3 layered beams. The theoretical
and experimental stability diagrams have been compared to assess the accuracy of the theoretical
results.
4.2 Description of the experimental set up
The main components are as follows:
1. Frame.
2. Adjustable Beam.
3. Electrodynamic shaker (EDS).
4. Load Cell.
5. Vibration Pick-ups.
6. Power Amplifier Panel.
7. Screw Jack.
8. Signal Analyzer (computer).
The schematic diagram of the equipments used for the experiment and photographic view of the
experimental set up is shown in the fig. The set up consists of a framework fabricated from steel
channel sections by welding. The frame is fixed in vertical position to the foundation by means of
foundation bolts and it has the provision to accommodate beams of different lengths. The periodic
axial load Pt cos Ωt is applied to the specimen by a 500N capacity electrodynamic shaker
(Saraswati Dynamics, India, Model no. SEV-005). The static load can be applied to the specimen
by means of a screw jack fixed to the frame at the upper end. The applied load on the specimen is
measured by a piezoelectric load cell (Brüel & Kjaer, model no. 2310-100), which is fixed
between the shaker and the specimen. The vibration response of the test specimen is measured by
means of vibration pick-ups (B&K type, model no. MM-0002). The signals from the pickups and
load cell are observed on a computer through a six-channel data acquisition system (B&K, 3560C), which works on Pulse software platform (B&K 7770, Version 9.0).
24
Fig 4.1 Schematic diagram of the test set up:1. Specimen, 2. Upper Pick-up, 3. Upper support,
4.
Screw Jack, 5. Load Cell, 6. Lower Pickup, 7.Lower support, 8. Acelerometer, 9. Vibration
generator, 10. oscillator and amplifier, 11. Data acquisition system.
25
Fig 4.2 Photograph of the experimental setup.
Fig 4.3 Photographs of attachments for clamped end
26
4.3 PREPARATION OF SPECIMEN:
For the Experimental validation of the dynamic stability of layered beams, multi-layered
specimens was made up of mild steel (M.S) and galvanized iron (G.I).The general preparation
procedure for the two type of materials were same expect change in the layer thickness. The
length of a single strip of beam was calculated from the EULER’S Equation for Buckling load:
L = ((4Л²EI)/ρ)½
Multi-layered beams were prepared by joining no.of strips with the help of nut, bolt and washer
assembly. The bolts were placed about 5cm apart and the torque on each bolt was assured to be
uniform with the help of a torquemeter.
The length of each layer was kept uniform. For a beam of thickness t mm, corresponding beam
was prepared having n no.of layers with thickness t1 mm.
where t = n*t1 (in mm)
Table 4.1 M.S, t = 0.98 mm(t
t1(mm)
0.16
0.32
0.5
No.of pieces
6
3
2
Corresponding Thickness
0.96
0.96
1
No.of pieces
8
4
2
Corresponding Thickness
1.28
1.28
1.26
No.of pieces
7
3
2
Corresponding Thickness
1.12
0.9
1.0
Table 4.2 M.S, t = 1.26 mm(t)
t1(mm)
0.16
0.32
0.63
Table 4.3 G.I, t = 1.16 mm(t)
t1(mm)
0.16
0.32
0.5
27
Fig4.4: Photograph showing the prepared specimen
28
4.4 Experimental procedure
An oscillator cum power amplifier unit drives the electrodynamic vibration shaker used for
providing the dynamic loading. The beam response was recorded by the non-contacting vibration
pickups. For straight beams with fixed-fixed end conditions two pickups, one at each end of the
beam were used.
Initially the beam was excited at certain frequency and the amplitude of excitation was increased
till the response was observed. Then the amplitude of excitation was kept constant and frequency
of excitation was changed in steps of 0.1 Hz. The excitation frequency was controlled with the
help of Pulse software. The generator module of the
Pulse software can produce an output signal of specified frequency; this signal is fed to the
shaker through its control unit. The experimental boundaries of instability regions were marked
by the parameters (Pt,Ω), which were measured just before a sudden increase of the amplitude of
the lateral vibration. The order of increase in amplitude of lateral vibration has been taken to be
around 4.0, to record the boundary frequency of instability. For accurate measurement of the
excitation frequency an accelerometer was fixed to the moving platform of the exciter, its
response was observed on computer in the frequency domain. The dynamic load component of
the applied load was measured from the response curve of the load cell. For ordinary beams the
excitation frequency was divided by the reference frequency ω1 to get the non-dimensional
excitation frequency (Ω/ω1).
29
CHAPTER
5
RESULTS AND CONCLUSIONS
30
RESULTS:
2
1 .8
1 .6
2ω
1
2ω
2ω
2
3
2ω
4
Dynamic Factor
β→
1 .4
1 .2
1
0 .8
0 .6
0 .4
0 .2
0
0
5
10
15
Fre quency Ratio Ω / ω →
F ig u r e - 1 In s t a b i lit y r e g i o n s f o r F ix e d - F i x e d C o n d i t i o n ,
' o ' W it h D a m p i n g ,' * ' W it h o u t D a m p in g
31
2
1 .8
2ω
1 .6
1
2ω
2ω
2
2ω
3
4
Dynamic Factor
β→
1 .4
1 .2
1
0 .8
0 .6
0 .4
0 .2
0
0
5
Fre quency Ra tio Ω / ω →
10
15
Figure-2 ,Instability Regions,for Fixed -Fixed condition,
,*,without damping,'o' with damping ξ 1=0.3,ξ 2=0.3
32
CONCLUSIONS:
From the obtained results it is seen that with increase in damping the area of instability
regions goes on decreasing. Moreover it is inferred that for a higher value of damping
instability regions start occurring after a comparatively higher value of applied load. The
damping effect produced varies increasingly with the no. of layers of beams used that is
more the number of layers of beam more is the damping effect. In the above graphs ξ
=0.1 was obtained from a two layered beam whereas a damping value of ξ=0.3 was
obtained in case of a four layered beam. By using multilayered beams the damping
increases and instability regions are produced after a comparatively higher value of
dynamic loading as is evident from the graphs.because of increased damping and hence
being less prone to vibrational damages multilayered beams can be trusted for practical
applications like in aerospace engineering against the usage of single layered beams.
33
CHAPTER
6
REFERENCES
34
REFERENCES
1. Abbas, B.A. H. and Thomas, J., Dynamic stability of Timoshenko beams resting on an
elastic foundation. Journal of sound and vibration, 60, 33 – 44, 1978.
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