Sukesh Chandra Mohanty DYNAMIC STABILITY OF BEAMS UNDER PARAMETRIC EXCITATION DOCTOR OF PHILOSOPHY
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DYNAMIC STABILITY OF BEAMS UNDER
PARAMETRIC EXCITATION
THESIS SUBMITTED TO THE NATIONAL INSTITUTE OF TECHNOLOGY,
ROURKELA FOR THE AWARD OF THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN
MECHANICAL ENGINEERING
BY
Sukesh Chandra Mohanty
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA769008, INDIA
SEPTEMBER, 2005
To my parents
Certificate
This is to certify that the thesis entitled “Dynamic stability of beams under
parametric excitation”, being submitted to the National Institute of Technology,
Rourkela (India) by Mr. Sukesh Chandra Mohanty for the award of the degree of
Doctor of Philosophy (Mechanical Engineering) is a record of bonafide research work
carried out by him under my supervision and guidance. Mr. Mohanty has worked for more than three years on the above problem and it has reached the standard fulfilling the requirements of the regulations of the degree. The results embodied in this thesis have not been submitted in part or full to any other university or institute for the award of any degree or diploma.
Rourkela (Dr.Niranjan Kavi)
Date: Professor
Department of Mechanical Engineering
National Institute of Technology
Rourkela769008
Orissa, India.
Acknowledgement
I express my deep sense of gratitude and reverence to my thesis supervisor
Dr.N.Kavi, Professor, Department of Mechanical Engineering, National Institute of
Technology, Rourkela, for his invaluable encouragement, helpful suggestions and supervision through out the course of this work.
I express my sincere thanks to the Director, N.I.T, Rourkela for providing me the necessary facilities for this study.
I would like to thank Prof. A.K.Behera and Prof. N.S.Dash exheads of the
Mechanical Engineering Department and Prof. B.K.Nanda present head of the department for their help and cooperation during the progress of this work.
My thanks are due to Prof. R.K.Sahoo and Dr.P.K.Roy of Mechanical
Engineering Department and Dr. M.R.Barik and Dr. S.K.Sahoo of Civil Engineering
Department for their valuable suggestions and help at various stages of the work.
I acknowledge with thanks the help rendered to me by my colleagues,
Dr. S.S.Mohaptra, Mr.R.K.Behera, Mr. Alok Satpathy and Dr. A.K.Satpathy.
The help extended by the staff of the Mechanical Engineering Department and
Central Workshop in fabrication of the experimental set up is acknowledged with thanks.
I am extremely grateful to my wife Madhumita and daughters Simun and Gungun for their loving support and sacrifice.
(Sukesh Chandra Mohanty)
ABSTRACT
The present investigation is an attempt to contribute towards the improved understanding of the dynamic stability of beams under parametric excitation. The dynamic stability of ordinary and sandwich beams subjected to longitudinal parametric excitation has been investigated theoretically and experiments have been carried out to validate some of the theoretical findings. The equations of motion have been derived using finite element method. For ordinary beams the instability regions have been established using Floquet’s theory and for sandwich beams modified Hsu’s method proposed by Saito and Otomi has been applied to determine the boundary frequencies of the instability regions.
The dynamic stability of a Timoshenko beam with localised damage and having fixedfree, pinnedpinned, fixedfixed and fixedpinned boundary conditions has been investigated to study the effects of parameters such as extent of damage, position of damage, static load factor and boundary conditions on its dynamic stability behaviour.
The parametric instability of a twisted cantilever beam with localised damage has been investigated to study the effect of angle of pretwist, location of damage, extent of damage and static load factor on its dynamic stability characteristics.
The governing equations of motion for a general multilayered symmetric cantilever sandwich beam subjected to parametric excitation have been derived.
Numerical results have been presented for three, five and seven layered beams to study the effects of various system parameters, such as core thickness parameter, core loss factor and number of layers on the stability behaviour of the sandwich beams. The dynamic stability behaviour of multilayered beams of various configurations based on some important criteria such as constant size, constant weight, constant flexural rigidity, constant size and flexural rigidity and constant weight and flexural rigidity has also been discussed.
Effects of static load factor and thermal gradient factor on the dynamic stability of a simply supported tapered beam with thermal gradient along its length have been studied.
Experimental work has been carried out to verify some of the theoretical results for beams with localised damage and multi layered symmetric cantilever sandwich beam.
Keywords: Parametric excitation, Dynamic stability, Localised damage, Boundary
condition, Pretwist, Thermal gradient, Sandwich beam, Simple resonance and combination resonance.
Contents
Abstract
Contents
iii
List of figures
List of tables
Nomenclature
vi xi xiii
1 Introduction
…………………………………………………………….………….
1
1.1 Introduction……………………………………………………………………..
1 i
1.2 Outline of the present work………………………………………………….….
4
2 Review of literature
……………………………………………………………...
6
2.1 Introduction……………………………………………………………………..
6
2.2 Methods of stability analysis of parametrically excited system………………… 7
2.3 Types of parametric resonance………………………………………………….
9
2.4 Effects of system parameters…………………………………………………… 9
2.4.1 Effect of localised damage……………………………………………… 9
2.4.2 Effect of pretwist angle………………………………………………...
10
2.5 Experimental 11
2.6 Sandwich 12
2.6.2
2.6.3
Static and dynamic analysis of sandwich beams…………………….....
Stability study of sandwich beams and columns…………………….....
14
16
2.6.4 Experimental 16
3 Dynamic stability of a Timoshenko beam with localised damage due to parametric excitation and to boundary conditions
……………………
18
3.1 Introduction……………………………………………………………………..
18
3.2 Formulation of the problem…………………………………………………….
19
3.2.1 Element 20
iii
3.2.2 Governing equations of motion………………………………………...
24
3.2.3 Regions of instability…………………………………………………..
25
3.3 Computational 29 and 31
3.5 Closure……………………………………………………………………….....
36
4 Dynamic stability of a pretwisted cantilever beam with localised damage under periodic axial force
…………………………………………..
49
4.1 Introduction…………………………………………………………………….
49
4.2 Formulation of the problem…………………………………………………….
50
4.2.1 Element 51
4.2.1 Governing equations of motion………………………………………...
55 and 57
4.4 Closure………………………………………………………………………….
60
5 Dynamic stability of a multilayered symmetric cantilever sandwich beam subjected to parametric excitation
…………………………………..
77
5.1 Introduction……………………………………………………………………. 77
5.2 Formulation of the problem…………………………………………………….
79
5.2.1 Element 80
5.2.1.1 Element matrix……………………………………..
80
5.2.1.2 Element matrix………………………………………...
83
5.2.2
5.2.3
5.2.1.3 Element geometric stiffness matrix………………………….
Governing equations of motion…………………………………….…..
Regions of instability……………………………………………….….
85
85
87
5.3 Computational 89 and 90
5.5 Closure………………………………………………………………………….
97
iv
6 Dynamic stability of a simply supported tapered beam with thermal gradient
….…………………………………………………………..……………...
111
6.1 Introduction………………………………………………………………….… 111
6.2 Formulation of the problem……………………………………………………. 112
6.2.2 Governing equations of motion………………………………………... 114
6.4 Closure…………………………………………………………………………. 119
7 Experimental
………………………………………………………………
125
7.1 Introduction……………………………………………………………………. 125
7.2 Description of the experimental set up………………………………………… 126
7.3 Preparation specimen……………………………………………………….. 127
7.6 Closure…………………………………………………………………….…… 133
8 Conclusion and scope for further research
.……………………………….
152
8.1 Conclusion……………………………………………………………………... 152
8.2 Scope for further research…………………………………………………….… 157
References
………………………………………………………………………….
AppendixA:
Flow chart for calculating the lower and upper boundary limits of instability regions based on Floquet’s theory…………………….…….
158
170
AppendixB:
Flow chart for calculating the lower and upper boundary limits of instability regions based on Hsu’s criteria……………………………...
171
List of publications
………………………………………………….………………. 172
v
LIST OF FIGURES
3.1 Beams with boundary conditions; (a) fixed free, (b) pinnedpinned,
(c) fixedfixed, (d) fixedpinned. 37
38 3.2 Timoshenko beam element.
3.3
Effect of damage position on first mode frequency,
ξ = 0.5, fixedfixed;, fixedpinned; pinnedpinned;  , fixed free;..
3.4
Effect of damage position on second mode frequency,
ξ = 0.5, key as fig.3.3
3.5
Effect of damage position on third mode frequency,
ξ = 0.5, key as fig.3.3
3.6
Effect of damage position on first mode buckling load,
ξ = 0.5, key a fig.3.3
3.7 Effect of damage position on instability regions, fixedfree end condition,
α=0.0, ξ==1.0;, ξ=0.5; ψ=0.1(+), ψ=0.3(*), ψ=0.5(o),
ψ= .7(
•
),
ψ=0.9(v).
3.8 Effect of damage position on instability regions, pinnedpinned end condition,
α= 0.0, ξ==1.0;, ξ=0.5; key as fig.3.7.
3.9 Effect of damage position on instability regions, fixedfixed end condition,
α= 0.0, ξ==1.0;, ξ=0.5; key as fig.3.7
3.10 Effect of damage position on instability regions, fixedpinned end condition,
α= 0.0, ξ==1.0;, ξ=0.5; key as fig.3.7
3.11 Effect of static load factor on instability regions, fixedfree end condition,
ξ=0.5, ψ=0.5; α= 0.0(o), ξ=0.5(+).
3.12 Effect of static load factor on instability regions, pinnedpinned end condition,
ξ=0.5, ψ=0.5; key as fig.3.11.
3.13 Effect of static load factor on instability regions, fixedfixed end condition,
ξ=0.5, ψ=0.5; key as fig.3.11.
3.14 Effect of static load factor on instability regions, fixedpinned end condition,
ξ=0.5, ψ=0.5; key as fig.3.11.
vi
39
39
40
40
41
42
43
44
45
46
47
48
4.1(a) Pretwisted beam and coordinate axes.
4.1(b) Angle of twist (
θ
t
)
4.2 Degrees of freedom of a beam element
4.3 Effect of pretwist angle on first mode natural frequency
4.4 Effect of pretwist angle on second mode natural frequency
4.5 Effect of pretwist angle on third mode natural frequency
4.6 Effect of damage location on first mode natural frequency
4.7 Effect of damage location on second mode natural frequency
4.8 Effect of damage location on third mode natural frequency
4.9 Effect of pretwist angle on critical buckling load
4.10
Effect of pretwist angle on instability regions,
ξ=0.5, ψ=0.3, θ
θ
1
=60
0
(+),
θ
1
=90
0
(o).
1
=30
0
(*),
4.11
Effect of pretwist angle on instability regions,
ξ=0.5, ψ=0.5, key as fig.
4.10.
4.12
Effect of pretwist angle on instability regions,
ξ=0.5, ψ=0.7, key as fig.
4.10.
4.13
Effect of damage location on instability regions,
ξ=0.5, θ
1
ψ= 0.1(*), ψ=0.3(+), ψ=0.5(o), ψ= .7(x), ψ=0.9(
•
).
=30
0
,
4.14
Effect of damage location on instability regions,
ξ=0.5, θ
1
=60
0
, key as fig.4.13.
4.15
Effect of damage location on instability regions,
ξ=0.5, θ
1
=90
0
, key as fig.4.13.
4.16
Effect of extent of damage on instability regions,
θ
ξ=0.75, (*), ξ=0.5, (o).
1
=30
0
,
ψ=0.1,
4.17
Effect of extent of damage on instability regions,
θ
1
=60
0
,
ψ= 0.1, key as fig.4.16.
64
65
65
66
62
62
62
63
63
64
66
67
67
68
69
70
71
72
vii
4.18
Effect of extent of damage on instability regions,
θ
1
=90
0
,
ψ=0.1, key as fig.4.16.
4.19
Effect of static load factor on instability regions,
θ
1
α= 0.5(*), α= 0.0(o).
=30
0
,
ψ=0.1,
4.20
Effect of static load factor on instability regions,
θ
1
=60
0
,
ψ=0.1, key as fig.4.19.
4.21
Effect of static load factor on instability regions,
θ
1
=90
0
,
ψ=0.1, key as fig.4.19.
5.1 Configuration of 2n+1 layered cantilever sandwich beam
73
74
75
76
99
99 5.2 Finite beam element for a 2n+1 layered cantilever sandwich beam
5.3 Effect of core thickness parameter on fundamental frequency parameter,
(g)
N
= 5.0,
η c
=0.18, N=3 ,
__
, N=5,
…
, N=7, ,.
5.5 Effect of core thickness parameter on fundamental frequency parameter constant size case (g)
3
= 1.0 ,
η c
=0.18, N=3 ,
__
, N=5,
….
, N=7, ,.
100
5.4 Effect of core thickness parameter on second mode frequency parameter, (g)
N
= 5.0 ,
η c
=0.18, key as fig.5.3
100
101
5.6 Effect of core thickness parameter on second mode frequency parameter constant size case (g)
3
= 1.0 ,
η c
=0.18, key as fig. 5.5
5.7 Effect of core thickness parameter on fundamental frequency parameter constant weight case (g)
3
= 1.0 ,
η c
=0.18, key as fig.5
5.8 Effect of core thickness parameter on second mode frequency parameter constant weight case (g)
3
= 1.0 ,
η c
=0.18, key as fig. 5.5
5.9 Effect of core thickness parameter on fundamental frequency parameter constant flexural rigidity case (g)
3
= 1.0 ,
η c
=0.18, key as fig. 5.5
101
102
102
103
5.10 Effect of core thickness parameter on second mode frequency parameter constant flexural rigidity case (g)
3
= 1.0 ,
η c
=0.18, key as fig. 5.5
5.11 Effect of core thickness parameter on fundamental frequency parameter constant size and flexural rigidity case (g)
3
= 0.2 ,
η c
=0.18, key as fig. 5.5
103
104
viii
5.12 Effect of core thickness parameter on second mode frequency parameter constant size and flexural rigidity case (g)
3
= 0.2 ,
η c
=0.18, key as fig. 5.5
104
5.13 Effect of core thickness parameter on fundamental frequency parameter constant weight and flexural rigidity case (g)
3
= 0.2 ,
η c
=0.18, key as fig. 5.5
105
5.14 Effect of core thickness parameter on second mode frequency parameter, constant weight and flexural rigidity case (g)
3
= 0.2 ,
η c
=0.18, key as fig. 5.5
5.15
Instability regions, (g)
N
= 5.0 ,(t
21
)
N
= 1/3,
η c
=0.18, N=3 ,
__
, N=5,
… .
,
N=7,  ,.
105
106
5.16
Instability regions, (g)
N
= 5.0 ,(t
21
)
N
= 2/3,
η c
=0.18, key as fig. 5.15.
5.17
Instability regions, (g)
N
= 5.0 ,(t
21
)
N
= 1/3,
η c
=0.3, key as fig. 5.15.
5.18
Instability regions, (g)
N
= 5.0 ,(t
21
)
N
= 2/3,
η c
=0.3, key as fig. 5.15.
106
107
107
5.19 Instability regions for constant size case, (g)
3
= 1.0 ,(t
21
)
3
= 1.0,
η c
=0.18, N=3 ,
__
, N=5,
… .
, N=7,  ,.
108
5.20 Instability regions for constant weight case, (g)
3
= 1.0 ,(t
21
)
3
= 1.0,
η c
=0.18, key as fig.5.19.
108
5.21 Instability regions for constant flexural rigidity case, (g)
3
= 1.0 ,
(t
21
)
3
= 1.0,
η c
=0.18, key as fig.5.19.
109
(t
21
)
3 regions constant size and flexural rigidity case, (g)
3
= 4.0,
η c
=0.18, key as fig.5.19.
= 0.2 ,
109
5.23 Instability regions for constant weight and flexural rigidity case,
(g)
3
= 0.2 , (t
21
)
3
= 4.0,
η c
=0.18, key as fig.5.19.
110
6.1 Simply supported tapered beam with thermal gradient 120
6.2 Tapered beam element
6.3 Effect of thermal gradient on fundamental frequency
6.4 Effect of thermal gradient on critical buckling load
6.5
Instability regions for
α =0.4, δ=0.0,,δ=0.3,,δ=0.6,…
ix
120
121
121
122
6.6
Instability regions for
α =0.5, key as fig.6.5
6.7
Instability regions for
α =0.8, key as fig.6.5
6.8
Instability regions for
δ=0.3, δ=0.4,,δ=0.5,,δ=0.8,…,
6.9
Instability regions for
δ=0.6, key as fig.6.8
6.10
Instability regions for
α =0.5, δ=0.0,, α =0.5, δ=0.3,, α =0.6138,
δ=0.0,…,
7.1 Schematic diagram of the test set up.
7.2
Instability regions, for fixedfree case,
ξ=0.5,ψ=0.1,τ=0.2,Theoretical boundary from FEM; , Experimental data;
•
.
7.3
Instability regions, for pinnedpinned case,
ξ=0.5,ψ=0.5, τ=0.2,
Theoretical boundary from FEM; , Experimental data;
•
.
7.4
Instability regions, for fixedfixed case,
ξ=0.5, ψ=0.5,τ=0.2,Theoretical boundary from FEM; , Experimental data;
•
.
7.5
Instability regions, for fixedpinned case,
ξ=0.5,ψ=0.7,τ=0.2,
Theoretical boundary from FEM; , Experimental data;
•
.
7.6
Instability regions,
θ
1
=30
0
,
ξ=0.5,ψ=0.1,φ=0.2, Theoretical boundary from FEM; , Experimental data;
•
.
7.7
Instability regions,
θ
1
=60
0
,
ξ=0.5,ψ=0.1,φ=0.2, Theoretical boundary from FEM; , Experimental data;
•
.
7.8
Instability regions,
θ
1
=90
0
,
ξ=0.5,ψ=0.1,φ=0.2, Theoretical boundary from FEM; , Experimental data;
•
.
7.9
Instability regions for three layer beam: t
21
=0.25,
η c
=0.55, Theoretical boundary from FEM; , Experimental data;
•
.
7.10
Instability regions for five layer beam: t
21
=0.25,
η c
=0.55, Theoretical boundary from FEM; , Experimental data;
•
.
7.11
Instability regions for seven layer beam: t
21
=0.5,
η c
=0.55, Theoretical boundary from FEM; , Experimental data;
•
.
122
123
123
124
124
134
135
136
137
138
139
140
141
142
142
143
x
LIST OF TABLES
3.1
3.2
Comparison of boundary frequencies of the first instability region obtained from present analysis with reference [16], for an undamaged Euler beam with pinnedpinned end conditions. Static load factor
α = 0.0.
Comparison of boundary frequencies of the first instability region obtained from the present analysis with reference [111], for an undamaged cantilever beam for static load factors
α=0.0 and
α=0.75.
4.1(a) Comparison of natural frequencies [Hz] of a twisted blade with those of references [32], [97] and [19].
4.1(b) Comparison of natural frequencies [rad/s] of a twisted blade with those of reference [9].
5.1 Comparison of the boundary frequencies for a three layer cantilever
6.1
7.1 sandwich beam from the present analysis with those of reference
[75].
Comparison of boundary frequencies of the first instability region obtained from present analysis with reference [29].
Physical and geometrical parameters of the test specimens for beams with localised damage and various boundary conditions.
7.2
7.3
32
32
57
57
91
117
Physical and geometrical parameters of the test specimens for pretwisted cantilever beams with localised damage.
144
Theoretical values of fundamental natural frequency and critical buckling load of uniform undamaged beams with various boundary conditions. 144
144
7.4 Theoretical values of natural frequency and critical buckling load of straight undamaged cantilever beam.
xi
144
7.5
7.6
7.7
7.8
7.9
Physical and geometrical data of sandwich beam specimens.
Theoretical and experimental resonant frequencies of Multilayer beams.
Experimental boundary frequencies of instability regions for uniform beam with localised damage, fixedfree end conditions.
Experimental boundary frequencies of instability regions for uniform beam with localised damage, pinnedpinned end conditions.
Experimental boundary frequencies of instability regions for uniform beam with localised damage, fixedfixed end conditions.
7.10 Experimental boundary frequencies of instability regions for uniform beam with localised damage, fixedpinned end conditions.
7.11
7.12
Experimental boundary frequencies of instability regions for twisted beam with localised damage.
Experimental boundary frequencies of instability regions for twisted beam with localised damage.
7.13 Experimental boundary frequencies of instability regions for twisted beam with localised damage.
7.14 Experimental boundary frequencies of instability regions for
3layered sandwich beam.
7.15 Experimental boundary frequencies of instability regions for
5layered sandwich beam.
7.16 Experimental boundary frequencies of instability regions for
7layered sandwich beam.
145
145
146
146
147
149
150
150
151
147
148
148
xii
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
Crosssectional area of the uniform beam.
A(x)
A
( 2
k
−
1 )
A
v(2j)
Crosssectional area of the 2jth viscoelastic elastic layer.
a
1
−
a
8
Set of generalised coordinates. b Width of the beam. b
1
Width of the tapered beam at the root. c Xcoordinate of the end of the damaged region. d
E
Area of crosssection of the tapered beam at any section x.
Crosssectional area of the (2k1)th constraining elastic layer.
Xcoordinate of the end of the damaged region.
Young’s modulus of the beam material.
E(x)
E
1
EK b
Modulus of elasticity for the beam material at any section x.
Young’s modulus for the tapered beam material at the reference temperature
T
o,
at the reference point, x = L.
Effective bending stiffness for the damaged region.
E
( 2
k
−
1 )
Young's modulus of the (2k1)th constraining elastic layer.
xiii
f
G
Xcoordinate of the center of the damage portion.
Shear modulus.
I
GK s
Effective shear stiffness for the damaged region.
G
v(2j)
Complex shear modulus of 2jth viscoelastic material layer.
G
∗
v(2j)
The inphase shear modulus of the 2jth viscoelastic material layer. g Shear parameter. h Height of the beam. h
1
Tapered beam height at the root,x = 0.
The second moment of inertia.
I
1
I
(
2 k
−
1
)
I(x)
Area moment of inertia of the tapered beam section at x = 0.
Area moment of inertia of the (2k1)th constraining elastic layer.
Area moment of inertia at any section x.
I yy
Crosssectional area moment of inertia of the beam about the Yaxis.
I zz
Crosssectional area moment of inertia of the beam about the Zaxis.
I yz
Product of inertia of the beam crosssection with respect to the Y and Zaxes.
y
' y
I
Moment of inertia of the beam cross section about principal inertia axis Y
′Y′.
'
z
' z'
I
Moment of inertia of the beam cross section about principal inertia axis Z
′Z′.
[ ]
Global elastic stiffness matrix.
k
′
Shear coefficient.
xiv
K b
K s
Constant representing the capacity of the damaged region to store bending strain energy.
Constant representing the capacity of the damaged region to store shear strain energy. l
Element elastic stiffness matrix.
[ ]
b
Element bending stiffness matrix.
[ ]
g
Global geometric stiffness matrix.
[ ]
g
Element geometric stiffness matrix.
[ ]
s
(e )
Element shear stiffness matrix.
Length of an element.
L Length of the beam.
[ ]
Global mass matrix.
[ ]
)
Element mass matrix.
[ ]
r
(e )
Element rotary inertia mass matrix.
[ ]
t
Element translational mass matrix.
[ ]
S
N
Shape function matrix for twisted beam.
[
N v
] Shape function matrix for lateral displacement,
v.
[
N
θ
]
Shape function matrix for rotation,
θ
.
P(t) Axial periodic load.
P s
Static component of the periodic load.
xv
P t
R
T t
(e)
T
T o
Time dependent component of the periodic load.
Flexural strain or curvature.
Kinetic energy of the beam.
Time coordinate.
Elemental kinetic energy.
Temperature at the reference point, x = L.
v t t
( 2
k
−
1 )
Thickness of the (2k1)th constraining elastic layer.
Thickness of the 2jth viscoelastic layer.
v (
2
j )
(e )
U d
Total strain energy of an element within the damaged region.
U
Total strain energy of the beam.
U
(e)
Elemental potential energy.
u
( 2
k
−
1 )
Axial displacement of the (2k1)th elastic layer.
u
v(2j)
Axial displacement of the 2jth viscoelastic layer.
Transverse displacement of the beam.
w Transverse displacement of the beam.
Greek symbols
α
Static load factor.
α
1
−
α
8
Set of generalised coordinates.
β
Dynamic load factor.
β
t
Slope of variation of Young’s modulus E with temperature T.
xvi
γ
γ v(2j)
{ }
δ
Shear strain.
Shear strain of the 2jth viscoelastic layer.
Global displacement vector.
Elemental nodal displacement vector.
Thermal gradient factor,
δ=
β
t
T
o.
ζ
= x/l
( ) c
( 2
j
)
Core loss factor of 2jth viscoelastic layer.
Θ
Nondimensional excitation frequency,
=
⎝
⎜⎜
⎛ Ω
ω
1
⎠
⎟⎟
⎞
2
.
θ
θ
1
Crosssectional rotation.
Angle of pretwist at the free end with respect to the fixed end.
θ
t
Angle of twist θ
t
at cross section x.
ξ b
Extent of damage in bending sense, =
EK b
/EI.
ξ s
Extent of damage in shear sense, =
GK s
/k'GA.
ρ
(2k1)
Mass density of the (2k1)th constraining elastic layer.
ρ
v(2j)
Mass density of the 2jth viscoelastic layer.
{ }
Set of generalised coordinates.
τ
Size parameter of the damaged region = (dc)/L.
[ ]
Normalized modal matrix corresponding to
−
1
. ϕ
Crosssectional rotation.
ψ = f/L
Nondimensional position of damage.
xvii
Ω
Excitation frequency of the dynamic load component.
Superscripts
(e) Element
Operators
‘·‘
‘
‘
′ ‘
″ ‘
∂
∂
t
∂
∂
x
∂
2
∂
x
2
xviii
Chapter
1
INTRODUCTION
1.1 Introduction
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
1
2
and the disturbing forces appear as inhomogeneity. 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
Ω =
2
ω
n
,
m
=
1 , 2 , 3 , 4
− − − −
n
The case
Ω =
2
ω
n
is known to be the most important in application and is called main or principal parametric resonance.
3
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 crosssection, 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 localised damage [90]. The modulus of elasticity of the material is greatly affected by the temperature. In highspeed 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, localised damage and thermal conditions have greater effect on the dynamic behaviour of the beams and hence need to be studied in depth.
Vibration control of machines and structures incorporating viscoelastic materials in suitable arrangement is an important aspect of investigation [87]. 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,
4
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.
1.2 Outline of the present work
The present work mainly deals with the parametric instability of ordinary and sandwich beams. The main objectives are to study the effects of various parameters on the dynamic stability of the beams. Theoretical investigations have been carried out and experimental verification of the theoretical findings has been done for some cases.
The equations of motion for the systems have been derived using finite element method. The stability boundary for ordinary beams have been established by using the
Floquet’s theory [17] and for sandwich beams the stability diagrams have been obtained by using the modified Hsu’s method proposed by Saito and Otomi [104]. For numerical computations computer programs have been developed using MATLAB [135]. For experimental work, an experimental set up capable of producing variable loading conditions has been designed and fabricated.
This thesis contains eight chapters including this chapter.
A detailed survey of relevant literature is reported in chapter 2.
In chapter 3 dynamic stability of a Timoshenko beam with a localised damage and with common boundary conditions, such as fixedfree, pinnedpinned, fixedfixed and fixedpinned has been analysed. Effects of parameters such as extent of damage, static load factor, position of damage and boundary conditions on the dynamic stability of the beam have been investigated.
5
Chapter 4 deals with the dynamic stability of a pretwisted cantilever beam with localised damage under periodic axial force. Effects of parameters namely angle of pretwist, position of damage, extent of damage and static load factor on the dynamic stability of the beam have been studied.
In chapter 5 dynamic stability of a multilayered symmetric cantilever sandwich beam subjected to parametric excitation has been reported. The effects of various system parameters, such as core thickness parameter, core loss factor and number of layers, as well as various beam configurations based on some important criteria on the stability behaviour of multilayered beams have been investigated.
Chapter 6 addresses to the problem of dynamic stability of a simply supported tapered beam with thermal gradient. Effects of thermal gradient and static load component on the stability behaviour of the beam have been studied.
In chapter 7 details of experimental set up and testing procedure have been outlined. Experimentally obtained stability diagrams for a few typical cases related to problems described in chapters (35) have been reported. The theoretical and experimental stability diagrams have been compared to make a qualitative assessment of the theoretical results.
Finally in chapter 8 important conclusions drawn from the present investigations reported in chapters 37 along with suggestions for further work have been presented.
Chapter
2
REVIEW OF LITERATURE
2.1 Introduction
Discovery of parametric resonance dates back to 1831. Faraday [40] was first to observe the phenomenon of parametric excitation, when he noticed that when a fluid filled container vibrates vertically, fluid surface oscillates at half the frequency of the container.
Melde [79] reported parametric resonance in the case of lateral vibration of a string.
Beliaev [13] was first to provide a theoretical analysis of parameter resonance while dealing with the stability of prismatic rods. These are a few early works.
Several review articles on parameter resonance have also been published. Evan
Iwanoski [39], Ibrahim and coworkers [5460], Ariaratnam [5] and Simitses [113] gave exhaustive account of literature on vibration and stability of parametrically excited systems. Review article of Habip [49] gives an account of developments in the analysis of sandwich structures. Articles of Nakra [8486] have extensively treated the aspect of vibration control with viscoelastic materials. Books by Bolotin [17], Schmidt [109] and
6
7
Neyfeh and Mook [88] deals extensively on the basic theory of dynamic stability of systems under parameter excitations .In this chapter further developments in subsequent years in the field of parametric excitation of system with specific reference 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 deal with the methods of stability analysis, types of resonance, study of different system parameters on the parameter instability of the system and experimental verification of the theoretical findings.
2.2 Methods of stability analysis of parametrically excited system
The governing equations for parametrically excited systems are second order differential equations with periodic coefficients, which have no exact solutions. 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 MatheiuHill type, a few well known solution methods those are 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 the asymptotic technique by Krylov, Bogoliubov and Mitroploskii.
Bolotin’s [17] method based on Floquet’s theory can be used to get satisfactory results for simple resonance only. Steven [115] later modified the Bolotin’s method for system with complex differentials equation of motion. Hsu [5253] proposed an approximate method of stability analysis of systems having small parameter excitations
8
Hsu’s method can be used to obtain instability zones of main, combination and difference types. Later Saito and Otomi [104] modified Hsu’s method to suit systems with complex differential equation of motion. Takahashi [122] 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 [131] and Zajaczkowski
[132] 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. [73] proposed a variable parameter incrementation method, which is free from limitations of small excitation parameters. It has the advantage of treating nonlinear systems.
Many investigators to study the dynamic stability of elastic systems have also applied finite element method. Brown et al. [18] studied the dynamic stability of uniform bars by applying this method. Abbas [2] studied the effect of rotational speed and root flexibility on the stability of a rotating Timoshenko beam by finite element method.
Abbas and Thomas [1] and Yokoyama [130] used finite element method to study the effect of support condition on the dynamic stability of Timoshenko beams. Shastry and
Rao by finite element method obtained critical frequencies [110] and the stability boundaries [111112] for a cantilever column under an intermediate periodic concentrated load for various load positions. Bauchau and Hong [11] studied the nonlinear response and stability of beams using finite element in time. Briseghella et al. [16] studied the dynamic stability problems of beams amd frames by using finite element
9
method. Svensson [121] by this method studied the stability properties of a periodically loaded nonlinear dynamic system, giving special attention to damping effects.
2.3 Types of parametric resonance
Multi degree freedom systems may exhibit simple resonance, resonance of sum type or resonance of difference type depending upon the type of loading, support conditions and system parameter.
Mettler [80] furnished a classification for various kinds of resonances exhibited by linear periodic system. Iwatsubo and his coworkers [6364] from their investigation on stability of columns found that uniform columns with simple supported ends do not exhibit combination type resonances. Saito and Otomi [104] on the basis of their investigation of stability of viscoelastic beams with viscoelastic support concluded that combination resonances of different type do not occur for axial loading, but it exits for tangential type of loading. Celep [21] found that for a simply supported pretwisted column, combination resonances of the sum type may exit or disappear depending on the pretwist angle and rigidity ratio of the crosssection. Ishida et al. [62] showed that an elastic shaft with a disk exhibits only difference type combination resonance. Chen and
Ku [23] from their investigations found that for a cantilever shaft disk system, the gyroscopic moment can enlarge the principal regions of dynamic instability.
2.4 Effects of system parameters
2.4.1 Effect of localised damage
Parekh and Carlson [90] developed a model to study the dynamic stability of a bar with a localised damage. They introduced the concept of effective stiffness for the damaged
10
region, which reflects the energy storing capacity of the damaged region. Later Datta and
Nagraj [31] used the concept of Parekh and Carlson and analysed the effect of foundation stiffness, extent of damage and position of damage on the dynamic stability of a tapered bar with flaws supported on elastic foundation. Datta and Lal [30] investigated the static stability of tapered beams with localized damage and subjected to intermediate concentrated load. Datta and Lal [34] work was extended by Mohanty and Kavi [81] by adding shear deformation to their analysis. Das and Dey [28] investigated the random vibration of beams with localised zones of damage under stochastic excitation.
2.4.2 Effect of pretwist angle
Dokumaci [37] investigated the effects of pretwist, ratio of bending rigidities and loading angle on the unstable zones of pretwisted blades under lateral parametric excitation.
Celep [21] studied the stability of a pretwisted simply supported column subjected to static and periodic axial loads. He found that combination resonance of sum type exists for small pretwist and vanishes for large pretwist angle. Gürgöze [47] studied the dynamic stability of pretwisted beams with hingedhinged, clampedclamped and clampedhinged boundary conditions and found that increasing the pretwist angle broadens the instability zones. Chen and Liao [24] carried out a dynamic analysis of a pretwisted spinning beam under constant axial compressive loads with elastic constraints using assumed mode method. Yang and Tsao [129] studied the vibration and stability of a pretwisted blade under nonconstant rotating speed and they found that instability can be minimised by increasing the pretwist angle. Tan et al. [123] investigated the parametric instability of spinning pretwisted beams using Euler beam theory and the method of multiple scales. They found that increasing the pretwist angle the first instability region
11
width decreases, while the second instability region becomes wider and both the regions move towards each other.
2.5 Experimental investigations
Experimental studies in the field of parametric instability are relatively few. Bolotin [17] experimentally verified his own theoretical findings. Stevens and EvanIwanowski [116] experimentally studied the effect of viscoelastic material of the parametric instability of the columns. Iwatsubo et al [63] carried out experiments to check their result obtained by finite difference method for the stability of clampedclamped and clampedsimply supported columns. Saito and Koizumi [105] from experiment obtained the amplitude frequency curves of a simply supported horizontal beam subjected to nonlinear parametric excitation. Sunakawa and Higuchi [120] through experiment studied the nonlinear response of thin columns under parametric excitations. The theoretical results obtained for a slender cantilever beam carrying a lumped mass was experimentally verified by Zovodney [133] and Zavodney and Nayfeh [134]. Dufour and Berlioz [38] carried out experiments to study the dynamic stability of a beam under axial periodic force and torque and compared their theoretical results with the experimental one.
Sugiyama et al [118] conducted experiments to verify the stabilizing effect of nonconservative follower force on the vibration and stability of cantilevered columns initially subjected to a conservative force due the rocket motor’s weight. Svensson [121] carried out experiments to study the effect of material damping as well as the damping effects at the hinges on the dynamic stability of a beam.
12
2.6 Sandwich Beams
The main objectives of the researchers dealing with sandwich beams may be grouped in the following categories. i) Prediction of resonant frequencies and loss factor ii) Static and dynamic analysis of sandwich beams iii) Stability study of sandwich beams and columns iv) Experimental investigations
2.6.1 Resonant frequencies and loss factor prediction
Kerwin [71] was the first to carry out a quantitative analysis of the damping effectiveness of a constrained viscoelastic layer and he obtained an expression to estimate the loss factor. Ungar [127] derived general expressions for the loss factors of uniform linear composites in terms of the properties of the constituting materials. Di Taranto [35] developed a theory to estimate natural frequencies, loss factors for a for a finite length sandwich beam. Jones et al. [67] theoretically and experimentally evaluated the damping capacity of a sandwich beam with viscoelastic core. Asnani and Nakra [6] analysed multilayer simply supported sandwich beams and estimated loss factors and displacement response effectiveness for beams of different number of layers. Chatterjee and
Baungarten [22] obtained for a simply supported sandwich beam, the damped natural frequencies and logarithmic decrement for the fundamental mode of vibration. They also conducted experiments to verify their theoretical results, which showed good agreement.
Nakra and Grootenhuis [83] studied theoretically as well as experimentally, the vibration characteristics of asymmetric dual core sandwich beams. They did not include the rotary and longitudinal inertia terms in their analysis. Later Rao [98] included both these effects
13
in his analysis. Asnani and Nakra [8] studied the effect of number of layers and thickness ratio on the system loss factors for a simply supported multilayer beam. Rao [92] investigated the influence of pretwist on resonant frequency and loss factor for a symmetric pretwisted simply supported sandwich beam and found that pretwisting reduces the loss factor and very soft thick cored beams is especially sensitive to even small changes of pretwist. Rao and St
ühler [93] analysed the damping effectiveness of tapered sandwich beam with simply supported and clamped free end conditions. Rao [98] investigated the free vibration of a short sandwich beam considering the higher order effects such as inertia, extension and shear of all the layers. He found that if these parameters are neglected for short sandwich beam there is an error as high as 45% in estimation of the loss factor and frequencies. Rubayi and Charoenee [103] carried theoretical and experimental investigations to obtain the natural frequencies of cantilever sandwich beams subjected to gravity force only. Rao [96] in another work gave graphs and equations to estimate frequencies and loss factors for sandwich beam under various boundary conditions. Johnson and his coworkers [6566] used the finite element method to solve frequencies and loss factors for beams and plates with constrained viscoelastic layer. Vaswani et al. [128] derived equations of motion for a multilayer curved sandwich beam subjected to harmonic excitation. Lall et al. [72] analysed the partially covered sandwich beams using three different methods and found that method by Marcus [77] estimates modal loss factor only, where as RayleighRitz and classical search method give both loss factor and resonant frequencies. Dewa et al. [33] studied the damping effectiveness of partially covered sandwich beams. They found that partially covered beams have better damping capacity than fully covered beams. Also through experiments
14
he validated his theoretical findings. Imaino and Harrison [61] adopted modal strain energy method and finite element technique to investigate damping of the first and second bending resonance of a sandwich beam with constrained damping layer. He and
Rao [50] developed an analytical model to carry out a parameter study of the coupled flexural and longitudinal vibration of a curved sandwich beam. Effects of parameters such as curvature, core thickness and adhesive shear modulus on the system loss factors and resonant frequencies were investigated. Same authors [51] in another work studied the vibration of multispan beams with arbitrary boundary condition. Effects of parameter like location of intermediate supports and adhesive thickness on the resonant frequencies and loss factors were investigated. Bhimaraddi [15] solved both the resonant frequencies and loss factors for a simply supported beam with constrained layer damping using a model, which accounted for the continuity of displacements and the transverse shear stresses across the interfaces of the layers. Sakiyama et al. [106] developed an analytical method for free vibration analysis of a three layer continuous sandwich beam and investigated the effect of shear parameter and core thickness on the resonant frequencies and loss factors. Fasana and Marchesiello [41] calculated the mode shapes, frequencies and loss factors for sandwich beams by RayleighRitz method. They choose polynomials, which satisfy the geometric boundary conditions as admissible function. Banerjee [10] studied the free vibration of a three layer sandwich beam using dynamic stiffness matrix method. He calculated the natural frequencies and mode shapes.
2.6.2 Static and dynamic analysis of sandwich beams
Mead and Markus [78] carried out the forced vibration analysis of a threelayered sandwich beam with viscoelastic core and with arbitrary boundary conditions. They
15
followed the method used by Di Taranto [35] in their analysis. Asnani and Nakra [7] carried out forced vibration analysis of sandwich beams with viscoelastic core and with fixedfixed and cantilever type end conditions. The forced vibration response obtained by applying Ritz method matched well with the experimental results. Rao [95] studied the forced vibration of a damped sandwich beam subjected to moving forces and found that increasing the shear stiffness of the core materials can reduce the dynamic magnification of the central deflection of the beam. Kapur [68] considered both rotary and longitudinal inertia in his analysis to study the dynamic response of two and threelayered viscoelastically damped beams subjected to halfsine shock excitation. Sharma and Rao
[108] determined static deflections and stresses in sandwich beams for both concentrated and distributed loads under various conditions. Frosting and Baruch [43] from their analysis of stresses in a sandwich beam with flexible core under concentrated and distributed loading found that transverse normal stresses at the interface between the skin and core in some cases are significant in determining the sudden failure of the beam. Sun et al. [119] developed a finite element model to study the effect of addon viscoelastic layer in damping and vibration control of unidirectional composite laminates. Their theoretical results compared well with the experimental findings. Qian and Demao [91] carried out modal analysis as well as response calculation in time domain using finite element technique. Salet and Hamelink [107] developed a numerical model based on finite difference method, for nonlinear analysis of sandwich beams with simply supported boundary conditions. Ha [48] suggested an exact analysis procedure for bending and buckling analysis of sandwich beam systems.
16
2.6.3 Stability study of sandwich beams and columns
Bauld [12] investigated the stability of sandwich columns with simply supported end conditions and subjected to pulsating axial loads. The stability of two layer sandwich cantilever beams with imperfect bonding was studied was studied by Chonan [26]. They obtained critical loads for divergence and flutter type instabilities and found that these are functions of shear and normal stiffness of the bond. In another work Chonan [27] studied the divergence and flutter type instabilities in symmetric sandwich beams with elastic bonding and found that critical divergence and flutter loads depends on the interface bond stiffness. Kar and Hauger [69] investigated the dynamic stability of a sandwich beam subjected to a direction controlled nonconservative force and determined the critical divergence and flutter loads. Ray and Kar [99] have investigated the dynamic stability of sandwich beams under various boundary conditions. The same authors [100102] also investigated the parameter stability of partially covered sandwich beams, dual cored sandwich beams and symmetric sandwich beams with higher order effects. Ray and Kar in these works derived the governing equation of motion by using Hamilton’s principle and converted the equations of motion to a set of coupled Hill’s equation in the time domain by Galerkin’s method. They assumed approximate series solutions, which satisfy majority of the boundary conditions. Lin and Chen [75] studied the effect of rotating speed, setting angle and hub radius on the dynamic stability of a rotating sandwich beam with a constrained damping layer.
2.6.4 Experimental investigations
The reported experimental works are mainly related to the experimental validation of theoretically predicted dynamic response, damping values, resonant frequencies and loss
17
factors of sandwich beams. Chatterjee and Baumgarten [22] experimentally determined the logarithmic decrement to validate their theoretically obtained values for damped natural frequencies and damping values for a simply supported sandwich beam. Asnani and Nakra [7] compared their theoretically obtained resonant frequencies by applying
Ritz method with experimental results for a threelayer sandwich beam. Trompette et al.
[125] carried out experiments to obtain resonant frequency and damping values and compared with their theoretical results, which showed good agreement. Mace [76] compared the frequency response curve obtained from experiment with his theoretical results and drew the conclusion that his predicted theory is efficient in predicting the dynamic response of beams that are damped by means of a thin viscoelastic film.
Gorrepati and Rao [45] measured from experiment, the natural frequencies and loss factor for a simply supported beam with adhesively bonded double strap joint to validate their results obtained by modal strain energy method. Chen and Chan [25] in order to establish their results obtained from integral finite element method experimentally obtained frequency response functions for elasticviscoelastic composite structures. In a recent work Nayfeh [89] conducted experiment to obtain resonant frequencies and loss factors and compared with values predicted by his developed model for vibration parallel to the plane of lamination of a symmetric elasticviscoelastic sandwich beam.
Chapter
3
DYNAMIC STABILITY OF A TIMOSHENKO
BEAM WITH LOCALISED DAMAGE DUE TO
PARAMETRIC EXCITATION AND TO
BOUNDARY CONDITIONS
3.1 Introduction
Dynamic analysis of many machine and structural components can be done by modeling them as uniform beams with different boundary conditions. These components quite often are subjected to time varying parametric excitation, which may lead to their instability. Advances in material science have contributed many alloys and composite materials having high strength to weight ratio. However during the manufacturing of these materials, inclusion of flaws affects their structural strength. Hence the effect of localised damage on the dynamic stability of beams with various common boundary conditions forms an important aspect of investigation.
Earlier studies on effect of localised damages on the stability behaviour of structural elements were mainly on static and fatigue strength consideration
18
19
[14,74]. Parekh and Carlson [90] introduced the concept of effective stiffness for the damaged region and analysed the dynamic stability of a bar with localised damage. They developed analytically an approximate solution for establishing principal regions of instability. Datta and Nagaraj [31] studied the dynamic stability of tapered bars with flaws and with simply supported end conditions resting on an elastic foundation. They considered Euler beam theory in their analysis. Datta and Lal [30] analysed the static stability behaviour of a tapered beam with localised damage subjected to an intermediate concentrated load, but shear deformation was not included in their analysis. The same work [30] was extended by Mohanty and Kavi [81] considering shear deformation.
This work is an attempt to study the dynamic stability of a uniform Timoshenko beam with localised damage subjected to parametric excitation under various boundary conditions. Finite element method along with Floquet’s theory has been used to carry out the analysis. Four parameters are used to characterise the damaged zone: location, size and effective bending and shear stiffness at the damaged region. Effective bending and shear stiffness at the damaged region is a measure of the extent of damage. Instability zones for different locations of the damage and for various boundary conditions of the beam have been established to study the effects of different parameters namely extent of damage, damage location, boundary conditions and static load factor.
3.2 Formulation of the problem
The beam is of uniform rectangular crosssection having a length L, width b and depth h.
The effect of the damage is represented by the presence of a flaw in the region c
< x < d.
Beams with end conditions such as fixedfree, pinnedpinned, fixed fixed and fixedpinned as shown in fig.–3.1 are considered. The beam is subjected to a pulsating axial
20
force P(t) = P
s
+ P
t cos
Ω
t
, acting along its undeformed axis.
Ω
is the excitation frequency of the dynamic load component, P
s
is the static and P
t
is the amplitude of the time dependent component of the load.
A typical finite element is shown in fig.3.2. The element consists of two nodes i and j with v
i ,
θ
i
, v
j and
θ
j
as the nodal displacements. v is the lateral displacement and
θ represents the crosssectional rotation.
The translation v consists of two displacement components, one due to bending and other due to transverse shear deformation
.
The rotation
θ is only due to bending deformation.
3.2.1 Element matrices
The total strain energy
( )
of an undamaged beam element of length l including the shear deformation is written in the form.
U
(
e
)
=
1
2
l
∫
0
E I
∂
θ
∂
x
2
dx
+
1
l
∫
2
0
k
′
G A
∂
v
∂
x
−
θ
2
dx
−
1
l
∫
2
0
P
(
t
)
∂
v
∂
x
2
dx
(3.1) where E is the Young’s modulus, I is the second moment of inertia, k
′ is the shear coefficient, G is the shear modulus and A is the crosssectional area.
The kinetic energy (
T
(e )
) of the beam element considering rotary inertia is given by
T
(
e
)
=
1
2
∫
l
0
ρ
A
∂
v
∂
t
2
dx
+
1
2
∫
l
0
ρ
I
⎛
⎜⎜
∂
θ
∂
t
⎞
⎟⎟
2
dx
(3.2)
A cubic displacement distribution for v is assumed over the element as
v
=
α
1
+
α
2
x
+
α
3
x
2
+
α
4
x
3
(3.3)
21
where
α
1
,
α
2
,
α
3
,
α
4
are called the generalised coordinates. The lateral displacement v and the cross sectional rotation
θ
within the element can be expressed in terms of the shape function matrix and nodal displacement vector {
∆
(e)
} respectively as,
v = [ N
v1
N v2
N v3
N v4
]
⎧
⎪
v i
θ
i
⎪
⎩
v
θ
j j
⎫
⎪
⎪
⎭
= [N
v
]
{ }
θ
= [N
θ
1
N
θ
2
N
θ
3
N
θ
4
]
⎧
⎪
v i
θ
i
⎫
⎪
⎪
⎩
v
θ
j j
⎪
⎭
(3.4)
=[N
θ
] (3.5) where
N
v1
= [13
ζ
2
+2
ζ
3
+ (1
ζ)Φ]/(1+Φ)
N
v2
= [
ζ2ζ
2
+
ζ
3
+ (
ζζ
2
)
Φ/2] l /(1+
Φ
)
N v3
= [3
ζ
2
2
ζ
3
+
ζΦ]/(1+Φ)
N
v4
= [
ζ
2
+
ζ
3
(
ζζ
2
)
Φ/2] l/(1+Φ)
N
θ
1
=6 [
ζ+ζ
2
]/[l (1+
Φ)]
N
θ
2
= [14
ζ
+3
ζ
2
+(1
ζ)Φ]/(1+Φ)
N
θ
3
=6 [
ζζ
2
]/[l (1+
Φ)]
N
θ
4
= [2
ζ
+3
ζ
2
+
ζΦ]/(1+Φ)
ζ= x/l
Φ =12 E I / [k′G A l
2
]
22
The flexural strain or curvature R and the shear strain
γ within the element can be written as
R =
d
θ
dx
= [B
b
]
∆
(e
(3.6)
γ =
dv dx
−
θ
= [B
s
]
∆
(e
(3.7) where
[B
b
] =
d dx
[
N
θ
]
[B
s
] =
d dx
[
N v
]
−
[
N
θ
]
(3.8)
= [B
v
] – [N
θ
] (3.9)
With the help of equations (3.43.9) the potential energy
( )
and the kinetic energy
(e )
of the element can be written in terms of nodal displacement vector,
{ }
as,
U
(e)
=
1
2
{ }
T
[ ]
b
{ } { }
2
T
[ ]
s
{ } { }
2
T
P
(
t
)
[ ]
g
{ }
=
1
2
{ }
T
(
[ ] [ ]
b s
(
e
)
){ } { }
2
T
P
(
t
)
[ ]
g
{ }
=
1
2
{ }
T
[ ]
)
{ } { }
2
T
P
(
t
)
[ ]
g
)
{ }
(3.10)
T
(e)
=
1
2
{ }
T
[ ]
t
{ } { }
2
T
[ ]
r
{ }
=
1
2
&
(
e
T
(
[ ] [ ]
t
M r e
)
) { }
=
1
2
{ }
T
[ ]
{
&
(
e
)
}
(3.11)
where
23
(
e
)
=
l
∫
0
[ ]
b
T
E I
[ ]
b dx
(3.12)
[ ]
s
=
l
∫
0
[ ]
s
T k
'
G A
[ ]
s dx
(3.13)
(
e
=
[ ]
b
) +
[ ]
s
(3.14)
[ ]
g
=
l
∫
0
[ ] [ ]
v v dx
(3.15)
[ ]
t
=
l
∫
0
[ ]
v
T
ρ
A
[ ]
v dx
(3.16)
[ ]
r
=
l
∫
0
[ ]
T
ρ
I
[ ]
dx
(3.17)
[ ] [ ] [
t
M r
(
e
)
]
(3.18)
K
,
K
,
[
K
(e )
] and
[
K g
(e )
] are element bending stiffness , element shear stiffness, element elastic stiffness and element geometric stiffness matrix respectively.
[ ]
t
,
[
M r
(e )
] and
[
M
(e )
] are element translational mass matrix and element rotary inertia mass matrix and element mass matrix respectively.
The total strain energy
( )
d
(e )
of an element within the damaged region including the shear deformation is written in the form,
U d
(
e
)
=
1
2
l
∫
0
E K b
⎝
⎜
⎛
∂
θ
∂
x
⎠
⎟
⎞
2
dx
+
1
l
∫
GK
2
0
s
⎝
⎜
⎛
∂
v
∂
x
−
θ
⎠
⎟
⎞
2
dx
−
1
l
∫
2
0
P
(
t
)
⎝
⎜
⎛
∂
v
∂
x
⎠
⎟
⎞
2
dx
(3.19)
The constants
K b
and
K s
are the measure of the deterioration within the damaged portion and they represent the capacity of the region to store strain energy.
The elastic
24
stiffness matrix for an element in the damaged portion of the beam can be calculated from eqs.( 3.123.14) by using the corresponding effective stiffness EK
b and GK
s
for the damaged region.
Under the assumption that the deterioration produced does not involve a loss of material, the expressions for the mass matrices of an element in the damaged region is same as those given in eqs.( 3.163.18 ).
3.2.2 Governing equations of motion
The total strain energy (
U
) of the beam with damaged portion can be written as
U
=
1
2
L
∫
0
E I
∂
θ
∂
x
2
dx
+
1
2
d
∫
c
(
E K b
−
E I
)
1
2
d
∫
c
(
GK s
−
k
′
G A
)
∂
v
∂
x
−
θ
∂
θ
∂
x
2
dx
−
1
2
L
∫
0
P
(
t
)
2
dx
+
1
2
L
∫
0
k
′
G A
∂
v
∂
x
2
dx
∂
v
∂
x
−
θ
2
dx
+
(3.20)
Under the assumption that the deterioration produced does not involve a loss of material, the expression for kinetic energy (T ) of the damaged beam is given as
T
=
1
2
∫
L
0
ρ
A
∂
v
∂
t
2
dx
+
1
2
∫
L
0
ρ
I
∂
θ
∂
t
2
dx
(3.21)
By dividing the beam in to several elements and assembling the element matrices, the potential energy (U) and the kinetic energy (T) for the damaged beam can be written in terms of global displacement vector {
∆} as,
U
=
1
2
{ }
T
[ ]
{ } { }
2
T
P
(
t
)
[ ]
g
{ }
(3.22)
T
=
1
2
{ }
T
[ ]
{ }
(3.23)
25
where
[
K
]
,
[
M
]
and
[ ]
g
are the global elastic stiffness matrix, global mass matrix and global geometric stiffness matrix respectively.
The equation of motion for the beam is obtained by using the Lagrangian, L=TU in the
Lagrange’s equation.
d dt
∂
L
∂
&
k
−
⎛
⎜⎜
∂
L
∂ ∆
k
⎞
⎟⎟
=
0
, For k=1 to n, n is the total number of coordinates. (3.24)
The equation of motion in matrix form for the axially loaded discretised system is,
[ ]
{ }
+
[ ]
{ }
−
P
(
t
)
[ ]
g
{ }
=
0
(3.25)
P
s,
the static and P
t
, the amplitude of time dependent component of the load, can be represented as the fraction of the fundamental static buckling load P
* of the beam without localised damage and having the similar end conditions. Hence substituting,
P(t) =
α
P
*
+
β
P
*
cos
Ω
t
, with
α and β
as called static and dynamic load factors respectively.
The eq. (3.25) becomes
{ }
+
[ ]
−
α
P
*
[ ]
g s
−
β
P
* cos
Ω
t
[ ]
g t
{ }
=
0
(3.26) where the matrices
[ ]
g s
and
[ ]
g t
reflect the influence of P
s
and P
t
respectively. If the static and time dependent components of the load are applied in the same manner, then
[ ]
g s
=
[ ]
g t
= .
3.2.3 Regions of instability
Equation (3.26) represents a system of second order differential equations with periodic coefficients of the MathieuHill type. From the theory of Mathieu functions [17], it is
26
evident that the nature of solution is dependent on the choice of load frequency and load amplitude. The frequency amplitude domain is divided in to regions, which give rise to stable solutions and to regions, which cause unstable solutions.
The eq.(3.26) does not change its form on addition of the period,
T
=
2
Π
Ω
to t.
This follows from the fact that cos
Ω
(
t
+
T
)
= cos
Ω
t
. Therefore if
∆(t) is a solution of the eq.(3.26), then
∆(t+T) is also its solution.
According to the Floquet solutions the kth solution of eq.(3.26) can be written as,
∆
k
(
t
+
T
)
=
ρ
k
∆
k
(t ) (3.27) where
ρ
k
is the characteristic constant.
These solutions which acquire a constant multiplier by the addition of the period T to t, can be represented in the form
∆
k
( )
=
χ
k
( )
e
( ) ln
ρ
k
(3.28) where
χ
k
is a periodic function of period T.
It follows from the eq.(3.28) that the behaviour of the solutions as t
→ ∞, depends on the value of the characteristic roots, more precisely, on the value of its moduli.
Taking in to account that ln
ρ
k
= ln
ρ
k
+
i
arg
ρ
k
,
∆
k
( )
= Φ
k
( )
e
( ) ln
ρ
k
(3.29) where
Φ
k
( )
=
χ
k
( )
e
(
i t T
) arg
ρ
k
(3.30)
If the characteristic number solution, eq.(3.29) will have an unbounded exponential multiplier, hence the solution is unlimited. If the same characteristic number is less than unity, then the corresponding
27
solution is damped as t increases. Finally, if the characteristic number is equal to unity, then the solution is periodic, i.e. it will be bounded in time. These are the conclusions of the Floquet theory.
Thus the periodic solutions characterise the boundary conditions between the dynamic stability and instability zones. So the periodic solution can be expressed as
Fourier series.
A solution with period 2T is represented by:
∆
(
t
)
=
K
∞
∑
=
1 , 3 ...
⎡
⎢⎣
{ }
k
sin
K
Ω
t
+
{ }
k
cos
2
K
Ω
2
t
⎤
⎥⎦
(3.31)
A solution with period T is represented by:
∆
(
t
)
=
{ }
o
+
K
=
∞
∑
2 , 4 ...
⎡
⎢⎣
{ }
k
sin
K
Ω
t
2
+
{ }
k
cos
K
Ω
2
t
⎤
⎥⎦
(3 .32)
The boundaries of the principal instability regions with period 2T are of practical importance [17]. If the series expansions of eq.(3.31) are used in eq.(3.26), term wise comparison of the sine and cosine coefficients will give infinite systems of homogeneous algebraic equations for the vectors
{ }
k
and
{ }
k
for the solutions on the stability borders. Nontrivial solutions exist if the determinant of the coefficient matrices of these equation systems of infinite order vanishes. When looking for numerical solutions, systems of finite order are required and as it is shown in reference [17], a sufficiently close approximation of the infinite eigenvalue problem is obtained by taking k=1 in the expansion in eq.(3.31) and putting the determinant of the coefficient matrices of the first order equal to zero. This technique is adopted originally in reference [17].
28
Substituting the first order (k=1) Fourier series expansion of eq.(3.31) in eq.(3.26) and comparing the coefficients of sin
Ω
t
2
and cos
Ω
t
terms, the condition for
2 existence of these boundary solutions with period 2T is given by
⎝
⎜⎜
⎛
[ ]
−
(
α
±
β
/ 2
)
P
*
[ ]
g
−
Ω
4
2
[ ]
⎠
⎟⎟
⎞
{ }
=
0
(3.33)
Equation(3.33) represents an eigenvalue problem for known values of
α
,
β
and P
*
. This equation gives two sets of eigenvalues (
Ω
) bounding the regions of instability due to the presence of plus and minus sign. The instability boundaries can be determined from the solution of the equation
[ ]
−
(
α
±
β
/ 2
)
P
*
[ ]
g
−
Ω
4
2
[ ]
=
0
(3.34)
Also the eq. (3.33) represents the solution to a number of related problems.
(i) For free vibration:
α
=
0 ,
β
=
0 &
ω
=
Ω
2
Equation (3.33) becomes
(
[ ]
−
ω
2
[ ]
)
{ }
=
0 (3.35)
(ii) For vibration with static axial load:
β
=
0 ,
α
≠
0 ,
ω
=
Ω
2
Equation (3.33) becomes
−
α
P
*
[ ]
−
2
[ ]
{ }
g
ω
=
0
(3.36)
(iii) For static stability:
α
=
1 ,
β
=
0 ,
Ω =
0
29
(
Equation (3.33) becomes
−
P
*
[ ]
g
)
{ }
=
0
(3.37)
(iv) For dynamic stability, when all terms are present
⎛ Ω
Let
ω
1
⎞
⎟⎟
ω
1 where
ω
1
is the fundamental natural frequency of the beam without damage and having similar boundary conditions. Equation (3.33) then becomes
−
α
β
2
P
*
[ ]
−
g
Θ
ω
2
1
4
[ ]
{ }
=
0
(3.38) where
Θ =
⎝
⎜⎜
⎛ Ω
ω
1
⎠
⎟⎟
⎞
2
The fundamental natural frequency
ω
1
and critical static buckling load P* can be solved using the eqs. (3.35) and (3.37) respectively. The regions of dynamic instability can be determined from eq.(3.38).
3.3 Computational procedure
In the initialisation phase, geometry and material parameters are specified. For example for a Timoshenko beam with localised damage, material parameters like Young’s modulus, mass density and shear modulus of the beam material and geometric parameters like dimensions of the beam, also the specifications of the damage like size of the damage, location of the damage and extent of damage are supplied as input data to the computer program. The beam is divided in to n number of elements and n+1 number of nodes. The elements of the element mass, elastic stiffness and geometric stiffness matrices are computed by GaussLegendre numerical integration method. The program
30
uses the built in Matlab function, ‘quadl’ to carry out the integration. Element matrices are assembled to obtain the global matrices. Boundary conditions are imposed by elimination method. For example for Timoshenko beam with fixedfree end conditions the first two rows and columns of the global matrices are eliminated to obtain the reduced matrices. The natural frequencies and buckling loads are calculated solving the eigenvalue problems in eq. (3.35) and eq.(3.37) respectively. The built in Matlab function
‘eig’ is used to calculate the eigenvalues and eigenvectors. The static load factor
α is set to the required value. The dynamic load factor
β
is increased in steps from zero and for each set of values of
α
,
β
and P
*
the excitation frequencies at the stability boundaries are obtained by solving for the eigenvalues of the eq.(3.34). The lower limit (boundary frequency ratios
Θ
1
) of the instability region is obtained; calculating the eigenvalues of
ω
4
1
2
⎡
⎢
−
1
−
α
β
2
⎠
P
*
[ ]
−
1
[ ]
g
⎤
⎥
− Θ Ι =
0
and upper limit (boundary frequency ratios
Θ
2
) of the instability regions are obtained; calculating the eigenvalues of
4
ω
1
2
⎡
⎢
[ ] [ ]
−
α
β
2
P
*
[ ]
−
1
[ ]
g
⎤
⎥
− Θ Ι =
0
. The plot of
Θ
1
versus
β
gives the lower boundary of the instability region, whereas plot of
Θ
2
versus
β
gives the upper boundary of the instability regions. The computational procedure to obtain the boundary frequencies of the instability regions is shown schematically in the flow chart given in
AppendixA. The entire computational process has been accomplished by computer codes developed in MATLAB [135].
31
3.4 Results and discussion
A beam of length 0.5m, width 20mm and thickness 6.0mm is considered for theoretical analysis. The Young’s modulus of the beam material is taken to be 2.07x10
11
N/m
2
. The following nondimensional parameters are used to study the effects of the damage; (i)
ξ b
=
EK b
/EI, the ratio of the effective stiffness of the damaged portion to that of the undamaged beam  this is a measure of the extent of damage in bending sense;(ii)
ξ s
=GK
s
/k'GA, ratio of the effective stiffness of the damaged portion to that of undamaged beam  this is a measure of the extent of damage in shear sense, (iii)
ψ = f/L, the nondimensional position of damage and (iv)
τ = (dc)/L, size parameter of the damaged region.
The extent of damage both in bending
ξ b
and shear
ξ s
has been taken equal and results are obtained for values of
ξ b
and
ξ s
equal to 0.5 and 1.0 (undamaged condition).
Position parameters (
ψ) of the damaged region are taken to be 0.1, 0.3, 0.5, 0.7 and 0.9 respectively. In the computations the shear coefficient
K ′
is taken as 0.85. The size parameter (
τ) of the damaged region is taken as 0.2. ξ b
=
ξ s
=
ξ has been taken to mark the extent of damage on the graphs. Convergence for the natural frequencies and buckling loads for the first five modes was obtained with a tenelement discretisation. This discretisation also gives satisfactory convergence for the boundary frequencies of the instability regions.
The boundary frequencies of the first instability region for an undamaged Euler beam with pinnedpinned end condition have been compared with those of reference [16] and they are found to be in very good agreement. This is presented in Table 3.1.
32
Table3.1 Comparison of boundary frequencies of the first instability region obtained from present analysis with reference [16], for an undamaged Euler beam with pinnedpinned end conditions. Static load factor
α = 0.0.
Dynamic load factor(
Lower limit of boundary frequency ratio
(
Ω
1
/
ω
1
)
β) Reference[16] Present
(
Upper limit of boundary frequency ratio
Ω
2
/
ω
1
)
Present
0 2.0006 2.0057 2.0006
0.2 1.8979 2.0971 2.0983
0.4 1.7893 2.1886 2.1917
0.6 1.6827 1.6736 2.28 2.2812
0.8 1.5494 2.3565 2.3674
1.0 1.4142 2.4476 2.4505
The boundary frequencies obtained from the present analysis for a cantilever column for different static load factors
α=0.0 and α=0.75, have been compared with reference [111] and they are found to be in very good agreement. The comparison is presented in Table 3.2.
Table3.2 Comparison of boundary frequencies of the first instability region obtained from the present analysis with reference [111], for an undamaged cantilever beam for static load factors
α=0.0 and α=0.75.
Dynamic load factor(
β)
α = 0.0 α = 0.75
Lower limit of boundary
Upper limit of boundary
Lower limit of boundary
Upper limit of boundary
( frequency ratio
Ω
1
/
ω
1
) ( frequency ratio
Ω
2
/
ω
1
) ( frequency ratio
Ω
1
/
ω
1
) ( frequency ratio
Ω
2
/
ω
1
)
Present Ref.[111] Present Ref.[111] Present Ref.[111] Present Ref.[111]
0 2.000 2.000 2.000 2.000 1.029 1.030 1.029 1.030
0.2 1.905 1.904 2.090 2.090 0.979 0.979 1.079 1.079
0.4 1.801 1.802 2.174 2.175 0.925 0.924 1.127 1.126
0.6 1.694 1.692 2.257 2.257 0.865 0.865 1.172 1.171
0.8 1.573 1.573 2.334 2.334 0.802 0.802 1.213 1.214
1.0 1.442 1.442 2.409 2.408 0.733 0.733 1.256 1.256
Figure (3.3) shows the effect of damage position parameter on the fundamental natural frequency of the beam under four boundary conditions considered. As expected for the fixedfixed case the frequency has the highest value and is minimum for the fixedfree end condition for any position of the damage. For fixedfree end condition the
33
fundamental frequency has the minimum value when the damage is located near to the fixed end and it increases as the damage moves towards the free end. For pinnedpinned and fixedfixed end conditions the minimum value occurs, when the damage is at the middle. For fixedpinned case the beam has minimum fundamental frequency when the damage position is in between middle and pinned end.
Figures (3.43.5) show the second and third mode frequencies, which shows that values of both the frequencies depend on the damage position for different boundary conditions.
Figure (3.6) shows the effect of damage position on the critical buckling load. The buckling load varies with damage position in the same manner as that of fundamental frequency for the four boundary conditions.
In figs.(3.7–3.10) the instability regions for a damaged beam for various locations of damage are shown along with the instability region for an undamaged beam(
ξ=1.0) for the four boundary conditions . It is seen that the instability regions for an undamaged beam occur at higher excitation frequency compared to a beam with damage at any location. This means that presence of localised damage or in other words extent of damage enhances the instability of the beam, since parametric instability occurs at lower frequency of excitation.
Figures 3.7(a)3.7(c) show the first three instability regions respectively for fixedfree end condition for different damage positions. It is seen from fig. 3.7(a) that as the damage moves from the fixed end to the free end the first principal instability region moves away from the dynamic load factor axis and the width of the instability regions reduces. When the damage is near to the free end the instability region almost coincides
34
with the instability region for the undamaged beam. This means that the damage near the fixed end is more severe on the dynamic instability behavior than that of the damage located at other positions, so far as first instability region is concerned. Figure 3.7(b) shows that the beam is most unstable so far as second instability region is concerned, when the damage is located at the middle. Figure 3.7(c) shows that the third principal instability region occurs at minimum frequency of excitation when the damage position is in between the middle and free end.
Figures 3.8(a)3.8(c) show the first three instability regions respectively for pinnedpinned end conditions. The first instability region occurs at minimum frequency of excitation when the damage is located at the middle. When the damage moves towards any of the pinned end from middle the first instability zone moves away from the dynamic load factor axis. The second instability region occurs at minimum frequency of excitation when the damage is located in between middle and any one of the pinned ends and at highest frequency when the damage is located at the middle. The third instability zone occurs at minimum frequency of excitation when the damage is at the middle. Because of symmetry, instability zones for
ψ =0.1 coincides with ψ =0.9 and ψ
=0.3 coincides with
ψ =0.7.
Figures 3.9(a)3.9(c) show the first three instability regions respectively for the beam with fixedfixed end conditions. The first instability region occurs at minimum frequency of excitation when the damage is at the middle and relocates itself at higher frequency when the damage is away from the middle position. The second instability region occurs at minimum frequency of excitation when the damage is in between middle and either of the fixed ends of the beam and at highest frequency of excitation for
ψ =0.5.
35
Third instability region occurs at minimum frequency of excitation when the damage is at the middle. Because of symmetry instability zones for
ψ =0.1 coincides with ψ =0.9 and
ψ =0.3 coincides with ψ =0.7.
Figures 3.10(a)3.10(c) show the first three instability regions respectively for fixedpinned end condition. From fig. 3.10(a) it is seen that the first instability region occurs at minimum frequency of excitation when the damage position is in between the middle and the pinned end and it occurs at highest frequency of excitation when the damage is in between the middle and fixed end. The second principal instability region occurs at minimum frequency of excitation when the damage is located in between the fixed end and the middle, fig. 3.10(b). Whereas the third principal instability region occurs at minimum frequency of excitation when the damage is located nearer to the pinned end, fig. 3.10(c).
Figures 3.11(a)3.11(c) show the first three instability regions respectively for fixedfree end condition for static load factor
α=0.0 and α=0.5. It is seen that the static load component shifts the instability regions towards the lower frequency of excitation and there is also increase in areas of the instability regions. The effect is more predominant on the first instability region than on other two regions. This means that the static load component has a destabilising effect in terms of the shifting of the instability regions towards lower frequencies of excitation and increase in areas of the instability regions.
Figures 3.12(a)3.12(c) show the first three instability regions respectively for pinnedpinned end condition for static load factor
α=0.0 and α=0.5. Similar behaviour as those for fixedfree end condition is also observed in this case.
36
Figures 3.13(a)3.13(c) show the effect of
α on the instability regions for fixedfixed end condition. Static load component has a destabilising effect in terms of shifting of the instability regions to lower frequencies of excitation and increase in areas.
Figures 3.14(a)3.14(c) show the effect of
α for fixedpinned case. The effect is same as those for other three end conditions discussed earlier.
3.5 Closure
The effect of localised damage on the stability of a beam with various boundary conditions has been analysed. The critical position of the damage for minimum values of natural frequencies depends on the boundary conditions and mode number. Critical buckling load also depends on the damage location and boundary conditions. Presence of damage always increases the instability of the beam. The critical position of the damage for maximum destabilising effect on the beam is different depending on the boundary conditions and the principal regions of instability of interest. Increase in static load component has a destabilising effect for all boundary conditions considered. It is observed that the dynamic stability behaviour of the beam depends not only upon the boundary conditions but also on the location of the damage.
c d
f
L
(a)
(b)
(c)
Fig 3.1
37
(d)
Fig.3.1, Beam with boundary conditions;
(a) Fixedfree, (b) Pinnedpinned, (c) Fixedfixed,
(d) Fixedpinned.
Y
v i v j
θ
i
θ
j
X
Fig.3.2, Timoshenko beam element.
38
39
140
120
100
80
60
40
20
0
0.1
0.2
0.3
0.4
0.5
Damage position
ψ
0.6
0.7
0.8
Fig. 3.3,Effect of damage position on first mode frequency,
ξ
=0.5,
fixedfixed; ,fixedpinned;...,pinnedpinned;  , fixedfree;. ,.
0.9
300
250
200
150
400
350
100
0.1
0.2
0.3
0.4
0.5
Damage position
ψ
0.6
0.7
0.8
0.9
Fig. 3.4,Effect of damage position on second mode frequency,
ξ
=0.5,key as fig. 3.3.
40
700
650
600
550
500
450
400
350
300
0.1
0.2
0.3
0.4
0.5
Damage position
ψ
0.6
0.7
0.8
Fig. 3.5,Effect of damage position on third mode frequency,
ξ
=0.5,key as fig. 3.3.
0.9
12
10
8
6
4
2
0
0.1
0.2
0.3
0.4
0.5
Damage position
0.6
0.7
0.8
0.9
ψ
Fig. 3.6,Effect of damage position on first mode buckling load,
ξ
=0.5,key as fig. 3.3.
41
1
0.8
0.6
0.4
0.2
(a)
0
0.8
Stable
1 2.2
Stable
2.4
1.2
1.4
1.6
1.8
Frequency ratio
Ω
/
ω
1
2 2.6
1
0.8
0.6
0.4
0.2
(b)
0
10
Stable
10.5
Stable
11 11.5
12
Frequency ratio
Ω
/
ω
1
12.5
13 13.5
1
0.8
0.6
0.4
0.2
(c)
Stable Stable
Stable
0
30 31 32 33
Frequency ratio
Ω
/
ω
1
34 35 36
Fig. 3.7,Effect of damage position on instability regions,fixedfree end condition,
α
=0.0,
ξ
=1.0;,
ξ
=0.5;
ψ
= 0.1(+),
ψ
= 0.3 (*),
ψ
= 0.5(o),
ψ
=0.7(
•
),
ψ
=0.9(
∨
).
42
1
0.8
0.6
0.4
0.2
(a)
0
0.8
Stable
1 1.2
Stable
1.4
1.6
1.8
Frequency ratio
Ω
/
ω
1
2 2.2
2.4
2.6
1
0.8
0.6
0.4
0.2
(b)
0
6.5
Stable
Stable
7 7.5
Frequency ratio
Ω
/
ω
1
8 8.5
1
0.8
0.6
0.4
0.2
(c)
Stable
Stable
Stable
0
15.5
16 16.5
17 17.5
18 18.5
Frequency ratio
Ω
/
ω
1
Fig. 3.8,Effect of damage position on instability regions, pinnedpinned end condition,
α
=0.0,
ξ
=1.0;,
ξ
=0.5;key as fig. 3.7.
19
43
1
0.8
0.6
0.4
0.2
(a)
0
1
Stable
Stable
1.5
Frequency ratio
Ω
/
ω
1
2 2.5
1
0.8
0.6
0.4
0.2
(b)
0
4.2
4.4
Stable
4.6
Stable
4.8
5 5.2
5.4
Frequency ratio
Ω
/
ω
1
5.6
5.8
6 6.2
1
0.8
0.6
0.4
0.2
(c)
0
9
Stable
Stable
9.5
10 10.5
Frequency ratio
Ω
/
ω
1
11 11.5
12
Fig. 3.9,Effect of damage position on instability regions,fixedfixed end condition,
α
=0.0,
ξ
=1.0;,
ξ
=0.5;key as fig. 3.7.
44
1
0.8
0.6
0.4
0.2
(a)
0
1
Stable
Stable
1.5
Frequency ratio
Ω
/
ω
1
2
1
0.8
0.6
0.4
0.2
(b)
0
5.2
5.4
Stable
5.6
Stable
5.8
6 6.2
6.4
Frequency ratio
Ω
/
ω
1
6.6
6.8
7
1
0.8
0.6
0.4
Stable
0.2
(c)
0
11.5
Stable
12 12.5
13
Frequency ratio
Ω
/
ω
1
13.5
14
Fig. 3.10,Effect of damage position on instability regions, fixedpinned end condition,
α
=0.0,
ξ
=1.0;,
ξ
=0.5;key as fig. 3.7.
2.5
7.2
14.5
45
1
0.8
0.6
0.4
0.2
(a)
0
0 0.5
1 1.5
Frequency ratio
Ω
/
ω
1
2 2.5
1
0.8
0.6
0.4
0.2
(b)
0
9.4
9.6
9.8
10 10.2
10.4
10.6
Frequency ratio
Ω
/
ω
1
10.8
11 11.2
11.4
1
0.8
0.6
0.4
0.2
(c)
0
33 33.5
34 34.5
Frequency ratio
Ω
/
ω
1
Fig. 3.11,Effect of static load factor on instability regions,fixedfree end condition,
ξ
=0.5,
ψ
=0.5;
α
=0.0(o);
α
=0.5(+).
1
0.8
0.6
0.4
0.2
(a)
0
0 0.5
1
Frequency ratio
Ω
/
1.5
ω
1
2
1
0.8
0.6
0.4
0.2
(b)
0
6.6
6.8
7 7.2
7.4
7.6
Frequency ratio
Ω
/
ω
1
7.8
8 8.2
1
0.8
0.6
0.4
0.2
(c)
0
15.2
15.4
15.6
15.8
16 16.2
Frequency ratio
Ω
/
ω
1
16.4
16.6
Fig. 3.12,Effect of static load factor on instability regions, pinnedpinned end condition,
ξ
=0.5,
ψ
=0.5;key as fig. 3.11.
16.8
2.5
8.4
17
46
47
1
0.8
0.6
0.4
0.2
(a)
0
0 0.5
1 1.5
Frequency ratio
Ω
/
ω
1
2 2.5
1
0.8
0.6
0.4
0.2
(b)
0
3.5
4 4.5
5
Frequency ratio
Ω
/
ω
1
5.5
6 6.5
1
0.8
0.6
0.4
0.2
(c)
0
8 8.5
9 9.5
Frequency ratio
Ω
/
ω
1
10 10.5
11
Fig. 3.13,Effect of static load factor on instability regions,fixedfixed end condition,
ξ
=0.5,
ψ
=0.5;key as fig. 3.11.
48
1
0.8
0.6
0.4
0.2
(a)
0
0 0.5
1 1.5
Frequency ratio
Ω
/
ω
1
2
1
0.8
0.6
0.4
0.2
(b)
0
4.5
5 5.5
Frequency ratio
Ω
/
ω
6
1
6.5
1
0.8
0.6
0.4
0.2
(c)
0
11.2
11.4
11.6
11.8
12 12.2
12.4
Frequency ratio
Ω
/
ω
1
12.6
12.8
Fig. 3.14,Effect of static load factor on instability regions, fixedpinned end condition,
ξ
=0.5,
ψ
=0.5;key as fig. 3.11.
13
2.5
7
13.2
Chapter
4
DYNAMIC STABILITY OF A PRETWISTED
CANTILEVER BEAM WITH LOCALISED
DAMAGE UNDER PERIODIC AXIAL FORCE
4.1 Introduction
Twisted beams have wide application in many industrial problems. Compressor blades, turbine blades, aircraft propeller blades, helicopter rotor blades, twist drill bits etc. can be modeled as twisted cantilever beams. The dynamic stability analysis of these elements is of considerable importance. Though there are many alloys and composite materials having high strength to weight ratio have been developed, during the manufacturing of these materials, inclusion of flaws affects their structural strength. Effect of localised damages on the stability behaviour of structural elements is of great importance.
There are relatively a few studies on the dynamic behaviour of pretwisted beams.
The earliest vibration analysis of pretwisted beams was reported by Troesch et al. [126] and Anlinker and Troesch [4]. Gupta and Rao [46] reported the natural frequencies of tapered Timoshenko beam using finite element method. Carniege Thomas [20] analysed
49
50
the coupled bendingbending vibration of pretwisted cantilever Euler beams.
Subrahmanyam et al. [117] studied the vibration of pretwisted cantilever beams using the Reissner method. Banerjee [9] studied the free vibration of twisted beams by dynamic stiffness method. The stability of pretwisted columns under compressive axial loads was reported by FrischFay [42]. Celep [21] analysed the dynamic stability of a simply supported pretwisted column. Gürgöze [47] studied the dynamic stability of pretwisted beams with hingedhinged, clampedclamped and clampedhinged boundary conditions.
Many investigators as mentioned earlier have studied the effect of localised damage on the dynamic stability of beams. Reported literature show that the effect of localised damage on the dynamic stability of a pretwisted cantilever beam has not been investigated.
In the present work the effects of a localised damage on the dynamic stability of a pretwisted cantilever beam subjected to a timedependent conservative end axial force is studied. The effects of parameters like the pretwist angle, the extent of damage, position of the damage and static load factor are studied. Three parameters are used to characterise the damaged region: location, size, and effective stiffness of the damaged region. Euler beam theory is used in the analysis. The equation of motion has been derived using finite element method. The principal instability regions are established by Floquet’s theory.
4.2 Formulation of the problem
The considered twisted beam configuration is shown in fig.4.1(a). The beam is of uniform rectangular crosssection and the angle of twist of the crosssection is assumed to be varying linearly along the length. The beam has a length L, width b and depth h. The angle of pretwist at the free end is
θ
1 with respect to the fixed end. The effect of the
51
damage is represented by the presence of a flaw in the region c
< x < d. The beam is subjected to a pulsating axial force P(t) = P
s
+ P
t cos
Ω
t
acting along its undeformed axis, where Ω is the excitation frequency of the dynamic load, P
s is the static and P
t is the amplitude of the time dependent component of the load.
The degrees of freedom of a typical finite beam element are shown in fig.4.2. The element consists of two nodes and each node has four degrees of freedom: two translations and two rotations. v and w are the two translations in Y and Z directions and
θ and ϕ
are the two rotations about
Z
and
Y
axes. The translations and rotations are due to bending only.
4.2.1 Element matrices
The total strain energy (U
(e)
) of an undamaged beam element of length l is expressed in the form
U
(
e
)
=
1
2 l
∫
0
E I zz
⎛
⎝
∂
2 v
∂
x
2
⎟
⎠
⎞
2 dx
+
1
2 l
∫
0
E I yy
⎛
⎝
∂
2 w
∂
x
2
⎟
⎠
⎞
2 dx
+
l
∫
0
E I yz
⎛
⎝
∂
2 v
∂
x
2
⎟
⎛
⎟
⎝
⎞
⎠
⎜
⎜
∂
2 w
⎞
∂
x
2
⎟
⎠
dx
−
1
2 l
∫
0
P
(
t
)
⎡
⎛
⎣
⎜⎜
⎢
⎢
⎝
∂
v
∂
x
⎞
⎟⎟
2
+
⎛
⎜⎜
∂
∂
w x
⎞
⎟⎟
2
⎤
⎥
⎦
dx
(4.1) where E is the Young’s modulus.
I yy
,
I zz
and
I yz
are the crosssectional area moments and product of inertia of the beam with respect to the Y and Z axes, respectively.
The kinetic energy (
T
(e)
) of an undamaged beam element of length l is given as
T
(
e
)
=
1
2
+
∫
l
0
ρ
1
2
∫
l
0
ρ
A
⎛
⎜⎜
∂
v
∂
t
⎞
⎟⎟
2
dx
I yy
∂
∂
x
2
w
∂
t
+
2
dx
1
2
+
∫
l
0
ρ
∫
l
0
ρ
A
⎛
⎜⎜
∂
w
∂
t
⎞
⎟⎟
2
dx
+
I yz
∂
∂
x
2
v
∂
t
1
2
∫
l
0
ρ
∂
∂
x
2
w
∂
t
I zz dx
∂
∂
x
2
v
∂
t
2
dx
(4.2)
As the element is twisted, the moment of inertia
I yy,
I
zz and
I
yz are function of x.
I yy,
I
zz and I
yz
at any cross section x, can be expressed in terms of moment of inertia
I y
'y '
52
and
I z
' z' of the cross section about principal inertia axes Y
′Y′ and Z′Z′ respectively and angle of twist
θ
t
at the cross section x, fig.4.1(b).
I zz
(x) =
I z
' z' cos
2 θ
t
+
I y
'
y
' sin
2 θ
t
I yy
(x) =
I y
'
y
' cos
2
θ
t
+
I z
'
z
' sin
2
θ
t
I
I
I yz
(x) =
1
2
(
I z
'
z
'
z
' z'
=
bh
3
/ 12
y
'
y
'
=
hb
3
/ 12

I y
'
y
'
) sin2
θ
t
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
(4.3)
A cubic displacement distribution for both v and w is assumed over the element as
v
=
a
1
+
a
2
x
+
a
3
x
2 +
a
4
x
3
w
=
a
5
+
a
6
x
+
a
7
x
2
+
a
8
x
3
(4.4) where
a
1
−
a
8
are the generalised coordinates. The lateral displacements v, w within the element can be expressed in terms of the nodal displacement vector {
∆ (e)
} and the shape function matrix .
{
∆
(e)
} = {
v i
θ
i v j
θ
j w i
ϕ
i w j
ϕ
j
}
T
=
[ ]
T
(4.5) where
{ }
=
{
v i
θ
i v j
θ
j
}
and
{ }
=
{
w i
ϕ
i w j
ϕ
j
}
v w
⎬
=
⎡
⎢
N
1
0
N
0
2
N
3
N
4
0 0
0 0 0
N
1
N
2
N
3
0
N
4
⎤
⎥
{ }
53
=
⎡
⎢
[ ] [ ]
[ ] [ ]
⎤
⎥
⎡
⎢
⎣
{ }
{ }
⎤
⎥
⎥
=
[ ]
{ }
(4.6) where
[ ]
=
[
N
1
N
2
N
3
N
4
]
,
[ ] [
0 0 0 0
]
N
1
= [13
ζ
2
+2
ζ
3
]
N
2
= [
ζ2ζ
2
+
ζ
3
] l
N
3
= [3
ζ 2
2
ζ 3
]
N
4
= [
ζ 2
+
ζ 3
] l
ζ = x/l
With the help of equation (4.6) the potential energy and kinetic energy
T
( ) of the element can be expressed in terms of nodal displacement vector as,
U
(
e
)
=
1
2
{ }
T
[ ]
)
{ } { }
2
T
P
(
t
)
[ ]
)
{ }
(4.7)
T
(
e
) =
1
2
{ }
T
[ ]
{ }
(4.8) where are element elastic stiffness matrix, geometric stiffness matrix and mass matrix respectively.
=
⎡
⎢
⎣
[ ] [ ]
[ ][ ]
⎥
⎤
⎥
(4.9)
[ ]
=
l
∫
0
[ ]
″
T
E I zz
[ ]
″
dx
(4.10)
[ ]
=
l
∫
0
[ ]
″
T
E I yy
[ ]
″
dx
(4.11)
54
[ ]
=
l
∫
0
[ ]
″
T
E I yz
[ ]
″
dx
(4.12)
[ ]
g e
=
⎡
⎢
⎣
[ ]
K
[ ]
4
g
⎥
⎤
⎥
(4.13)
[ ]
g
=
∫
l
0
[ ] [ ]
′
dx
(4.14)
=
⎡
⎢
⎣
[ ] [ ] [ ]
[ ] [ ] [ ]
⎥
⎤
⎥
(4.15)
[ ]
=
l
∫
o
[ ]
T
ρ
A
[ ]
dx
(4.16)
[ ]
=
l
∫
o
[ ]
′
T
I zz
[ ]
′
dx
(4.17)
[ ]
=
l
∫
o
[ ]
′
T
I yy
[ ]
′
dx
(4.18)
[ ]
=
l
∫
o
[ ]
′
T
I yz
[ ]
′
dx
(4.19)
[ ]
″
=
∂
2
∂
x
2
[ ] [ ]
′
=
∂
∂
x
[ ]
The total strain energy
( )
of an element within the damaged region is expressed in the form.
U d
(
e
)
=
1
2
l
∫
0
(
ξ
b
E I zz
)
⎛
⎜
⎝
∂
2
∂
x v
2
⎟
⎠
2
dx
+
1
2
l
∫
0
(
ξ
b
E I yy
)
⎛
⎜
⎝
∂
2
∂
x w
2
⎟
⎠
2
dx
+
l
∫
0
(
ξ
b
E I yz
)
⎛
⎜
⎝
∂
2
v
∂
x
2
⎟
⎝ ⎠
⎜
⎜
⎛
∂
2
∂
x w
⎞
2
⎟
⎠
dx
−
1
2
l
∫
0
P
(
t
)
⎡
⎛
⎣
⎜⎜
⎢
⎢
⎝
∂
∂
v x
⎞
⎟⎟
2
+
⎛
⎜⎜
∂
∂
w x
⎞
⎟⎟
2
⎤
⎥
⎦
dx
(4.20)
55
The constant elastic stiffness matrix of an element in the damaged region of the beam can be calculated from eqs. (4.94.12) by using the corresponding effective stiffness of the damaged region.
Under the assumption that the deterioration produced does not involve a loss of material, the expression for the mass matrix of an element in the damaged region is same as that given in eq.(4.15).
4.2.2 Governing equations of motion
The potential energy (U) of the beam with damaged portion can be written as,
U
=
1
2
L
∫
0
E I zz
⎛
⎝
∂
2
∂
x
2 v
⎟
⎠
⎞
2 dx
+
1
2
L
∫
0
E I yy
⎛
⎝
∂
2
∂
x
2 w
⎟
⎠
2 dx
+
L
∫
0
E I yz
⎛
⎝
∂
∂
2 v x
2
⎞
⎠
⎛
⎝
∂
2 w
∂
x
2
⎞
⎠
+
1
2 d
∫
c
E I yy
(
ξ
b
−
1
)
⎛
⎜
⎝
∂
∂
2 w x
2
⎟
⎠
2 dx
+
d
∫
c
E I yz
(
ξ
b
−
1
)
⎛
⎜
⎝
∂
∂
2 v x
2
⎞
⎠
⎛
⎝
∂
∂
2 w x
2
⎞
⎠
dx dx
−
+
1
2 d
∫
c
(
E I zz
(
ξ
b
−
1
)
⎛
⎜
⎝
∂
∂
2 v x
2
⎟
⎠
⎞
2 dx
1
2
L
∫
0
P
(
t
)
⎡
⎛
⎣
⎜⎜
⎢
⎢
⎝
∂
v
∂
x
⎞
⎟⎟
2
+
⎛
⎜⎜
∂
∂
w x
⎞
⎟⎟
2
⎤
⎥
⎦
dx
(4.21)
Under the assumption that the deterioration produced does not involve a loss of material, the expression for kinetic energy ( T) of the damaged beam is given as
T
=
1
2
∫
L
0
ρ
A
⎛
⎜⎜
∂
∂
t v
⎠
⎞
2
dx
+
1
2
∫
L
0
ρ
A
⎛
⎜⎜
∂
∂
t w
⎠
⎞
2
dx
+
1
2
∫
L
0
ρ
I zz
∂
2
v
∂
x
∂
t
2
dx
+
1
2
∫
L
0
ρ
I yy
∂
2
w
∂
x
∂
t
2
dx
+
∫
L
0
ρ
I yz
∂
2
v
∂
x
∂
t
∂
2
w
∂
x
∂
t dx
(4.22)
By dividing the beam in to several elements, the potential energy ( U) and the kinetic energy ( T) for the damaged beam can be written in terms of global displacement vector, {
∆} as
U
=
1
2
{ }
T
[ ] { } { }
2
T
P
(
t
)
[ ]
{ }
(4.23)
56
T
=
1
2
{ }
T
[ ]
{ }
(4.24) where
[ ]
,
[ ]
and
[
K g
]
are the global elastic stiffness, global mass and global geometric stiffness matrices respectively.
Substituting Lagrangian, L=TU in the Lagrange’s equation, the final equation of motion for the damaged beam in matrix form is written as,
[ ]
{ }
+
[ ]
{ }
−
P
(
t
)
[ ]
g
{ }
=
0
(4.25)
P s
the static and
P t
the amplitude of time dependent component of the load, can be represented as the fraction of the fundamental static buckling load
P
* of a geometrically similar undamaged straight cantilever beam. Hence substituting, P(t) =
α
P
*
+
β
P
*
cos
Ω with
α and
β
as static and dynamic load factors respectively.
The eq. (4.25) becomes
{ }
+
[ ]
−
α
P
*
[ ]
s
−
β
P
* cos
Ω
t
[ ]
t
{ }
=
0
(4.26)
If the static and time dependent component of loads are applied in the same manner, then
[ ]
s
=
[ ]
t
=
[ ]
.
Equation (4.26) represents a system of second order differential equations with periodic coefficients of the MathieuHill type. The regions of instability are established using Floquet’s theory which reduces eq.(4.26) to an eigenvalue problem given below, eq.(4.27).
57
−
α
β
2 ⎠
P
*
[ ]
− Θ
ω
4
2
1
[ ]
{ }
=
0
(4.27)
ω
1
is the fundamental frequency of the geometrically similar undamaged straight cantilever beam. Following the same computational procedure as described in chapter 3 the boundaries of the instability regions can be established.
4.3 Results and discussion
To check the accuracy of the present computation, natural frequencies for the first four modes were compared with those of numerical results reported relating to pretwisted blade treated experimentally and by theoretical means, by Dawson [32], Rao [97],
Carnegie [19] and Banerjee [9].
Table 4.1(a), Comparison of natural frequencies [Hz] with those of [32], [97] and [19]
Mode
No.
Present FEM analysis
(7 Elements)
Dawson [32] Rao [97] Carnegie [19]
Experimental
Table 4.1(b), Comparison of natural frequencies [rad/s] with those of [9]
Mode No.
Present FEM analysis
(7 Elements)
1 2 3 4
3.4742 13.3604 25.1918 56.4764
Banerjee [9] 3.47173 13.3465 25.1707 56.3716
The above results show close matching of the present one with the available results.
58
The following properties of the beam are taken for numerical computations:
Length of the beam = 0.152m; cross – sectional dimension = 25.4 x 4.6 mm; material mass density of the beam = 8000 kg /m
3
. The size parameter
τ is equal to 0.2 unless stated otherwise.
Figures (4.34.5) show the effect of pretwist angle (
θ
1
) on the first three natural frequencies for a beam with an extent of damage
ξ b
=
ξ= 0.5 and for different positions of the damage ( ψ ). On the same figures, variation of natural frequencies for an undamaged beam (
ξ=1.0) with pretwist angle is also shown. It is seen that the first and third mode natural frequencies increase with increase in pretwist angle, whereas the second mode natural frequency decreases with increase in pretwist angle. This is due to the fact that with increase in pretwist angle, moment of inertia about the axis of first and third mode vibration increases, whereas it decreases about the axis of second mode vibration. All the three frequencies have minimum value when the damage is located nearest to the fixed end. The variation in first mode natural frequency with pretwist angle is very less compared to the change in second and third mode natural frequencies. When the damage is very near to the free end (
ψ=0.9) the frequency curves for the damaged beam are very close to those for the undamaged beam.
Figures (4.64.8) show the variation of first three natural frequencies with damage location for angles of pretwist 30
0
, 60
0 and 90
0 respectively
.
The first mode natural frequency increases as the damage position moves from the fixed end towards the free end for all values of
θ
1
. As the damage moves towards the free end, the rate of increase of frequency with respect to pretwist angle reduces. The second and third mode frequencies have minimum and maximum values when the damage is near the fixed end and free end
59
respectively. For intermediate positions of the damage the values of these two frequencies fluctuate between these maximum and minimum limits.
Figure (4.9) shows the effect of pretwist angle on the first mode buckling load. It is observed that the first modebuckling load increases with pretwist angle irrespective of damage position. The fundamental buckling load is minimum when damage is near the fixed end and as the damage moves towards the free end, the buckling load increases and approaches the value for an undamaged beam when it is nearest to the free end.
Figures (4.104.12) show the effect of pretwist angle on the first three instability zones for a beam with extent of damage
ξ = 0.5 and damage position ψ = 0.3, 0.5 and 0.7
respectively. It is seen that with increase in pretwist angle the first and third instability zones shift to higher frequencies of excitation, whereas the second instability zone shifts to lower frequency of excitation. The effect is more pronounced on second and third zones as compared to first one. This is due to the fact that the first mode natural frequency is least affected by the change in pretwist angle. There is also change in areas of the instability zones. There is very small change in the area of the first instability zone, whereas there is a marked increase in area of the second zone and decrease in third zone area.
Figures 4.13(a)4.13(c) show the effect of location of damage (
ψ) on the first three instability regions respectively for
ξ=0.5 and
θ
1
= 30
0
. It is seen that all the three instability regions occur at minimum frequency of excitation when the damage is nearest to the fixed end. When the damage is at the free end the instability regions occur at maximum frequency of excitation. As the damage moves from the fixed end to free end the first instability region moves away from the dynamic factor axis. There is reversal of
60
positions of second and third instability zones as the damage moves from fixed end to the free end. This is due the reason that when damage moves from the fixed end to the free end, the second and third mode natural frequencies may increase or decrease depending on the damage position (Figs. 4.7 and 4.8). Figures (4.14 and 4.15) show the instability regions for
θ
1
= 60
0
and 90
0 respectively. Similar trends as those for
θ
1
= 30
0
are observed for these two cases also.
Figures (4.164.18) show the effect of extent of damage
ξ on instability zones for angles of pretwist 30
0
,60
0 and 90
0
respectively for damage position
ψ = 0.1. The values of
ξ considered are 0.5 and 0.75. It is noticed that with increase in extent of damage the instability zones shift to lower frequencies of excitation irrespective of pretwist angle.
Also the areas of the instability zones increase with increase in extent of damage. That is as the extent of damage increases the instability increases.
Figures (4.194.21) show the effect of static load factor
α on instability zones for angle of pretwist 30
0
,60
0 and 90
0
respectively for damage position
ψ = 0.1. The values of
α considered are 0.0 and 0.5. It is noticed that with increase in static load component the instability regions are relocated at lower frequencies of excitation and their areas also increase. The effect is more pronounced on first instability region. This means that increase in static load component enhances the instability of the beam.
4.4 Closure
From the study of dynamic stability of a pretwisted cantilever beam with localised damage subjected to end parametric excitation it is revealed that angle of pretwist has significant effect on second and third instability zones. Increase in pretwist angle has a
61
stabilising effect on the third instability zone and destabilising effect on the second instability zone. Variation in pretwist angle does not have significant effect on the first instability region. Extent of damage has always a destabilising effect for any pretwist angle. Localised damage has a greater destabilising effect when it is located near the fixed end than when it is towards the free end, irrespective of the pretwist angle. Increase in static load component has a destablising effect irrespective of the value of pretwist angle.
c
d
Y
θ
1
Y' f
L
Fig. 4.1(a), Pretwisted beam and coordinate axes
Z
Z'
X, X '
P s
+
P t cos Ω t
Y
Y ′
Z
Z′
θ
t
Y
Y ′
Fig. 4.1(b), Angle of twist (
θ
t
)
Z ′
Z v i
,
ϕ
i v j
,
ϕ
j w j ,
θ
j w i ,
θ
i
62
Fig. 4.2, Degrees of freedom of a beam element
63
170
165
160
155
150
145
140
135
130
125
120
30
ξ=1.0
Ψ=0.3,ξ=0.5
Ψ=0.7,ξ=0.5
Ψ=0.1,ξ=0.5
Ψ=0.5,ξ=0.5
Ψ=0.9,ξ=0.5
40 50 60 70 80
Pretwist angle(
θ
1
) in degree
Fig. 4.3, Effect of pretwist angle on first mode natural frequency
90 100
900
850
800
750
700
650
600
550
500
450
400
30
ξ=1.0
Ψ=0.1,ξ=0.5
Ψ=0.3,ξ=0.5
Ψ=0.5,ξ=0.5
Ψ=0.7,ξ=0.5
Ψ=0.9,ξ=0.5
40 50 60 70
Pretwist angle (
θ
1
) in degree
80
Fig. 4.4,Effect of pretwist angle on second mode natural frequency
90 100
64
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
ξ=1.0
Ψ=0.3,ξ=0.5
Ψ=0.7,ξ=0.5
Ψ=0.1,ξ=0.5
Ψ=0.5,ξ=0.5
Ψ=0.9,ξ=0.5
800
30 40 50 60 70
Pretw ist angle(
θ
1
) in degree
80
Fig. 4.5, Effect of pretwist angle on third mode natural frequency
90
170
165
160
155
150
θ
1
=30
0
θ
1
=60
0
θ
1
=90
0
145
140
135
130
125
120
0 0.1
0.2
0.3
0.4
0.5
0.6
ψ
)
0.7
0.8
0.9
Fig. 4.6, Effect of damage location on first mode natural frequency
100
1
65
800
600
550
500
450
750
700
650
θ
1
=30
0
θ
1
=60
0
θ
1
=90
0
400
0 0.1
0.2
0.3
0.4
0.5
0.6
Damage location (
ψ
0.7
0.8
0.9
Fig. 4.7,Effect of damage location on the second mode natural frequency
1
1800
1700
1600
1500
1400
1300
1200
1100
θ
1
=30
0
θ
1
=60
θ
1
=90
0
0
1000
900
800
0 0.1
0.2
0.3
0.4
0.5
0.6
Damage location (
ψ
)
0.7
0.8
Fig. 4.8, Effect of damage location on third mode natural frequency
0.9
1
6
66
ξ=1.0
Ψ=0.1,ξ=0.5
Ψ=0.3,ξ=0.5
Ψ=0.5,ξ=0.5
Ψ=0.7,ξ=0.5
Ψ=0.9,ξ=0.5
5
4
3
30 40 50 60 70
Pretw ist angle (
θ
1
) in degree
80
Fig. 4.9,Effect of pretwist angle on critical buckling load
90 100
0.6
0.5
0.4
0.3
0.2
0.1
1
0.9
0.8
0.7
Stable
2
ω
2
Stable
Stable
0
0
ξ
=0.5,
5
ψ
=0.3,
θ
1
= 30
10
Frequency ratio
Ω
/
1
Fig. 4.10,Effect of pretwist angle on instability regions,
0
(*),
θ
1
= 60
0
(+),
θ
15
ω
1
= 90
0
(o).
2
ω
3
Stable
20 25
67
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Stable
2
ω
2
Stable
Stable
2
ω
3
Stable
0
0 5 10 15 20 25
Frequency ratio
Ω
/
ω
1
Fig. 4.11,Effect of pretwist angle on instability regions,
ξ
=0.5,
ψ
=0.5,key as fig. 4.10.
0.7
0.6
0.5
1
0.9
0.8
0.4
0.3
0.2
0.1
2
ω
2
2
ω
3
0
0 5 10
Frequency ratio
Ω
/
15
ω
1
20 25
Fig. 4.12,Effect of pretwist angle on instability regions,
ξ
=0.5,
ψ
=0.7,key as fig. 4.10.
1
0.8
0.6
0.4
0.2
0
0.8
(a)
1 1.2
1.4
1.6
1.8
Frequency ratio
Ω
/
ω
1
2 2.2
2.4
1
0.8
0.6
0.4
0.2
(b)
0
7.5
8 8.5
Frequency ratio
Ω
/
ω
9
1
9.5
1
0.8
0.6
0.4
0.2
(c)
0
12 12.5
13 13.5
14 14.5
Frequency ratio
Ω
/
ω
1
Fig. 4.13,Effect of damage location on instability regions,
15
ξ
=0.5,
θ
1
=30
0
,
ψ
= 0.1(*),
ψ
= 0.3 (+),
ψ
= 0.5(o),
ψ
=0.7(x),
ψ
=0.9(
•
).
2.6
10
15.5
68
1
0.8
0.6
0.4
0.2
(a)
0
0.8
1 1.2
1.4
1.6
1.8
Frequency ratio
Ω
/
ω
1
2 2.2
2.4
1
0.8
0.6
0.4
0.2
(b)
0
6 6.2
6.4
6.6
6.8
7 7.2
Frequency ratio
Ω
/
ω
1
7.4
7.6
7.8
1
0.8
0.6
0.4
0.2
(c)
0
15 15.5
16 16.5
17
Frequency ratio
Ω
/
ω
1
17.5
18
Fig. 4.14,Effect of damage location on instability regions,
ξ
=0.5,
θ
1
=60
0
, key as fig. 4.13.
2.6
8
18.5
69
1
0.8
0.6
0.4
0.2
(a)
0
0.8
1 1.2
1.4
1.6
1.8
Frequency ratio
Ω
/
ω
1
2 2.2
2.4
1
0.8
0.6
0.4
0.2
(b)
0
5 5.2
5.4
5.6
5.8
6
Frequency ratio
Ω
/
ω
1
6.2
6.4
6.6
1
0.8
0.6
0.4
0.2
(c)
0
17.5
18 18.5
19 19.5
20
Frequency ratio
Ω
/
ω
1
20.5
21 21.5
Fig. 4.15,Effect of damage location on instability regions,
ξ
=0.5,
θ
1
=90
0
, key as fig. 4.13.
2.6
6.8
22
70
1
0.8
0.6
0.4
0.2
(a)
0
0.8
1 1.2
1.4
1.6
1.8
Frequency ratio
Ω
/
ω
1
2 2.2
1
0.8
0.6
0.4
0.2
(b)
0
7.5
8
Frequency ratio
Ω
/
ω
1
8.5
1
0.8
0.6
0.4
0.2
(c)
0
12.4
12.6
12.8
13 13.2
13.4
13.6
Frequency ratio
Ω
/
ω
1
13.8
Fig.  4.16,Effect of extent of damage on instability regions,
θ
1
= 30
0
,
ψ
=0.1,
ξ
=0.75,(*),
ξ
=0.5,(o).
14
2.4
2.6
9
14.2
14.4
71
1
0.8
0.6
0.4
0.2
(a)
0
0.8
1 1.2
1.4
1.6
1.8
Frequency ratio
Ω
/
ω
1
2 2.2
2.4
2.6
1
0.8
0.6
0.4
0.2
(b)
0
6 6.5
Frequency ratio
Ω
/
ω
1
7 7.5
1
0.8
0.6
0.4
0.2
(c)
0
15 15.5
16
Frequency ratio
Ω
/
ω
16.5
1
17
Fig. 4.17,Effect of extent of damage on instability regions,
θ
1
= 60 o
,
ψ
=0.1, key as fig. 4.16.
17.5
72
1
0.8
0.6
0.4
0.2
0
0.8
(a)
1 1.2
1.4
1.6
1.8
Frequency ratio
Ω
/
ω
1
2 2.2
2.4
2.6
1
0.8
0.6
0.4
0.2
(b)
0
5 5.5
Frequency ratio
Ω
/
ω
1
6 6.5
1
0.8
0.6
0.4
0.2
(c)
0
17.5
18 18.5
19
Frequency ratio
Ω
/
ω
1
19.5
20
Fig. 4.18,Effect of extent of damage on instability regions,
θ
1
= 90 o
,
ψ
=0.1, key as fig. 4.16.
20.5
73
1
0.8
0.6
0.4
0.2
0
0
(a)
0.5
1 1.5
Frequency ratio
Ω
/
ω
1
1
0.8
0.6
0.4
0.2
(b)
0
7.2
7.3
7.4
7.5
7.6
Frequency ratio
Ω
/
ω
1
7.7
1
0.8
0.6
0.4
0.2
(c)
0
12 12.5
Frequency ratio
Ω
/
ω
1
13
Fig. 4.19,Effect of static load factor on instability regions,
θ
1
= 30
0
,
ψ
=0.1,
α
=0.5,(*),
α
=0.0,(o).
7.8
2
7.9
2.5
8
13.5
74
1
0.8
0.6
0.4
0.2
(a)
0
0 0.5
1 1.5
Frequency ratio
Ω
/
ω
1
2
1
0.8
0.6
0.4
0.2
(b)
0
5.8
5.9
6 6.1
6.2
6.3
6.4
Frequency ratio
Ω
/
ω
1
6.5
6.6
6.7
1
0.8
0.6
0.4
0.2
(c)
0
14.6
14.8
15 15.2
15.4
Frequency ratio
Ω
/
ω
1
15.6
15.8
Fig. 4.20,Effect of static load factor on instability regions,
θ
1
= 60
0
,
ψ
=0.1, key as fig.4.19.
2.5
6.8
16
75
1
0.8
0.6
0.4
0.2
(a)
0
0 0.5
1 1.5
Frequency ratio
Ω
/
ω
1
2
1
0.8
0.6
0.4
0.2
(b)
0
4.8
5 5.2
5.4
5.6
Frequency ratio
Ω
/
ω
1
5.8
6
1
0.8
0.6
0.4
0.2
(c)
0
17.4
17.6
17.8
18 18.2
18.4
Frequency ratio
Ω
/
ω
1
Fig. 4.21,Effect of static load factor on instability regions,
θ
key as fig.4.19.
1
= 90
0
,
ψ
=0.1,
2.5
6.2
18.6
76
Chapter
5
DYNAMIC STABILITY OF A MULTILAYERED
SYMMETRIC CANTILEVER SANDWICH BEAM
SUBJECTED TO PARAMETRIC EXCITATION
5.1 Introduction
Vibration control of machines and structures incorporating viscoelastic materials in suitable arrangement is an important aspect of investigation [87]. The use of viscoelastic layers constrained between elastic layers is known to be effective for damping of flexural vibrations of structures, over a wide frequency range. 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 aircraft structures and other applications, such as robot arms for effective vibration control. Such members may experience parametric instability when subjected to time dependant forces.
DiTaranto and Balsingame [36] obtained composite loss factor for selected laminated beams. Mead and Markus [78] studied the forced vibration of a three layer
77
78
damped sandwich beam with arbitrary boundary conditions. Rao [96] calculated the frequency parameters and loss factors of sandwich beams under various boundary conditions and presented them in the form of equations and graphs. Kavi and Asnani [70] studied the forced vibrations of stinger stiffened damped sandwich panel under point and line excitations. They determined the resonant frequencies, loss factors and frequency response by transfer matrix method. Banerjee [10] carried out free vibration analysis of three layered symmetric sandwich beams using dynamic stiffness method.
The theory of flexural vibration of symmetric multilayered beams was analysed by Agbasiere and Grootenhuis [3]. Asnani and Nakra [6] investigated the flexural vibration of multilayered unsymmetrical beams. Asnani and Nakra in their later work [8] explored the damping effectiveness during flexural vibration of multilayered beams with number of layers up to 15, with simply supported end conditions. Vaswani, et al. [128] obtained resonant frequency and system loss factor for general multilayered curved beams.
The purpose of the present work is to study the parametric instability of a multilayered cantilever sandwich beam subjected to end periodic axial load. Equation of motion for a general 2n+1 layered beam is derived using finite element method in conjunction with Hamilton’s principle. The regions of instability for simple and combination resonance are established using modified Hsu's method proposed by Saito and Otomi [104].
79
5.2 Formulation of the problem
A 2n+1 layer sandwich beam, incorporating viscoelastic damping material is shown in fig.(5.1). There are n number of viscoelastic layers and n+1 numbers of elastic layers. A layer of viscoelastic material separates two adjacent stiff elastic material layers. The beam is symmetric and has a length L and width b respectively. It is subjected to a pulsating axial force P(t) = P
s
+ P
t
cos
Ω
t
, where P
s
is the static, P
t is the amplitude of the time dependent component of the load acting along its undeformed axis at the free end and
Ω
is the excitation frequency of the dynamic load component. The finite element model is developed based on the following assumptions:
(1) The transverse displacement w of the beam is small and is same for all points of a given cross section.
(2) The rotary inertia and shear deformation in the constraining layers are neglected.
That is the elastic face layers obey the Euler Bernoulli beam theory.
(3) Linear theories of elasticity and viscoelasticity are applicable to the layers.
(4) The layers are perfectly bonded so that displacements are continuous across the interfaces
(5) Damping in the viscoelastic layer is primarily due to shear. Bending and extensional effects in the core are neglected.
(6) Young's modulus of the viscoelastic material is negligible compared to the elastic material.
80
5.2.1 Element matrices
As shown in fig.(5.2) the beam element model presented here consists of two nodes and each node has n+3 degrees of freedom. Nodal displacements are given by
{
∆
(e )
}
= {w
p ϕ
p
u
1p
u
3p ….
u
(2n+1)p
w
q ϕ
q
u
1q
u
3q…
u
(2n+1)q
}
T
(5.1) where p and q are elemental nodal numbers. The axial displacement, (u
(2k1)
) of the constraining layers, the transverse displacement,(w) and the rotational angle,( ϕ
) can be expressed in terms of nodal displacements and finite element shape functions.
u
(2k1)
= [ N
(2k1)
]
{
∆
(e )
}
, k=1,2 …n+1,
w = [N
w
]
{ }
, ϕ
=
[
N w
]
′
(5.2) where the prime denotes differentiation with respect to axial coordinate x. The shape functions are given as below.
The shape function matrices, [N
(2k1)
] are of 1
×(2n+6) size with the elements
[N
(2k1)
]
(1,(k+2))
= 1
ζ and [N
(2k1)
]
(1,(n+k+5))
=
ζ respectively and all other elements are zero.
The size of the shape function matrix [N
w
] is 1
×(2n+6) with the elements
[N
w
]
(1,1)
= (13
ζ
2
+2
ζ
3
); [N
w
]
(1,2)
= (
ζ 2ζ
2
+
ζ
3
) l ; [N
w
]
(1,(n+4))
= (3
ζ
2
2
ζ
3
);
[N
w
]
(1,(n+5))
= (
ζ
2
+
ζ
3
) l and all other elements are zero. where
ζ = x / l
,
l is the length of the element.
5.2.1.1 Element stiffness matrix
Elemental potential energy (
U
(e)
) is equal to the sum of the potential energy of the constraining layers and viscoelastic layers.
U
(e)
=
U c
(e)
+
U v
(e)
(5.3)
81
(i) Potential energy of the constraining layers
The potential energy of the constraining layers due to axial extension and bending is given as
U
(e) c
=
n k
+
1
∑ ∫
=
1
l
0
E
( 2
k
−
1 )
I
( 2
k
−
1 )
⎛
⎜⎜
∂
2
w
∂
x
2
⎞
⎟⎟
2
dx
+
1
2
l
∫
0
E
( 2
k
−
1 )
A
( 2
k
−
1 )
⎛
⎜⎜
∂
u
( 2
k
−
1 )
∂
x
⎞
⎟⎟
2
dx
(5.4) where E
( 2
k
−
1 )
, A
( 2
k
−
1 )
=
b
{
t
( 2
k
−
1 )
}
and I
( 2
k
−
1 )
=
b
{
t
( 2
k
−
1 )
}
3
12
are the Young's modulus, crosssectional area and area moment of inertia of the (2k1)th constraining layer respectively.
By substituting eq.(5.2) in to eq. (5.4) the element potential energy of the constraining layers can be written as
U
(
c e
)
=
n k
+
1
∑
=
1
1
2
{ }
)
T
(
[
K
(
(
e
)
2
k
−
1 )
w
] [
K
(
(
e
)
2
k
−
1 )
u
]
)
{ }
)
(5.5) where
[
K
(
e
)
k u
]
( 2
−
1 )
=
E
( 2
k
−
1 )
[
K
(
e
)
k w
]
( 2
−
1 )
=
E
( 2
k
−
1 )
A
( 2
k
−
1 )
l
∫
0
[
N
( 2
k
−
1 )
′
] [
N
( 2
k
−
1 )
′
]
I
( 2
k
−
1 )
l
∫
0
[ ] [ ]
″
dx dx
⎫
⎪
⎪
⎭
(5.6)
(ii) Potential energy of the viscoelastic layers
The potential energy of the viscoelastic layers due to shear deformation is given as
U
(
e v
)
=
n
∑ ∫
j
=
1
2
l
0
G v
( 2
j
)
A v
( 2
j
)
γ
2
v
( 2
j
)
dx
(5.7)
82
where A
v(2j) is the crosssectional area and G
v(2j)
is the complex shear modulus of 2jth layer.
G v(2j)
=
G
∗ v(2j)
[
1
+
i
( )
( 2
j
)
]
, where
G
∗
v(2j)
is the inphase shear modulus of the 2jth viscoelastic material layer,
( )
( 2
j
) is the associated core loss factor and
i
= −
1
.
The shear strain
γ
v(2j) of the 2jth viscoelastic layer from kinematic relationships between the constraining layers[78] is expressed as follows:
γ
v (2j)
=
u
2
j
+
1
−
t v (2j) u
2
j
−
1
+
(
t
2
j
+
1
+
2
t v (2j)
2
t v (2j)
+
t
2
j
−
1
)
∂
w
∂
x
(5.8)
Substituting eq. (5.2) in to eq.(5.8)
γ
v
( 2
j
)
can be expressed in terms of nodal displacements and element shape functions:
γ
v
(2j)
=
[ ]
(2j)
{ }
(5.9) where
[
N
γ
(2j)
(
t v (2j)
)
+
(
t
2
j
+
1
+
2
t v (2j)
2
t v (2j)
+
t
2
j
−
1
)
[ ]
′
(5.10)
Substituting eq. (5.9) in to eq. (5.7), the potential energy of the viscoelastic material layers is given by
U v
(
e
)
=
n
∑
j
=
1
1
2
{ }
T
(
[
K
(
v
γ
e
)
( 2
j
)
]
)
{ }
(5.11) where
(
e
)
v
γ
=
j n
∑
=
1
G v
( 2
j
)
A v
( 2
j
)
l
∫
0
[
N
γ
T
] [
N
γ
]
(2j) (2j)
dx
(5.12)
From eq. (5.3) elemental potential energy
U
(e)
=
n k
+
1
∑
=
1
1
2
{ }
T
(
[
K
(
(
e
2
)
k
−
1
)
u
] [
K
(
(
e
2
)
k
−
1
)
w
]
){ }
+
j n
∑
=
1
1
2
{ }
T
(
[
K v
(
γ
e
)
(
2 j
)
]
){
∆
(
e
)
}
83
=
1
2
{ }
T
(
[ ]
) {
∆
(
e
)
}
(5.13) where
[ ]
=
n k
+
1
∑
=
1
(
[
K
(
(
e
)
2
k
−
1 )
u
] [
K
(
(
e
)
2
k
−
1 )
w
]
)
+
j n
∑
=
1
(
[
K
(
e v
γ
)
( 2
j
)
]
)
(5.14)
is the element stiffness matrix
5.2.1.2 Element mass matrix
Elemental kinetic energy (
T
(e)
) is equal to the sum of the kinetic energy of the constraining layers and viscoelastic layers.
T
(e)
=
T
c
(e)
+
T v
(e)
(5.15)
(i) Kinetic energy of the constraining layers is written as
T
c
(e)
=
n
+
1
∑
k
=
1
1
2
0
∫
l
ρ
( 2
k
−
1 )
A
( 2
k
−
1 )
⎛
⎜⎜
∂
w
∂
t
⎞
⎟⎟
2
dx
+
1
2
0
∫
l
ρ
( 2
k
−
1 )
A
( 2
k
−
1 )
⎛
⎜⎜
∂
u
( 2
k
−
1 )
∂
t
⎞
⎟⎟
2
dx
(5.16) where
ρ
(2k1)
is the mass density of the (2k1)th constraining layer.
By substituting eq. (5.2) in to eq. (5.16), the element kinetic energy of the constraining layers can be written as
T c
(
e
)
=
n
+
1
k
∑
=
1
1
2
{ }
T
(
[
M
(
e
)
( 2
k
−
1 )
u
] [
M
(
(
e
)
2
k
−
1 )
w
]
(5.17) where
[
M
(
e
)
k u
]
( 2
−
1 )
=
[
M
(
e
)
k w
]
( 2
−
1 )
=
ρ
( 2
k
−
1 )
ρ
( 2
k
−
1 )
A
( 2
k
−
1 )
A
( 2
k
−
1 )
l
∫
0
[
N
( 2
k
−
1 )
u
T
] [
N
( 2
k
−
1 )
u
]
dx
⎫
(5.18)
l
∫
0
[ ] [ ]
dx
⎪
⎭ and the dot denotes differentiation with respect to time t.
84
(ii) Kinetic energy of the viscoelastic layers is written as
T v
(
e
)
=
j n
∑ ∫
=
1
l
0
ρ
v
( 2
j
)
A v
( 2
j
)
⎧
⎪⎩
⎜
∂
w
∂
t
2
+
⎛
⎜⎜
∂
u v
( 2
j
)
∂
t
⎞
⎟⎟
2
⎫
⎪⎭
dx
(5.19) where A
v(2j) is the crosssectional area and
ρ
v(2j)
is the mass density of the 2jth viscoelastic layer
The axial displacement u
v(2j) of the 2jth viscoelastic layer derived from kinematic relationships between the constraining layers[78] is expressed as follows:
u v
(2j)
=
u
2
j
+
1
+
u
2
j
−
1
2
+
(
t
2
j
+
1
−
t
2
j
−
1
)
4
∂
w
∂
x
(5.20)
Substituting eq. (5.2) in to eq. (5.20)
u v
can be expressed in terms of nodal displacements and element shape functions:
u v
(2j)
=
[ ]
(2j)
{
∆
(
e
)
}
(5.21) where
[
N v
(2j)
]
=
1
2
(
[
N
2
j
+
1
] [
N
2
j
−
1
]
)
(
t
2
j
+
1
−
t
2
j
−
1
)
[ ]
′
4
(5.22)
Substituting eq. (5.2) in to eqs. (5.21) and (5.19), the kinetic energy of viscoelastic material layers is given by
T v
(
e
)
=
j n
∑
=
1
1
2
(
[
M v
(
e
)
( 2
j
)
]
) {
(
e
)
}
(5.23) where
[
M
(
e
)
v
( 2
j
)
]
=
j n
∑
=
1
ρ
v
( 2
j
)
A v
( 2
j
)
l
∫
0
N v
( 2 )
N v
( 2
j
)
dx
+
ρ
v
( 2
j
)
A v
( 2
j
)
l
∫
0
[ ] [ ]
dx
(5.24)
From eq.(5.15)
T
(e)
=
n k
+
1
∑
=
1
1
2
{ }
T
(
[
M
(
e
)
( 2
k
−
1 )
u
] [
M
(
(
e
)
2
k
−
1 )
w
]
) { }
+
n
∑
j
=
1
1
2
{ }
T
(
[
M v
(
e
)
( 2
j
)
]
) {
(
e
)
}
(5.25)
85
T
(e)
=
1
2
{ }
T
(
[ ]
) {
&
(
e
)
}
(5.26) where
[ ]
=
n
+
1
∑
k
=
1
(
[
M
(
(
e
)
2
k
−
1 )
u
] [
M
(
e
)
( 2
k
−
1 )
w
]
)
+
j n
∑
=
1
(
[
M
(
e
)
v
( 2
j
)
]
)
(5.27) and
[
M
(e )
] is the elemental mass matrix.
5.2.1.3 Element geometric stiffness matrix
The elemental work done by axial periodic force P(t) is written as
W p
(
e
)
=
1
2
l
∫
0
P
(
t
)
(5.28)
∂
w
∂
x
2
dx
Substituting eq.(5.2) in to eq.(5.28), the work done by the axial periodic load can be rewritten as
W p
(
e
)
=
1
2
{ }
T
P
(
t
)
K
(
e
)
g
∆
(
(5.29) where
[
K
(
e g
)
]
=
∫
l
0
[ ] [ ]
′
dx
, the elemental geometric stiffness matrix.
5.2.2 Governing equations of motions
The element equation of motion for a sandwich beam is obtained by using Hamilton's principle.
δ
∫
2
t t
1
(
T
(
e
)
−
U
(
e
)
+
W p
(
e
)
)
dt
=
0 (5.30)
86
Substituting eqs.(5.13), (5.26) and (5.29 ) in to eq. (5.30) the equation of motion for the sandwich beam element is obtained as follows:
[ ]
)
{ }
+
[ ]
)
{ }
−
P
(
t
)
[ ]
)
{ }
=
0
(5.31)
Assembling mass, elastic stiffness and geometric stiffness matrices of individual element, the equation of motion for the beam is written as
[ ]
{ }
+
[ ]
{ }
−
P
(
t
)
[ ]
g
{ }
=
0
(5.32) where
{
∆
}
is the global displacement matrix.
The static component P
s
and dynamic component P
t of the load P(t), can be represented in terms of P
*
as P
s
=
α
P
*
and P
t
=
β
P
*
and, where P
*
=D/L
2 and
D
=
n
+
1
k
∑
=
1
E
( 2
k
−
1 )
I
( 2
k
−
1 )
.
Hence substituting
P(t)
=
α
P *
+
β
P * cos
Ω
t
, where
α
and
β
are static and dynamic load factors respectively.
Substituting P(t), eq.(5.32) becomes
[ ]
{ }
+
[ ]
{ }
−
( P s
+ P t cos
Ω t )
[ ]
{ }
=
0
(5.33)
[ ]
{ }
+
(
[ ]
−
P s
[ ]
S
)
{ }
−
P t cos
Ω t
[ ]
t
{ }
=
0
(5.34) where the matrices
[
K g s
] and
[ ]
t
reflect the influence of P
s
and P
t
respectively. If the static and time dependent component of loads are applied in the same manner, then
[ ] [ ] [
s t
K g
]
.
{ }
+
[ ]
{ }
−
(
β
P
∗ cos
Ω t
) [ ]
g
{ }
=
0
(5.35) where
[ ]
=
[ ]
−
P s
(5.36)
87
The global displacement matrix
{ }
can be assumed as
=
[ ]
{ }
(5.37) where
[
Φ
]
is the normalized modal matrix corresponding to
[ ]
{ }
+
[ ]
{ }
=
0
(5.38) and
{
Γ
}
is a new set of generalised coordinates .
Substituting eq.5.37 in eq.5.35, eq.5.35 is transformed to the following set of N
c
coupled
Mathieu equations.
Γ
&&
m
+
( )
Γ
m
+
β
P
∗ cos
Ω
t n
N c
∑
=
1
b mn
Γ
n
=
0
m
=
1 , 2 , ........
N c
.
(5.39) where
( )
m
are the distinct eigenvalues of
[ ]
−
1 complex matrix
= − −
1
[ ]
[ ]
and
and
b mn
are the elements of the
ω
m
=
ω
m
.
R
+
i
ω
m
.
I
,
b mn
=
b mn
.
R
+
i b mn
.
I
and i
=
1
5.2.3 Regions of instability
Hsu [52] developed a procedure for obtaining the regions of instability for simple and combination resonances by applying the series expansion of perturbation method and method of constant variation of parameters to a system having multiple degrees of freedom. Saito and Otomi [104] applied the Hsu’s method to the Matheiu equation (5.39), which has complex coefficients and obtained the conditions for the existence of regions of instability for simple and combination resonances of sum and difference types. The conditions are as given below.
Case (A): Simple resonance
88
The boundaries of the instability regions are given by
Ω
2
ω
0
−
ω
µ
,
R
⟨
1
4
⎡
⎢
β
2
(
b
2
µµ
,
R
ω
+
2
µ
,
R b
2
µµ
,
I
)
−
16
ω
2
µ
,
I
⎤
⎥
1 / 2
,
µ
=
1 , 2 .........
N c
.
(5.40) where
ω
0
=
D mL
4
,
ω
µ
,
R
=
ω
µ
,
R
/
ω
0
,
ω
µ
,
I
=
ω
µ
,
I
/
ω
0
, m is mass per unit length of the multilayered sandwich beam.
When damping is neglected, the regions of instability are given by
Ω
2
ω
0
−
ω
µ
,
R
⟨
1
4
⎡
⎢
⎣
β
(
b
µµ
,
R
ω
µ
,
R
)
⎤
⎥
⎦
,
µ
=
1 , 2 ,......
N c
.
(5.41)
Case (B): Combination resonance of sum type
The boundaries of the regions of instability of sum type are given by
Ω
2
ω
0
−
1
2
(
__
ω
µ
,
R
+
__
ω
ν
,
R
)
⟨
1
8
(
ω
µ
,
I
(
ω
µ
,
I
+
ω
ν
,
I
ω
ν
,
I
)
1 / 2
)
⎢
⎢
⎣
⎡ β
µ ≠ ν
,
µ
,
ν =
1 , 2 ,...
N c
.
2
(
b
µν
,
R b
νµ
,
R
ω
µ
,
R
ω
ν
+
,
R b
µν
,
I b
νµ
,
I
)
−
16
ω
µ
,
I
ω
ν
,
I
⎥
⎥
⎦
⎤
1 / 2
(5.42)
When damping is neglected, the regions of instability are given by
Ω
2
ω
0
−
1
2
(
__
ω
µ
,
R
+
__
ω
ν
,
R
)
⟨
1
4
⎡
⎢
β
2
(
b
µν
ω
µ
,
R
,
R b
νµ
ω
ν
,
R
,
R
)
⎤
⎥
1 / 2
,
µ
≠
ν
,
µ
,
ν
=
1 , 2 ,...
N c
.
(5.43)
Case (C): Combination resonance of difference type
The boundaries of the regions of instability of difference type are given by
Ω
2
ω
0
−
1
2
(
__
ω
ν
, R
−
__
ω
µ
, R
)
⟨
1
8
(
ω
µ
, I
(
ω
µ
.
I
+
ω
ω
ν
.
I
ν
, I
)
)
1
/
2
⎢
⎢
⎣
⎡ β
2
( b
µν
, I b
νµ
, I
ω
µ
, R
−
b
µν
, R b
νµ
, R
ω
ν
, R
)
−
16
ω
µ
, I
ω
ν
, I
⎥
⎥
⎦
⎤
1
/
2
,
ν
⟩
µ
,
µ
,
ν
=
1 , 2 ,...
N c
.
(5.44)
When damping is neglected, the unstable regions are given by
89
Ω
2
ω
0
−
1
2
(
__
ω
ν
.
R
−
__
ω
µ
, R
)
⟨
1
4
⎡
⎢
⎣
−
β
2
( b
µν
.
, R b
νµ
., R
ω
µ
, R
ω
ν
, R
)
⎥
⎦
⎤
1
/
2
,
ν
⟩
µ
,
µ
,
ν
=
1
,
2
,...
N c
.
(5.45)
For a combination of excitation frequency (
Ω/2
ω o
), natural frequency/frequencies
(
ω
µ or/and
ω
ν
), and dynamic load factor (
β
), when one or more of the above conditions are satisfied, instability of the system is indicated. For example if for a particular pair of
Ω/2
ω o and
β
, the condition given by eq.(5.40) is satisfied for
µ
= 1(
ω
µ
=
ω
1
), then this will lead to instability of simple resonance of type 2
ω
1 of the system. Similarly if for a pair of
Ω/2
ω o and
β
, the condition given by eq.(5.40) is satisfied for
µ
= 2(
ω
µ
=
ω
2
), this will lead to simple resonance of type 2
ω
2.
For a pair of
Ω/2
ω o and
β
, if eq.(5.42) is satisfied when
µ
= 1(
ω
µ
=
ω
1
) and
ν = 2 (
ω
ν
=
ω
2
)
,
then the system will have combination resonance of sum type (
ω
1
+
ω
2
). If eq.(5.44) is satisfied when
µ
=1, (
ω
µ
=
ω
1
) and
ν
= 2, (
ω
ν
=
ω
2
)
,
then this will lead to combination resonance of difference type, (
ω
2

ω
1
). When pairs of
Ω/2
ω o and
β
, satisfying a particular instability condition is plotted in the parameter space of dynamic load component (
β
) and excitation frequency (
Ω/2
ω o
), the corresponding instability region is obtained.
5.3 Computational procedure
In the initialisation phase, geometry and material parameters are specified. The input data to the computer program are material parameters like Young’s modulus of the elastic layers, in phase shear modulus and core loss factor of the viscoelastic layers, mass density of both the elastic and viscoelastic layers and geometric parameters like length and width of the beam and thickness of the elastic and viscoelastic layers. The beam is divided in to n s
number of elements and n s
+1 number of nodes. The elements of the
element mass, elastic stiffness and geometric stiffness matrices
(
[ ] [ ] and
90
) are computed by GaussLegendre numerical integration method. The program uses the built in Matlab function, ‘quadl’ to carry out the integration. Element matrices are assembled to obtain global matrices
(
[ ] [ ] and
)
. Boundary conditions are imposed by elimination method. For example for fixedfree end condition, the first n+3 rows and columns of the global matrices are eliminated to obtain the reduced matrices. The static load factor (
α) is set to the required value. The normalised modal matrix corresponding to eq.(5.38) is obtained by solving for the associated eigenvectors of
[ ]
−
1 matrix. The built in Matlab function ‘eig’ is used to calculate the eigenvectors.
Coefficient matrices of eq.(5.35) are premultiplied by
[ ]
−
1 and
[ ]
−
1
and postmultiplied by
[ ]
to obtain the eq.(5.39). The dynamic load factor
β
and the excitation frequency
Ω/2
ω o
are increased in steps. For a particular pair of values of
β
and
Ω/2
ω o
the eqs.(5.40, 5.42, 5.44) are checked and if at least one of the conditions is satisfied, the program stores the pair,
β
and
Ω/2
ω o
. When pairs of
Ω/2
ω o and
β
, satisfying a particular instability condition is plotted in the parameter space of dynamic load component (
β
) and excitation frequency (
Ω/2
ω o
), the corresponding instability region is obtained. The detailed computational procedure is shown schematically in the flow chart given in AppendixB. The entire computational process has been implemented by computer programs developed in MATLAB [135].
5.4 Results and discussion
91
The multilayered beam considered being symmetric with identical viscoelastic and elastic layers, for the viscoelastic layers t
v(2j)
= t
2 ,
G
∗
v(2j)
=
G
∗
v
,
( )
( 2
j
)
=
η c
and
ρ
v
( 2
j
)
=
ρ
2
for all
j and for the elastic layers t
(2k1)
= t
1,
E
( 2
K
−
1 )
=
E
1 and
ρ
( 2
k
−
1 )
=
ρ
1
for all k have been taken. For calculation purpose the young’s modulus E
1
of the elastic layers and the in phase shear modulus of the viscoelastic material layers
G v
∗
are taken as 70x10
9
N/m
2
and
2.6x10
5
N/m
2 respectively. The ratio of mass density
ρ
21
of the viscoelastic material layer and elastic material layer is taken to be 0.4.With a ten element discretisation of the beam, the resonant frequencies and modal system loss factors obtained for a threelayer beam are compared with those of Rao [96] and results are found to be in good agreement.
Resonant frequencies obtained from experiment for beams of three, five and seven numbers of layers, also compare well with the calculated values and these data are given in Table7.6. The boundary frequencies of a three layer cantilever sandwich beam, obtained from the present analysis have been compared with those of [75] and these are found to be in good agreement. This is shown in Table 5.1.
Table 5.1: Comparison of the boundary frequencies for a three layer cantilever sandwich beam from the present analysis with those of reference [75].
92
To study the effect of various system parameters, such as core thickness parameter t
21,
(defined as the ratio of the thickness of the viscoelastic core to the thickness of the elastic layer), number of layers and core loss factor, numerical results have been obtained for three, five and seven layered beams. As well as five special cases for which beams with different number of layers and all having (i) constant size or (ii) constant weight or (iii) constant flexural rigidity or (iv) constant size and flexural rigidity or (v)
Dynamic load factor
(
β)
0.05
Ref
Simple resonance instability region (2
ω
1
)
Simple resonance instability region (2
ω
2
)
Combination resonance of difference type instability region (
ω
2

ω
1
)
Lower Upper Lower Upper Lower Upper
3.869 4.000
0.1
0.2
Ref 3.672 4.262
Present 12.811
Ref 3.443 4.590 20 25.333 12.15 12.85
0.3
Ref 3.118 4.787 19.6 26.933 11.8 13.1
0.4
Ref 2.787 5.082 19.2 29.227 11.55 13.35
0.5
Ref [75] 2.623 5.41 19.013 31.493 11.32 13.69 constant weight and flexural rigidity have also been investigated. For first three cases core thickness parameter (t
21
) is same for beams of any number of layers.
In the following discussion, N=2n+1 is the total number of layers of the beam and
(g)
N
=
G
(
t
21
∗
v
)
N
⎝
⎜⎜
⎛
t
L
1
⎠
⎟⎟
⎞ 2
N
⎜
⎜
⎝
⎛
2
E
1
⎟
⎟
⎠
⎞
as defined in Rao [96], is the shear parameter for an N layered beam.
93
n e n e
′ are the number of viscoelastic layers in (2
n e
+1) and (2
n′
e
+1) layered beams with
t
1 and t
'
1
being the thickness of elastic layers in corresponding beams the relation between
t
1 and t
'
1
for above mentioned cases are;
(i) Constant size
( )
( )
(
(
2
2
n e n e
′
+
1 )
+
1 )
= [
[
(
(
n e n e
′
+
1 )
+
1 )
+
+
n e
′
n e
]
]
(ii) Constant weight
( )
( )
(
(
2
2
n n e
′
e
+
1 )
+
1 )
=
(
[
[
(
n n e
′
e
+
+
1 )
1 )
+
+
(
ρ
21
)( )
(
ρ
21
)( )
n n
′
e e
]
]
(iii)Constant flexural rigidity
( )
( )
(
(
2
2
n n e e
′
+
1 )
+
1 )
=
⎛
⎜⎜
n n e e
′
+
1
+
1
⎞
⎟⎟
1
3
(iv) Constant size and flexural rigidity
21
( 2
n
′
e
+
1 )
=
⎛
⎜⎜
n
′
e n e
+
+
1
1
⎞
⎟⎟
1 / 3
(
[
n e
+
1 )
+
( )
( 2
n e
+
1 )
[
n
′
e
]
(v) Constant weight and flexural rigidity
( )
( 2
n
′
e
+
1 )
=
⎛
⎜⎜
n
′
e n e
+
+
1
1
⎞
⎟⎟
1 / 3
(
[
n e
+
1 )
+
(
ρ
21
)
(
[
(
ρ
21
)
( 2
2
n
′
e n e
+
+
1 )
1 )
(
( )
(
n
′
e
)
]
2
n e
+
1 )
]
−
(
n
′
e
+
1 )
]
−
(
n
′
e
+
1 )
94
Figures (5.35.4) show the effect of core thickness parameter (t
21
) on first and second modal frequency parameters
η
1 and
η
2 of the sandwich beam with three, five and seven layers
.
Modal frequency parameter is defined as the ratio of resonant frequency of the corresponding mode for the sandwich beam to ω o
. For all the three types of beam shear parameter (g)
N
has been taken as 5.0. It is observed from the graphs that for a beam of particular number of layers, both the frequency parameters increase linearly with increase in core thickness parameter. Also with increase in number of layers when core thickness parameter is same for all types of beam, both the frequency parameters increase. The increase in resonant frequencies with number of layers is more for higher values of core thickness parameter.
Figures (5.55.6) show the effect of core thickness parameter on
η
1 and
η
2 for constant size case. Both the frequency parameters increase with increase in core thickness parameter. For constant size case with increase in number of layers both the frequency parameters decrease with increase in number of layers when the core thickness parameter is kept unchanged.
Figures (5.75.8) show the variation of
η
1 and
η
2
for constant weight case. The behaviours are same as those for constant size case.
Figures (5.95.10) show the variation of
η
1 and
η
2 for constant flexural rigidity case. The first mode frequency parameter shows a linear variation with core thickness parameter. Irrespective of the value of core thickness parameter with increase in number of layers there is an increase in fundamental frequency parameter
η
1
. Second mode frequency parameter decreases with increase in number of layers for lower values of core thickness parameter. For higher values of core thickness parameter the trend is reverse.
95
Figures (5.115.12) show the effect of core thickness parameter on
η
1 and
η
2
for constant size and flexural rigidity case. Both the frequency parameters increase with increase in core thickness parameter. For lower value of core thickness parameter with increase in number of layers both the frequency parameters increase. For higher values of core thickness parameter the effect of increase in number of layers becomes less dominant on resonant frequencies and frequency parameters converge to the same value irrespective of number of layers.
Figures (5.135.14) show the same curves for constant weight and flexural rigidity case. The trend of the curves is same as those for constant size and flexural rigidity case, but the effect is more pronounced compared to constant size and flexural rigidity case.
Figures (5.155.16) show the instability zones for three, five and seven layered beams with
ρ
c
equal to 0.18 and t
21 equal to 1/3 and 2/3 respectively. For beams with the same number of layers, instability zones are also shown in figs. (5.17 and 5.18) for
ρ
c
equal to 0.3 and core thickness parameter t
21 equal to 1/3 and 2/3 respectively.
Comparison of fig.(5.15) with fig.(5.16) and fig.(5.17) with fig.(5.18) shows that with increase in t
21 with
ρ
c
and N fixed, the instability regions shift to higher frequencies of excitation along the frequency axis and they also move vertically up along the dynamic load factor axis. More over their areas decrease and some of the instability regions also vanish with increase in t
21.
This shows improved stability with increase in core thickness parameter.
From figs. (5.155.18), it can be observed that with increase in number of layers with t
21 and
ρ
c
fixed, the instability zones shift to higher frequencies of excitation and
96
they start at higher values of dynamic load factor. There is reduction in areas of the instability zones and some instability zones vanish. This indicates improved stability with increase in number of layers, keeping the core thickness parameter fixed.
Figures (5.15 and 5.17) show the instability zones for three, five and seven layer beams with core thickness parameter 1/3 and core loss factor 0.18 and 0.3 respectively.
Instability zones for beams with similar number of layers are also shown in figs. (5.16 and 5.18) for core thickness parameter 2/3 and same values of core loss factor. It is seen that for beams with same t
21,
with increase in core loss factor the instability zones shift to higher values of dynamic load component and there is reduction in their areas also. Some instability zones also disappear with increase in core loss factor. This shows improvement in stability with increase in core loss factor. However there is no marked shift in the positions of the instability regions along the frequency axis with increase in core loss factor.
Figure (5.19) shows the instability zones for the constant size case with (g)
3
= 1.0 and core thickness parameter (t
21
)
3
= 1.0 for the reference three layer beam. It is observed that with increase in number of layers the instability zones shift to lower frequencies of excitation. This is due to decrease in resonant frequencies of the beam with increase in number of layers for constant size case. But with increase in number of layers the instability regions of sum type vanishes, also other principal instability regions start at higher values of dynamic load component. So there may be deterioration or improvement in stability depending on the applied excitation frequency and dynamic load component.
In fig. (5.20) the instability regions for constant weight case are shown for same values of (g)
3
and (t
21
)
3
. Similar behaviour as those for constant size case is observed.
97
Figure (5.21) shows the instability zones for the constant flexural rigidity case. It is seen that with increase in number of layers the first instability zone shifts to higher frequency of excitation, where as the second instability zone shifts to lower frequency of excitation. This is due to the fact that the fundamental resonant frequency for constant flexural rigidity case increases with increase in number of layers for any value of core thickness parameter between 0.01 and 5.0. But with increase in number of layers there is decrease in second modal frequency parameter for lower values of t
21, for higher values of
t
21
the trend is reverse (fig5.10). With increase in number of layers, there is reduction in areas of both the first and second simple instability zones, also the instability region of combination resonance type vanishes and instability commences with higher dynamic load component, indicating improved stability.
Figure (5.22) shows the instability zones for constant size and flexural rigidity case for parameters (g)
3
= 0.2 and core thickness parameter (t
21
)
3
= 4.0. It is seen that with increase in number of layers the instability zones shift to higher frequency of excitation, there is reduction in areas of the instability zones and instability starts with higher value of dynamic load component, this indicates improvement in stability with increase in number of layers.
Figure (5.23) shows the instability zones for constant weight and flexural rigidity case for parameters (g)
3
=0.2 and core thickness parameter (t
21
)
3
= 4.0. Similar trends as those for constant size and flexural rigidity case are observed, but the effect is more pronounced in this case compared to constant size and flexural rigidity case. This is because of large change in resonant frequencies with increase in number of layers for
98
constant weight and flexural rigidity case compared to constant size and flexural rigidity case, figs. (5.115.14).
5.5 Closure
The first and second resonant frequencies increase with increase in core thickness parameter and number of layers. For constant size and constant weight cases with increase in core thickness parameter the resonant frequencies increase and frequencies decrease with increase in number of layers for a particular core thickness parameter. For the case of constant flexural rigidity the fundamental resonant frequency increases with increase in number of layers or core thickness parameter. The second resonant frequency may increase or decrease with increase in number of layers depending on the core thickness parameter. For constant size and flexural rigidity case increase in number of layers or core thickness parameter increases the resonant frequencies. For higher values of core thickness parameter increase in number of layers has marginal effect on the resonant frequencies. For constant weight and flexural rigidity case, increase in core thickness parameter or number of layers increase the resonant frequencies. The effect of these two factors on resonant frequencies for this case is more pronounced than that of constant size and flexural rigidity case. For higher values of core thickness parameter the effect of increase in number of layers also becomes less effective as in the case of constant size and flexural rigidity case.
Increase in core thickness parameter, number of layers and core loss factor has stabilizing effect. For constant size and constant weight case, increase in number of layers deteriorates the stability with instability zones relocating them at lower frequencies
99
of excitation, improves the stability with instability zones shifting to higher values on the dynamic load axis and disappearance of some instability regions. Constant flexural rigidity criterion improves stability with increase in number of layers. Among constant size, constant weight and constant flexural rigidity, constant flexural rigidity criterion is the most effective in improving the stability. Both constant size and flexural rigidity and constant weight and flexural rigidity criteria improve the stability with increase in number of layers. Constant weight and flexural rigidity case is more effective in improving the stability than constant size and flexural rigidity case.
t
2 t
1
Elastic Layer (2n+1)
Viscoelastic
Layer (2n)
Elastic Layer
(2j+1)
100
101
25
20
15
10
40
35
30
5
0.01
0.5
1 1.5
2 2.5
3
Core thickness parameter (t
21
)
3.5
4 4.5
5
Fig. 5. 3,Effect of core thickness parameter on fundamental frequency parameter,
(g)
N
= 5.0,
η
c
= 0.18;N=3; ,N=5; ... ,N=7; ,.
120
110
100
90
80
70
60
50
40
30
20
0.01
0.5
1 1.5
2 2.5
3 3.5
4 4.5
5
Core thickness parameter (t
21
)
Fig. 5.4 ,Effect of core thickness parameter on second mode frequency parameter
(g)
N
= 5.0,
η
c
= 0.18,key as fig. 5.3.
102
18
16
14
12
10
8
6
4
2
0.01
0.5
1 1.5
2 2.5
3 3.5
4 4.5
5
Core thickness parameter (t
21
)
Fig. 5.5 ,Effect of core thickness parameter on fundamental frequency parameter,
constant size case,(g)
3
=1.0,
η
c
= 0.18;N=3; ,N=5; ... ,N=7; ,.
45
40
35
30
25
60
55
50
20
15
10
0.01
0.5
1 1.5
2 2.5
3
Core thickness parameter (t
21
)
3.5
4 4.5
5
Fig. 5.6 ,Effect of core thickness parameter on second mode frequency parameter,
constant size case,(g)
3
=1.0,
η
c
=0.18,key as fig. 5.5.
103
12
10
8
6
18
16
14
4
2
0.01
0.5
1 1.5
2 2.5
3
Core thickness parameter (t
21
)
3.5
4 4.5
5
Fig. 5.7,Effect of core thickness parameter on fundamental frequency parameter,
constant weight case,(g)
3
=1.0,
η
c
= 0.18,key as fig. 5.5.
35
30
25
20
15
45
40
60
55
50
10
0.01
0.5
1 1.5
2 2.5
3 3.5
4 4.5
5
Core thickness parameter (t
21
)
Fig. 5.8,Effect of core thickness parameter on second mode frequency parameter,
constant weight case,(g)
3
=1.0,
η
c
= 0.18,key as fig. 5.5.
104
12
10
8
6
4
2
20
18
16
14
0
0.01
0.5
1 1.5
2 2.5
3
Core thickness parameter (t
21
)
3.5
4 4.5
5
Fig. 5.9,Effect of core thickness parameter on fundamental frequency parameter,
constant flexural rigidity case,(g)
3
=1.0,
η
c
= 0.18,key as fig. 5.5.
50
45
40
35
60
55
30
25
20
15
10
0.01
0.5
1 1.5
2 2.5
3 3.5
4 4.5
5
Core thickness parameter (t
21
)
Fig. 5.10,Effect of core thickness parameter on second mode frequency parameter,
constant flexural rigidity case,(g)
3
=1.0,
η
c
= 0.18,key as fig. 5.5.
105
14
13
12
11
16
15
10
9
8
7
6
4 5 6 7 8 9 10
Core thickness parameter (t
21
)
Fig. 5.11,Effect of core thickness parameter on fundamental frequency parameter,
constant size and flexural rigidity case, (g)
3
= 0.2,
η
c
= 0.3,key as fig. 5.5.
50
48
46
44
42
40
38
36
34
32
30
4 5 6 7 8 9 10
Core thickness parameter (t
21
)
Fig. 5.12 ,Effect of core thickness parameter on second mode frequency parameter,
constant size and flexural rigidity,(g)
3
= 0.2,
η
c
= 0.18,key as fig. 5.5.
106
14
13
12
11
16
15
10
9
8
7
6
4 5 6 7 8 9 10
Core thickness parameter (t
21
)
Fig. 5.13,Effect of core thickness parameter on fundamental frequency parameter,
constant weight and flexural rigidity case, (g)
3
= 0.2,
η
c
= 0.3,key as fig.5.5.
46
44
42
40
50
48
38
36
34
32
30
4 5 6 7 8 9 10
Core thickness parameter (t
21
)
Fig. 5.14,Effect of core thickness parameter on second mode frequency parameter,
constant weight and flexural rigidity case,(g)
3
= 0.2,
η
c
= 0.18,key as fig.5.5.
107
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
2
ω
1
ω
1
+
ω
2
2
ω
2
0
0 5 10 15 20 25 30 35
Frequency ratio
Ω
/2
ω
Fig. 5.15,Instability regions,(g)
N
= 5.0,(t
21
)
N
=1/3,
η
o c
= 0.18;N=3; ,N=5; ...,N=7; ,.
40
3
2.5
2
1.5
1
0.5
5
4.5
4
3.5
2
ω
1
ω
1
+
ω
2
2
ω
2
0
0 5 10 15 20 25
Frequency ratio
Ω
/2
ω
o
Fig. 5.16,Instability regions,(g)
N
= 5.0,(t
21
)
N
=2/3,
η
c
30 35 40
= 0.18,key as fig. 5.15.
45
108
3
2.5
2
1.5
1
0.5
5
4.5
4
3.5
2
ω
1
ω
1
+
ω
2
2
ω
2
0
0 5 10 15 20
Frequency ratio
Ω
/2
ω
Fig. 5.17,Instability regions,(g)
N
= 5.0,(t
21
)
N
=1/3, o
η
c
25 30 35
= 0.3,key as fig. 5.15.
3
2.5
2
1.5
1
0.5
5
4.5
4
3.5
2
ω
1
ω
1
+
ω
2
2
ω
2
0
0 5 10 15 20 25
Frequency ratio
Ω
/2
ω
o
Fig. 5.18,Instability regions,(g)
N
= 5.0,(t
21
)
N
=2/3,
η
c
30 35
= 0.3,key as fig. 5.15.
40
40
45
109
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
2
ω
2
0
0 5 10 15 20 25 30
N=3; ,N=5; ... ,N=7; ,.
Frequency ratio
Ω
/2
ω
o
Fig. 5.19,Instability regions for constant size case ,(g)
3
=1.0,(t
21
)
3
=1.0,
η
c
= 0.18;
35
1.5
1
0.5
3
2.5
2
5
4.5
4
3.5
2
ω
2
0
5 10 15 20 25 30 35
Frequency ratio
Ω
/2
ω
o
Fig. 5.20,Instability regions for constant weight case,(g)
3 key as fig. 5.19.
=1.0,(t
21
)
3
= 1.0,
η
c
= 0.18,
110
3.5
3
2.5
2
1.5
1
0.5
5
4.5
4
2
ω
2
0
0 5 10 15 20
Frequency ratio
Ω
/2
ω
o
25 30
Fig. 5.21,Instability regions for constant flexural rigidity case,(g)
η
c
= 0.18,key as fig. 5.19.
3
=1.0,(t
21
)
3
=1.0,
35
2.5
2
1.5
1
0.5
5
4.5
4
3.5
3
2
ω
2
0
0 5 10 15 20
Frequency ratio
Ω
/2
ω
o
25 30 35
Fig. 5.22 ,Instability regions for constant size and flexural rigidity case,(g)
3
=0.2,
(t
21
)
3
= 4.0,
η
c
=0.18,key as fig. 5.19.
40
111
3
2.5
2
1.5
1
0.5
5
4.5
4
3.5
2
ω
2
0
0 5 10 15 20
Frequency ratio
Ω
/2
ω
o
25 30 35 40
Fig. 5.23,Instability regions for constant weight and flexural rigidity case,(g)
3
=0.2,
(t
21
)
3
= 4.0,
η
c
= 0.18,key as fig. 5.19.
Chapter
6
DYNAMIC STABILITY OF A
SIMPLY SUPPORTED TAPERED BEAM
WITH THERMAL GRADIENT
6.1 Introduction
The modulus of elasticity of the material is greatly affected by the temperature. In highspeed atmospheric flights, nuclear engineering applications, drilling operations and steam and gas turbines, the mechanical and structural parts are subjected to high temperature.
For most of the engineering materials the Young’s modulus varies linearly with temperature [44,114]. Such machine and structural components may be subjected to time varying forces, which may lead to their parametric instability. Thermal effect on the dynamic stability is an important aspect of investigation. Many structural and machine components may be modeled as tapered beams.
Brown et al. [18] studied the dynamic stability of uniform bars using finite element method. Dhotarad and Ganeshan [34] carried out vibration analysis of a rectangular plate subjected to a uniform thermal gradient. Datta and Chakrabarty [29]
111
112
investigated the stability of a tapered beam by finite element method. Tomar and Jain
[124] studied the effect of thermal gradient on the frequencies of rotating tapered beams.
Mohanty and Kavi [82] in their other work studied the effect of thermal gradient on the dynamic stability of a Timoshenko beam.
The present work deals with the principal resonance of a parametrically excited simply supported tapered Euler beam subjected to thermal gradient along its length.
Finite element method is employed to carry out the analysis. The principal regions of instability are determined by Floquet’s theory.
6.2 Formulation of the problem
The simply supported beam of length L, tapered both in breadth and depth wise is shown in fig.6.1. A typical finite element is shown in fig.6.2. The element consists of two nodes i and j with v
i ,
θ
i
, v
j and
θ
j
as the nodal displacements. v is the lateral displacement and
θ represents the crosssectional rotation.
The lateral displacement v consists of only the bending displacement. The beam is subjected to a pulsating axial force P(t) = P
s
+ P
t cos
Ω
t
, where
Ω
is the excitation frequency of the dynamic load component, P
s
is the static and P
t is the amplitude of the time dependent component of the load acting along its undeformed axis.
6.2.1 Element matrices
The potential energy
(e )
of an element of length ‘l’ subjected to an axial force ‘P(t)’ is written as
U
(
e
)
=
1
2
l
∫
0
E
(
x
)
I d
2
v dx
2
2
dx
−
1
2
l
∫
0
P
(
t
)
⎛
⎝
dv dx
⎟
⎠
⎞
2
dx
(6.1)
113
The kinetic energy (T
(e)
) for an element is given by
T
(
e
)
=
1
2
l
∫
0
ρ
A
( )
{ }
dx
(6.2)
A cubic displacement distribution for v is assumed over the element as
v
=
a
1
+
a
2
x
+
a
3
x
2
+
a
4
x
3
(6.3) where
a
1
,
a
2
,
a
3
,
a
4
are called the generalised coordinates. Shape functions can be derived by using the expression of the static deflection of an Euler beam. The lateral displacement v within the element can be expressed in terms of the nodal displacement vector
{
∆
(e )
} as,
v = [N
1
N
2
N
3
N
4
]
⎧
⎪
v i
θ
i
⎫
⎪
⎪
⎩
v j
θ
j
⎪
⎪
= [N]
∆
(e
(6.4) where
[
N
] is the element shape function matrix,
N
1
= [13
ζ
2
+2
ζ
3
]
N
2
= [
ζ2ζ
2
+
ζ
3
] l
N
3
= [3
ζ
2
2
ζ
3
]
N
4
= [
ζ
2
+
ζ
3
] l
ζ = x/l
114
With the help of eq. (6.4) the potential energy (
U
(e)
)
and the kinetic energy(
T
(e )
) can be written in terms of nodal displacement vector as, respectively,
U
(e)
=
1
2
{ }
T
[ ]
{ } { }
2
T
P
(
t
)
[ ]
g
{ }
(6.5)
T
(e )
=
1
2
{ }
T
[ ]
{ }
(6.6)
Element mass matrix, element elastic stiffness matrix and element stability matrix are given by the expressions
[ ]
=
l
∫
o
[ ]
T
ρ
A
(
x
)
[ ]
dx
(6.7)
[ ]
)
=
l
∫
0
[ ] ″
T
E
(
x
)
I
(
x
)
[ ] ″
dx
(6.8)
[ ]
g
=
∫
l
0
[ ] [ ]
′
dx
(6.9) where
[ ]
″
=
∂
2
∂
x
2
[ ]
and
[ ]
′
=
∂
∂
x
[ ]
6.2.2 Governing equations of motion
The element equation of motion for the beam is derived by using Hamilton's principle.
δ
t
2
∫
t
1
(
T
(
e
)
−
U
(
e
)
)
dt
=
0
(6.10)
Substituting eqs. (6.5), (6.6) in to eq. (6.10) the equation of motion for the beam element is obtained as follows:
[ ]
)
{ }
+
[ ]
)
{ }
−
P
(
t
)
[ ]
g
)
{ }
=
0
(6.11)
115
Assembling mass, elastic stiffness and geometric stiffness matrices of individual element, the equation of motion for the global system is written as
[ ]
{ }
+
[ ]
{ }
−
P
(
t
)
[ ]
g
{ }
=
0
(6.12) where
{
∆
}
= Global displacement vector.
P s
the static and
P t
the amplitude of time dependent component of the load can be represented as the fraction of the fundamental static buckling load
P
*
of the beam without thermal gradient. Hence substituting, P(t) =
α
P
*
+
β
P
*
cos
t
α and
β
as static and dynamic load factors respectively.
The eq. (6.12) becomes
{ }
+
[ ]
−
α
P
*
[ ]
g s
−
β
P
* cos
Ω
t
[ ]
g t
{ }
=
0
(6.13)
If the static and time dependent component of loads are applied in the same manner, then
[ ]
g s
=
[ ]
g t
=
[ ]
g
.
Equation (6.13) represents a system of second order differential equations with periodic coefficients of the MathieuHill type. The instability regions are established using the Floquet’s theory and the same computational method as described in the chapter (3).
The Young’s modulus E is assumed to vary linearly along the length of beam due to thermal gradient and is given by [34].
E (x)= E
1
[1
β
t
T o
(1x/L)]
=
E
1
[1
δ(1x/L)] (6.14)
116
E
1
is the Young’s modulus of the material at the reference temperature
T o
,
which is at the reference point, x = L(tip),
β
t
is the slope of variation of E with T and
δ=
β
t
T o
is the thermal gradient factor.
The width ,height and area moment of inertia of the crosssection at any position x measured from the root side, is given as
b
(
x
)
=
b
1
−
δ
b x
⎫
I h
(
A
(
(
x x x
)
)
)
=
=
=
h
1
b
(
−
x
)
δ
h x
{ }{ }
{ }{ }
3
/ 12
⎪
⎪
⎪
(6.15)
δ
b
and
δ
h
are the taper per unit length in width and height respectively. b
1 and h
1
are the width and height at the root,x=0.
The overall matrices, [K], [K
g
] and [M] are obtained by assembling the corresponding element matrices. The displacement vector consists of only active nodal displacements.
6.3 Results and discussion
The following properties of the beam are taken for numerical computations: Length of the beam = 1m; cross – sectional dimension at the tip = 20 x 20 mm; cross – sectional dimension at the root = 120 x 120 mm; material mass density of the aluminum beam =
2800 kg /m
3
; E
1
= 70 x 10
3
MPa.
A tenelement discretisation of the beam gives satisfactory convergence of natural frequencies and buckling loads for the first five modes. Also the instability regions obtained without thermal gradient match well with those of [29]. Comparison is given in
Table 6.1.
117
Table6.1 Comparison of boundary frequencies of the first instability region obtained from present analysis with reference [29]. Static load factor
α=0.5. ω
1 is the fundamental frequency of the beam without thermal gradient
Dynamic load factor
(
β)
Lower limit of boundary frequency ratio (
Ω
1
/
ω
1
)
Reference [29] Present
Upper limit of boundary frequency ratio (
Ω
2
/
ω
1
)
Reference [29] Present
0 1.4551 1.4724 1.4551 1.4724
Figures (6.3) and (6.4) show the variation of natural frequency and buckling load respectively of the first three modes with thermal gradient factor
δ. The values of both the frequency and buckling load decrease with increase in the value of
δ. This decrease is negligible for fundamental natural frequency and buckling load. The rate of decrease is more for higher modes.
In order to study the stability of the beam, instability regions have been obtained for different values of thermal gradient factor
δ and static load factor α. These are shown in figs. (6.56.9). First three regions of primary resonance have been shown in the figures.
In these figures
ω
1 is the fundamental frequency of the beam without thermal gradient.
Figure (6.5) shows the instability regions for
α=0.4 and δ = 0, 0.3 and 0.6. It is seen that with thermal gradient present, the lower boundary of the first instability region curls towards the dynamic load factor axis, thereby making the first instability region wider. For higher values of thermal gradient factor the lower boundary truncates on the
β
axis at values of
β
less than one. For example, for
α = 0.4 and δ= 0.6 the lower boundary of the first instability region truncates on the
β
axis at
β
= 0.8. Truncation of lower boundary on the
β
axis at values of
β
less than one indicates instability for
118
amplitude of time dependent component of the load less than the fundamental static buckling load. There is also increase in areas of the other two instabilities regions.
Moreover increase in thermal gradient factor (
δ) shifts the instability regions towards the dynamic load factor axis, that is the instability occurs at lower frequency of excitation.
This shift is less for the first instability region but for other two regions it is relatively large.
Figure (6.6) shows the instability regions for
α=0.5 and δ = 0, 0.3 and 0.6 respectively. Increase of thermal gradient factor has the same effect on the instability regions as discussed above.
Figure (6.7) shows the instability regions for
α=0.8 and δ = 0, 0.3 and 0.6 respectively. The instability regions show the same behaviour with increase in
δ as discussed earlier. For
α=0.8 and δ = 0.6 the lower boundary of the first instability region vanishes.
Figure (6.8) shows the instability regions for
δ = 0.3 and α=0.4, 0.5 and 0.8. It is seen that with increase in static load factor, the instability region shifts towards dynamic factor axis. Also there is increase in area of the instability region with increase in
α. The increase in area of the first instability region is quite large compared to second and third one.
Figure (6.9) shows the instability regions for
δ = 0.6 and α =0.4, 0.5 and 0.8. The effect of
α on the instability regions is same as discussed earlier.
To study the relative effect of
α and δ on the instability regions three sets of values of
α and δ, i)α=0.5 and δ= 0.0 ii) α=0.5 and δ= 0.3 iii) α=0.6118 and δ= 0.0 have been considered. The additional load factor 0.1183, in addition to 0.5, in case (iii)
119
corresponds to the difference in fundamental buckling load for the beam without thermal gradient and with thermal gradient (
α=0.3). This is shown in fig. (6.10). It is seen that with
α=0.5 and δ= 0.0, when in one case the load factor is increased to 0.6183 with δ=0.0 and in other case with
α=0.5 there is a thermal gradient, δ= 0.3, all the three instability regions shifts towards lower frequency of excitation. Because of thermal gradient the shift of regions towards lower frequency is more than compared to an equivalent increase in static load factor. The relative effect on the first primary resonance zone is not that much pronounced. But the thermal gradient factor shifts the second and third instability regions considerably more towards the dynamic factor axis as compared to the static load factor. This means that the thermal gradient factor has relatively more destabilising effect than static load factor.
6.4 Closure
Effect of thermal gradient and static load factor on the dynamic stability of a simply supported tapered beam is analysed by finite element method in conjunction with
Floquet’s theory. Present investigation reveals that an increase in thermal gradient factor decreases the natural frequency and the static buckling load. The effect is more pronounced on second and third modes than on first mode. Increase in static load component or thermal gradient factor has a destabilising effect. Thermal gradient factor has a greater destablising effect than static load factor, for an equivalent change in static load component.
L
Fig. – 6.1, Simply supported tapered beam with
thermal gradient
θ
i v i v j
θ
j
Fig
.
6.2, Tapered beam element
120
121
600
400
200
1200
1000
800
35
30
25
20
15
10
5
50
45
40
0
0 0.1
0.2
0.3
0.4
0.5
Thermal gradient factor(
δ)
0.6
0.7
Fig.  6.3, Effect of thermal gradient factor on fundamental frequency
0.8
mode1 mode2 mode3
0.9
mode1 mode2 mode3
0
0 0.1 0.2
0.3
0.4
0.5
0.6
0.7
Thermal gradient factor (
δ)
Fig. 6.4, Effect of thermal gradient factor on critical buckling load
0.
8
0.9
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Stable
2
ω
2
Stable
2
ω
3
0
0 5 10 15
Frequency ratio
Ω
/
ω
1
Fig. 6.5,Instability regions for,
α
=0.4,
δ
=0.0; ,
δ
=0.3; ,
δ
=0.6;... ,.
20
0.6
0.5
0.4
0.3
0.2
0.1
1
0.9
0.8
0.7
2
ω
2
Stable
Stable
0
0 5 10
Frequency ratio
Ω
/
15
ω
1
Fig. 6.6,Instability regions for,
α
=0.5,key as fig.6.5.
2
ω
3
20
122
25
25
123
0.6
0.5
0.4
0.3
0.2
0.1
1
0.9
0.8
0.7
Stable
2
ω
2
Stable
0
0 5 10
Frequency ratio
Ω
/
15
ω
1
Fig. 6.7,Instability regions for,
α
=0.8,key as fig.6.5.
2
ω
3
20 25
0.6
0.5
0.4
0.3
0.2
0.1
1
0.9
0.8
0.7
Stable
Stable
0
0 5 10
Frequency ratio
Ω
/
15
ω
1
20
Fig. 6.8,Instability regions for,
δ
=0.3 ,
α
=0.4; ,
α
=0.5; ,
α
=0.8;... ,.
Stable
25
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Stable
Stable
0
0 2 4 6 8 10 12
Frequency ratio
Ω
/
ω
1
Fig. 6.9,Instability regions for,
δ
=0.6,key as fig. 6.8.
14 16 18 20
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Stable
2
ω
2
Stable
2
ω
3
0
0 5
Fig. 6.10,Instability regions for,
10
Frequency ratio
Ω
/
15
ω
1
20
α
=0.5,
δ
=0.0;,
α
=0.5,
δ
=0.3;,
α
=0.6138,
δ
=0.0;... ,.
25
124
Chapter
7
EXPERIMENTAL WORK
7.1 Introduction
The aim of the experimental work is to establish experimentally the stability diagrams for a few typical cases related to beams with localised damage and for sandwich beams, described in chapters (35). For uniform Timoshenko beam and pretwisted cantilever beam with localised damage, the instability regions have been experimentally determined for the positions of the damage for which the first instability region occurs at minimum frequency of excitation. For uniform beam with localised damage these positions of the damage for fixedfree, pinnedpinned, fixedfixed and fixedpinned boundary conditions, correspond to position parameters
ψ = 0.1, ψ = 0.5, ψ = 0.5 and ψ = 0.7 respectively. For uniform twisted cantilever beam the instability regions have been experimentally determined for
ψ=0.1. For sandwich beams, the stability diagrams have been experimentally established for 3, 5 and 7 layered beams. The theoretical and experimental stability diagrams have been compared to assess the accuracy of the theoretical results.
7.2 Description of the experimental set up
The schematic diagram of the equipments used for the experiment and photographic view of the experimental set up are shown in the fig.7.1 and Plate1 respectively. 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 P
t cos
Ωt is applied to the specimen by a 500N capacity electrodynamic shaker (Saraswati Dynamics,
India, Model no. SEV005). 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. 2310100), which is fixed between the shaker and the specimen. The vibration response of the test specimen is measured by means of vibration pickups (B&K type, model no. MM0002). The signals from the pickups and load cell are observed on a computer through a sixchannel data acquisition system (B&K, 3560C), which works on Pulse software platform (B&K 7770,
Version 9.0).
In order to achieve pin end condition for the beam end, an attachment is provided.
The attachment has a pin mounted on two ball bearings as shown in Plate2. The beam end is adhered to the pin. For pin end condition at both the ends of the beam, two such attachments, one fixed to the load cell and other attached to the screw jack are provided.
For fixed end condition the beam end is rigidly clamped to the steel angle bars by means of screws (Plate 2).
126
7.3 Preparation of specimens
i) Beams with localised damage
The specimens were prepared from aluminum strips of 1.612mm thickness and 25.4mm wide. Rectangular cutouts of suitable dimension were made on the strip at required places to produce artificial damage of specified extent of damage and size. For example to produce a damage with extent of damage
ξ = 0.5, size parameter τ = 0.2 and position parameter
ψ = 0.5, a slot of width equal to the half the width of the strip, length equal to
0.2 times the length of the specimen was cut symmetrically about the longitudinal axis at the middle of the beam. The cutouts were filled with suitable filler material (Mseal) of weight equal the weight of the material removed to produce the slot. The filler material was uniformly distributed over the slot area so that mass per unit length of the beam was same as that of the original beam. To fabricate twisted beam with damage, the aluminum strips were twisted, clamping one end and twisting the other end to the required angle.
Care was taken that the angle of twist varies linearly from clamped end to the free end.
The slot of required size was cut at the required position and filled with filler material.
The details of the physical and geometric data of the specimens are given in Tables 7.1
7.4. The photographs of the specimens are also shown in Plate3. ii) Sandwich beams
Sandwich beams were fabricated from strips cut from mild steel sheets of suitable thickness. The viscoelastic core of the sandwich beam is P.V.C. In preparing the sandwich beams the face layers were made free from dirt, grease etc. by cleaning their surfaces with acetone and carbon tetrachloride. The adhesive used for bonding the layers is commercially available dendrite. After application of thin layer of adhesive on the
127
surfaces, the layers were bonded and the sandwich beams were kept under load for about six hours. Slipping of the layers was avoided by providing positioning guides at all the edges of the specimen during the setting time. The Young’s modulus of the specimen materials was determined by measuring the static deflection of a test specimen under known load. Mass density of the specimen materials was measured by measuring the weight and volume of a piece of specimen material. The details of the physical and geometric data of the specimens are given in Tables 7.57.6 and the photographs of the specimen are shown in Plate3.
7.4 Testing 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 noncontacting vibration pickups. For straight beams with pinnedpinned, fixedfixed and fixedpinned end conditions two pickups, one at each end of the beam were used, where as for fixedfree end condition only one pick up was placed near to the fixed end to sense the vibration response of the beam. For twisted cantilever to sense the response of the beam in two fixed perpendicular directions, two pickups mutually perpendicular to each other were placed near the fixed end. The air gap between the pickups and the vibrating surface were so adjusted that the measurements were in the linear range. The amplitude of the signal gives no absolute displacement since it is not calibrated.
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
128
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 (P
t
,
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
ω
1 to get the nondimensional excitation frequency
(
Ω/
ω
1
). For sandwich beams the excitation frequency was divided by 2
ω o
to get the nondimensional excitation frequency (
Ω/2
ω o
).
Similarly the dynamic load amplitude was divided by the reference load P
* to get the dynamic load factor
β
. . The details of observations are given in Tables 7.77.16.
7.5 Results and discussion
The boundary frequencies obtained from experiments are shown in Tables 7.77.16 and also in figs.7.27.11. In Tables 7.77.16, the second column shows the experimentally measured amplitude of the applied dynamic load (P
t
). The columns 3 rd
, 5 th
and 7 th
of these tables show the experimentally measured lower limits (excitation frequencies) of the instability regions, whereas frequencies shown in the columns 4 th
, 6 th
and 8 th of the same tables are the experimentally determined upper limits of the instability regions. In the 9 th
column, the calculated dynamic load factor (
β
), which is obtained by dividing the
129
dynamic load amplitude (P
t
) by P
* has been recorded. In the columns 10 to 15 the calculated boundary frequency ratios corresponding to the lower and upper limits of the instability regions are shown. For example in the row, corresponding to sl.no.1 in the
Table 7.7, the experimentally measured amplitude of the dynamic load component is
1.4N. The experimentally obtained excitation frequencies corresponding to the lower boundary of the first, second and third instability regions are 17.0, 133.7 and 383.8 Hz respectively. The experimentally obtained excitation frequencies corresponding to the upper boundary of the first, second and third instability regions are 18.8, 135 and 386.0
Hz respectively. The calculated corresponding dynamic load factor is 0.118. The calculated frequency ratios corresponding to the lower limits of the first, second and third instability regions are 1.45, 11.4 and 32.72 respectively, whereas corresponding upper limits are 1.6, 11.51 and 32.91 respectively. i) Timoshenko beam with localised damage.
Figures (7.27.5) show the theoretical instability regions along with the experimentally obtained boundary frequencies of the instability regions for uniform Timoshenko beam with localised damage and with fixedfree, pinnedpinned, fixedfixed and fixedpinned boundary conditions respectively. Figures 7.2 (a)7.2(c) show the instability regions for fixedfree case. It is seen that the experimental boundary frequencies are close to the theoretical boundaries. The maximum differences between the theoretical and experimental values of the excitation frequency ratio for the first three instability regions are 0.11, 0.15 and 0.2 respectively. In terms of the absolute value of the excitation frequency, these values correspond to a difference of 1.3Hz, 1.8Hz and 2.3Hz respectively, which show reasonable agreement between the theoretical and experimental
130
results. Figures 7.3(a)7.3(c) show the instability regions for pinnedpinned case. The maximum differences between the theoretical and experimental values of the excitation frequency ratio for the three instability regions are 0.27, 0.36 and 0.84 respectively, In terms of the absolute value of the excitation frequencies, these values correspond to a difference of 4.3Hz, 5.7 Hz and 13.4 Hz respectively, which show considerable deviation between the theoretical and experimental results. From figs. 7.4(a)7.4(c), it is seen that the experimental boundary frequencies are very close to the theoretical boundaries for fixedfixed case. The maximum differences between the theoretical and experimental values of the excitation frequency ratio for the first three instability regions are 0.05, 0.09 and 0.1 respectively. In terms of the absolute value of the excitation frequency these values correspond to a difference of 0.9Hz, 1.6 Hz and 1.8 Hz respectively, which show close agreement between the theoretical and experimental results. Figures 7.5(a)7.5(c) show the instability regions for fixedpinned case. The maximum differences between the theoretical and experimental values of the excitation frequency ratio for the first three instability regions are 0.12, 0.14 and 0.18 respectively. In terms of the absolute value of the excitation frequency these values correspond to a difference of 3.0Hz, 3.5 Hz and 4.5
Hz respectively. It is seen that for fixedfree and fixedfixed cases there is reasonable agreement between the theoretical and experimental results. For pinnedpinned and fixedpinned case there is deviation in the theoretical and experimental results, particularly in case of pinnedpinned end condition the difference is quite considerable. Svensson [121] has observed that damping in hinges shifts the instability regions to lower frequencies. In this case also the experimentally observed points are at lower values compared to the
131
theoretical ones. So this difference may be due to the damping present in the hinged ends, which has not been taken in to account in the theoretical analysis. ii) Twisted cantilever beam with localised damage.
Figures 7.6(a)7.6(c) show the theoretical and the experimental instability regions for the twisted beam with an angle of pretwist 30
0
. It is seen that there is close agreement between these two results. The maximum deviation in frequency ratio occurs in case of the second instability zone and this is equal to 0.1. This deviation in frequency ratio is equivalent to a difference of 1.3 Hz in the theoretical and experimental values of the excitation frequency. Figures7.7 (a)7.7(c) show the instability regions for twisted beam with an angle of pretwist 60
0
. In this case there is also close agreement between theoretical and experimental results, maximum deviation in excitation frequency being
1.72 Hz, which occurs in case of third instability region. Instability regions for twisted beam with an angle of pretwist 90
0 are shown in figs.7.8 (a)7.8(c). In this case there is also close agreement between theoretical and experimental results, maximum deviation in excitation frequency being 1.1 Hz, which occurs in case of second instability region. iii) Multilayer cantilever sandwich beams.
The theoretical and experimentally obtained first two resonant frequencies for the three, five and seven layer beams having the physical and geometrical parameters as given in
Table7.5 are presented in Table7.6. These values show reasonable agreement.
Figure (7.9) shows the theoretical and experimental instability diagrams for a three layer cantilever sandwich beam. It is seen that first simple instability region and the instability region of combination type from theoretical analysis are fairly close to the experimental ones. There is marked deviation between the theoretical and experimental second simple
132
instability region. This may be due to the deviation in the theoretical and actual values of the second resonant frequency, Table7.6. Figure (7.10) shows the theoretical and experimental instability regions for a five layer cantilever sandwich beam. In this case for the geometrical and physical parameters of the beam, the instability regions of combination resonance of sum type or difference type and second simple resonance do not occur within the value of dynamic load factor
β
=5.0. The figure shows reasonable agreement between the experimental and theoretical results. Figure (7.11) shows the theoretical and experimental instability diagrams for a seven layer cantilever sandwich beam. In this case also first simple instability region and the instability region of combination type from theoretical analysis are fairly close to the experimental ones.
There is marked deviation between the theoretical and experimental second simple instability region, which may be due to the deviation in the theoretical and actual value of the second resonant frequency, Table 7.6.
7.6 Closure
The theoretical and experimental instability diagrams have been compared to assess the accuracy of the theoretical results. It has been found that for Timoshenko beam with localised damage, the theoretical and experimental results are in close agreement for fixedfree and fixedfixed boundary conditions. But there is deviation in case of pinnedpinned and fixedpinned end conditions. This deviation may be due to the presence of damping in the hinges. For pretwisted cantilever beams there is close agreement between the theoretical and experimental instability regions. For sandwich beams there is reasonable agreement between the theoretical and experimental results.
133
11
8
6
7
5
9
4
3
2
1
10
PC
Fig. 7.1, Schematic diagram of the test set up, 1Specimen, 2Upper pickup,
3Upper support (Pinned end), 4Screw jack, 5Load cell, 6Lower pickup,
7Lower support (Clamped end),8Accelerometer, 9Vibration generator,
10Oscillator and amplifier,11Data acquisition system.
134
1
0.8
0.6
0.4
0.2
(a)
1 1.2
1.4
1.6
Frequency ratio
Ω
/
ω
1
1.8
2 2.2
1
0.8
0.6
0.4
0.2
(b)
0
10.8
11 11.2
11.4
11.6
Frequency ratio
Ω
/
ω
1
11.8
12 12.2
12.4
1
0.8
0.6
0.4
0.2
(c)
0
32.2
32.4
32.6
32.8
33 33.2
33.4
33.6
Frequency ratio
Ω
/
Fig.7.2,Instability regions,for fixedfree case,
ξ
ω
1
=0.5,
ψ
=0.1,
τ
=0.2,
Theoretical cal boundary from FEM;, Experimental data;
•
.
135
1
0.8
0.6
0.4
0.2
(a)
0
0.8
1 1.2
1.4
1.6
1.8
Frequency ratio
Ω
/
ω
1
2 2.2
2.4
1
0.8
0.6
0.4
0.2
(b)
0
7 7.5
Frequency ratio
Ω
/
ω
1
8
1
0.8
0.6
0.4
0.2
(c)
0
15 15.2
15.4
15.6
15.8
16 16.2
16.4
16.6
Frequency ratio
Ω
/
ω
1
Fig.7.3,Instability regions,for pinnedpinned case,
ξ
=0.5,
ψ
=0.5,
τ
=0.2,
Theoretical cal boundary from FEM;, Experimental data;
•
.
16.8
2.6
8.5
17
136
1
0.8
0.6
0.4
0.2
(a)
0
1 1.5
Frequency ratio
Ω
/
ω
1
2 2.5
1
0.8
0.6
0.4
0.2
(b)
0
4.4
4.6
4.8
5 5.2
5.4
Frequency ratio
Ω
/
ω
1
5.6
5.8
6 6.2
1
0.8
0.6
0.4
0.2
(c)
0
9 9.2
9.4
9.6
9.8
10 10.2
Frequency ratio
Ω
/
ω
1
10.4
10.6
10.8
11
Fig.7.4,Instability regions,for fixedfixed case,
ξ
=0.5,
ψ
=0.5,
τ
=0.2,
Theoretical cal boundary from FEM;, Experimental data;
•
.
137
1
0.8
0.6
0.4
0.2
(a)
0
1 1.5
Frequency ratio
Ω
/
ω
1
2 2.5
1
0.8
0.6
0.4
0.2
(b)
0
5.2
5.4
5.6
5.8
6
Frequency ratio
Ω
/
ω
1
6.2
6.4
6.6
6.8
1
0.8
0.6
0.4
0.2
(c)
0
12.2
12.4
12.6
12.8
13
Frequency ratio
Ω
/
ω
1
13.2
13.4
13.6
13.8
Fig.7.5,Instability regions,for fixedpinned case,
ξ
=0.5,
ψ
=0.7,
τ
=0.2,
Theoretical cal boundary from FEM;, Experimental data;
•
.
138
1
0.8
0.6
0.4
0.2
(a)
0
0.8
1 1.2
1.4
1.6
Frequency ratio
Ω
/
ω
1
1.8
2
1
0.8
0.6
0.4
0.2
(b)
0
9.6
9.8
10 10.2
Frequency ratio
Ω
/
ω
1
10.4
10.6
1
0.8
0.6
0.4
0.2
(c)
0
27.2
27.25
27.3
27.35
27.4
27.45
Frequency ratio
Ω
/
ω
1
27.5
Fig. 7.6 ,Instability regions,
θ
1
=30
0
,
ξ
=0.5,
ψ
=0.1,
φ
=0.2,
Theoretical Boundary from FEM; , Experimental data;
•
.
27.55
2.2
10.8
27.6
139
1
0.8
0.6
0.4
0.2
(a)
0
0.8
1 1.2
1.4
1.6
Frequency ratio
Ω
/
ω
1
1.8
1
0.8
0.6
0.4
0.2
(b)
0
7.7
7.8
7.9
8 8.1
8.2
8.3
Frequency ratio
Ω
/
ω
1
8.4
8.5
1
0.8
0.6
0.4
0.2
(c)
0
28.6
28.7
28.8
28.9
29 29.1
29.2
29.3
29.4
Fig. 7.7 ,Instability regions,
θ
1
Frequency ratio
Ω
/
ω
1
=60
0
,
ξ
=0.5,
ψ
=0.1,
φ
=0.2,
Theoretical Boundary from FEM; , Experimental data;
•
.
2
8.6
29.5
2.2
8.7
29.6
140
1
0.8
0.6
0.4
0.2
0
0.8
(a)
1 1.2
1.4
1.6
Frequency ratio
Ω
/
ω
1
1.8
1
0.8
0.6
0.4
0.2
(b)
0
6.2
6.3
6.4
6.5
6.6
6.7
Frequency ratio
Ω
/
ω
6.8
1
6.9
7
1
0.8
0.6
0.4
0.2
(c)
0
27.3
27.4
27.5
27.6
27.7
27.8
Frequency ratio
Ω
/
ω
1
27.9
28 28.1
Fig.7.8 ,Instability regions,
θ
1
=90
0
,
ξ
=0.5,
ψ
=0.1,
φ
=0.2,
Theoretical Boundary from FEM; , Experimental data;
•
.
2
7.1
28.2
2.2
7.2
28.3
141
3.5
3
2.5
2
1.5
1
5
4.5
4
3
2.5
2
1.5
1
0.5
5
4.5
4
3.5
2
ω
1
ω
1
+
ω
2
2
ω
2
0
0 5 10 15 20 25
Frequency ratio
Ω
/2
30
ω
o
Fig.  7.9 ,Instability regions for three layer beam:,t
21
35
=0.25,
η
c
40 45
=0.55,
Theoretical Boundary from FEM;, Experimental data;
•
.
50
2
ω
1
10 11 12 13 14 15
Frequency ratio
Ω
/2
16
ω
o
Fig.  7.10,Instability regions for five layer beam:,t
17
21
=0.25,
η
c
18 19
=0.55,
Theoretical Boundary from FEM;, Experimental data;
•
.
20
142
3.5
3
2.5
2
1.5
1
0.5
5
4.5
4
2
ω
1
ω
1
+
ω
2
2
ω
2
0
0 20 40 60
Frequency ratio
Ω
/2
ω
o
80
Fig.  7.11 ,Instability regions for seven layer beam:,t
21
=0.5,
η
c
100
=0.55,
Theoretical Boundary from FEM;, Experimental data;
•
.
120
143
PHYSICAL AND GEOMETRICAL PARAMETERS OF THE TEST SPECIMENS.
Table7.1, Beams with localised damage and various boundary conditions.
Material of the specimens: Aluminium
Young’s Modulus of the Aluminium material,
E
Al
=
55 .
583
×
10
9
N
/
m
2
and
Mass density of Aluminium,
ρ
Al
=
2613
Kg
/
m
3
.
CASE Length of the beam in mm
Width of the beam in mm
Thickness of the beam in mm
Pinnedpinned 460 do do
Fixedfixed 650 do do
Fixedpinned 460 do do
Table7.2, Theoretical values of fundamental natural frequency and critical buckling load
of uniform undamaged beams with various boundary conditions.
CASE Fundamental frequency in Hz. Critical buckling load in N.
Fixedfree 11.73
Pinnedpinned 15.93
11.87
22.99
Fixedfixed 18.09
Fixedpinned 24.89
Table7.3, Pretwisted cantilever beams with localised damage.
Physical and geometrical parameters of the test specimens.
46.05
45.97
Material of the specimens: Aluminium
Young’s Modulus of the Aluminium material,
E
Al
=
55 .
583
×
10
9
N
/
m
2
and
Mass density of Aluminium,
ρ
Al
=
2613
Kg
/
m
3
.
Length of the beams in mm
Width of the beams in mm
Thickness of the beam in mm
300 25.4 1.612
Table7.4, Theoretical values of natural frequency and critical buckling load of straight
undamaged cantilever beam.
Fundamental frequency in Hz. Critical buckling load in N.
13.34 13.511
144
Table 7.5, Physical and geometrical data of sandwich beam specimens.
Three layer beam
L
t
1 t
2
G
*
v
0.00025
×10
6
η
c
0.55
Five layer beam
L
t
1 t
2
G
*
v
0.00025
×10
6
η
c
0.55
Seven layer beam
L
t
1 t
2
G
*
v
0.00025
×10
6
η
c
0.55
L =Length of the beam in m.
E
1
2.08
×10
11
E
1
2.08
×10
11
E
1
2.08
×10
11
t
1=
Thickness of the elastic layer in m.
t
2=
Thickness of the viscoelastic layer in m .
G v
*
=Shear modulus of viscoelastic core layer in N/m
2
.
η
c
=Core loss factor .
E
1=
Young’s modulus of the elastic layer in N/m
2
.
Elastic layer is made up of mild steel sheet and Viscoelastic core material is of PVC.
Table 7.6, Theoretical and experimental resonant frequencies of Multilayer beams.
Resonant frequency of threelayer beam in Hz
Second 134.43 145 +7.9
Resonant frequency of Fivelayer beam in Hz
First 13.32 12
Second 71.21 80.6
Resonant frequency of Sevenlayer beam in Hz
145
Experimental boundary frequencies of instability regions
Uniform beam with localised damage and different boundary conditions.
Table7.7, Experimental boundary frequencies of instability regions for uniform beam with localised damage,
Fixedfree end conditions, P
*
=11.87 N,
ω
1
=11.73 Hz.
Sl
No
1
Dynamic load
Amplitude
(P t
) in N
1.4
1 st
Zone
Lower limit
(
Ω
11
)
17.0
Upper limit
(
Ω
12
)
18.8
Excitation Frequency (
Ω)
2 nd
Zone 3 rd
Zone
Lower limit
(
Ω
21
)
133.7
Upper limit
(
Ω
22
)
135.0
Lower limit
(
Ω
31
)
383.8
Upper limit
(
Ω
32
)
386.0
Dynamic
Load
Factor
β=
P t
/ P
*
0.118
1 st
Zone
Lower limit
(
Ω
11
/
ω
1
)
1.45
Excitation frequency ratio
Ω/ω
1
Upper limit
(
Ω
12
/
ω
1
)
1.6
2 nd
Zone
Lower
Upper limit
(
Ω
22
/
ω
1
)
3rd Zone
Lower limit
(
Ω
31/
ω
1
)
Upper limit
(
Ω
32
/
ω
1
) limit
(
Ω
21
/
ω
1
)
11.4 11.51
32.72
32.91
2 14.3
21.0 136.9
381.2
387.7 1.22
1.79
11.11
11.67
32.5
33.05
3 7.89
4 9.61
12.4
22.6
129.8
138.9
380.5
388.6
0.665
1.06
1.93
11.2
23.6
127.7
139.4
378.4
389.4
0.81 0.95
2.01
11.07
11.84
32.44
33.13
10.89
11.88
32.26
33.2
Table7.8, Experimental boundary frequencies of instability regions for uniform beam with localised damage,
Pinnedpinned end conditions, P
*
= 22.99 N,
ω
1
= 15.93 Hz.
Sl
No
Dynamic load
Amplitude
(P t
) in N
1
1 st
Zone
Lower limit
(
Ω
11
)
Upper limit
(
Ω
12
)
Excitation Frequency (
Ω)
2 nd
Zone 3 rd
Zone
Lower limit
(
Ω
21
)
Upper limit
(
Ω
22
)
Lower limit
(
Ω
31
)
Upper limit
(
Ω
32
)
Dynamic
Load
Factor
β=
P t
/ P
*
25.6
27.4 124.2
253.0
256.6
Excitation frequency ratio
Ω/ω
1
1 st
Zone
Lower limit
(
Ω
11
/
ω
1
)
Upper limit
(
Ω
12
/
ω
1
)
2 nd
Zone
Lower limit
(
Ω
21
/
ω
1
)
Upper limit
(
Ω
22
/
ω
1
)
3rd Zone
Lower limit
(
Ω
31/
ω
1
)
Upper limit
(
Ω
32
/
ω
1
)
1.61
1.72
7.71 7.80
15.90
16.10
3 17.6
122.6 252.5
258.6
32.8 126.6
247.0
260.0
4 12.8
40.4 240.6
0.87
1.58
1.83
7.70
7.82
15.85
16.23
1.10
2.05
7.20
7.95
15.50
16.3
0.8
2.54
7.00
8.10
15.10
16.5
146
Table7.9, Experimental boundary frequencies of instability regions for uniform beam with localised damage,
Fixedfixed end conditions, P
*
= 46.05 N,
ω
1
= 18.09 Hz.
Sl
No
Dynamic load
Amplitude
(P t
) in N
1 st
Zone
Lower limit
(
Ω
11
)
Upper limit
(
Ω
12
)
Excitation Frequency (
Ω)
2 nd
Zone 3 rd
Zone
Lower limit
(
Ω
21
)
Upper limit
(
Ω
22
)
Lower limit
(
Ω
31
)
Upper limit
(
Ω
32
)
32.2
95
Dynamic
Load
Factor
β=
P t
/ P
*
98.0
179.8
182.4
Excitation frequency ratio
Ω/ω
1
1 st
Zone
2 nd
Zone
3rd Zone
Lower limit
(
Ω
11
/
ω
1
)
Upper limit
(
Ω
12
/
ω
1
)
Lower
Upper limit
(
Ω
22
/
ω
1
)
Lower limit
(
Ω
31/
ω
1
)
Upper limit
(
Ω
32
/
ω
1
) limit
(
Ω
21
/
ω
1
)
1.78
1.86
5.25 5.42
9.94
10.08
2
3
4
5
29.8
35.6 100.0
177.2
186.2
26.0
39.0 104.4
173.4
189.4
1.65
1.97
1.44
2.15
24.8
40.6 106.4
169.2
191.8 1.37
2.24
21.8
41.2 108.4
166.4
192.6 1.21
2.28
5.16
5.53
4.85
5.77
4.73
5.88
4.64
5.99
9.8
10.29
9.58
10.47
9.35
10.6
9.2
10.65
Table7.10, Experimental boundary frequencies of instability regions for uniform beam with localised damage,
Fixedpinned end conditions. P
*
= 45.97 N,
ω
1
= 24.89Hz.
Sl
No
Dynamic load
Amplitude
(P t
) in N
1
1 st
Zone
Lower limit
(
Ω
11
)
Upper limit
(
Ω
12
)
Excitation Frequency (
Ω)
2 nd
Zone 3 rd
Zone
Lower limit
(
Ω
21
)
Upper limit
(
Ω
22
)
Lower limit
(
Ω
31
)
Upper limit
(
Ω
32
)
Dynamic
Load
Factor
β=
P t
/ P
*
42.1
45.7 149.6
319.6
323.0
Excitation frequency ratio
Ω/ω
1
1 st
Zone
2 nd
Zone
Lower limit
(
Ω
11
/
ω
1
)
Upper limit
(
Ω
12
/
ω
1
)
Lower limit
(
Ω
21
/
ω
1
)
Upper limit
(
Ω
22
/
ω
1
)
1.69
1.84
5.75 6.0
3rd Zone
Lower limit
(
Ω
31/
ω
1
)
Upper limit
(
Ω
32
/
ω
1
)
12.84
12.98
2
3 28.36
4
34.9
48.8 153.0
313.6
326.1
32.4
52.3
134.4
157.0
307.4
328.5
0.617
25.4
55.0 160.5
304.9
305.0
1.4
1.96
5.6
1.33
2.1 5.4
6.15
6.31
12.6
13.1
12.35
13.2
1.02
2.21
5.25
6.45
12.25
13.4
147
Cantilever twisted beam with localised damage.
Table7.11, Experimental boundary frequencies of instability regions for twisted beam with localised damage
L=0.3m,
θ
1
= 30
0
,
ψ=0.1,τ=0.2, P
*
= 13.511 N,
ω
1
= 13.34Hz.
Sl
No
Dynamic load
Amplitude
(P t
) in N
1
2
1 st
Zone
Lower limit
(
Ω
11
)
Upper limit
(
Ω
12
)
Excitation Frequency (
Ω)
2 nd
Zone 3 rd
Zone
Lower limit
(
Ω
21
)
Upper limit
(
Ω
22
)
Lower limit
(
Ω
31
)
Upper limit
(
Ω
32
)
Dynamic
Load
Factor
β=
P t
/ P
*
20.2
22.8 136.8
365.6
366.4
1 st
Zone
Lower limit
(
Ω
11
/
ω
1
)
Excitation frequency ratio
Upper limit
(
Ω
12
/
ω
1
)
1.51
1.7
Ω/ω
1
2 nd
Zone
3rd Zone
Lower limit
(
Ω
21
/
ω
1
)
Upper limit
(
Ω
22
/
ω
1
)
Lower limit
(
Ω
31/
ω
1
)
Upper limit
(
Ω
32
/
ω
1
)
10.1 10.25
27.4
27.46
19.4
23.4 137.4
364.8
366.2
17.4
24.0 368.0
1.45
1.3
1.75
1.8
10.07
10.3
27.35
27.45
9.9 10.4
27.32
27.5
4 15.4
25.4 140.0
364.2
367.0
5 13.4
26.8 368.2
1.15
1.0
1.9
2.0
9.85
9.7
10.5
10.6
27.3
27.2
27.51
27.6
Table7.12, Experimental boundary frequencies of instability regions for twisted beam with localised damage
L= 0.3m,
θ
1
= 60
0
,
ψ=0.1, τ=0.2, P
*
= 13.511 N,
ω
1
= 13.34Hz.
Sl
No
Dynamic load
Amplitude
(P t
) in N Lower limit
(
Ω
11
)
1 st
Zone
Upper limit
(
Ω
12
)
Excitation Frequency (
Ω)
2 nd
Zone 3 rd
Zone
Lower limit
(
Ω
21
)
Upper limit
(
Ω
22
)
Lower limit
(
Ω
31
)
Upper limit
(
Ω
32
)
Dynamic
Load
Factor
β=
P t
/ P
*
1 st
Zone
Lower limit
(
Ω
11
/
ω
1
)
Excitation frequency ratio
Ω/ω
1
Upper limit
(
Ω
12
/
ω
1
)
2 nd
Zone
Lower limit
(
Ω
21
/
ω
1
)
Upper limit
(
Ω
22
/
ω
1
)
3rd Zone
Lower limit
(
Ω
31/
ω
1
)
Upper limit
(
Ω
32
/
ω
1
)
20.0
22.7
109.4
26.2
22.7
108.1
389.5 1.5
1.7
8.2 8.3
29.0
29.2
390.9 1.4
1.7
8.1
8.4
28.95
29.3
5
17.4
14.6
24.7
26.0
106.7
105.4
13.4
26.7
104.7
390.9 1.3
1.85
8.0
8.5
28.8
29.3
390.2 1.1
2.0
7.9
8.5
28.8
29.4
390.5 1.0
2.0
7.85
8.6
28.6
29.5
148
Table7.13, Experimental boundary frequencies of instability regions for twisted beam with localised damage
L=0.3m,
θ
1
= 90
0
,
ψ=0.1, τ=0.2, P
*
= 13.511 N,
ω
1
= 13.34Hz.
Sl
No
Dynamic load
Amplitude
(P t)
1 st
Zone
Lower limit
(
Ω
11
)
Upper limit
(
Ω
12
)
Excitation Frequency (
Ω)
2 nd
Zone 3 rd
Zone
Lower limit
(
Ω
21
)
Upper limit
(
Ω
22
)
Lower limit
(
Ω
31
)
Upper limit
(
Ω
32
)
Dynamic
Load
Factor
β=
P t
/ P
*
1 st
Zone
Lower limit
(
Ω
11
/
ω
1
)
Excitation frequency ratio
Ω/ω
1
Upper limit
(
Ω
12
/
ω
1
)
2 nd
Zone
Lower limit
(
Ω
21
/
ω
1
)
Upper limit
(
Ω
22
/
ω
1
)
3rd Zone
Lower limit
(
Ω
31/
ω
1
)
Upper limit
(
Ω
32
/
ω
1
)
1 1.49
20.0
22.7
89.4 370.2 0.11
1.5
1.7
6.7 6.8
27.75
27.9
2
3
3.65
5.0
18.7
23.3
88.0
91.4
369.5
373.5
18.7
24.7
87.4
92.0
368.2
373.5
1.4
1.4
1.75
1.85
6.6
6.55
6.85
27.7
28.0
6.9
27.6
28.0
4 8.65
16.0
25.4
85.4 366.9 0.64
1.2
1.9
6.4
7.0
27.5
28.1
5 11.35
14.7
26.7
84.0 365.5 0.84
1.1
2.0
6.3
7.1
27.4
28.2
149
Multilayered sandwich beams
Table7.14, Experimental boundary frequencies of instability regions for 3layered sandwich beam,
L= 0.3m, t
1
=0.001m, t
21
= 0.25, P
*
=7.7 N,
ω
o
=2.96 Hz.
Sl
No
Dynamic load
Amplitude
(P t)
Lower limit
(
Ω
11
)
2
ω
1
Upper limit
(
Ω
12
)
Excitation Frequency (
Ω)
Lower limit
(
Ω
21
)
ω
1+
ω
2
Upper limit
(
Ω
22
)
5.62
42.6
2
ω
2
Lower limit
(
Ω
31
)
Upper limit
(
Ω
32
)
Dynamic
Load
Factor
β=
P t
/ P
*
0.73
7.2
Excitation frequency ratio
Lower limit
(
Ω
11
/2
ω o
)
2
ω
1
Upper limit
(
Ω
12
/2
ω o
)
8.29
Lower limit
ω
1+
ω
2
(
Ω
21
/ 2
ω o
)
Upper limit
(
Ω
22
/ 2
ω o
)

Ω/2ω o
Lower limit
(
Ω
31/
2
ω o
)
2
ω
2
Upper limit
(
Ω
32
/2
ω o
)

1.82
6.3
9.0
24 26.94

2.8
5.51
9.97
22.5
28.5

278.2 4.59
21 30 42.0
46.0
3 32.6
59.0 168.7
4 27.2
63.4
5 24.3
68.1 284.2 4.1
11.5
20 31 47.0
48.0
Table7.15, Experimental boundary frequencies of instability regions for 5layered sandwich beam,
L= 0.5m, t
1
Sl Dynamic
=0.001m, t
21
= 0.25, P
*
=4.16 N,
Excitation Frequency (
Ω)
ω
o
=1.07Hz.
Dynamic
Excitation frequency ratio
Ω/2ω o
No load
Amplitude
(P t)
Lower limit (
Ω
11
)
2
ω
1
Upper limit (
Ω
12
)
ω
1+
ω
2
Lower limit
(
Ω
21
)
Upper limit
(
Ω
22
)
Lower limit
(
Ω
31
)
2
ω
2
Upper limit
(
Ω
32
)
Load
Factor
β=
P t
/ P
*
Lower limit
(
Ω
11
/2
ω o
)
2
ω
1
Upper limit
(
Ω
12
/2
ω o
)
ω
1+
ω
2
Lower limit
(
Ω
21
/ 2
ω o
)
Upper limit
(
Ω
22
/ 2
ω o
)
Lower limit
(
Ω
31/
2
ω o
)
2
ω
2
Upper limit
(
Ω
32
/2
ω o
)
1 26.3
27.3
1.55

2 25.3
28.5
2.81
11.8 
3 24.6
28.9
3.81
11.5 
4 24.2
28.9
4.2
11.3
13.5


5 24.4
29.5
4.7
11.4 




150
Table7.16, Experimental boundary frequencies of instability regions for 7layered sandwich beam,
L= 0.3m, t
1
=0.0005m, t
21
= 0.5, P
*
=1.9259 N,
ω
o
=2.05Hz.
Sl
No
Dynamic load
Amplitude
(P t)
Lower limit
(
Ω
11
)
Excitation Frequency (
Ω)
2
ω
1
Upper limit (
Ω
12
)
Lower
ω
1+
ω
2
Upper limit
(
Ω
21
) limit
(
Ω
22
)
Lower limit
(
Ω
31
)
2
ω
2
Upper limit
(
Ω
32
)
Dynamic
Load
Factor
β=
P t
/ P
*
Excitation frequency ratio
Ω/2ω o
Lower limit
(
Ω
11
/2
ω o
)
2
ω
1
Upper limit
(
Ω
12
/2
ω o
)
ω
1+
ω
2
Lower
Upper limit
(
Ω
21
/ 2
ω o
) limit
(
Ω
22
/ 2
ω o
)
Lower limit
(
Ω
31/
2
ω o
)
2
ω
2
Upper limit
(
Ω
32
/2
ω o
)
82.0
82.0
0.54
17.5
20 
0.73
17.0
20 

3 57.4
53.7 246.0
39.0 260.4
348.5
397.7
32.8
160.0
160.0
276.8
324.0
418.0
1.9
2.8
3.7
4.6
14.0
22.5
13.1
25.0
51.0
54.0
45.5
60.0




9.5
27.0
42.0
63.5
85.0
97.0
8.0
29.0
39.02
67.5
79.0
101.95
5
6 8.86
151
Plate1: Photograph of experimental setup
146
Plate2, Photograph of attachments for,
(A)Pinned end (B) Clamped end.
146
147
Chapter
8
CONCLUSION AND
SCOPE FOR FURTHER RESEARCH
8.1 Conclusion
The present work deals with the theoretical and experimental investigation of dynamic stability of ordinary and sandwich beams under parametric excitation. The governing equations of motion for the system have been derived using finite element method. The primary instability regions for ordinary beams have been established by applying the
Floquet’s theory developed by Bolotin. For sandwich beam, stability criteria proposed by
Saito and Otomi have been applied to determine the instability regions. The following important conclusions have been drawn from the various studies.
In chapter 3, the dynamic stability of a uniform Timoshenko beam with localised damage and different boundary conditions have been studied. The beam is subjected to an axial periodic load along its undeformed axis. Fixedfree, pinnedpinned, fixedfixed and fixedpinned boundary conditions have been considered in the analysis.
152
153
It is observed that presence of localised damage or in other words extent of damage enhances the instability of the beam for all the boundary conditions.
For fixedfree end condition the damage near the fixed end is more severe on the dynamic instability behavior than that of the damage located at other positions, so far as first instability region is concerned. The beam is most susceptible to instability so far as second instability region is concerned, when the damage is located at the middle. The third principal instability region relocates itself at minimum frequency of excitation when the damage position is in between the middle and free end.
For pinnedpinned end condition the first instability region occurs at minimum frequency of excitation when the damage is located at the middle. The beam is more stable when the damage is located towards any of the pinned end from middle. The second instability region occurs at minimum frequency of excitation when the damage is located in between the middle and any one of the pin ends and at highest frequency when the damage is located at the middle. The third instability zone occurs at minimum frequency of excitation when the damage is at the middle.
For the beam with fixedfixed end condition the presence of damage at the middle makes the occurrence of first mode primary resonance most probable. The second instability region occurs at minimum frequency of excitation when the damage is in between the middle and either of the fixed ends of the beam. Third instability region occurs at minimum frequency of excitation when the damage is at the middle.
For fixedpinned end condition the first instability region occurs at minimum frequency of excitation when the damage position is in between the middle and the pinned end. The second principal instability region occurs at minimum frequency of
154
excitation when the damage is located in between the fixed end and the middle. Whereas the third principal instability region occurs at minimum frequency of excitation when the damage is located nearer to the pinned end.
Increase in static load component has a destabilising effect for all the four boundary conditions.
Chapter 4 deals with the dynamic stability of a pretwisted cantilever beam with localised damage and subjected to end parametric excitation. Angle of pretwist has significant effect on second and third instability zones. Increase in pretwist angle has a stabilising effect on the third instability zone and destabilising effect on the second instability zone. Change in pretwist angle does not have significant effect on the first instability region. Localised damage has a greater destabilising effect when it is located near the fixed end than when it is towards the free end, irrespective of the magnitude of pretwist angle. Increase in extent of damage increases the instability in terms of the shifting of instability zones to lower frequencies of excitation and increase in the areas of the instability zones. Increase in static load component enhances the instability of the beam. The effect is more pronounced on first instability region.
In chapter 5 dynamic stability of a multilayered symmetric cantilever sandwich beam subjected to parametric excitation has been studied. It is observed that increase in core thickness parameter, core loss factor and number of layers has stabilising effect. For constant size and constant weight cases, increase in number of layers deteriorates the stability with instability zones relocating them at lower frequencies of excitation, improves the stability with stability zones shifting to higher values on the dynamic load axis and disappearance of some instability regions. Constant flexural rigidity criterion
155
improves stability with increase in number of layers. Among constant size, constant weight and constant flexural rigidity criteria, constant flexural rigidity is the most effective in improving the stability. Both constant size and flexural rigidity and constant weight and flexural rigidity criteria improve the stability with increase in number of layers. Constant weight and flexural rigidity case is more effective in improving the stability than constant size and flexural rigidity case.
Chapter 6 addresses the problem of dynamic stability of a simply supported tapered beam with uniform thermal gradient. It is observed that increase in thermal gradient decreases the natural frequency and the static buckling load. The effect is more pronounced on second and third modes than on first mode. Increase in static load component or thermal gradient has a destabilising effect. Thermal gradient has a greater destabilising effect than static load factor, for an equivalent change in static load component.
In chapter 7 experimental results are reported. Experimental results corroborate the theoretical findings except for the case of uniform Timoshenko beam with localised damage and with pinnedpinned and fixedpinned end conditions. The discrepancies in these two cases may be due to the damping in the hinged ends.
A summary of the important conclusions may be stated as follows:
A) For a uniform Timoshenko beam with localised damage,
i) presence of damage always increases the instability of the beam,
ii) the critical position of the damage for maximum destabilising effect on the beam is different depending on the boundary conditions and the principal regions of instability of interest,
156
iii) increase in static load component has a destabilising effect for all boundary conditions considered and iv) dynamic behaviour of the beam depends upon the boundary conditions as well as the location of the damage. Such mechanical components should be carefully designed with respect to the applied frequency of excitation, dynamic and static load components of the excitation force.
B) For pretwisted cantilever beam with localised damage, i) angle of pretwist has significant effect on second and third instability zones. Increase in pretwist angle has a stabilising effect on the third instability zone and destabilising effect on the second instability zone. Change in pretwist angle does not have significant effect on the first instability region, ii) extent of damage has always a destabilising effect on all the instability regions, iii) localised damage has a greater destabilising effect when it is located near the fixed end than when it is towards the free end, irrespective of pretwist angle and iv) increase in static load component has a destabilising effect on all the instability regions irrespective of pretwist angle.
C) For multilayered cantilever sandwich beam, i) increase in core thickness parameter, core loss factor and number of layers has stabilising effect, ii) for constant size and constant weight cases, increase in number of layers may deteriorate or improve the stability, iii) constant flexural rigidity criterion improves stability with increase in number of layers,
157
iv) among constant size, constant weight and constant flexural rigidity criteria, constant flexural rigidity is the most effective one in improving the stability and v) both constant size and flexural rigidity and constant weight and flexural rigidity criteria improve the stability. Constant weight and flexural rigidity case is more effective in improving the stability than constant size and flexural rigidity case.
D) For simply supported tapered beam with thermal gradient i) increase in static load component or thermal gradient worsens the stability of the beam.
8.2 Scope for further research
Some of the possible areas for further research on dynamic stability of beams are described below. i) Effort was made to experimentally establish the stability diagram for the simply supported tapered beam with thermal gradient. It was realised that substantial redesign of the experimental setup is required to achieve uniform thermal gradient along the length of the beam. This could be the subject of considerable further work. ii) In actual practice the beams may be subjected to loads like combination of forced and parametric excitation, parametric excitations consisting of more than one frequency i.e. multifrequency excitation and random excitation etc. The dynamic behaviour of beams under these types of excitations may be explored. iii) Study of dynamic stability of beams considering material and geometric nonlinearities may also be an aspect of investigation.
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Start
Read input data (Material and geometric parameters)
Generate mesh
(e )
,
(e )
,
Generate element matrices,
[ ]
[ ]
g
)
,
[ ]
t
,
[ ]
r
and
[ ]
)
AppendixA
170
Assemble element matrices to global matrices
[ ]
, and
[ ]
g
Generate reduced global matrices for specific boundary conditions.
Read
α,ω
1,
P
*
and
β
Determine the lower limit of
Θ bounding the instability region, solving the eigenvalue problem: eq.(3.38).
[ ]
−
⎝
⎜
α
β
2
⎠
⎟
⎞
P
*
− Θ
ω
1
2
4
[ ]
=
0
Determine the upper limit of
Θ bounding the instability region, solving the eigenvalue problem: eq.
(3.38).
−
⎝
⎜
α
β
2
⎠
⎟
⎞
P
*
− Θ
ω
1
2
4
[ ]
=
0
End
Flow chart for calculating the lower and upper boundary limits of instability
regions based on Floquet’s theory.
AppendixB
171
Start
Read input data (Material and geometric parameters).
Generate mesh.
Generate element matrices, ,
K g
and .
Assemble element matrices to global matrices, , and .
Generate reduced global matrices for specific boundary conditions.
Read
α, and P
*
.
Calculate the normalised modal matrix, [
Φ], solving for the eigen vectors of
[ ]
−
1 matrix.
Obtain the eq.(5.39) premultiplying the coefficient matrices of eq.(5.35) by
[ ]
−
1 and
[ ]
−
1
and postmultiplying by
[ ]
Read
ω o
and
β.
Obtain the boundaries of instability regions from eqs. (5.40,5.42,5.44).
End
Flow chart for calculating the lower and upper boundary limits of instability
regions based on Hsu’s criteria.
BioData
Name:
Mr. Sukesh Chandra Mohanty.
Date of birth: 04071965.
Permanent address: At: Pathuria Sahi, Po: Puri, Dist: Puri, Orissa, India,
Pin752001.
Educational qualifications:
Degree University
B.Sc. (Engineering)
(Mechanical)
M.Tech.(Machine design specialisation)
Sambalpur University
Beneras Hindu University
Year of passing
1987
1990
Class/Division
First div.
do
Teaching experience: Faculty in Mechanical Engineering Department, N.I.T, Rourkela
since 16091991.(Lecturer16.09.91 to27.09.97, Senior lecturer
28.09.97 to14.02.2004, Lecturer (Selection grade)  15.0204 to
270905, Asst. Professor280905 till date)
Merit certificate received: Received the certificate of merit from the Institution of
Engineers, India for 20032004 for the published paper titled “Tooth Load Sharing and
Contact Stress Analysis of High Contact Ratio Spur Gears in Mesh.”
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