ABSTRACT
The dynamic analysis of discontinuous structures has gained the attention of researchers over past decades because of their ability to resist dynamic loading such as wind, waves, earthquakes, traffic, blasts and explosion, as well as the increasing demand for better performance, assisting in weight reduction and easy assembling of structure. Due to reduction in computational time and cost, advancement in numerical methods and the availability of powerful computing facilities, analytical models have become the most popular technique for dynamic analysis of such structure. In this study, an improved analytical model that uses receptance model based on Timoshenko beam model for the dynamic of discontinuous beam in order to obtain the modal parameter was proposed. The Timoshenko beam model was modified based on transverse response of beam at first three modes in order to overcome its shortcomings. To clarify the implementation of the proposed model, two types of discontinuities, namely, stepped beam with aligned and misaligned neutral axes, were considered and numerical simulations provided to demonstrate the modal parameter and frequency response of the structure. Additionally, a flexible multi-configuration beam vibration testing rig was modified and experimental modal analysis (EMA) of uniform and discontinuous beams provided. Results obtained were compared to other analytical techniques and experiments. The results of the proposed analytical model showed identical result with experiment with at most 0.99% deviation from the experimental results for all frequency modes. The result when compared with other analytical technique shows at most 92% frequency deviation with reference to the proposed analytical model for all frequency modes. It was also found that increase in the discontinuity depth and discontinuity location for the natural frequencies produced irregular curves for free-free boundary configurations. Application of the modified model to other designed model analysis as observed, produced a good correlation between experimental and analytical results. The proposed model can also be extended easily to deal with various discontinuous structures as well as a helpful tool in different structural analysis, thereby increasing the quality of the product, safety, reliability and performance of structures.
TABLE OF CONTENTS
Cover page i
Title page ii
Declaration iii
Dedication iv
Certification v
Acknowledgements vi
Table of Contents vii
List of Tables x
List of Figures xiii
Nomenclature xix
Abstract xxii
CHAPTER 1: INTRODUCTION
1.1
Background of the Study 1
1.2
Statement of Problem 6
1.3
Aim and Objectives of Study 8
1.4
Scope of Study 9
1.5
Justification of the Study 9
CHAPTER 2: LITERATURE REVIEW
2.1 Review
of Beam Theories and Continuous Beam Analysis 10
2.2 Overview
of Discontinuous Beam Analysis 16
2.3 Review
of Models and Simulation of Dynamic Analysis of Beam 19
2.3.1 Dynamic
Analysis of Continuous Beam 19
2.3.2 Dynamic
Analysis of Discontinuous Beam 22
2.4 Review
of Dynamic Analysis Techniques 27
2.4.1 Modal
Superposition Method 27
2.4.2 Receptance
Method 28
2.4.3 Discrete
Singular Convolution Method 28
2.4.4 Differential
Quadrature Method 29
2.4.5 Hierarchical
Function Method 30
2.4.6 Spectral-Techebychev
Method 31
2.4.7 Stokes
Transformation Method 31
2.5 Theory
of Modal Analysis 32
2.6 Overview
of Experimental Modal Analysis (EMA) 36
2.6.1 Preparation
of Test Structure 37
2.6.2 Excitation
Mechanism 37
2.6.3 Mechanism
of Sensing 38
2.6.4 Data
Acquisition and Processing Mechanisms 39
2.6.5 Modal
Parameter Extraction and Identification 40
2.7 Overview of
Receptance Method and Receptance Coupling Substructure Analysis (RCSA) 50
CHAPTER 3: MATERIALS AND METHODS
3.1 Materials 56
3.2 Methods
and Model Development 56
3.2.1 Spatial
Solution for the Timoshenko beam Theory 56
3.2.2 Development
of Receptance Model for Uniform Beams 62
3.2.3 Development of Receptance Model for Discontinuous Beams with
Aligned
Neutral Axes 73
3.2.4 Development of
Receptance Model for Discontinuous Beams with Misaligned
Neutral Axis 91
3.2.5 Development of
Receptance Model for discontinuous beams using receptance
coupling substructure analysis (RCSA) 115
3.3 Analysis and
experimental procedure of flexible multi-configuration beam vibration test rig 120
3.3.1 Analysis
of slider base and wheel diameter 123
3.3.2 Analysis
of fixed and adjustable handle and pulling force prediction 124
3.3.3 Experimental
procedure/setup 126
3.4 Numerical
investigations 127
3.5 Model
validation 130
CHAPTER 4: RESULTS AND DISCUSSION
4.1 Comparative
performance analysis of modified flexible multi-configuration beam vibration
testing 132
4.1.1 Experimental results
of natural frequencies for beam specimens 132
4.1.2 Effects of beam
length on modal parameters 137
4.1.3 Effect of modulus of
Elasticity on modal parameter 139
4.1.4 Effect of mass
density on modal parameters 141
4.2 Modal
Validation/Analysis 142
4.2.1 Validation of
uniform beam model 142
4.2.2 Validation of
discontinuous beam with aligned neutral axes 149
4.2.3 Validation of
discontinuous beam with misaligned neutral axes 155
4.2.4 Comparative analysis
of beam with aligned and misaligned neutral axes 161
4.2.5 Validation of
discontinuous beam based on receptance coupling substructure
analysis (RCSA) 163
4.2.6 Effect of discontinuity
on modal parameter 165
4.2.7 Effect of
discontinuity depth and location on natural frequency 167
4.3 Application of
modified model 170
4.3.1 Beam with circular
cross-sectional area 170
4.3.2 Beam with crack or
damage 172
4.3.3 Composite beams 175
CHAPTER 5: CONCLUSION AND RECOMMENDATIONS
5.1 Conclusions 178
5.1.1 Contributions to
Knowledge 178
5.2 Recommendations 179
References 180
Appendices 195
LIST OF TABLES
2.1
The beam models
with the corresponding effect 12
2.2 The
Percentage deviates from experimental values obtained by
Trail-Nash and Collar (1953) 13
2.3
Transfer Functions
for vibration responses 34
3.1 Geometric and
material properties of aluminium alloy (uniform beam) employed in the
simulation models 127
3.2 Geometric and
material properties of aluminium alloy (beam with aligned neutral axes) employed in the
simulation models 128
3.3 Geometric and
material properties of aluminium alloy (beam with misaligned neutral axes) employed
in the simulation models 129
4.1 Experimental
results of natural frequencies for 350mm length beam specimen 132
4.2 Experimental
results of natural frequencies for 450mm length beam specimen 132
4.3 The NFD
and PEMS values between MUTR and EVTR for 350mm beam specimen 133
4.4 The NFD
and PEMS values between MUTR and EVTR for 450mm beam specimen 133
4.5 Percentage
derivation for all frequency modes on experimental results of EVTR 137
4.6 Percentage
derivation for all frequency modes on experimental results of MVTR 137
4.7 Comparison
of natural frequencies by different experimental results of uniform beam model 143
4.8 The
NFD values for different experimental results of uniform beam 143
4.9 The
PEMS values for different experimental results of uniform beam 143
4.10 Comparison
of analytical model and experimental results of uniform beam model 145
4.11 The NFD
values for different analytical models and experimental results of uniform beam 145
4.12 The PEMS
values for different analytical model and experimental results of uniform beam 145
4.13 The
receptance function as a function of normalized displacement for all frequency modes of
uniform beam 148
4.14 Comparison
of natural frequencies by different experimental values for beam with aligned neural axes 149
4.15 The NFD
values for different experimental results of beam with aligned neutral axes 149
4.16 The PEMS
values for different experimental results of beam with aligned neutral axes 149
4.17 Comparison
of analytical models and experimental results of beam with aligned neutral axes 151
4.18 The NFD
values for different analytical results of beam with aligned neutral axes 152
4.19 The PEMS
values for different analytical results of beam with aligned neutral axes 152
4.20 The
receptance function as a function of normalized displacement for all frequency model of
beam with aligned neutral axes 154
4.21 Comparison
of natural frequencies by different experimental values for beam with misaligned neutral axes 156
4.22 The NFD
values for different experimental results of beam with misaligned neutral axes 156
4.23 The PEMS
values for different experimental results of beam with misaligned neutral axes 156
4.24 Comparison
of analytical model and experimental result of beam with misaligned neutral axes 158
4.25 The NFD
values for different analytical results of beam with misaligned neutral axes 158
4.26 The PEMS
values for different analytical results of beam with misaligned neutral
axes 158
4.27 The
receptance function as a function of normalized displacement for all frequency models of
beam with misaligned neutral axes 161
4.28 Comparison
of analytical results of beam with aligned and misaligned neutral axes 162
4.29 Comparison
of analytical result based on RCSA and experimental results of discontinuous beam 164
4.30 Comparison
of modal parameter of continuous and discontinuous beam models 165
4.31 Variation
of natural frequency and discontinuity depth at constant discontinuity
location 168
4.32 Variation
of natural frequency with discontinuity location at constant discontinuity depth 168
4.33 Geometric
and material properties of free-free circular steel beam employed in the simulation model 170
4.34 The
modal parameters of experimental result presented by Peter et al (2016)
and present analytical result 170
4.35 Geometric
and material properties of free-free circular steel beam (rectangular cross section)
with crack employed in the simulation model 173
4.36 The
modal parameters of experimental result for crack beam presented by Swamy et al
(2017) and present analytical result 173
4.37 Geometric
and material properties of free-free FFU Composite beam (rectangular cross section)
employed in the simulation model 175
4.38 The
modal parameters of experimental result for FFU composite beam presented by
Pasakorn et al (2020) and present analytical result 176
LIST OF FIGURES
1.1 Discontinuous
structures in Engineering practices 2
1.2 Material removal
process by milling 3
2.1 Theoretical and Experimental route to vibration analysis 34
2.2 Graphical display of a frequency response function 35
2.3 The process of the Experimental modal analysis 37
2.4 Impact Hammer 38
2.5 Accelerometer 39
2.6 Various signals, and their fourier representation 40
2.7 FRF representation
for modal parameter identification 41
2.8 Experimental Modal analysis for cantilever beam presented by
Chaphalkar et al. (2015) 42
2.9 Experimental Modal analysis for fiber Reinforced plastic (FRP)
presented by Mishra et al. (2018) 43
2.10 Experimental modal
analysis arrangement for cantilever, simply
supported and fixed beam, presented by Kumar et al.
(2015) 44
2.11 Experimental modal
analysis for cantilever beam presented by
Kamble et al (2016) 45
2.12 Experimental modal analysis
for cantilever beam presented by
Vazir et al.
(2013) 45
2.13 Experimental modal
analysis for cantilever beam presented by
Chinka et al
(2018) 46
2.14 Experimental modal
analysis for damaged beam structure 47
2.15 Experimental Modal analysis for free-free
stepped beam presented
by Koplow et al.
(2006) and Lu et al. (2009) 48
2.16 Multi-configuration
Beam Vibration Testing rig 49
2.17 RCSA flowchart for Spring-mass-Damper System. 51
2.18 RCSA Flowchart for Compound beam System 52
2.19 Schematic of
vertical spindle milling machine 52
2.20 Two-component assembly, with component
responses coupled through
a rigid
connection to get the assembly receptance(s). 53
3.1 Simple beams in transverse vibration and a free-body diagram of
the effect of
shear deformation on an element of a bending beam. 57
3.2 Schematic of a uniform beam subjected to a
force of amplitude F
and frequency
62
3.3 Stepped beams with aligned neutral axis subjected to a force of
amplitude F and
frequency
. 74
3.4 Stepped beam with misaligned neutral axis subjected to a force of
amplitude F and
frequency
. 91
3.5 Free body diagram
of (a) force and (b) displacement for a discontinuous
beam with a
misaligned neutral axis. 93
3.6 Receptance coupling
of two beam substructures with displacement and
rotations as
described coordinate locations 115
3.7 Subassembly I-II
consisting of beam, the generalize force
is applied
to
115
3.8(a) Isometric view of
the proposed testing rig in colored details 122
3.8 (b) Isometric view of
the proposed testing rig in plain details 122
3.9 Exploded view of
the modified testing rig. 123
3.10 Demonstration of a
wheel rolling over a sliding surface. 123
3.11 Free body diagram of fixed and adjustable handle 125
3.12 Beams dimensions for
numerical investigation. Dimensions are
in (mm) 130
3.13 Plots of measured vs
predicted shape vector 131
4.1 The comparison
between MVTR and EVTR natural frequency results for 350mm beam specimen 133
4.2 The comparison
between MVTR and EVTR natural frequency results for 450mm beam specimen 134
4.3 MAC-matrix between
EVTR and MVTR experimental modes for 350mm beam specimen 135
4.4 MAC-matrix between
EVTR and MVTR experimental modes for 450mm beam specimen 135
4.5 MAC-Plot between EVTR and MVTR experimental modes for 350mm beam specimen 136
4.6 MAC-Plot between EVTR and MVTR experimental modes for 450mm beam specimen 136
4.7 Graphical
representation of percentage derivation for all frequency modes on experimental results of
EVTR 138
4.8 Graphical representation of percentage
derivation for all frequency modes on experimental results of MVTR 138
4.9 Effect
of Young’s modulus on natural frequencies for 350mm length
on EVTR 139
4.10 Effect
of Young’s modulus on natural frequencies for 450mm length on EVTR 140
4.11 Effect
of Young’s modulus on natural frequencies for 350mm length on MVTR 140
4.12 Effect
of Young’s modulus on natural frequencies for 450mm length on MVTR 140
4.13 The
effect of mass density on natural frequencies for 350mm length on EVTR 141
4.14 The
effect of mass density on natural
frequencies for 450 length on EVTR 141
4.15 The
effect of mass density on natural frequency for 350 length on MVTR 141
4.16 The
effect of mass density on natural frequencies for 450 length on MVTR 142
4.17 Comparison of modal parameter by different experimental results
for uniform beam. 143
4.18 MAC-Matrix between present experimental and reference models for
uniform beam. 144
4.19 MAC-Plot between the present experimental and reference models for
uniform beam. 144
4.20 The frequency difference/deviation between analytical and
experimental results of uniform
beam model. 146
4.21 MAC-Matrix between experimental and proposed analytical results of
uniform beam model. 147
4.22 MAC-Plot between the experimental and proposed analytical results
of uniform beam model. 147
4.23 Mode shape of uniform beam for all frequency modes. 148
4.24 Comparison of modal parameter by different experimental results
for Beam with aligned neutral axes. 150
4.25 MAC-Matrix between present experimental and reference models for
beam with aligned neutral axes. 150
4.26 MAC-plot between present experimental and reference models for
beam with aligned neutral axes. 151
4.27 Frequency differences/deviation between analytical and
experimental results of beam with
aligned neutral axes. 152
4.28 MAC-Matrix between
experimental and proposed analytical results of beam with aligned neutral axes. 153
4.29 MAC-Plot between the experimental and
proposed analytical results of beam with aligned neutral axes. 154
4.30 Mode shape of beam with aligned neutral axes. 155
4.31 Comparison of modal parameter by different experimental results
for Beam with misaligned neutral axes. 156
4.32 MAC-Matrix between present experimental and reference models for
beam with misaligned
neutral axes. 157
4.33 MAC-Plot between present experimental and reference models for
beam with misaligned neutral axes. 157
4.34 Frequency differences/deviation between analytical and
experimental results of beam with
misaligned neutral axes. 159
4.35 MAC Matrix between
experimental and proposed analytical results of beam with misaligned neutral
axes. 160
4.36 MAC-Plot between the experimental and proposed analytical results
of beam with misaligned neutral
axes. 160
4.37 Mode shape of beam with misaligned neutral axes. 161
4.38 Graphical representation of natural frequency
of beam with aligned and misaligned neutral axes for all frequency modes. 162
4.39 MAC-Matrix between beam with aligned and misaligned neutral axes 163
4.40 MAC-Plot between beam with aligned and misaligned neutral axes 163
4.41 Graphical representation of natural frequency based on RCSA 164
4.42 MAC-Matrix
between present experimental and proposed analytical result based on RCSA 164.
4.43 MAC-plot
between the experimental result and proposed analytical result based on RCSA 165
4.44 Graphical
representation of natural frequencies of continuous and discontinuous beam for all frequency modes 166
4.45 MAC-Matrix between
continuous and discontinuous beams for all frequency modes 166
4.46 MAC-plot between continuous and discontinuous beams for all
frequency modes 167
4.47 Effect of normalized discontinuity depth on normalized natural
frequency for all frequency modes. 169
4.48 Effect of normalized discontinuity location on normalized natural
frequency for all frequency modes. 169
4.49 Graphical representation of natural frequencies for experimental
result presented by Peter et al.
(2016) and present analytical result. 171
4.50 MAC-Matrix between experimental result presented by Peter et al. (2016) and present analytical
result. 172
4.51 MAC-Plot between experimental result presented by Peter et a.l (2016) and present analytical
result. 172
4.52 Graphical representation of natural frequencies for experimental
result presented by Swamy et al. (2017) and present analytical
result. 174
4.53 MAC-Matrix between experimental result presented by Swamy et al. (2017) and present analytical
result. 174
4.54 MAC-Plot between experimental result presented by Swamy et al. (2017) and present analytical
result. 175
4.55 Graphical representation of natural frequencies for experimental
result presented by Pasakurn et al.
(2020) and present analytical result. 176
4.56 MAC-matrix between experimental result presented by Pasakorn et al. (2020) and present analytical
result. 177
4.57 MAC plot between experimental result presented by Pasakorn et al. (2020)and present analytical
result. 177
G.1 Detailed
drawing of the slider base. 213
G.2 Detailed drawing of the fixed handle. 213
G.3 Round
bar for specimen locking 214
G.4
Beam hook 214
G.5
Testing rig support table 215
G.6
Rectangular beam specimen 215
G.7 Specimen storage drawer 216
G.8
Sliding pipe 216
G.9 Round
beam specimen 216
G.10 Dampers 217
G.11 Slider
lock 217
H.1 Frequency
response function (FRFs) for uniform beam 227
H.2 Frequency
response function (FRFs) for beam with aligned neutral axes 227
H.3 Frequency
response function (FRFs) for beam with misaligned neutral axes 228
H.4 Frequency response
function (FRFs) for beam based on receptance coupling substructure analysis 228
NOMENCLATURE
|
|
Area of the beam section, m2
|
|
|
Aluminium
|
|
|
Distance between the mass and the bottom of the
handle, m
|
|
|
Degree of freedom
|
|
|
Distance between beams neutral axis
|
|
|
Experimental Modal analysis
|
|
|
Existing beam vibration test rig.
|
|
|
Force applied
|
|
|
Frequency response function
|
|
|
Shear modulus
|
|
|
Weight of the handle, m
|
|
|
Moment of inertia of the beam section
|
|
|
Shear coefficient
|
|
|
Length of the beam section, in metre
|
|
|
Moment applied
|
|
|
Modal assurance criterion
|
|
|
Modified beam vibration test rig
|
|
|
Normal force exerted by the slider, N
|
|
|
Number of modes
|
|
|
Natural frequency difference
|
|
|
Percentage error of each mode shape
|
|
|
Load distributed over the contact, N/m
|
|
|
Wheel radius, m
|
|
|
Receptance coupling substructure analysis
|
|
|
Time in sec.
|
|
|
Tool – holder – spindle – machine
|
|
|
Total weight of the handle, N
|
|
|
Assembly or normalized displacement
|
|
|
Width of the contact area between a wheel and plane,
m
|
|
|
Young’s modulus, N/m2
|
|
|
Non-homogenous forcing function of both space and
time, N.
|
|
|
Input function
|
|
|
Assembly receptance function
|
|
|
Irreversible function
|
|
|
Frequency response function
|
|
|
Shear force
|
|
|
Substructure receptance function
|
|
|
Rotational degree of freedom
|
|
|
Time function
|
|
|
Axial displacement at position x and time t
|
|
|
Spectral – Tehebycher function
|
|
|
Nth natural mode of beam
|
|
|
Substructure displacement
|
|
|
Techebycher polynomial
|
|
|
Response function
|
|
|
Level of discontinuing at beams
|
|
|
Ratio of inner radius to outer radius of the hollow
circle
|
|
|
Stokes Transformation Function
|
|
|
Transverse displacement at position 𝑥 and
time t,m
|
|
|
Transverse displacement/position in metres
|
|
|
Assembly displacement in the direction of coordinate
j
|
|
|
Assembly rotation in the direction of coordinate j
|
|
|
Axial force
|
|
|
Beam height or depth
|
|
|
Beam rectangular width
|
|
|
Coefficient of rolling friction
|
|
|
Critical or cutoff frequency
|
|
|
Discrete grid point
|
|
A
|
Distance between the center of the mass and the side
of the handle, m
|
|
|
Effective elastic modulus, N/m2
|
|
|
First differential coefficient of legendre polynomial
|
|
|
Force applied in coordinate k
|
|
|
Force applied to the substructure in coordinate k
|
|
|
Hierarchical function
|
|
|
Horizontal force, N
|
|
|
Maximum pressure, N/m2
|
|
|
Mode shape or eigenvectors
|
|
|
Moment applied in coordinate k
|
|
|
Moment applied to the substructure in coordinate k
|
|
|
Nth order legendre polynomial
|
|
|
Poison’s ratio
|
|
|
Relative displacement between beam sections in x
direction.
|
|
|
Shear force
|
|
|
Substructure rotation in coordinate j
|
|
|
Wave number
|
|
|
Wave number
|
|
|
Weighing coefficients
|
|
|
Wave number
|
|
|
Modified wave number
|
|
|
Angle due to shear
|
|
|
Dirac delta function
|
|
|
Assembly rotation
|
|
|
Angle of rotation of the cross-section due to the
bending moment at position
, and time t,
rad
|
|
|
Eigen value of the nth mode
|
|
|
Coefficient of friction
|
|
|
Mass density of the beam, kg/m2
|
|
|
Angular frequency
|
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND
OF THE STUDY
A
beam is an elongated member, usually slender, intended to resist lateral loads
by bending (Cook and Young, 1999). They are continuous structural systems with
distributed properties and which, consequently, have an infinite number of
degrees of freedom (Jabboor, 2011). Beam plays a vital role in structural
dynamics, research and industry to make accurate prediction of the response of
many different structures (Inman, 2001). They have a wide variety of
applications in engineering science, such as in civil and mechanical
engineering, automobile industries, aerospace industries, marine engineering
and off-shore structures (Sahu, 2014). Beams are associated with end conditions
at the two ends and the combination of these gives classical boundary
conditions, namely, free-free, fixed-free, fixed-pinned, fixed- sliding,
fixed-fixed and pinned-pinned. They are also classified according to geometric
configuration as continuous (uniform) or discontinuous beams. Continuous beams
have constant thickness and width along the longitudinal direction, whereas
discontinuous beams are non-uniform structures, with complex geometry that
contains joints, connections or notches (Koplow et al., 2006).
Due
to increasing demands for better performance and the use of lighter and more flexible
structures in modern machinery, discontinuous structures are widely used in the
manufacturing industries for structural dynamic problems (Jung, 1992; He and
Fu, 2001; Samala, 2013). It has been established that discontinuous beams have
a better distribution of strength and mass as compared to uniform beam thereby
being capable of meeting special functional requirement in Engineering (Shukla,
2013). The design of such structures is necessary to resist dynamic forces,
such as wind, earthquakes, waves, traffic, blasts and explosion (Chao and
Leina, 2016; Ece et al., 2007). In addition,
a study conducted by Huang (2007), Kim et al. (2017) and Jabboor (2011) showed that at least 70% of
engineering structures are discontinuous, thereby assisting in weight reduction
of every possible gram and easy assembling of structures, in order to minimize
their inertial property during operation.
In most cases, there are lots of difficulties attributed to modeling
such structures as it becomes more complicated, consisting of more segments and
joints (Celcus, 2012; Koplow et al.,
2006; Huang, 2007).
In
many practical engineering applications, some of the structures which can be
modeled as a discontinuous beam-like element include: helicopter rotor blades,
robot arms, spacecraft antenna, flexible satellites, airplane wings, high-rise
building, long-span bridges, vibratory drilling and many other structures.
There are also other discontinuous structures such as; beam with crack, beam
carrying attached masses or support and spindle-holder-tool-assemblies. A
classic example of different discontinuities is illustrated in Fig. 1.1.
Fig. 1.1:Discontinuous structures in engineering
practices
According
to Koplow, (2005), the effect of discontinuity on structure can also be
demonstrated during milling operation (material removal process). This is
illustrated in Fig. 1.2.
Fig. 1.2: Material removal process by milling
Source: Koplow, (2005)
Interestingly,
this material removal process also presented two important challenges: (i) discontinuity in beam
structure which interfered with direct application of general beam theory and
(ii) change in beam’s dynamic responses as each layer of materials was
gradually removed, which leads to shift in beam modal parameters.
To
have a good understanding of the above-mentioned industrial process, the
dynamic behaviour of a structure in a given frequency range can be modeled as a
set of individual modes of vibration through the process of modal analysis
(Patwari et al., 2009; Kien, 2001).
The terms that described each mode are natural frequencies, damping ratios and
mode shapes, which are collectively referred to as modal parameters (Inman,
2001). More generally, an essential step
required for response and load prediction, stability analysis, structural
health monitoring (SHM), damage detection, stress analysis, system design and
structural coupling is provided by the identification of the dynamic modal
parameter of the structures. Moreover,
in order to understand any structural vibration problem, the resonances of a
structure need to be identified and quantified (Ibsen and Liingaard, 2006).
This simply can be achieved by defining the modal parameter of the
structure.
In
fact, having reliable dynamic mathematical models are essential for the design
and analysis of complex structures whose operation, integrity, safety, and
control depend on their dynamic characteristics or modal parameter (Arras,
2016). For this purpose, there are three ways of achieving a mathematical model
for a dynamic behavior of structure with the help of good approximation
concept, namely by analytical prediction, finite element analysis (FEM) and by
experimental measurement. According to
Huang (2007) and Altintas et al.
(2005), the demand for high-quality structural models and analytical processing
techniques is growing fast and the reasons for such a demand are as follows:
(i) the importance of reducing the time factor for simulation and testing
because of the fact that product development cycle is reducing due to
competition growth in industry. (ii) the expensiveness of the physical test as
result of the product becoming more complex and delicate cannot be compared to
the analytical model which is a cheaper alternative. (iii) the fast growth in technology
advancement creates more opportunity for the analytical simulation replacing
the idea of experiments and test as long as full use of computing power are
made,. and (iv) analytical models are increasingly used in real-time and more
critical application, in which both accuracy and efficiency of the model need
to be at the highest standard. Moreover,
this is in line with Erturk et al.
(2006), which revealed that analytical model reduces the computational time
considerably by a factor of forty-five (45) compared with using finite element
analysis and experiment. Koplow et al.
(2006) noted that it is cumbersome and expensive to model structures with
finite element package and tested with experimental work pieces. Thus, it
becomes useful to have analytical model for structural response. On the
contrary, Miguel et al. (2008), noted
that analytical and experimental procedures are complemented for the dynamic
behaviour of a structure, one not being able to be substituted by the other.
This suggests that experimental analysis should be carried out in order to
confirm that proposed analytical model can reliably represent real structure
and be used in further design stages.
Basically,
the modeling and dynamic response predictions for continuous beams have been
well investigated by many researchers using analytical and experimental
approaches (Laszek, 2009; Yavari et al.,
2002; Ersin et al., 2011;
Moallemi-Oreh and Karkon, 2013; Li et al.,
2014; Lee and Park, 2013). In the case of discontinuous structures, there are
lots of difficulties encountered in modeling such structures due to the
discontinuities along the longitudinal direction. In fact, this can complicate
the vibration responses significantly. The approximated methods have been
employed to study the dynamic analysis of such structures. These methods
include, partitioning or substructuring method (Koplow et al., 2006; Schmitz and Donaldson, 2000), finite difference
approach (Krishman et al., 1998),
shear deformation theory (Ju et al.,
1994) and transfer matrices approach (Mihail et al. 2016). According to DeKlerk
et al. (2008) and Park et al. (2003) substructuring allows the
following potential advantages: (i) evaluating the dynamic behavior of
structures that are too large or complex to be analyzed as a whole (ii) elimination of local subsystem behavior
which has no significant impact on the assembled structure (iii) possibility of
combining modeling parts analytically and experimentally identifying components
and (iv) sharing and combining substructures from different project groups.
However, the principal drawback to the use of substructures is the ability to
link and handle substructure’s boundaries in stress as well as displacement
analysis, which usually leads to loss of accuracy for dynamic responses
(Campbell and Pray, 2005), Nevertheless, with these disadvantages,
substructuring method has persisted as an important concept and most efficient
techniques for reducing the complexity of a discontinuous structures (Schmitz and
Donaldson 2000; Park et al, 2003).
Accordingly,
a variety of approximate or numerical solution techniques have been employed
using the above methods to solve beam vibrations problems under different
boundary conditions, which include, but are not limited to, modal superposition
method (Naguleswaran, 2003a, 2003b; Lin, 2009), receptance method (Abu-Hilal
2003; Koplow et al., 2006), discrete
singular convolution (Wei, 1999, 2001; Secgin and Sarigul, 2009). Differential
element quadrature method (Bert and Malik, 1996; Shu, 2000), hierarchical
function method (Bardell, 1991; Beslin and Nicolas, 1997), Stokes
transformation method (Wang and Lin,
1996; Kim and Kim, 2005), Adomian
decomposition (Mao and Pietrzko, 2010)
and spectral-tchebychev method (Yagciet
al., 2009). It should be noted that
the receptance technique has shown its potential in dealing with solution for
the mode shape function by direct application of forces into the boundary
conditions, which is not applicable to other methods (Koplow, 2005; Zhang,
2012). It takes advantage of the fact
that the transverse displacement of the complex structure is in proportion with
the forcing function of both space and time (Kim et al; 2017). However,
discrete singular convolution and differential element quadrature methods, just
as they require experience and use of trial and error method, also require the
controversy with the selection of the test function and grid point. On the
other hand, hierarchical function method is used on simply supported beams and
the application of the method to other general boundary conditions needs
further investigation (Zhang, 2012). So, it becomes necessary to study complex
structures using receptance method.
Generally,
researchers have associated the study of complex structures with
discontinuities through the use of distributional approach (Yavari et al., 2000; Falsone, 2002; Binodi and Caddemi,
2007; Pameri and Cicirellow, 2011; Chalishajar et al., 2016). To be precise, various types of discontinuities have
been modeled as singularity functions, expressed in terms of the Heaviside’s
and Dirac’s delta distributions (generalized functions) resulting to
straight-forward closed form solutions. Although these researchers came out with
excellent job at modeling the discontinuities in the static perspective, there
is the absence of their formulations for dynamic loading. To bridge this gap, a
substantial amount of research work dealing with dynamic analysis of
discontinuous beams using different end conditions and applying different
methodologies are in existence (Jang and Bert, 1989a, 1989b; Naguleswaran,
2002a, 2002b; Duan and Wang, 2013; Jowerski and Dowell, 2008; Faille and
Santini, 2008; Xing-Jianet al. 2005;
Lee, 2015; Wang and Wang, 2013; Meher, 2014; Beput and Bhutani, 1994). While
all these previous works have concentrated on the free vibration analysis, very
little work has been presented on the forced vibration case. Moreover, it is
also well known that the free vibration analysis strongly effect the accuracy
of modal parameter in complex structures (Koplow et al., 2006; Lu et al.,
2009). Nevertheless, the analysis of forced responses of discontinuous beams
using different techniques have been carried out by few researchers (Koplow et al., 2006; Lu et al., 2009; Bashash et al.,
2008; Schmitz et al., 2001a, 2001b;
Gupta and Sharma, 1998).
It
is indeed peculiar that despite the above research progresses, the analytical
model still yields inaccurate and over estimated modal parameter. Perhaps more attention should be given to the
development of improved analytical model that is capable of predicting
accurately and efficiently the dynamic response of complex structures. The question remains, though as with other
models, can quality, reliable as well as timely and cost-effective structural
responses be obtained through analytical models? The challenges to researchers and structural
engineers in industries are how to develop accurate analytical model with
better reliability and performance (Presas et
al., 2017; Presas and Seidel, 2015).
1.2 STATEMENT
OF PROBLEM
As
stated earlier, it is preferable to have an analytical model of the complex
structure which can reduce the amount of design, testing and manufacturing of
the product (Koplow et al., 2006;
Huang, 2007; Altinatas et al.,
2005). Nevertheless, as the structure
becomes more complicated, consisting of more segments and joints, the accuracy
and efficiency of the analytical model deteriorate fast due to the difficulties
in modeling complex structures.
Moreover, it is of paramount importance to accurately determine the
dynamic structural response of structure or assembly in order to avoid dynamic
problems. Such undesirable noise and vibration, have been a great challenge to
researchers and structural engineers in industry as it can drastically effect
quality of product/service, safety, reliability and performance (Keir, 2004;
Shaik, 2017; Presas et al., 2017;
Presas and Seidel, 2015); Ashory, 1999; Sani et al.,2010, 2011).
Generally, investigations reveal that the problems of inaccuracy in
analytical models contribute to rapid increase in time and cost during product
development cycle, and consequently, result in researchers moving their
attention to the development of modeling methods based on experimental
observation (Huang, 2007; Presas et al.,
2017; Motterhead and Friswell 1992; Emory and Zhu, 2006). As a result, most of
the research efforts to improve the modal parameter through analytical method
have been based on the use of model updating techniques (Arras, 2016; Haapaniemi
et al., 2003). However, it is still time consuming and
costly, and cannot accurately represents the dynamic response of structures in
all frequency range. Moreover, most of
these studies concentrated on the use of free vibration analysis.
So
far, very little work has been done on dynamic analysis of discontinuous beam
with respect to forced vibration analysis. Koplow et al. (2006) analytically investigated the spatial dynamic
response of discontinuous beams using receptance technique based on Euler-Bernoulli
beam theory without numerical simulation. They modeled a stepped beam with a
rectangular cross-section as free-free boundary conditions. The study also
applied experimental modal analysis and receptance coupling substructure
analysis (RCSA) to validate the analytical model. Analytical predictions showed
error of 7 percent, with respect to the fundamental mode shape and this
increased rapidly at higher modes.
Generally, results showed that analytical models were higher than
experimental measurements. Furthermore,
the work mentioned the necessity of developing an improved analytical model
that can match or correlate effectively with experimental results. Lu et al. (2009) restudied his work and
presented a new method to analyze the forced vibration of beam with either a
single step change or multiple step change using the composite element method
(CEM) based on finite element techniques and results compared with those in
Koplow et al. (2006). Bashash et al.(2008) also extended his work and
investigated forced vibration analysis of beams with multiple jumped
discontinuities of their cross-section using partitioning method based on modal
superposition techniques. The model extended the beam presented by Koplow et al. (2006) to beams with n-uniform
sectional area.
However,
even though the above works have reported successful results in modeling
discontinuous beam in dynamic sense, the models ignored the effects of shear
deformation and rotary inertia. Consequently, they tend to yield inaccurate and
over estimated modal parameter, whose problem increases rapidly for responses
at higher modes. So, the effect of shear deformation and rotary inertia are
significant and should be considered by the model. This suggests that
Timoshenko beam theory should be utilized for the modeling and dynamic analysis
of discontinuous beams and the general forms of spatial solution using the separation
of variables as given by equations (1.1) and (1.2):
In
this theory, transverse and rotational modes are coupled making it difficult to
obtain closed form analytical solutions. The short comings are attributed to
the existence of unknown parameters (wave numbers)
in particular sinusoidal and
hyperbolic terms. In this work, a modified Timoshenko model based on transverse
response of beam at first three modes or spatial location was developed in
order to overcome aforementioned shortcomings.
The analytical solution for the dynamic analysis of discontinuous beams
using the proposed model was presented and subsequently, the accuracy of the
proposed model verified based on results obtained using experimental modal
analysis and the existing literature presented by Koplow et al. (2006); Lu et al.
(2009), Wang and Wang (2013)and Meher, (2014) respectively.
1.3 AIM AND
OBJECTIVES OF STUDY
The
aim of this work is development of improved models for analyzing modal
parameters of discontinuous beams. The
specific objectives are:
i.
Modification of Timoshenko
beam model for effective evaluation of closed-form receptance model and
receptance coupling substructure analysis in all boundary configurations.
ii.
Numerical analysis of
aligned, misaligned and uniform beams using the improved models.
iii.
Experimental analysis of
the improved model using a flexible multi-configuration beam vibration test
rig.
iv.
Comparative analysis of
the modified Timoshenko beam models.
1.4 SCOPE
OF STUDY
This study involves analytical and
experimental study for predicting modal parameter of discontinuous beams using
modified Timoshenko Beam model. Also, this study considers beams with two types
of discontinuities, namely:
i.
Stepped beam with aligned
neutral axis and
ii.
Stepped beam with
misalignment of the individual beam segments neutral axis
In
addition, the model prediction in this work was based on concepts of first
spectrum Timoshenkobeam (frequencies below the critical frequency). This is in
line with Stephen and Puchegger (2006) and Iker et al. (2011), which revealed that Timoshenko prediction above the
critical frequency should be disregarded for those end conditions for which the
frequency equation does not factorize. Moreover, it is generally sufficient to
measure the transverse response of the beam at first three modes or spatial
location (Smivnova, 2008; Ewins, 2000; Naguleswaran, 2002a, 2002b).
1.5
JUSTIFICATION
OF THE STUDY
This work presented the modal
parameters for discontinuous beams using analytical and experimental models.
The purpose of developing analytical model is economically driven. It minimizes
the design cycle time, and reduces capital spending on fabrication and testing
of prototypes. However, in the absence of the validation of analytical model,
reliability may not be guaranteed especially in the usage of further design
stage, since it might not represent real structure. Thus, experimental model is
useful in order to confirm the accuracy of analytical model.
The
modal parameter helps the designer to make necessary change in the design to
avoid undesirable noise and vibration, which eventually lead to resonance
condition of extreme amplitude of vibration. Accuracy in modal parameter will
boost quality, safety, reliability and performance of the prospective product.
Again, they will serve as a helpful tool in different subsequent structural
analyses; damage detection (Swamy et al.,
2017; Quali et al., 2017; Dash,
2012), crack detection (Nguyen, 2014; Karandikar et al., 2016; Chinka et al.,
2018), aeroustical radiation (Lee, 2003), structural health monitoring (Scott et al., 1996; Hossiotis and Jeong, 1995;
Zaher, 2002), structural model optimization (Li, 2013), structural modification
(Ravi, 1994; Gibbons, 2017), structural
intensity, sensitivity and stability analysis (Naguleswaran, 2002a, 2003b) and
receptance coupling substructure analysis (Schmitz and Donaldson, 2000; Schmitz and Burns, 2003; Park et al., 2003; Mancisidor et al., 2011).
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