Vibration Analysis of Cracked Aluminium Plates

Vibration Analysis of Cracked Aluminium Plates
اسم المؤلف
Asif Israr
التاريخ
20 نوفمبر 2018
المشاهدات
التقييم
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رسالة دكتوراة بعنوان
Vibration Analysis of Cracked Aluminium Plates
Asif Israr
LIST OF FIGURES
Figure 3-1: Isotropic plate loaded by uniform pressure and a small crack at the
centre 41
Figure 3-2: In-plane forces and a crack of length 2a at the centre of the plate
element .45
Figure 3-3: Two sided constraints and plate deformation having a part-through
crack at the centre 46
Figure 3-4: Line Spring Model (LSM) representing the bending and tensile
stresses for a part-through crack of length 2a, after Rice and Levy, 1972 48
Figure 3-5: Isotropic plate loaded by arbitrary located concentrated force and
small crack at the centre, and parallel to the x-axis 52
Figure 3-6: Plate first mode natural frequency of aspect ratio 0.5/1 as a function
of half-crack length 61
Figure 3-7: Plate first mode natural frequency as a function of the thickness of
the plate for the half-crack length of 0.01 m 62
Figure 4-1: Linear and nonlinear response curves for three different cases of
boundary conditions and half-crack length of 0.01 m 73
Figure 4-2: Comparison between linear and nonlinear model of the cracked
rectangular plate and the boundary condition CCFF .74
Figure 4-3: The amplitude of the response as a function of the detuning
parameter, ?mn [rad/s] and the point load at different locations [m] of the plate
element .74
Figure 4-4: Nonlinear overhang in the form of the softening spring characteristic
by the use of NDSolve within Mathematica™ for an aspect ratio of 0.5/1, and
initial conditions zero .76
Figure 4-5: Three modes of vibration for an aluminium plate without a crack 80
Figure 4-6: Three modes of vibration for an aluminium plate of sides 0.5×1 m
and a half-crack length of 0.01 m 80List of Figures
viii
Figure 4-7: Three modes of vibration for an aluminium plate of sides 0.5×1 m
and a half-crack length of 0.025 m 80
Figure 5-1: Dynamics 2 program code for the cracked plate model and a
boundary condition SSSS .88
Figure 5-2: Explanation of Lyapunov exponent 89
Figure 5-3: Bifurcation diagrams for three different cases of boundary
conditions and a half-crack length of 0.01 m for the normalised control
parameter, ? .93
Figure 5-4: Bifurcation diagrams for three different cases of boundary
conditions and a half-crack length of 0.025 m for the normalised control
parameter, ? .93
Figure 5-5: Bifurcation diagrams for three different cases of boundary
conditions and a half-crack length of 0.01 m for the normalised excitation
acceleration in the x-direction .97
Figure 5-6: Bifurcation diagrams for three different cases of boundary
conditions and a half-crack length of 0.025 m for the normalised excitation
acceleration in the x-direction .98
Figure 5-7: Dynamical system analysis for a half-crack length of 0.01 m and the
boundary condition SSSS .102
Figure 5-8: Dynamical system analysis for a half-crack length of 0.01 m and the
boundary condition CCSS .103
Figure 5-9: Dynamical system analysis for a half-crack length of 0.025 m and
the boundary condition SSSS .104
Figure 5-10: Dynamical system analysis for a half-crack length of 0.025 m and
the boundary condition CCSS .105
Figure 5-11: Poincaré map for the three boundary condition cases and a halfcrack length of 0.01 m from the use of specialised code written in
Mathematica™ 106
Figure 5-12: Time plots, and phase planes for a half-crack length of 0.01 m by
the use of specialised code written in Mathematica™ and the boundary
condition SSSS .107List of Figures
ix
Figure 5-13: Time plots, and phase planes for a half-crack length 0.01 m by the
use of specialised code written in Mathematica™ and the boundary condition
CCSS 108
Figure 5-14: Time Plots, and phase planes for a half-crack length 0.01 m by the
use of specialised code written in Mathematica™ and the boundary condition
CCFF 109
Figure 6-1: Simple layout of the experimental setup 112
Figure 6-2: Aluminium plate with crack at the centre 113
Figure 6-3: Complete assembly of the test rig 114
Figure 6-4: First mode shape for an un-cracked aluminium plate of aspect ratio
0.5/1 116
Figure 6-5: First mode shape for cracked aluminium plate of crack length 50
mm, and aspect ratio of the plate 0.5/1 .116
Figure 6-6: Amplitude responses (in terms of voltage signals) for first mode of
an un-cracked (white mesh area) and cracked (coloured mesh area) aluminium
plates of aspect ratio 0.5/1 117
Figure 7-1: Comparison between method of multiple scales and that of
numerical integration .120List of Tables
x
LIST OF TABLES
Table 3-1: Natural frequencies of the cracked plate model for different boundary
conditions and aspect ratios .60
Table 3-2: Relative changes of the natural frequencies of the cracked simply
supported plate for the first mode only 63
Table 4-1: Peak amplitudes for three sets of boundary conditions and two sets
of half-crack length for a plate aspect ratio of 0.5/1 75
Table 4-2: Finite element analysis results 81
Table 5-1: Data used for dynamical system analysis for various cases of
boundary conditions and half-crack lengths 87
Table 5-2: Dynamics 2 command values for plotting 91
Table 6-1: Experimental results of first mode of vibration for an un-cracked and
cracked aluminium plates .117Tables of Contents
xi
TABLE OF CONTENTS
AUTHOR’S DECLARATION iii
ABSTRACT iv
ACKNOWLEDGEMENTS . v
LIST OF FIGURES vii
LIST OF TABLES x
TABLE OF CONTENTS . xi
NOMENCLATURE . xiv
Chapter 1 1
INTRODUCTION . 1
1.1 Motivation 1
1.2 Overview 2
1.3 Research Objectives . 4
1.4 Outline and Methodology . 5
Chapter 2 7
LITERATURE REVIEW . 7
2.1 Historical Background 7
2.2 Nonlinear Deflection Theory 10
2.3 Damage in Plate Structures 13
2.4 Finite Element Approach in Cracked Plate Structures 21
2.5 Damage Detection Methodologies in Plate Structures 23
2.6 Damaged Plates of Various Sizes and Shapes . 26
2.7 Solution Methodologies 27
2.7.1 Perturbation Theory 31
2.7.2 The Method of Multiple Scales .33
2.8 Dynamical Systems and Nonlinear Transitions to Chaos . 36Table of Contents
xii
Chapter 3 39
FORMULATION OF CRACKED PLATES . 39
3.1 Governing Equation of Cracked Rectangular Plate . 39
3.2 Addition of In-plane or Membrane Forces 44
3.3 Crack Terms Formulation . 47
3.4 Galerkin’s Method for a Vibrating Cracked Plate . 51
3.5 Linear Viscous Damping 58
3.6 Investigation of Natural Frequencies for Cracked Plate Model . 59
Chapter 4 64
APPROXIMATE ANALYTICAL TECHNIQUES 64
4.1 The Method of Multiple Scales . 65
4.2 Analytical Results . 72
4.2.1 Linear and Nonlinear Response Curves .72
4.2.2 Comparison of Linear and Nonlinear Cracked Plate Model 73
4.2.3 Nonlinearity affects by changing the location of the Point Load 74
4.2.4 Damping Coefficient influences Nonlinearity 75
4.2.5 Peak Amplitude 75
4.3 Direct Integration – NDSolve within Mathematica™ 75
4.4 Finite Element (FE) Technique – ABAQUS/CAE 6.7-1 76
4.4.1 Steps Taken to Perform FE Analysis 77
4.4.2 ABAQUS/CAE Results .79
Chapter 5 82
DYNAMICAL SYSTEM ANALYSIS . 82
5.1 Dynamical System Theory 83
5.2 A Model of the Cracked Plate for Dynamical System Analysis . 85
5.2.1 Nondimensionalisation .85
5.3 Dynamics 2 – A Tool for Bifurcation Analysis . 87
5.3.1 Bifurcation Analysis 89Table of Contents
xiii
5.3.2 Lyapunov Exponents 89
5.3.3 Amplitude of response, x, as a function of normalised excitation
frequency, ? and the Lyapunov exponent 91
5.3.4 Amplitude of response, x, as a function of normalised excitation
acceleration and the Lyapunov exponent .94
5.3.5 Time plots, Phase planes, and Poincaré maps 99
5.4 Specialised Numerical Calculation Code written in Mathematica™
… . 106
Chapter 6 111
EXPERIMENTAL INVESTIGATIONS 111
6.1 Instrumentation . 111
6.2 Machining Procedure of the Crack in the Aluminium Plate . 112
6.3 Test Setup 113
6.4 Experimental Results 115
Chapter 7 118
RESULTS AND DISCUSSION 118
7.1 Analytical Results . 118
7.2 Numerical Results . 121
7.3 Experimental Results 123
7.4 Conclusions . 124
Chapter 8 125
CONCLUSIONS AND FUTURE RECOMMENDATIONS 125
8.1 Summary 125
8.2 Future Recommendations 128
LIST OF REFERENCES 12
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