رسالة دكتوراة بعنوان Vibration Analysis of Cracked Aluminium Plates
منتدى هندسة الإنتاج والتصميم الميكانيكى
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 رسالة دكتوراة بعنوان Vibration Analysis of Cracked Aluminium Plates

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تاريخ التسجيل : 01/07/2009
العمر : 29
الدولة : مصر
العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
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مُساهمةموضوع: رسالة دكتوراة بعنوان Vibration Analysis of Cracked Aluminium Plates   الإثنين 29 أبريل 2013, 6:53 pm

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أحضرت لكم رسالة دكتوراة بعنوان

Vibration Analysis of Cracked Aluminium Plates
Asif Israr

ويتناول الموضوعات الأتية :

LIST OF FIGURES
Figure 3-1: Isotropic plate loaded by uniform pressure and a small crack at the
centre41
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, 197248
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 05/1 as a function
of half-crack length61
Figure 3-7: Plate first mode natural frequency as a function of the thickness of
the plate for the half-crack length of 001 m62
Figure 4-1: Linear and nonlinear response curves for three different cases of
boundary conditions and half-crack length of 001 m73
Figure 4-2: Comparison between linear and nonlinear model of the cracked
rectangular plate and the boundary condition CCFF74
Figure 4-3: The amplitude of the response as a function of the detuning
parameter, σmn [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 05/1, and
initial conditions zero76
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 05x1 m
and a half-crack length of 001 m80
List of Figures
Figure 4-7: Three modes of vibration for an aluminium plate of sides 05x1 m
and a half-crack length of 0025 m80
Figure 5-1: Dynamics 2 program code for the cracked plate model and a
boundary condition SSSS88
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 001 m for the normalised control
parameter, ω93
Figure 5-4: Bifurcation diagrams for three different cases of boundary
conditions and a half-crack length of 0025 m for the normalised control
parameter, ω93
Figure 5-5: Bifurcation diagrams for three different cases of boundary
conditions and a half-crack length of 001 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 0025 m for the normalised excitation
acceleration in the x-direction 98
Figure 5-7: Dynamical system analysis for a half-crack length of 001 m and the
boundary condition SSSS102
Figure 5-8: Dynamical system analysis for a half-crack length of 001 m and the
boundary condition CCSS103
Figure 5-9: Dynamical system analysis for a half-crack length of 0025 m and
the boundary condition SSSS104
Figure 5-10: Dynamical system analysis for a half-crack length of 0025 m and
the boundary condition CCSS105
Figure 5-11: Poincaré map for the three boundary condition cases and a halfcrack
length of 001 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 001 m by
the use of specialised code written in Mathematica™ and the boundary
condition SSSS107
List of Figures
Figure 5-13: Time plots, and phase planes for a half-crack length 001 m by the
use of specialised code written in Mathematica™ and the boundary condition
CCSS108
Figure 5-14: Time Plots, and phase planes for a half-crack length 001 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 centre113
Figure 6-3: Complete assembly of the test rig114
Figure 6-4: First mode shape for an un-cracked aluminium plate of aspect ratio
05/1116
Figure 6-5: First mode shape for cracked aluminium plate of crack length 50
mm, and aspect ratio of the plate 05/1116
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 05/1117
Figure 7-1: Comparison between method of multiple scales and that of
numerical integration
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 only63
Table 4-1: Peak amplitudes for three sets of boundary conditions and two sets
of half-crack length for a plate aspect ratio of 05/1 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 lengths87
Table 5-2: Dynamics 2 command values for plotting91
Table 6-1: Experimental results of first mode of vibration for an un-cracked and
cracked aluminium plates
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
11 Motivation 1
12 Overview 2
13 Research Objectives 4
14 Outline and Methodology 5
Chapter 2 7
LITERATURE REVIEW 7
21 Historical Background 7
22 Nonlinear Deflection Theory 10
23 Damage in Plate Structures 13
24 Finite Element Approach in Cracked Plate Structures 21
25 Damage Detection Methodologies in Plate Structures 23
26 Damaged Plates of Various Sizes and Shapes 26
27 Solution Methodologies 27
271 Perturbation Theory31
272 The Method of Multiple Scales 33
28 Dynamical Systems and Nonlinear Transitions to Chaos 36
Table of Contents
xii
Chapter 3 39
FORMULATION OF CRACKED PLATES 39
31 Governing Equation of Cracked Rectangular Plate 39
32 Addition of In-plane or Membrane Forces 44
33 Crack Terms Formulation 47
34 Galerkin’s Method for a Vibrating Cracked Plate 51
35 Linear Viscous Damping 58
36 Investigation of Natural Frequencies for Cracked Plate Model 59
Chapter 4 64
APPROXIMATE ANALYTICAL TECHNIQUES 64
41 The Method of Multiple Scales 65
42 Analytical Results 72
421 Linear and Nonlinear Response Curves72
422 Comparison of Linear and Nonlinear Cracked Plate Model 73
423 Nonlinearity affects by changing the location of the Point Load74
424 Damping Coefficient influences Nonlinearity 75
425 Peak Amplitude 75
43 Direct Integration – NDSolve within Mathematica™ 75
44 Finite Element (FE) Technique – ABAQUS/CAE 67-1 76
441 Steps Taken to Perform FE Analysis77
442 ABAQUS/CAE Results 79
Chapter 5 82
DYNAMICAL SYSTEM ANALYSIS 82
51 Dynamical System Theory 83
52 A Model of the Cracked Plate for Dynamical System Analysis 85
521 Nondimensionalisation 85
53 Dynamics 2 - A Tool for Bifurcation Analysis 87
531 Bifurcation Analysis 89
Table of Contents
xiii
532 Lyapunov Exponents89
533 Amplitude of response, x, as a function of normalised excitation
frequency, ω and the Lyapunov exponent91
534 Amplitude of response, x, as a function of normalised excitation
acceleration and the Lyapunov exponent 94
535 Time plots, Phase planes, and Poincaré maps 99
54 Specialised Numerical Calculation Code written in Mathematica™
… 106
Chapter 6 111
EXPERIMENTAL INVESTIGATIONS 111
61 Instrumentation 111
62 Machining Procedure of the Crack in the Aluminium Plate 112
63 Test Setup 113
64 Experimental Results 115
Chapter 7 118
RESULTS AND DISCUSSION 118
71 Analytical Results 118
72 Numerical Results 121
73 Experimental Results 123
74 Conclusions 124
Chapter 8 125
CONCLUSIONS AND FUTURE RECOMMENDATIONS 125
81 Summary 125
82 Future Recommendations 128
LIST OF REFERENCES 129
PUBLICATIONS 150
LIST OF FIGURES IN APPENDICES A-1
CONTENT OF APPENDICES

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رسالة دكتوراة بعنوان Vibration Analysis of Cracked Aluminium Plates

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