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| موضوع: رسالة دكتوراة بعنوان Vibration Analysis of Cracked Aluminium Plates الإثنين 29 أبريل 2013, 6:53 pm | |
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أخوانى فى الله أحضرت لكم رسالة دكتوراة بعنوان Vibration Analysis of Cracked Aluminium Plates Asif Israr
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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 0.5x1 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 0.5x1 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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