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CHAPTER 1 Some Basics of the Theory of Vibrations 1 1.1 A Single Degree of Freedom System 1 1.2 Response of a SDOF to (Any) Transient Load 7 1.3 Multiple-Degrees-of-Freedom (MDOF) System 11 1.4 Infinite-Degrees-of-Freedom (Continuous) System 18 1.5 Mounted Mass 19 CHAPTER 2 Dynamic Response of Beams and Other Structures to Deterministic Excitation 21 2.1 A Generic Example of a Cantilever Beam 21 2.2 Some Basics of the Slender Beam Theory 21 2.3 Modal Analysis of a Slender Cantilever Beam 25 2.4 Stress Modes of a Slender Cantilever Beam 29 2.5 Response of a Slender Beam to Harmonic Excitation 35 2.5.1 Response of beams to base excitation 37 2.5.2 Response of a cantilever beam to harmonic tip force 38 2.5.3 Response of a cantilever beam to harmonic base excitation 41 2.5.4 Two external forces 44 2.6 Response of a Structure with Mounted Mass to Harmonic Excitation 45 2.7 Symmetric and Anti-symmetric Modes and Loads 52 2.8 Response of a Simply Supported Plate to Harmonic Excitation 57 2.9 Vibrations of Shells 66 CHAPTER 3 Dynamic Response of a Structure to Random Excitation 70 3.1 Random Excitation and Response 70 3.2 Response of an Elastic Structure to Random Excitation 78 3.2.1 Closed form solution 85 3.3 Response of a Cantilever Beam to Clamp Displacement Excitation 88 3.4 Response of a Cantilever Beam to Tip Displacement Excitation 92 3.5 Simulation of an Important Structural Parameter in a Vibration Test 96 3.5.1 Two examples 97 3.6 Response of a Structure to Acoustic Excitation 116 3.7 An Example of a Frame Structure 125 3.8 Response of a Structure with Mounted Mass to Random Excitation 132 3.9 Response of a Simply Supported Plate to Random Excitation 140 CHAPTER 4 Contacts In Structural Systems 146 4.1 Static Contact 146 4.1.1 An example of a static contact problem 147 4.2 Analytical Solution for a Dynamic Contact Problem 150 4.3 The Two DOF Contact Problem 157 4.4 Numerical Solution of a Dynamic Contact Problem-Force Excitation 162 4.5 Numerical Solution of a Dynamic Contact Problem-Base Excitation 167 CHAPTER 5 Nondeterministic Behavior of Structures 173 5.1 Probabilistic Analysis of Structures 173 5.1.1 The basic stress-strength case 177 5.2 Solutions for the Probability of Failure 180 5.2.1 Analytical solution-the Lagrange multiplier method 180 5.2.2 The monte carlo simulation 183 5.2.3 Solution with a probabilistic analysis program 184 5.2.4 Solutions for cases where no closed-form expressions exist 185 5.3 Solution with a Commercial Finite Element Program 191 5.4 Probability of Failure of Dynamically Excited Structures 192 5.5 Structural Systems 198 5.6 Model Uncertainties 208 CHAPTER 6 Random Crack Propagation 217 6.1 Crack Propagation in a Structural Element 217 6.2 Effects of a Static Bias on the Dynamic Crack Growth 223 6.3 Stochastic Crack Growth and the Probability of Failure for Harmonic Excitation 228 6.4 Initial Cracks and Flaws 237 6.5 Probabilistic Crack Growth Using the "Unified" Approach 246 6.6 Stochastic Crack Growth and the Probability of Failure for Random Excitation 266 CHAPTER 7 Design Criteria 275 7.1 Dynamic Design Criteria 275 7.1.1 Case of under-design 276 7.1.2 Case of over-design 277 7.2 The Factor of Safety 283 7.2.1 Factor of safety 283 7.3 Reliability Demonstration of Structural Systems 289 7.3.1 Reliability demonstration (verification) is integrated into the design process 293 7.3.2 Analysis of failure mechanism and failure modes 293 7.3.3 Modeling the structural behavior; and verifying the model by tests 294 7.3.4 Design of structural development tests to surface failure modes 294 73.5 Design of structural development tests to surface unpredicted failure modes 295 73.6 "Cleaning"failure mechanism and failure modes 295 73.7 Determination of the required safety margins, the confidence in the models and the relevant parameters 296 73.8 Determination of the demonstrated reliability by "orders of magnitude" 296 CHAPTER 8 Some Important Computer Programs for Structural Analysis 300 8.1 Finite Elements Programs 300 8.2 Probabilistic Analysis Programs 301 8.3 Crack Propagation Programs 304 8.4 Mathematical Solvers 305 CHAPTER 9 Conclusions-Do and Don't Do in Dynamic and Probabilistic Analyses 307 APPENDIX Computer Files for the Demonstration Problems 313 1 Introduction 313 2 List of Files 314 3 Files Listing 317 References 420 Index 427

Library of Congress subject headings for this publication: Structural dynamics, Structural analysis (Engineering)Probabilities