Vibrations of Engineering Structures

The increasing size and complexity of new structural forces in engineering have made it necessary for designers to be aware of their dynamic behaviour. Dynamics is a subject which has traditionally been poorly taught in most engineering courses. This book was conceived as a way of providing engineers with a deeper knowledge of dynamic analysis and of indicating to them how some of the new vibrations problems can be solved. The authors start from basic principles to end up with the latest random vibration applications. The book originated 1n a week course given annually by the authors at the Computational Mechanics Centre, Ashurst Lodge, Southampton, England. Special care was taken to ensure continuity in the text and notations. Southampton 1984 CONTENTS Page Foreword Chapter 1 Introduction to Vibration 1. Introductory Remarks 1 2. Single Degree of Freedom Systems: Equations of Motion and Types of Problem 2 3. Response 6 4. General Structures: Equations of Motion 11 5. Response 15 6. Dynamic Interaction Problems 20 Chapter 2 Free Vibration, Resonance and Damping l. Introduction 25 25 2. Spring-Mass System 3. Simple Pendulum 27 4. Beam with Central Load 28 5. Rolling of a Ship 28 6. Springs in Parallel 30 7. Springs in Series 30 8. Free Vibration 31 9. Energy of Vibrating System 33 10. Damped Free Vibration 34 11. Undamped Forced Response 38 12. Damped Forced Response 39 13. Undamped Transient Vibration 42 14. Damped Transient Vibration 43 15.

1 Introduction to Vibration.- 1. Introductory Remarks.- 2. Single Degree of Freedom Systems: Equations of Motion and Types of Problem.- 3. Response.- 4. General Structures: Equations of Motion.- 5. Response.- 6. Dynamic Interaction Problems.- 2 Free Vibration, Resonance and Damping.- 1. Introduction.- 2. Spring-Mass System.- 3. Simple Pendulum.- 4. Beam with Central Load.- 5. Rolling of a Ship.- 6. Springs in Parallel.- 7. Springs in Series.- 8. Free Vibration.- 9. Energy of Vibrating System.- 10. Damped Free Vibration.- 11. Undamped Forced Response.- 12. Damped Forced Response.- 13. Undamped Transient Vibration.- 14. Damped Transient Vibration.- 15. Summary of Results.- 3 Vibrations of Multi-Degree of Freedom Systems.- 1. Introduction.- 2. Free Vibrations of Two Degree of Freedom Systems.- 3. Free Vibrations of a Multi-Degree of Freedom System.- 4. Orthogonality of Mode Shapes.- 5. Modal Decomposition.- 6. Damped Free Vibrations of Multi-Degree of Freedom Systems.- 7. Forced Vibrations of Multi-Degree of Freedom Systems.- 4 Eigenvalue-Eigenvector Solution.- 1. Introduction.- 2. Three Degree of Freedom System.- 3. Zeros of Determinants.- 4. Banded and Symmetric Matrices.- 5. Reduction of Eigenvalue Equation to Standard Form.- 6. Solution of Standard Eigenvalue Equations by Stürm Sequence Technique.- 7. Solution of the Original Equations using Stürm Sequence Technique.- 8. Simultaneous Iteration.- 9. Comparison of Eigenvalue Solution Methods.- 10. Node Condensation.- 11. Substructure Analysis.- 12. Rate of Change of Eigenvalues.- 5 Approximate Methods for Calculating Natural Frequencies and Dynamic Response of Elastic Systems.- 1. Equivalent One Degree of Freedom Systems.- 2. Continuous Beams.- 3. Distribution Methods.- 4. Multi-Storey Frames.- 6 Determination of Response.- 1. Introductory Remarks.- 2. Steady State Response.- 3. Damping.- 4. Truncation of Series Solution.- 5. Response Spectrum Methods.- 7 The Finite Element Technique.- 1. Introduction.- 2. The Principle of Virtual Displacements.- 3. Finite Element Discretization and Element Matrices.- 4. System Equations.- 5. Solution.- 8 Two Dimensional and Plate Bending Applications.- 1. Introduction.- 2. In-Plane Plate Elements.- 3. In-Plane Vibration of Plates.- 4. Plate Bending Elements.- 5. Transverse Vibration of Plates.- 6. Combination of Plate and Beam Elements.- 9 Transient Response of Structures.- 1. Introduction.- 2. Transient Response without Damping.- 3. Damping.- 4. Damped Transient Response.- 5. Numerical Methods.- 10 Machine Foundations.- 1. Introduction.- 2. Transmissibility of a Foundation on a Rigid Base.- 3. Transmissibility of a Foundation on a Flexible Base.- 4. Low Tuned and High Tuned Foundations.- 5. Dynamic Absorber.- 6. Damped Dynamic Absorber.- 7. Design Codes.- 8. Steel Foundations for Turbo-Alternators.- 9. Conclusions.- 11 Vibration of Axi-Symmetric Shells.- 1. Introduction.- 2. Novozhilov’s Thin Shell Theory.- 3. Finite Element Displacement Method applied to Axi-Symmetric Shells.- 4. Vibration Applications.- 5. Example.- Appendix. Matrices used in the Text.- 12 Some Recent Advances in Structural Vibration.- 1. Introductory Remarks.- 2. Direct Integration Methods.- 3. Accuracy.- 4. Non-linear Problems.- 5. Partitioning.- 13 Fluid Structure Interaction Problems.- 1. Introduction.- 2. The Mechanics of Drag, Inertia and Lift.- 3. Total Hydrodynamic Forces.- 4. Final Remarks.- 14 Introduction to Random Vibrations.- 1. Random Processes.- 2. Spectral Density Function.- 3. The Weiner-Khinchin Relationship.- 4. Response of a Single Spring System to Random Load.- 15 Earthquake Response of Structures.- 1. Introduction.- 2. Beam Analysis.- 3. Spectral Density of Response.- 16 Response of Structures to Wind Loading.- 1. Introduction.- 2. Response of Shells.- 17 Random Response Analysis of Off-Shore Structures.- 1. Introduction.- 2. One Degree of Freedom System.- 3. Multi-Degree of Freedom System.- 4. Closing Remarks.