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A presentation of the theory behind the Rayleigh-Ritz (R-R) method, as well as a discussion of the choice of admissible functions and the use of penalty methods, including recent developments such as using negative inertia and bi-penalty terms. While presenting the mathematical basis of the R-R method, the authors also give simple explanations and analogies to make it easier to understand. Examples include calculation of natural frequencies and critical loads of structures and structural components, such as beams, plates, shells and solids. MATLAB codes for some common problems are also supplied.
List of contents
PREFACE xi
INTRODUCTION AND HISTORICAL NOTES xiii
CHAPTER 1. PRINCIPLE OF CONSERVATION OF ENERGY AND RAYLEIGH'S PRINCIPLE 1
CHAPTER 2. RAYLEIGH'S PRINCIPLE AND ITS IMPLICATIONS 11
CHAPTER 3. THE RAYLEIGH-RITZ METHOD AND SIMPLE APPLICATIONS 21
CHAPTER 4. LAGRANGIAN MULTIPLIER METHOD 33
CHAPTER 5. COURANT'S PENALTY METHOD INCLUDING NEGATIVE STIFFNESS AND MASS TERMS 39
CHAPTER 6. SOME USEFUL MATHEMATICAL DERIVATIONS AND APPLICATIONS 55
CHAPTER 7. THE THEOREM OF SEPARATION AND ASYMPTOTIC MODELING THEOREMS 67
CHAPTER 8. ADMISSIBLE FUNCTIONS 81
CHAPTER 9. NATURAL FREQUENCIES AND MODES OF BEAMS 89
CHAPTER 10. NATURAL FREQUENCIES AND MODES OF PLATES OF RECTANGULAR PLANFORM 113
CHAPTER 11. NATURAL FREQUENCIES AND MODES OF SHALLOW SHELLS OF RECTANGULAR PLANFORM 133
CHAPTER 12. NATURAL FREQUENCIES AND MODES OF THREE-DIMENSIONAL BODIES 149
CHAPTER 13. VIBRATION OF AXIALLY LOADED BEAMS AND GEOMETRIC STIFFNESS 161
CHAPTER 14. THE RRM IN FINITE ELEMENTS METHOD 181
BIBLIOGRAPHY 197
APPENDIX 203
INDEX 229
About the author
Sinniah Ilanko, Professor of Mechanical Engineering, Department of Engineering, The University of Waikato, Te Whare Wananga o Waikato, Hamilton, New Zealand.
Summary
A presentation of the theory behind the Rayleigh-Ritz (R-R) method, as well as a discussion of the choice of admissible functions and the use of penalty methods, including recent developments such as using negative inertia and bi-penalty terms.
