Cover -- Title Page -- Copyright -- Contents -- Preface -- Introduction and Historical Notes -- 1: Principle of Conservation of Energy and Rayleigh's Principle -- 1.1. A simple pendulum -- 1.2. A spring-mass system -- 1.3. A two degree of freedom system -- 2: Rayleigh's Principle and Its Implications -- 2.1. Rayleigh's principle -- 2.2. Proof -- 2.3. Example: a simply supported beam -- 2.4. Admissible functions: examples -- 3: The Rayleigh-Ritz Method and Simple Applications -- 3.1. The Rayleigh-Ritz method -- 3.2. Application of the Rayleigh-Ritz method -- 3.2.1.1. Short cut to setting up the stiffness and mass matrices -- 4: Lagrangian Multiplier Method -- 4.1. Handling constraints -- 4.2. Application to vibration of a constrained cantilever -- 5: Courant's Penalty Method Including Negative Stiffness and Mass Terms -- 5.1. Background -- 5.2. Penalty method for vibration analysis -- 5.3. Penalty method with negative stiffness -- 5.4. Inertial penalty and eigenpenalty methods -- 5.5. The bipenalty method -- 6: Some Useful Mathematical Derivations and Applications -- 6.1. Derivation of stiffness and mass matrix terms -- 6.2. Frequently used potential and kinetic energy terms -- 6.3. Rigid body connected to a beam -- 6.4. Finding the critical loads of a beam -- 7: The Theorem of Separation and Asymptotic Modeling Theorems -- 7.1. Rayleigh's theorem of separation and the basis of the Ritz method -- 7.2. Proof of convergence in asymptotic modeling -- 7.2.1. The natural frequencies of an n DOF system with one additional positive or negative restraint -- 7.2.2. The natural frequencies of an n DOF system with h additional positive or negative restraints -- 7.3. Applicability of theorems (1) and (2) for continuous systems -- 8: Admissible Functions -- 8.1. Choosing the best functions -- 8.2. Strategy for choosing the functions.

8.3. Admissible functions for an Euler-Bernoulli beam -- 8.4. Proof of convergence -- 9: Natural Frequencies and Modes of Beams -- 9.1. Introduction -- 9.2. Theoretical derivations of the eigenvalue problems -- 9.3. Derivation of the eigenvalue problem for beams -- 9.4. Building the stiffness, mass matrices and penalty matrices -- 9.4.1. Terms Kij of the non-dimensional stiffness matrix K -- 9.4.2. Terms Mij of the non-dimensional mass matrix M -- 9.4.3. Terms Pij of the non-dimensional penalty matrix P -- 9.5. Modes of vibration -- 9.6. Results -- 9.6.1. Free-free beam -- 9.6.2. Clamped-clamped beam using 250 terms -- 9.6.3. Beam with classical and sliding boundary conditions using inertial restraints to model constraints at the edges of the beam -- 9.7. Modes of vibration -- 10: Natural Frequencies and Modes of Plates of Rectangular Planform -- 10.1. Introduction -- 10.2. Theoretical derivations of the eigenvalue problems -- 10.3. Derivation of the eigenvalue problem for plates containing classical constraints along its edges -- 10.4. Modes of vibration -- 10.5. Results -- 11: Natural Frequencies and Modes of Shallow Shells of Rectangular Planform -- 11.1. Theoretical derivations of the eigenvalue problems -- 11.2. Frequency parameters of constrained shallow shells -- 11.3. Results and discussion -- 12: Natural Frequencies and Modes of Three-dimensional Bodies -- 12.1. Theoretical derivations of the eigenvalue problems -- 12.2. Results -- 13: Vibration of Axially Loaded Beams and Geometric Stiffness -- 13.1. Introduction -- 13.2. The potential energy due to a static axial force in a vibrating beam -- 13.3. Determination of natural frequencies -- 13.3.1. The effect of partial lateral restraints -- 13.3.2. Summary -- 13.3.3. Limitations of the above derivations -- 13.4. Natural frequencies and critical loads of an Euler-Bernoulli beam.

13.5. The point of no return: zero natural frequency -- 13.5.1. Natural frequency -- 13.5.2. Why not forever? -- 13.5.3. Point of no return -- 14: The RRM in Finite Elements Method -- 14.1. Discretization of structures -- 14.2. Theoretical basis -- 14.3. Essential conditions at the boundaries and nodes -- 14.4. Derivation of interpolation functions (shape functions) -- 14.5. Derivation of element matrix equations using the Rayleigh-Ritz method -- 14.5.1. Uniform distributed load -- 14.5.2. Point load -- 14.5.3. Concentrated moment -- 14.5.4. External loads at the nodes -- 14.6. Assembly of element matrices -- 14.7. Eigenvalue problems: geometric stiffness matrix for calculating critical loads -- 14.8. Eigenvalue problems: vibration analysis -- 14.9. Consistent mass matrix for a beam element -- 14.10. Lumped mass matrix for a beam element -- 14.11. The Rayleigh-Ritz and the Galerkin methods -- Bibliography -- Appendix -- A.1. Rayleigh-Ritz with stiffness and mass penalty for a cantilever beam with polynomial admissible functions -- A.2. Lagrangian Multiplier Method for a cantilever beam with polynomial admissible functions -- A.3. RRM for calculating the critical loads and modes of cantilever or propped cantilever beams with polynomial admissible functions -- A.4. RRM for calculating the natural frequencies and modes of vibration of beams using admissible functions presented in Chapter 8 -- A.5. RRM for calculating the natural frequencies and modes of vibration of plates using admissible functions presented in Chapter 8 -- A.6. RRM for calculating the natural frequencies and modes of vibration of shells using admissible functions presented in Chapter 8 -- Index.