| Description |
1 online resource (254 pages) |
| Series |
Mechanical engineering and solid mechanics series |
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Mechanical engineering and solid mechanics series
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| Contents |
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 |
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4: Lagrangian Multiplier Method4.1. 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 |
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6.4. Finding the critical loads of a beam7: 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 |
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8.3. Admissible functions for an Euler-Bernoulli beam8.4. 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 |
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9.6.3. Beam with classical and sliding boundary conditions using inertial restraints to model constraints at the edges of the beam9.7. 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 |
| 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. 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 sup |
| Notes |
11.2. Frequency parameters of constrained shallow shells |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Vibration.
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Calculus of variations.
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Differential equations.
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Vibration
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vibration (physical)
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Calculus of variations
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Differential equations
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Vibration
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| Form |
Electronic book
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| Author |
Monterrubio, Luis
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Mochida, Yusuke
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| ISBN |
9781118984437 |
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1118984439 |
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9781848216389 |
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1848216386 |
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