## Vibration and Damping in Distributed Systems, Volume 2Vibration and Damping in Distributed Systems, Volume II discusses asymptotic methods, including equations with variable coefficients, asymptotic estimates of eigenfrequencies of membranes and plates, WKB approximations and the wave propagation method of Keller and Rubinow, which are developed and applied to scattering problems. The book provides data on the Rayleigh and max-min methods, Courant's nodal domain theorem, the numerical methods of finite-element, boundary-element and spectral types, and an asymptotic method due to Bolotin. Computer graphics are used to enhance understanding and motivate intuition concerning vibration phenomena. The book exhibits a collection of eigenmodes of membranes and plates. It illustrates special effects associated with focusing, whispering gallery and bouncing ball, as well as dynamic motion sequences of a membrane and a plate. Issues involved in experimental determination of internal damping rates and mechanisms in elastic beams are discussed. |

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### Contents

Space | 1 |

frequency domain test 269 | 5 |

Multidimensional Spaces | 59 |

Miscellaneous Methods | 158 |

Distributions Sobolev Spaces and Boundary Value | 161 |

Visualization | 184 |

Strongly Continuous Semigroups of Evolution | 198 |

tion Cosemigroups in a Banach space | 234 |

puted by the boundary element method | 250 |

the boundary element method | 258 |

The Method of Energy Identities | 309 |

Remarks on Experimental Determination of Modal | 321 |

Appendix II | 348 |

361 | |

364 | |

puted by the boundary element method | 243 |

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### Common terms and phrases

Airy function amplitude analysis approximation assume asymptotic expansion beam boundary damping boundary element method caustic chart Chen clamped B.C. coefficients condition consider constant coordinates corresponding covering space curvature curve damping rates defined denote determined differential equations Dirichlet B.C. FIGURE disk dumbbell-shaped domain eigenfrequencies Eigenfunction Eigenmode eigenvalue problem energy example finite frequency geometrical optics geometrical optics expansion geometry Helmholtz equation impedance B.C. integral Keller and Rubinow Lagrangian manifold Lemma linear Maslov Maxwell's equations motion nodal domains nodal line normal congruence obtain parameter partial differential phase space plate equation plate with clamped rays satisfying scattering shown in Fig Snapshots spherical wave subject to zero Table Theorem trajectories variables vector vibrating circular plate vibrating membrane subject wave equation wave front wave method wave number wave propagation wavefunction whispering gallery mode zero B.C. zero Dirichlet B.C. zero Neumann B.C. Zhou