On a curve veering aberration
01 Jan 1974-Zeitschrift für Angewandte Mathematik und Physik (Birkhäuser-Verlag)-Vol. 25, Iss: 1, pp 99-111
TL;DR: In this article, the Ritz-Galerkin method is applied to the classical eigenvalue problem of the free vibration of a rectangular membrane, and it is shown that the behavior of curves which approach each other and suddenly veer away may be the result of approximation in the representation of physical reality.
Abstract: In numerous places in the literature of eigenvalue problems of mathematical physics one finds curves which approach each other and suddenly veer away. The author postulates that this ugly behavior may be the result of approximation in the representation of physical reality. In the present paper such behavior is demonstrated to arise from the application of the well-known Ritz-Galerkin method to the classical eigenvalue problem of the free vibration of a rectangular membrane.
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TL;DR: In this article, the effects of disorder on the modes of vibration of nearly periodic structures are investigated, and perturbation methods for the eigenvalue problem are applied to predict the occurrence of strong localization and eigen value loci veering.
388 citations
TL;DR: Criteria are established to distinguish veerings from crossings in both continuous and discretized models and the existence of curve veering in continuous models is illustrated by presenting the exact solution of an elementary eigenvalue problem.
345 citations
TL;DR: The literature on reduced-order modeling, simulation, and analysis of the vibration of bladed disks found in gas-turbine engines is reviewed in this paper, where an emphasis is placed on key developments in the last decade that have enabled better prediction and understanding of the forced response of mistuned bladed disk, especially with respect to assessing and mitigating the harmful impact of mistuning on blade vibration, stress increases, and attendant high cycle fatigue.
Abstract: The literature on reduced-order modeling, simulation, and analysis of the vibration of bladed disks found in gas-turbine engines is reviewed. Applications to system identification and design are also considered. In selectively surveying the literature, an emphasis is placed on key developments in the last decade that have enabled better prediction and understanding of the forced response of mistuned bladed disks, especially with respect to assessing and mitigating the harmful impact of mistuning on blade vibration, stress increases, and attendant high cycle fatigue. Important developments and emerging directions in this research area are highlighted.
340 citations
242 citations
TL;DR: In this article, the authors review a recent technology development based on coupled MEMS resonators that has the potential of fundamentally transforming MEMS Resonant sensors, including the mode localization effect.
Abstract: In this paper, we review a recent technology development based on coupled MEMS resonators that has the potential of fundamentally transforming MEMS resonant sensors. Conventionally MEMS resonant sensors use only a single resonator as the sensing element, and the output of the sensor is typically a frequency shift caused by the external stimulus altering the mechanical properties, i.e. the mass or stiffness, of the resonator. Recently, transduction techniques utilizing additional coupled resonators have emerged. The mode-localized resonant sensor is one example of such a technique. If the mode localization effect is utilized, the vibrational amplitude pattern of the resonators changes as a function of the quantity to be measured. Compared to using frequency shift as an output signal, the sensitivity can be improved by several orders of magnitude. Another feature of the mode-localized sensors is the common mode rejection abilities due to the differential structure. These advantages have opened doors for new sensors with unprecedented sensitivity.
168 citations
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1,736 citations
TL;DR: In this article, the free vibration of rectangular plates is analyzed using the Ritz method with 36 terms containing the products of beam functions, including clamped, simply-supported, and free edge conditions.
1,076 citations
Book•
01 Jan 1938
TL;DR: In this article, the Fourier-Bessel series has been used to solve boundary value problems in the context of Fourier Cosine and Sine Equations of physics, including the following:
Abstract: Preface 1 Fourier Series Piecewise Continuous Functions Fourier Cosine Series Examples Fourier Sine Series Examples Fourier Series Examples Adaptations to Other Intervals 2 Convergence of Fourier Series One-Sided Derivatives A Property of Fourier Coefficients Two Lemmas A Fourier Theorem A Related Fourier Theorem Examples Convergence on Other Intervals A Lemma Absolute and Uniform Convergence of Fourier Series The Gibbs Phenomenon Differentiation of Fourier Series Integration of Fourier Series 3 Partial Differential Equations of Physics Linear Boundary Value Problems One-Dimensional Heat Equation Related Equations Laplacian in Cylindrical and Spherical Coordinates Derivations Boundary Conditions Duhamel's Principle A Vibrating String Vibrations of Bars and Membranes General Solution of the Wave Equation Types of Equations and Boundary Conditions 4 The Fourier Method Linear Operators Principle of Superposition Examples Eigenvalues and Eigenfunctions A Temperature Problem A Vibrating String Problem Historical Development 5 Boundary Value Problems A Slab with Faces at Prescribed Temperatures Related Temperature Problems Temperatures in a Sphere A Slab with Internally Generated Heat Steady Temperatures in Rectangular Coordinates Steady Temperatures in Cylindrical Coordinates A String with Prescribed Initial Conditions Resonance An Elastic Bar Double Fourier Series Periodic Boundary Conditions 6 Fourier Integrals and Applications The Fourier Integral Formula Dirichlet's Integral Two Lemmas A Fourier Integral Theorem The Cosine and Sine Integrals Some Eigenvalue Problems on Undounded Intervals More on Superposition of Solutions Steady Temperatures in a Semi-Infinite Strip Temperatures in a Semi-Infinite Solid Temperatures in an Unlimited Medium 7 Orthonormal Sets Inner Products and Orthonormal Sets Examples Generalized Fourier Series Examples Best Approximation in the Mean Bessel's Inequality and Parseval's Equation Applications to Fourier Series 8 Sturm-Liouville Problems and Applications Regular Sturm-Liouville Problems Modifications Orthogonality of Eigenfunctions adn Real Eigenvalues Real-Valued Eigenfunctions Nonnegative Eigenvalues Methods of Solution Examples of Eigenfunction Expansions A Temperature Problem in Rectangular Coordinates Steady Temperatures Other Coordinates A Modification of the Method Another Modification A Vertically Hung Elastic Bar 9 Bessel Functions and Applications The Gamma Function Bessel Functions Jn(x) Solutions When v = 0,1,2,... Recurrence Relations Bessel's Integral Form Some Consequences of the Integral Forms The Zeros of Jn(x) Zeros of Related Functions Orthogonal Sets of Bessel Functions Proof of the Theorems Two Lemmas Fourier-Bessel Series Examples Temperatures in a Long Cylinder A Temperature Problem in Shrunken Fittings Internally Generated Heat Temperatures in a Long Cylindrical Wedge Vibration of a Circular Membrane 10 Legendre Polynomials and Applications Solutions of Legendre's Equation Legendre Polynomials Rodrigues' Formula Laplace's Integral Form Some Consequences of the Integral Form Orthogonality of Legendre Polynomials Normalized Legendre Polynomials Legendre Series The Eigenfunctions Pn(cos theta) Dirichlet Problems in Spherical Regions Steady Temperatures in a Hemisphere 11 Verification of Solutions and Uniqueness Abel's Test for Uniform Convergence Verification of Solution of Temperature Problem Uniqueness of Solutions of the Heat Equation Verification of Solution of Vibrating String Problem Uniqueness of Solutions of the Wave Equation Appendixes Bibliography Some Fourier Series Expansions Solutions of Some Regular Sturm-Liouville Problems Some Fourier-Bessel Series Expansions Index
537 citations
01 Jun 1954
TL;DR: In this article, the free transverse vibrations of rectangular plates with all possible boundary conditions obtained by combining free, freely-supported, and fixed edges were considered, and the effect on frequency of an edge being supported and partially restrained is discussed.
Abstract: In this paper are considered the free transverse vibrations of rectangular plates with all possible boundary conditions obtained by combining free, freely-supported, and fixed edges. The Rayleigh method, assuming waveforms similar to those of beams, is used to derive a simple approximate frequency expression for all modes of vibration. The terms in this expression depend on the nodal pattern and the boundary conditions; they are tabulated for fifteen boundary conditions—all four edges free, freely-supported, or fixed and the twelve cases in which some of the edges have one condition and the rest another. The expression can also be used to obtain frequencies for a plate which has a combination of all three boundary conditions. The effect on frequency of an edge being supported and partially restrained is discussed.For some boundary conditions it is possible to compare derived frequencies with those obtained by various methods of accurate analysis and by experiment; except for a few cases, the results from ...
443 citations