A selective review of direct, semi-inverse and inverse eigenvalue problems for structures described by differential equations with variable coefficients

DOI10.1007/BF02736214zbMath0987.74003WikidataQ113327856 ScholiaQ113327856MaRDI QIDQ5933952

Isaac Elishakoff

Publication date: 21 August 2001

Published in: Archives of Computational Methods in Engineering (Search for Journal in Brave)

zbMATH Keywords

closed-form solutionsinverse problemseigenvalue problemsdirect problemsinverse vibration problemssemi-inverse problems

Mathematics Subject Classification ID

Vibrations in dynamical problems in solid mechanics (74H45) Bifurcation and buckling (74G60) Inverse problems in equilibrium solid mechanics (74G75) Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids (74-02) General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to mechanics of deformable solids (74-00) Thin bodies, structures (74Kxx)

Related Items

Solving an inverse eigenvalue problem with triple constraints on eigenvalues, singular values, and diagonal elements, Buckling of Standing Tapered Timoshenko Columns with Varying Flexural Rigidity Under Combined Loadings, Visco-piezo-elastic parameter estimation in laminated plate structures, Class of analytical closed-form polynomial solutions for guided--pinned inhomogeneous beams

Uses Software

• PFVIBAT

Cites Work

• Explicit solution of the inverse problem for a vibrating string
• Some exact solutions for the bending of beams with spatially stochastic stiffness
• Perturbation method in the problem of large deflections of circular plates with nonuniform thickness
• The central limit theorem for current-valued processes induced by geodesic flows
• Buckling analysis by influence function
• Some observations on the dynamic equations of prismatic members in compression
• Comparison of Rayleigh's noninteger-power method with Rayleigh-Ritz method
• Inverse spectral theory using nodal points as data - A uniqueness result
• Lower bounds to eigenvalues for vibrations and buckling of structures by analytical applications of a priori known qualitative information
• Transverse vibrations of linearly tapered beams with ends restrained elastically against rotation subjected to axial force
• Buckling of masonry pier under its own weight
• Influence of transverse shear and rotatory inertia on axisymmetric vibrations of polar orthotropic annular plates of parabolically varying thickness
• Exact analysis of beams on two-parameter elastic foundations
• Transverse vibration of circular and elliptic plates with variable thickness
• The shape of the ideal column
• Transverse vibration of circular and elliptic plates with quadratically varying thickness
• Isospectral plane domains and surfaces via Riemannian orbifolds
• On the existence of strong liftings in second countable topological spaces
• The method of weighted residuals and variational principles. With application in fluid mechanics, heat and mass transfer
• On the determination of the density of a vibrating string from spectral data
• Inverse problem for a vibrating beam
• On single and bimodal optimum buckling loads of clamped columns
• Optimal shape and non-homogeneity of a non-uniformly compressed column
• Bounds for the maximum deflection and curvature of nonuniform beam- columns
• When you can't hear the shape of a manifold
• On the direct determination of invariant parameters governing anisotropic plate bending problems
• Exact solution for the transverse vibration of a beam a part of which is a taper beam and other part is a uniform beam
• The tallest column -- optimality revisited
• Static and free vibrational analysis of beams and plates by differential quadrature method
• Optimal design of elastic rods under axial gravitational load using the maximum principle
• Über die Wahl der Ansatzfunktionen bei der Durchführung des Verfahrens von Galerkin
• A remarkable nature of the effect of boundary conditions on closed-form solutions for vibrating inhomogeneous Bernoulli-Euler beams
• Novel closed-form solutions in buckling of inhomogeneous columns under distributed variable loading
• A fifth-order polynomial that serves as both buckling and vibration mode of an inhomogeneous structure
• Inverse spectral problems for inhomogeneous elastic cylinders
• Transverse vibration of a circular plate with arbitrary thickness variation
• Three-dimensional elasticity solutions for free vibrations of circular plates: A polynomials-Ritz analysis
• Efficient inverse frequency design of tapered beams including shear deformations
• Bimodal optimisation of a column in an elastic medium, with respect to buckling or vibration
• Application of integral equations to the flexural vibration of a wedge with rotary inertia and shear
• On the stability of a strut under uniformly distributed axial forces
• The convergence of finite element method in solving linear elastic problems
• A direct matrix method for elastic stability analysis of struts
• On the extremal fundamental frequencies of vibrating beams
• On the creep buckling of shells
• Optimal design of vibrating circular plates
• On the inverse problem of natural vibrations of elastic shells
• Differential quadrature and longterm integration
• Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations
• Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte
• On equivalence of the Galerkin and Fourier series methods for one class of problems
• On the analysis of laterally vibrating slender beams subject to various complicating effects
• EXACT SOLUTIONS FOR THE LONGITUDINAL VIBRATION OF NON-UNIFORM RODS
• An unusual exact, closed-form solution for axisymmetric vibration of inhomogeneous simply supported circular plates
• Optimization Of An Axially Loaded Beam On A Foundation
• A solution of the inverse nodal problem
• Influence of linear depth taper on the frequencies of free torsional vibration of a cantilever beam
• On the Nodal Sets of the Eigenfunctions of Certain Homogeneous and Nonhomogeneous Membranes
• Power series expansions of dynamic stiffness matrices for tapered bars and shafts
• ON THE INVERSE PROBLEM OF THE VIBRATING STRING OR ROD
• An Analogy Between the Flexural Vibrations of a Cone and a Disc of Linearly Varying Thickness
• The Second Approximation for Buckling Loads of Tapered Struts
• Some special solutions for the flexural vibration of discs of varying thickness
• Fundamental solutions of the dynamical equations of elasticity for nonhomogeneous media
• Free Vibration of a Simply Supported Bar With a Linearly Variable Height of Cross Section
• Strongest Columns and Isoperimetric Inequalities for Eigenvalues
• The inverse problem for the vibrating beam
• Apparently First Closed-Form Solution for Frequencies of Deterministically and/or Stochastically Inhomogeneous Simply Supported Beams
• The shape of the strongest column is arbitrarily close to the shape of the weakest column
• HOW CAN WE RECOGNIZE AN ACCEPTABLE MODE SHAPE FOR A VIBRATING BEAM?
• Exact static and dynamic stiffness matrices for general variable cross section members
• Exact Bernoulli-Euler dynamic stiffness matrix for a range of tapered beams
• Inverse Problems in Vibration
• Bimodal Optimization of Compressed Columns on Elastic Foundations
• Exact Bernoulli‐Euler static stiffness matrix for a range of tapered beam‐columns
• A note on the strongest fixed-fixed column
• Structural Parameter Identification in the Frequency Domain: The Use of Overdetermined Systems
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• A survey of matrix inverse eigenvalue problems
• Two new approximate methods for analyzing free vibration of structural components
• Solution of inverse nodal problems
• Application of differential quadrature to static analysis of structural components
• [https://portal.mardi4nfdi.de/wiki/Publication:3844906 Eine Anwendung des Verfahrens der schrittweisen N�herung auf den Knickstab mit stufenweise ver�nderlichem Querschnitt]
• The Elastic Plane With a Circular Insert, Loaded by a Radial Force
• Lateral Vibrations of a Beam Under Initial Linear Axial Stress
• An exact, closed form, solution for the flexural vibration of a thin annular plate having a parabolic thickness variation
• Natural frequencies of simply supported circular plates
• On the Multiplicity of Solutions of the Inverse Problem for a Vibrating Beam
• Finite difference buckling analysis of non-uniform columns
• Inverse Problem for the Vibrating Beam in the Free-Clamped Configuration
• Vibration of curved plate assemblies subjected to membrane stresses
• Inverse problems in structural dynamics—I. Theory
• Inverse problems in structural dynamics—II. Applications
• Static, dynamic and stability analysis of structures composed of tapered beams
• Differential quadrature method in the buckling analysis of beams and composite plates
• Exact solution for general variable cross-section members
• One cannot hear the shape of a drum
• Axisymmetric isospectral annular plates and membranes
• On the Optimal Design of Columns Against Buckling
• Axisymmetric vibrations of radially polarized piezoelectric ceramic cylinders
• Axisymmetric vibrations of annular plates of variable thickness
• A note on the vibration and stability of a circular plate elastically restrained against rotation
• Transverse vibrations of tapered beams
• PFVIBAT—a computer program for plane frame vibration analysis by an exact method
• The shape of the strongest column and some related extremal eigenvalue problems
• Finite element eigenvalue analysis of tapered and twisted Timoshenko beams
• Reference frequencies for the validation of numerical solutions of transverse vibrations of non-uniform beams
• Buckling of a Stochastically Imperfect Finite Column on a Nonlinear Elastic Foundation: A Reliability Study
• A general solution to vibrations of beams on variable winkler elastic foundation
• Free vibrations of foundation beams on two-parameter elastic soil
• On Standard Eigenvalues of Variable-Coefficient Heat and Rod Equations
• Vibration of an Euler-Bernoulli beam of constant depth and with linearly varying breadth
• Pole assignment for the vibrating rod
• Constructing the shape of a rod from eigenvalues
• Buckling instabilities of thick elastic beams subjected to axial stresses
• Static and dynamic analysis of straight bars with variable cross-section
• Free vibrations of tapered beams with flexible ends
• Statics, Stability And Vibration Of Non-prismatic Beams And Columns
• The Vibration of a Circular Plate With Varying Thickness
• Axial Vibrations of a Stochastic Rod
• Transverse Vibrations of a Linearly Tapered Cantilever Beam With Constraining Springs
• Constructing a Finite Element Model of a Vibratory Rod from Eigendata
• Inverse Mode Problems for the Discrete Model of a Vibrating Beam
• The calculation of critical buckling loads for externally constrained structures
• On the Reconstruction of a String from Spectral Data
• A Direct Solution for the Transverse Vibration of Euler-Bernoulli Wedge and Cone Beams
• Vibration in the Two Principal Planes of a Non-Uniform Beam of Rectangular Cross-Section, One Side of Which Varies as the Square Root of the Axial Co-ordinate
• Analysis of High-Speed Rotating Disks With Variable Thickness and Inhomogeneity
• An Inverse Mode Problem for the Continuous Model of an Axially Vibrating Rod
• Eigenmodes of Isospectral Drums
• Vibrations and buckling of cross-ply nonsymmetric laminated composite beams
• Free vibrations of Timoshenko beams on two-parameter elastic foundation
• The Shape of the Tallest Column
• Inverse problems: recovery of BV coefficients from nodes
• Enlarging A Spectral Gap By Structural Modification
• Vibration of non-uniform rods and beams
• A closed-form solution for the generalized Euler problem
• Inverse Problems at the Boundary for an Elastic Medium
• Modeling and control for vibration suppression of a flexible active structure
• Some planar isospectral domains
• On isospectral rods, horns and strings
• On isospectral spring-mass systems
• Dynamic stiffness matrix for variable cross‐section Timoshenko beams
• On a uniform approximation of the density function of a string equation using eigenvalues and nodal points and some related inverse nodal problems
• Drums That Sound the Same
• Inverse nodal problems: finding the potential from nodal lines
• Vibration Frequencies of Tapered Bars and Circular Plates
• On the optimal design of a vibrating beam
• FREE VIBRATION OF POLAR ORTHOTROPIC CIRCULAR PLATES OF VARIABLE THICKNESS WITH ELASTICALLY RESTRAINED EDGE
• New Closed-Form Solutions for Buckling of a Variable Stiffness Column by Mathematica®
• ON THE VIBRATIONS OF AN AXIALLY VIBRATING ELASTIC ROD WITH DISTRIBUTED MASS ADDED IN-SPAN
• Consistent matrix formulations for structural analysis using finite-element techniques.
• Random Transverse Vibrations of Elastic Beams
• Can One Hear the Shape of a Drum?
• A Modified Ritz Method
• Die Äquivalenz der Methode <scp>Galerkins</scp> mit der Methode von Ritz für nichtlineare Operatoren
• Application of Galerkin's method to the dynamic analysis of structures.
• [https://portal.mardi4nfdi.de/wiki/Publication:5536142 �ber die Befreiung der Ansatzfunktionen des Ritzschen und Galerkinschen Verfahrens von den Randbedingungen]
• THE NORMAL MODES OF A CIRCULAR PLATE OF VARIABLE THICKNESS
• Generalized Hypergeometric Function Solutions on the Transverse Vibration of a Class of Nonuniform Beams
• The Buckling of a Column With Random Initial Deflections
• The eigenvalue problem for structural systems with statistical properties.
• Fourier Analysis of the Finite Element Method in Ritz‐Galerkin Theory
• An integral equation approach to the fundamental frequency of vibrating beams
• Accuracy of finite element eigenproblems
• Random Eigenvalue Problems in Structural Analysis
• A method for selection of significant terms in the assumed solution in a Rayleigh-Ritz analysis
• Vibrations of circular plates of variable thickness under an inplane force
• Constant curvature beam finite elements for in-plane vibration
• Convergence studies of eigenvalue solutions using two finite plate bending elements
• Solution methods for eigenvalue problems in structural mechanics
• Matrix dynamic and instability analysis with non‐uniform elements
• The free vibration of rectangular plates
• [https://portal.mardi4nfdi.de/wiki/Publication:5720784 Eine Anwendung der Methode der Problemumkehrung auf die Knickung des geraden Stabes mit ver�nderlichem Querschnitt]
• A Generalized Minimum Principle and Its Application to the Vibration of a Wedge With Rotatory Inertia and Shear
• Über eine Methode zur angenäherten Lösung nichtlinearer Differentialgleichungen mit Anwendung auf die Berechnung der Durchbiegung bei der Knickung gerader Stäbe
• Das Knicken schwerer Gestänge
• Elastische Knickung gerader Stäbe von exponentiell veränderlichem Querschnitt unter dem Einfluß ihres Eigengewichtes
• Vibration Modes of Tapered Beams
• The Kolmogorov-Smirnov Test for Goodness of Fit
• Introduction to Finite Element Vibration Analysis
• Eine Näherungslösung für die Knicklast eines Stabes, dessen Druckkraft und dessen Trägheitsmoment sich nach Potenzgesetzen ändern
• Elements of structural optimization.
• Inverse buckling problem for inhomogeneous columns