Improved finite element analysis of beam vibration
About: This article is published in Journal of Sound and Vibration.The article was published on 1986-02-22. It has received 23 citations till now. The article focuses on the topics: Finite element method.
Citations
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TL;DR: In this paper, the complementary energy method is applied to the free vibration analysis of various structural components, including prismatic and tapered bars, prismatic beams, and axisymmetric motion of circular membranes.
Abstract: Two approximate methods, which have not previously been used for structural dynamics problems, are applied to the free vibration analysis of various structural components. The first method is a new version of the complementary energy method. It is shown to be considerably more accurate than the conventional Rayleigh and Rayleigh-Schmidt methods when applied to spatially one-dimensional free vibration problems: prismatic and tapered bars, prismatic beams, and axisymmetric motion of circular membranes. The second method is the differential quadrature method introduced by Bellman and his associates. It is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates.
307 citations
103 citations
TL;DR: In this article, a finite element method is developed and applied to treat the free vibration analysis of beams supported on elastic foundations, and the entire analysis is programmed to run on a microcomputer and with few elements modelling the beam, gives quick and reliable results.
83 citations
TL;DR: In this article, a review of the application of a little known version of the Rayleigh technique to a variety of problems in solid and structural mechanics is presented, together with some new material.
28 citations
TL;DR: In this article, the eigenvalue problem of Euler-Bernoulli discontinuous beams is addressed, and a formulation of well-established lumped-mass methods employing exact influence coefficients is readily feasible, based on appropriate Green's functions yielding the response of the discontinuous beam to a static unit force.
23 citations
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TL;DR: In this article, ausnahmsweise gelingt hier eine Entwicklung nach Potenzreihen, and noch seltener ist dieselbe im ganzen Bereich numerisch brauchbar.
Abstract: JJie Randwertaufgaben der mathematischen Physik erfordern durchweg die Darstellung endlicher, stetiger Funktionen in vorgeschriebenen endlichen Bereichen. Nur ausnahmsweise gelingt hier eine Entwicklung nach Potenzreihen, und noch seltener ist dieselbe im ganzen Bereich numerisch brauchbar. Endlich scheitert, selbst in Fällen, wo die Entwicklung prinzipiell möglich wäre, ihre Berechnung häufig an dem Umstand, daß sie die Lösung unendlich vieler linearen Gleichungen mit unendlich vielen Unbekannten erfordert. Sehr 'viel besser eignen sich Entwicklungen nach Polynomen, fownersche Reihen usw. für die Darstellung einer reellen Funktion w ( , y,...) in einem gegebenen Bereich, da hier für die Konvergenz im ganzen Bereich nur Eigenschaften der Stetigkeit usw. gefordert werden, die bei den Randwertaufgaben meist erfüllt sind. Bei numerisch gegebenem w bietet die Bestimmung der Koeffizienten eines Polynoms ton = au+&!#+'·* von gegebenem Grade n, derart, daß wn als Approximation von w gelten könne, keinerlei Schwierigkeit, und es kann die Genauigkeit bei genügend großem n unbegrenzt gesteigert werden. Ist aber w als Integral einer Differentialgleichung, unter gewissen Nebenbedingungen, definiert, so gelingt die Berechnung der Koeffizienten at zunächst nur in dem sehr speziellen Fall, wo eine Integration durch rasch konvergente Potenzreihen möglich ist Es erhebt sich die Forderung, die angenäherte Darstellung des Integrals im ganzen vorgeschriebenen Bereich durch ein Polynom von gegebenem Grade n auch in diesem Falle allgemein durchzuführen, in der Art, daß bei wachsen
616 citations
TL;DR: In this article, the authors present a general deductive investigation of the open pipe problem under certain restrictions, free from any arbitrary assumptions as to what takes place at the open end.
Abstract: Introduction . A lthough the theory of aerial vibrations has been treated by more than one generation of mathematicians and experimenters, comparatively little has been done towards obtaining a clear view of what goes on in any but the more simple cases. The extreme difficulty of anything like a general deductive investigation of the question is no doubt one reason. On the other hand, experimenters on this, as on other subjects, have too often observed and measured blindly without taking sufficient care to simplify the conditions of their experiments, so as to attack as few difficulties as possible at a time. The result has been vast accumulations of isolated facts and measurements which lie as a sort of dead weight on the scientific stomach, and which must remain undigested until theory supplies a more powerful solvent than any now at our command. The motion of the air in cylindrical organ-pipes was successfully investigated by Bernoulli and Euler, at least in its main features; but their treatment of the question of the open pipe was incomplete, or even erroneous, on account of the assumption that at the open end the air remains of invariable density during the vibration. Although attacked by many others, this difficulty was not finally overcome until Helmholtz, in a paper which I shall have repeated occasion to refer to, gave a solution of the problem under certain restrictions, free from any arbitrary assumptions as to what takes place at the open end. Poisson and Stokes have solved the problem of the vibrations communicated to an infinite mass of air from the surface of a sphere or circular cylinder.
181 citations
TL;DR: In this article, the Cauchy function is used in the solution, by successive approximations, for the lowest frequency and modal configuration of a stepped beam, and the results are then com- pared to the exact solution based on the differential equation of motion.
Abstract: Summary Some results from the theory of integral equations, applicable to the problem of free lateral vibration of beams, are presented and the Cauchy function method is utilized in the solution, by successive approximations, for the lowest frequency and modal configuration of a stepped beam. The results are then com pared to the exact solution based on the differential equation of motion.
30 citations
TL;DR: In this paper, the authors generalized the Rayleigh-Ritz direct variational method for the determination of constant coefficients of assumed shape functions by the introduction of arbitrary parameters in the shape functions and illustrated by several simple examples of freely vibrating beams and a plate.
Abstract: The so-called Rayleigh-Ritz direct variational method for the determination of constant coefficients of assumed shape functions has been generalized by the introduction of arbitrary parameters in the shape functions and then illustrated by means of several simple examples of freely vibrating beams and a plate. The arbitrary parameters are determined by minimizing the natural frequency by the rules of differential calculus or by a number of trial calculations, in difficult problems. Although the calculational labor is greater in the applications of the proposed technique than in the common Rayleigh-Ritz method, the technique offers some advantages in regard to accuracy and flexibility. The proposed technique can also be used for estimating Saint-Venant's torques, bifurcation loads on columns and plates, and other eigenvalues.
30 citations