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Approximate Determination of Eigenfrequencies in Damped Lumped-Mass Elastic Systems

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The application of Graeffe's method or root-squaring process for the approximate determination of the eigenfrequencies of lumped-mass multi-degree-of-freedom elastic systems with damping has been investigated. The various procedures of solution were examined by means of specific examples, involving various combinations of real roots and complex pairs of the characteristic algebraic equation. It was found that eigenfrequencies can be determined at the desired accuracy for any number of degrees of freedom by simple algebraic means and also that suitable algorithms based on the present method can be constructed, leading to minimal computer effort, as compared with other conventional techniques.

Zusammenfassung

Die Anwendbarkeit der Methode von Graeffe oder der Quadrierung von Wurzeln zur näherungsweisen Bestimmung der Eigenfrequenzen von gedämpften elastischen Systemen aus konzentrierten Massen mit vielen Freiheitsgraden wurde untersucht. Die verschiedenen Lösungsverfahren wurden anhand von speziellen Beispielen getestet, welche verschiedene Kombinationen von reellen Wurzeln und Wurzelpaaren der charakteristischen algebraischen Gleichung enthalten. Es zeigte sich, daß die Eigenfrequenzen mit der gewünschten Genauigkeit für jede Zahl von Preiheitsgraden mit einfachen algebraischen Mitteln bestimmt werden können, und daß ein geeigneter Algorithmus aus den vorhandenen Methoden gebildet werden kann, der, verglichen mit anderen konventionellen Techniken, zu minimalem Rechenaufwand führt.

Sommaire

On a cherché à appliquer la méthode de Graeffe (ou le procédé de quadrature des racines) pour déterminer approximativement les fréquences propres des systèmes élastiques à masses localisées et à plusieurs degrés de liberté avec amortissement. Divers procédés de résolution ont été examinés sur des exemples spécifiques comportant différentes combinaisons de racines réelles ou de paires de racines complexes pour l'équation algébrique caractéristique. On a trouvé qu'on pouvait déterminer les fréquences propres à la précision désirée pour n'importe quel nombre de degrés de liberté avec des moyens algébriques simples. Les algorithmes correspondant aux méthodes présentées peuvent se construire sans difficulté majeure et conduisent à un minimum de calcul machine comparativement aux techniques classiques.
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Document Type: Research Article

Publication date: October 1, 1983

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  • Acta Acustica united with Acustica, published together with the European Acoustics Association (EAA), is an international, peer-reviewed journal on acoustics. It publishes original articles on all subjects in the field of acoustics, such as general linear acoustics, nonlinear acoustics, macrosonics, flow acoustics, atmospheric sound, underwater sound, ultrasonics, physical acoustics, structural acoustics, noise control, active control, environmental noise, building acoustics, room acoustics, acoustic materials, acoustic signal processing, computational and numerical acoustics, hearing, audiology and psychoacoustics, speech, musical acoustics, electroacoustics, auditory quality of systems. It reports on original scientific research in acoustics and on engineering applications. The journal considers scientific papers, technical and applied papers, book reviews, short communications, doctoral thesis abstracts, etc. In irregular intervals also special issues and review articles are published.
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