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The Transverse Modes of a String with Variable Mass Density

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The normal modes of transverse oscillation of a string of varying linear mass density are investigated by approximating the density function with segments of constant value. The computer implementation of the resulting 2 × 2 matrix formalism yielding the characteristic frequencies and associated standing-wave patterns, leads to the straightforward mode analysis of strings with arbitrary density profiles. The method also supplies a connection between the perturbations in characteristic frequency and the small changes in string density which cause them, thereby allowing mode frequencies to be manipulated in a controlled way through the density function. Computed examples are given to demonstrate all results.

Zusammenfassung

Es wurden die Normalmoden der transversalen Schwingungen einer Saite veränderlicher Massendichte untersucht, indem die Dichtefunktion durch Abschnitte konstanter Dichte angenähert wurde. Die Durchführung eines 2 × 2-Matrix-Formalismus mit dem Computer, welche die Eigenfrequenzen und die zugehörigen Schwingungsverteilungen ergibt, führt direkt zur Modenanalyse von Saiten mit beliebigen Dichteprofilen. Die Methode liefert auch eine Verknüpfung zwischen den Störungen der Eigenfrequenzen aufgrund kleiner Änderungen der Saitendichte und ermöglicht damit die kontrollierte Manipulation der Eigenfrequenzen mittels der Dichtefunktion. Alle Ergebnisse werden durch Rechenbeispiele veranschaulicht.

Sommaire

Une étude des modes normaux de vibration transversale d'une corde à variation linéaire de densité de masse a été effectuée en simulant la fonction de densité par des segments de valeurs constantes. La résolution par ordinateur de la matrice 2 × 2 résultant de la mise en équation, et fournissant les fréquences propres et les modes de vibration propres associés, rend possible une analyse modale immédiate des cordes à profil de densité quelconque. La méthode fournit également la relation entre les perturbations fines de fréquences propres et les variations légères de densité qui en sont la cause, ce qui permet de manipuler les fréquences propres en jouant sur la fonction de densité. On fournit des exemples de calcul pour démontrer ces résultats.
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Document Type: Research Article

Publication date: February 1, 1992

More about this publication?
  • 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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