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The Modes of a Circular Membrane of Non-Uniform Radial Density

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The normal modes of a circular symmetric clamped membrane of arbitrary non-uniform radial mass density are analysed with the aid of the method of constant density radial segments. Using a 2 × 2 matrix formalism, a generic mode condition is derived from which the characteristic frequencies and standing wave patterns are directly computed. For the purpose of manipulating these mode frequencies through small changes in membrane density, an explicit first order perturbational connection between these quantities is established. Computed examples are given to demonstrate the results.

Zusammenfassung

Die Eigenschwingungen einer kreissymmetrischen eingespannten Membran von beliebiger Radialverteilung der Massendichte werden mit der Methode der Radialsegmente konstanter Dichte untersucht. Unter Benutzung eines 2 × 2-Matrixformalismus wird eine Bedingung für die Moden abgeleitet, aus der die charakteristischen Frequenzen und Eigenschwingungen direkt berechnet werden. Um diese Eigenfrequenzen durch kleine Veränderungen der Massendichte manipulieren zu können, wird aufgrund einer Störungsrechnung erster Ordnung eine Verknüpfung zwischen diesen Größen hergestellt. Zur Demonstration der Ergebnisse werden berechnete Beispiele vorgestellt.

Sommaire

Les modes normaux d'une membrane encastrée à symétrie circulaire et de densité de masse radiale arbitraire non uniforme fait l'objet d'une analyse basée sur la méthode des segments de densité radiale constante. A l'aide d'un formalisme matriciel 2 × 2 on détermine un mode générique qui permet le calcul direct des fréquences propres et des diagrammes de vibration entretenue. On établit également une méthode de perturbation au premier ordre pour décrire l'effet que produiraient sur ces fréquences propres de légères modifications de densité de la membrane. On donne quelques exemples de calculs pour illustrer les résultats.
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Document Type: Research Article

Publication date: May 1, 1993

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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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