Circular membrane with a centrally-located opening – analytical model using parameterisation of the non-empty torurs

Aleksandra Waszczuk-Młyńska , Stanisław Radkowski


This paper discusses a model of the circular membrane using the second order basic partial differential equation. The aim of this work is to present a new analytical model of a membrane with an opening in its central part (damage-simulating opening). Parametrisation applied in the model is based on the parametrisation of the torus with one difference, i.e. the radius defining the torus’s circle is not a constant quantity , but it varies within the range , where is a maximal radius of the circle defining the torus. In order to build a model of a damaged membrane, only one surface of a non-empty torus is needed, that is a surface created by the torus’s circle rotation. A new partial differential equation is computed by means of the Fourier method of separation of variables, and next, by applying the Bessel substitution. The model’s correctness has been verified based on the experiment. The obtained data resulting from the model (in this case the frequency of natural vibrations) was verified as a result of the experiment.
Author Aleksandra Waszczuk-Młyńska (FACME / IAE)
Aleksandra Waszczuk-Młyńska,,
- Institute of Automotive Engineering
, Stanisław Radkowski (FACME / IAE)
Stanisław Radkowski,,
- Institute of Automotive Engineering
Journal seriesZAMM-Zeitschrift fur Angewandte Mathematik und Mechanik, ISSN 0044-2267
Issue year2019
Publication size in sheets0.5
Keywords in Englishcircular membrane, the Bessel substitution, the second order partial differential equation, torus
ASJC Classification2604 Applied Mathematics; 2206 Computational Mechanics
Languageen angielski
Score (nominal)70
Score sourcejournalList
ScoreMinisterial score = 70.0, 29-01-2020, ArticleFromJournal
Publication indicators WoS Citations = 1; Scopus Citations = 1; Scopus SNIP (Source Normalised Impact per Paper): 2017 = 1.085; WoS Impact Factor: 2018 = 1.467 (2) - 2018=1.351 (5)
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