Funding informationWarsaw University of Technology, Faculty of Automotive and Construction Machinery Engineering * Work carried out within the project nr PBS3/A9/30/2015 "Autonomous reconfiguration technology of materials in vehicles"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 ∈ (0, 0 ), where 0 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 correctness of the analytical model and the obtained data resulting from the model (in this case the natural frequency) were verified based on the experiment and the numerical model. K E Y W O R D Scircular membrane, the Bessel substitution, the second order partial differential equation, torus

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Fault diagnostics present a crucial technical issue in the areas of both the condition monitoring of machines and the monitoring of structural health. The identification of faults at an early stage in their development has an immense effect on the safety of monitored structures. Correct identification allows for the monitoring of the development of faults and the choosing of optimal operation strategies. This article discusses a method of monitoring structural health, based on the application of the Hilbert transforms (GHT and FrHT) for detecting fault formations and their development in membrane structures. A signal resulting from the HT is analyzed using spectral analysis to identify features indicating the technical state.