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MODELING OF MEMBRANE SURFACE STATE UNDER POINT IMPACT

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Subject of Research. We consider the problem of finding an analytical solution to a mathematical model and computer- generated simulation of the concentrated mechanical stress impact at a given point of a loaded elastic body. An adequate mathematical model and a method providing minimum computer-time solution is proposed. An abstract rectangular homogeneous plate-strip is chosen as the object of study. Its edges are fixed and it is loaded by a concentrated action and has a negligible bending stiffness. Abstraction lies in the concentration of membrane basic properties in one parameter — the propagation velocity of elastic waves with no consideration for their attenuation. The object mathematical model is a homogeneous two-dimensional wave equation with inhomogeneous initial and homogeneous boundary conditions. The concentrated action is defined by the Dirac delta function in the initial conditions. Method. The proposed mathematical solution was performed by Fourier method taking into account the orthogonality in l2 space of sinusoidal functions, the properties of Dirac delta functions and zero boundary conditions. The solution provides minimum calculation time applying computer programs. Main Results. The paper presents the process of deriving an analytical solution for the selected mathematical model of concentrated exposure at a specific point of a rectangular homogeneous plate-strip with fixed edges. The resulting solution can be easily programmed. The model gives the possibility to simulate the object behavior with different input data. Modeling is performed using Maple computer algebra system. The model estimated values of a given point effect on a specific homogeneous membrane-strip are presented in a graphical form. The graphs show how the membrane surface state changes over time if the exposure point is not in the membrane center. Practical Relevance. The presented results in the form of an analytical solution make it possible to study real-time dynamics of the membrane surface states under the impact of a known load depending on the input data. The modeling process is characterized by the lack of necessity to search for a solution to the two-dimensional wave equation.

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