Improved Mathematical Relation of The Modal Shapes of Thin Rectangular Plates

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Gilbert Rainer GILLICH
Codruța HAMAT
Mihaela Dorica STROIA
Marius Florian PREDUS


In this paper a new analytical solution for solving the issues that arise in determining the correct form of the eigenmodes, for the case of simply supported on two opposite edges and clamped on the other two of a rectangular plate, is described. The case of a homogenous plate, isotropic, with constant thickness and uniformly distributed weight, and dynamically driven in the direction normal to the plane of the plate with a median surface in the xOy plane is considered. The deformed shape of the middle area Z(x, y) is analyzed. The theory of elasticity states that this phenomenon can be described by a bi-harmonic differential equation of smooth plates having constant thickness. However, the eigenfunctions obtained for this scenario, lead to eigenmodes representations which are accurate only for modes n=1 and m=1. In this paper it is shown that the proposed analytical solution accurately describes the eigenmodes for the above mentioned plate, with the given limit constraints, for every m and n modes. Moreover, this new solution stands out for its simplicity. MatLab environment was used for simulations. The results obtained using the proposed solution were compared with the results obtained by means of classical method. From the analysis we carried out, we observed a symmetrical distribution of the eigenmodes, on both sides of the equilibrium position.


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HATIEGAN, C., GILLICH, G., VASILE, O., HAMAT, C., STROIA, M. and PREDUS, M. 2023. Improved Mathematical Relation of The Modal Shapes of Thin Rectangular Plates. Romanian Journal of Acoustics and Vibration. 19, 2 (Mar. 2023), 157-163.