Geometrically Non-Linear Dynamic Behavior of Simply Supported Rectangular Plates Carrying a Concentrated Mass

Simply supported plates carrying an added point mass are encountered in many engineering fields, like circuit boards or slabs carrying machines at different locations. Determination of the plate modified dynamic characteristics is a quite laborious task, especially in the non-linear regime, which is rarely treated in the literature. The added mass effect on the plate linear parameters was first examined using Hamilton’s principle and spectral analysis. The modified plate's non-linear fundamental mode was then calculated and its non-linear response to high levels of harmonic excitation was determined. The non-linear formulation, involving a fourth order tensor due to the membrane forces induced in the plate mid-plane by large vibration amplitudes, led to a non-linear algebraic amplitude equation. The iterative solution gave the free vibration case a better qualitative understanding and a quantitative evaluation of the effect of the added mass. The non-linear forced response of the modified plate, examined for a wide frequency range, shows that the added eccentric mass induces changes in the area between the mass location and the simple supports and decreases the non-linear hardening effect. The numerical results, covering new situations, are expected to be useful in engineering applications necessitating for some reason the addition to the plate of a point mass or an adaptation of the plate frequencies in order to avoid the occurrence of undesirable resonances.