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Monday, July 8 • 16:10 - 16:30
LINEAR AND GEOMETRICALLY NON-LINEAR FREE AND FORCED VIBRATION OF FULLY CLAMPED RECTANGULAR PLATES CARRYING MULTIPLE MASSES

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In many engineering fields, plates carrying added masses are present. For example, in a circuit board or a slab carrying machines at different locations, the solid components fixed on the surface can be simulated as concentrated masses due to their small dimensions compared to those of the plate. This may make the calculations of the natural frequencies and their associated mode shapes quite a laborious task, especially in the nonlinear regime. In this paper, a semi-analytical method is developed in order to investigate the linear and geometrically nonlinear vibrations of fully clamped rectangular plates carrying multiple masses at various locations. The theoretical model, based on Hamilton's principle and spectral analysis, used previously to examine the non-linear structural dynamic behavior of various beams, plates and shells, is adapted here. The non-linear formulation, involving a fourth order tensor due to the membrane forces induced by the large vibration amplitudes, has been used to analyze the geometrically non-linear case after the preformation of the necessary change of bases from the BFB (beam functions basis) to the MFB (modal functions basis) which is needed in the present case due to the distortion in the mode shapes caused by the presence of the added masses. The numerical results obtained are compared with the literature in different cases of an added mass at the central line of the plate. A focus is then made on the non-linear case in order to get a better qualitative understanding and quantitative evaluation of the effect of the added masses on the mode shapes and the corresponding backbone curves at large vibration amplitudes. Then, the single-mode approach is used to investigate the effects of the geometrical non-linearity on the forced response of fully clamped rectangular plates carrying one concentrated mass at its center under a harmonic distributed force.

Moderators
avatar for Simon Jones

Simon Jones

Associate Professor, Rose-Hulman Institute of Technology
Professional interests include undergraduate engineering education, finite element modeling, ground-borne vibrations, vibrations of musical instruments, and dynamics of toys.
GM

Guilhem Michon

Professor, ISAE-SUPAERO

Authors

Monday July 8, 2019 16:10 - 16:30 EDT
St-Laurent 7
  T07 Struct. dyn. & nonlin. vib., RS03 Struct ac & vibr