Analysis of Free Flexural Vibrations of Composite Mindlin Base Plates or Panels Reinforced by Three Bonded Plate Strips

The present study is about the theoretical analysis and the numerical solutions of the free vibrations of a certain type of the “Bonded and Stiffened System”. In other words, the “Free Bending Vibrations of Composite Mindlin Base Plates or Panels Reinforced by Three Bonded Stiffening Plate Strips” are investigated. The plate elements of the entire “System” are considered as the dissimilar “Orthotropic Mindlin Plates” with the transverse shear deformations and the transverse and the rotatory moments of inertia. The relatively very thin and linearly elastic adhesive layers between the “Stiffening Plate Strips” and the “Base Plate or Panel” are assumed to be subjected to transverse normal and shear stresses. Also, the dissimilar adhesive layers with unequal thicknesses and the different material characteristics are to be taken into account in the theoretical formulation. The dynamic equations, the stress resultant-displacement expressions and the adhesive layer equations of the “System” under study are combined together in each “Part” (or “Region”) of the “Bonded and Stiffened System”. Following some manipulations and combinations, they are, eventually, reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in “state vector” forms. The aforementioned system of equations are very suitable for the direct application of the present “Method of Solution” developed for the problem. This semi-analytical and the numerical solution technique is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)”. The system of equations in the “state vector” forms are numerically integrated and the mode shapes and the natural frequencies are presented for various sets of the “Boundary Conditions”. It was observed that, in the “Hard” (or relatively “Stiff) adhesive layers case, the regions with the “Bonded Stiffening Plate Strips” are “almost stationary” in all modes up to the sixth mode, while the rest of the “System” vibrates in a way dependent on the “Support Conditions” of the “Base Plate or Panel”. However, in the “Soft” (or relatively “Flexible”) adhesive layer cases, there is no “almost stationary” region anywhere in the “System”. Additionally, some important parameters such as “Thickness Ratio” on the natural frequencies are investigated and graphically presented for a set of the “Support Conditions”.