Interpretation of boundary conditions in the analytical and numerical shell solutions for mode analysis of multilayered structures

The paper proposes the first 18 vibration modes for plates, and the first 14 vibration modes for cylinders and cylindrical shells. All the edges of these structures are simply supported and the free frequencies are calculated using an exact three-dimensional shell model. A comparison is proposed using two different numerical models such as a classical two-dimensional finite element model and a refined two-dimensional generalized differential quadrature model. The 3D exact model gives all types of vibration modes, when the four edges are simply supported, changing the imposed half-wave numbers m and n in the two in-plane directions α and β. Some of these modes have one of the two half-wave numbers equals zero. When this condition is simultaneously combined with the condition of transverse displacement different from zero, the resulting vibration mode is defined as cylindrical bending mode. The cylindrical bending case has all the derivatives made in the direction where m=0 or n=0 equal zero. This feature means that the vibrational behavior does not change along this particular direction. The numerical models with the simply supported boundary conditions for all the edges do not achieve these results. These cylindrical bending numerical results are obtained modifying the boundary conditions. Proposed results will demonstrate the validity of this idea and how to modify the mathematical models in order to obtain and improve the cylindrical bending solutions.