Issue 38

G. S. Serovaev et alii, Frattura ed Integrità Strutturale, 38 (2016) 392-398; DOI: 10.3221/IGF-ESIS.38.48 394 Figure 1 : Geometrical representation of the plate. The boundary condition corresponds to the cantilever fixing of the plate on the border x = L The free mode model allows the mutual penetration of volumes (Fig. 2), hence surfaces S1 and S2 are free from stresses and boundary conditions for this models have the following form: zz zx zy S S S S S S , , , 1 2 1 2 1 2 0       For the constrained model, the coincident nodes associated with the surfaces S 1 and S 2 have equal displacements in the z axis direction while other components of the displacement remain independent therefore the boundary conditions can be written as follows zx zy S S S S z z S S U U , , 1 2 1 2 1 2 0      A numerical solution of the dynamic problem of vibration of a plate was found by the finite element method using the commercial package ANSYS. The finite-element formulation in matrix form can be written as follows           M u K u F t ( )    (2) where   M - mass matrix of the system,   K - stiffness matrix of the system,   F t ( ) - time dependent load function,   u - the vector of nodal displacements,   u  - the vector of nodal accelerations. In the absence of external influences, the problem is reduced to the typical problem of finding eigenvalues and eigenvectors i t i i u x t e x ( , ) ( )     (3) where  - natural frequency, i  - eigenvector or mode shape. The finite element formulation of the problem has the following form           K M u 2 0 0    (4) where   u 0 - the vector of nodal values of natural vibration modes.