Transverse Vibration of an Orthotropic Plate with Holes and Inclusions
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TRANSVERSE VIBRATION OF AN ORTHOTROPIC PLATE WITH HOLES AND INCLUSIONS T. V. Shopa
UDC 539.3
Within the framework of the refined theory taking into account the transverse shear strains and inertial components, we consider the problem of stationary vibration of an orthotropic plate with holes and absolutely rigid inclusions. The inclusions may have different types of bonds with the plate. We analyze the case of translational motion of the inclusions along the direction of normal to the median surface of the plate. We study various harmonic (in time) boundary conditions on the outer boundary of the plate and on the contours of the holes. On the basis of the indirect method of boundary elements and a sequential approach to the representation of Green’s functions, the boundary-value problem is reduced to a system of integral equations and relations, which is solved by the collocation method. The numerical results are presented for a rectangular plate containing a circular hole and a circular inclusion. Keywords: vibration, orthotropic plate, holes, inclusions, indirect method of boundary elements.
Thin-walled structural elements of complex shapes interacting with the other bodies are widely used in the engineering practice. Quite frequently they operate under the action of loads variable as functions of time. Therefore, it is necessary to study their dynamic behavior. At present, there are numerous works devoted to the analysis of vibration of rectangular or round plates with one or two round or rectangular holes [1–15]. The analysis of vibration of plates of complex shapes with holes and inclusions of any configuration encounters significant mathematical difficulties. Some numerical schemes for the analysis of transverse vibration of the plates whose inner and outer boundaries have complex shapes were proposed in [16–21]. The results of investigations of the process of vibration of plates with masses concentrated at points or associated masses are well known. However, the available data on vibration of the plates with through massive inclusions of certain shapes are quite poor [16, 22]. Moreover, there are no works dealing with the analysis of vibration of plates containing both holes and inclusions. Statement of the Problem Consider a problem of stationary vibration of an orthotropic plate containing N holes and absolutely rigid inclusions. Assume that the set of inclusions includes N1 inclusions with arbitrary shapes and locations inter-
acting with the plate via thin elastic Winkler-type interlayers with the stiffness coefficients k ( j ) (α), ( j = 1, N1 ), N 2 inclusions that are rigidly fastened to the plate, and N 3 hinged inclusions. The plate also contains N 4 , N 5 , and N 6 holes. On the contours of these holes, we specify the components of displacements, forces, and displacements and forces, respectively. The contours of the inclusions and holes are, respectively, described by the curves L( j ) , j = 1, N1 + N 2 + N 3 , and L( j ) , j = N1 + N 2 + N 3 + 1, N . Pidstryhach Institute for Applied Problems i
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