Studying the free vibrations of multilayer plates with a complex planform

Abstract

An efficient method is developed to solve the free-vibration problems for arbitrarily shaped orthotropic multilayer plates in a refined formulation. The method is based on the R-function and Ritz methods. Sequences of coordinate functions satisfying kinematic boundary conditions are constructed in an analytic form. The method is used to solve the vibration problem for multilayer square and arbitrarily shaped plates. The results obtained for square plates are analyzed. A comparison of these results with those available in the literature demonstrates the efficiency of the method Keywords: free vibrations, orthotropic multilayer plate, refined formulation, Ritz method, thickness variability, square plate Introduction. Multilayer plates and shells are widely used to model composite elements of many modern structures. The design of multilayer elements should take into account the physical characteristics of layers, which may be essentially dissimilar. To this end, refined theories that account for the shear strains of layers are generally used, which drastically complicates the governing equations. There are a great many studies [3, 4, 8-13, etc.] that propose methods for analyzing the stress state and dynamic behavior of composite plates and shells. However, most of them address shallow shells and plates with a circular or rectangular planform. Much fewer publications consider plates or shells with complex planforms and boundary conditions other than hinged support. Therefore, nonclassical methods for design of composite plates of complex geometry are of theoretical and applied interest. In the present paper, we propose a method for studying the free vibrations of multilayer plates. The method is based on a refined Timoshenko theory, the R-function method, and variational methods. The key advantage of our method is the possibility of describing complex geometries and constructing analytic solutions. The R-function method allows us to resolve a great many challenges in the field of vibrations of multilayer plates and shells.

Identification of nonstationary axisymmetric load distributed along a cylindrical shell

“…Publications on nonstationary vibrations of elastic members are very numerous. Noteworthy are recent interesting studies in this field [7][8][9][10][11]14]. There are far fewer publications concerned with so-called inverse problems.…”

Vibrations of polygonal plates with various boundary conditions

“…Intermediate between these alternative approaches are numerical-and-analytic methods. They considerably extend the range of solvable problems, keeping the continuum nature of the problem formulation [2,7,16]. Such a numerical-and-analytic approach based on the generalized Kantorovich-Vlasov method in the form of the method of complete systems [1] is used here to solve stationary problems of bending of L-shaped plates.…”

On one dynamic problem for structurally inhomogeneous beams

“…For example, bimetallic beams are used as a basic element in instrument making, composite columns, floors, etc. Gradient changes in the mass and mechanical characteristics across the cross-section necessitate a search for generalized solutions to relevant problems of mechanics [7,11], which can be found using integral principles, the virtual-displacement principle being the most general among them.This paper studies the dynamic deformation of structurally inhomogeneous beams and proposes a method for analyzing dynamic processes based on the virtual-displacement principle. We will derive equations of motion for a cross-section of arbitrary shape with one or two axes of symmetry, boundary conditions, and types of loading and discuss a solution describing the dynamic deformation of a hinged layered beam of rectangular cross-section under harmonic loading.…”

“…For example, bimetallic beams are used as a basic element in instrument making, composite columns, floors, etc. Gradient changes in the mass and mechanical characteristics across the cross-section necessitate a search for generalized solutions to relevant problems of mechanics [7,11], which can be found using integral principles, the virtual-displacement principle being the most general among them.…”