Dynamics of a Viscoelastic Plate Carrying Concentrated Mass with Account of Physical Nonlinearity of Material. Part 1. Mathematical Model, Solution Method and Computational Algorithm
In dynamic calculations of thin-walled structures, an account of nonlinear viscoelastic properties of material plays an important role in a reliable assessment of the strength capability of structures. In this regard, in the mechanics of a deformable rigid body, much attention is paid to the description of nonlinear material properties and the methods to solve specific problems for various thin-walled structures under static and dynamic loads. Thin-walled structures such as plates and shells often play the role of a bearing surface, to which lining, fasteners, various instrument assemblies and other structural elements are attached. In dynamic calculation, the attached elements having an inertial character are considered as additional mass rigidly connected to the systems and concentrated in points. The effect of concentrated mass is introduced using the Dirac delta function. In this paper, a mathematical model has been constructed, a solution method has been proposed, and a computational algorithm has been developed for the problem of oscillations of a viscoelastic plate carrying concentrated mass, with account of physically nonlinear strain of material under different conditions of fixing the plate contours within the Kirchhoff-Love hypothesis. The physical relationship between stresses and strains, with account of nonlinearity, is taken in the form of the Boltzmann-Volterra integral model, where the weakly singular Koltunov-Rzhanitsyn kernel is taken in calculations as the relaxation kernel. Discretization on spatial variables has been conducted by the Bubnov-Galerkin method, and non-decaying systems of integro-differential equations (IDE) with respect to time function of the problem have been obtained in a general case. To solve the IDE, a numerical method was proposed based on the use of quadrature formulas, which eliminate the features in the relaxation kernel. A unified computational algorithm to determine the deflection of a viscoelastic plate with concentrated masses has been developed.