Calculating lateral deflection of plates subjected to heterogeneous forces of inertia using a variation-difference method

The deformation of plates is reduced to the generalized problem of eigenvalues based on the stability criterion, which establishes equilibrium in the mechanical system. A method for calculating stability for thin plates under inertia loads exerting force on the basal plane is proposed. Having a differential formulation of the problem, the matrices are formed. The first is a stiffness matrix: it is based on Marie-Sophie Germain’s biharmonic equation. The second matrix represents the change of internal stresses or internal forces in the plate. The stiffness matrix is always symmetric and positive definite for the fixed plate. The matrix of the internal forces in the approximation of derivatives of functions under central differences, from the action of inertial forces can be asymmetric with respect to the main diagonal, can also degenerate and rows of this matrix is the feature of inertia loads. The finite difference method allows us to form a system of large dimensional equations. However, difficulties may arise at the free edges and corners of the plate, which complicates the calculation procedure. Therefore, a transition from the differential formulation of the problem to an integral formulation discretization with variational-difference method is performed. In this case a second row of nodes is not formed during the formation of the stiffness matrix at the free edge. The matrix of the internal forces is always symmetric; it can be ill-conditioned, however, this factor does not affect the problem of determining eigenvalues. Scientific literature provides many theoretical studies and solutions to practical tasks of calculating the stability of structures, including the calculation of longitudinal-transverse bending of thin plates. However, this is a task that has positively certain operators. We have conducted a research of the application of the variational-difference method for calculating the stability of structures. The differential formulation of the boundary value problem is transformed into a variational formulation; the stability criterion is solved; the issue of approximating differential operators for discrete problems with a finite number of variables is addressed in the paper. The paper also describes a developed set of algorithms for the Maple mathematical system and a compilation of calculating programs. Examples of the calculation are considered. We have studied a plate that is rigidly secured at one side while the other three sides are left unfixed. Values of the critical accelerations have been obtained. Problems are assigned to the generalized problem of eigenvalues in which the acceleration parameter, such as the parameter of load, is the only unknown property. Paper objective: the development of a method for calculating inertia loads on plates.