Voropay O. Using of Volterra integral equations in nonstationary problems of plate's dynamics

The object of the study: rectangular plates impacted by non-stationary loads, taking into account the features of the loading and additional supports. The subject of the study: Volterra integral equations to which the problems of nonstationary deformation of plate structural elements are reduced. The inverse problem of simultaneous action on the plate of several unknown independent nonstationary loads is solved. Stable with respect to input data perturbations computational algorithms for recovering complex impulse loads are developed and implemented. The solutions to the ill-posed problem of controlling nonstationary vibrations of a rectangular plate taking into account various concentrated features and the problem of controlling vibrations in a region (чтобы не было двух area) small relative to the area of the plate are constructed. Methods and algorithms for discretizing Volterra integral equation systems and solving block systems of linear algebraic equations using the Tikhonov regularization algorithm and generalized Cramer or Gauss algorithms are developed. A new approach to modeling the dissipative properties of the material (models of internal viscous and internal hysteresis friction) at the nonstationary deforming of structural elements using linear smoothing integral operators is formulated. The identification results are shown to improve when applying a new approach that uses the specified integral operators. An experimental study of the impact loading of a medium thickness rectangular isotropic plate was carried out in order to substantiate (prove) the theoretical results. The results are of theoretical and applied importance and can be used in research institutes, design organizations and other research institutions related to the calculations of plate-shaped structural elements under complex impulse loading, as well as to control of nonstationary vibrations and vibration protection of structural elements.