EXTRACTION OF ELASTIC, VISCOUS AND INERTIAL COMPONENTS FROM THE TOTAL REACTION OF AN ADDITIONAL SUPPORT ATTACHED TO A RECTANGULAR PLATE

The paper deals with a mechanical system consisting of a hinged rectangular plate and an additional viscoelastic support with considering its mass-inertia. The impact of the characteristics of additional support on the plate strained state is studied by an original approach of extracting elastic, viscous and inertial components from the total reaction. The plate is assumed to be medium thickness, elastic and isotropic. The Timoshenko hypothesis is used for deformation equations. The external non-stationary force initiates plate vibrations. The impact of the additional support is replaced by the action of three unknown independent non-stationary concentrated forces. The basic formulas for deriving system of three Volterra integral equations are proposed. The system is then solved by numerical and analytical method. By discretizing in time the system of Volterra integral equations is reduced to a system of matrix equations. The system of matrix equations is solved with using generalized Kramer’s algorithm for block matrices and Tikhonov’s regularization method. Note that the approach proposed is applicable for other objects with additional supports, such as beams, plates and shells having various boundary contour and boundary supporting. The results of computing elastic, viscous and inertial components of total reactions on the plate are given. The approach proposed is verified by matching the results of computations by two different methods, namely numerical and analytical for one total reaction and numerical for the total reaction obtained by adding elastic, viscous and inertial components.

Bending-gravity waves in the sea with an ice cover from moving disturbances

В статье проводятся исследования колебаний плавающего ледяного покрова под действием движущихся возмущений. В основу колебаний плавающего ледяного покрова положены линеаризованные уравнения гидромеханики и линейная классическая теория колебаний пластин. Ледяной покров рассматривается как тонкая упругая изотропная пластинка. Анализируются образующиеся при этом трехмерные изгибно-гравитационные волны. Показано, что при движении источника возмущений со скоростью 0<v<v0 изгибно-гравитационные волны не образуются, а наблюдается статический прогиб. Здесь v0 – минимальное значение фазовой скорости. При скорости движения v0<v<v1 образуется одна система изгибно-гравитационных волн. Эти волны распространяются как впереди, так и за источником возмущений. Волны, бегущие впереди источника, обусловлены упругими и массовыми силами пластинки. Волны, распространяющиеся за источником, имеют характер гравитационной волны для чистой воды. При v1<v<(gH)1/2 образуется три системы волн. Упругие волны распространяются впереди источника. Две другие волны распространяются за источником и носят характер поперечной и продольной корабельных волн. При v>(gH)1/2 образуются впереди источника упругие волны, а за источником продольные корабельные волны. Исследовано влияние скорости перемещения нагрузки на амплитуды образующихся волн. The article studies the fluctuations of the floating ice cover under the action of moving perturbations. The vibrations of the floating ice cover are based on the linearized equations of hydro-mechanics and the linear classical theory of plate vibrations. The ice sheet is considered as a thin elastic isotropic plate. The resulting three-dimensional bending-gravity waves are analyzed. It is shown that when the source of disturbances moves at a speed of 0<v<v0, bending-gravitational waves are not formed, but a static deflection is observed. Here v0is the minimum value of the phase velocity. At the speed of motion v0<v<v1, a single system of bending-gravitational waves is formed. These waves propagate both ahead and behind the source of the disturbances. The waves traveling ahead of the source are caused by the elastic and mass forces of the plate. The waves propagating behind the source have the character of a gravitational wave for pure water. At v1<v<(gH)1/2, three wave systems are formed. Elastic waves propagate ahead of the source. The other two waves propagate behind the source and have the character of transverse and longitudinal ship waves. At v>(gH)1/2, elastic waves are formed in front of the source, and longitudinal ship waves are formed behind the source. The influence of the load displacement velocity on the amplitudes of the generated waves is investigated.

Low-frequency vibration damping of a plate with high vibration losses

Object and purpose of research. This work was an experimental check of the possibility to improve the efficiency of low-frequency vibration damping for structural plates with high initial losses of vibration energy. Materials and methods. Vibration dampers used in this study were resonant anti-vibration straps partially tuned to the frequency of intense plate vibrations. Main results. This work demonstrated the possibility of additional efficient vibration damping of a plate with high initial losses of energy at the lower frequencies of its bending resonances. Conclusion. It is shown that vibration dampers suggested by the authors are highly efficient, as well as feature lighter weight and lower thickness than conventional vibration dampers.

GEOMETRIC MODELING OF A SHAPE OF PARALLELOGRAM PLATES IN A PROBLEM OF FREE VIBRATIONS USING CONFORMAL RADII

The paper considers a method of geometric modeling applied when solving basic twodimensional problems of the theory of elasticity and structural mechanics, in particular the applied problems of engineering. The subject of this study is vibrations of thin elastic parallelogram plates of constant thickness. To determine a basic frequency of vibrations, the interpolation method based on the geometric characteristic of the shape of plates (membrane, cross sections of a rod) is proposed. This characteristic represents a ratio of interior and exterior conformal radii of the plate. As is known from the theory of conformal mappings, conformal radii are those obtained by mapping of a plate onto the interior and exterior of a unit disk. The paper presents basic terms, tables, and formulas related to the considered geometric method with a comparative analysis of the curve diagrams obtained using various interpolation formulas. The original computer program is also developed. The main advantage of the proposed method of determining the basic frequency of plate vibrations is a graphic representation of results that allows one to accurately determine the required solution on the graph among the other solutions corresponding to the considered case of parallelogram plates. Although there are many known approximate approaches, which are used to solve the considered problems, only geometric modeling technique based on the conformal radii ratio gives such an opportunity.

Problem formulation of a rectangular orthotropic elastodiffusive Kirchhoff plate vibrations

Исследуются нестационарные упругодиффузионные колебания ортотропной пластины Кирхгофа с учетом релаксации диффузионных потоков. В общем случае пластина находится под действием растягивающих усилий, изгибающих и крутящих моментов и перерезывающих сил. Здесь же заданы плотности диффузионных потоков. Для постановки задачи используется модель нестационарного плоского изгиба упругодиффузионной пластины Кирхгофа, полученная с помощью вариационного принципа Даламбера. We study unsteady elastic diffusion vibrations of a rectangular ortotropic Kirchhoff plate in the presence of diffusion fluxes relaxation. In general formulation the plate is subjected to tensile and shear forces as well as bending moments and torque. The unsteady model of an elastodiffusive Kirchhoff plate is obtained using the d’Alembert variational principle.