Background In this article, to the development of the theory and methods for calculating vibrations of dissipative mechanical systems consisting of solids, which include both deformable and non-deformable bodies is discussed. Theoretically, the problem in the mathematical aspect, there are not enough developed solution methods and algorithms for dissipative mechanical systems has not yet been posed. The problems of choosing a nucleus and its rheological parameters, their influence on the frequency and damping coefficient systems have not been studied Purpose The goal of the work is to formulate the statement, develop the solution methods and the algorithm for studying the problems of the dynamics of dissipative mechanical systems consisting of thin-walled plates (or shells) with attached masses and point development theory. Methods A method and algorithm for solving problems of eigen and forced vibrations of dissipative mechanical systems consisting of rigid and deformable bodies, based on the methods of Muller, Gauss, Laplace and Runge integral transform, are developed. Results To describe the dissipative properties of the system as a whole, the concept of a global damping coefficient (GDC) is introduced. In the case of a dissipatively homogeneous system of the GDC, it is determined by the imaginary part of the first modulo complex natural frequency. In the role of GDC in the case of a dissipatively inhomogeneous system, are the imaginary parts of both the first and second frequencies. Moreover, the “Change of Roles” occurs with the characteristic value of the stiffness coefficient of the deformable elements; the real parts of the first and second frequencies are closest. At the indicated characteristic value of the deformable elements, the global damping coefficient has a pronounced maximum. A change in the parameter, on which the global damping coefficient so substantially depends, can be achieved by varying physical properties or geometrical dimensions, thereby opening up the promising possibility of effectively controlling the damping properties of dissipative-inhomogeneous mechanical systems. Conclusions The developed solution methods, algorithms and programs allow determining the dissipative properties of a mechanical system depending on various physic-mechanical parameters, geometrical dimensions and boundary conditions. A method for estimating the dissipative properties of the system as a whole (with forced vibrations) depending on the instantaneous values of the deformable elements (shock absorbers) has been developed. The developed method allows to reduce (several times) the amplitudes of displacements and stresses.
Multilayer viscoelastic cylindrical shells have found their wide application in construction, mechanical engineering, aircraft and rocket engineering. The aim of the work is to investigation the reaction of an infinitely long three-layer cylindrical shell to the action of a normal load moving along an axis with a constant to resonant velocity. The paper presents a mathematical formulation of the problem, developed solution methods and obtained numerical results for the problems of stationary deformation of an infinitely long three-layered cylindrical shell under normal loading. The equations of motion of the bearing layers satisfy the Kirchhoff-Love hypotheses. The solution methods are based on the joint application of the integral Fourier transform in the axial coordinate and the decomposition of all given and desired quantities into Fourier series in the angular coordinate. The outer and inner shells satisfy the Kirchhoff-Love hypotheses. The Lame viscoelasticity equation is used as a linear equation of the filler motion. An effective algorithm for solving the problem of osculations of a three-layer viscoelastic cylindrical shell under normal loading has been developed on a computer. Critical velocities of wave propagation in a three-layer shell under the influence of moving loads are found.
The fundamentals of theoretical methods for engineering calculations of bending vibrations of thin plates, the material of which has hereditary properties, are presented. The manifestation of the hereditary properties of the material of the plate under consideration at a given stress-strain state can be judged by the relaxation core of the material. The dependence of the creep and relaxation kernels on the time difference corresponds to the fact that the “memory” of the material about the force effect produced at a given moment is determined by the elapsed time interval. In particular, this means that if the force action on the elastic-viscous body is cyclic, then the deformation of the body will also be cyclic with some phase shift. A technique for solving forced vibrations of plates under the influence of harmonic loads is proposed. An exact solution to the problem of forced vibrations of a supported rectangular plate, whose dissipative properties are described by memory functions, is constructed.
Increased machine efficiency, increased speeds of working bodies, reduced material consumption, increased load, the need to ensure reliable operation of equipment and safe working conditions are the main factors, that determine attention to vibration protection problems. This constitutes the current trend in the modern dynamics of machines. In this work, we propose a method for the dynamic synthesis of dynamic vibration dampers of a viscoelastic mechanical system having the structure of several (two) bodies elastically attached to the object. The aim of the work is to develop methods of mathematical and computer modeling of the processes of dynamics of active and passive vibration protection systems, as well as to increase the efficiency of using passive and active vibration protection of objects. When developing a system for protecting radio-electronic devices (RED) at resonant frequencies, elements of the theory of automatic control and mathematical modeling were used. An algorithm for generating feedback signals for the information-measuring system for active control of vibration-protective radio-electronic devices is proposed. A method is developed for numerically - analytical study of oscillatory processes of nonlinear mechanical systems consisting of spatial rigid bodies interconnected by massless viscoelastic elements.
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