scholarly journals Investigation of the influence of an additional viscoelastic support mass-inertial characteristics on the rectangular plate non-stationary deformation

2021 ◽  
pp. 190
Author(s):  
Alexey Voropay ◽  
Pavel Yegorov
Keyword(s):  
Mechanical System ◽  
Rectangular Plate ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Oscillatory Process ◽  
Structural Elements ◽  
Noticeable Effect ◽  
Additional Support ◽  
Many Sources

Modeling additional supports that affect the non-stationary deformation of lamellar structural elements is associated with a number of idealizations and assumptions. Many sources describe the deformation of supported structural elements using absolutely rigid additional supports or stiffeners. In reality, additional supports have viscoelastic properties (viscous and elastic components). When studying non-stationary vibrations, one should also take into account the mass-inertial properties of additional supports. Goal. The goal of the work is: 1) refinement of the existing mathematical model of an additional viscoelastic support by taking into account the influence of its mass-inertial characteristics; 2) study of the influence of these characteristics on the non-stationary deformation of a rectangular plate. Methodology. The non-stationary deformation of beams or plates is described by systems of partial differential equations. For these objects, good results are given by models based on the hypotheses of S.P. Timoshenko, taking into account the inertia of rotation and shear. Such systems of equations can be solved by expanding the sought functions (displacements and angles of rotation) in the corresponding series and using the direct and inverse integral Laplace transform. The determination of the unknown reaction of the additional viscoelastic support, taking into account its mass-inertial characteristics, is carried out on the basis of solving the Volterra integral equations. Results. In this work, an analytical and numerical solution in a general form is obtained, which makes it possible to determine the dependence of the change in time of reaction between the plate and the additional support for various parameters of the mechanical system. Originality. The solution to this problem is based on the further development by the authors of an approach to modeling additional supports in the form of additional unknown non-stationary loads, which are determined from the analysis of Volterra integral equations. Practical value. Examples of calculations for the considered mechanical system at three different values of mass are given. It is shown that the mass-inertial characteristics of the additional support cause a noticeable effect on the oscillatory process, and the changes concern both amplitude and phase characteristics.

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

2021 ◽  
pp. 20-28
Author(s):  
Alexey Voropay ◽  
Pavel Yegorov
Keyword(s):  
Rectangular Plate ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Matrix Equations ◽  
Total Reaction ◽  
Plate Vibrations ◽  
System Of Matrix Equations ◽  
Plates And Shells ◽  
Additional Support ◽  
The Impact

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.

scholarly journals THE INFLUENCE OF MASS AND INERTIAL CHARACTERISTICS OF AN ADDITIONAL VISCOELASTIC SUPPORT IN THE NONSTATIONARY DEFORMING OF A RECTANGULAR PLATE

2021 ◽  
pp. 15-23
Author(s):  
Alexey Voropay ◽  
Pavel Yegorov
Keyword(s):  
Isotropic Plate ◽  
Main Attention ◽  
Concentrated Force ◽  
Noticeable Effect ◽  
Nonstationary Deformation ◽  
Plates And Shells ◽  
Different Shapes ◽  
Analytical Relations ◽  
Additional Support

The nonstationary loading of a mechanical system consisting of a rectangular elastic isotropic plate and an additional viscoelastic support is investigated. The main attention is devoted to taking into account the mass and inertial characteristics of the additional viscoelastic support during modeling. As the main object, to which an additional support is attached, a plate of medium thickness within the framework of Timoshenko's hypotheses is considered. Since the focus of the paper is on the influence of the additional support, the plate itself is assumed to be hinged for simplicity of its model. We point out that the results presented are applicable to other objects that have additional supports (beams, plates and shells, which can have different supports along the contour and different shapes in plan). Nonstationary deformation is caused by the application of an external transverse disturbing load to the plate. The influence of the additional support on the deformation of the plate is replaced by the application of an unknown additional variable concentrated force, which, in fact, is the reaction of interaction between the plate and the additional support. The determination of this unknown reaction is reduced to solving the first kind Volterra integral equation. In this work, the main analytical relations for obtaining integral equations or their systems are derived, and an algorithm for their solving is presented. The results of calculations for specific numerical values are described. Moreover, the effect of an additional viscoelastic support on the plate is considered, both with and without taking into account the mass and inertial characteristics of the support. It is shown that for small masses the effect is practically absent, which can serve as an indirect proof of the correctness of the model obtained. As the main conclusion, it can be pointed out that the mass and inertial characteristics of the additional viscoelastic support have a noticeable effect on the vibration process, on both the amplitude and phase characteristics.

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