scholarly journals DIVIDING OF THE FULL REACTION OF THE ADDITIONAL SUPPORT CONTACTING WITH THE PLATE INTO VISCOUS, ELASTIC AND INERTIAL COMPONENTS

2021 ◽  
pp. 80-86
Author(s):  
A. V. Voropay ◽  
P. A. Yehorov
Keyword(s):  
Tikhonov Regularization Method ◽  
Volterra Equations ◽  
Matrix Equations ◽  
Structural Elements ◽  
System Of Matrix Equations ◽  
Deformed State ◽  
Plates And Shells ◽  
Stationary Loading ◽  
Analytical Relations ◽  
Additional Support

An original approach for dividing the reaction of a viscoelastic support into inertial, viscous and elastic components is proposed to assess the effect of various characteristics of additional supports on the deformed state of structural elements. The effectiveness of the proposed approach was tested for a mechanical system consisting of a rectangular isotropic plate of medium thickness, hinged-supported along the contour, and an additional concentrated viscoelastic support, taking into account its mass-inertial characteristics. The deformation of the plate is considered within the framework of Timoshenko's hypotheses. Vibrations of the plate are caused by the applying of an external non-stationary loading. The influence of the additional support is modeled by three independent non-stationary concentrated forces. The paper presents the main analytical relations for obtaining a system of three integral Volterra equations, which is solved numerically and analytically. After performing discretization in time, the system of integral equations is transformed into a system of matrix equations. The resulting system of matrix equations is solved using the generalized Cramer algorithm for block matrices and the Tikhonov regularization method. We point out that the material described is 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). The results of a numerical experiment to determine the components (viscous, elastic and inertial) of the full reaction to the plate, arising due to the presence of an additional support, are presented. The reliability of the proposed approach is confirmed by the coincidence of the results of comparing the reactions found by two methods: numerical-analytical for one complete reaction, as in work [1], and numerical for the full reaction (obtained by adding three components).

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.

A system of matrix equations and its applications

2013 ◽  
Vol 56 (9) ◽  
pp. 1795-1820 ◽  
Author(s):  
QingWen Wang ◽  
ZhuoHeng He
Keyword(s):  
Matrix Equations ◽  
System Of Matrix Equations

A System of Matrix Equations over the Quaternion Algebra with Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 233-253 ◽  
Author(s):  
Xiangrong Nie ◽  
Qingwen Wang ◽  
Yang Zhang
Keyword(s):  
General Solution ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Electronic Networks ◽  
Consistency Conditions ◽  
Numerical Example ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations [Formula: see text] and [Formula: see text] over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations [Formula: see text] over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.

A system of matrix equations with five variables

2015 ◽  
Vol 271 ◽  
pp. 805-819 ◽  
Author(s):  
Abdur Rehman ◽  
Qing-Wen Wang
Keyword(s):  
Matrix Equations ◽  
System Of Matrix Equations

scholarly journals Finite Element Vibration Analysis of Beams, Plates and Shells

1999 ◽  
Vol 6 (2) ◽  
pp. 97-109 ◽  
Author(s):  
Jaroslav Mackerle
Keyword(s):  
Finite Element ◽  
Composite Materials ◽  
Conference Proceedings ◽  
Vibration Analysis ◽  
Structural Elements ◽  
Plates And Shells

This bibliography lists references to papers, conference proceedings and theses/dissertations dealing with finite element vibration analysis of beams, plates and shells that were published in 1994–1998. It contains 361 citations. Also included, as separated subsections, are vibration analysis of composite materials and vibration analysis of structural elements with cracks/contacts.

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