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.

Solving Complex Fuzzy Linear Matrix Equations

In this paper, a kind of complex fuzzy linear matrix equation A X ˜ B = C ˜ , in which C ˜ is a complex fuzzy matrix and A and B are crisp matrices, is investigated by using a matrix method. The complex fuzzy matrix equation is extended into a crisp system of matrix equations by means of arithmetic operations of fuzzy numbers. Two brand new and simplified procedures for solving the original fuzzy equation are proposed and the correspondingly sufficient condition for strong fuzzy solution are analysed. Some examples are calculated in detail to illustrate our proposed method.

Solvability Conditions and General Solution of a System of Matrix Equations Involving η-Skew-Hermitian Quaternion Matrices

In this article, we study the solvability conditions and the general solution of a system of matrix equations involving η-skew-Hermitian quaternion matrices. Several special cases of this system are discussed, and we recover some well-known results in the literature. An algorithm and a numerical example for the validation of our main result are also provided.

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

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).

All Solutions of the Yang–Baxter-Like Matrix Equation for Nilpotent Matrices of Index Two

Let A be a nilpotent matrix of index two, and consider the Yang–Baxter-like matrix equation AXA=XAX. We first obtain a system of matrix equations of smaller sizes to find all the solutions of the original matrix equation. When A is a nilpotent matrix with rank 1 and rank 2, we get all solutions of the Yang–Baxter-like matrix equation.

Explicit formulas and determinantal representation for η-skew-Hermitian solution to a system of quaternion matrix equations

Some necessary and sufficient conditions for the existence of the ?-skew-Hermitian solution quaternion matrix equations the system of matrix equations with ?-skew-Hermicity, A1X = C1, XB1 = C2, A2Y = C3, YB2 = C4, X = -X?*; Y=-Y?*, A3XA?*3 + B3YB?*3=C5, are established in this paper by using rank equalities of the coefficient matrices. The general solutions to the system and its special cases are provided when they are consistent. Within the framework of the theory of noncommutative row-column determinants, we also give determinantal representation formulas of finding their exact solutions that are analogs of Cramer?s rule. A numerical example is also given to demonstrate the main results.