scholarly journals On the theory of linear micropolar hemitropic media

2020 ◽  
pp. 16-24
Author(s):  
Евгений Валерьевич Мурашкин ◽  
Юрий Николаевич Радаев
Keyword(s):  
Differential Equations ◽  
Elastic Potential ◽  
Dynamic Equations ◽  
Crucial Importance ◽  
Micropolar Continuum ◽  
Micropolar Media ◽  
Basic Equations ◽  
Principle Of Virtual Displacements

В статье рассматриваются вопросы применения относительных тензоров при моделирвоании гемитропных микрополярных сред. Вводится определяющая форма микрополярного упругого потенциала. С помощью принципа виртуальной работы получаются определяющие уравнения для силовых и моментных характеристик микрополярного континуума в терминах относительных тензоров. Приводятся уравнения движения микрополярного континуума в терминах относительных тензоров. Выводится финальная форма динамических уравнений для перемещений и микровращений в случае полуизотропной (гемитропной) симметрии. The paper deals with the application of relative tensors to modeling hemitropic micropolar media. The latter is of crucial importance for biomechanics, mechanics of growing solids and mechanics of metamaterials. The constitutive form of the micropolar elastic potential is discussed. The basic equations of micropolar continuum are derived due to the principle of virtual displacements. Differential equations of the micropolar continuum are given in terms of relative tensors. The final form of dynamic equations for displacements and microrotations in the case of semi-isotropic (hemitropic) micropolar continuum is derived and discussed.

scholarly journals PSEUDOTENSOR FORMULATION OF THE MECHANICS OF HEMITROPIC MICROPOLAR MEDIA

2020 ◽  
Vol 82 (4) ◽  
pp. 399-412
Author(s):  
Yu.N. Radayev ◽  
E.V. Murashkin
Keyword(s):  
Dimensional Space ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Elastic Potential ◽  
Micropolar Continuum ◽  
Micropolar Continua ◽  
Statics And Dynamics ◽  
Elasticity Tensors ◽  
Micropolar Media ◽  
Differential Properties

The possibility of applications of relative tensors concepts to the mechanics of micropolar continuum and, in particular, for the hemitropic micropolar continua is considered. The fundamental tensors and orienting relative scalars in three-dimensional space are introduced. Permutation symbols and absolute Levi-Civita tensors are investigated in further details. Algebraic and differential properties of the relative tensors are discussed. The weights of the fundamental kinematic tensors are determined. The wryness tensor and the asymmetric strain tensor are constructed in terms of the vectors of micro-rotation and displacements. Notions of force and couple traction vectors, associated force and associated couple stress vector, force and couple stresses tensors are discussed in the frameworks of relative tensors algebra. The weights of the basic micropolar elasticity tensors are determined and discussed. The constitutive form of the micropolar elastic potential is introduced as an absolute scalar in order to obtain micropolar constitutive equations. In the linear case, the elastic potential is a quadratic form whose coefficients are pseudoscalars. The weights of the constitutive pseudoscalars are calculated. The dimensionless constitutive micropolar constants and constitutive constants with physical dimensions are discriminated. Statics and dynamics of micropolar elastic continua are developed in terms of relative tensors. Dynamic equations involving displacements and microrotations in the case of semi-isotropic (hemitropic) symmetry are derived and represented by the pseudotensor technique. The paper can be considered as a script of fundamental formulas and concepts related to the algebra and differentiation of relative tensors of arbitrary rank.

scholarly journals Plane thermoelastic harmonic waves in hemitropic micropolar media

2020 ◽  
pp. 174-179
Author(s):  
Евгений Валерьевич Мурашкин ◽  
Юрий Николаевич Радаев
Keyword(s):  
Harmonic Wave ◽  
Dynamic Equations ◽  
Harmonic Waves ◽  
Micropolar Continuum ◽  
Coupled Thermoelasticity ◽  
Cubic Equation ◽  
Wave Solutions ◽  
Coupled Thermoelasticity Problems ◽  
Wave Numbers ◽  
Micropolar Media

В работе рассматривается решение задачи о распространении плоской термоупругой гармонической волны в гемитропной микрополярной среде. Приводятся два варианта динамических уравнений гемитропного микрополярного континуума. Определены пространственные поляризации волн перемещений и микровращений относительно волнового вектора плоской волны. Обсуждается качественный характер возможных волновых решений уравнений связанной термоупругости. Отдельно рассматривается случай атермической волны. Вычисление волновых чисел приводится к исследованию одного кубического уравнения с вещественными коэффициентами. The paper is devoted to the problem of a plane thermoelastic harmonic wave propagation in hemitropic micropolar media. Two versions of the dynamic equations of the hemitropic micropolar continuum are presented. The spatial polarizations of the displacements and microrotations waves relative to the wave vector of a plane wave are determined. The characterictic features of possible wave solutions of the coupled thermoelasticity problems are discussed. The case of athermal waves is separately considered. Computation of wave numbers is reduced to the analysis of a cubic equation with real coefficients.

scholarly journals Li–Yorke Chaos in Hybrid Systems on a Time Scale

2015 ◽  
Vol 25 (14) ◽  
pp. 1540024 ◽  
Author(s):  
Marat Akhmet ◽  
Mehmet Onur Fen
Keyword(s):  
Differential Equations ◽  
Hybrid Systems ◽  
Time Scales ◽  
Time Scale ◽  
Impulsive Differential Equations ◽  
Reduction Technique ◽  
Duffing Equation ◽  
Dynamic Equations ◽  
Definition Of ◽  
First Time

By using the reduction technique to impulsive differential equations [Akhmet & Turan, 2006], we rigorously prove the presence of chaos in dynamic equations on time scales (DETS). The results of the present study are based on the Li–Yorke definition of chaos. This is the first time in the literature that chaos is obtained for DETS. An illustrative example is presented by means of a Duffing equation on a time scale.

Lyapunov stability for measure differential equations and dynamic equations on time scales

2019 ◽  
Vol 267 (7) ◽  
pp. 4192-4223 ◽  
Author(s):  
M. Federson ◽  
R. Grau ◽  
J.G. Mesquita ◽  
E. Toon
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Lyapunov Stability ◽  
Dynamic Equations

Artificial neural networks and adaptive neuro-fuzzy inference systems for parameter identification of dynamic systems

2020 ◽  
Vol 39 (5) ◽  
pp. 6145-6155
Author(s):  
Ramin Vatankhah ◽  
Mohammad Ghanatian
Keyword(s):  
Differential Equations ◽  
Dynamic Systems ◽  
Fuzzy Inference ◽  
Dynamic Equations ◽  
Control Analysis ◽  
Unknown Parameters ◽  
Inference System ◽  
Neuro Fuzzy ◽  
Artificial Neural ◽  
Inference Systems

There would always be some unknown geometric, inertial or any other kinds of parameters in governing differential equations of dynamic systems. These parameters are needed to be numerically specified in order to make these dynamic equations usable for dynamic and control analysis. In this study, two powerful techniques in the field of Artificial Intelligence (AI), namely Artificial Neural Network (ANN) and Adaptive Neuro-Fuzzy Inference System (ANFIS) are utilized to explain how unknown parameters in differential equations of dynamic systems can be identified. The data required for training and testing the ANN and the ANFIS are obtained by solving the direct problem i.e. solving the dynamic equations with different known parameters and input stimulations. The governing ordinary differential equations of the system is numerically solved and the output values in different time steps are obtained. The output values of the system and their derivatives, the time and the inputs are given to the ANN and the ANFIS as their inputs and the unknown parameters in the dynamic equations are estimated as the outputs. Finally, the performances of the ANN and the ANFIS for identifying parameters of the system are compared based on the test data Percent Root Mean Square Error (% RMSE) values.

Discontinuous failure in micropolar elastic-degrading models

2017 ◽  
Vol 27 (10) ◽  
pp. 1482-1515 ◽  
Author(s):  
Lapo Gori ◽  
Samuel S Penna ◽  
Roque L da Silva Pitangueira
Keyword(s):  
Material Model ◽  
Element Model ◽  
Additional Material ◽  
Micropolar Continuum ◽  
Starting Point ◽  
Localization Analysis ◽  
Acceleration Waves ◽  
Micropolar Media ◽  
Waves Propagation ◽  
Bending Length

The present paper investigates the phenomenon of discontinuous failure (or localization) in elastic-degrading micropolar media. A recently proposed unified formulation for elastic degradation in micropolar media, defined in terms of secant tensors, loading functions and degradation rules, is used as a starting point for the localization analysis. Well-known concepts on acceleration waves propagation, such as the Maxwell compatibility condition and the Fresnel–Hadamard propagation condition, are derived for the considered material model in order to obtain a proper failure indicator. Peculiar problems are investigated analytically in details, in order to evaluate the effects on the onset of localization of two of the additional material parameters of the micropolar continuum, the Cosserat’s shear modulus and the internal bending length. Numerical simulations with a finite element model are also presented, in order to show the regularization behaviour of the micropolar formulation on the pathological effects due to the localization phenomenon.

Dynamic Characteristics Analysis of Flexible Mechanism Using Three Parameters Damping Model

Volume 2 ◽  
2004 ◽  
Author(s):  
Hongzhao Liu ◽  
Baixi Liu ◽  
Daning Yuan
Keyword(s):  
Differential Equations ◽  
Dynamic Characteristics ◽  
Virtual Work ◽  
Dynamic Equations ◽  
Linkage Mechanism ◽  
Varying Coefficients ◽  
Characteristics Analysis ◽  
Time Varying Coefficients ◽  
Stiffness And Damping ◽  
Damping Model

In this paper, a practical structural damping model for the dynamic analysis of damping alloy is presented. The differential equations of the beam element with three parameters representing stiffness and damping characteristics are deduced through the three parameters constitution and the virtual work principle. By means of the Kineto-Elastodynamic theory, the established element dynamic equations are assembled into the system dynamic equations of flexible linkage mechanism. In order to solve the high order differential equations with time-varying coefficients, a closed form numerical algorithm is constructed. Lastly, an example of a four-bar elastic linkage mechanism is given to show the effectiveness of the proposed method in studying dynamic characteristics of structure containing damping alloy components.

scholarly journals Solvability and Stability of Impulsive Set Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Shihuang Hong ◽  
Jing Gao ◽  
Yingzi Peng
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Difference Equations ◽  
Time Scale ◽  
Dynamic Equations ◽  
Stability Of Solutions ◽  
Real Numbers ◽  
Generalized Derivative ◽  
Special Cases ◽  
Existence And Stability

A class of new nonlinear impulsive set dynamic equations is considered based on a new generalized derivative of set-valued functions developed on time scales in this paper. Some novel criteria are established for the existence and stability of solutions of such model. The approaches generalize and incorporate as special cases many known results for set (or fuzzy) differential equations and difference equations when the time scale is the set of the real numbers or the integers, respectively. Finally, some examples show the applicability of our results.

Trajectory Generation for Three-Degree-of-Freedom Cable-Suspended Parallel Robots Based on Analytical Integration of the Dynamic Equations

2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Xiaoling Jiang ◽  
Clément Gosselin
Keyword(s):  
Differential Equations ◽  
Trajectory Generation ◽  
Parallel Robots ◽  
Degree Of Freedom ◽  
Dynamic Equations ◽  
Special Cases ◽  
Linear Differential ◽  
Pass Through ◽  
Cable Forces ◽  
Three Degree Of Freedom

This paper proposes a trajectory generation technique for three degree-of-freedom (3-dof) planar cable-suspended parallel robots. Based on the kinematic and dynamic modeling of the robot, positive constant ratios between cable tensions and cable lengths are assumed. This assumption allows the transformation of the dynamic equations into linear differential equations with constant coefficients for the positioning part, while the orientation equation becomes a pendulum-like differential equation for which accurate solutions can be found in the literature. The integration of the differential equations is shown to yield families of translational trajectories and associated special frequencies. This result generalizes the special cases previously identified in the literature. Combining the results obtained with translational trajectories and rotational trajectories, more general combined motions are analyzed. Examples are given in order to demonstrate the results. Because of the initial assumption on which the proposed method is based, the ratio between cable forces and cable lengths is constant and hence always positive, which ensures that all cables remain under tension. Therefore, the acceleration vector remains in the column space of the Jacobian matrix, which means that the mechanism can smoothly pass through kinematic singularities. The proposed trajectory planning approach can be used to plan dynamic trajectories that extend beyond the static workspace of the mechanism.

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