Wyniki wyszukiwania - BazTech

1

Using stability analysis of discrete elastic systems to study the buckling of nanostructures

Korobeynikov S. N., Alyokhin V. V., Annin B. D., Babichev A. V.

Archives of Mechanics

|

2012

|

Vol. 64, nr 4

367-404

EN

Stability/instability criteria of discrete elastic systems are used to study the buckling of nanostructures. The deformation of nanostructures is simulated by solving the nonlinear equations of molecular mechanics. The external forces applied to the nanostructure are assumed to be dead (that is the directions of their action remain constant throughout nanostructure deformation). We note that the positive-definiteness property of the tangential stiffness matrix of a nanostructure is a universal sufficient stability criterion for both equilibrium states and quasi-static/dynamic motions of the nanostructure. The equilibrium configurations are stable in Lyapunov's sense, and quasi-static/dynamic motions are stable in a finite time interval t "isin" (0, Tcr) in which the positive-definiteness property of this matrix is preserved. For dynamic motions of nanostructures, the stability property in this time interval follows from Lee’s criterion of quasi-bifurcation of solutions of second order ODEs. The non-positive definiteness of the tangential stiffness matrix of nanostructures at a time t > Tcr corresponds to both unstable equilibrium configurations and unstable dynamic motions. Computer procedures for determining the critical time and buckling mode(s) are developed using this criterion and are implemented in the PIONER FE code. This code is used to obtain new solutions for the deformation and buckling of twisted (10, 10) armchair and (10, 0) zigzag single-walled carbon nanotubes

2

Brief review of tasks and results of programm of creation of unified geometrical theory of control (UGTC)

Butkovskiy A. G., Babichev A. V.

Systems Science

|

2004

|

Vol. 30, no 1

37-42

EN

The purpose of this paper is in very much compressed thesis form to depict the results and problems that were obtained during development of "Unified geometrical theory of control (UGTC)", or "Theory of control structures (TCS)". It is emphasized that UGTC deals with control of structures and symmetries, and a number of structures are considered. Because the UGTC treats the basic concepts of control theory, some main philosophical and methodological principles related to the control science and mathematics are discussed. Concrete theoretical results are given in geometrical form, which permits to show the invariance and generality of these statements. Particularly, the geometrical construction of foliation extends to the synthesis in the control theory as well as to the choice axiom in the set theory, demonstrating an important connection between appropriate problems. Also the problems of axiomatic control theory and control in metric and topological spaces are mentioned. For systems with differential structure the various representations in terms of differential forms, partial differential equations and phase portraits of differential inclusions are considered. The relation between optimality principle and physical variable of energy tensor is determined.

3

An axiomatical approach to the structure of control and optimal control

Buktkovskiy A.G., Babichev A.V.

Systems Science

|

2001

|

Vol. 27, nr 1

49-58

EN

The paper deals with an axiomatical approach to the problem of determining and studying the control structures. A number of abstract mathematical structures are considered to define the concept of "space of admissibles" for control systems. It is the basic concept, which covers the convenient notions of spaces of admissible motions, controls, etc. Also the notion of a partition for the admissible and the notion of generalized boundary are defined. In such terms the general optimal control problem is stated. The metric approach to this problem is developed, which yields the necessary and sufficient optimality conditions. On the basis of the concepts of secondary and k-fold metrics the optimality conditions are represented as the structural identities. These results are obtained without using any notion of state space for the system. Such a notion proves to be well defined under the additional condition in terms of the lattice theory. This indicates the dependence of hierarchy of the formal control structures on the axiomatics involved.