ὅδε οἶκος, ὦ ἑταῖρε, μνημεῖον ἐστιν ζῴων τῶν σοφῶν ἀνδρῶν, καὶ τῶν ἔργων αὐτῶν

**Seminar
MECHANICS OF MACHINES AND MECHANISMS - MODELS AND MATHEMATICAL METHODS**

**PROGRAM**

**Plan rada Seminara Mehanika mašina i mehanizama - modeli i matematičke metode za NOVEMBAR 2019.**

**UTORAK, 05.11.2019. u 17:00, Sala 301f, MI SANU, Kneza Mihaila 36**

*Ekaterina Podolskaya, Laboratory of Discrete models in Mechanics, Institute for Problems in Mechanical Engineering, St. Petersburg, Russia; REC Gazpromneft-Polytech, Department of Theoretical Mechanics, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia*

**LOSS OF ELLIPTICITY AND STRUCTURAL TRANSFORMATIONS IN PLANAR SIMPLE CRYSTAL LATTICE**

This work focuses on investigation of structural (phase) transformations in crystal lattices from continuum and discrete points of view. Namely, the continuum, which is equivalent to a simple lattice in the sense of the Cauchy–Born energy, is constructed using long-wave approximation, and its strong ellipticity domains in finite strain space are obtained. It is shown that various domains correspond to variants of triangular and square lattices, and the number of the domains depends on the interaction potential parameters. Non-convex energy profiles and stress–strain diagrams, which are typical for materials allowing twinning and phase transformations, are obtained on the straining paths which connect the domains and cross nonellipticity zones. The procedures of the lattice stability examinations and estimation of energy relaxation by means of molecular dynamical (MD) simulation are developed, and experimental construction of the envelope of the energy profiles, corresponding to the energy minimizer, is done on several straining paths. The MD experiment also allows to observe the energy minimizing microstructures, such as twins and two-phase structures.

*Zbog porodičnih razloga predavača predavanje neće biti održano. Novi termin predavanja biće naknadno utvrđen*

This lecture aims at presenting a brief overview of the main analytical, numerical and experimental results achieved by the presenter and her co-authors Michael J Brennan (Brazil, ex UK) and Gianluca Gatti (Italy) during the previous decade in the field of nonlinear dynamics of vibration isolators and coupled oscillators. Motivated by the experimental phenomena observed in the dynamics of a test-rig consisting of a shaker and an attached nonlinear oscillator, a first attempt was done to derive an analytical solution for the system response in terms of the frequency response curve (FRC), whose theoretical solution was validated numerically and experimentally. The corresponding FRC was found to be characterized by the existence of an inner detached part. However, the main assumption was that the mass of the attached nonlinear system was much less than the moving mass of the shaker. As a result, the effect of the shaker dynamics on the nonlinear system behaviour was significant, but the counter-effect was negligible, so that the mathematical formulation did not allow, for example, an insight on the performance of the nonlinear oscillator as a mass damper for the primary systems. To overcome such limitation, a subsequent work was carried out, where the assumption for a small mass ratio was removed, but the limitation was that the natural frequency of the shaker was considered much lower than the linear natural frequency of the attachment. A recent experimental work provided a final evidence of the theoretical findings of the creation and existence of the inner detached part of the FRC reported previously.

The lecture is aimed at younger researchers who are focused on exploring the dynamics of complex structures. Various methods for determining the influence coefficients of beams with different bearings will be presented:

* a Influence coefficient of displacement of cross section “i” of elastic light beam as well as console under the action of the unit force in section “k”;

* a Influence coefficient of angle of tangent to elastic line of elastic light beam as well as console at cross section “i” of under the action of the unit force in section “k”;

* a Influence coefficient of displacement of cross section “i” of elastic light beam as well as console under the action of the unit couple in section “k”;

* a Influence coefficient of angle of tangent to elastic line of elastic light beam as well as console at cross section “i” of under the action of the unit couple in section “k”

The examples of setting systems of ordinary linear and nonlinear differential equations of dynamics of discrete systems of complex structures with multiple degrees of freedom of oscillations are precursors. Also, few examples of setting systems of ordinary fractional order differential equations of dynamics of discrete systems of complex structures, fractional type, with multiple degrees of freedom of oscillations will be presented. The possibility of independent main modes of fractional properties of the dynamics of a complex distress system is pointed out.

In the last fifteen years, new methods that better solve complicated optimization problems appear. All of these methods are inspired by natural phenomena, and are called biologically inspired methods. All of these methods are inspired by natural phenomena and they are called biologically inspired methods. The most famous are: Genetic Algorithm - GA, Differential Evolution - DE, Particle Swarm Optimization - PSO, Ant Colony Optimization - ACO, Cuckoo Search - CS, Firefly Algorithm - FA, Bat Algorithm - BA, Krill Herd Algorithm - KHA, Grey Wolf Optimizer - GWO and others. All these algorithms can be applied to a number of problems, given the possibility of setting up a wide range of initial values of the design variables - so you do not need experience in setting close to the initial value, a function that optimizes by application of these methods may not be differentiable and continuous, there are no restrictions in terms of the number of variables being optimized. These methods are very simple and offer great upgrade capabilities - making the algorithm efficient by simple modifications. At the same time, methods are very powerful for finding the global optimum for very complex optimization engineering problems. This lecture gives an overview of the application of these algorithms for solving some optimization problems in mechanical engineering based on the results obtained in the papers below.

Seminar Mehanika mašina i mehanizama - modeli i matematičke metode započeo je sa radom u junu 2018.god. Seminar se održava do dva puta mesečno, utorkom u periodu od 17.00 - 19.00 u Matematičkom institutu SANU.

Prof. dr Katica (Stevanović) Hedrih

Rukovodilac seminara

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dr Ivana Atanasovska

Korukovodilac seminara

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Milan Cajić

Sekretar seminara

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