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1

Determining Frequency-Energy Dependencyof Nonlinear Normal Modes and Internal ResonancesUsing a Numerical Independent Approach

Ghariblou Arash, Mahmoudi Mussa

Computer Assisted Methods in Engineering and Science

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2022

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Vol. 29, no. 3

261--292

EN

Application of linear normal modes to the nonlinear area provides an in-depth investigation of structures. In this paper, a straightforward approach is proposed to investigatenonlinear normal modes (NNMs) thoroughly and focus on all possible solutions and bifurcations, independent of all initial assumptions and prior solutions. In this context, afterdiscretization of the response domain over an appropriate resolution, a periodicity algorithm is suggested to capture the solutions that meet the NNMs criteria. Afterward, thefrequency and energy of the system during accepted responses and degrees of freedom(DOFs)’ relations are derived. Finally, after verifying the proposed approach and acquir-ing new internal resonances, the frequency-energy plots and NNMs of a nonlinear elasticsystem with more substantial nonlinearities and a two-story steel structure with nonlinearmaterial are studied. It is worth noting that the periodicity algorithm and capturing allpossible solutions and bifurcations are among the apparent novelties of the current paper.

2

Natural convection in a partially heated cylinder: A numerical study

Núñez José, Beltrán Alberto, Rivero Michel

Journal of Theoretical and Applied Mechanics

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2021

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Vol. 59 nr 4

623--636

EN

This work presents a numerical study on a natural convective flow in a cylindrical container heated from below, cooled from above, and partially heated from the lateral wall. Mass, momentum and energy equations were solved with a developed hybrid Fourier-finite volume code and validated with the commercial software COMSOL Multiphysics for steady-state solutions. The primary solutions correspond to steady-states Cm with azimuthal wavenumbers m. The results show mode competition between different states leading to many flow solutions including steady axisymmetric, steady non-axisymmetric, time-dependent pulsating wave solutions, and other flow states with a variety of spatiotemporal symmetries.

3

Area of the unstable solution of rolling equation : jumps of the oscillations amplitude

Wawrzyński W.

Journal of KONES

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2018

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Vol. 25, No. 3

489--496

EN

The most interesting motion of the ship is rolling. This is because the rolling amplitudes are much bigger than amplitudes of other degrees of freedom and under resonance conditions, which can exceed 40º. In such a case, when the maximum of the righting arm curve is placed at relatively small angles, the roll equation reveals a strongly nonlinear character and bistability areas as well as an area of unstable solutions of the roll equation occurs. Together with the appearance of the above-mentioned areas, amplitude jumps are possible. In the study, the case of strongly nonlinear rolling is analysed. For the purpose of numerical simulations, the 1DOF mathematical model of rolling with damping dependent on amplitude and frequency is used. The article presents the roll spectrum including the bistability areas and the area of unstable solutions for one loading condition of the offshore support vessel. It is demonstrated that for strongly nonlinear rolling, rolling with two different amplitudes for the same value of excitation is possible. It is also shown that transitions (jumps) between these amplitudes are possible too. A few scenarios of jumps of the rolling amplitude within the region of unstable solutions of the rolling equation are presented. The presented rolling scenarios show that under some circumstances rolling can be observed as chaotic.

4

Numerical studies of nonlinear gearing models using bond graph method

Margielewicz J., Wojnarowski J., Zawiślak S.

International Journal of Applied Mechanics and Engineering

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2018

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Vol. 23, no. 4

885--896

EN

The present paper is dedicated to computer simulations performed using a numerical model of a one-stage gear. The motion equations were derived utilizing the bond graph method. The formulated model takes into consideration the variable stiffness of toothings as well as an inter-tooth clearance which has been represented via discontinuous elements with so called dead zones. As a result of these assumptions, the nonlinear model was obtained which enables representation of the dynamic phenomena of the considered gear. In the paper, an influence of errors of gear wheels’ co-operation on the character of excited dynamic phenomena was studied. The methodology of the analyses consists in utilization of the following tools: color maps of distribution of the maximal Lapunov coefficient and bifurcation diagrams. Based upon them, the parameters were determined, for which the Poincare portrait represents a structure of the chaotic attractor. For the identified attractors, the initial attractors were calculated numerically - which along with the changes of the control parameters are subjected to multiplication, stretching or rotation.

5

Susceptibility of the roll equation to the bifurcation phenomenon depending on the damping coefficient value and form of the roll damping formula

Wawrzyński W.

Zeszyty Naukowe Akademii Morskiej w Szczecinie

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2017

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nr 52 (124)

161--169

EN

This paper deals with the susceptibility of the roll equation to the bifurcation phenomenon depending on the damping coefficient value and form of the roll damping formula. Generally, the bifurcation phenomenon depends mainly on the shape of the righting arm curve (GZ curve), but roll damping also has a significant impact. The commonly used formulas for roll damping are presented, as well as values of the linear equivalent roll damping coefficient, calculated according to the simple Ikeda method. Values of the linear equivalent roll damping coefficient were calculated for a wide spectrum of roll amplitudes and roll frequencies for two ships. The loading conditions for these ships were selected to show different GZ curve characteristics. One ship has a softening spring characteristic and the second has a hardening spring characteristic. For these two ships, a number of calculations of roll spectra are presented where the bifurcation phenomenon occurs. Calculations were made for different damping coefficient values and forms of the roll damping formula.

6

A nonlinear electromechanical pendulum arm with a nonlinear energy sink control (NES) approach

Alişverişçi G. F., Bayıroluǧlu H., Felix J. L. P., Balthazar J. M., Brasil R. M. L. R. F.

Journal of Theoretical and Applied Mechanics

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2016

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Vol. 54 nr 3

975--986

EN

This paper considers the nonlinear dynamics of an electromechanical device with a pendulum arm and a Nonlinear Energy Sink (NES) put on the point of the pendulum suspension. It is shown that the (NES) is capable of absorbing energy from the system. The numerical results are shown in a bifurcation diagram, phase plane, Poincar´e map and Lyapunov exponents.

7

Direct numerical simulation of the Taylor–Couette flow with the asymmetric end-wall boundary conditions

Tuliszka-Sznitko E., Kiełczewski K.

Archives of Mechanics

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2016

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Vol. 68, nr 5

395--418

EN

In the paper the authors present the results obtained during a direct numerical simulation of the transitional Taylor–Couette flow in closed cavity. The spectral vanishing viscosity method is used to stabilize computations for higher Reynolds numbers. The Taylor–Couette flow is widely used for studying the primary pattern formation, transitional flows and fully turbulent flows. The Taylor–Couette flow is also important from engineering point of view: the results can be interesting for engineers dealing with gas turbines and axial compressors. In the paper the attention is focused on the influence of the end-wall boundary conditions on the flow structures and on statistics (i.e. the radial profiles of the angular velocity, angular momentum, torque, the Reynolds stress tensor components). The results are discussed in the light of experimental and numerical data published in literature (F. Wendt, Ing.-Arch., 4, 1933; H. Brauckmann, B. Eckhardt, J. Fluid Mech., 718, 2013).

8

Numerical and analytical approaches for roll motion analysis in regular longitudinal waves

Peşman E., Kolukısa D. C., Taylan M.

Polish Maritime Research

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2015

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nr 3

28--35

EN

In this study numerical and analytical approaches were investigated in terms of accuracy of their results, practicality of solution and ability to reproduce the main features of the parametric roll phenomenon such as loss of stability and bifurcations in parametric roll motion analysis of ships. In general, single-degree-of-freedom analytical approach is based on reducing number of degrees of freedom from 3 to 1 by using the quasi-static Froude-Krylov assumption, incorporating heave and pitch effects by means of a time varying restoring moment. On the other hand, numerical approaches to motion of six and four degrees of freedom are based on three dimensional diffraction/radiation and potential flow theories. In summary, this paper reveals that analytical approaches are sufficiently adequate to obtain accurate practical results for this relatively complex phenomenon.

9

Bifurkacje i chaos w układach dynamicznych

Basiński R., Trzaska Z.

Elektronika : konstrukcje, technologie, zastosowania

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2008

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Vol. 49, nr 2

7-14

PL

Przedstawiono charakterystykę szczególnych procesów dynamicznych, które mogą pojawiać się w silnie nieliniowych układach o specyficznej strukturze. Zdefiniowano podstawowe kryteria pozwalające na identyfikację zjawisk chaotycznych. Duży nacisk położono na analizę drgań w obwodzie Chua'y w którym można zaobserwować pełny zestaw zachowań charakterystycznych dla dynamicznych układów nieliniowych. W pracy omówiono również możliwość występowania dynamiki chaotycznej w układzie statecznika lampy fluorescencyjnej. Wyznaczono baseny przyciągania dziwnych atraktorów. Uzyskane wyniki symulacji komputerowych poddano szczegółowej analizie.

EN

It profile in work was introduced of special dynamic processes, which can appear in factorials non-linear systems about specific structure. The basic permissive on identification of chaotic phenomena criterions were defined. It the large pressure was put on analysis the oscillations in circuit Chua's, in which one it was possible to observe the full scale of characteristic behaviours for dynamic non-linear systems. The possibility of occurrence in arrangement of stabilizer of fluorescent lamp the chaotic dynamics in work was introduced also. The pools of attraction were marked strange attractors. The detailed analysis was subjected the got results of computer simulations.

10

Bifurcations and chaos in a DC-DC buck converter

Kocewiak Ł.

Elektronika : konstrukcje, technologie, zastosowania

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2008

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Vol. 49, nr 10

105-110

EN

The dynamics of power electronic system with pulse width modulation (PWM) control is studied in this paper. Behaviour characteristic for a nonlinear dynamical system is observed and theoretically explained. The set of system parameters influenced on periodic and chaotic oscillations existence are established and presented. For periodic motions, the regularities of their origin are studied and their possible bifurcations are established. A DC-DC buck converter controlled by a voltage feedback is taken as a representative system. The studied system is described by a system of piecewise-smooth non-autonomous differential equations. The research are focused on chaotic oscillations analysis and analytical search for bifurcations dependent on parameter. The most frequent route to chaos by the period doubling is observed in the second order DC-DC buck converter. Other bifurcations as a complex behaviour in power electronic system evidence are also described. The system sensitive dependence to initial conditions variation is studied and a positive largest Lyapunov exponent as a chaos indicator calculated. Appropriate methods of analysis are applied and used to observe chaotic phenomena. In order to verify theoretical investigation the experimental DC-DC buck converter was built. The results were obtained from three sources: mathematical model numerical calculation, electrical circuit computer simulation and experimental verification from the practical buck converter circuit. A very good agreement between theory and experiment was reached.

PL

Przedmiotem badań jest układ energoelektroniczny sterowany przy użyciu modulacji szerokości impulsu (PWM - Pulse Width Modulation). Zachowanie charakterystyczne dla nieliniowych układów dynamicznych zostało zaobserwowane oraz teoretycznie wytłumaczone. Przedziały parametrów, przy których występują. okresowe i chaotyczne oscylacje w układzie zostały określone. Regularności w występowaniu zachowania okresowego zostały zaobserwowane i towarzyszące nagłym zmianom wie bifurkacje odkryte. Przekształtnik DC-DC obniżający napięcie zamknięty w pętli napięciowej jest analizowany. Rozpatrywany nieautonomiczny układ jest opisany przy użyciu układu równań różniczkowych kawałkami gładkich. Badania są skoncentrowane na badaniu oscylacji występujących w przekształtniku oraz bifurkacji występujących na skutek zmian pewnych parametrów. Chaos pojawiający się na drodze kaskady podwojenia okresu został zaobserwowany w analizowanym przekształtniku DC-DC. Inne bifurkacje jako dowód złożonych zachowań w układach energoelektronicznych również zostały zaobserwowane i opisane. Charakterystyczna dla nieliniowego układu dynamicznego w stanie chaotycznym wrażliwość na zmianę warunków początkowych została potwierdzona na drodze symulacji komputerowych. Największy wykładnik Lyapunova jako wyznacznik zachowania chaotycznego również został wyznaczony. Odpowiednie metody stosowane do analizy złożonych zachowań w układach nieliniowych zostały użyte z powodzeniem. W celu zweryfikowania analizy teoretycznej układ obniżający napięcie został zbudowany i przeanalizowany pod kątem występowania zjawisk chaotycznych. Wyniki zostały otrzymane z trzech niezależnych źródeł: analiza numeryczna modelu matematycznego przekształtnika, symulacja komputerowa układu elektrycznego i pomiary przeprowadzone w laboratorium. Bardzo dobra zgodność między teorią oraz eksperymentem laboratoryjnym została osiągnięta.

11

Analysis of continuous bifurcations of periodic solutions exhibited by three coupled rods with horizontal obstacle

Awrejcewicz J., Kudra G., Lamarque C.-H.

Vibrations in Physical Systems

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2002

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Vol. 20

92--93

12

Supression of chaotic oscillations of gas bubbles by polymer additives

Brujan E. A.

Applied Mechanics and Engineering

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2000

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Vol. 5, no 2

485-498

EN

The effect of polymer concentration on the bifurcation structure of a bubble oscillator is theoretically examined in terms of bifurcation diagrams and phase diagrams. The equation describing the driven spherical oscillations of a gas bubble in CMC aqueous solutions is obtained by using a Williamson model fluid. The numerical results reveal the suppression of chaotic oscillations of bubbles by polymer additives.

13

Stability analysis of nonlinear cellular neural networks with hysteresis in the output dynamics

Slavova A.

Applied Mechanics and Engineering

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1998

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Vol. 3, no 2

305-322

EN

In this work, hysteresis of cellular neural network is investigated. The general state equation with linear and nonlinear templates is introduced. Moreover, the output equation is own dynamics where the nonlinearity in the feedback is allowed to exibit hysteresis. Stability analysis of this cellular neural network is made. New approach for studying such dynamical systems is introduced using Lyapunov's majorizing equations.