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1

Nonhomogeneous nonlinear oscillator with damping: asymptotic analysis in continuous and discrete time

Rouhani Behzad Djafari, Piranfar Mohsen Rahimi

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

274--282

EN

We consider the following second order evolution equation modelling a nonlinear oscillator with damping ü(t) +𝛾 ů(t) + Au(t) = f(t), where A is a maximal monotone andα-inverse strongly monotone operator in a real Hilbert space H. With suitable assumptions on 𝛾 and f(t) we show that A-1(0) ≠ ∅, if and only if (SEE) has a bounded solution and in this case we provide approximation results for elements of A-1(0) by proving weak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A-1(0). As a discrete version of (SEE), we consider the following second order difference equation un+1-un-αn(un-un-1)+λnAun+1 ∋ f(t), where A is assumed to be only maximal monotone (possibly multivalued). By using the results in [Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411-417], we prove ergodic, weak and strong convergence theorems for the sequence un, and show that the limit is the asymptotic center of un and belongs to A−1(0). This again shows that A−1(0) ≠ ∅ if and only if un is bounded. Also these results solve an open problem raised in [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3-11], namely the study of the convergence results for the inexact inertial proximal algorithm. Our paper is motivated by the previous results in [Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465-476; Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226–235; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289-1311; Djafari Rouhani B., Khatibzadeh H., A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl., 2010, 363, 648-654; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal., 2009, 70, 4369-4376; Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411-417] and significantly improves upon the results of [Attouch H., Maingé P. E., Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 2011, 17(3), 836-857], and [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3-11].

2

The stability conditions of the cubic damping Van der Pol-Duffing oscillator using the HPM with the frequency-expansion technology

El-Dib Y. O.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 3

31--44

EN

In this paper, we perform the frequency-expansion formula for the nonlinear cubic damping van der Pol’s equation, and the nonlinear frequency is derived. Stability conditions are performed, for the first time ever, by the nonlinear frequency technology and for the nonlinear oscillator. In terms of the van der Pol’s coefficients the stability conditions have been performed. Further, the stability conditions are performed in the case of the complex damping coefficients. Moreover, the study has been extended to include the influence of a forcing van der Pol’ oscillator. Stability conditions have been derived at each resonance case. Redoing the perturbation theory for the van der Pol oscillator illustrates more of a resonance formulation such as sub-harmonic resonance and super-harmonic resonance. More approximate nonlinear dispersion relations of quartic and quintic forms in the squaring of the extended frequency are derived, respectively.

3

Impact and chaotic phenomena in nonlinear nonsmooth electrical dynamical systems

Trzaska Z.

Przegląd Elektrotechniczny

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2015

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R. 91, nr 4

77--85

EN

The article presents results of the analysis of nonlinear oscillators with nonsmooth elements and nonlinear systems with nonsmooth excitations. Main attention has been focused on highlighting specific properties of nonsmooth systems compared to their smooth counterparts. Nonsmooth transformation of the time variable and the replacement of initial issues by boundary problems have been taken as the base for the analytical method. Results of numerical simulations and computing in the form of graphs of voltage and current waveforms and attractors are presented.

PL

W artykule przedstawiono wyniki analizy nieliniowych oscylatorów z niegładkimi elementami oraz układów nieliniowych z niegładkimi wymuszeniami. Zasadnicza uwaga została skupiona na uwypukleniu specyficznych właściwości układów niegładkich w porównaniu do ich odpowiedników gładkich. Niegładką transformację zmiennej czasowej i zastąpienie zagadnień początkowych zagadnieniami brzegowymi przyjęto jako podstawę metody analitycznej. Przedstawiono wyniki symulacji numerycznych i obliczeń komputerowych w postaci wykresów przebiegów napięć i prądów oraz atraktorów.

4

Wpływ efektów nieliniowych na parametry ultradźwiękowego układu drgającego dużej mocy

Kogut P., Milewski A., Kiełbasiński M., Kluk P., Nafalski L., Krzemiński Ł., Kardyś W., Król K.

Elektronika : konstrukcje, technologie, zastosowania

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2015

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Vol. 56, nr 11

16-18

PL

W artykule przedstawiono wpływ efektów nieliniowych na parametry pracy ultradźwiękowego układu drgającego dużej mocy. W pracy zaprezentowano układ do pomiaru zależności pomiędzy napięciem zasilania amplitudą i mocą wydzielaną w przetworniku oraz impedancją. Omówiono i zaprezentowano wyniki pomiarów dla dwóch układów drgających złożonych z samego przetwornika ultradźwiękowego dużej mocy i przetwornika z dołączonym koncentratorem. W wyniku przeprowadzonego eksperymentu zaobserwowano efekt nasycenia się mechanicznego układu drgającego w rezonansie szeregowym.

EN

In the paper impact of the nonlinear effects on the high power ultrasonic vibrating system parameters was examined and shown. Authors presents the measurement stand for the ultrasonic stack nonlinear behavior measurements. The results of conducted experiment included measurements of the mechanical displacement amplitude and electrical power characteristics in the function of supplied voltage excitation amplitude. Experimental setup included two configurations of the vibrating systems: High power Langevin type transducer, and Langevin transducer connected to the mechanical amplitude booster. Measurement results have shown that mechanical displacement amplitude saturates for high voltage excitation.

5

Modelowanie i analiza nieliniowych kontaktowych mikrodrgań normalnych wymuszonych siłą harmoniczną

Grudziński K., Kostek R.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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2003

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z. 20

141-146

PL

W pracy przedstawiono wyniki symulacyjnych badań nieliniowych kontaktowych mikrodrgań normalnych, wymuszonych siłą harmoniczną w układzie dwóch ciał stałych, modelujących płaskie połączenia prowadnicowe obrabiarek. Celem badań było wyznaczenie przebiegu i parametrów drgań kontaktowych jako funkcji częstotliwości i amplitudy siły wymuszającej. Z przeprowadzonych badań wynika, że przy częstotliwościach wymuszeń f[w] ≤ f[o] w rozważanym układzie występują ciekawe efekty: ruchy drgające o dużej amplitudzie i złożonej kinematyce, piki rezonansowe, skokowe zmiany amplitud, bifurkacje typu widły, drgania chaotyczne.

EN

This work presents the results of numerical investigations of nonlinear normal contact micro-vibrations caused by a harmonic force in a system of two bodies in plane contact, that models the slideway connections of machine tools. The aim at the computational analysis was to determine plots and parameters of the focesed vibrations as a function of frequency and amplitude. From the calculation results one can see that at excitation frequencies f[w] ≤ f[o] a number of interesting phenomena can occur in the investigated system such as: high amplitude vibration with complex motion, resonance peaks, jumping changes of amplitude, bifurcations, chaotic vibrations.

6

A comparison of statistical and equivalent linearization for stochastic dynamic systems

Socha L.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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2001

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z. 15

323-328

EN

The objective of this paper is to show differences between statistical and equivalent linearization on an example of a nonlinear oscillator under Gaussian white noise external excitations and to establish assumptions under what conditions the both linearization techniques are the same. Two types of equivalency criteria are considered, namely mean-square criterion with Gaussian closure and square metric in probability density functions space.

PL

Celem niniejszego artykułu jest pokazanie różnic między statystyczną i równoważną linearyzacją na przykładzie oscylatora nieliniowego oraz ustalenie założeń, przy których obie metody są identyczne. Rozpatrywane będą dwa typy kryteriów: kryterium średnio-kwadratowe z Gaussowską techniką zamykania oraz kryterium kwadratowe w przestrzeni funkcji gęstości prawdopodobieństw.

7

On the stability "in the large'' and unsafe disturbances in a nonlinear oscillator

Szemplińska-Stupnicka W., Tyrkiel E., Zubrzycki A.

Computer Assisted Mechanics and Engineering Sciences

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2001

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Vol. 8, No. 1

155-168

EN

The problem of the stability "in the large'' and the unsafe disturbances of the equilibrium position is studied for the structures whose dynamics is governed by the equation of motion of the pendulum with parametric excitation. The system displays a variety of nonlinear and chaotic phenomena, so that the study requires the use of theoretical concepts of the mathematics of chaos. Detailed explorations are performed by the aid of the nonlinear software package Dynamics.

8

On the boundary crises of chaotic attractors

Tyrkiel E., Szemplińska-Stupnicka W.

Computer Assisted Mechanics and Engineering Sciences

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2000

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Vol. 7, No. 4

743-755

EN

In nonlinear dissipative mechanical systems, bifurcations of chaotic attractors called boundary crises appear to be the cause of most sudden changes in chaotic dynamics. They result in a sudden loss of stability of chaotic attractor, together with destruction of its basin of attraction and its disappearance from the phase portrait. Chaotic attractor is destroyed in the collision with an unstable orbit (destroyer saddle) sitting on its basin boundary, and the structure of the saddle defines the type of the crisis - regular or chaotic one. In the paper we exemplify both types of the boundary crisis by using a mathematical model of the symmetric twin-well Duffing oscillator; we consider the regular boundary crisis of the cross-well chaotic attractor, and the chaotic boundary crisis of the single-well chaotic attractor. Our numerical analysis makes use of the underlying topological structure of the phase space, namely the geometry of relevant invariant manifolds, as well as the structure of basins of attraction of the coexisting attractors. The study allows us to establish some relevant relations between the properties of the regular and chaotic boundary crisis, and to outline the differences that result mainly in the post-crisis