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1

Non-linear stochastic dynamics of a cable-mass system with finite bending stiffness via an equivalent linearization technique

Weber Hanna, Kaczmarczyk Stefan, Iwankiewicz Radosław

Journal of Theoretical and Applied Mechanics

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2020

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Vol. 58 nr 2

483--497

EN

The non-linear stochastic dynamic behaviour of a high-rise vertical transportation system modelled as a concentrated mass and a cable with finite bending stiffness is considered. The slow time scale is defined and lateral cable displacements coupled with transverse motions are expanded in terms of approximating functions. The excitation of the high-rise building is assumed in the form of a narrow-band mean-square process equivalent to the harmonic process. The equivalent linearization technique is used to replace the original non-linear system with a linear approximation whose coefficients are determined from minimization of the mean-square equation difference between both systems.

2

Equivalent linearization technique in non-linear stochastic dynamics of a cable-mass system with time-varying length

Weber H., Iwankiewicz R., Kaczmarczyk S.

Archives of Mechanics

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2019

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Vol. 71, nr 4-5

393--416

EN

In this paper the transverse vibrations of a vertical cable carrying a concentrated mass at its lower end and moving slowly vertically within the host structure are considered. It is assumed that longitudinal inertia of the cable can be neglected, with the longitudinal motion of the concentrated mass coupled with the lateral motion of the cable. An expansion of the lateral displacement of a cable in terms of approximating functions is used. The excitation acting upon the cable-mass system is a base-motion excitation due to the sway motion of a host tall structure. Such a motion of a structure often results due to action of the wind, hence it may be adequately idealized as a narrow-band random process. The narrow-band process is represented as the output of a system of two linear filters to the input in a form of a Gaussian white noise process. The non-linear problem is dealt with by an equivalent linearization technique, where the original non-linear system is replaced with an equivalent linear one, whose coefficients are determined from the condition of minimization of a mean-square error between the non-linear and the linear systems. The mean value and variance of the transverse displacement of the cable as well as those of the longitudinal motion of the lumped mass are determined with the aid of an equivalent linear system and compared with the response of the original non-linear system subjected to the deterministic harmonic excitation.

3

Onsager principle for nonlinear mechanical systems modeled by stochastic dissipative equations

Battezzati M.

Archives of Mechanics

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2012

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Vol. 64, nr 2

177-177

EN

A classical mechanical system subjected to frictional forces is considered in the limit of large frictional coefficient. Random white noise is also introduced in conformity to the fluctuation-dissipation theorem. The velocity is split into a deterministic component plus a random stochastic component consequently, the evolution operator (generator) for the probability density in configuration space is evaluated recalling previous work by the same author, by stochastically averaging the flux of particles. The averages depend upon the history of the system, but memory may be eliminated by suitably defining the drift, in the limit of large time. The fundamental solution of the diffusion equation is recast into the form of a Feynman path integral, and subsequently transformed into an Onsager–Machlup path integral, whose regressive stationary solutions satisfy the minimum entropy production principle. It is focused upon the role played by the appropriate definition of drift velocity adopted in this approach, allowing for interpretation of the Onsager–Machlup potential

4

Comparison of the aluminium versus steel telecommunication towers in stochastic finite element method eigenvibrations analysis

Solecka M., Kamiński M., Szafran J.

Mechanics and Mechanical Engineering

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2011

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Vol. 15, nr 2

101-116

EN

The main aim of this paper is to make a comparison of the eigenfrequencies of the high telecommunication towers alternatively manufactured using the stainless steel and the aluminium components. It is provided each time assuming that the Young modulus of the applied materialź is the Gaussian input random variable and using the generalized stochastic perturbation method using the global version of the Response Function Method. Up to the fourth order probabilistic moments and characteristics are computed in the three dimensional Finite Element Method model of the tower composed from the continuous linear elastic edge beams spanned by the large number of the linear elastic bars. A computational part of the work is made using the hybrid usage of the computer algebra system MAPLE and the FEM engineering package ROBOT used widely in the civil engineering practice.

5

Nieliniowa dynamika indeksów giełdowych WIG20 i ATX: analiza porównawcza

Gurgul H., Suder M

Ekonomia Menedżerska

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2010

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nr 7

103-120

PL

W pracy przedstawiono wyniki badań nad identyfikacją struktur nieliniowych oraz chaosu deterministycznego występujących na giełdzie austriackiej i polskiej. Biorąc za przykład indeksy ATX oraz WIG20 z okresu I 2001-VIIl 2008 oraz wykorzystując najbardziej popularne metody statystyczne wykrywania chaosu, wykazano, że w badanych szeregach występują struktury nieliniowe. Uzyskane wyniki wskazały, że w obydwu przypadkach mamy do czynienia z układami chaotycznymi, jednak liczbę zmiennych dynamicznych (wymiar korelacyjny), potrzebnych do opisania układu dla obydwu indeksów różnią się nieznacznie. Dla ATX wynosi ona 6, a dla WIG20 7. Przeprowadzona analiza wykazała również niewielkie różnice w długości cykli nieokresowych.

EN

This paper contains results of research concentrated at identification of nonlinear structures and deterministic chaos occurring on Austrian and Polish stock market. Taking into consideration the ATX and WIG20 indices from period I 2001-VIII 2008 and using the most popular statistical methods of uncovering the existence of chaos in nonlinear structures was indicated. Our results confirmed that in both considered cases chaos systems exist, however the number of dynamic variables required to describe the system for each index vary subtly. This value equals 6 for ATX and 7 for WIG20. Conducted analysis also brought evidence of small differences in length of nonperiodic cycles.

6

Eigenvalue analysis for high telecommunication towers with lognormal stiffness by the response function method and SFEM

Kamiński M. M., Szafran J.

Computer Assisted Mechanics and Engineering Sciences

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2009

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Vol. 16, No. 3/4

279-290

EN

The main aim of this paper is to demonstrate the application of the generalized stochastic perturbation techniąue to model the lognormal random variables in structural mechanics. This is done to study probabilistic characteristics of the eigenvibrations for the high telecommunication towers with random stiffness, which are modeled as the linear elastic 3D trusses. The generalized perturbation technique based on the Taylor expansion is implemented using the Stochastic Finite Element Method in its Response Function version. The main difficulty here, in a comparison to this techniąue previous applications, is a necessity of both odd and even order terms inclusion in all the Taylor expansions. The hybrid numerical approach combines the traditional FEM advantages with the symbolic computing and its visualization power and it enables for a verification of probabilistic convergence of the entire computational procedure.

7

Random eigenvibrations of elastic structures by the response function method and the generalized stochastic perturbation technique

Kamiński M. M., Szfran J.

Archives of Civil and Mechanical Engineering

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2009

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Vol. 9, no 4

5-32

EN

This paper addresses the important question in structural analysis how to efficiently model the eigenvibrations of the spatial structures with random physical and/or geometrical parameters. The entire computational methodology is based on the traditional Finite Element Method enriched with the stochastic perturbation technique in its generalized nth order approach, while the computational implementation is performed by the use of the academic FEM software in conjunction with the symbolic algebra computer system MAPLE. Contrary to the previous straightforward solution techniques, now the response function method is applied to compute any order probabilistic moments and coefficients of the structural eigenvalues. The response function is assumed in the polynomial form, the coefficients of which are computed from the several solutions of the deterministic problem around the mean value of the given input random parameter. This method is illustrated with the stochastic eigenvibrations of the simple single degree of freedom system and small steel tower modelled as the 3D truss structure with random mass density and Young modulus. This technique may find its wide application in reliability analysis of the real existing engineering structures using the commercial Finite Element Method packages as well as the other discrete computational techniques like the Finite Difference Method at least.

PL

Artykuł ukazuje metody analizy konstrukcji pozwalające efektywnie modelować drgania własne konstrukcji przestrzennych z losowym parametrem fizycznym bądź geometrycznym. Całkowita metodologia komputerowa jest oparta na tradycyjnej Metodzie Elementów Skończonych, wzbogaconej metodą perturbacji stochastycznej i jej podejściem n-tego rzędu. Komputerowa implementacja została wykonana w programie Metody Elementów Skończonych w powiązaniu z systemem komputerowym algebry symbolicznej MAPLE. W przeciwieństwie do poprzednich rozwiązań bezpośrednich, metoda funkcji odpowiedzi jest zastosowana do obliczeń probabilistycznych momentów dowolnego rzędu i współczynników wartości własnych konstrukcji. Funkcja odpowiedzi jest przyjęta w formie wielomianowej, a współczynniki zostały wyznaczone na podstawie kilku rozwiązań zagadnienia deterministycznego w otoczeniu wartości średniej odpowiedniego parametru losowego. Metoda ta jest zilustrowana na przykładzie stochastycznych drgań własnych prostego układu z jednym stopniem swobody i małej wieży stalowej modelowanej, jako kratowa konstrukcja 3D z losową gęstością masy, a także losowym modułem Younga. Metoda może zostać szeroko zastosowana w analizach niezawodności istniejących konstrukcji inżynierskich przy użyciu komercyjnych programów MES, jak również innych dyskretnych metod obliczeniowych, np. Metody Różnic Skończonych, czy Metody Elementów Brzegowych.

8

Anisotropic gas transfer through the composite membranes

Kurchatov I. M., Laguntsov N. I., Okunev A. Y., Pisarev G. I., Tronin V. N., Uvarov V. I.

Ars Separatoria Acta

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2007

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No. 5

45-54

EN

Using of stochastic dynamics methods, the probability distribution function of molecules by their moving directions in arbitrary porous media, where free molecular flow takes place, was determined. It was shown that, in some cases, the molecules in the channel can generally move athwart the channel, while an average velocity of molecules moving along the channel can significantly decrease. The anisotropic phenomenon and the hysteresis of permeability through composite asymmetric membranes were qualitatively explained.

9

Stochastic dynamics and reliability of degrading systems

Sobczyk K.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2006

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Vol. 54, nr 1

125-137

EN

In this paper the basic methodology of the coupled response-degradation modelling of stochastic dynamical systems is presented along with the effective analysis of selected problems. First, the general formulation of the problems of stochastic dynamic s coupled with the evolution of deterioration process is given. Then same specific degrading oscillatory systems under random excitation are analyzed with a special attention on the systems with fatigue-induced stiffness degradation. Both, the general discussion and the analysis of seleeted exemplary problems indicate how the reliability of deteriorating stochastic dynamical systems can be assessed.

10

Stochastyczna dynamika belki na podłożu Pasternaka

Rystwej A.

Zeszyty Naukowe. Budownictwo / Politechnika Śląska

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2004

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z. 102

425-432

PL

W pracy rozpatruje się drgania belki nieskończenie długiej spoczywającej na podłożu Pasternaka obciążonej ciągiem sił skupionych poruszających się w jednym kierunku ze stałą prędkością. Przyjęto, że wartości sił, jak również ich rozmieszczenie na długości belki są zmiennymi losowymi. Podano rozwiązanie na kumulantę dla ogólnego i maksymalnego ugięcia belki.

EN

The paper presents an infinite beam resting on a Pasternak foundation is subject to vibration under a series of concentrated forces moving at a constant velocity. The force values and their spacing along the beam are assumed to be random variables. The solution for the n-th order cumulant of the general and maximum beam deflection is given.