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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.

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Stochastic methods in mechanics: status and challenges

Iwankiewicz R., Kotulski Z.

Prace Instytutu Podstawowych Problemów Techniki PAN

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2009

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nr 1

3-72

EN

The purpose of the Conference is to summarize the recent advances, to discuss the status, the development trends and also to share views on the current and futurę challenges in the wide subject area of stochastic mechanics and, morę generally, in applied stochastics. The development of stochastic methods in mechanics and of stochastic dynamics in particular, has been achieved over the past four decades owing to the research and collective, persistent efforts of many people. However, with no doubt there can be identified a group of world leaders who madę mile stone, or corner stone, contributions to the subject area, having laid out the avenues for pursuing research and having inspired the research of many others. One of those leaders is Professor Kazimierz Sobczyk, who and whose research work are worldwide known. It happens that 2009 is the year of his seventieth birthday. This has presented a very special occasion for convening a conference aimed at taking a broader view on the status and futurę challenges of the subject area of stochastic mechanics. The present book contains the extended abstracts of 26 papers to be presented at the conference. The papers cover the rangę of up-to-date topics in applied stochastics and stochastic mechanics as well as some topics in mechanics of materials. These topics are of relevance to Professor Sobczyk areas of research and his pioneering contributions. This volume of abstracts is complemented with an essay by Professor Kazimierz Sobczyk presenting his way through stochastic mechanics. We also present a list of publications by Professor Kazimierz Sobczyk. The editors would like to thank all the authors and the attendees of the Conference for their contributions to this event.

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Equations for the Response Probability Density of Dynamic Systems under Multi-Component Non-Poisson Impulse Process Excitations

Iwankiewicz R.

Machine Dynamics Problems

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2008

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Vol. 32, No. 4

24-36

EN

The excitation considered in the present paper consists of n statistically independent random trains of impulses, each of whom is driven by a non-Poisson, renewal process with inter-arrival times being the sum of two independent negative-exponential distributed random variables with parameters vv, Vs, µs (S = 1, 2, ..., n). Each of the original impulse processes is recast into a Poisson driven impulse process with the aid of an auxiliary, purely jump stochastic variable. Each auxiliary variable is governed by the stochastic differential equation driven by two independent Poisson processes, with parameters Vs, µs, thus it is tantamount to two Markov states. The Markov chain for the whole problem is constructed by considering the coincidences of the states of the individual jump processes. The necessary so-called jump probability intensity functions are determined for all state variables and all possible jumps. The equations governing the joint probability density-distribution function of the response and of the Markov states of the auxiliary variables are derived from the general integro-differential forward Chapman-Kolmogorov equation. The resulting equations form a set of integro-partial differential equations.