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

Hanna Weber*, Radoslaw Iwankiewicz, Stefan Kaczmarczyk

*Corresponding author for this work

Research output: Contribution to JournalArticle

Abstract

In this paper the transverse vibrations of a vertical cable carrying at its lower end a concentrated mass and moving slowly vertically within the host structure are considered. It is assumed that longitudinal inertia of a 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 of vibrations of a cable-mass system is a base-motion excitation due to the sway motion of a host tall structure. Such a motion of a structure is assumed to result from 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 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 system. The mean value and variance of the transverse displacement of the cable as well as those of a 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.
Original languageEnglish
Pages (from-to)393-416
JournalArchives of Mechanics
Volume71
Issue number4-5
DOIs
Publication statusPublished - 25 Oct 2019

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Linearization
Cables
Linear systems
Nonlinear systems
White noise
Random processes
Mean square error

Keywords

  • cable-mass system
  • stochastic dynamics
  • equivalent linearization technique

Cite this

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title = "Equivalent linearization technique in non-linear stochastic dynamics of a cable-mass system with time-varying length",
abstract = "In this paper the transverse vibrations of a vertical cable carrying at its lower end a concentrated mass and moving slowly vertically within the host structure are considered. It is assumed that longitudinal inertia of a 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 of vibrations of a cable-mass system is a base-motion excitation due to the sway motion of a host tall structure. Such a motion of a structure is assumed to result from 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 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 system. The mean value and variance of the transverse displacement of the cable as well as those of a 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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author = "Hanna Weber and Radoslaw Iwankiewicz and Stefan Kaczmarczyk",
year = "2019",
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language = "English",
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}

Equivalent linearization technique in non-linear stochastic dynamics of a cable-mass system with time-varying length. / Weber, Hanna; Iwankiewicz, Radoslaw; Kaczmarczyk, Stefan.

In: Archives of Mechanics, Vol. 71, No. 4-5, 25.10.2019, p. 393-416.

Research output: Contribution to JournalArticle

TY - JOUR

T1 - Equivalent linearization technique in non-linear stochastic dynamics of a cable-mass system with time-varying length

AU - Weber, Hanna

AU - Iwankiewicz, Radoslaw

AU - Kaczmarczyk, Stefan

PY - 2019/10/25

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N2 - In this paper the transverse vibrations of a vertical cable carrying at its lower end a concentrated mass and moving slowly vertically within the host structure are considered. It is assumed that longitudinal inertia of a 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 of vibrations of a cable-mass system is a base-motion excitation due to the sway motion of a host tall structure. Such a motion of a structure is assumed to result from 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 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 system. The mean value and variance of the transverse displacement of the cable as well as those of a 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.

AB - In this paper the transverse vibrations of a vertical cable carrying at its lower end a concentrated mass and moving slowly vertically within the host structure are considered. It is assumed that longitudinal inertia of a 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 of vibrations of a cable-mass system is a base-motion excitation due to the sway motion of a host tall structure. Such a motion of a structure is assumed to result from 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 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 system. The mean value and variance of the transverse displacement of the cable as well as those of a 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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