Transient and forced oscillations of systems with constant hysteretic damping
type: Item
instance of: scholarly article
title: Transient and forced oscillations of systems with constant hysteretic damping
zbMATH Open document ID: 0588.70016
DOI: 10.1016/0093-6413(85)90057-6
author: B. George
published in: Mechanics Research Communications
publication date: 1985-01-01T00:00:00Z
review text: The title problem is investigated in connection with a linear model with one degree of freedom. The hysteretic damping is modelled with a force N(t) whose Fourier-transform is: \(\bar N(\omega)=k(1+i\eta sgn \omega)\), where k is the rigidity of a spring connected parallel with the massless member incorporating the dissipative effect, \(\eta\) is the loss factor considered as independent from the circular frequency \(\omega\), i is the imaginary unit. According to the authors, \textit{S. H. Crandall} published [(*) in: S. Lees (ed.), Air, space and instruments, Draper Anniversary Volume, McGraw-Hill (1962) on p. 183] calculations for the impulse response function of this model; he used asymptotic methods, but a closed form result was not given. The authors, utilyzing the residuum-theorem and using a suitable curve on the complex plane, get the result in a closed form. In this solution a non-elementary residuum integral occurs which vanishes exponentially as a function of time. An upper bound for this integral is calculated as function of \(\eta\) ; for sufficiently small \(\eta\) this bound is approximately \(\eta/4.\) The authors state that \textit{E. G. Goloskokow} and \textit{A. P. Filippov} [Unstationary oscillations of mechanical systems (1966; Zbl 0173.350)] started to solve this problem, but solve, in the end, another one. Difficulties of physical interpretability of this model - treated already by several authors - are emphasized by the present authors too, and illustrating this point they compute initial conditions given by this method and deviating from the known ones. For \(t<0\)- Crandall (*) realized - this calculation does not give an identically zero response violating in this way the causality principle. This response is shown on a diagram for two \(\eta\) values. Crandall approximated the above model by a viscous one in two different ways. The first approximation demands equality of the hysteretic and the viscous force by a circular frequency which …
Mathematics Subject Classification ID: 70J30
Mathematics Subject Classification ID: 70-08
zbMATH DE Number: 3945401
zbMATH Keywords: Fourier integral representation
zbMATH Keywords: transient free vibration
zbMATH Keywords: contour integration
zbMATH Keywords: non-causal behaviour
zbMATH Keywords: viscous approximation
zbMATH Keywords: asymptotic solution for small loss-factors
zbMATH Keywords: structural damping response
zbMATH Keywords: damping properties of a vibrating single degree of freedom system
zbMATH Keywords: hysteretic damping
zbMATH Keywords: impulse response function
zbMATH Keywords: eigen circular frequency
reviewed by: Á. P. Bosznay
MaRDI profile type: Publication
full work available at URL: 0093-6413(85)90057-6
OpenAlex ID: W2010237843
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description: scientific article; zbMATH DE number 3945401