Transient and forced oscillations of systems with constant hysteretic damping
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 is equal to the eigen circular frequency with vanishing damping; in this way the ''equivalent'' damping coefficient can be obtained. The second approximation enables determination of the rigidity and damping coefficient of the viscous member by demanding the same poles in the complex responses of both models. This paper gives a comparison of the impulse response functions stipulated with the hysteretic damping and of the above mentioned two approximations for two \(\eta\) values.
asymptotic solution for small loss-factors
damping properties of a vibrating single degree of freedom system
scientific article; zbMATH DE number 3945401