Abstract
Our first approach to the phenomenon of hysteresis is a direct one. We consider the hysteresis diagrams and loops as they present themselves, without trying to understand how their respective forms might follow from general physical principles such as, for example, the universal balance laws for mass, linear momentum and internal energy1; instead, we assume the form of the hysteresis loops as given, and we study them from a purely mathematical point of view. The central notion will be that of a hysteresis operator, usually denoted by W. Defined in accordance with the rules and functions that accompany a given hysteresis model, a hysteresis operator W maps input functions v = v(t) into output functions w = w(t) (we denote the independent variable by t since it will always represent a time variable). Using this formulation, we will examine the structures and the resulting memory effects of various kinds of hysteresis models. In addition, we establish the relevant connections between the different types of hysteresis operators and basic notions of calculus such as continuity and function spaces.
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References
We also refer to the recent collection of lecture notes edited by Visintin (1994b).
In the literature, this operator is often referred to as the Ishlinskii operator or the Ishlinshii transducer, see Krasnoselskii-Pokrovskii (1989). We prefer to name it after Prandtl whose paper appeared more than a decade before Ishlinskii’s who studied this operator in the early forties.
For an account of Preisach’s life, see Vajda-Della Torre (1995).
See Krasnoselskii-Pokrovskii (1989), Section 34.2.
See Krasnoselskii-Pokrovskii (1989), Section 2.
See Krasnoselskii-Pokrovskii (1989), p. 158, in their discussion of vector hysteresis models. Actually, this has only recently been used in the analysis of scalar hysteresis operators.
The estimate (3.21) constitutes the first result published by the group around Krasnoselskii on hysteresis operators, see Krasnoselskii et al. (1970).
See Krasnoselskii-Pokrovskii (1989), p. 365.
See Brokate (1989a), Proposition 2.6, and Krejčí-Lovicar (1990), Lemma 2.
This result is due to Visintin (1994a).
See Hönig (1975) for an exposition of some basic properties of regulated functions.
See Krasnoselskii-Pokrovskii (1989), Section 6.6.
See also Section 2.8 below.
See Krejčí (1989), Brokate (1989a).
Cf. (4.23).
See Brokate-Visintin (1989).
The generalized play operator serves as the basic hysteresis model in Krasnoselskii-Pokrovskii (1989), where an extensive discussion under minimal assumptions can be found. The connection with the Preisach model has been noted in Krasnoselskii-Pokrovskii (1976).
See the treatments in Visintin (1994a) and Alt (1985).
Accordingly, such hysteresis models are often referred to as independent domain models, see Everett et al. (1952, 1954, 1955).
See e.g. Woodward-Delia Torre (1960).
See, for example, Section II.8 in Mayergoyz (1991).
See also Brokate et al. (1995c).
For a different approach, see also Rychlik (1987, 1992, 1993).
See Matsuishi-Endo (1968).
See Murakami (1992), McDowell-Ellis (1993), Dreßler et al. (1994).
After Wöhler (1870).
Cf. Basquin (1910).
See Palmgren (1924) and Miner (1945).
See Krüger et al. (1985).
See Krejčí (1991a).
See also Iwan (1966, 1967).
See Masing (1926).
This function has been introduced in Everett (1955).
This feature is often referred to as the congruency property of the Preisach model, see Mayergoyz (1985).
See Remark 2.4.15.
This result is due to Mayergoyz (1985), see also Brokate (1990a).
For treatments of the aspects of numerical approximation and identification, we refer the interested reader to the works of Verdi-Visintin (1985,1989), Hoffmann-Sprekels-Visintin (1988), Hoffmann-Meyer (1989), Brokate (1990c), Mayergoyz (1991), Hütter (1991) and Verdi (1994).
See Mayergoyz-Priedman (1988) and Mayergoyz (1991).
The original contribution in this connection is Krejčí (1991a), see Remark 2.10.14.
For the strictly monotone case, part (ii) of the proposition is due to Krejčí (1991a).
See Proposition 2.2.16.
See Remark 2.4.15.
See Krejčí (1991a).
This result is originally due to Krejčí (1986a).
See Delia Torre (1966). For recent refinements, we refer to Vajda et al. (1992) and Vajda-Della Torre (1993a, 1993b).
This result is due to Krejčí (1991a) and Brokate (1992a).
See Brokate-Visintin (1989).
See Brokate-Visintin (1989).
The results of this section are due to Brokate-Dreßler-Krejčí (1995b).
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© 1996 Springer-Verlag New York, Inc.
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Brokate, M., Sprekels, J. (1996). Hysteresis Operators. In: Hysteresis and Phase Transitions. Applied Mathematical Sciences, vol 121. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4048-8_3
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DOI: https://doi.org/10.1007/978-1-4612-4048-8_3
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