Phenomenological Hysteresis Modeling
PKP-type hysteresis operator with wiping out property (see "Study of a play-like operator" for the definition).
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Contributions on phenomenological hysteresis modeling.
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For nearly a century, scientists have been developing hysteresis models based either on fundamental physical laws or on observations or experimental results. For specific problems, models describing hysteretic systems can be derived from an understanding of the physical laws related to the system. This process is usually extremely challenging and the resulting models are too complex to be used for practical applications. Due to many limitations, the usefulness of physical models has been limited to qualitative studies.
Engineering designers seek alternative, simpler models which, although not derived from fundamental theory, do exhibit sufficiently correct input–output features, and are useful for characterization, design and control purposes. Since these phenomenological models are utilized for generalized hysteresis modeling, they are typically referred to as hysteresis operators. They describe hysteretic relationships as mappings between input and output function spaces.
The mathematical development of hysteresis operators started with a representation by differential equations, which dates back to Duhem's model in 1897 [1]. These operators reflect the observation that hysteresis curves for physical systems are smoothly monotone except when the input changes direction. However, it was not until the 1970s that differential equation models were given a comprehensive study [2,3]. The primary focus of these studies was to describe the forced vibrations of a hysteretic system under periodic excitation. Since then, the model related to these studies has been referred to as the Bouc–Wen model and has been used in civil and mechanical engineering.
Many others followed Duhem's approach [4–6]. A notable model is the Jiles–Atherton model for ferromagnetic hysteresis [7,8], which was developed to describe the dynamic process of magnetization in terms of the energy in the magnetic materials. The Jiles–Atherton model is based on physical observations of magnetic domains and the pining of the domains' walls in ferromagnetic materials. The model is used to describe the stress effect on the magnetization process [9–11] and is useful primarily for simulations of dynamic magnetization curves of ferromagnetic materials. Even in the early 1900s, scientists observed the dependency of the output on the previous evolution of hysteresis input. For example, Madelung's rules for magnetic hysteresis were developed in 1905 [12]. The output-dependency effect is referred to as the memory effect of hysteresis operators. Differential operator models have local memory; however, they have a shortcoming in that their trajectories do not depend on previous extrema, as is observed in ferromagnetic hysteresis.
In the 1930s F. Preisach developed a hysteresis operator (referred to as the Preisach operator) based on hypotheses corresponding to physical mechanisms of magnetization [13]. The model retains much of the intuition associated with physical models of magnetic hysteresis and was initially regarded as a physical model of hysteresis. In the 1950s, D.H. Everett independently developed the same operator for the absorption of hysteresis, thereby providing applications of the Preisach operator beyond magnetization. In 1970, M. Krasnoselskii developed a purely mathematical formulation of the Preisach operator using relay operators and, later, a systematic analysis of the mathematical properties of these operators was conducted. This study reveals the phenomenological nature of the Preisach operator. In the 1980s, several other mathematicians also began to study the generalization of Preisach operators. Operators of Preisach type are thoroughly described in [12]. In recent decades, the Preisach operator has been successfully applied for modeling in other areas: ferromagnetic hysteresis, shape memory alloy, piezoelasticity, magnetostriction, plasticity–elasticity and soil hydrology. Broad applications of the Preisach model demonstrate its generality as a hysteresis operator. Today the Preisach operator is considered one of the best phenomenological hysteresis models. See Mayergoyz's monograph [14] for further information.
References:
[1] P. Duhem, ``Die dauernden aenderungen und die thermodynamik," I. Z. Phys. Chem., vol. 22, pp. 543–589, 1897.
[2] R. Bouc, ``Modele & Mathematique d`hysteresis," Acustica, vol. 4, pp. 16–-25, 1971.
[3] Y.K. Wen, ``Method for random vibration of hysteretic systems," Journal of the Engineering Mechanics Division, vol. 102(EM2), pp. 246-–263, 1976.
[4] P.R. Dahl, ``A magnetic hysteresis model," Guidance and control 1992; Proceedings of the 15th Annual AAS Rocky Mountain Conference, Keystone, CO, pp. 119--134, 1992.
[5] B.D. Coleman and M.L. Hodgdon, ``A Constitutive Relation for Rate-Independent Hysteresis in Ferromagnetically Soft Materials," Int. J. Engng. Sei., vol. 24, no. 6, pp. 897-–919, 1986.
[6] M.L. Hodgdon. ``Mathematical theory and calculations of magnetic hysteresis curves," IEEE Trans. on Magnetics, vol. 24, no. 6, pp. 3120--3122, 1988.
[7] D.C. Jiles and D.L. Atherton, ``Theory of magnetization process in ferromagnets and its applications to the magnetomechanical effect," J. Phys. D.: Appl. Phys., vol. 17, pp. 1265--1281, 1984.
[8] D.C. Jiles and D.L. Atherton, ``Ferromagnetic hysteresis a", IEEE Trans. on Magnetics, vol. 19, pp. 2183--2185, 1983.
[9] D.C. Jiles and S. Hariharan, ``Interpretation of the magnetization mechanism in Terfenol-D using Barkhausen pulse-height analysis and irreversible magnetostriction," J of Appl. Phys., vol. 67, pp. 5013--5015, 1990.
[10] I.M. Markar and D.L. Atherton, ``Effect of the stress of the magnetostriction of 2% Mn pipeline steel," IEEE Trans. on Magnetics, vol. 30, pp. 1383--1393, 1994.
[11] P. Garikepati, T.T. Chang, and D.C. Jiles, ``Theory of ferromagnetic hysteresis: Evaluation of stress from hysteresis curve," IEEE Trans. on Magnetics, vol. 24, pp. 2922--2924, 1988.
[12] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Applied Mathematical Sciences, New York, NY: Springer Verlag, 1996.
[13] F. Preisach, ``Ber die magnetische nachwirkung," Zeitschrift fur Fhyszk, vol. 94, pp.277–-302, 1935.
[14] I.D. Mayergoyz, Mathematical Models of Hysteresis, New York, NY: Springer Verlag, 1991.
Engineering designers seek alternative, simpler models which, although not derived from fundamental theory, do exhibit sufficiently correct input–output features, and are useful for characterization, design and control purposes. Since these phenomenological models are utilized for generalized hysteresis modeling, they are typically referred to as hysteresis operators. They describe hysteretic relationships as mappings between input and output function spaces.
The mathematical development of hysteresis operators started with a representation by differential equations, which dates back to Duhem's model in 1897 [1]. These operators reflect the observation that hysteresis curves for physical systems are smoothly monotone except when the input changes direction. However, it was not until the 1970s that differential equation models were given a comprehensive study [2,3]. The primary focus of these studies was to describe the forced vibrations of a hysteretic system under periodic excitation. Since then, the model related to these studies has been referred to as the Bouc–Wen model and has been used in civil and mechanical engineering.
Many others followed Duhem's approach [4–6]. A notable model is the Jiles–Atherton model for ferromagnetic hysteresis [7,8], which was developed to describe the dynamic process of magnetization in terms of the energy in the magnetic materials. The Jiles–Atherton model is based on physical observations of magnetic domains and the pining of the domains' walls in ferromagnetic materials. The model is used to describe the stress effect on the magnetization process [9–11] and is useful primarily for simulations of dynamic magnetization curves of ferromagnetic materials. Even in the early 1900s, scientists observed the dependency of the output on the previous evolution of hysteresis input. For example, Madelung's rules for magnetic hysteresis were developed in 1905 [12]. The output-dependency effect is referred to as the memory effect of hysteresis operators. Differential operator models have local memory; however, they have a shortcoming in that their trajectories do not depend on previous extrema, as is observed in ferromagnetic hysteresis.
In the 1930s F. Preisach developed a hysteresis operator (referred to as the Preisach operator) based on hypotheses corresponding to physical mechanisms of magnetization [13]. The model retains much of the intuition associated with physical models of magnetic hysteresis and was initially regarded as a physical model of hysteresis. In the 1950s, D.H. Everett independently developed the same operator for the absorption of hysteresis, thereby providing applications of the Preisach operator beyond magnetization. In 1970, M. Krasnoselskii developed a purely mathematical formulation of the Preisach operator using relay operators and, later, a systematic analysis of the mathematical properties of these operators was conducted. This study reveals the phenomenological nature of the Preisach operator. In the 1980s, several other mathematicians also began to study the generalization of Preisach operators. Operators of Preisach type are thoroughly described in [12]. In recent decades, the Preisach operator has been successfully applied for modeling in other areas: ferromagnetic hysteresis, shape memory alloy, piezoelasticity, magnetostriction, plasticity–elasticity and soil hydrology. Broad applications of the Preisach model demonstrate its generality as a hysteresis operator. Today the Preisach operator is considered one of the best phenomenological hysteresis models. See Mayergoyz's monograph [14] for further information.
References:
[1] P. Duhem, ``Die dauernden aenderungen und die thermodynamik," I. Z. Phys. Chem., vol. 22, pp. 543–589, 1897.
[2] R. Bouc, ``Modele & Mathematique d`hysteresis," Acustica, vol. 4, pp. 16–-25, 1971.
[3] Y.K. Wen, ``Method for random vibration of hysteretic systems," Journal of the Engineering Mechanics Division, vol. 102(EM2), pp. 246-–263, 1976.
[4] P.R. Dahl, ``A magnetic hysteresis model," Guidance and control 1992; Proceedings of the 15th Annual AAS Rocky Mountain Conference, Keystone, CO, pp. 119--134, 1992.
[5] B.D. Coleman and M.L. Hodgdon, ``A Constitutive Relation for Rate-Independent Hysteresis in Ferromagnetically Soft Materials," Int. J. Engng. Sei., vol. 24, no. 6, pp. 897-–919, 1986.
[6] M.L. Hodgdon. ``Mathematical theory and calculations of magnetic hysteresis curves," IEEE Trans. on Magnetics, vol. 24, no. 6, pp. 3120--3122, 1988.
[7] D.C. Jiles and D.L. Atherton, ``Theory of magnetization process in ferromagnets and its applications to the magnetomechanical effect," J. Phys. D.: Appl. Phys., vol. 17, pp. 1265--1281, 1984.
[8] D.C. Jiles and D.L. Atherton, ``Ferromagnetic hysteresis a", IEEE Trans. on Magnetics, vol. 19, pp. 2183--2185, 1983.
[9] D.C. Jiles and S. Hariharan, ``Interpretation of the magnetization mechanism in Terfenol-D using Barkhausen pulse-height analysis and irreversible magnetostriction," J of Appl. Phys., vol. 67, pp. 5013--5015, 1990.
[10] I.M. Markar and D.L. Atherton, ``Effect of the stress of the magnetostriction of 2% Mn pipeline steel," IEEE Trans. on Magnetics, vol. 30, pp. 1383--1393, 1994.
[11] P. Garikepati, T.T. Chang, and D.C. Jiles, ``Theory of ferromagnetic hysteresis: Evaluation of stress from hysteresis curve," IEEE Trans. on Magnetics, vol. 24, pp. 2922--2924, 1988.
[12] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Applied Mathematical Sciences, New York, NY: Springer Verlag, 1996.
[13] F. Preisach, ``Ber die magnetische nachwirkung," Zeitschrift fur Fhyszk, vol. 94, pp.277–-302, 1935.
[14] I.D. Mayergoyz, Mathematical Models of Hysteresis, New York, NY: Springer Verlag, 1991.