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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bouc–Wen model of hysteresis ： ウィキペディア英語版 Bouc–Wen model of hysteresis In structural engineering, the Bouc–Wen model of hysteresis is used to describe non-linear hysteretic systems. It was introduced by Bouc and extended by Wen, who demonstrated its versatility by producing a variety of hysteretic patterns. This model is able to capture, in analytical form, a range of hysteretic cycle shapes matching the behaviour of a wide class of hysteretical systems. Due to its versatility and mathematical tractability, the Bouc–Wen model has gained popularity. It has been extended and applied to a wide variety of engineering problems, including multi-degree-of-freedom (MDOF) systems, buildings, frames, bidirectional and torsional response of hysteretic systems, two- and three-dimensional continua, soil liquefaction and base isolation systems. The Bouc–Wen model, its variants and extensions have been used in structural control—in particular, in the modeling of behaviour of magneto-rheological dampers, base-isolation devices for buildings and other kinds of damping devices. It has also been used in the modelling and analysis of structures built of reinforced concrete, steel, masonry, and timber. ==Model formulation== Consider the equation of motion of a single-degree-of-freedom (sdof) system: here, $\backslash textstyle\; m$ represents the mass, $\backslash textstyle\; u(t)$ is the displacement, $\backslash textstyle\; c$ the linear viscous damping coefficient, $\backslash textstyle\; F(t)$ the restoring force and $\backslash textstyle\; f(t)$ the excitation force while the overdot denotes the derivative with respect to time. According to the Bouc–Wen model, the restoring force is expressed as: where $\backslash textstyle\; a:=\backslash frac$ is the ratio of post-yield $\backslash textstyle\; k\_f$ to pre-yield (elastic) $\backslash textstyle\; k\_i:=\backslash frac$ stiffness, $\backslash textstyle\; F\_y$ is the yield force, $\backslash textstyle\; u\_y$ the yield displacement, and $\backslash textstyle\; z(t)$ a non-observable hysteretic parameter (usually called the ''hysteretic displacement'') that obeys the following nonlinear differential equation with zero initial condition ($\backslash textstyle\; z(0)\; =\; 0$), and that has dimensions of length: or simply as: where $\backslash textstyle\; \backslash operatorname$ denotes the signum function, and $\backslash textstyle\; A$, $\backslash textstyle\; \backslash beta>0$, $\backslash textstyle\; \backslash gamma$ and $\backslash textstyle\; n$ are dimensionless quantities controlling the behaviour of the model ($\backslash textstyle\; n=\backslash infty$ retrieves the elastoplastic hysteresis). Take into account that in the original paper of Wen (1976),〔 $\backslash textstyle\; \backslash beta$ is called $\backslash textstyle\; \backslash alpha$, and $\backslash textstyle\; \backslash gamma$ is called $\backslash textstyle\; \backslash beta$. Nowadays the notation varies from paper to paper and very often the places of $\backslash textstyle\; \backslash beta$ and $\backslash textstyle\; \backslash gamma$ are exchanged. Here the notation used by Ref.〔Song J. and Der Kiureghian A. (2006) Generalized Bouc–Wen model for highly asymmetric hysteresis. Journal of Engineering Mechanics. ASCE. Vol 132, No. 6 pp. 610-618〕 is implemented. The restoring force $\backslash textstyle\; F(t)$ can be decomposed into an elastic and a hysteretic part as follows: and therefore, the restoring force can be visualized as two springs connected in parallel. For small values of the positive exponential parameter $\backslash textstyle\; n$ the transition from elastic to the post-elastic branch is smooth, while for large values that transition is abrupt. Parameters $\backslash textstyle\; A$, $\backslash textstyle\; \backslash beta$ and $\backslash textstyle\; \backslash gamma$ control the size and shape of the hysteretic loop. It has been found〔Ma F., Zhang H., Bockstedte A., Foliente G.C. and Paevere P. (2004). Parameter analysis of the differential model of hysteresis. Journal of applied mechanics ASME, 71, pp. 342–349〕 that the parameters of the Bouc–Wen model are functionally redundant. Removing this redundancy is best achieved by setting $\backslash textstyle\; A=1$. Wen〔 assumed integer values for $\backslash textstyle\; n$; however, all real positive values of $\backslash textstyle\; n$ are admissible. The parameter $\backslash textstyle\; \backslash beta$ is positive by assumption, while the admissible values for $\backslash textstyle\; \backslash gamma$, that is $\backslash textstyle\; \backslash gamma:=()$, can be derived from a thermodynamical analysis (Baber and Wen (1981)〔Baber T.T. and Wen Y.K. (1981). Random vibrations of hysteretic degrading systems. Journal of Engineering Mechanics. ASCE. 107(EM6), pp. 1069–1089〕). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bouc–Wen model of hysteresis」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.