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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 Newmark-beta method ： ウィキペディア英語版 Newmark-beta method The Newmark-beta method is a method of numerical integration used to solve differential equations. It is widely used in numerical evaluation of the dynamic response of structures and solids such as in finite element analysis to model dynamic systems. The method is named after Nathan M. Newmark,〔Newmark, N. M. (1959) A method of computation for structural dynamics. Journal of Engineering Mechanics, ASCE, 85 (EM3) 67-94.〕 former Professor of Civil Engineering at the University of Illinois, who developed it in 1959 for use in structural dynamics. Using Taylor's theorem with Lagrange remainder, along with the intermediate value theorem, the Newmark-''β'' method states that the first time derivative (velocity in the equation of motion) can be solved as, :$\backslash dot\_=\backslash dot\_n+\; \backslash Delta\; t~\backslash ddot\_\backslash gamma\; \backslash ,$ where :$\backslash ddot\_\backslash gamma\; =\; (1\; -\; \backslash gamma)\backslash ddot\_n\; +\; \backslash gamma\; \backslash ddot\_~~~~0\backslash leq\; \backslash gamma\; \backslash leq\; 1$ therefore :$\backslash dot\_=\backslash dot\_n\; +\; (1\; -\; \backslash gamma)\; \backslash Delta\; t~\backslash ddot\_n\; +\; \backslash gamma\; \backslash Delta\; t~\backslash ddot\_.$ Because acceleration also varies with time, however, those two theorems must also be applied to the second time derivative to obtain the correct displacement. Thus, :$u\_=u\_n\; +\; \backslash Delta\; t~\backslash dot\_n+\backslash begin\; \backslash frac\; 1\; 2\; \backslash end\; \backslash Delta\; t^2~\backslash ddot\_\backslash beta$ where again :$\backslash ddot\_\backslash beta\; =\; (1\; -\; 2\backslash beta)\backslash ddot\_n\; +\; 2\backslash beta\backslash ddot\_~~~~0\backslash leq\; 2\backslash beta\backslash leq\; 1$ Newmark showed that a reasonable value of $\backslash gamma$ is 0.5, therefore the update rules are, :$\backslash dot\_=\backslash dot\_n\; +\; \backslash begin\backslash frac\backslash end~(\backslash ddot\_n\; +\; \backslash ddot\_)$ :$u\_=u\_n\; +\; t~\backslash dot\_n\; +\; \backslash begin\backslash frac\backslash end\; \backslash Delta\; t^2\; \backslash ddot\_n\; +\; \backslash beta\; t^2\; \backslash ddot\_$ Setting ''β'' to various values between 0 and 1 can give a wide range of results. Typically ''β'' = 1/4, which yields the constant average acceleration method, is used. ==References== 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「Newmark-beta method」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.