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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 Euler's equations (rigid body dynamics) ： ウィキペディア英語版 Euler's equations (rigid body dynamics) In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with its axes fixed to the body and parallel to the body's principal axes of inertia. Their general form is: :$$ \mathbf \cdot \dot + \boldsymbol\omega \times \left( \mathbf \cdot \boldsymbol\omega \right) = \mathbf. where ''M'' is the applied torques, ''I'' is the inertia matrix, and ω is the angular velocity about the principal axes. In 3D principal orthogonal coordinates, they become: :$$ \begin I_1\dot_+(I_3-I_2)\omega_2\omega_3 &= M_\\ I_2\dot_+(I_1-I_3)\omega_3\omega_1 &= M_\\ I_3\dot_+(I_2-I_1)\omega_1\omega_2 &= M_ \end where ''Mk'' are the components of the applied torques, ''Ik'' are the principal moments of inertia and ω''k'' are the components of the angular velocity about the principal axes. ==Motivation and derivation== Starting from Euler's second law, in an inertial frame of reference (subscripted "in"), the time derivative of the angular momentum L equals the applied torque :$$ \frac}} \ \stackrel\ \frac \left( \mathbf__ \ \stackrel\ L_\mathbf_ + L_\mathbf_ + L_\mathbf_ = I_\omega_\mathbf_ + I_\omega_\mathbf_ + I_\omega_\mathbf_ where ''Mk'', ''Ik'' and ω''k'' are as above. In a ''rotating'' reference frame, the time derivative must be replaced with (see time derivative in rotating reference frame) :$$ \left(\frac\right)_\mathrm+ \boldsymbol\omega\times\mathbf=\mathbf where the subscript "rot" indicates that it is taken in the rotating reference frame. The expressions for the torque in the rotating and inertial frames are related by :$$ \mathbf_\mathbf, where Q is the rotation tensor(not rotation matrix) , an orthogonal tensor related to the angular velocity vector by :$\backslash boldsymbol\backslash omega\; \backslash times\; \backslash boldsymbol\; =\; \backslash dot^\backslash boldsymbol$ for any vector v. In general, L = I·ω is substituted and the time derivatives are taken realizing that the inertia tensor, and so also the principal moments, do not depend on time. This leads to the general vector form of Euler's equations :$$ \mathbf \cdot \dot + \boldsymbol\omega \times \left( \mathbf \cdot \boldsymbol\omega \right) = \mathbf. If principal axis rotation :$L\_\; \backslash \; \backslash stackrel\backslash \; I\_\backslash omega\_$ is substituted, and then taking the cross product and using the fact that the principal moments do not change with time, we arrive at the Euler equations in components at the beginning of the article. 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「Euler's equations (rigid body dynamics)」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.