网站换程序 搜索引擎,企业网站托管外包方式,公司网页图片,2022拉新推广赚钱的app本文仅供学习使用#xff0c;总结很多本现有讲述运动学或动力学书籍后的总结#xff0c;从矢量的角度进行分析#xff0c;方法比较传统#xff0c;但更易理解#xff0c;并且现有的看似抽象方法#xff0c;两者本质上并无不同。 2024年底本人学位论文发表后方可摘抄 若有… 本文仅供学习使用总结很多本现有讲述运动学或动力学书籍后的总结从矢量的角度进行分析方法比较传统但更易理解并且现有的看似抽象方法两者本质上并无不同。 2024年底本人学位论文发表后方可摘抄 若有帮助请引用 本文参考 黎 旭,陈 强 洪,甄 文 强 等.惯 性 张 量 平 移 和 旋 转 复 合 变 换 的 一 般 形 式 及 其 应 用[J].工 程 数 学 学 报,2022,39(06):1005-1011. 食用方法 质量点的动量与角动量 刚体的动量与角动量——力与力矩的关系 惯性矩阵的表达与推导——在刚体运动过程中的作用 惯性矩阵在不同坐标系下的表达 务必自己推导全部公式并理解每个符号的含义 机构运动学与动力学分析与建模 Ch00-2质量刚体的在坐标系下运动Part2 2.2.3 欧拉方程 Euler equation 2.2.3 欧拉方程 Euler equation
对式 H ⃗ Σ M / O F \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F} H ΣM/OF进一步分析有 H ⃗ Σ M / O F ∫ R ⃗ O P i F × ( d m i ⋅ d R ⃗ P i F d t ) ∫ ( ( R ⃗ P i F − R ⃗ O F ) × V ⃗ P i F ) d m i ∫ ( R ⃗ P i F × V ⃗ P i F ) d m i − ∫ ( R ⃗ O F × V ⃗ P i F ) d m i H ⃗ Σ M F − R ⃗ O F × P ⃗ G F \begin{split} \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F}\int{\vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F}\times \left( \mathrm{d}m_i\cdot \frac{\mathrm{d}\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F}}{\mathrm{d}t} \right)}\int{\left( \left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F}-\vec{R}_{\mathrm{O}}^{F} \right) \times \vec{V}_{\mathrm{P}_{\mathrm{i}}}^{F} \right) \mathrm{d}m_i} \\ \int{\left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F}\times \vec{V}_{\mathrm{P}_{\mathrm{i}}}^{F} \right) \mathrm{d}m_i}-\int{\left( \vec{R}_{\mathrm{O}}^{F}\times \vec{V}_{\mathrm{P}_{\mathrm{i}}}^{F} \right) \mathrm{d}m_i} \\ \vec{H}_{\Sigma _{\mathrm{M}}}^{F}-\vec{R}_{\mathrm{O}}^{F}\times \vec{P}_{\mathrm{G}}^{F} \end{split} H ΣM/OF∫R OPiF×(dmi⋅dtdR PiF)∫((R PiF−R OF)×V PiF)dmi∫(R PiF×V PiF)dmi−∫(R OF×V PiF)dmiH ΣMF−R OF×P GF 对上式进一步求导则有 d H ⃗ Σ M / O F d t d H ⃗ Σ M F d t − d ( R ⃗ O F × P ⃗ G F ) d t d H ⃗ Σ M F d t − V ⃗ O F × P ⃗ G F − m t o t a l ⋅ R ⃗ O F × a ⃗ G F \frac{\mathrm{d}\vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F}}{\mathrm{d}t}\frac{\mathrm{d}\vec{H}_{\Sigma _{\mathrm{M}}}^{F}}{\mathrm{d}t}-\frac{\mathrm{d}\left( \vec{R}_{\mathrm{O}}^{F}\times \vec{P}_{\mathrm{G}}^{F} \right)}{\mathrm{d}t}\frac{\mathrm{d}\vec{H}_{\Sigma _{\mathrm{M}}}^{F}}{\mathrm{d}t}-\vec{V}_{\mathrm{O}}^{F}\times \vec{P}_{\mathrm{G}}^{F}-m_{\mathrm{total}}\cdot \vec{R}_{\mathrm{O}}^{F}\times \vec{a}_{\mathrm{G}}^{F} dtdH ΣM/OFdtdH ΣMF−dtd(R OF×P GF)dtdH ΣMF−V OF×P GF−mtotal⋅R OF×a GF 其中 H ⃗ Σ M F ∫ R ⃗ P i F × p ⃗ P i F ∫ ( R ⃗ G F R ⃗ G P i F ) × ( d m i ⋅ ( V ⃗ G F V ⃗ G P i F ) ) ∫ R ⃗ G F × V ⃗ G F d m i ⏟ m t o t a l ⋅ R ⃗ G F × V ⃗ G F ∫ R ⃗ G F × V ⃗ G P i F d m i ⏟ 0 ∫ R ⃗ G P i F × V ⃗ G F d m i ⏟ 0 ∫ R ⃗ G P i F × V ⃗ G P i F d m i ⏟ ∫ R ⃗ G P i F × ( ω ⃗ M F × R ⃗ G P i F ) d m i m t o t a l ⋅ R ⃗ G F × V ⃗ G F ∫ R ⃗ G P i F × ( ω ⃗ M F × R ⃗ G P i F ) d m i m t o t a l ⋅ R ⃗ G F × V ⃗ G F ∫ ( R ⃗ G P i F ⋅ R ⃗ G P i F ) ω ⃗ M F d m i − ∫ ( R ⃗ G P i F ⋅ ω ⃗ M F ) R ⃗ G P i F d m i \begin{split} \vec{H}_{\Sigma _{\mathrm{M}}}^{F}\int{\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F}\times \vec{p}_{\mathrm{P}_{\mathrm{i}}}^{F}}\int{\left( \vec{R}_{\mathrm{G}}^{F}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \times \left( \mathrm{d}m_i\cdot \left( \vec{V}_{\mathrm{G}}^{F}\vec{V}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \right)} \\ \begin{array}{c} \underbrace{\int{\vec{R}_{\mathrm{G}}^{F}\times \vec{V}_{\mathrm{G}}^{F}}\mathrm{d}m_i}\\ m_{\mathrm{total}}\cdot \vec{R}_{\mathrm{G}}^{F}\times \vec{V}_{\mathrm{G}}^{F}\\ \end{array}\begin{array}{c} \underbrace{\int{\vec{R}_{\mathrm{G}}^{F}\times \vec{V}_{\mathrm{GP}_{\mathrm{i}}}^{F}}\mathrm{d}m_i}\\ 0\\ \end{array}\begin{array}{c} \underbrace{\int{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\times \vec{V}_{\mathrm{G}}^{F}}\mathrm{d}m_i}\\ 0\\ \end{array}\begin{array}{c} \underbrace{\int{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\times \vec{V}_{\mathrm{GP}_{\mathrm{i}}}^{F}}\mathrm{d}m_i}\\ \int{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\times \left( \vec{\omega}_{\mathrm{M}}^{F}\times \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right)}\mathrm{d}m_i\\ \end{array} \\ m_{\mathrm{total}}\cdot \vec{R}_{\mathrm{G}}^{F}\times \vec{V}_{\mathrm{G}}^{F}\int{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\times \left( \vec{\omega}_{\mathrm{M}}^{F}\times \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right)}\mathrm{d}m_i \\ m_{\mathrm{total}}\cdot \vec{R}_{\mathrm{G}}^{F}\times \vec{V}_{\mathrm{G}}^{F}\int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \vec{\omega}_{\mathrm{M}}^{F}}\mathrm{d}m_i-\int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}\mathrm{d}m_i \end{split} H ΣMF∫R PiF×p PiF∫(R GFR GPiF)×(dmi⋅(V GFV GPiF)) ∫R GF×V GFdmimtotal⋅R GF×V GF ∫R GF×V GPiFdmi0 ∫R GPiF×V GFdmi0 ∫R GPiF×V GPiFdmi∫R GPiF×(ω MF×R GPiF)dmimtotal⋅R GF×V GF∫R GPiF×(ω MF×R GPiF)dmimtotal⋅R GF×V GF∫(R GPiF⋅R GPiF)ω MFdmi−∫(R GPiF⋅ω MF)R GPiFdmi 将 H ⃗ Σ M F \vec{H}_{\Sigma _{\mathrm{M}}}^{F} H ΣMF进一步求导则有 d H ⃗ Σ M F d t { R ⃗ G F × m t o t a l ⋅ a ⃗ G F 2 ∫ ( V ⃗ P i F ⋅ R ⃗ G P i F ) ω ⃗ M F d m i ∫ ( R ⃗ G P i F ⋅ R ⃗ G P i F ) α ⃗ M F d m i − ∫ ( V ⃗ G P i F ⋅ ω ⃗ M F ) R ⃗ G P i F d m i − ∫ ( R ⃗ G P i F ⋅ α ⃗ M F ) R ⃗ G P i F d m i − ∫ ( R ⃗ G P i F ⋅ ω ⃗ M F ) V ⃗ G P i F d m i { R ⃗ G F × m t o t a l ⋅ a ⃗ G F ( ∫ ( R ⃗ G P i F ⋅ R ⃗ G P i F ) α ⃗ M F d m i − ∫ ( R ⃗ G P i F ⋅ α ⃗ M F ) R ⃗ G P i F d m i ) − ∫ ( R ⃗ G P i F ⋅ ω ⃗ M F ) ( ω ⃗ M F × R ⃗ G P i F ) d m i { R ⃗ G F × m t o t a l ⋅ a ⃗ G F ( ∫ ( R ⃗ G P i F T R ⃗ G P i F ) ⋅ E 3 × 3 α ⃗ M F d m i − ∫ ( R ⃗ G P i F T α ⃗ M F ) R ⃗ G P i F d m i ) − ∫ ( R ⃗ G P i F T ω ⃗ M F ) ( ω ⃗ M F × R ⃗ G P i F ) d m i { R ⃗ G F × m t o t a l ⋅ a ⃗ G F α ⃗ M F ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 − R ⃗ G P i F R ⃗ G P i F T ) d m i − ω ⃗ M F × ( ∫ ( R ⃗ G P i F R ⃗ G P i F T ) d m i ⋅ ω ⃗ M F ) \begin{split} \frac{\mathrm{d}\vec{H}_{\Sigma _{\mathrm{M}}}^{F}}{\mathrm{d}t}\begin{cases} \vec{R}_{\mathrm{G}}^{F}\times m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F}2\int{\left( \vec{V}_{\mathrm{P}_{\mathrm{i}}}^{F}\cdot \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \vec{\omega}_{\mathrm{M}}^{F}}\mathrm{d}m_{\mathrm{i}}\int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \vec{\alpha}_{\mathrm{M}}^{F}}\mathrm{d}m_{\mathrm{i}}\\ -\int{\left( \vec{V}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}\mathrm{d}m_{\mathrm{i}}-\int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{\alpha}_{\mathrm{M}}^{F} \right) \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}\mathrm{d}m_{\mathrm{i}}-\int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) \vec{V}_{\mathrm{GP}_{\mathrm{i}}}^{F}}\mathrm{d}m_{\mathrm{i}}\\ \end{cases} \\ \begin{cases} \vec{R}_{\mathrm{G}}^{F}\times m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F}\left( \int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \vec{\alpha}_{\mathrm{M}}^{F}}\mathrm{d}m_{\mathrm{i}}-\int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{\alpha}_{\mathrm{M}}^{F} \right) \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}\mathrm{d}m_{\mathrm{i}} \right)\\ -\int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) \left( \vec{\omega}_{\mathrm{M}}^{F}\times \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \mathrm{d}m_{\mathrm{i}}}\\ \end{cases} \\ \begin{cases} \vec{R}_{\mathrm{G}}^{F}\times m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F}\left( \int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \cdot E^{3\times 3}\vec{\alpha}_{\mathrm{M}}^{F}}\mathrm{d}m_{\mathrm{i}}-\int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{\alpha}_{\mathrm{M}}^{F} \right) \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}\mathrm{d}m_{\mathrm{i}} \right)\\ -\int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{\omega}_{\mathrm{M}}^{F} \right) \left( \vec{\omega}_{\mathrm{M}}^{F}\times \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \mathrm{d}m_{\mathrm{i}}}\\ \end{cases} \\ \begin{cases} \vec{R}_{\mathrm{G}}^{F}\times m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F}\vec{\alpha}_{\mathrm{M}}^{F}\int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot E^{3\times 3}-\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m_{\mathrm{i}}\\ -\vec{\omega}_{\mathrm{M}}^{F}\times \left( \int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m_{\mathrm{i}}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right)\\ \end{cases} \end{split} dtdH ΣMF⎩ ⎨ ⎧R GF×mtotal⋅a GF2∫(V PiF⋅R GPiF)ω MFdmi∫(R GPiF⋅R GPiF)α MFdmi−∫(V GPiF⋅ω MF)R GPiFdmi−∫(R GPiF⋅α MF)R GPiFdmi−∫(R GPiF⋅ω MF)V GPiFdmi⎩ ⎨ ⎧R GF×mtotal⋅a GF(∫(R GPiF⋅R GPiF)α MFdmi−∫(R GPiF⋅α MF)R GPiFdmi)−∫(R GPiF⋅ω MF)(ω MF×R GPiF)dmi⎩ ⎨ ⎧R GF×mtotal⋅a GF(∫(R GPiFTR GPiF)⋅E3×3α MFdmi−∫(R GPiFTα MF)R GPiFdmi)−∫(R GPiFTω MF)(ω MF×R GPiF)dmi⎩ ⎨ ⎧R GF×mtotal⋅a GFα MF∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi−ω MF×(∫(R GPiFR GPiFT)dmi⋅ω MF) 其中 ⇒ − ω ⃗ M F × ∫ ( R ⃗ G P i F R ⃗ G P i F T ) d m i ⋅ ω ⃗ M F ω ⃗ M F × ( ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 − R ⃗ G P i F R ⃗ G P i F T − R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 ) d m i ⋅ ω ⃗ M F ) ω ⃗ M F × ( ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 − R ⃗ G P i F R ⃗ G P i F T ) d m i ⋅ ω ⃗ M F ) − ω ⃗ M F × ( ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 ) d m i ⋅ ω ⃗ M F ) ⏟ 0 \begin{split} \Rightarrow -\vec{\omega}_{\mathrm{M}}^{F}\times \int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m_{\mathrm{i}}\cdot \vec{\omega}_{\mathrm{M}}^{F} \\ \vec{\omega}_{\mathrm{M}}^{F}\times \left( \int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot E^{3\times 3}-\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}-{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot E^{3\times 3} \right)}\mathrm{d}m_{\mathrm{i}}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) \\ \vec{\omega}_{\mathrm{M}}^{F}\times \left( \int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot E^{3\times 3}-\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m_{\mathrm{i}}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) -\begin{array}{c} \underbrace{\vec{\omega}_{\mathrm{M}}^{F}\times \left( \int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot E^{3\times 3} \right)}\mathrm{d}m_{\mathrm{i}}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) }\\ 0\\ \end{array} \end{split} ⇒−ω MF×∫(R GPiFR GPiFT)dmi⋅ω MFω MF×(∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT−R GPiFTR GPiF⋅E3×3)dmi⋅ω MF)ω MF×(∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi⋅ω MF)− ω MF×(∫(R GPiFTR GPiF⋅E3×3)dmi⋅ω MF)0
将上两式进行汇总可得 ⇒ d H ⃗ Σ M F d t { R ⃗ G F × m t o t a l ⋅ a ⃗ G F ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 − R ⃗ G P i F R ⃗ G P i F T ) d m i α ⃗ M F ω ⃗ M F × ( ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 − R ⃗ G P i F R ⃗ G P i F T ) d m i ⋅ ω ⃗ M F ) R ⃗ G F × m t o t a l ⋅ a ⃗ G F [ I ] Σ M / G F α ⃗ M F ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) \begin{split} \Rightarrow \frac{\mathrm{d}\vec{H}_{\Sigma _{\mathrm{M}}}^{F}}{\mathrm{d}t}\begin{cases} \vec{R}_{\mathrm{G}}^{F}\times m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F}\int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot E^{3\times 3}-\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m_{\mathrm{i}}\vec{\alpha}_{\mathrm{M}}^{F}\\ \vec{\omega}_{\mathrm{M}}^{F}\times \left( \int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot E^{3\times 3}-\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m_{\mathrm{i}}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right)\\ \end{cases} \\ \vec{R}_{\mathrm{G}}^{F}\times m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F}\left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\vec{\alpha}_{\mathrm{M}}^{F}\vec{\omega}_{\mathrm{M}}^{F}\times \left( \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) \end{split} ⇒dtdH ΣMF⎩ ⎨ ⎧R GF×mtotal⋅a GF∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmiα MFω MF×(∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi⋅ω MF)R GF×mtotal⋅a GF[I]ΣM/GFα MFω MF×([I]ΣM/GF⋅ω MF)
其中 [ I ] Σ M / G F ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 − R ⃗ G P i F R ⃗ G P i F T ) d m i \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot E^{3\times 3}-\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m_i [I]ΣM/GF∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi [ I ] Σ M / G F \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F} [I]ΣM/GF被称为惯性矩阵inertia matrix或称为惯量矩阵为该物体在固定坐标系下相对于质心点 G G G的惯性张量。
进而可知 d H ⃗ Σ M F d t M ⃗ Σ M F ∫ R ⃗ P i F × d F ⃗ P i F R ⃗ G F × m t o t a l ⋅ a ⃗ G F [ I ] Σ M / G F α ⃗ M F ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) \frac{\mathrm{d}\vec{H}_{\Sigma _{\mathrm{M}}}^{F}}{\mathrm{d}t}\vec{M}_{\Sigma _{\mathrm{M}}}^{F}\int{\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F}\times \mathrm{d}\vec{F}_{\mathrm{P}_{\mathrm{i}}}^{F}}\vec{R}_{\mathrm{G}}^{F}\times m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F}\left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\vec{\alpha}_{\mathrm{M}}^{F}\vec{\omega}_{\mathrm{M}}^{F}\times \left( \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) dtdH ΣMFM ΣMF∫R PiF×dF PiFR GF×mtotal⋅a GF[I]ΣM/GFα MFω MF×([I]ΣM/GF⋅ω MF) 上式被称为欧拉方程在惯性坐标系下相对固定点的表达式当固定点与质心点重合时(此时G点为固定点)则有 M ⃗ Σ M / G F M ⃗ Σ M F − R ⃗ G F × ( m t o t a l ⋅ a ⃗ G F ) R ⃗ G F × ( m t o t a l ⋅ a ⃗ G F ) [ I ] Σ M / G F α ⃗ M F ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) − R ⃗ G F × ( m t o t a l ⋅ a ⃗ G F ) [ I ] Σ M / G F α ⃗ M F ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) \begin{split} \vec{M}_{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\vec{M}_{\Sigma _{\mathrm{M}}}^{F}-\vec{R}_{\mathrm{G}}^{F}\times \left( m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F} \right) \\ \vec{R}_{\mathrm{G}}^{F}\times \left( m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F} \right) \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\vec{\alpha}_{\mathrm{M}}^{F}\vec{\omega}_{\mathrm{M}}^{F}\times \left( \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) -\vec{R}_{\mathrm{G}}^{F}\times \left( m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F} \right) \\ \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\vec{\alpha}_{\mathrm{M}}^{F}\vec{\omega}_{\mathrm{M}}^{F}\times \left( \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) \end{split} M ΣM/GFM ΣMF−R GF×(mtotal⋅a GF)R GF×(mtotal⋅a GF)[I]ΣM/GFα MFω MF×([I]ΣM/GF⋅ω MF)−R GF×(mtotal⋅a GF)[I]ΣM/GFα MFω MF×([I]ΣM/GF⋅ω MF) 此时为固定坐标系下相对固定点质心 G G G求解的欧拉方程。
