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
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\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}
HΣM/OF=∫ROPiF×(dmi⋅dtdRPiF)=∫((RPiF−ROF)×VPiF)dmi=∫(RPiF×VPiF)dmi−∫(ROF×VPiF)dmi=HΣMF−ROF×PGF
对上式进一步求导,则有:
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\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}
dtdHΣM/OF=dtdHΣMF−dtd(ROF×PGF)=dtdHΣMF−VOF×PGF−mtotal⋅ROF×aGF
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\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}
HΣMF=∫RPiF×pPiF=∫(RGF+RGPiF)×(dmi⋅(VGF+VGPiF))=
∫RGF×VGFdmimtotal⋅RGF×VGF+
∫RGF×VGPiFdmi0+
∫RGPiF×VGFdmi0+
∫RGPiF×VGPiFdmi∫RGPiF×(ωMF×RGPiF)dmi=mtotal⋅RGF×VGF+∫RGPiF×(ωMF×RGPiF)dmi=mtotal⋅RGF×VGF+∫(RGPiF⋅RGPiF)ωMFdmi−∫(RGPiF⋅ωMF)RGPiFdmi
将
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\vec{H}_{\Sigma _{\mathrm{M}}}^{F}
HΣMF进一步求导,则有:
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\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}
dtdHΣMF=⎩
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⎧RGF×mtotal⋅aGF+2∫(VPiF⋅RGPiF)ωMFdmi+∫(RGPiF⋅RGPiF)αMFdmi−∫(VGPiF⋅ωMF)RGPiFdmi−∫(RGPiF⋅αMF)RGPiFdmi−∫(RGPiF⋅ωMF)VGPiFdmi=⎩
⎨
⎧RGF×mtotal⋅aGF+(∫(RGPiF⋅RGPiF)αMFdmi−∫(RGPiF⋅αMF)RGPiFdmi)−∫(RGPiF⋅ωMF)(ωMF×RGPiF)dmi=⎩
⎨
⎧RGF×mtotal⋅aGF+(∫(RGPiFTRGPiF)⋅E3×3αMFdmi−∫(RGPiFTαMF)RGPiFdmi)−∫(RGPiFTωMF)(ωMF×RGPiF)dmi=⎩
⎨
⎧RGF×mtotal⋅aGF+αMF∫(RGPiFTRGPiF⋅E3×3−RGPiFRGPiFT)dmi−ωMF×(∫(RGPiFRGPiFT)dmi⋅ωMF)
其中:
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\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}
⇒−ωMF×∫(RGPiFRGPiFT)dmi⋅ωMF=ωMF×(∫(RGPiFTRGPiF⋅E3×3−RGPiFRGPiFT−RGPiFTRGPiF⋅E3×3)dmi⋅ωMF)=ωMF×(∫(RGPiFTRGPiF⋅E3×3−RGPiFRGPiFT)dmi⋅ωMF)−
ωMF×(∫(RGPiFTRGPiF⋅E3×3)dmi⋅ωMF)0
将上两式进行汇总,可得:
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\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}
⇒dtdHΣMF=⎩
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⎧RGF×mtotal⋅aGF+∫(RGPiFTRGPiF⋅E3×3−RGPiFRGPiFT)dmiαMF+ωMF×(∫(RGPiFTRGPiF⋅E3×3−RGPiFRGPiFT)dmi⋅ωMF)=RGF×mtotal⋅aGF+[I]ΣM/GFαMF+ωMF×([I]ΣM/GF⋅ωMF)
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\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=∫(RGPiFTRGPiF⋅E3×3−RGPiFRGPiFT)dmi
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\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
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G的惯性张量。
进而可知:
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\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)
dtdHΣMF=MΣMF=∫RPiF×dFPiF=RGF×mtotal⋅aGF+[I]ΣM/GFαMF+ωMF×([I]ΣM/GF⋅ωMF)
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:
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\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}
MΣM/GF=MΣMF−RGF×(mtotal⋅aGF)=RGF×(mtotal⋅aGF)+[I]ΣM/GFαMF+ωMF×([I]ΣM/GF⋅ωMF)−RGF×(mtotal⋅aGF)=[I]ΣM/GFαMF+ωMF×([I]ΣM/GF⋅ωMF)
此时为固定坐标系下相对固定点质心
G
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G求解的欧拉方程。