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The Euler equations

Euler equation inertial frame at the CM \begin{align*} &\frac{d}{dt}\left(\mathbf I\,\mathbf\omega\right)=\mathbf\tau\\ &\Rightarrow\\ &\mathbf I\,\mathbf{\dot{\omega}}+\mathbf{\dot{I}}\,\mathbf\omega=\mathbf\tau\quad ,\text{with}\\ &\mathbf I=\mathbf R\,\mathbf I_B\quad ,\mathbf{\dot{I}}=\mathbf{\dot{R}}\,\mathbf I_B = \underbrace{\left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}} \\ \omega_{{z}}&0&-\omega_{{x}}\\ -\omega_{{y}}&\omega_{{x}}&0\end {array} \right]}_{\mathbf{\omega}^\times} \,\mathbf R\,\mathbf I_B\quad\Rightarrow\\\\ &\boxed{\quad\mathbf I\,\mathbf{\dot{\omega}}+\mathbf\omega\times\,\mathbf I\,\mathbf\omega=\mathbf\tau\quad~Eq. (1)} \end{align*}

Euler equation body frame at the CM

\begin{align*} &\text{with}\\ &\mathbf I=\mathbf{R}\,\mathbf I_B\,\mathbf R^T\quad,\text{you obtain Eq. (1)}\\\\ &\mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf\tau\\&\text{ multiply from the left with}~\mathbf R^T\quad,\Rightarrow\\\\ &\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf R^T\,\mathbf\omega\times\, \mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf R^T\,\mathbf\tau\\\\ &\boxed{\quad\mathbf I_B\,\mathbf{\dot{\omega}}_B+\mathbf\omega_B\times\,\mathbf I_B\,\mathbf\omega_B=\mathbf\tau_B\quad~Eq. (2)} \end{align*}

Euler equation body frame (parallel to B frame) at body point $~P$ \begin{align*} &\text{with }\\ &\mathbf I_B=\mathbf I_P+\underbrace{m\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right] \, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right]}_{\mathbf I_{CP}}\\ &\mathbf\omega_P=\mathbf\omega_B\\ &\mathbf\tau_P=\mathbf\tau_B+\mathbf r_{CP}\times\mathbf F_F\quad\Rightarrow\\\\ &\boxed{\quad\left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf{\dot{\omega}}_P+\mathbf\omega_P\times\, \left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf\omega_P=\mathbf\tau_P\quad~Eq. (3)} \end{align*}\begin{align*} &\text{with }\\ &\mathbf I_P=\mathbf I_B\underbrace{-m\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right] \, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right]}_{\mathbf I_{CP}}\\ &\mathbf\omega_P=\mathbf\omega_B\\ &\mathbf\tau_P=\mathbf\tau_B+\mathbf r_{CP}\times\mathbf F_F\quad\Rightarrow\\\\ &\boxed{\quad\left(\mathbf I_B+\mathbf I_{CP}\right)\,\mathbf{\dot{\omega}}_P+\mathbf\omega_P\times\, \left(\mathbf I_B+\mathbf I_{CP}\right)\,\mathbf\omega_P=\mathbf\tau_P\quad~Eq. (3)} \end{align*}

  • $~\mathbf I~$ Intertia tensor
  • $~\mathbf\tau~$ External torgue
  • $~\mathbf R~$ transformation matrix between body frame and inertial frame
  • $~\mathbf r_{CP}~$ body fixed vector from the center of mass to point p
  • $~\mathbf F_F~$ fictitious forces at the center of mass
  • $~$ subscript B body frame at the center of mass
  • $~$ subscript P body frame at body point P

The Euler equations

Euler equation inertial frame at the CM \begin{align*} &\frac{d}{dt}\left(\mathbf I\,\mathbf\omega\right)=\mathbf\tau\\ &\Rightarrow\\ &\mathbf I\,\mathbf{\dot{\omega}}+\mathbf{\dot{I}}\,\mathbf\omega=\mathbf\tau\quad ,\text{with}\\ &\mathbf I=\mathbf R\,\mathbf I_B\quad ,\mathbf{\dot{I}}=\mathbf{\dot{R}}\,\mathbf I_B = \underbrace{\left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}} \\ \omega_{{z}}&0&-\omega_{{x}}\\ -\omega_{{y}}&\omega_{{x}}&0\end {array} \right]}_{\mathbf{\omega}^\times} \,\mathbf R\,\mathbf I_B\quad\Rightarrow\\\\ &\boxed{\quad\mathbf I\,\mathbf{\dot{\omega}}+\mathbf\omega\times\,\mathbf I\,\mathbf\omega=\mathbf\tau\quad~Eq. (1)} \end{align*}

Euler equation body frame at the CM

\begin{align*} &\text{with}\\ &\mathbf I=\mathbf{R}\,\mathbf I_B\,\mathbf R^T\quad,\text{you obtain Eq. (1)}\\\\ &\mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf\tau\\&\text{ multiply from the left with}~\mathbf R^T\quad,\Rightarrow\\\\ &\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf R^T\,\mathbf\omega\times\, \mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf R^T\,\mathbf\tau\\\\ &\boxed{\quad\mathbf I_B\,\mathbf{\dot{\omega}}_B+\mathbf\omega_B\times\,\mathbf I_B\,\mathbf\omega_B=\mathbf\tau_B\quad~Eq. (2)} \end{align*}

Euler equation body frame (parallel to B frame) at body point $~P$ \begin{align*} &\text{with }\\ &\mathbf I_B=\mathbf I_P+\underbrace{m\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right] \, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right]}_{\mathbf I_{CP}}\\ &\mathbf\omega_P=\mathbf\omega_B\\ &\mathbf\tau_P=\mathbf\tau_B+\mathbf r_{CP}\times\mathbf F_F\quad\Rightarrow\\\\ &\boxed{\quad\left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf{\dot{\omega}}_P+\mathbf\omega_P\times\, \left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf\omega_P=\mathbf\tau_P\quad~Eq. (3)} \end{align*}

  • $~\mathbf I~$ Intertia tensor
  • $~\mathbf\tau~$ External torgue
  • $~\mathbf R~$ transformation matrix between body frame and inertial frame
  • $~\mathbf r_{CP}~$ body fixed vector from the center of mass to point p
  • $~\mathbf F_F~$ fictitious forces at the center of mass
  • $~$ subscript B body frame at the center of mass
  • $~$ subscript P body frame at body point P

The Euler equations

Euler equation inertial frame at the CM \begin{align*} &\frac{d}{dt}\left(\mathbf I\,\mathbf\omega\right)=\mathbf\tau\\ &\Rightarrow\\ &\mathbf I\,\mathbf{\dot{\omega}}+\mathbf{\dot{I}}\,\mathbf\omega=\mathbf\tau\quad ,\text{with}\\ &\mathbf I=\mathbf R\,\mathbf I_B\quad ,\mathbf{\dot{I}}=\mathbf{\dot{R}}\,\mathbf I_B = \underbrace{\left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}} \\ \omega_{{z}}&0&-\omega_{{x}}\\ -\omega_{{y}}&\omega_{{x}}&0\end {array} \right]}_{\mathbf{\omega}^\times} \,\mathbf R\,\mathbf I_B\quad\Rightarrow\\\\ &\boxed{\quad\mathbf I\,\mathbf{\dot{\omega}}+\mathbf\omega\times\,\mathbf I\,\mathbf\omega=\mathbf\tau\quad~Eq. (1)} \end{align*}

Euler equation body frame at the CM

\begin{align*} &\text{with}\\ &\mathbf I=\mathbf{R}\,\mathbf I_B\,\mathbf R^T\quad,\text{you obtain Eq. (1)}\\\\ &\mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf\tau\\&\text{ multiply from the left with}~\mathbf R^T\quad,\Rightarrow\\\\ &\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf R^T\,\mathbf\omega\times\, \mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf R^T\,\mathbf\tau\\\\ &\boxed{\quad\mathbf I_B\,\mathbf{\dot{\omega}}_B+\mathbf\omega_B\times\,\mathbf I_B\,\mathbf\omega_B=\mathbf\tau_B\quad~Eq. (2)} \end{align*}

Euler equation body frame (parallel to B frame) at body point $~P$ \begin{align*} &\text{with }\\ &\mathbf I_P=\mathbf I_B\underbrace{-m\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right] \, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right]}_{\mathbf I_{CP}}\\ &\mathbf\omega_P=\mathbf\omega_B\\ &\mathbf\tau_P=\mathbf\tau_B+\mathbf r_{CP}\times\mathbf F_F\quad\Rightarrow\\\\ &\boxed{\quad\left(\mathbf I_B+\mathbf I_{CP}\right)\,\mathbf{\dot{\omega}}_P+\mathbf\omega_P\times\, \left(\mathbf I_B+\mathbf I_{CP}\right)\,\mathbf\omega_P=\mathbf\tau_P\quad~Eq. (3)} \end{align*}

  • $~\mathbf I~$ Intertia tensor
  • $~\mathbf\tau~$ External torgue
  • $~\mathbf R~$ transformation matrix between body frame and inertial frame
  • $~\mathbf r_{CP}~$ body fixed vector from the center of mass to point p
  • $~\mathbf F_F~$ fictitious forces at the center of mass
  • $~$ subscript B body frame at the center of mass
  • $~$ subscript P body frame at body point P
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Eli
Eli
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The Euler equations

Euler equation inertial frame at the CM \begin{align*} &\frac{d}{dt}\left(\mathbf I\,\mathbf\omega\right)=\mathbf\tau\\ &\Rightarrow\\ &\mathbf I\,\mathbf{\dot{\omega}}+\mathbf{\dot{I}}\,\mathbf\omega=\mathbf\tau\quad ,\text{with}\\ &\mathbf I=\mathbf R\,\mathbf I_B\quad ,\mathbf{\dot{I}}=\mathbf{\dot{R}}\,\mathbf I_B = \underbrace{\left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}} \\ \omega_{{z}}&0&-\omega_{{x}}\\ -\omega_{{y}}&\omega_{{x}}&0\end {array} \right]}_{\mathbf{\omega}^\times} \,\mathbf R\,\mathbf I_B\quad\Rightarrow\\\\ &\boxed{\quad\mathbf I\,\mathbf{\dot{\omega}}+\mathbf\omega\times\,\mathbf I\,\mathbf\omega=\mathbf\tau\quad~Eq. (1)} \end{align*}

Euler equation body frame at the CM

\begin{align*} &\text{with}\\ &\mathbf I=\mathbf{R}\,\mathbf I_B\,\mathbf R^T\quad,\text{you obtain Eq. (1)}\\ &\mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf\tau\quad,\text{ multiply from the left with}~\mathbf R^T\quad,\Rightarrow\\ &\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf R^T\,\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf R^T\,\mathbf\tau\\\\ &\boxed{\quad\mathbf I_B\,\mathbf{\dot{\omega}}_B+\mathbf\omega_B\times\,\mathbf I_B\,\mathbf\omega_B=\mathbf\tau_B\quad~Eq. (2)} \end{align*}\begin{align*} &\text{with}\\ &\mathbf I=\mathbf{R}\,\mathbf I_B\,\mathbf R^T\quad,\text{you obtain Eq. (1)}\\\\ &\mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf\tau\\&\text{ multiply from the left with}~\mathbf R^T\quad,\Rightarrow\\\\ &\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf R^T\,\mathbf\omega\times\, \mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf R^T\,\mathbf\tau\\\\ &\boxed{\quad\mathbf I_B\,\mathbf{\dot{\omega}}_B+\mathbf\omega_B\times\,\mathbf I_B\,\mathbf\omega_B=\mathbf\tau_B\quad~Eq. (2)} \end{align*}

Euler equation body frame at body point $~P$Euler equation body frame (parallel to B frame) at body point $~P$ \begin{align*} &\text{with }\\ &\mathbf I_B=\mathbf I_P+\underbrace{m\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right] \, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right]}_{\mathbf I_{CP}}\\ &\mathbf\omega_P=\mathbf\omega_B\\ &\mathbf\tau_P=\mathbf\tau_B+\mathbf r_{CP}\times\mathbf F_F\quad\Rightarrow\\\\ &\boxed{\quad\left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf{\dot{\omega}}_P+\mathbf\omega_P\times\, \left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf\omega_P=\mathbf\tau_P\quad~Eq. (3)} \end{align*}

  • $~\mathbf I~$ Intertia tensor
  • $~\mathbf\tau~$ External torgue
  • $~\mathbf R~$ transformation matrix between body frame and inertial frame
  • $~\mathbf r_{CP}~$ body fixed vector from the center of mass to point p
  • $~\mathbf F_F~$ fictitious forces at the center of mass
  • $~$ subscript B body frame at the center of mass
  • $~$ subscript P body frame at body point P

The Euler equations

Euler equation inertial frame at the CM \begin{align*} &\frac{d}{dt}\left(\mathbf I\,\mathbf\omega\right)=\mathbf\tau\\ &\Rightarrow\\ &\mathbf I\,\mathbf{\dot{\omega}}+\mathbf{\dot{I}}\,\mathbf\omega=\mathbf\tau\quad ,\text{with}\\ &\mathbf I=\mathbf R\,\mathbf I_B\quad ,\mathbf{\dot{I}}=\mathbf{\dot{R}}\,\mathbf I_B = \underbrace{\left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}} \\ \omega_{{z}}&0&-\omega_{{x}}\\ -\omega_{{y}}&\omega_{{x}}&0\end {array} \right]}_{\mathbf{\omega}^\times} \,\mathbf R\,\mathbf I_B\quad\Rightarrow\\\\ &\boxed{\quad\mathbf I\,\mathbf{\dot{\omega}}+\mathbf\omega\times\,\mathbf I\,\mathbf\omega=\mathbf\tau\quad~Eq. (1)} \end{align*}

Euler equation body frame at the CM

\begin{align*} &\text{with}\\ &\mathbf I=\mathbf{R}\,\mathbf I_B\,\mathbf R^T\quad,\text{you obtain Eq. (1)}\\ &\mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf\tau\quad,\text{ multiply from the left with}~\mathbf R^T\quad,\Rightarrow\\ &\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf R^T\,\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf R^T\,\mathbf\tau\\\\ &\boxed{\quad\mathbf I_B\,\mathbf{\dot{\omega}}_B+\mathbf\omega_B\times\,\mathbf I_B\,\mathbf\omega_B=\mathbf\tau_B\quad~Eq. (2)} \end{align*}

Euler equation body frame at body point $~P$ \begin{align*} &\text{with }\\ &\mathbf I_B=\mathbf I_P+\underbrace{m\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right] \, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right]}_{\mathbf I_{CP}}\\ &\mathbf\omega_P=\mathbf\omega_B\\ &\mathbf\tau_P=\mathbf\tau_B+\mathbf r_{CP}\times\mathbf F_F\quad\Rightarrow\\\\ &\boxed{\quad\left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf{\dot{\omega}}_P+\mathbf\omega_P\times\, \left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf\omega_P=\mathbf\tau_P\quad~Eq. (3)} \end{align*}

  • $~\mathbf I~$ Intertia tensor
  • $~\mathbf\tau~$ External torgue
  • $~\mathbf R~$ transformation matrix between body frame and inertial frame
  • $~\mathbf r_{CP}~$ body fixed vector from the center of mass to point p
  • $~\mathbf F_F~$ fictitious forces at the center of mass
  • $~$ subscript B body frame at the center of mass
  • $~$ subscript P body frame at body point P

The Euler equations

Euler equation inertial frame at the CM \begin{align*} &\frac{d}{dt}\left(\mathbf I\,\mathbf\omega\right)=\mathbf\tau\\ &\Rightarrow\\ &\mathbf I\,\mathbf{\dot{\omega}}+\mathbf{\dot{I}}\,\mathbf\omega=\mathbf\tau\quad ,\text{with}\\ &\mathbf I=\mathbf R\,\mathbf I_B\quad ,\mathbf{\dot{I}}=\mathbf{\dot{R}}\,\mathbf I_B = \underbrace{\left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}} \\ \omega_{{z}}&0&-\omega_{{x}}\\ -\omega_{{y}}&\omega_{{x}}&0\end {array} \right]}_{\mathbf{\omega}^\times} \,\mathbf R\,\mathbf I_B\quad\Rightarrow\\\\ &\boxed{\quad\mathbf I\,\mathbf{\dot{\omega}}+\mathbf\omega\times\,\mathbf I\,\mathbf\omega=\mathbf\tau\quad~Eq. (1)} \end{align*}

Euler equation body frame at the CM

\begin{align*} &\text{with}\\ &\mathbf I=\mathbf{R}\,\mathbf I_B\,\mathbf R^T\quad,\text{you obtain Eq. (1)}\\\\ &\mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf\tau\\&\text{ multiply from the left with}~\mathbf R^T\quad,\Rightarrow\\\\ &\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf R^T\,\mathbf\omega\times\, \mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf R^T\,\mathbf\tau\\\\ &\boxed{\quad\mathbf I_B\,\mathbf{\dot{\omega}}_B+\mathbf\omega_B\times\,\mathbf I_B\,\mathbf\omega_B=\mathbf\tau_B\quad~Eq. (2)} \end{align*}

Euler equation body frame (parallel to B frame) at body point $~P$ \begin{align*} &\text{with }\\ &\mathbf I_B=\mathbf I_P+\underbrace{m\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right] \, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right]}_{\mathbf I_{CP}}\\ &\mathbf\omega_P=\mathbf\omega_B\\ &\mathbf\tau_P=\mathbf\tau_B+\mathbf r_{CP}\times\mathbf F_F\quad\Rightarrow\\\\ &\boxed{\quad\left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf{\dot{\omega}}_P+\mathbf\omega_P\times\, \left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf\omega_P=\mathbf\tau_P\quad~Eq. (3)} \end{align*}

  • $~\mathbf I~$ Intertia tensor
  • $~\mathbf\tau~$ External torgue
  • $~\mathbf R~$ transformation matrix between body frame and inertial frame
  • $~\mathbf r_{CP}~$ body fixed vector from the center of mass to point p
  • $~\mathbf F_F~$ fictitious forces at the center of mass
  • $~$ subscript B body frame at the center of mass
  • $~$ subscript P body frame at body point P
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The Euler equations

Euler equation inertial frame at the CM \begin{align*} &\frac{d}{dt}\left(\mathbf I\,\mathbf\omega\right)=\mathbf\tau\\ &\Rightarrow\\ &\mathbf I\,\mathbf{\dot{\omega}}+\mathbf{\dot{I}}\,\mathbf\omega=\mathbf\tau\quad ,\text{with}\\ &\mathbf I=\mathbf R\,\mathbf I_B\quad ,\mathbf{\dot{I}}=\mathbf{\dot{R}}\,\mathbf I_B = \underbrace{\left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}} \\ \omega_{{z}}&0&-\omega_{{x}}\\ -\omega_{{y}}&\omega_{{x}}&0\end {array} \right]}_{\mathbf{\omega}^\times} \,\mathbf R\,\mathbf I_B\quad\Rightarrow\\\\ &\boxed{\quad\mathbf I\,\mathbf{\dot{\omega}}+\mathbf\omega\times\,\mathbf I\,\mathbf\omega=\mathbf\tau\quad~Eq. (1)} \end{align*}

Euler equation body frame at the CM

\begin{align*} &\text{with}\\ &\mathbf I=\mathbf{R}\,\mathbf I_B\,\mathbf R^T\quad,\text{you obtain Eq. (1)}\\ &\mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf\tau\quad,\text{ multiply from the left with}~\mathbf R^T\quad,\Rightarrow\\ &\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B} +\mathbf R^T\,\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf R^T\,\mathbf\tau\\\\ &\boxed{\quad\mathbf I_B\,\mathbf{\dot{\omega}}_B+\mathbf\omega_B\times\,\mathbf I_B\,\mathbf\omega_B=\mathbf\tau_B\quad~Eq. (2)} \end{align*}

Euler equation body frame at body point $~P$ \begin{align*} &\text{with }\\ &\mathbf I_B=\mathbf I_P+\underbrace{m\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right] \, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}} \\ z_{{{\it CP}}}&0&-x_{{{\it CP}}} \\ -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array} \right]}_{\mathbf I_{CP}}\\ &\mathbf\omega_P=\mathbf\omega_B\\ &\mathbf\tau_P=\mathbf\tau_B+\mathbf r_{CP}\times\mathbf F_F\quad\Rightarrow\\\\ &\boxed{\quad\left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf{\dot{\omega}}_P+\mathbf\omega_P\times\, \left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf\omega_P=\mathbf\tau_P\quad~Eq. (3)} \end{align*}

  • $~\mathbf I~$ Intertia tensor
  • $~\mathbf\tau~$ External torgue
  • $~\mathbf R~$ transformation matrix between body frame and inertial frame
  • $~\mathbf r_{CP}~$ body fixed vector from the center of mass to point p
  • $~\mathbf F_F~$ fictitious forces at the center of mass
  • $~$ subscript B body frame at the center of mass
  • $~$ subscript P body frame at body point P