物理

❴栖岸计划❵刚体进动章动知识点

进动，可理解为在一自转-公转体系中  

外力矩只存在重力力矩，引起的公转进动角动量  

$\vec{L}_s = \tilde{I}_s \vec{\omega}_s = I_s \vec{\omega} \hat{y}$

转系：  

$\vec{L}_s = I_s \vec{\omega} \hat{y}, \quad \Omega = \Omega \hat{z}$ 

$\frac{d(\vec{L}_s)}{dt} = 0$

换地系：  

$\frac{d(\vec{L}_{\text{地}})}{dt} = \frac{d(\vec{L}_s)}{dt} + \vec{\Omega} \times \vec{L}_s$

等价矢量换系公式：  

$\frac{d(\vec{r})}{dt} = \frac{d(\vec{r}')}{dt} + \vec{\omega} \times \vec{r}$

$\frac{d(\vec{L}_{\text{地}})}{dt} = \vec{M}_{\text{外力}} = mg l_{oc}$

$mg l_{oc} = \vec{\Omega} \times \vec{L}_s = \Omega \vec{L}_s \hat{z} \times \hat{y} = \Omega I_s \vec{\omega} \hat{x}$

$\Omega = \frac{mg l_{oc}}{I_s \vec{\omega}}$

转系：  

$\vec{\omega}_s = \omega_s \hat{j}' = \omega_s \hat{y}'$ 

$\vec{\Omega} = \Omega \cos\phi \hat{z}_{\text{地}} = \Omega \cos\phi \hat{z}$

地系：  

$\vec{\omega}_{\text{地}} = (\omega_s + \Omega \sin\phi) \hat{y}'$ $\quad \vec{\Omega}_{\text{地}} = \Omega \cos\phi \hat{z}$

$\vec{L}_s = I_s (\omega_s + \Omega \sin\phi) \hat{y}', \quad \vec{L}_n = I_{zz} \Omega \cos\phi \hat{z}$

用右边矢量转换即可