**Generalized Coordinates** are variables chosen to fully describe a system's position **Degrees of Freedom**: the number of generalized coordinates $ \text{\# of DOFs} = 6 \cdot \text{\# of bodies} - \text{\# of constraints} $ **Rigid Bodies**: sets of particles that move and carry forces together **Holonomic Systems**: systems where GCs are synchronized with DOFs $ \text{\# of DOFs}=\text{\# of GCs} $ **Instant Center of Rotation** (ICR): the point $Q$ where each point $P$ of a rotating rigid body has velocity orthogonal to its radius; $\vec{r}_{P/Q} \cdot \vec{v}_{P/Q} = 0$ ![[Pasted image 20250630160100.png]] ### Frame of References Newtonian mechanics require an inertial frame of reference. Often an intermediate frame of reference is used to work out the equations of motion. ![[Pasted image 20250630155605.png]] $P:$ fixed in $F_{1}$, rotating in the inertial frame $F$ $F_{1}$: intermediate frame $O_{1}$: origin point of $F_{1}$ #### Velocity $ \vec{v}_{P/F}=\vec{v}_{O_{1}/F}+\vec{\omega}\times \vec{r}_{P/O_{1}}+\vec{v}_{P/F_{1}} $ #### Acceleration $ \vec{a}_{P/F} = \vec{a}_{O_{1}/F}+\underbrace{ \vec{\omega}\times(\vec{\omega}\times \vec{r}_{P/O_{1}}) }_{\text{centripetal} }+\underbrace{ 2\vec{\omega}\times \vec{v}_{P/F_{1}} }_{\text{coriolis} }+\underbrace{ \dot{\vec{\omega}}\times \vec{r}_{P/O_{1}} }_{ \text{Euler} }+\vec{a}_{P/F_{1}} $ - **Centripetal acceleration** maintains a circle: $R(\dot{\theta})^{2}$ - **Coriolis force** is caused by change in radius of rotation: $2 \dot{r} \dot{\theta}$ - **Euler force** is caused by change in rate of rotation: $R \ddot{\theta}$