Direct Dynamics: Newton–Euler Equations of Motion

Abstract

The Newton–Euler equations of motion for a rigid body in plane motion are

$$ m \mathbf{\ddot{r}}_C = \mathbf{\sum F} \quad \text{and} \quad I_{Czz} \mathbf{\alpha} = \mathbf{\sum} \mathbf{M}_{C} \text{,} $$

or using the Cartesian components

$$ m\ddot{x}_C = \mathbf{\sum}F_{x}\text{,} \quad m\ddot{y}_C = \mathbf{\sum}F_{y}\text{, and} \quad I_{Czz} \ddot{\Theta} = \mathbf{\sum} M_{C} \text{.} $$

The forces and moments are known and the differential equations are solved for the motion of the rigid body (direct dynamics).