Newton-Euler Equations

Author: CislunarSpace
Site: https://cislunarspace.cn

Definition

Newton-Euler equations describe complete dynamics of rigid body translation (Newton's second law) and rotation (Euler equations). For aircraft, the equations relate external forces to center-of-mass motion and external moments to attitude motion.

Newton Equation (Translation)

Basic Form

\vec{F} = m\vec{a}

Euler Equation (Rotation)

Angular Momentum

\vec{H} = \mathbf{I} \vec{\omega}

Basic Form

\vec{M} = \frac{d\vec{H}}{dt} = \mathbf{I}\dot{\vec{\omega}} + \vec{\omega} \times (\mathbf{I}\vec{\omega})

Stratospheric Airship Characteristics

Due to large volume and light weight, the inertia matrix has distinct characteristics:

Parameter	Characteristic
$I_{xx}$	Small (slim body)
$I_{yy} = I_{zz}$	Large
$I_{xy} = I_{xz} = I_{yz}$	Approximately zero (symmetry)

Six-DOF Motion Equations
Aerodynamic Force/Coefficient

References

Goldstein H, Poole C, Safko J. Classical Mechanics[M]. Addison-Wesley, 2023.
Etkin B, Reid L D. Dynamics of Flight: Stability and Control[M]. Wiley, 2024.