Six-DOF Equations of Motion

Rigid-body translation, rotation, and the coordinated turn

Aerospace EngineeringFlight Dynamics, Avionics & ControlFree preview

⏱️ About 16 min

Six-DOF Equations of Motion — illustration

Decorative illustration.

How do Newton's and Euler's laws govern an aircraft twisting and turning through three-dimensional space?

💡

The big idea: Aircraft motion resolves into three translational and three rotational degrees of freedom, coupling inertia, aerodynamics, and gravity.

🎯 By the end, you'll be able to

Write the 6-DOF rigid-body equations in body axes.
Identify the gyroscopic cross-coupling term in the rotational equations.
Derive and compute the coordinated turn rate and radius.

📎 Helpful to know first

Familiarity with vectors, cross products, and basic aircraft axes.

Rigid-Body Dynamics

Flight dynamics begins with Newton's second law applied to a rigid airframe. In body axes, translation is governed by \(\mathbf{F} = m(\dot{\mathbf{V}} + \boldsymbol{\omega} \times \mathbf{V})\), accounting for moving frames.

Rotational Motion

Rotation follows Euler's equation. The angular momentum changes according to applied moments, but the inertia tensor and body rates introduce inertial coupling.

\[ \mathbf{M} = \mathbf{I}\dot{\boldsymbol{\omega}} + \boldsymbol{\omega}\times(\mathbf{I}\boldsymbol{\omega}) \]

Euler Angles and Coordinated Turns

For a level coordinated turn, lift balances weight and provides centripetal force. Bank angle \(\phi\) and true airspeed \(V\) dictate the yaw rate \(\dot{\psi}\).

\[ \dot{\psi} = \dfrac{g\tan\phi}{V} \]

🔑 Gyroscopic Coupling

The term \(\boldsymbol{\omega}\times(\mathbf{I}\boldsymbol{\omega})\) causes gyroscopic moments. In aircraft with large spinning engines or rapid rates, pitching induces yawing and vice versa.

🎮 Coordinated Turn Explorer LIVE

Predict first: Set bank angle and true airspeed to see the steady turn rate and radius.

In a level coordinated turn the six-DOF equations reduce to a balance of lift, weight and yaw rate.

📝 Worked example: A jet flies a level coordinated turn at bank angle \(\phi = 30^\circ\) and true airspeed \(V = 100\) m/s (\(g = 9.81\)). Find the turn rate \(\dot{\psi}\) and turn radius \(R\).

1. tan(30 deg) = 0.5774
2. psi_dot = g tan(phi)/V = 9.81*0.5774/100 = 0.0566 rad/s = 3.25 deg/s
3. R = V^2/(g tan phi) = 10000/(9.81*0.5774) = 1766 m

✓ The turn rate is about 0.0566 rad/s (3.25 deg/s) and the radius is about 1766 m.

✏️ Practice: Compute the turn rate and radius for \(\phi = 45^\circ\) and \(V = 120\) m/s.

Solution

Hint: psi_dot = g tan phi / V, R = V^2/(g tan phi).
Answer: psi_dot = 0.0818 rad/s (4.68 deg/s); R = 1468 m.

Check your understanding

1. In the rotational equation M = I omega_dot + omega x (I omega), the term omega x (I omega) represents:

The cross-product term couples the body-axis rates, producing gyroscopic coupling between roll, pitch and yaw.

2. For a fixed airspeed, increasing the bank angle in a coordinated turn makes the turn radius:

R = V^2/(g tan phi): a larger bank angle increases tan phi and shrinks the radius.

✅ Key takeaways

The 6-DOF equations couple translational and rotational dynamics in body axes.
The \(\boldsymbol{\omega} \times (\mathbf{I}\boldsymbol{\omega})\) term creates inertial cross-coupling between axes.
Coordinated turn relations yield simple formulas for turn rate and radius from bank angle and speed.

➡️ Having established the full nonlinear 6-DOF model, we next linearise these equations around trim to study longitudinal modes.

Want to test yourself on this? Try the Aerospace Engineering Aptitude test →