1. Lecture #4 Topics Translational Equations of Motion
Kinematic Equations
Rotational Equations of Motion
Discussion of Inertial Coupling

3. Rotational Equations

4. Rotational Equations of Motion
𝐼 𝑥𝑥 𝑃 − 𝐼 𝑥𝑧 𝑅 +𝑃𝑄 − 𝐼 𝑦𝑧 𝑄 2 − 𝑅 2 − 𝐼 𝑥𝑦 𝑄 −𝑅𝑃 𝑄 −𝑅𝑃 + 𝐼 𝑧𝑧 − 𝐼 𝑦𝑦 𝑅𝑄= 𝐿 𝐴 + 𝐿 𝑃
𝐼 𝑦𝑦 𝑄 + 𝐼 𝑥𝑥 − 𝐼 𝑧𝑧 𝑃𝑅− 𝐼 𝑥𝑦 𝑃 +𝑄𝑅 − 𝐼 𝑦𝑧 𝑅 −𝑃𝑄 + 𝐼 𝑥𝑧 𝑃 2 − 𝑅 2 = 𝑀 𝐴 + 𝑀 𝑃
𝐼 𝑧𝑧 𝑅 − 𝐼 𝑥𝑧 𝑃 −𝑄𝑅 − 𝐼 𝑥𝑦 𝑃 2 − 𝑄 2 − 𝐼 𝑦𝑧 𝑄 +𝑅𝑃 𝑄 +𝑅𝑃 + 𝐼 𝑦𝑦 − 𝐼 𝑥𝑥 𝑃𝑄= 𝑁 𝐴 + 𝑁 𝑃
Inertia important for design
The L, M, N refers to rolling, pitching and yawing moment.
Given the moment induced at the center of mass, the equations allow prediction of the roll, yaw and pitch rates. (Remind that it is wrt to inertial)
Not in traditional form:- Since the left hand side derivatives. Not in convenient form to solve (integrate the equations). Next couple of slides will be focused on bringing the equations in traditional form.

5. Moments of Inertia

6. Simplification: Symmetric Aircraft
𝐼 𝑥𝑦 = 𝐼 𝑦𝑥 = 𝐼 𝑦𝑧 = 𝐼 𝑧𝑦 =0
𝐼 𝑥𝑥 𝑃 − 𝐼 𝑥𝑧 𝑅 +𝑃𝑄 + 𝐼 𝑧𝑧 − 𝐼 𝑦𝑦 𝑅𝑄= 𝐿 𝐴 + 𝐿 𝑃
𝐼 𝑦𝑦 𝑄 + 𝐼 𝑥𝑥 − 𝐼 𝑧𝑧 𝑃𝑅+ 𝐼 𝑥𝑧 𝑃 2 − 𝑅 2 = 𝑀 𝐴 + 𝑀 𝑃
𝐼 𝑧𝑧 𝑅 − 𝐼 𝑥𝑧 𝑃 −𝑄𝑅 + 𝐼 𝑦𝑦 − 𝐼 𝑥𝑥 𝑃𝑄= 𝑁 𝐴 + 𝑁 𝑃

7. Special Case: Fuselage Heavy Aircraft
𝐼 𝑥𝑧 = 𝐼 𝑧𝑥 ≈0

8. Fuselage Heavy Aircraft Rotational Motion
𝐼 𝑥𝑦 = 𝐼 𝑦𝑥 = 𝐼 𝑦𝑧 = 𝐼 𝑧𝑦 =0; 𝐼 𝑥𝑧 = 𝐼 𝑧𝑥 ≈0
𝐼 𝑥𝑥 𝑃 + 𝐼 𝑧𝑧 − 𝐼 𝑦𝑦 𝑅𝑄= 𝐿 𝐴 + 𝐿 𝑃
𝐼 𝑦𝑦 𝑄 + 𝐼 𝑥𝑥 − 𝐼 𝑧𝑧 𝑃𝑅= 𝑀 𝐴 + 𝑀 𝑃
𝐼 𝑧𝑧 𝑅 + 𝐼 𝑦𝑦 − 𝐼 𝑥𝑥 𝑃𝑄= 𝑁 𝐴 + 𝑁 𝑃

9. Inertial Coupling

10. Inertial Coupling
A potentially lethal phenomenon of high-speed flight in which the inertia of the heavier fuselage overpowers the aerodynamic stabilizing forces of the wing and empennage
The problem became apparent post Wwar II in airplanes with short wing span.

11. Inertial Coupling
Occurs when the aircraft is maneuvering at steady roll rate and low angle of attack (small aerodynamic and propulsive moment flight condition). 𝑃 =0 ⟹𝑃= 𝑃 0

12. Rearrange Equations: Rotational Motion
𝑃 =− 𝐼 𝑧𝑧 − 𝐼 𝑦𝑦 𝐼 𝑥𝑥 𝑅𝑄+ 𝐿 𝐴 + 𝐿 𝑃
𝑄 =− 𝐼 𝑥𝑥 − 𝐼 𝑧𝑧 𝐼 𝑦𝑦 𝑃𝑅+ 𝑀 𝐴 + 𝑀 𝑃
𝑅 =− 𝐼 𝑦𝑦 − 𝐼 𝑥𝑥 𝐼 𝑧𝑧 𝑃𝑄+ 𝑁 𝐴 + 𝑁 𝑃

13. X-2
𝐼 𝑥𝑥 =5043< 𝐼 𝑦𝑦 =25474≈ 𝐼 𝑧𝑧 =29106
𝑃 =− 𝐼 𝑧𝑧 − 𝐼 𝑦𝑦 𝐼 𝑥𝑥 𝑅𝑄+ 𝐿 𝐴 + 𝐿 𝑃
𝑄 =− 𝐼 𝑥𝑥 − 𝐼 𝑧𝑧 𝐼 𝑦𝑦 𝑃𝑅+ 𝑀 𝐴 + 𝑀 𝑃
𝑅 =− 𝐼 𝑦𝑦 − 𝐼 𝑥𝑥 𝐼 𝑧𝑧 𝑃𝑄+ 𝑁 𝐴 + 𝑁 𝑃
Izz-Iyy =0; Pdot =0
Ixx-Izz/Iyy =1 Qdot =PR
Rdot = -PQ

15. References
“On the Roll-Coupling Instabilities of High- Performance Aircraft”, Craig C. Jahnke, Vol. 356, No , Nonlinear Flight Dynamics of High- Performance Aircraft (Oct. 15, 1998), pp , Royal Society,
“Coupling Dynamics in Aircraft: A Historical Perspective”, Richard E. Day, Dryden Flight Research Center, Edwards, NASA Special Publication 532, 1997
“Mechanics of Flight”, Warren Phillips, ISBN-13: