This File includes the very basics of airplane dynamic stability, rigid body equation of motion, and moment/torque equation of motion of an airplane.

1. Introduction to Aircraft Dynamic
Stability & Equations of motion
-Dhruv Panchal

2. Dynamic Stability
• Purpose is to Understand of
Dynamic Stability and Response of
the airplane
• Behavior or Response of an airplane
when it disturbs (Small
Perturbation) from Steady State
• Dynamic stability is how an airplane
responds over time to a disturbance
and How transient is behaving w.r.t
time whether it is coming back to its
equilibrium position or not
X(t)= Disturb/perturb variable

3. Dynamically Stable System
After the Disturbance, an aircraft comes back to its equilibrium state.

4. Dynamically Unstable System
After the Disturbance, an aircraft does not come back to its equilibrium state.

5. Dynamically Neutrally Stable System
After disruption, the amplitude remains unchanged and with constant oscillation.

6. Example
Aircraft is flying at cruise with 5° AOA and given 0.5° AOA disturbance
𝛼 = 5°

7. Mass-Damper-Spring
System

8. Response
It is defined as the change with time of motion variable related to some
steady-state flight condition resulting from an externally or internally
generated disturbance.

9. K= Spring Coefficient
C= Damping Coefficient
X= Displacement
Restoring force ∝ Displacement
Rate of Change of Distrubnace ∝ cx’
Why Mass Spring
and Damper
System?

10. Analogy of Spring
Stiffness with
flight dynamics
• 𝐶 𝑚 = 𝐶 𝑚0
+ 𝑪 𝒎 𝜶
𝜶
• 𝐶 𝑚 𝛼
𝛼 = Restoring moment = 𝑘𝑥(Restoring moment
of spring in case of m.s.d.s)
• 𝐶 𝑚 𝛼
𝛼 = 𝑘𝑥(Analogous )
ⅆ𝐶 𝑚
ⅆ𝛼
< 0
Pitching Moment Coefficient =𝐶 𝑚 =
𝑃𝑀
1
2
𝜌𝑣2 𝑠 ҧ𝑐
𝐶 𝑚
𝛼

11. Analogy of Damping
with Flight Dynamics
• q = Pitching Rate
• 𝑙 𝑡 = Distance between CG to A.C(Tail)
• Δ𝛼 =
𝑞𝑙 𝑡
𝑣
• M ∝
𝑞𝑙 𝑡
𝑣
= x’c = Analogy with Damping
𝑞𝑙 𝑡
𝑣

13. Equation of Motion of
M.S.D System Using
N.L.M
Restoring Force ∝ 𝑘𝑥
(Linear Damping)Damping Force ∝ c𝑥′(𝑡)
𝐹 = 𝑚𝑎
−𝑘𝑥 − 𝑐𝑥′
= 𝑀
ⅆ2 𝑥
ⅆ𝑡2

14. Rigid Body Equation of
Motion
• (𝑥 𝑏 , 𝑦 𝑏 , 𝑧 𝑏 ) = Body Fixed Axes
System
• 6 Degree of Freedom – 3
translation and 3 Rotational Motion
• Need to do all measurement w.r.t
inertial frame of reference.

15. Inertial Frame of Reference
• Inertial Frame : No Acceleration
• For Aircraft taking Earth as inertial frame
• For our transient study we made some
assumption
1. Neglect the rotation of Earth
2. We take flat earth
• Challenges
1. Different Orientation of Aircraft, Moment of
inertia of an airplane w.r.t to inertial frame go
on changing
2. Aerodynamic Forces ≅ Body Frame

16. Rigid Body Equation of Motion
• ത𝐹 =
ⅆ
ⅆ𝑡
𝑀 ҧ𝑣 = 𝑀
ⅆത𝑣
ⅆ𝑡
+ ҧ𝑣
ⅆ𝑀
ⅆ𝑡
•
ⅆ𝑀
ⅆ𝑡
=0 (For Dynamic Stability analysis and
small duration)
• ത𝐹= 𝑀
ⅆത𝑣
ⅆ𝑡
, ҧ𝑣= Velocity Vector w.r.t. Inertial
Frame of Reference
• ത𝐹 = 𝑀
ⅆത𝑣
ⅆ𝑡

17. Developing The Equation
of Motion
• σ 𝛿𝑀 = 𝑀
𝛿𝑀

18. Rigid body Equation of Motion
• 𝛿𝑀 = Elemental Mass
• 𝛿 ത𝐹 = 𝛿𝑀
ⅆത𝑣
ⅆ𝑡
• Now ҧ𝑣 w.r.t to inertial frame
• ҧ𝑣 = ҧ𝑣𝑐 +
𝑑 ҧ𝑟
ⅆ𝑡
• 𝛴𝛿 ത𝐹 = ത𝐹 =
ⅆ
ⅆ𝑡
𝛴 ҧ𝑣𝑐 +
𝑑 ҧ𝑟
ⅆ𝑡
𝛿𝑀
𝜹𝑴

19. Rigid body Equation of Motion
• ത𝐹 = 𝛿𝑀
ⅆത𝑣 𝑐
ⅆ𝑡
+
𝑑2
ⅆ𝑡2 𝛴 ҧ𝑟𝛿𝑀
• ത𝐹 = 𝛿𝑀
ⅆത𝑣 𝑐
ⅆ𝑡
(B.A.S Located at C.G.)
• ത𝐹 = 𝛿𝑀
ⅆത𝑣 𝑐
ⅆ𝑡
• ҧ𝑣𝑐=𝑢 𝐸
ሶ𝑙 𝐸 + 𝑣 𝐸 ሶ𝑗 𝐸+ 𝑤 𝐸 𝑘 𝐸

20. Moment Equation of Motion
• External Moment(m) = Rate of change of
angular momentum(H)
• 𝛿𝑚 =
ⅆ
ⅆ𝑡
𝛿 ഥ𝐻 =
ⅆ
ⅆ𝑡
( ҧr × ҧ𝑣) 𝛿𝑀
• ҧ𝑣 = ҧ𝑣𝑐 +
𝑑 ҧ𝑟
ⅆ𝑡
= ҧ𝑣𝑐 + (ഥ𝜔 × ҧ𝑣)
• ഥ𝜔= Angular velocity w.r.t. Inertial Frame
• ഥ𝐻= 𝛴( ҧr × ҧ𝑣) 𝛿𝑀
Angular momentum = ҧr × 𝑀𝑣
M

21. Moment Equation of Motion
• ഥ𝐻 = 𝛴( ҧr × ҧ𝑣𝑐 + (ഥ𝜔 × 𝑣) 𝛿𝑀
• ഥ𝐻= 𝛴 ҧr𝛿𝑀 × ҧ𝑣𝑐 + ҧr × (ഥ𝜔 × 𝑣) 𝛿𝑀
• ഥ𝐻 = ҧr × (ഥ𝜔 × 𝑣) 𝛿𝑀
• ҧr= 𝑥 Ƹ𝑖 𝐸 + 𝑦 ሶƸ𝑗 𝐸+ z෠𝑘 𝐸
• ഥ𝜔= 𝑃 Ƹ𝑖 𝐸 + Q ሶƸ𝑗 𝐸+ R෠𝑘 𝐸

22. Moment Equation of Motion
ഥ𝜔 × ҧr =
Ƹ𝑖 𝐸
ሶƸ𝑗 𝐸
෠𝑘 𝐸
𝑃 Q R
𝑥 𝑦 z
= (𝑄𝑧 − 𝑅𝑦) Ƹ𝑖 𝐸 − (𝑃𝑧 − 𝑅𝑥) ሶƸ𝑗 𝐸 + ෠𝑘 𝐸(Py − 𝑄𝑥)
ഥ𝐻 = ҧr × (ഥ𝜔 × 𝑣) =
Ƹ𝑖 𝐸
ሶƸ𝑗 𝐸
෠𝑘 𝐸
𝑥 y z
𝑄𝑧 − 𝑅𝑦 𝑃𝑧 − 𝑅𝑥 Py − 𝑄𝑥

23. Moment Equation of Motion
ഥ𝐻 = 𝛴 𝑃(𝑦2+𝑧2) − 𝑄𝑦𝑥 − 𝑅𝑧𝑥 𝛿𝑀 − 𝛴(
)
𝑃𝑥𝑦 − 𝑄𝑥2 − 𝑄𝑧2 +
𝑅𝑦𝑧 𝛿𝑀 + 𝛴(𝑅𝑥2 − 𝑃𝑥𝑧 − 𝑄𝑧𝑦 + 𝑦2 𝑅) 𝛿𝑀
𝑃(𝑦2+𝑧2) − 𝑄𝑦𝑥 − 𝑅𝑧𝑥 = Ƹ𝑖 𝐸 Component
𝑃𝑥𝑦 − 𝑄𝑥2 − 𝑄𝑧2 + 𝑅𝑦𝑧 = ሶƸ𝑗 𝐸 Component
𝑅𝑥2 − 𝑃𝑥𝑧 − 𝑄𝑧𝑦 + 𝑦2 𝑅 = ෠𝑘 𝐸 Componen
𝐼 𝑥𝑥 = 𝛴𝑃(𝑦2+ 𝑧2)𝛿𝑀, Similarly for 𝐼 𝑦𝑦 𝑎𝑛𝑑 𝐼𝑧𝑧.

24. Moment Equation of Motion
ഥ𝐻 = 𝐻 𝑥 Ƹ𝑖 𝐸+ 𝐻 𝑦
ሶƸ𝑗 𝐸+ 𝐻𝑧
෠𝑘 𝐸
𝐻 𝑥 = 𝑃𝐼 𝑥𝑥 − 𝑄𝐼 𝑥𝑦 − 𝑍𝐼 𝑥𝑧
𝐻 𝑦 = −𝑃 𝐼 𝑥𝑦 + 𝑄𝐼 𝑦𝑦 + 𝑅𝐼 𝑦𝑧
𝐻𝑧 = −𝑃𝐼 𝑥𝑧 − 𝑄𝐼 𝑦𝑧 + 𝑅𝐼𝑧𝑧