Stability
What the aircraft initially tends to do, and how its disturbed motion develops, without a corrective pilot command.
M0 · Modeling language and readiness
How does an aircraft become a model?
Today we build the shared language for the entire semester: motion, frames, states, stability and the path from physics to feedback.
Supports official CLO1 and CLO4Conceptual orientationNo graded checkpoint
Dr. Ayman M. AbdallahAerospace Engineering Department, KFUPMDirector for Aviation and Space Exploration Research Center
Temporary lesson status. This is an independently authored classroom/study synthesis for Semester 261. The official syllabus controls course facts. No textbook figures or protected textbook text are reproduced.
1
Observe the aircraftMotion, forces, moments and controls
2
Build the nonlinear modelRigid aircraft · 6 degrees of freedom
3
Choose an operating pointEquilibrium and trim
4
Study local behaviorLinearization · states · modes
5
Shape the responseFeedback · SAS · autopilot
01 · The field
Flight dynamics asks how motion evolves
Given an aircraft, its current state, the forces acting on it and the control inputs—what happens next?
Flight dynamics studies the short-term motion of a flight vehicle and the characteristics of that motion. It connects rigid-body mechanics, aerodynamics, propulsion, computation and control.
Dynamics
How position, velocity, attitude and rates change with time under forces and moments.
Control
How deliberate inputs—and later feedback—change the motion toward a desired response.
02 · The integration hub
Flight Dynamics brings the aircraft disciplines together
Aircraft motion is never produced by one subject in isolation. Flight Dynamics integrates the vehicle, its environment and its control system to predict what the aircraft will do next.
Aerodynamics
Aircraft Structures
Propulsion
Flight Control
Aeroelasticity
Flight Dynamicsmotion · stability · response · control
Guidance & Navigation
Aircraft Design
Simulation & Modeling
Flight Testing
Systems Integration
geometry+mass & inertia+aerodynamic forces & moments+propulsion+structural behavior+controls+atmosphere & disturbances+feedback
Predict motion
How will the aircraft move, and is that motion stable after a disturbance?
Predict response
How will it respond to pilot commands, control effectors and propulsion?
Shape response
Can feedback improve damping, stability, tracking or workload?
03 · Semester narrative
One connected engineering chain
The sequence is cumulative. Each stage supplies the model, assumptions or evidence needed by the next.
NL-6DOFAircraft physicsforces, moments, kinematics
TRIMOperating pointequilibrium residuals and constraints
LIN-FULL / LIN-LON / LIN-LATLocal modelslinearization and state space
MODE-APPROXDynamic modeseigenstructure and response
CL-CONTROLFeedback systemrobustness, SAS and autopilot
These model-level labels follow the proposed Phase 2 notation philosophy. They are orientation labels in this temporary lesson, not a formal Phase 3 implementation.
04 · Coordinate language
A component has meaning only after its frame is declared
For the body frame b, this temporary lesson uses the aircraft center of mass as origin: xb forward, yb right, zb down. The axes form a right-handed triad.
Passive component transformation
va = Cab vb
The physical vector is unchanged; its components are re-expressed from frame b into frame a.
+xb forward+yb right+zb downOriginal AE 426 teaching graphic. The three arrows share the center-of-mass origin and form a right-handed body triad.
i
Inertial frame, used only when its origin and orientation are declared.
n
Local navigation frame: North, East, Down under the local-Earth approximation.
b
Aircraft-fixed body frame: forward, right, down.
w / s
Wind follows instantaneous air-relative velocity; stability is tied to the reference condition. They are not interchangeable.
05 · Motion variables
Motion uses three distinct scalar-component groups
Translations describe motion along the body axes. Rotations describe angular velocity about those axes. Attitude angles describe orientation relative to another frame.
| Scalar components | Meaning | Positive body direction / convention |
|---|---|---|
| (u, v, w) | Air-relative velocity components | forward, right, down |
| (p, q, r) | Body angular-rate components | right-hand rule about xb, yb, zb |
| (ϕ, θ, ψ) | 3-2-1 roll, pitch and yaw/heading angles | declared yaw–pitch–roll attitude sequence |
Translational velocity components(u, v, w)
forward · right · down, resolved in body axes
Body angular-rate components(p, q, r)
roll-rate · pitch-rate · yaw-rate components
Attitude / Euler angles(ϕ, θ, ψ)
roll · pitch · yaw/heading in the declared 3-2-1 sequence
06 · A crucial distinction
p, q, r are not generally ϕ̇, θ̇, ψ̇
Body angular rates[p, q, r]T
Components of one angular-velocity vector resolved in body axes.
≠
Euler-angle rates[ϕ̇, θ̇, ψ̇]T
Rates of three sequential attitude coordinates.
They become approximately equal only near small roll and pitch angles. The inverse mapping becomes singular when cos θ = 0.
p = ϕ̇ − ψ̇ sin θ
q = θ̇ cos ϕ + ψ̇ sin ϕ cos θ
r = −θ̇ sin ϕ + ψ̇ cos ϕ cos θ
q = θ̇ cos ϕ + ψ̇ sin ϕ cos θ
r = −θ̇ sin ϕ + ψ̇ cos ϕ cos θ
Safe takeaway for today
Never relabel body rates as Euler-angle rates. First declare the 3-2-1 convention, then use the kinematic mapping.
07 · Air-relative geometry
α and β describe how air-relative velocity meets the aircraft
The angles are geometric properties of the air-relative velocity vector—not decorative attitude angles and not ground-track angles. The complementary projections below connect the physics to the scalar components.
Side view · angle of attack α
α+w downu lies along +xb; positive w tilts VA below +xb in the forward–right–down body frame.Top view · sideslip angle β
β+v rightv is the lateral body-axis component; positive β follows positive v under the adopted convention.VAb = [u, v, w]T VA = √(u² + v² + w²)
α = atan2(w, u) β = atan2(v, √(u² + w²))
α = atan2(w, u) β = atan2(v, √(u² + w²))
Ground speed and air-relative speed are different vectors whenever wind is present. Aerodynamic forces are tied to air-relative motion; the green arrows depict VA, while the darker references depict body-axis directions.
08 · Control effectors
What moves—and what does it primarily influence?
Aerodynamic surfaces change forces and moments by changing the local flow. Throttle is different: it commands the propulsion system and therefore the thrust force and flight condition.
Ailerons · δadifferential wing surfaces → primarily rolling moment
Elevator · δehorizontal-tail surface → primarily pitching moment
Rudder · δrvertical-tail surface → primarily yawing moment
Throttle / thrust command · δTpropulsion command → changes thrust and operating condition
Primary does not mean isolated
Aerodynamic and propulsion effects are coupled. This overview does not assign an unresolved course-wide positive surface-deflection sign; later quantitative diagrams must show each physical deflection explicitly.
thrust / δT
Ailerons δaElevator δeRudder δrPropulsion / thrust
09 · Response to disturbance
Static tendency and dynamic history answer different questions
Static stability concerns the initial restoring or departing tendency after a small disturbance from a steady condition. Dynamic stability concerns the full time history—whether the motion decays, persists or grows.
Important relationship
In the ordinary linear context used here, dynamic stability requires the appropriate static tendency, but a statically stable aircraft is not automatically dynamically stable.
10 · The operating point
Trim is a reference operating condition
A trim condition is a selected steady or equilibrium motion in which the required force, moment and kinematic residuals satisfy declared constraints. It supplies the reference state x0 and input u0.
ẋ = f(x, u, parameters)
0 = R(x0, u0; constraints)
0 = R(x0, u0; constraints)
EquilibriumSelected states do not change under the declared model.
Steady motionRelevant dynamic quantities remain constant, though some coordinates may keep changing.
TrimState, attitude, controls and thrust close the selected residuals and constraints.
Steady level flight
Steady climb
Steady descent
Steady coordinated turn
Not automatic: trim does not require level flight, zero pitch angle, zero heading change or zero motion. The declared constraints determine which operating condition is being solved.
11 · From nonlinear physics to a local model
We linearize to expose structure near trim
The nonlinear equations describe broad behavior but can hide the local couplings that govern modes and feedback. Write each variable as trim plus a perturbation, keep first-order terms, and record what was neglected.
x = x0 + Δx, u = u0 + Δu
Δẋ = A Δx + B Δu
Δy = C Δx + D Δu
Δẋ = A Δx + B Δu
Δy = C Δx + D Δu
What the matrices mean
A links state perturbations to state-rate perturbations. B links input perturbations to state-rate perturbations. C and D define the measured or reported outputs.
A = (∂f/∂x)0 B = (∂f/∂u)0
Local means local
The model is only trustworthy near the declared trim condition and under the assumptions used to derive it. Smaller perturbations should generally improve agreement with a correct first-order model.
12 · Natural response patterns
Modes are coordinated patterns of aircraft motion
Eigenvalues describe growth/decay and oscillation; eigenvectors help reveal which states participate. Named aircraft modes are physical interpretations of this coupled structure—not labels assigned from a pole alone.
A v = λ v
λ = σ + jωd stable continuous-time mode: σ < 0
λ = σ + jωd stable continuous-time mode: σ < 0
Longitudinal
- Short-period: faster pitch/angle-of-attack-dominated response.
- Phugoid: slower exchange involving speed and flight-path/attitude behavior.
Lateral-directional
- Roll subsidence: primarily roll-rate decay.
- Dutch roll: coupled yaw–roll–sideslip oscillation.
- Spiral: slow bank/heading tendency that may be stable or unstable.
These descriptions are intentionally qualitative. Exact time scales, damping targets and handling-quality claims require a declared aircraft, condition, model and applicable source; no universal values are asserted here.
13 · Shaping the response
Feedback watches the aircraft—and adjusts what happens next
A command is converted into control action. The aircraft moves; sensors measure selected motion; that measurement returns to the controller. The dynamic aircraft is the physical system at the center of the loop.
Command
→
Controller
→
Actuator / control effector
→
Aircraft dynamics
→
Motion + sensors
measured output / feedback returns to the controller
Feedback
Uses measured motion to modify control action. Signal definitions and feedback sign must be explicit.
Stability Augmentation System
An SAS uses inner feedback to reshape selected aircraft modes or improve damping.
Autopilot
An autopilot commands higher-level variables such as attitude, altitude or flight path through a declared loop architecture.
Function is not implementation. Fly-by-wire is an implementation architecture; it may contain stability augmentation, command augmentation and autopilot functions. “Inner loop faster than outer loop” is a design heuristic, not proof of stability.
14 · How the pieces support your learning
Two references, one AE 426 learning path
Continuity / foundational reference
Yechout
- frames and nonlinear equations
- aircraft mechanics and derivative foundations
- continuity with existing AE 426 examples
Principal integration / enrichment reference
Nelson
- stability and state-space aircraft models
- dynamic modes and classical feedback
- SAS and autopilot integration
The lesson remains self-contained at its intended level. The books provide authoritative depth and alternative explanations; their notation is translated carefully rather than copied wholesale.
Ungraded · diagnostic only
15 · Prerequisite check
Six questions before we build the model
Answer from memory first. Reveal the reasoning after you commit. These questions identify what to refresh; they do not contribute to your grade.
1
A vector is transformed from body components to navigation components. What physical quantity changes?
A) The vector itself · B) Only its components · C) Its magnitude must change
Answer: B. A passive coordinate transformation re-expresses the same physical vector. For a proper orthogonal DCM, its Euclidean norm is preserved.
2
For a right-handed body frame with x forward and y right, which direction completes the triad?
Use x × y = z.
Answer: z points down. Forward × right gives downward for the adopted aircraft body-axis convention.
3
When can p, q, r be approximated by ϕ̇, θ̇, ψ̇?
A) Always · B) Near small roll and pitch angles · C) Only at high airspeed
Answer: B. The exact 3-2-1 mapping contains trigonometric coupling. The component sets are approximately equal near ϕ ≈ 0 and θ ≈ 0.
4
For a continuous linear mode λ = σ + jω, what sign of σ indicates decay?
Answer: σ < 0. The modal envelope contains eσt, so a negative real part decays with time.
5
Why must a trim point be defined before linearization?
Reason: the Jacobians A and B are evaluated at a specific reference state and input. Change the operating point and the local model can change.
6
If the states have mixed units, do all entries of A have units of s−1?
Answer: not necessarily. Each entry satisfies unit(Aij) = unit(ẋi) / unit(xj). Mixed state units produce mixed matrix-entry units.
16 · Your input
Help Shape Your AE 426 Learning Experience
Your responses will help guide the examples, review material, simulations, videos, learning resources, and project activities used in AE 426.
The academic rigor and learning requirements of AE 426 remain unchanged. The survey helps improve how the course supports you in meeting them.
17 · Takeaway
Carry these four habits into every equation
Declare
frame, axes, signs, units and model level
Anchor
the model to a feasible equilibrium or trim condition
Interpret
states, modes and feedback through aircraft motion
Verify
with dimensions, limiting cases and independent checks
Next connection
Frames and kinematics lead directly into the nonlinear rigid-aircraft equations of motion.
Source anchors
- Current AE 426 CH0 Course Overview, slides 1–35: scope, 6-DOF, axes, stability and control baseline.
- Yechout, 2nd ed., Ch. 4 §§4.1–4.2, printed pp. 163–170: frames and 3-2-1 kinematics.
- Nelson, 2nd ed., Ch. 3 §§3.2–3.4, printed pp. 97–104: frames, Euler angles and body-rate distinction.
- AE 426 Phase 2 Master Course Map, Notation/Convention Standard, Learning Experience Specification, Textbook Integration Map, Technical Correction Register and Copyright/Licensing Register.
Technical boundary: Established frame/rate/angle relations are used where they have strong textbook and independent support. Unresolved control-surface signs, numerical datasets, universal handling-quality values and unapproved technical corrections are omitted or explicitly flagged.