Module 7
7.1 Static stability
7.2 Longitudinal Stability
7.3 Stick free stability
7.4 Aerodynamic Balancing
7.5 Numerical Problems
7.1 Static stability
Aircraft Static Stability (Longitudinal)
Aircraft Stability and Control – Overview
Aircraft stability and control determine how well an airplane maintains equilibrium and responds to disturbances such as turbulence or pilot inputs. For successful flight, the aircraft must:
- Maintain equilibrium (balanced forces and moments)
- Be capable of controlled maneuvering
- Provide safe and manageable handling qualities
Equilibrium and Trim Condition
An aircraft is in equilibrium (trimmed flight) when:
- Resultant forces = 0
- Resultant moments = 0
Static Stability
Static stability describes the initial tendency of an aircraft to return to its equilibrium condition after a disturbance. "Static stability is the initial tendency of an aircraft to return to equilibrium after disturbance." For a statically stable system:
A restoring force or moment is required for stability.
- A restoring force or moment must act to bring the aircraft back to equilibrium.
- Stability refers only to the initial response.
- Dynamic stability describes how motion evolves with time.
CMcg = 0
If angle of attack increases, a restoring moment must pitch the nose downward. | Type | Description |
|---|---|
| Stable | Returns toward equilibrium |
| Unstable | Moves further away |
| Neutral | Stays in new position |
Static and Dynamic Stability
| Parameter | Static Stability | Dynamic Stability |
|---|---|---|
| Time Consideration | Immediate response | Time evolution |
| Focus | Initial tendency | Complete motion behavior |
| Mathematical Condition | Sign of stability derivative | Solution of differential equation |
| Can Oscillate? | Not considered | Yes |
| Can Grow With Time? | Not described | Yes |
Definition of Longitudinal Static Stability
Longitudinal static stability refers to the ability of an aircraft to develop a restoring pitching moment when its angle of attack is disturbed from the trim condition.
If an external disturbance increases the angle of attack, a statically stable aircraft produces a nose-down moment that returns it toward equilibrium. If the angle of attack decreases, the aircraft produces a nose-up moment restoring the original condition.
Mathematical condition for stability:
- The slope of pitching moment coefficient (Cm) vs angle of attack (α) must be negative
- dCm / dα < 0
Only when this restoring tendency exists is the aircraft considered longitudinally statically stable.
Condition for Static Stability
For longitudinal static stability:
∂CM / ∂α < 0
This means pitching moment must decrease when angle of attack increases. Equivalent condition using lift coefficient: ∂CM / ∂CL < 0
Therefore, stable aircraft generate restoring pitching moments when disturbed. - Stable aircraft → negative slope
- Neutral aircraft → zero slope
- Unstable aircraft → positive slope
Wing Contribution to Stability
The wing produces lift and pitching moment about the center of gravity. The aerodynamic center is approximately located at 25% of the mean aerodynamic chord. Pitching moment depends on the relative positions of:
- Center of gravity (CG)
- Aerodynamic center (AC)
CL = L / (1/2 ρ V² S)
CM = M / (1/2 ρ V² S c̄)
If CG moves behind AC, stability decreases. This is why aircraft use additional stabilizing surfaces such as the tail. Tail Contribution to Stability
The horizontal tail produces lift that generates a stabilizing pitching moment. Key features:
- Tail angle of attack depends on wing downwash.
- Tail moment arm strongly influences stability.
- Tail volume ratio determines effectiveness.
VH = (lt St) / (S c̄)
Tail pitching moment: CMt = − VH η CLt
The tail generally produces a restoring moment that improves longitudinal stability. Tail lift: CLt = CLαt (αw − iw + it − ε)
Downwash approximation: ε = ε0 + (dε/dα) αw
Total pitching moment: CM = CM0 + CMα αw
Control Effects – Elevator Trim
Elevators produce additional lift and pitching moment for trim control. Lift with elevator deflection:
CL = CLα (α − α0) + CLδe δe
Pitching moment: Cm = Cm0 + Cmα α + Cmδe δe
Pitching moment increment: ΔCm = Cmδe δe Trim condition: Cm = 0
Elevator deflection required for trim depends on aircraft speed because lift required for level flight changes. Elevator required for trim:
(δe)TRIM = [ CLα Cm0 + Cmα (CLTRIM + CLα α0) ] / (Cmα CLδe − CLα Cmδe)
Neutral Point and CG Location
The Neutral Point (NP) is the aerodynamic center of the complete aircraft where the pitching moment coefficient does not change with angle of attack.In other words, Neutral Point is defined as the location of the CG at which the pitch stability derivative is zero. Neutral Point Condition:
∂Cm / ∂α = 0
The neutral point is the aerodynamic location where the aircraft becomes neutrally stable. If: - CG is ahead of neutral point → stable
- CG equals neutral point → neutral stability
- CG behind neutral point → unstable
CMα = CLαT (Xcg - Xnp)
For stability: Xcg < Xnp
This is why aircraft CG limits are strictly controlled. Static Margin
Static margin (SM)is a measure of an aircraft's longitudinal static stability. It represents the distance between the Neutral Point (NP) and the Center of Gravity (CG), expressed as a fraction of the mean aerodynamic chord.
SM = (Xnp − Xcg)/MAC
Where • Xcg = CG location • Xnp = Neutral Point Typical aircraft static margin:
5% – 15% of Mean Aerodynamic Chord
| Static Margin | Condition |
|---|---|
| SM > 0 | Stable |
| SM = 0 | Neutral |
| SM < 0 | Unstable |
CG vs Neutral Point Stability Simulator
Neutral Point (hNP)
Dynamic Stability Interactive Module
Motion Description
7.2 Longitudinal Stability
Aircraft Stability & Control Equations
Stick-fixed stability
Stick-fixed stability refers to the aircraft's stability when the elevator is held in a fixed position and is not free to move with the relative wind. In this state, the total pitching moment (Cm) is the sum of the moments generated by individual components like the wing, fuselage, nacelles, and powerplant.
Formula: Cm_total = Cm_{wing} + Cm_{fus} + Cm_{nac} + Cm_{tail} + Cm_{pwr}The wing aerodynamic center is the point where the pitching moment remains constant despite changes in lift
.- Stable Moment: Occurs when the center of gravity (c.g.) is ahead of the a.c.
- Unstable Moment: Occurs when the c.g. is aft of the a.c.
- Location: The a.c. is typically near 25% MAC, while the c.g. range spans 10-40% MAC.
Pitching Moment Model
\[ C_m = C_{m0} + C_{m\alpha}\alpha \] Where:
- Cm: Total pitching moment coefficient.
- Cm0: Pitching moment at zero angle of attack (must be positive for trim).
- Cmα: Longitudinal static stability derivative (slope). For a stable aircraft, this value must be negative.
- α: Angle of attack.
Total Aircraft Lift
\[ L = L_w + L_t \] Tail lift: \[ L_t = \eta q S_t C_{L_t} \]
Tail Angle of Attack
The horizontal tail is the primary component used to counteract the unstable moments of the fuselage and wing[cite: 133]. To ensure static longitudinal stability, the aircraft must have a negative pitching moment slope (dC_m / dC_L < 0). \[ \alpha_t=\alpha - \epsilon + i_t \]
Tail Lift Coefficient
\[ C_{L_t} = C_{L\alpha t}(\alpha - \epsilon + i_t) \]
Downwash Relations
General: \[ \epsilon = \epsilon_0 + \frac{d\epsilon}{d\alpha}\alpha \] Elliptic lift distribution: \[ \epsilon = \frac{2C_L}{\pi AR} \] Rate of change: \[ \frac{d\epsilon}{d\alpha} = \frac{2C_{L\alpha}}{\pi AR} \]
Horizontal Tail Volume Ratio
\[ V_H = \frac{l_t S_t}{S\bar{c}} \]
Tail Pitching Moment Contribution
\[ C_{m,t} = - \eta V_H C_{L_t} \]
Total Aircraft Stability
\[ C_{m\alpha} = C_{m\alpha,w} + C_{m\alpha,t} \] Stability requirement: \[ C_{m\alpha} < 0 \]
Fuselage and Powerplant Effects
Most additional components tend to contribute to instability:
| Component | Stability Effect |
|---|---|
| Fuselage | Generally unstable due to shape and wing airflow (upwash/downwash). |
| Tractor Powerplant | Generally destabilizing because of the force created by turning air at the propeller or intake. |
| Pusher Powerplant | Can reverse destabilizing effects and create stabilizing moments. |
Contribution of Aircraft Components to Stability
The total pitching moment of the airplane is the summation of the moments generated by its individual parts:
Cmtotal = Cmwing + Cmfus + Cmtail + Cmpwr
- Wing: Typically provides an unstable (positive) slope when the C.G. is aft.
- Fuselage: Almost always destabilizing due to its shape and airflow interaction.
- Horizontal Tail: Provides a strong negative slope to pull the total sum into the Stable Region.
Note: For a stable aircraft, the "Total" line must have a negative slope (tilting downwards).
7.3 Stick free stability
Stick-Fixed and Stick-Free Stability in Aircraft
Stabilty
Aircraft longitudinal stability depends not only on the center of gravity (CG) and aerodynamic forces, but also on the behavior of control surfaces, especially the elevator.
Two important stability conditions are analyzed:
- Stick-Fixed Stability
- Stick-Free Stability
These conditions describe whether the control stick (elevator control) is held fixed or allowed to move freely.
Stick-Fixed Stability
Definition: Stick-fixed stability refers to the longitudinal static stability of an aircraft when the elevator is held rigidly in a fixed position.
In this condition:
- The control stick does not move
- The elevator deflection remains constant
- Aircraft response depends only on aerodynamic forces
Mathematical Stability Condition
dCm / dα < 0
If the derivative is negative:
- Increase in angle of attack produces a nose-down moment
- The aircraft returns to equilibrium
- The aircraft is statically stable
Factors Affecting Stick-Fixed Stability
Center of Gravity
Forward CG increases stability, while aft CG reduces stability.
Tail Volume Ratio
VH = (St × Lt) / (S × c)
| Parameter | Description |
|---|---|
| St | Tail Area |
| Lt | Tail Arm |
| S | Wing Area |
| c | Mean Aerodynamic Chord |
Stick-Free Stability
Stick-free stability occurs when the elevator is allowed to float freely without being restrained by the pilot.
The elevator moves under aerodynamic hinge moments until equilibrium is reached.
Zero Hinge Moment Condition
H = 0
When the elevator is free, it rotates until aerodynamic forces balance, causing the hinge moment to become zero.
Effect of Elevator Float
When the elevator floats:
- The elevator deflects automatically
- The tail lift effectiveness reduces
- The restoring pitching moment decreases
Result: Stick-Free Stability < Stick-Fixed Stability
Effect of Static Margin
SM = (NP − CG) / c
Stick Fixed Static Margin
SM_fixed = (NP_fixed − CG) / c
Stick Free Static Margin
SM_free = (NP_free − CG) / c
Since NP_free is forward of NP_fixed:
SM_free < SM_fixed
Stick Force Gradient
A stick force gradient is the change in yoke force as the aircraft’s airspeed deviates from trim condition. An aircraft’s yoke force gives pilots haptic feedback which improves their situational awareness without the need to assess instrument readings. A stick force gradient must provide a positive (pull) force when airspeed is lower than the trim airspeed and a negative (push) force when airspeed is higher than the trim speed. A stick force gradient that is zero provides no haptic feedback forces to the pilot at any airspeed regardless of trim. A positive slope stick force gradient creates an unstable aircraft.
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Y-axis: Stick Force (+ Pull / − Push)
X-axis: Airspeed
X-axis: Airspeed
7.4 Aerodynamic Balancing
Aerodynamic Balancing in Aircraft
What is Aerodynamic Balancing?
- Aerodynamic balancing is the control of hinge moment characteristics of aircraft control surfaces.
- The floating characteristics and stick force depend on hinge moment parameters.
- Two important parameters:
- Chα – Hinge moment variation with angle of attack.
- Chδ – Hinge moment variation with control surface deflection.
- Too low hinge moment → Controls become highly sensitive and unstable.
- Too high hinge moment → Controls become heavy and sluggish.
- Proper balancing ensures stability, controllability, and pilot comfort.
Need for Aerodynamic Balancing
- To reduce excessive stick force.
- To avoid over-sensitive control response.
- To improve flight stability.
- To minimize pilot fatigue during long flights.
- To ensure safe and smooth aircraft operation at different speeds.
Methods Used for Aerodynamic Balancing
1. Set-Back Hinge Balance
2. Horn Balance
3. Internal Balance
4. Beveled Trailing Edge
5. Tab (Balance / Trim / Servo Tab)
Set-Back Hinge Balance
- Hinge line positioned slightly aft of the leading edge of control surface.
- A small portion of the surface lies ahead of the hinge line.
- Produces aerodynamic force that reduces hinge moment.
- Commonly used in light aircraft elevators and rudders.
- Examples: Cessna 172, Piper PA-28 Cherokee, Beechcraft Bonanza.
Horn Balance
- Horn-shaped projection added ahead of hinge line.
- Usually located at the tip of rudder or elevator.
- Air pressure on horn produces counteracting hinge moment.
- Reduces pilot stick force.
- Examples: de Havilland Tiger Moth, Cessna 152, Piper J-3 Cub.
Internal Balance
- Uses sealed internal chambers within control surface.
- Pressure difference generates balancing moment.
- Maintains clean aerodynamic profile.
- Suitable for high-speed and jet aircraft.
- Examples: Supermarine Spitfire, Hawker Hunter, MiG-21.
Beveled Trailing Edge
- Trailing edge cut at an angle.
- Alters pressure distribution over control surface.
- Provides minor hinge moment reduction.
- Simple and economical method.
- Examples: DHC-1 Chipmunk, Cessna 150, Piper PA-18 Super Cub.
Tab (Trim / Servo / Balance Tab)
- Small auxiliary surface attached to trailing edge.
- Produces balancing aerodynamic force.
- Types: Trim Tab, Servo Tab, Balance Tab.
- Highly effective for large control surfaces.
- Examples: Boeing 747, Airbus A320, Douglas DC-3.
Advantages of Aerodynamic Balancing
- Reduces pilot effort and fatigue.
- Improves aircraft controllability.
- Enhances stability at high speeds.
- Prevents over-control and oscillations.
- Essential for safe and efficient aircraft operation.
Comparison Table
| Method | Balancing Location | Effectiveness | Common Usage |
|---|---|---|---|
| Set-Back Hinge | Portion ahead of hinge | Moderate | Light aircraft |
| Horn Balance | External horn projection | Moderate to High | Trainers & small aircraft |
| Internal Balance | Inside control surface | High | High-speed jets |
| Beveled Trailing Edge | Trailing edge | Low to Moderate | Minor corrections |
| Tab | Auxiliary trailing surface | High | Large transport aircraft |
Aileron Reversal
What is Aileron Reversal?
- Aileron reversal is an aeroelastic phenomenon occurring at high flight speeds.
- Aircraft wings are flexible due to aluminum and composite structures.
- Flexibility causes wing twisting under aerodynamic loads.
- This twist reduces or reverses the intended rolling effect of the aileron.
- It negatively affects aileron effectiveness and roll control.
Explanation of the Phenomenon
- Consider a right wing with a downward-deflected aileron.
- At subsonic speeds, aerodynamic load acts near mid-chord.
- At supersonic speeds, load shifts toward the rear of the wing.
- If the load centroid lies behind the elastic axis, the wing twists nose-down.
- This twist reduces the angle of attack of the wing section.
- Lift decreases instead of increasing as intended.
- In extreme cases, the lift direction reverses.
- The roll control derivative (ClδA) changes sign.
- This loss of effectiveness is called aileron reversal.
Effects and Design Considerations
- Occurs mainly at high speeds.
- Limits the operational flight envelope.
- Requires careful structural and aerodynamic design.
- Wing stiffness plays a major role in preventing reversal.
- High-performance aircraft have a defined aileron reversal speed.
- The F-14 fighter aircraft experiences aileron reversal at high speed.
Methods to Prevent Aileron Reversal
- Increase wing structural stiffness.
- Limit aileron deflection at high speeds.
- Use two sets of ailerons (inboard for high speed, outboard for low speed).
- Reduce aileron chord length.
- Use spoilers for roll control.
- Move ailerons toward the inboard wing section.
Practical Example
- The Boeing 747 uses three roll control devices: inboard ailerons, outboard ailerons, and spoilers.
- Outboard ailerons are disabled at high speeds.
- Spoilers (10–15% chord plates) create local lift loss for roll control.
- Proper wing stiffness ensures reversal does not occur within operational limits.
7.5 Numerical Problems
Solved Numerical Example — Aircraft Stability
Given Aircraft Data
Wing area: \[ S = 30 \; m^2 \] Mean aerodynamic chord: \[ \bar{c} = 2.5 \; m \] Tail area: \[ S_t = 8 \; m^2 \] Tail arm: \[ l_t = 6 \; m \] Tail lift curve slope: \[ C_{L\alpha t} = 4.5 \; /rad \] Angle of attack: \[ \alpha = 5^\circ = 0.0873 \; rad \] Tail efficiency: \[ \eta = 0.9 \] Downwash angle: \[ \epsilon = 2^\circ = 0.0349 \; rad \] Tail incidence: \[ i_t = 0^\circ \] Wing stability slope: \[ C_{m\alpha,w} = -0.40 \]
Step 1 — Tail Volume Ratio
\[ V_H = \frac{l_t S_t}{S \bar{c}} \] \[ V_H = \frac{6 \times 8}{30 \times 2.5} \] \[ V_H = \frac{48}{75} = 0.64 \]
Step 2 — Tail Angle of Attack
\[ \alpha_t = \alpha - \epsilon + i_t \] \[ \alpha_t = 0.0873 - 0.0349 + 0 \] \[ \alpha_t = 0.0524 \; rad \]
Step 3 — Tail Lift Coefficient
\[ C_{L_t} = C_{L\alpha t} \alpha_t \] \[ C_{L_t} = 4.5 \times 0.0524 \] \[ C_{L_t} = 0.236 \]
Step 4 — Tail Pitching Moment Contribution
\[ C_{m,t} = - \eta V_H C_{L_t} \] \[ C_{m,t} = - (0.9)(0.64)(0.236) \] \[ C_{m,t} = -0.136 \]
Step 5 — Total Aircraft Stability
\[ C_{m\alpha} = C_{m\alpha,w} + C_{m,t} \] \[ C_{m\alpha} = -0.40 + (-0.136) \] \[ C_{m\alpha} = -0.536 \]
Final Result — Stability Check
Stability requirement: \[ C_{m\alpha} < 0 \] Computed: \[ C_{m\alpha}=-0.536 \] Since value is negative,Aircraft is longitudinally statically stable.
Aircraft Longitudinal Stability Calculator
Enter Aircraft Data
System Stability & control by Dr Aishwarya Dhara