Flight Dynamics & Control

Flight dynamics is the study of how aircraft move through the air and how they respond to control inputs. Understanding flight dynamics is essential for safe flight operation, aircraft design, and flight control system development.

Aircraft Axes and Motion

An aircraft has three primary axes of rotation:

Longitudinal Axis (Roll)

• Runs from nose to tail
• Motion: Roll (banking left or right)
• Control surface: Ailerons

Lateral Axis (Pitch)

• Runs from wingtip to wingtip
• Motion: Pitch (nose up or down)
• Control surface: Elevator (or stabilator)

Vertical Axis (Yaw)

• Runs vertically through the aircraft
• Motion: Yaw (nose left or right)
• Control surface: Rudder

Reference Frame and Euler Angles

Body-Fixed Reference Frame

The aircraft-fixed coordinate system (x, y, z) is typically defined as:

• x-axis: Points forward (longitudinal axis)
• y-axis: Points to the right (lateral axis)
• z-axis: Points downward (vertical axis)

Euler Angles

Euler angles describe aircraft attitude:

• $\phi$ (phi): Roll angle
• $\theta$ (theta): Pitch angle
• $\psi$ (psi): Yaw angle (heading)

Equations of Motion

Force Equations

The six-degree-of-freedom equations of motion for an aircraft are:

Where:

• $m$ = aircraft mass
• $\vec{V}$ = velocity vector
• $\vec{F}$ = applied forces (lift, drag, thrust, weight)
• $\vec{H}$ = angular momentum vector
• $\vec{M}$ = applied moments

Linearized Equations

For small disturbances from steady flight, equations can be linearized:

Where $\delta\vec{x}$ is the state vector and $\delta\vec{u}$ is the control input vector.

Static Stability

Longitudinal Stability

Longitudinal stability refers to the aircraft's tendency to return to its trim angle of attack after a disturbance.

The pitching moment coefficient is:

For static stability, $C_{m\alpha} < 0$ (stabilizing moment).

Neutral Point

The neutral point is the center of gravity location where the aircraft has neutral stability:

Where $V_H$ is the horizontal tail volume coefficient.

Static Margin

Where $h$ is the center of gravity location and $c_{mean}$ is the mean aerodynamic chord.

Dynamic Modes

Longitudinal Modes

Short Period Mode

• High frequency oscillation
• Involves rapid pitching motion with little change in airspeed
• Well-damped in stable aircraft
• Natural frequency: 1-5 rad/s

Phugoid Mode

• Long-period oscillation between kinetic and potential energy
• Airspeed and altitude exchange
• Lower frequency and lightly damped
• Period: 30-200 seconds

Lateral-Directional Modes

Roll Mode

• Pure rolling motion
• Usually very stable and quickly damped
• Time constant: < 1 second for well-designed aircraft

Dutch Roll

• Coupled yawing and rolling oscillation
• Common in swept-wing aircraft
• Damping depends on dihedral effect and vertical tail

Spiral Mode

• Slow divergence or convergence of flight path
• Can be unstable in some aircraft configurations
• Usually well-damped in well-designed aircraft

Control Systems

Primary Controls

• Ailerons: Create differential lift for roll control
• Elevator: Changes tail lift for pitch control
• Rudder: Creates side force for yaw control

Secondary Controls

• Flaps: Increase lift and drag for landing/low-speed flight
• Slats: Increase stall angle of attack
• Spoilers: Reduce lift and increase drag

Control Power

The moment generated per control surface deflection:

Control Authority

The maximum moment available from control surfaces.

Aircraft Handling Qualities

MIL-F-8785C Specifications

Aircraft are categorized by mission requirements:

Category I: Normal flight tasks with minimal pilot attention Category II: Moderate pilot attention, quick response not required Category III: Active pilot attention, precise control required

Cooper-Harper Scale

A rating scale from 1-10 for handling qualities:

• 1: Excellent
• 2: Good
• 3: Adequate
• 4-10: Inadequate for various reasons

Real-World Application: Aircraft Stability Analysis

Consider a general aviation aircraft with the following characteristics:

• Weight: 2,500 lbs (11,120 N)
• Wing area: 180 ft² (16.7 m²)
• Mean aerodynamic chord: 5.0 ft (1.52 m)
• Center of gravity: 25% MAC

Stability Analysis Example

import math

# Aircraft parameters
weight = 11120    # N
wing_area = 16.7  # m²
c_bar = 1.52      # m (mean aerodynamic chord)
rho = 1.225       # kg/m³ (air density at sea level)

# Cruise conditions
airspeed = 60     # m/s
alpha_trim = 4    # degrees

# Aerodynamic coefficients (simplified)
CL_alpha = 0.1    # per degree (lift curve slope)
Cm_alpha = -0.02  # per degree (pitching moment slope)
Cm_delta_e = -0.5 # elevator effectiveness

# Calculate lift coefficient needed for level flight
CL_required = weight / (0.5 * rho * airspeed**2 * wing_area)

# Check static stability (Cm_alpha < 0 indicates stability)
is_stable = Cm_alpha < 0

print(f"Lift coefficient required: {CL_required:.3f}")
print(f"Static margin (stability): {'Stable' if is_stable else 'Unstable'}")
print(f"Required angle of attack: {alpha_trim:.1f} degrees")

# Calculate required elevator deflection for trim
delta_e_trim = -Cm_alpha * alpha_trim / Cm_delta_e
print(f"Required elevator deflection: {delta_e_trim:.2f} degrees")

Your Challenge: Aircraft Response Analysis

Analyze the response of an aircraft to a sudden control input and understand the resulting motion.

Goal: Calculate the aircraft's response to a step input in elevator deflection and understand the resulting dynamics.

Aircraft Parameters

# Simplified aircraft model parameters
mass = 1500      # kg
inertia_y = 35000  # kg·m² (pitch moment of inertia)

# Aerodynamic derivatives (typical for transport aircraft)
CZ_alpha = -0.45   # Lift slope (approximate)
Cm_alpha = -0.4    # Pitch stiffness
Cm_alpha_dot = -8.0 # Pitch damping derivative
Cm_q = -12.0       # Pitch damping
Cm_delta_e = -1.2  # Control effectiveness

# Convert to dimensional stability derivatives
S = 30.0           # m² (wing area)
c_bar = 3.0        # m (mean aerodynamic chord)
rho = 0.8          # kg/m³ (density at altitude)
V = 100            # m/s (cruise speed)

# Convert to body-axis stability derivatives
Z_alpha = CZ_alpha * 0.5 * rho * V * S
M_alpha = Cm_alpha * 0.5 * rho * V**2 * S * c_bar
M_alpha_dot = Cm_alpha_dot * 0.5 * rho * V * S * c_bar**2
M_q = Cm_q * 0.5 * rho * S * c_bar**2
M_delta_e = Cm_delta_e * 0.5 * rho * V**2 * S * c_bar

# Define step input (sudden elevator deflection)
elevator_deflection = 5  # degrees

Use the aircraft's equations of motion to estimate the response to the control input.

Hint:

• The short period mode can be approximated using the linearized equations
• Natural frequency: $\omega_n = \sqrt{\frac{M_\alpha}{I_y}}$
• Damping ratio: $\zeta = \frac{M_q + M_{\alpha\dot{}}}{2\sqrt{M_\alpha I_y}}$

# TODO: Calculate natural frequency and damping ratio of short period mode
short_period_freq = 0  # Natural frequency of short period mode (rad/s)
short_period_damping = 0  # Damping ratio of short period mode

# Calculate initial response parameters
initial_theta_accel = 0  # Initial pitch acceleration (rad/s²)
initial_alpha_rate = 0   # Initial rate of angle of attack change (rad/s)

# Print results
print(f"Short period natural frequency: {short_period_freq:.2f} rad/s")
print(f"Short period damping ratio: {short_period_damping:.2f}")
print(f"Initial pitch acceleration: {initial_theta_accel:.4f} rad/s²")
print(f"Initial alpha rate: {initial_alpha_rate:.4f} rad/s")

How would you design a control system to improve the aircraft's handling qualities?