Dynamic Stability

Module 9

9.1 Euler angles

9.2 Dynamic modes

9.3 Mode Equations

9.4 Numericals

9.1 Euler angles

Euler’s Equations of Aircraft Motion

Euler’s equations describe the rotational motion of an aircraft about its center of gravity. They relate the aerodynamic moments to the angular velocities about the body axes.

$ I_x \dot{p} - (I_y - I_z)qr = L $

$ I_y \dot{q} - (I_z - I_x)pr = M $

$ I_z \dot{r} - (I_x - I_y)pq = N $

Where:

p – Roll rate
q – Pitch rate
r – Yaw rate
L – Rolling moment
M – Pitching moment
N – Yawing moment
I_x, I_y, I_z – Moments of inertia about body axes

Stability Derivatives

Stability derivatives represent the change in aerodynamic forces and moments due to small changes in motion variables. They are obtained by linearizing the aerodynamic forces and moments about an equilibrium flight condition.

Examples of Stability Derivatives:

$ C_{m\alpha} = \frac{\partial C_m}{\partial \alpha} $ → Pitch stability derivative
$ C_{l\beta} = \frac{\partial C_l}{\partial \beta} $ → Lateral stability derivative
$ C_{n\beta} = \frac{\partial C_n}{\partial \beta} $ → Directional stability derivative
$ C_{lp} = \frac{\partial C_l}{\partial p} $ → Roll damping derivative
$ C_{nr} = \frac{\partial C_n}{\partial r} $ → Yaw damping derivative

Linearized Moment Equations

Using stability derivatives, aerodynamic moments can be expressed as:

$ L = \frac{1}{2} \rho V^2 S b (C_{l\beta}\beta + C_{lp}p + C_{lr}r) $

$ M = \frac{1}{2} \rho V^2 S c (C_{m\alpha}\alpha + C_{mq}q) $

$ N = \frac{1}{2} \rho V^2 S b (C_{n\beta}\beta + C_{np}p + C_{nr}r) $

Aircraft Equation of Motion (Longitudinal)

Equations of Motion

m u̇ = X

m ẇ = Z

I_y q̇ = M

Second-Order Dynamic Response

Standard form of the characteristic equation:

\lambda^2 + 2\zeta\omega_{n}\lambda + \omega_{n}^2 = 0

The roots (eigenvalues) are defined as:

\lambda_{1,2} = -\zeta\omega_{n} \pm i\omega_{n}\sqrt{1 - \zeta^2} = \eta \pm i\omega

Where the damping components are:

\eta = -\zeta\omega_{n}, \quad \omega = \omega_{n}\sqrt{1 - \zeta^2}

Damping Ratio ($\zeta$)	Type of Root	Time Response
$\zeta < -1$	Two positive real roots	Exponentially growing motion
$0 > \zeta > -1$	Complex roots (pos. real part)	Growing sinusoidal motion
$\zeta = 0$	Complex roots (real part 0)	Undamped harmonic motion
$0 < \zeta< 1$	Complex roots (neg. real part)	Underdamped decaying sinusoidal

Damping Ratio ($\zeta$): 0.5

Underdamped exponentially decaying sinusoidal motion

System Characteristics

Type of Root:Complex roots with a negative real part

Time Response:Decaying sinusoidal motion

$$\lambda_{1,2} = -\zeta\omega_n \pm i\omega_n\sqrt{1-\zeta^2}$$

9.2 Dynamic modes

Aircraft Dynamic Stability Modes

Dynamic stability describes the time response of an aircraft after a disturbance. It determines whether oscillations decay, remain constant, or grow with time.

Longitudinal dynamic mode

1. Short Period Mode

The short-period mode is an oscillation with a period of only a few seconds that is usually heavily damped by the existence of lifting surfaces far from the aircraft’s center of gravity, such as a horizontal tail or canard. The time to damp the amplitude to one-half of its value is usually on the order of 1 second.

Motion: Rapid pitch oscillation involving angle of attack and pitch rate.

Equation:

Natural Frequency:
ω_n,sp = √(-M_α / I_y)

or,

ω_n,sp = √( (Z_α M_q / u₀) − M_α )

Damping Ratio:
ζ_sp = -M_q / (2 √(-M_α I_y))

or,

ζ_sp = -( M_q + M_α + (Z_α / u₀) ) / (2 ω_nsp)

$ \ddot{\theta} + 2\zeta\omega_n \dot{\theta} + \omega_n^2 \theta = 0 $

High frequency oscillation
Strongly damped
Controlled mainly by horizontal tail

2. Phugoid Mode

Phugoid motion is a long-period, low-frequency longitudinal oscillation in aircraft characterized by alternating climbing and descending paths, where altitude and airspeed exchange periodically while pitch angle changes slightly. It is a stable, gentle motion caused by a disturbance in equilibrium, typically lasting 20–60 seconds, and is usually corrected automatically by the pilot or damped naturally.

Motion: Long-period oscillation between altitude and velocity.

ω_n,p = √((-Z_u g) / u₀)

ω_n,p = g / V

T = 2π / ω_p

$ \ddot{h} + g \theta = 0 $

Slow oscillation
Energy exchange between kinetic and potential energy
Light damping

Comparison between longitudinal modes

Feature	Short Period Mode	Phugoid Mode
Duration / Period	Very short (seconds)	Long (minutes)
Damping	Usually heavily damped	Lightly damped
Airspeed (V)	Nearly constant	Varies significantly
Angle of Attack (α)	Varies significantly	Nearly constant
Primary Driver	Pitch stiffness and pitch damping	Exchange of potential and kinetic energy

Lateral-directional Mode

1. Dutch Roll Mode

A dutch roll is a combination of yawing and rolling motions. It can happen at any speed, developing from the use of the stick (i.e., aileron) and rudder, which generate a rolling action when in yaw. If a sideslip disturbance occurs, the aircraft yaws in one direction and, with strong roll stability, then rolls away in a countermotion. The aircraft “wags its tail” from side to side, so to speak. The term Dutch roll derives from the rhythmic motion of Dutch iceskaters swinging their arms and bodies from side to side as they skate over wide frozen areas.

Motion: Coupled yaw and roll oscillation.

Natural Frequency:
ω_dr = √(-N_β / I_z)

Damping Ratio:
ζ_dr = -N_r / (2 √(-N_β I_z))

$ \ddot{\beta} + C_{nr}\dot{\beta} + C_{n\beta}\beta = 0 $

Oscillatory yawing and rolling motion
Controlled by vertical tail and dihedral effect
Often damped using yaw damper

2. Spiral Mode

Motion: Slow divergence in bank angle.

Spiral Stability Condition:
(C_lβ C_nr − C_nβ C_lr) / C_lp < 0

$ \dot{\phi} = C_{l\beta}\beta + C_{lp}p $

Very slow instability
Aircraft gradually enters spiral dive

Directional Divergence

This results from directional (i.e, yaw) instability. The fuselage is a destabilizing body, and if an aircraft does not have a sufﬁciently large V-tail to provide stability, then sideslip increases accompanied by some roll, with the extent depending on the roll stability. The condition can continue until the aircraft is broadside to the relative wind.

Spiral Divergence

Spiral divergence or spiral instability is a condition where an aircraft is directionally very stable, but laterally very unstable. It is characterized by low angle of attack and high airspeed. When the lateral equilibrium of the aircraft is disturbed by a gust of air and a sideslip is introduced, the strong directional stability tends to yaw the nose into the resultant relative wind while the comparatively weak dihedral lags in restoring the lateral balance.

3. Roll Mode

Roll Subsidence:A pilot commands the roll rate by application of the aileron. Deﬂection of the ailerons generates a rolling moment, but the aircraft has a roll inertia and the roll rate builds up. Very quickly, a steady roll rate is achieved when the rolling moment generated by the ailerons is balanced by an equal and opposite moment proportional to the roll rate. When a pilot has achieved the desired bank angle, the ailerons are neutralized and the resisting rolling moment very rapidly damps out the roll rate. The damping effect of the wings is called roll subsidence.

Motion: Pure roll damping motion.

Roll Time Constant:
τ_roll = − I_x / L_p

$ \dot{p} = C_{lp}p $

Highly damped
Aircraft quickly stops rolling after disturbance

Comparison

Stability Mode	Type of Motion	Axis of Motion	Speed of Response	Main Characteristics
Short Period Mode	Pitch oscillation	Longitudinal axis	Fast	Rapid change in angle of attack and pitch rate, strongly damped
Phugoid Mode	Altitude and velocity oscillation	Longitudinal axis	Slow	Exchange between kinetic and potential energy, lightly damped
Dutch Roll Mode	Yaw and roll oscillation	Lateral–Directional	Moderate	Oscillatory yawing and rolling motion, common in swept-wing aircraft
Spiral Mode	Gradual bank divergence	Lateral–Directional	Very slow	Aircraft slowly enters spiral dive if not corrected
Roll Mode	Pure rolling motion	Lateral axis	Very fast	Strong roll damping due to wings

9.3 Mode Equations

Longitudinal Stability Numerical Analysis

The state-space matrix ẋ = Ax for the UAV is defined as:

$$ \begin{bmatrix} \Delta\dot{u} \\ \Delta\dot{w} \\ \Delta\dot{q} \\ \Delta\dot{\theta} \end{bmatrix} = \begin{bmatrix} -0.045 & 0.036 & 0.0000 & -32.2 \\ -0.369 & -2.02 & 176 & 0.0000 \\ 0.0019 & -0.0396 & -2.948 & 0.000 \\ 0.0000 & 0.0000 & 1.0000 & 0.0000 \end{bmatrix} \begin{bmatrix} \Delta u \\ \Delta w \\ \Delta q \\ \Delta \theta \end{bmatrix} $$

The characteristic equation is found via $|\lambda\mathbf{I} - \mathbf{A}| = 0$:

$$\lambda^4 + 5.05\lambda^3 + 13.2\lambda^2 + 0.67\lambda + 0.59 = 0$$

Phugoid (Long Period)

Eigenvalues:
$\lambda_{1,2} = -0.0171 \pm i(0.213)$

Time to half amp ($t_{1/2}$):
$40.3 \text{ s}$

Period: $29.5 \text{ s}$

Cycles ($N_{1/2}$): $1.37$

Short Period

Eigenvalues:
$\lambda_{3,4} = -2.5 \pm i(2.59)$

Time to half amp ($t_{1/2}$):
$0.28 \text{ s}$

Period: $2.42 \text{ s}$

Cycles ($N_{1/2}$): $0.11$

Lateral-Directional State-Space Model

The lateral-directional linearized equations of motion in state-space form ($\dot{\mathbf{x}} = \mathbf{Ax} + \mathbf{Bu}$) are expressed as:

$$ \begin{bmatrix} \Delta \dot{\beta} \\ \Delta \dot{p} \\ \Delta \dot{r} \\ \Delta \dot{\phi} \end{bmatrix} = \begin{bmatrix} \frac{Y_{\beta}}{u_0} & \frac{Y_{p}}{u_0} & -\left( 1 - \frac{Y_{r}}{u_0} \right) & \frac{g \cos \theta_0}{u_0} \\ L_{\beta} & L_{p} & L_{r} & 0 \\ N_{\beta} & N_{p} & N_{r} & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} \Delta \beta \\ \Delta p \\ \Delta r \\ \Delta \phi \end{bmatrix} + \begin{bmatrix} 0 & \frac{Y_{\delta_r}}{u_0} \\ L_{\delta_a} & L_{\delta_r} \\ N_{\delta_a} & N_{\delta_r} \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \Delta \delta_a \\ \Delta \delta_r \end{bmatrix} $$

Where:
$\beta$: Sideslip angle | $p, r$: Roll and Yaw rates | $\phi$: Roll angle | $\delta_a, \delta_r$: Aileron and Rudder deflections

Dutch Roll Approximation

If we consider the Dutch roll mode to consist primarily of sideslipping and yawing motions, then we can neglect the rolling moment equation. With these assumptions, the system reduces to:

$$ \begin{bmatrix} \Delta \dot{\beta} \\ \Delta \dot{r} \end{bmatrix} = \begin{bmatrix} \frac{Y_{\beta}}{u_0} & -\left( 1 - \frac{Y_{r}}{u_0} \right) \\ N_{\beta} & N_{r} \end{bmatrix} \begin{bmatrix} \Delta \beta \\ \Delta r \end{bmatrix} $$

Solving for the characteristic equation yields:

$$ \lambda^2 - \left( \frac{Y_{\beta} + u_0 N_{r}}{u_0} \right) \lambda + \frac{Y_{\beta} N_{r} - N_{\beta} Y_{r} + u_0 N_{\beta}}{u_0} = 0 $$

From this expression, we can determine the undamped natural frequency and the damping ratio as follows:

$$ \omega_{n_{DR}} = \sqrt{ \frac{Y_{\beta} N_{r} - N_{\beta} Y_{r} + u_0 N_{\beta}}{u_0} } $$

$$ \zeta_{DR} = -\frac{1}{2\omega_{n_{DR}}} \left( \frac{Y_{\beta} + u_0 N_{r}}{u_0} \right) $$

Spiral Mode Stability Analysis

The characteristic root for this equation is defined as:

\lambda_{\text{spiral}} = \frac{L_{\beta}N_{r} - L_{r}N_{\beta}}{L_{\beta}}

The stability derivatives $L_{\beta}$ (dihedral effect) and $N_{r}$ (yaw rate damping) usually are negative quantities. On the other hand, $N_{\beta}$ (directional stability) and $L_{r}$ (roll moment due to yaw rate) generally are positive quantities.

 If the derivatives have the usual sign, then the condition for a stable spiral model is: 

L_{\beta}N_{r} - N_{\beta}L_{r} > 0

L_{\beta}N_{r} > N_{\beta}L_{r}

Increasing the dihedral effect $L_{\beta}$ or the yaw damping $N_{r}$ (or both) can make the spiral mode stable.

Aircraft Dynamic Stability – Sample Stepwise Solutions

1. Short Period Mode

Mw = -1.2 Zw = -5.5 Mq = -8 Mass parameter μ = 120

Natural Frequency ωₙ = √( Mw × Zw / μ )

Damping Ratio ζ = -Mq / (2√(Mw × Zw))

Step 1: Mw × Zw = (-1.2)(-5.5) = 6.6

Step 2: 6.6 / 120 = 0.055

Step 3: ωₙ = √0.055 = 0.235 rad/s

Step 4: √(MwZw) = √6.6 = 2.57

Step 5: ζ = 8 / (2 × 2.57) = 1.56

Short-period motion is **overdamped and stable**.

2. Phugoid Mode

Cruise velocity V = 60 m/s Gravity g = 9.81 m/s²

Natural frequency (Phugoid) ωₙ = g / V

Period of oscillation T = 2π / ωₙ

Step 1: ωₙ = 9.81 / 60

ωₙ = 0.1635 rad/s

Step 2: T = 2π / 0.1635

T = 38.4 seconds

Phugoid motion has **long period oscillations with low damping**.

3. Roll Mode

Rolling moment derivative Lp = -4.5 Inertia parameter Ix = 2

Roll time constant τ = - Ix / Lp

Step 1: τ = -2 / (-4.5)

τ = 0.44 seconds

Roll mode is **very fast and strongly stable**.

4. Dutch Roll Mode

Directional derivative Nr = -0.3 Yaw damping Nv = -0.8 Aircraft inertia Iz = 5

Natural frequency ωₙ = √(|Nr × Nv| / Iz)

Step 1: Nr × Nv = (-0.3)(-0.8)

= 0.24

Step 2: 0.24 / 5 = 0.048

Step 3: ωₙ = √0.048

ωₙ = 0.219 rad/s

Dutch roll shows **oscillatory yaw-roll coupling**.

5. Spiral Mode

Rolling derivative Lβ = -0.05 Yaw derivative Nβ = 0.02

Spiral stability condition Nβ / Lβ

Step 1: 0.02 / (-0.05)

= -0.4

Negative value indicates **spiral stability**.

9.4 Numericals

Dynamic Numerical Analysis

1. Longitudinal Stability Analysis

The state-space matrix for longitudinal motion is defined as:

\begin{bmatrix} \Delta\dot{u} \\ \Delta\dot{w} \\ \Delta\dot{q} \\ \Delta\dot{\theta} \end{bmatrix} = \begin{bmatrix} -0.045 & 0.036 & 0.0000 & -32.2 \\ -0.369 & -2.02 & 176 & 0.0000 \\ 0.0019 & -0.0396 & -2.948 & 0.000 \\ 0.0000 & 0.0000 & 1.0000 & 0.0000 \end{bmatrix} \begin{bmatrix} \Delta u \\ \Delta w \\ \Delta q \\ \Delta \theta \end{bmatrix}

Characteristic Equation: $\lambda^4 + 5.05\lambda^3 + 13.2\lambda^2 + 0.67\lambda + 0.59 = 0$

Phugoid Mode

$\lambda_{1,2} = -0.0171 \pm i(0.213)$
$t_{1/2} = 40.3 \text{ s}$
$P = 29.5 \text{ s}$
$N_{1/2} = 1.37 \text{ cycles}$

Short Period Mode

$\lambda_{3,4} = -2.5 \pm i(2.59)$
$t_{1/2} = 0.28 \text{ s}$
$P = 2.42 \text{ s}$
$N_{1/2} = 0.11 \text{ cycles}$

2. Lateral-Directional Numerical Solution

Numerical State-Space Matrix for Lateral Motion:

\begin{bmatrix} \Delta\dot{\beta} \\ \Delta\dot{p} \\ \Delta\dot{r} \\ \Delta\dot{\phi} \end{bmatrix} = \begin{bmatrix} -0.254 & 0 & -1.0 & 0.182 \\ -16.02 & -8.40 & 2.19 & 0 \\ 4.488 & -0.350 & -0.760 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} \Delta\beta \\ \Delta p \\ \Delta r \\ \Delta \phi \end{bmatrix}

The resulting characteristic equation is:

$$ \lambda^4 + 9.417\lambda^3 + 13.982\lambda^2 + 48.102\lambda + 0.4205 = 0 $$

Lateral Eigenvalues (Exact):

Spiral Mode: $\lambda = -0.00877$
Roll Mode: $\lambda = -8.435$
Dutch Roll: $\lambda = -0.487 \pm i(2.335)$

Approximate Modal Results:

1. Spiral Mode:

$$ \lambda_{\text{spiral}} = \frac{L_{\beta}N_{r} - L_{r}N_{\beta}}{L_{\beta}} $$ $$ \lambda_{\text{spiral}} = \frac{(-16.02 \text{ s}^{-2})(-0.76 \text{ s}^{-1}) - (2.19 \text{ s}^{-1})(4.49 \text{ s}^{-2})}{-16.02 \text{ s}^{-2}} $$ $$ \lambda_{\text{spiral}} = -0.144 \text{ s}^{-1} $$

2. Roll Mode:

$$ \lambda_{\text{roll}} = L_{p} = -8.4 \text{ s}^{-1} $$

3. Dutch Roll Mode:

The roots are determined from the characteristic equation:

$$ \lambda^2 - \left( \frac{Y_{\beta} + u_0 N_{r}}{u_0} \right) \lambda + \frac{Y_{\beta} N_{r} - N_{\beta} Y_{r} + u_0 N_{\beta}}{u_0} = 0 $$ $$ \text{or } \lambda^2 + 1.102\lambda + 4.71 = 0 $$

Which yields the following roots and parameters:

$$ \lambda_{\text{DR}} = -0.51 \pm 2.109i $$ $$ \omega_{n_{\text{DR}}} = 2.17 \text{ rad/s} $$ $$ \zeta_{\text{DR}} = 0.254 $$

Lateral-Directional Approximation Calculator

u₀ (Velocity m/s)

Y_β (Side Force)

Y_r (Yaw Force)

L_β (Dihedral)

N_β (Directional)

N_r (Yaw Damping)

L_r (Roll due to Yaw)

L_p (Roll Damping)

Spiral & Roll Modes

Spiral Mode ($\lambda_{spiral}$):0

Roll Mode ($\lambda_{roll}$):0

Dutch Roll Mode

Natural Freq ($\omega_{n_{DR}}$):0

Damping Ratio ($\zeta_{DR}$):0

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