6.2 Dynamic stability

Dynamic stability fundamentals

Dynamic stability describes how an aircraft behaves over time after being disturbed from equilibrium. Where static stability asks "does the aircraft tend to return?" dynamic stability asks "what does the motion look like as it returns?" An aircraft can be statically stable but dynamically unstable if its oscillations grow in amplitude rather than dying out.

Analyzing dynamic stability requires understanding how aerodynamic forces, moments, and the aircraft's inertial properties interact during perturbed motion.

Stability vs instability

Positive dynamic stability means the aircraft's oscillations decrease in amplitude over time, eventually converging back to the original equilibrium state.

Dynamic instability means oscillations grow in amplitude after a disturbance. An unstable aircraft requires constant pilot input or active control systems to maintain steady flight.

Neutral dynamic stability falls between these two: the aircraft oscillates at constant amplitude indefinitely, neither converging nor diverging.

Equations of motion

The equations of motion for a rigid aircraft come from Newton's second law, relating linear and angular accelerations to applied forces and moments. They're expressed in a body-fixed coordinate system with six degrees of freedom.

Translational equations:

$m(\dot{u} + qw - rv) = X$

$m(\dot{v} + ru - pw) = Y$

$m(\dot{w} + pv - qu) = Z$

Rotational equations:

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

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

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

Here, $u, v, w$ are body-axis velocity components, $p, q, r$ are roll/pitch/yaw rates, $I_x, I_y, I_z$ are moments of inertia, and $X, Y, Z, L, M, N$ are the total applied forces and moments. The cross-coupling terms (like $qw$ and $rv$) arise because the equations are written in a rotating reference frame.

Linearization of equations

The full equations of motion are nonlinear, which makes them difficult to solve analytically. Linearization simplifies them by assuming small perturbations from a steady-state trim condition and dropping higher-order terms.

This produces a set of linear differential equations that can be solved with standard tools from linear systems theory, including Laplace transforms, transfer functions, and state-space methods. The linearized system takes the familiar form:

$\dot{x} = Ax + Bu$

where $A$ is the stability matrix (built from stability derivatives), $x$ is the state vector of perturbation variables, and $B$ captures control input effects.

Linearization is valid only for small perturbations. Large-amplitude maneuvers or post-stall behavior require the full nonlinear equations.

Longitudinal dynamic stability

Longitudinal dynamic stability covers motion in the pitch plane: changes in pitch angle, angle of attack, and forward velocity. Two distinct modes emerge from the linearized longitudinal equations, each with very different characteristics.

Short period mode

The short period mode is a relatively high-frequency, heavily damped oscillation in pitch. It primarily involves changes in angle of attack ($\alpha$) and pitch rate ($q$), with very little change in airspeed.

Think of it as the aircraft quickly "settling" its nose after a gust hits. In most well-designed aircraft, this mode damps out within a few seconds.

Key factors that influence the short period mode:

• Pitch moment of inertia ($I_y$): higher inertia lowers the natural frequency
• Pitch damping ($C_{m_q}$): more negative values increase damping
• CG position relative to the aerodynamic center: affects pitch stiffness ($C_{m_\alpha}$)

Pilots interact with this mode constantly during maneuvering, so adequate damping and a natural frequency that matches pilot expectations are critical for good handling qualities.

Long period (phugoid) mode

The phugoid is a slow, lightly damped oscillation involving a trade-off between kinetic and potential energy. The aircraft gently climbs (trading speed for altitude) and then descends (trading altitude for speed) in long, shallow waves. Angle of attack stays roughly constant throughout.

Typical phugoid periods range from 30 seconds to several minutes, depending on the aircraft and flight speed. Because it's slow, pilots can usually control it manually even when damping is poor.

Key factors influencing the phugoid:

• Lift-to-drag ratio ($L/D$): higher $L/D$ tends to reduce phugoid damping
• Thrust variation with speed: affects the energy exchange
• CG position relative to the neutral point

Lateral-directional dynamic stability

Lateral-directional stability involves coupled motion in roll and yaw, plus sideslip. The linearized lateral-directional equations produce three distinct modes.

Roll mode

The roll mode is a non-oscillatory (purely real eigenvalue) mode describing how quickly the aircraft reaches a steady roll rate after a disturbance. It's typically very fast and heavily damped.

The roll mode time constant ($\tau_R$) governs how quickly the roll rate converges. It depends primarily on:

• Roll moment of inertia ($I_x$): higher inertia means a slower response
• Roll damping ($C_{l_p}$): more negative values give a shorter time constant

There's minimal coupling to yaw or sideslip in this mode.

Spiral mode

The spiral mode is a slow, non-oscillatory mode describing the aircraft's tendency to gradually enter a banked, turning descent after a small lateral disturbance. It involves coupling between roll and yaw.

The spiral mode can be:

• Stable: the aircraft slowly returns to wings-level flight
• Unstable: the aircraft slowly diverges into an ever-steepening bank and turn
• Neutral: the bank angle remains constant

Many aircraft have a mildly unstable spiral mode, which is acceptable as long as the divergence is slow enough for the pilot to correct. The balance between dihedral effect ($C_{l_\beta}$) and weathercock stability ($C_{n_\beta}$) largely determines spiral mode stability.

Dutch roll mode

Dutch roll is an oscillatory mode coupling yaw, roll, and sideslip. The aircraft's nose traces a roughly elliptical path relative to the flight direction, with the wings rocking side to side.

Key influences on Dutch roll:

• Yaw damping ($C_{n_r}$): more negative values improve damping
• Weathercock stability ($C_{n_\beta}$): affects the natural frequency
• Yaw moment of inertia ($I_z$)
• Vertical tail size and placement

Poor Dutch roll damping creates objectionable handling qualities and passenger discomfort. This is why many aircraft, especially swept-wing transports, require yaw dampers.

Stability derivatives

Stability derivatives are partial derivatives that quantify how aerodynamic forces and moments change when the aircraft's motion variables or control surface deflections are perturbed. They form the coefficients of the linearized equations of motion and are the building blocks of dynamic stability analysis.

Dimensional vs nondimensional

Dimensional derivatives have units tied to the specific force or moment (e.g., $\text{N/rad}$ or $\text{N·m·s/rad}$). They depend on the specific flight condition.

Nondimensional derivatives are normalized by reference quantities like dynamic pressure ($\bar{q}$), wing area ($S$), and characteristic lengths (span $b$ or chord $\bar{c}$). For example, $C_{m_\alpha}$ is nondimensional pitch stiffness.

Nondimensionalization makes it possible to compare stability characteristics across different aircraft and flight conditions on equal footing.

Contribution to dynamic modes

Different stability derivatives drive different modes. Here are the most important relationships:

Longitudinal modes:

• Short period: dominated by $C_{m_\alpha}$ (pitch stiffness) and $C_{m_q}$ (pitch damping), with $C_{L_\alpha}$ (lift curve slope) setting the angle-of-attack response
• Phugoid: influenced by $C_{L_u}$ (lift sensitivity to speed changes) and $C_{D_u}$; drag-related derivatives affect the energy dissipation that damps the phugoid

Lateral-directional modes:

• Roll mode: primarily $C_{l_p}$ (roll damping)
• Spiral mode: balance between $C_{l_\beta}$ (dihedral effect) and $C_{n_\beta}$ (weathercock stability), along with $C_{l_r}$ and $C_{n_r}$
• Dutch roll: $C_{n_\beta}$ (weathercock stability), $C_{n_r}$ (yaw damping), and $C_{l_\beta}$ (dihedral effect)

Dynamic stability analysis techniques

Two widely used methods for analyzing the linearized equations of motion are eigenvalue analysis and root locus plots. Both work with the state-space representation of the aircraft's dynamics.

Eigenvalues and eigenvectors

Eigenvalue analysis solves the characteristic equation of the system matrix $A$ to find the eigenvalues ($\lambda$).

• Complex eigenvalues ($\lambda = \sigma \pm j\omega$) correspond to oscillatory modes. The real part $\sigma$ gives the damping (negative = stable), and the imaginary part $\omega$ gives the damped natural frequency.
• Real eigenvalues correspond to non-oscillatory modes (like the roll and spiral modes). A negative real eigenvalue is stable; positive is unstable.

From complex eigenvalues, you can extract the natural frequency $\omega_n = |\lambda|$ and damping ratio $\zeta = -\sigma / \omega_n$.

Eigenvectors describe the "shape" of each mode: the relative amplitudes and phase relationships among the state variables. For example, the short period eigenvector shows large $\alpha$ and $q$ components with small velocity change.

Root locus plots

Root locus plots show how eigenvalues move in the complex plane as a single parameter is varied (e.g., a feedback gain, a stability derivative, or a geometric parameter).

How to read a root locus plot:

1. Each branch traces the path of one eigenvalue as the parameter changes
2. Eigenvalues in the left half-plane (negative real part) are stable
3. Eigenvalues crossing the imaginary axis into the right half-plane indicate the onset of instability
4. Distance from the real axis indicates oscillation frequency; distance from the imaginary axis indicates damping

Root locus plots are especially useful for designing feedback control laws and understanding how parameter changes affect all modes simultaneously.

Factors affecting dynamic stability

Aircraft geometry and configuration

The aircraft's shape and layout directly determine its stability derivatives and therefore its dynamic modes.

• Wing planform, aspect ratio, and sweep affect lift distribution, aerodynamic damping derivatives ($C_{l_p}$, $C_{n_r}$), and the lift curve slope
• Horizontal tail size and moment arm are primary drivers of longitudinal pitch stiffness and damping
• Vertical tail size and placement control weathercock stability ($C_{n_\beta}$) and yaw damping ($C_{n_r}$)
• Fuselage length and cross-section influence moments of inertia and can contribute destabilizing yaw and pitch moments

Configuration changes during flight (deploying flaps, extending landing gear, opening weapons bays) alter the stability derivatives and can shift modal characteristics noticeably.

Flight conditions and environment

Dynamic stability is not fixed; it varies across the flight envelope.

• Airspeed and altitude change dynamic pressure, which scales all aerodynamic forces and moments. Control surface effectiveness also varies.
• Mach number introduces compressibility effects. Transonic flight can cause significant shifts in stability derivatives (e.g., aft movement of the aerodynamic center leading to Mach tuck).
• Atmospheric turbulence and wind shear can excite dynamic modes and increase pilot workload.

Aircraft must demonstrate adequate stability and controllability across the entire certified flight envelope, from low-speed approach to high-speed cruise.

Dynamic stability augmentation systems

When an aircraft's natural dynamic stability doesn't meet handling qualities requirements, stability augmentation systems (SAS) provide artificial damping or stiffness through feedback control.

Yaw dampers

Yaw dampers target the Dutch roll mode, which is often poorly damped in swept-wing aircraft.

How a yaw damper works:

1. A yaw rate gyro senses the aircraft's yaw rate ($r$)
2. The signal is processed through a control law (typically a washout filter plus gain)
3. The system commands rudder deflections proportional to yaw rate, opposing the Dutch roll oscillation

Modern yaw dampers often blend yaw rate with lateral acceleration feedback for better performance. The washout filter prevents the yaw damper from fighting steady turns.

Pitch dampers

Pitch dampers improve short period mode damping, which is especially important for:

• Aircraft with relaxed static stability (common in modern fighters for enhanced maneuverability)
• Large, flexible aircraft where structural modes can couple with the short period
• High-performance aircraft requiring precise pitch control

The system senses pitch rate ($q$) and commands corrective elevator deflections. In some large aircraft, pitch dampers also augment phugoid damping to reduce long-period oscillations during cruise.

Testing dynamic stability

Flight testing and wind tunnel testing are complementary approaches to validating an aircraft's dynamic stability characteristics. Both are essential parts of the design and certification process.

Flight testing methods

Flight testing evaluates dynamic stability under real-world conditions using instrumented aircraft.

Common flight test techniques:

• Pulse inputs: a brief, sharp control input excites a specific mode, and the subsequent free response is recorded to measure damping and frequency
• Doublet inputs: a pair of opposing pulses that efficiently excite oscillatory modes while minimizing steady-state offsets
• Steady-heading sideslips: used to evaluate lateral-directional static stability and extract related derivatives
• Frequency sweeps: pilot applies sinusoidal inputs across a range of frequencies to map the aircraft's frequency response

Flight test data serves to validate mathematical models, refine stability derivative estimates, and assess handling qualities against pilot rating criteria (e.g., Cooper-Harper ratings).

Wind tunnel testing methods

Wind tunnel testing measures stability derivatives and dynamic response in a controlled, repeatable environment.

Key techniques include:

• Forced oscillation testing: the model is mechanically oscillated at known frequencies and amplitudes while aerodynamic forces and moments are measured. This directly yields dynamic derivatives like $C_{m_q}$ and $C_{n_r}$.
• Free oscillation testing: the model is displaced from equilibrium and released. The resulting motion is recorded to extract damping ratios and natural frequencies.
• Rotary balance testing: the model undergoes steady-state rotation to measure damping derivatives associated with sustained angular rates.

Wind tunnel results feed into design refinement, numerical model validation, and flight test planning. Testing can range from small-scale models to full-scale components, though Reynolds number scaling must be carefully considered.

Dynamic stability in aircraft design

Design considerations for stability

Achieving good dynamic stability requires coordinated decisions across the entire airframe:

• Wing and tail geometry must provide adequate aerodynamic damping and stiffness for all modes
• CG and aerodynamic center positioning determine the static margin, which directly affects short period frequency and phugoid characteristics
• Control surface sizing must provide enough authority to stabilize all dynamic modes, including at the edges of the flight envelope
• Stability augmentation systems are incorporated when the bare airframe can't meet handling qualities requirements on its own

Designers iterate using analytical predictions, CFD simulations, wind tunnel data, and eventually flight test results.

Trade-offs with other performance aspects

Dynamic stability requirements frequently conflict with other design goals:

• Stability vs. maneuverability: larger static margins improve dynamic stability but reduce agility. Modern fighters deliberately use reduced or negative static margins (with fly-by-wire augmentation) to gain maneuverability.
• Tail sizing vs. weight and drag: larger tails improve damping and stiffness but add weight and drag, reducing range and payload capacity.
• Augmentation systems vs. complexity: SAS can solve stability deficiencies, but they add weight, cost, power consumption, and potential failure modes that must be addressed through redundancy.

The right balance depends on the aircraft's mission. A commercial transport prioritizes passenger comfort and safety margins. A fighter prioritizes agility and accepts the complexity of full-authority flight control systems.