M10-T1 — Aircraft Dynamics and Their Five Modes

Learning objectives

After this lesson, you should be able to:

• Write the longitudinal and lateral-directional small-disturbance models of a symmetric aircraft in stability axes.
• Identify the short period, phugoid, roll subsidence, spiral and Dutch roll from the poles of the model.
• Approximate the frequency and damping of each mode from the stability derivatives that govern it.
• Predict how a change of static margin moves the short period.

The intuition

UAV M1-T2: Multirotor, fixed-wing and VTOL architectures sets out how a fixed-wing vehicle differs from a multirotor, and UAV M5-T1: UAV equations of motion gives the six-degree-of-freedom equations and the wind triangle. This lesson starts from the linearization of module 2 and does what a flight control engineer does first with a new airframe: finds its modes.

Two sets of equations

For an aircraft that is symmetric about its centreline, flying straight and level, a disturbance in the plane of symmetry (pitch, forward speed, vertical speed) does not produce a rolling or yawing disturbance, to first order, and vice versa. So the linear model splits. The longitudinal states are the forward-speed perturbation $u$, the vertical-speed perturbation $w$, the pitch rate $q$ and the pitch angle $\theta$; the input is the elevator (and throttle). The lateral-directional states are the sideslip velocity $v$, roll rate $p$, yaw rate $r$ and bank angle $\phi$; the inputs are the aileron and rudder. The coefficients are stability derivatives: the change in a force or moment per unit change of a state, such as $M_w$ (pitching moment per unit vertical velocity, the static stability).

The five modes

The longitudinal model has two oscillatory modes. The short period is a fast pitching oscillation in which the angle of attack and pitch rate trade off at nearly constant speed: for the example aircraft its natural frequency is 4.97 rad/s (period 1.8 s) and damping ratio 0.71. The phugoid is a slow exchange of height and speed at almost constant angle of attack: 0.40 rad/s (period 15.8 s), with a damping ratio of only 0.018, so it persists for many cycles.

The lateral-directional model has one oscillation and two real modes. The roll subsidence is a heavily damped rolling response with a time constant of 0.12 s, set almost entirely by the roll damping $L_p = -8$ per second. The Dutch roll is a coupled yawing and rolling oscillation with the nose and wings swaying out of step: 3.63 rad/s and a damping ratio of only 0.11. The spiral is a very slow real mode, here stable with a time constant of 31.6 s, in which a bank angle slowly builds or decays together with the heading.

One derivative, one mode

The modes are not independent of the design, and each is mostly governed by one or two derivatives. The static margin appears as $M_w$: making it more negative (more stable) raises the short-period frequency and lowers its damping. For the example, $M_w$ = −0.1, −0.3, −0.5 and −1.0 give short-period frequencies of 3.83, 4.44, 4.97 and 6.09 rad/s and damping ratios of 0.92, 0.79, 0.71 and 0.58. The phugoid, in contrast, is set by the lift and drag change with speed, and the roll mode by roll damping. This separation is what lets the autopilot of the next lesson close each loop at its own bandwidth.

A symmetric aircraft has a longitudinal model (short period and phugoid) and a lateral-directional model (roll, spiral, Dutch roll). Each mode is governed mainly by one or two stability derivatives, so the response tells which derivative to change.

Definitions

Stability axes

Body axes with the x axis along the projection of the velocity in the plane of symmetry in the trim condition, so that the trim has $w_0 = 0$.

Stability derivatives

The partial derivatives of force and moment per unit mass with respect to a state or control, divided by the inertia where appropriate: $X_u,\ Z_w,\ M_w,\ M_q,\ Y_v,\ L_p,\ N_v,\ N_r$ and so on.

Static margin

The distance of the centre of gravity ahead of the neutral point in chords; it sets $M_w$, which is negative when the aircraft is statically stable in pitch.

Short period, phugoid

The fast and the slow oscillatory modes of the longitudinal model.

Roll subsidence, spiral, Dutch roll

The fast real, slow real and oscillatory modes of the lateral-directional model.

Derivation: approximations for the five modes

Each approximation freezes the fast variables of the mode being studied.

1. $$s^2 - (Z_w + M_q)\,s + (Z_wM_q - U_0M_w) = 0$$

   Short period: the speed changes slowly, so set $u \approx 0$, and the pitch angle does not enter the pitching moment; the $(w, q)$ block has this characteristic polynomial.

2. $$\omega_{sp} = \sqrt{Z_wM_q - U_0M_w},\qquad \zeta_{sp} = -\dfrac{Z_w + M_q}{2\omega_{sp}}$$

   Read off the natural frequency and damping. For the example, $Z_w = -4,\ M_q = -3,\ M_w = -0.5,\ U_0 = 25$ give $\omega_{sp} = \sqrt{12 + 12.5} = 4.95$ rad/s and $\zeta_{sp} = 7/(2\cdot4.95) = 0.71$, matching the computed 4.97 and 0.71. The static stability enters through $-U_0M_w$, which grows the frequency, and not through the damping.

Phugoid and the lateral modes

1. $$\omega_p^2 \approx \dfrac{-Z_u\,g}{U_{\mathrm{eff}}},\quad U_{\mathrm{eff}} = U_0 - \dfrac{Z_wM_q}{M_w}$$

   Phugoid: the short period is fast, so $\dot q = 0$ and $w = -(M_q/M_w)q$ follow the pitch rate; substituting into the vertical equation replaces $U_0$ by an effective speed. With the Lanchester value $Z_u = -2g/U_0$ and a stiff short period, $U_{\mathrm{eff}} \to U_0$ and $\omega_p = \sqrt2 g/U_0$. In the example $U_{\mathrm{eff}} = 49$ m/s and $\omega_p = 0.40$ rad/s (the Lanchester formula alone gives 0.56).

2. $$\lambda_{\mathrm{roll}} \approx L_p$$

   Roll subsidence: the roll rate decays at the roll damping, since $\dot p = L_p p + L_v v + L_r r$ and the fast roll equation settles before sideslip and yaw respond: $\lambda = -8$ per second here, a time constant of 0.12 s.

3. $$\lambda_{\mathrm{spiral}} \approx \dfrac{g}{U_0}\,\dfrac{L_vN_r - L_rN_v}{L_pN_v - L_vN_p}$$

   Spiral: set the derivatives of the fast variables to zero (quasi-steady roll and sideslip) and keep the slow bank angle. Stable when $L_vN_r > L_rN_v$: enough dihedral and yaw damping against cross-coupling and weathercock. For the example 0.80 against 0.45, giving $-0.035$ per second (computed −0.032).

4. $$\omega_{dr} \approx \sqrt{N_vU_0 + Y_vN_r}$$

   Dutch roll: yaw and sideslip dominate, with the weathercock stiffness $N_vU_0$ as the spring and yaw damping $N_r$ (plus sideslip damping) as the damper: 3.4 rad/s against the computed 3.6.

Each mode is a different balance of forces: the short period balances lift and pitching moment at constant speed, the phugoid balances kinetic and potential energy, roll subsidence is roll damping, the spiral is a slow competition of dihedral against weathercock, and the Dutch roll is a yaw oscillation with a weathercock spring.

A worked example

An illustrative small aircraft at $U_0 = 25$ m/s with round-number derivatives chosen for realistic modes (not a particular airframe): $X_u = -0.06,\ Z_u = -0.8,\ Z_w = -4,\ M_w = -0.5,\ M_q = -3$; $Y_v = -0.25,\ L_v = -1,\ L_p = -8,\ L_r = 1,\ N_v = 0.45,\ N_p = -0.3,\ N_r = -0.8$.

1. Longitudinal modes. The eigenvalues are −3.52 ± 3.50j (frequency 4.97 rad/s, damping 0.709, period 1.80 s: the short period) and −0.007 ± 0.399j (0.399 rad/s, damping 0.018, period 15.75 s: the phugoid).
2. Lateral-directional modes. −0.406 ± 3.61j (3.63 rad/s, damping 0.112: Dutch roll), −8.21 (time constant 0.12 s: roll subsidence) and −0.0317 (time constant 31.6 s: spiral, stable).
3. Approximations. Short period 4.95 rad/s and 0.707; phugoid 0.560 (Lanchester) or 0.400 with the effective speed 49 m/s; roll −8.0; spiral −0.0352; Dutch roll 3.38 rad/s.
4. Static margin. $M_w$ = −0.1, −0.3, −0.5, −1.0 give short-period frequencies 3.83, 4.44, 4.97, 6.09 rad/s and damping 0.92, 0.79, 0.71, 0.58; the formula $\sqrt{Z_wM_q - U_0M_w}$ gives 3.81, 4.42, 4.95, 6.08.
5. From the response. An elevator doublet gives pitch-rate extrema 0.46, 0.99, 1.55 and 2.40 s after the start (half-period of the short period 0.90 s) and pitch-angle extrema at 12.5, 20.4 and 28.3 s (half-period of the phugoid 7.8 s, from $\pi/0.399$).

import math
import gnckit as g

U0, G = 25.0, 9.81
# an illustrative small fixed-wing aircraft in stability axes (round numbers chosen to give realistic modes, not a particular airframe)
Xu, Xw, Zu, Zw, Mw, Mq = -0.06, 0.0, -0.80, -4.0, -0.5, -3.0
# longitudinal states (u, w, q, theta), input elevator; lateral states (v, p, r, phi), inputs aileron and rudder
Alon = [[Xu, Xw, 0.0, -G], [Zu, Zw, U0, 0.0], [0.0, Mw, Mq, 0.0], [0.0, 0.0, 1.0, 0.0]]
Yv, Lv, Lp, Lr, Nv, Np, Nr = -0.25, -1.0, -8.0, 1.0, 0.45, -0.3, -0.8
Alat = [[Yv, 0.0, -U0, G], [Lv, Lp, Lr, 0.0], [Nv, Np, Nr, 0.0], [0.0, 1.0, 0.0, 0.0]]

def describe(name, evs):
    out = []
    for e in sorted(evs, key=lambda c: (abs(complex(c).imag) < 1e-9, abs(complex(c)))):
        e = complex(e)
        if e.imag > 1e-9:
            out.append(f'  {name:11s} pair {e.real:+.4f} +- {e.imag:.4f}j: natural frequency {abs(e):.3f} rad/s, damping ratio {-e.real / abs(e):.3f}, period {2 * math.pi / e.imag:6.2f} s')
        elif abs(e.imag) < 1e-9:
            out.append(f'  {name:11s} real root {e.real:+.4f}: time constant {abs(1 / e.real):6.2f} s' + (' (unstable, doubling time %.1f s)' % (math.log(2) / e.real) if e.real > 0 else ''))
    return out

print('longitudinal modes:')
for l in describe('', g.mat_eigvals(Alon)): print(l)
print('lateral-directional modes:')
for l in describe('', g.mat_eigvals(Alat)): print(l)

print('approximations: short period from the (w, q) block, phugoid from the Lanchester formula, roll from L_p, spiral from the quasi-steady condition')
a11, a12, a21, a22 = Zw, U0, Mw, Mq
tr, det = a11 + a22, a11 * a22 - a12 * a21
wsp = math.sqrt(det); zsp = -tr / (2 * wsp)
print(f'  short period: wn = {wsp:.3f} rad/s, zeta = {zsp:.3f}')
wp_l = math.sqrt(-Zu * G / U0)
Ueff = U0 - Zw * Mq / Mw                     # the short period is quasi-steady: w = -(Mq / Mw) q feeds back into the vertical equation
wp = math.sqrt(-Zu * G / Ueff)
zp = -Xu / (2 * wp)
print(f'  phugoid: Lanchester-type wn = sqrt(-Zu g / U0) = {wp_l:.3f} rad/s; with the quasi-steady short period, U0 is replaced by {Ueff:.1f} m/s and wn = {wp:.3f} rad/s, zeta = -Xu / (2 wn) = {zp:.3f}')
print(f'  roll subsidence: lambda = L_p = {Lp:.2f} 1/s')
spiral = (G / U0) * (Lv * Nr - Lr * Nv) / (Lp * Nv - Lv * Np)
print(f'  spiral (quasi-steady): lambda = {spiral:+.4f} 1/s (stable while L_v N_r > L_r N_v: {Lv * Nr:.2f} against {Lr * Nv:.2f})')
wd = math.sqrt(Nv * U0 + Yv * Nr)
print(f'  Dutch roll (rough): wn = sqrt(N_v U0 + ...) = {wd:.2f} rad/s')

print('changing the static margin: M_w from -0.1 to -1.0 (more negative = more stable) moves the short period')
for mw in (-0.1, -0.3, -0.5, -1.0):
    Al = [[Xu, Xw, 0.0, -G], [Zu, Zw, U0, 0.0], [0.0, mw, Mq, 0.0], [0.0, 0.0, 1.0, 0.0]]
    ev = [complex(e) for e in g.mat_eigvals(Al) if complex(e).imag > 1e-9]
    ev.sort(key=lambda c: -abs(c))
    e = ev[0]
    print(f'  M_w = {mw:5.2f}: short period wn {abs(e):.3f} rad/s, zeta {-e.real / abs(e):.3f}; sqrt(Zw Mq - U0 Mw) = {math.sqrt(Zw * Mq - U0 * mw):.3f}')

print('response to an elevator doublet is dominated by the short period; the phugoid appears over tens of seconds. Measured from a simulated record:')
Be = [0.0, -9.0, -14.0, 0.0]
def sim(A, B, ufun, T, dt=0.005):
    x = [0.0] * 4; rec = []
    for i in range(int(T / dt)):
        t = i * dt
        def f(t_, s): 
            d = g.mat_vec(A, s)
            return [d[k] + B[k] * ufun(t_) for k in range(4)]
        x = g.rk4_step(f, t, x, dt)
        rec.append((t, list(x)))
    return rec
dbl = lambda t: 0.02 if t < 0.5 else (-0.02 if t < 1.0 else 0.0)
rec = sim(Alon, Be, dbl, 60.0)
q = [(t, x[2]) for t, x in rec]
peaks = [(q[i][0], q[i][1]) for i in range(1, len(q) - 1) if (q[i][1] - q[i - 1][1]) * (q[i + 1][1] - q[i][1]) < 0 and abs(q[i][1]) > 1e-4]
early = [p for p in peaks if p[0] < 4.0]
print(f'  pitch-rate extrema in the first 4 s at t = ' + ', '.join(f'{p[0]:.2f}' for p in early[:5]) + f' s; half-period of the short period {math.pi / math.sqrt(1 - zsp ** 2) / wsp:.2f} s')
th = [(t, x[3]) for t, x in rec]
tp = [th[i][0] for i in range(1000, len(th) - 1) if (th[i][1] - th[i - 1][1]) * (th[i + 1][1] - th[i][1]) < 0][:3]
print(f'  pitch-angle extrema at t = ' + ', '.join(f'{t:.1f}' for t in tp) + f' s; half-period of the phugoid {math.pi / wp:.1f} s')

longitudinal modes:
              pair -0.0071 +- 0.3988j: natural frequency 0.399 rad/s, damping ratio 0.018, period  15.75 s
              pair -3.5229 +- 3.5000j: natural frequency 4.966 rad/s, damping ratio 0.709, period   1.80 s
lateral-directional modes:
              pair -0.4058 +- 3.6109j: natural frequency 3.634 rad/s, damping ratio 0.112, period   1.74 s
              real root -0.0317: time constant  31.56 s
              real root -8.2067: time constant   0.12 s
approximations: short period from the (w, q) block, phugoid from the Lanchester formula, roll from L_p, spiral from the quasi-steady condition
  short period: wn = 4.950 rad/s, zeta = 0.707
  phugoid: Lanchester-type wn = sqrt(-Zu g / U0) = 0.560 rad/s; with the quasi-steady short period, U0 is replaced by 49.0 m/s and wn = 0.400 rad/s, zeta = -Xu / (2 wn) = 0.075
  roll subsidence: lambda = L_p = -8.00 1/s
  spiral (quasi-steady): lambda = -0.0352 1/s (stable while L_v N_r > L_r N_v: 0.80 against 0.45)
  Dutch roll (rough): wn = sqrt(N_v U0 + ...) = 3.38 rad/s
changing the static margin: M_w from -0.1 to -1.0 (more negative = more stable) moves the short period
  M_w = -0.10: short period wn 3.825 rad/s, zeta 0.919; sqrt(Zw Mq - U0 Mw) = 3.808
  M_w = -0.30: short period wn 4.437 rad/s, zeta 0.794; sqrt(Zw Mq - U0 Mw) = 4.416
  M_w = -0.50: short period wn 4.966 rad/s, zeta 0.709; sqrt(Zw Mq - U0 Mw) = 4.950
  M_w = -1.00: short period wn 6.088 rad/s, zeta 0.578; sqrt(Zw Mq - U0 Mw) = 6.083
response to an elevator doublet is dominated by the short period; the phugoid appears over tens of seconds. Measured from a simulated record:
  pitch-rate extrema in the first 4 s at t = 0.46, 0.99, 1.55, 2.40 s; half-period of the short period 0.90 s
  pitch-angle extrema at t = 12.5, 20.4, 28.3 s; half-period of the phugoid 7.8 s

In robotics

The modes are the vocabulary of aircraft control.

• Design requirements. Specifications state limits on the frequency and damping of each mode (short-period damping above about 0.35, a minimum time to double for an unstable spiral, a minimum Dutch-roll damping), so the first analysis of a new design is its eigenvalues and mode shapes.
• Which derivative to change. A short period too lively or too sluggish is fixed by the static margin (moving the centre of gravity or the tail size); a poorly damped Dutch roll by the fin size or a yaw damper; a short spiral time constant by dihedral. The next lesson shows the damper that handles the Dutch roll without redesigning the airframe.
• Identification from flight. A doublet on the elevator or rudder excites a mode and the response gives its frequency and damping by fitting a decaying sinusoid; this is how derivatives are checked in flight test, and the lab M10-L1: Name the Mode does it from simulated records.
• Time scales. The modes span three orders of magnitude (0.12 s to 31.6 s): this is why the autopilot has nested loops, and why the phugoid and spiral are left to the pilot or to slow outer loops.
• Beyond the linear model. The model is valid for small disturbances about one trim. Large angles of attack, stalls, inertia coupling in rolling manoeuvres and flexibility change the modes, and the gain-scheduled design of module 5 covers the variation over the envelope.

The phugoid, short period and Dutch roll are closed-loop targets for M10-T2: The Autopilot by Successive Loop Closure. The linearization used here is that of M2-T4: Trim and Linearization applied to the aircraft instead of Hopper.

Common misconceptions

Key takeaways

1. A symmetric aircraft has uncoupled longitudinal (u, w, q, θ) and lateral-directional (v, p, r, φ) models.
2. The longitudinal modes are the short period (4.97 rad/s, ζ 0.71) and the phugoid (0.40 rad/s, ζ 0.018) for the example.
3. The lateral modes are the roll subsidence (time constant 0.12 s), the spiral (31.6 s) and the Dutch roll (3.63 rad/s, ζ 0.11).
4. Short period: $\omega_{sp} = \sqrt{Z_wM_q - U_0M_w}$; roll: $\lambda \approx L_p$; spiral stable if $L_vN_r > L_rN_v$; Dutch roll: $\sqrt{N_vU_0 + Y_vN_r}$.
5. Making $M_w$ more negative raises the short-period frequency and lowers its damping.
6. The modes span 0.12 s to 31.6 s, which is why the autopilot is built from nested loops.

References and further reading

Each source is marked free or paid. Free to read does not mean free to reuse: the course resources page lists every source with its licence.

• B. Etkin and L. D. Reid, Dynamics of Flight: Stability and Control, 3rd edition, Wiley, 1996, the standard text on the modes and derivatives. Paid.
• R. C. Nelson, Flight Stability and Automatic Control, 2nd edition, McGraw-Hill, 1998. Paid.
• R. W. Beard and T. W. McLain, Small Unmanned Aircraft: Theory and Practice, Princeton University Press, 2012, the chapters on the dynamics of small fixed-wing aircraft. Paid.
• MIT 16.333 Aircraft Stability and Control (OpenCourseWare) , the lectures on the longitudinal and lateral-directional modes. Free; licence CC BY-NC-SA 4.0.

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What’s next

Related content

Nothing below is required to finish this lesson.

• Lab
  M10-L1: Name the Mode

  From the responses to an elevator and a rudder doublet, estimate the frequency and damping of each mode, name it, and say which stability derivative to change to fix the worst one.

Other courses on this site treat neighbouring topics from their own angle:

• UAV M1-T2: Multirotor, fixed-wing and VTOL architectures
• UAV M5-T1: UAV equations of motion (wind triangle)
• UAV M6-T6: Flight modes
• UAV M6-T3: PX4 architecture
• Navigation M9-T5: Other Platforms: Legged, Aerial, Marine and Off-Road

This course leaves the following to the pages that already teach them:

• Multirotor flight control
• The wind triangle
• PX4 and ArduPilot software, flight modes and tooling