FLIGHT CONTROL & AUTOPILOT / CHAPTER 05

Vertical Speed & Flight-path Angle

Command the climb. Watch the energy and speed respond.

Coupled small-disturbance longitudinal dynamics · original synthetic teaching example · small perturbations around fixed trim

02 / LIVE B · 3D ATTITUDE

0 s

Body direction · flight direction · level reference

A · no saved case

CONTROL → RESPONSE → FEEDBACK

Preparing the model

03 / THE CALCULATED TRAJECTORY

Rate of climb and path angle share the same motion

A · dashed amber B · solid turquoise

Calculated height versus distance. Axes have different scales. Dotted line: integrated rate command, not an altitude-hold target. Each case stops at its own validity boundary.

Body attitude and the energy balance

Local mechanical-energy change

Δ E = m V_{0} Δ V + m g Δ h

Net power follows the same first-order aircraft equations.

Symbols & units

$θ, γ, α$: Pitch attitude, flight-path angle and angle of attack. Angles are radians internally; α = θ − γ.
$Δ E$: First-order aircraft mechanical-energy change, J. The kinetic term is a linear perturbation, not exact finite-amplitude kinetic energy.
$m, V_{0}$: Fixed mass 1800 kg; trim speed 60 m/s.

Model context and notation

Pitch, flight-path angle and angle of attack remain distinct. ΔE = mV₀ΔV + mgΔh; net power follows the same first-order aircraft equations.

Rate and angle

The local model uses vertical speed ≈ V₀γ, with V₀ = 60 m/s. A 1° path angle corresponds to about 1.047 m/s. The neglected products of perturbations are second order.

Pitch is not path angle

Angle of attack is the difference between body pitch and flight-path angle. The pitch loop must move the body to change the velocity direction.

Climbing needs energy

Without extra thrust, a sustained climb can reduce speed. The simulation stops at its model boundary; it never silently extends into stall or impossible thrust.

04 / ONE PLAYBACK CLOCK

Follow the whole response

A · dashed amber / B · turquoise
Dotted: command. Click or use arrow keys to scrub.

Compare A and B

Quantity	A · saved	B · live

A stays fixed for this page session. Export includes settings and sampled histories.

Equations, assumptions and sources

Model contract

The same synthetic 1,800 kg wing–tail aircraft as chapters 01–02 is linearised at 60 m/s and 1,500 m. Targets ramp over 1–6 s. Vertical-speed feedback requests a pitch change; the unchanged inner pitch loop drives an elevator actuator. Autothrottle is a PI speed loop with engine lag. VS mode and FPA mode are equivalent at matching local commands: w = V₀γ. This local model cannot demonstrate large-speed changes at a fixed angle. Density is frozen. The moment pulse is prescribed, not a wind gust.

Vertical-speed feedback, pitch control and thrust

w = V_{0} Δ γ

e_{w} = w_{c} - w

{Δ θ}_{c} = \frac{w_{c}}{V_{0}} + R (0.02 e_{w} + 0.003 \int e_{w} d t)

{Δδ}_{e,c} = - 1.2 ({Δ θ}_{c} - Δ θ) + 0.5 q

{Δ T}_{c} = 400 e_{v} + 40 \int e_{v} d t

The relation w = V₀Δγ is first order about level flight. Elevator and thrust retain the existing actuator and engine lag; this is not an altitude-hold target.

Symbols & units

$Δ$: Change from the initial trim state. It does not indicate a different aircraft or an absolute angle.
$w, w_{c}, e_{w}$: Rates in m/s. w is vertical speed here, not wind. V₀ = 60 m/s. FPA mode uses wc = V₀γc.
$R$: Dimensionless response multiplier. The vertical gains are 0.02 rad s/m and 0.003 rad/m.
$e_{v}, {Δ T}_{c}$: eᵥ: m/s; thrust command change: N. Thrust gains are 400 N s/m and 40 N/m; autothrottle OFF commands zero change.

Model context and notation

Limits and sources

Stop before leaving the existing aircraft bounds: speed change 10%; angle change 5°; height change 150 m; pitch command 6°; elevator and thrust constraints as in chapter 02. No saturation recovery, stall or anti-windup simulation.

Teaching connections: reference frames, turning performance, flight control and flight management systems. Original extension with synthetic parameters.

MIT OCW · longitudinal autopilots supports the control concepts; its aircraft coefficients are not used.

Numerical verification does not establish flight-test validity.