Library / Flight dynamics and control / FLT-02
Flight dynamics and control
Pitch-rate response to an elevator step
Pitch rate after a step elevator deflection, for a reduced first-order model of the aircraft near trim.
- Ref
- FLT-02
- Rev
- v1
- Field
- Flight dynamics and control
- Result
- q(t) [°/s]
The reduced model q̇ = −λqq + beδe describes the pitch rate near trim. λq = −aq is the rate at which the aircraft's own damping removes pitch rate, and be is the elevator control derivative. For a step elevator deflection the pitch rate rises exponentially to the steady value beδe/λq with time constant 1/λq. Angle-of-attack coupling is left out, so this is not the full short-period model.
Pitch rate after a step elevator deflection, for a reduced first-order model of the aircraft near trim. The model is q̇ = −λqq + beδe, where λq = −aq, and the rate rises exponentially to the steady value beδe/λq.
Calculator
- Steady-state pitch rate, beδe/λq6 °/s
- Time constant, 1/λq0.5 s
- Time to reach 95 % of the steady rate, 3/λq1.5 s
- Fraction of the steady rate reached at t86.47 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The pitch rate heads for 6 °/s with a time constant of 0.5 s, so it has reached 86.5 % of that after 1 s.
- Halving the damping to λq = 1 doubles the steady pitch rate to 12 °/s but doubles the time constant to 1 s. The pitch rate after 1 s is 7.585 °/s.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| δe Elevator deflection | 0.5 to 5 ° | 1.729 | 17.29 | 15.56 | +10% | +0.519 °/s |
| be Elevator effectiveness (nose-up positive) | 2 to 20 1/s² | 1.297 | 12.97 | 11.67 | +10% | +0.519 °/s |
| λq Pitch-rate damping rate (minus the rate derivative) | 0.5 to 6 1/s | 1.995 | 9.443 | 7.448 | +10% | −0.338 °/s |
| t Time since the step | 0 to 5 s | 0 | 6 | 6 | +10% | +0.147 °/s |
Sweep of Time since the step
| Time since the step (s) | Pitch rate (°/s) |
|---|---|
| 0 | 0 |
| 0.625 | 4.281 |
| 1.25 | 5.507 |
| 1.875 | 5.859 |
| 2.5 | 5.96 |
| 3.125 | 5.988 |
| 3.75 | 5.997 |
| 4.375 | 5.999 |
| 5 | 6 |
Sweep one variable
Variables and how each one contributes
Pitch rate
What the equation gives you. Everything else feeds it.
Pitch-rate damping rate (minus the rate derivative)
The pitch-rate damping rate in 1/s: λq = −aq. Larger means more damping and a faster response. More damping both lowers the steady pitch rate and shortens the time constant. At a given time the rate can be higher or lower, since the two effects compete. Raising it by 10 % (to 2.2) moves the result by −6.51 %, from 5.188 to 4.85 °/s.
Elevator effectiveness (nose-up positive)
The elevator effectiveness be in 1/s², positive for a nose-up response to positive deflection. Directly proportional: a more effective elevator gives a proportionally larger pitch rate at every instant. Raising it by 10 % (to 8.8) moves the result by +10 %, from 5.188 to 5.707 °/s.
Elevator deflection
The elevator deflection in degrees, held constant after the step. Directly proportional, as long as the linear model holds. Raising it by 10 % (to 1.65) moves the result by +10 %, from 5.188 to 5.707 °/s.
Time since the step
The time since the step, in s. The rate rises towards its steady value, gaining less each second. Raising it by 10 % (to 1.1) moves the result by +2.84 %, from 5.188 to 5.335 °/s.
Worked example with the values above
- Steady rate: beδe/λq = 8 × 1.5 / 2 = 6 °/s.
- Fraction reached: 1 − e−2 = 1 − 0.1353 = 0.8647.
- q(t) = 6 × 0.8647 = 5.188 °/s.
Assumptions and limits
- A reduced linear model near trim: constant airspeed, small deflections, and no coupling to angle of attack.
- A step input in elevator from rest, with the pitch rate zero at the start.
Common mistakes
- Confusing the pitch rate q with the rate of change of pitch angle θ̇. They are equal only with the wings level.
- Assuming the sign convention without checking it. The sign of the elevator effect depends on the axes and deflection convention.
Statistics and uncertainty
The steady pitch rate scales with be/λq, so independent errors of 10 % in each give about 14 % in the steady rate (√(10² + 10²)). Control derivatives are usually identified from flight test or a validated model, and they change with speed, altitude and centre of gravity, so a single value is valid only near one trim point.
Where it is used
- Estimating how quickly an aircraft responds to an elevator command.
- Sizing the inner pitch-rate loop and checking the model against a flight-test step.
Maths you need
- First-order response: exponential approach with time constant 1/λ
- Control derivative: the change in moment per unit deflection