Aircraft Dynamics and Control Aircraft motion is more complicated than spacecraft motion due to the dependence of forces and moments on attitude and velocity Typical first investigations of aircraft dynamics and control focuses on the longitudinal and lateral motions of symmetric (left-to-right) aircraft near straight and level flight

Motion Equations for a Rigid Aircraft Fb : v˙

=

Fb : ω˙

=

Fi : r˙ θ˙

¤ 1 £ −ω v + faero + fthrust + Rbi agrav m ¤ £ −1 × I −ω Iω + gaero + gthrust ×

=

Rib v

=

S−1 (θ)ω agrav = [0 0 g]T

where

Note that v and ω are expressed in Fb , and r is expressed in Fi Usually, aircraft d & c folks write v

=

[u v w]T ,

ω = [p q r]T

Assuming left-right symmetry (but not up-down symmetry) implies that Ixy = Iyz = 0, but in general Ixz 6= 0

Reference Frames for Aircraft D&C Inertial Frame, Fi . Typically assume flat Earth, but rotating Earth is used for some problems. The inertial frame has 3-axis down, and origin at aircraft mass center Velocity Frame, Fv . A 3-2 rotation from Fi through the horizontal flight path angle ξ and the vertical flight path angle γ: Rvi = R2 (γ)R3 (ξ) Wind Frame, Fw . A 1 rotation from Fv through bank angle μ: Rwv = R1 (μ) Body Frame, Fb . From Fw , a -3 rotation through sideslip angle β, followed by a 2 rotation through α: Rbw = R2 (α)R3 (−β) Body Frame, Fb . A 3-2-1 rotation from Fi through yaw (ψ), pitch (θ), and roll (φ) angles: Rbi = R1 (φ)R2 (θ)R3 (ψ)

Reference Frames for Aircraft D&C (2) Illustration (from R. Stengel’s Flight Dynamics) shows relationship between pitch, yaw, angle of attack, sideslip, and flight path angles Note that the illustration is based on a zero-roll and zerobank angles Roll and bank are similar to each other: roll is about the body 1-axis, and bank is about the velocity vector

Motion Equations for a Rigid Aircraft (2) Typical kinematics variables are yaw (ψ), pitch (θ), and roll (φ) angles, a 3-2-1 sequence (could use quaternions or other Euler angle sequence) The aerodynamic forces and moments depend on airspeed (Va ), angle of attack (α), and sideslip angle (β) Relationship between velocity vector components (u, v, s) and airflow angles (α, β): ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ √ 2 2 2 u cαcβ V u +v +w ⎣ v ⎦ = V ⎣ sβ ⎦ ⎣ α ⎦ = ⎣ tan−1 (w/u) ⎦ w sαcβ β sin−1 (v/V )

Straight and Level Flight

The state vector is h iT x = v T ω T rT θ T =

[u v w p q r x y z φ θ ψ]

T

For straight and level flight, the aircraft must have a non-zero angle of attack (to generate lift=weight), and hence non-zero pitch angle The straight and level state vector is x∗

=

T

[Vo cos αo 0 Vo sin αo 0 0 0 xo yo zo 0 θo 0]

The control to achieve this steady motion is u∗

=

T

[δEo δAo δRo δTo ]

(Elevator angle, Aileron angle, Rudder angle, Thrust)

Linearized Motion Linearizing about the straight and level flight leads to a linear system with n = 12, p = 4 (12 states, 4 inputs) The equations are block-decoupled into two systems: a longitudinal motion system and a lateral motion system, each with six states and two inputs Longitudinal Motion: x

=

[∆u ∆w ∆q ∆x ∆z ∆θ]

u

=

[∆δE ∆δT ]

T

T

Lateral Motion: T

x

=

[∆v ∆p ∆r ∆y ∆φ ∆ψ]

u

=

[∆δA ∆δR]

T

Longitudinal Motion Longitudinal Motion: Elevator and Thrust control position, velocity in vertical plane, pitch angle and pitch rate

⎡ A

=

x

=

[∆u ∆w ∆q ∆x ∆z ∆θ]

u

=

[∆δE ∆δT ]

Xu

⎢ Zu ⎢ 1−Zw˙ ⎢ M + aZ u u ⎢ ⎢ cos θ o ⎢ ⎣ − sin θo 0

Xw Zw 1−Zw˙

Mw + aZw sin θo cos θo 0

T

T

Xq − wo Zq +uo 1−Zw˙

Mq + a(Zq + uo ) 0 0 1

0 0 0 0 0 0

Xz Zz 1−Zw˙

0 0 0 0

−go cos θo

⎤

⎥ ⎥ Mθ ago sin θo ⎥ ⎥ ⎥ b ⎥ ⎦ c −go sin θo 1−Zw˙

0

The Xu , Mu , Mθ etc. terms represent the partial derivatives of the forces and moments with respect to velocity and angular velocity; a, b, c also depend on aerodynamic properties All terms are evaluated at the reference flight condition, typically using extensive tabulated aerodynamic data for a particular aircraft

Longitudinal Motion (2) Longitudinal Motion: Elevator and Thrust control position, velocity in vertical plane, pitch angle and pitch rate x

=

[∆u ∆w ∆q ∆x ∆z ∆θ]

u

=

[∆δE ∆δT ] ⎡

B

=

XδE

T

T

⎢ ZδE ⎢ 1−Zw˙ ⎢ MδE + aZδE ⎢ ⎢ 0 ⎢ ⎣ 0 0

XδT ZδT 1−Zw˙

MδT + aZδT 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

All terms are evaluated at the reference flight condition, typically using extensive tabulated aerodynamic data for a particular aircraft Because the coupling of the position components with the velocity and angular components is light, a fourth-order model is frequently used by omitting x and z

Longitudinal Motion (3) Longitudinal Motion (4th order model): Elevator and Thrust control velocity in vertical plane, pitch angle and pitch rate

A

=

⎡

x

=

[∆u ∆w ∆q ∆θ]

u

=

[∆δE ∆δT ]

Xu

⎢ Zu ⎢ 1−Zw˙ ⎣ Mu + aZu 0

Xw Zw 1−Zw˙

Mw + aZw 0

T

T

Xq − wo Zq +uo 1−Zw˙

Mq + a(Zq + uo ) 1

The B matrix simply loses the 4th and 5th rows

−go cos θo

⎤

⎥ ⎥ Mθ ago sin θo ⎦ 0 −go sin θo 1−Zw˙

Longitudinal Motion (4) Business Jet Example of Longitudinal Motion (4th order model): Elevator and Thrust control velocity in vertical plane, pitch angle and pitch rate ⎡ ⎤ −0.0121 0.0960 −6.4500 −9.7870 ⎢ −0.1160 −1.2770 100.8000 −0.6202 ⎥ ⎥ A = ⎢ ⎣ 0.0050 −0.0781 −1.2790 0.0000 ⎦ 0.0000 0.0000 1.0000 0.0000 ⎡ ⎤ 0.0065 4.6740 ⎢ −13.1700 0.0000 ⎥ ⎥ B = ⎢ ⎣ −9.0690 0.0000 ⎦ 0.0000 0.0000 Stability check:

eig(A) ⇒ λ ∈ (−1.2767 ± 2.8101i, −0.0074 ± 0.1256i) Stable, and block-diagonalizable

Longitudinal Motion (5) Business Jet Example of Longitudinal Motion (4th order model) continued.... Controllability

Mc rank Mc

⎡

0.0000 0.0047 0.0572 ⎢ −0.0132 0 −0.8973 = ⎢ ⎣ −0.0091 0 0.0126 0 0 −0.0091 = 4 ⇒ Controllable

−0.0001 −0.0005 0.0000 0

−0.0795 2.4178 0.0542 0.0126

⎤ −0.0002 0.0031 ⎥ ⎥ × 103 0.0000 ⎦ 0.0000

Observability If any state is observable (C is a row of the identity matrix), then the system is observable. Modal Analysis

Eb

Λb

⎡

−0.0444 0.0145 ⎢ 0.9985 0 = ⎢ ⎣ 0.0000 0.0278 0.0082 −0.0037 ⎡ −1.2767 2.8101 ⎢ −2.8101 −1.2767 = ⎢ ⎣ 0.0000 −0.0000 −0.0000 0.0000

⎤ −0.9992 0 −0.0375 0.0017 ⎥ ⎥ −0.0016 0.0001 ⎦ 0.0012 0.0128 −0.0000 −0.0000 −0.0074 −0.1256

⎤ 0.0000 0.0000 ⎥ ⎥ 0.1256 ⎦ −0.0074

Longitudinal Motion (6) Business Jet Example of Longitudinal Motion (4th order model) continued.... Res ponse to Initial Conditions

To: Out(1 )

0.05

0

-0.05

Amplitud e

To: Out( 2)

1 0.5 0 -0.5

To: Out(3 )

0.01 0 -0.01 -0.02

To: Ou t(4)

10

x 10

-3

Note Final Time

5 0 -5 0

0.5

1

1.5

2

2.5

3

3. 5

4

4.5

Time (sec )

Short-period, heavily damped motion resulting from initial conditions in first pair of modes, with λ = −1.2767 ± 2.8101i

Longitudinal Motion (7) Business Jet Example of Longitudinal Motion (4th order model) continued.... Res ponse to Initial Conditions

To: Out( 1)

1 0.5 0 -0.5 -1

Amplitud e

To: Out(2 )

0.05

0

-0.05

To: Ou t(3)

2

x 10

-3

1 0 -1 -2

To: Out(4 )

0.01

Note Final Time

0 -0.01 -0.02

0

100

200

300

400

500

600

700

800

Time (sec )

Long-period, lightly damped motion resulting from initial conditions in second pair of modes, with λ = −0.0074 ± 0.1256i. This mode is called the phugoid mode

Lateral Motion Lateral Motion: Aileron and Rudder control position, velocity perpendicular to vertical plane, roll and yaw angles and rates x

=

[∆v ∆p ∆r ∆y ∆φ ∆ψ]

u

=

[∆δA ∆δR]

T

T

As with longitudinal motion, A and B depend on numerous aerodynamic force and moment terms that depend on the reference state Typically, there are two zero eigenvalues, one real positive eigenvalue, one real negative eigenvalue, and a complex conjugate pair The two zero eigenvalues correspond to crossrange and yaw modes; the unstable eigenvalue corresponds to a spiral divergence mode, the stable real eigenvalue corresponds to a roll mode, and the complex pair corresponds to the so-called Dutch roll mode