10. Aircraft Equations of Motion

Home , Angular momentum, Equations of motion, Flight dynamics, Inertia, Moment (physics), Rotating reference frame

Aircraft Equations of Motion: Translation and Rotation Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Learning Objectives • What use are the equations of motion? • How is the angular orientation of the airplane described? • What is a cross-product-equivalent matrix? • What is angular momentum? • How are the inertial properties of the airplane described? • How is the rate of change of angular momentum calculated? Lockheed F-104 Reading: Flight Dynamics 155-161

Review Questions § What characteristic(s) provide maximum gliding range? § Do gliding heavy airplanes fall out of the sky faster than light airplanes? § Are the factors for maximum gliding range and minimum sink rate the same? § How does the maximum climb rate vary with altitude? § What are “energy height” and “specific excess power”? § What is an “energy climb”? § How is the “maneuvering envelope” defined? § What factors determine the maximum steady turning rate?

1 Dynamic Systems

Actuators

Sensors

Dynamic Process: Current state depends on Observation Process: Measurement may prior state contain error or be incomplete x = dynamic state y = output (error-free) u = input z = measurement w = exogenous disturbance n = measurement error p = parameter t or k = time or event index

dx(t) y(t) = h[x(t),u(t)] = f [x(t),u(t),w(t),p(t),t] dt z(t) = y(t) + n(t)

Ordinary Differential Equations Fall Into 4 Categories

dx(t) dx(t) = f [x(t),u(t),w(t),p(t),t] = F(t)x(t) + G(t)u(t) + L(t)w(t) dt dt

dx(t) dx(t) = f [x(t),u(t),w(t)] = Fx(t) + Gu(t) + Lw(t) dt dt 4

2 What Use are the Equations of Motion?

• Nonlinear equations of motion – Compute exact flight paths and dx(t) = f [x(t),u(t),w(t),p(t),t] motions dt • Simulate flight motions • Optimize flight paths dx(t) • Predict performance = Fx(t) + Gu(t) + Lw(t) – Provide basis for approximate dt solutions

• Linear equations of motion – Simplify computation of flight paths and solutions – Define modes of motion – Provide basis for control system design and flying qualities analysis

Examples of Airplane Dynamic System Models

• Nonlinear, Time-Varying • Nonlinear, Time-Invariant – Large amplitude motions – Large amplitude motions – Negligible change in mass – Significant change in mass

• Linear, Time-Varying • Linear, Time-Invariant – Small amplitude motions – Small amplitude motions – Perturbations from a dynamic flight path – Perturbations from an equilibrium flight path

3 Translational Position

Position of a Particle Projections of vector magnitude on three axes

⎡ ⎤ ⎡ x ⎤ cosδ x ⎢ ⎥ r ⎢ y ⎥ r cos = ⎢ ⎥ = ⎢ δ y ⎥ ⎢ z ⎥ ⎢ cosδ ⎥ ⎣ ⎦ ⎣⎢ z ⎦⎥

⎡ cosδ ⎤ ⎢ x ⎥ ⎢ cosδ y ⎥ = Direction cosines ⎢ cosδ ⎥ ⎣⎢ z ⎦⎥

4 Cartesian Frames of Reference

• Translation • Two reference frames of interest – Relative linear positions of origins – I: Inertial frame (fixed to inertial space) • Rotation – B: Body frame (fixed to body) – Orientation of the body frame with respect to the inertial frame

Common convention (z up) Aircraft convention (z down)

Measurement of Position in Alternative Frames - 1

• Two reference frames of interest – I: Inertial frame (fixed to inertial space) – B: Body frame (fixed to body) Inertial-axis view

⎡ x ⎤ r ⎢ y ⎥ = ⎢ ⎥ ⎣⎢ z ⎦⎥

rparticle = rorigin + Δrw.r.t. origin

• Differences in frame orientations must be taken into account in adding vector components

Body-axis view 10

5 Measurement of Position in Alternative Frames - 2

Inertial-axis view r r HI r particleI = origin−BI + BΔ B

Body-axis view r r HB r particleB = origin−IB + I Δ I

Rotational Orientation

6 Direction Cosine Matrix

⎡ ⎤ cosδ11 cosδ 21 cosδ 31 B ⎢ ⎥ HI = ⎢ cosδ12 cosδ 22 cosδ 32 ⎥ ⎢ cosδ cosδ cosδ ⎥ ⎣ 13 23 33 ⎦ • Projections of unit vector components of one reference frame on another • Rotational orientation of one reference frame with respect to another B • Cosines of angles between rB = HI rI each I axis and each B axis

Properties of the Rotation Matrix

B B ⎡ ⎤ ⎡ ⎤ cosδ11 cosδ 21 cosδ 31 h11 h12 h13 B ⎢ ⎥ ⎢ ⎥ HI = ⎢ cosδ12 cosδ 22 cosδ 32 ⎥ = ⎢ h21 h22 h23 ⎥ ⎢ ⎥ ⎢ ⎥ cosδ13 cosδ 23 cosδ 33 h31 h32 h33 ⎣ ⎦I ⎣ ⎦I

B B rB = HI rI sB = HI sI Orthonormal transformation Angles between vectors are preserved r s Lengths are preserved

rI = rB ; sI = sB

∠(rI ,sI ) = ∠(rB ,sB ) = x deg

7 Euler Angles • Body attitude measured with respect to inertial frame • Three-angle orientation expressed by sequence of three orthogonal single-angle rotations

Inertial ⇒ Intermediate1 ⇒ Intermediate2 ⇒ Body

• 24 (12) possible sequences of single-axis rotations • Aircraft convention: 3-2- 1, z positive down ψ : Yaw angle θ : Pitch angle φ : Roll angle 15

Euler Angles Measure the Orientation of One Frame with Respect to the Other

• Conventional sequence of rotations from inertial to body frame – Each rotation is about a single axis – Right-hand rule – Yaw, then pitch, then roll – These are called Euler Angles

Yaw rotation (ψ) about zI Pitch rotation (θ) about y1 Roll rotation (ϕ) about x2

Other sequences of 3 rotations can be chosen; however, once sequence is chosen, it must be retained 16

8 Reference Frame Rotation from Inertial to Body: Aircraft Convention (3-2-1)

Yaw rotation (ψ) about zI axis

⎡ ⎤ ⎡ x ⎤ ⎡ cosψ sinψ 0 ⎤⎡ x ⎤ xI cosψ + yI sinψ ⎢ ⎥ ⎢ ⎥ 1 ⎢ y ⎥ ⎢ y ⎥ x sin y cos r = H r ⎢ ⎥ = ⎢ −sinψ cosψ 0 ⎥⎢ ⎥ = ⎢ − I ψ + I ψ ⎥ 1 I I ⎢ z ⎥ ⎢ 0 0 1 ⎥⎢ z ⎥ ⎢ z ⎥ ⎣ ⎦1 ⎣ ⎦⎣ ⎦I ⎣ I ⎦

Pitch rotation (θ) about y1 axis

⎡ x ⎤ ⎡ cosθ 0 −sinθ ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥ 2 2 1 2 y = ⎢ ⎥ y r = H r = ⎡H H ⎤r = H r ⎢ ⎥ ⎢ 0 1 0 ⎥⎢ ⎥ 2 1 1 ⎣ 1 I ⎦ I I I ⎢ z ⎥ ⎢ sinθ 0 cosθ ⎥⎢ z ⎥ ⎣ ⎦2 ⎣ ⎦⎣ ⎦1

Roll rotation (ϕ) about x2 axis

⎡ x ⎤ ⎡ 1 0 0 ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ B B 2 1 B y 0 cosφ sinφ y rB = H2 r2 = ⎣⎡H2 H1 HI ⎦⎤rI = HI rI ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ z ⎥ ⎢ 0 −sinφ cosφ ⎥⎢ z ⎥ ⎣ ⎦B ⎣ ⎦2 ⎣ ⎦ 17

The Rotation Matrix The three-angle rotation matrix is the product of 3 single-angle rotation matrices:

B B 2 1 HI (φ,θ,ψ) = H2 (φ)H1 (θ)HI (ψ)

⎡ 1 0 0 ⎤⎡ cosθ 0 −sinθ ⎤⎡ cosψ sinψ 0 ⎤ ⎢ ⎥ ⎢ ⎥ = 0 cosφ sinφ ⎢ ⎥ ⎢ ⎥⎢ 0 1 0 ⎥⎢ −sinψ cosψ 0 ⎥ ⎢ 0 −sinφ cosφ ⎥ sinθ 0 cosθ ⎢ ⎥ ⎣ ⎦⎣⎢ ⎦⎥⎣ 0 0 1 ⎦

⎡ cosθ cosψ cosθ sinψ −sinθ ⎤ ⎢ ⎥ = ⎢ − cosφ sinψ + sinφ sinθ cosψ cosφ cosψ + sinφ sinθ sinψ sinφ cosθ ⎥ ⎢ sinφ sinψ + cosφ sinθ cosψ −sinφ cosψ + cosφ sinθ sinψ cosφ cosθ ⎥ ⎣ ⎦ an expression of the Direction Cosine Matrix 18

9 Rotation Matrix Inverse Inverse relationship: interchange sub- and superscripts B rB = HI rI −1 r = HB r = HI r I ( I ) B B B Because transformation is orthonormal Inverse = transpose Rotation matrix is always non-singular

B −1 B T I ⎣⎡HI (φ,θ,ψ )⎦⎤ = ⎣⎡HI (φ,θ,ψ )⎦⎤ = HB (ψ ,θ,φ)

I B −1 B T I 1 2 HB = (HI ) = (HI ) = H1H2HB

I B B I 19 HB HI = HI HB = I

Checklist q What are direction cosines? q What are Euler angles? q What rotation sequence is used to describe airplane attitude? q What are properties of the rotation matrix?

10 Angular Momentum

Angular Momentum of a Particle • Moment of linear momentum of differential particles that make up the body – (Differential masses) x components of the velocity that are perpendicular to the moment arms

" ω x % dh = (r × dm v) = (r × v m )dm $ ' ω = $ ω y ' $ ' = ⎣⎡r × (vo + ω × r)⎦⎤dm ω #$ z &' • Cross Product: Evaluation of a determinant with unit vectors (i, j, k) along axes, (x, y, z) and (vx, vy, vz) projections on to axes

i j k x y z r × v = = (yvz − zvy )i + (zvx − xvz ) j + (xvy − yvx )k

vx vy vz 22

11 Cross-Product- Equivalent Matrix

i j k Cross product x y z r × v = = (yvz − zvy )i + (zvx − xvz ) j + (xvy − yvx )k

vx vy vz

# yv − zv & # & % ( z y ) ( # 0 −z y & vx % ( % ( % ( = zv − xv = rv = z 0 x vy % ( x z ) ( % − ( % ( % ( % ( % ( −y x 0 vz (xvy − yvx ) $ ' $% '( $% '( " 0 −z y % Cross-product-equivalent $ ' matrix r = $ z 0 −x ' $ −y x 0 ' # & 23

Angular Momentum of the Aircraft

• Integrate moment of linear momentum of differential particles over the body

⎡ ⎤ hx xmax ymax zmax ⎢ ⎥ h = ⎡r × v + ω × r ⎤dm = r × v ρ(x, y,z)dx dydz = h ∫ ⎣ ( o )⎦ ∫ ∫ ∫ ( ) ⎢ y ⎥ Body xmin ymin zmin ⎢ h ⎥ ⎣⎢ z ⎦⎥ ρ(x, y,z) = Density of the body

• Choose the center of mass as the rotational center Supermarine Spitfire h = r × v dm + r × ω × r dm ∫ ( o ) ∫ ⎣⎡ ( )⎦⎤ Body Body = 0 − ∫ ⎣⎡r × (r × ω)⎦⎤dm Body = − (r × r)dm × ω ≡ − (r!r!)dmω ∫ ∫ 24 Body Body

12 Location of the Center of Mass

# & xcm 1 xmax ymax zmax % ( r = r dm = rρ(x, y,z)dx dydz = % y ( cm m ∫ ∫ ∫ ∫ cm Body xmin ymin zmin % z ( $ cm '

The Inertia Matrix

13 The Inertia Matrix " ω % $ x ' h = − r! r! ω dm = − r! r! dm ω = ω ω = ω ∫ ∫ I $ y ' $ ' Body Body ω #$ z &' where ⎡ 0 −z y ⎤⎡ 0 −z y ⎤ ⎢ ⎥⎢ ⎥ I = − ∫ r! r! dm = − ∫ ⎢ z 0 −x ⎥⎢ z 0 −x ⎥ dm Body Body ⎢ −y x 0 ⎥⎢ −y x 0 ⎥ ⎣ ⎦⎣ ⎦ ⎡ (y2 + z2 ) −xy −xz ⎤ ⎢ ⎥ = ∫ ⎢ −xy (x2 + z2 ) −yz ⎥dm Body ⎢ 2 2 ⎥ ⎢ −xz −yz (x + y ) ⎥ ⎣ ⎦ Inertia matrix derives from equal effect of angular rate on all particles of the aircraft 27

Moments and Products of Inertia

2 2 ⎡ ⎤ ⎡ (y + z ) −xy −xz ⎤ I xx −I xy −I xz ⎢ ⎥ ⎢ ⎥ 2 2 − − I = ∫ ⎢ −xy (x + z ) −yz ⎥dm = ⎢ I xy I yy I yz ⎥ Body ⎢ 2 2 ⎥ ⎢ ⎥ −xz −yz (x + y ) − xz − yz zz ⎣⎢ ⎦⎥ ⎣⎢ I I I ⎦⎥

Inertia matrix • Moments of inertia on the diagonal • Products of inertia off the diagonal ⎡ ⎤ • If products of inertia are zero, (x, y, z) I xx 0 0 are principal axes ---> ⎢ ⎥ 0 0 • All rigid bodies have a set of principal ⎢ I yy ⎥ ⎢ 0 0 ⎥ axes ⎢ I zz ⎥ ⎣ ⎦

Ellipsoid of Inertia

2 2 2 I xx x + I yy y + I zzz = 1

14 Inertia Matrix of an Aircraft with Mirror Symmetry

⎡ 2 2 ⎤ ⎡ ⎤ (y + z ) 0 −xz I xx 0 −I xz ⎢ ⎥ ⎢ ⎥ 2 2 0 0 I = ∫ ⎢ 0 (x + z ) 0 ⎥dm = ⎢ I yy ⎥ Body ⎢ ⎥ ⎢ ⎥ xz 0 (x2 y2 ) − 0 ⎢ − + ⎥ ⎣⎢ I xz I zz ⎦⎥ ⎣ ⎦ Nose high/low product of inertia, Ixz

Nominal Configuration Tips folded, 50% fuel, W = 38,524 lb

xcm @0.218 c 6 2 I xx = 1.8 ×10 slug-ft 6 2 North American XB-70 I yy = 19.9 ×10 slug-ft 6 2 I xx = 22.1×10 slug-ft = −0.88 ×106 slug-ft2 29 I xz

Checklist q How is the location of the center of mass found? q What is a cross-product-equivalent matrix? q What is the inertia matrix? q What is an ellipsoid of inertia? q What does the “nose-high” product of inertia represent?

15 Historical Factoids Technology of World War II Aviation • 1938-45: Analytical and experimental approach to design – Many configurations designed and Spitfire flight-tested – Increased specialization; radar, navigation, and communication – Approaching the "sonic barrier • Aircraft Design – Large, powerful, high-flying aircraft – Turbocharged engines – Oxygen and Pressurization B-17

P-51D 31

Power Effects on Stability and Control

Brewster Buffalo F2A • Brewster Buffalo: over-armored and under-powered • During W.W.II, the size of fighters remained about the same, but installed horsepower GrummanWildcat F4F doubled (F4F vs. F8F) • Use of flaps means high power at low speed, increasing relative significance of thrust effects Grumman Bearcat F8F

16 World War II Carrier-Based Airplanes

Douglas TBD • Takeoff without catapult, relatively low landing speed http://www.youtube.com/watch?v=4dySbhK 1vNk • Tailhook and arresting gear Grumman TBF • Carrier steams into wind • Design for storage (short tail length, folding wings) affects stability and control

Chance-Vought F4U Corsair

F4U

Multi-Engine Aircraft of World War II

Consolidated B-24 Boeing B-29 Boeing B-17

Douglas A-26 • Large W.W.II aircraft had unpowered controls: – High foot-pedal force North American B-25 – Rudder stability problems arising from balancing to reduce pedal force

• Severe engine-out problem Martin B-26 for twin-engine aircraft

17 WW II Military Flying Boats Seaplanes proved useful during World War II

Martin PB2M Mars Lockheed PBY Catalina

Martin PBM Mariner Boeing XPBB Sea Ranger

Saunders-Roe SR.36 Grumman JRF-1 Goose Lerwick

Rate of Change of Angular Momentum

18 Newtons 2nd Law, Applied to Rotational Motion In inertial frame, rate of change of angular momentum = applied moment (or torque), M

⎡ ⎤ mx d ⎢ ⎥ dh (Iω) dI dω ! = = ω + I = Iω + Iω! = M = ⎢ my ⎥ dt dt dt dt ⎢ m ⎥ ⎢ z ⎥ ⎣ ⎦

Angular Momentum and Rate

Angular momentum and rate vectors are not necessarily aligned

h = Iω

19 How Do We Get Rid of dI/dt in the Angular Momentum Equation?

Chain Rule ... and in an inertial frame d( ω) I = !ω + ω! ! I I I ≠ 0 dt • Dynamic equation in a body-referenced frame – Inertial properties of a constant-mass, rigid body are unchanging in a body frame of reference – ... but a body-referenced frame is non-Newtonian or non-inertial – Therefore, dynamic equation must be modified for expression in a rotating frame 39

Angular Momentum Expressed in Two Frames of Reference • Angular momentum and rate are vectors – Expressed in either the inertial or body frame – Two frames related algebraically by the rotation matrix

B I hB (t) = HI (t)hI (t); hI (t) = HB (t)hB (t)

B I ω B (t) = HI (t)ω I (t); ω I (t) = HB (t)ω B (t)

20 Vector Derivative Expressed in a Rotating Frame Chain Rule Effect of body-frame rotation h HI h H I h I = B B + B B Rate of change Alternatively expressed in body frame h HI h h HI h  h I = B B + ω I × I = B B + ω I I Consequently, the 2nd term is H I h  h  HI h B B = ω I I = ω I B B # 0 −ω ω & ... where the cross-product % z y ( equivalent matrix of angular rate is ω = % ω z 0 −ω x ( % ( 0 % −ω y ω x ( $ ' 41

External Moment Causes Change in Angular Rate

Positive rotation of Frame B w.r.t. Frame A is a negative rotation of Frame A w.r.t. Frame B

In the body frame of reference, the angular momentum change is

! B ! ! B B ! B ! hB = HI hI + HI hI = HI hI − ω B × hB = HI hI − ω" BhB = HBM − ω" ω = M − ω" ω I I BIB B B BIB B

Moment = torque = force x moment arm ! $ ! $ mx mx ! L $ # & # & # & m B m MI = # y & ; MB = HI MI = # y & = # M & # & # & "# N %& # mz & # mz & " %I " %B 42

21 Rate of Change of Body- Referenced Angular Rate due to External Moment In the body frame of reference, the angular momentum change is ! B ! ! B B ! hB = HI hI + HI hI = HI hI − ω B × hB B ! B = HI hI − ω" BhB = HI MI − ω" BIBω B = M − ω" ω B BIB B For constant body-axis inertia matrix h! = ω! = M − ω" ω B IB B B BIB B Consequently, the differential equation for angular rate of change is

−1 ω! B = B (MB − ω" B Bω B ) I I 43

Checklist

q Why is it inconvenient to solve momentum rate equations in an inertial reference frame? q Are angular rate and momentum vectors aligned? q How are angular rate equations transformed from an inertial to a body frame?

22 Next Time: Aircraft Equations of Motion: Flight Path Computation Reading: Flight Dynamics 161-180

Learning Objectives How is a rotating reference frame described in an inertial reference frame? Is the transformation singular? What adjustments must be made to expressions for forces and moments in a non-inertial frame? How are the 6-DOF equations implemented in a computer? Damping effects 45

Supplemental Material

23 Moments and Products of Inertia (Bedford & Fowler) Moments and products of inertia tabulated for geometric shapes with uniform density Construct aircraft moments and products of inertia from components using parallel-axis theorem Model in CREO, etc.