Ch 9 Longitudinal Dynamics

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M A 3703 F light Dynamics Ai rcraft Stabi li ty & Controls By T. G. Pai

W

L W

L T

Chapter 9

L ongitudinal Dynamics 1



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M A 3703 Fl ight Dynamics

2

Longitudinal Dynamics

State Space EquationsCharacteristics Equations and Modes of Aircraft

Longitudinal :2 pairs of Complex Conjugate Roots -Phugoidand Short PeriodLateral : 2 Real Roots and one pair of Complex Conjugate

Stability Derivatives – Longitudinal and Lateral/Directional

One Degree of Freedom (DOF) Approximation to PitchingMotionTwo DOF approximation for Phugoid Motion and Short

Period Oscillations



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3

T eqw w u

eqw u w

T ew u

T e

T e

T e

M M qM w M w M u M qZ g qu Z w Z u Z w Z

X X g w X u X u

sin)()1(

cos

00

0

ar pv zz xz

r ar pv xx xz

r r pv

r a

r a

r

N r N pN v N pI I r LLr L pLv Lr I I p

g Y r Y pY v Y r u v

)()(

c 00

Axial Force

Normal Force

Rolling Moment

Yawing Moment

Pitching Moment

Side Force

Longitudinal and Lateral/Directional EOM



State-Space Longitudinal EOM

T

e

qq

uu

qu

u

Z uZ M M

Z uZ M M

Z u

Z

Z u

Z

X X

q

u

Z ug M u

Z u

uZ M M

Z uZ M

M Z u

Z M M

Z ug

Z u

uZ

Z uZ

Z uZ

g X X

q

u

T

T

e

e

T e

T e

00

0100

sin)(

sincos0

00

00

0

00

0

0

00

0

0

0

0

00

0

●

X A X

B u

X = A X + B u

●

M A 3703 F li ght Dynamics



Longitudinal Stability DerivativesM A 3703 Fl ight Dynamics



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6

BAxx

r pv x

qw u x

r a u

T e u

Solution of EOM in State Space Form

State Space Form of EOM:

Longitudinal:

Lateral /Directional

Where x is response vector and u is control inputs. A and

B depend on stability derivatives, inertia parameters andcontrol parameters .





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Longitudinal Aircraft DynamicsFrom Longitudinal State Space Equations we getLongitudinal Quartic in λ having normally two pairs of complex conjugate as its roots corresponding to ShortPeriod Oscillations (SPO) and Phugoid (long periodoscillations) modes:

A1λ 4+ B 1 λ 3+ C 1 λ 2+ D 1 λ + E 1 = 0(λ –λ 1)(λ -λ 2)(λ -λ 3)(λ -λ 4) = 0λ 1,2 = σ1 ± jω1 λ 3,4 =σ2 ± jω2

SPO Roots: λ 1,2

Complex Pair of ConjugatePhugoid Roots: λ 3,4 Complex pair of conjugate

Roots λ 1,2 , and λ 3,4 hence frequency/damping of SPO andPhugoid motion depend on initial flight conditions, stabilityderivatives and inertia parameters



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9

Longitudinal Characteristics Equation (Quartic)

a) Short Period Oscillation (SPO): λ 1,2 = ηspo ± iωspob) Phugoid (Long Period Oscillation): λ 3,4 = ηPh ± iωPh

SPO :

Period ~ 3 – 6 sec; Highly Damped. Aircraft inertia is veryhigh to respond; hence velocity changes are negligible

Phugoid:

Period ~ 50 - 100 sec or higher; Lightly DampedAngle of attack remains nearly constant and pitchingmoment does not change. Interchange of KE ( flight speed)and PE (altitude).

4



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10

Longitudinal Modes of Aircraft

PHUGOID orLong Period

Short Period Oscillation(SPO)



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11

Approximations of Longitudinal Dynamics (Δu,Δα, Δ

Short Period Approximation:Velocity remains constant : Δu = 0(Ai rcraf t I ner tia H igh to respond)Neglect Axial Force EquationSolve Normal Force & Pitching Moment Equations for(Δu, Δα, Δq)

Long Period or PHUGOID Approximation:

α nearly constant: Δα= 0 (or Δw = 0)Neglect Pitching Moment EquationSolve Axial & Normal Force Eqns for (Δu, Δα, Δq)



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T eqw w u

eqw u w

T ew u

T e

T e

T e


X X g w X u X u

sin)()1(

cos 00

0Axial Force

Normal Force

Pitching Moment

Phugoid Approx to Long EOM for Level FlightCos θ0 = 1

Δu -- X u Δu + g Δθ = 0 Δu = X u Δu -- g ΔθZ u Δu + u 0 Δq = 0 Δθ = -- (Z u/u 0 )Δu

Δu Xu -- gx = A =Δθ -- (Z u /u 0 ) 0

λI -A = 0 λ ² - X u λ -- (gZ u /u 0) = 0

● ●

●

CharacteristicsEquation for Phugoid

motion is :λ - Xu g= 0

(Z u/u 0 ) λ



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Longitudinal Stability Derivatives X u and Z uThe axial and normal force of aerodynamic and propulsive

origin may be written for level flight at small α in body axissystem as:

X = – D + TZ = - L

Taking the derivative of the above wrt u we get∂X/∂ u = – ∂D/∂u + ∂T/ ∂uand ∂Z/ ∂ u = – ∂L/ ∂u

Using D = ½ ρu²S C D, L = ½ ρu²S C L

∂D/∂u = ρu 0S (C D)0 + ½ ρu 0²S ( ∂C D/∂u) 0

= (ρu 0S/2) [2 (C D)0 + (C Du )0 ]where (C D)0 = C D and (C Du )0 = ∂C D/∂(u/u 0) at u = u 0

We denote ∂T/ ∂u at u= u 0 as ( ∂T/ ∂u) 0



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Longitudinal Stability Derivatives X u and Z u(contd)Starting from D = ½ ρu²S C D, we obtained

∂D/∂u = ( ρu 0S/2) [2 (C D)0 + (C Du )0 ]

Similarly differentiating L = ½ ρu²S C L

∂L/ ∂u = ( ρu 0S/2) [2C L0 + (C Lu )0 ]where

C L0 = C L and (C Lu ) 0 = ∂C L /∂(u/u 0) at u = u 0

Substituting these values we get∂X/∂ u = – ∂D/∂u + ∂T/ ∂u

= (ρu 0S/2) [2 (C D)0 + (C Du )0 ] + ( ∂T/ ∂u) 0

∂Z/ ∂ u = – ∂L/ ∂u = -- ( ρu 0S/2) [2C L0 + (C Lu )0 ]With no compressibility effects the quantities C Du , C Lu ,∂T/ ∂u will be zero and we have

∂X/∂u = -- ρu 0S (C D)0

∂Z/ ∂u = -- ρu 0SC L0



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Frequency and Damping for Phugoid

The characteristics equation for Phugoid motion:λ ² - X uλ -- (gZ u/u 0 ) = 0The roots of this equation are

λ p = [X u ±√ {Xu² + 4 (gZ u/u 0 )}]/2We know for low speed regime (no compressibility effects)

∂X/∂u = -- ρu 0S (C D)0 and ∂Z/ ∂u = -- ρu 0SC L0

Following subscript notation we haveXu = (1/m) ∂X/∂u = -- ρu 0S (C D)0 /m

and Z u = (1/m) ∂Z/ ∂u = -- ρu 0SC L0 /m

Using initial level flight conditionW = mg = ½ ρu 0²S C L0

Xu = --2g/{ u 0(L/D) 0}and Z u = -- 2g/u 0



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Frequency and Damping for Phugoid (contd)Two stability derivatives X u and Z u appearing in Phugoid

equation:Xu = --2g/{ u 0(L/D) 0}Z u = -- 2g/u 0

Phugoid ζp

and ωnp

:ω np = √ - (gZ u/u 0 ) ζp = - X u/ 2 ω np

= √2 (g/u 0) = 1/{ √2 (L/D) 0}

T = √2 πu 0/gWith increase in flight speed - Phugoid period T increases

ζp = 1/{ √2 (L/D) 0}Higher the aerodynamic efficiency L/D -the poorer will be

Phugoid damping ζp



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Phogoid Trajectory

Short Period Longitudinal Oscillation(Δαvariation)



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Approximations of Longitudinal Dynamics (Δu,Δα, Δ

Short Period Approximation:Velocity remains constant : Δu = 0(Ai rcraft I ner tia H igh to respond)

Neglect Axial Force EquationSolve Normal Force & Pitching Moment Equations for(Δu, Δα, Δq)

T eqw w u

eqw u w

T ew u

T e

T e

T e


X X g w X u X u

sin)()1(

cos

00

0

Pitching Mom

Axial Force

Normal Force

Δα = Δw/u0; Z w = 0; Z q = 0.





Normal Force Eqn

Δα = (Zα/u 0)Δα + Δq)Pitching Mom Eqn

Δq = (M α + M αZ α/u 0) Δα +(Mα+ M q)Δq

State Space Equation for SPO approx:Δα Z α/u 0 1x = A =

Δq (M α + M αZ α/u 0) (M α + M q )

λI -A = 0

Characteristic Equation for SPO approximation:λ ² - (M α + M q + Z α/u 0 )λ + (M q Z α/u 0 - M α) = 0


SPO Approximation to Long. EOM for Level Flight

•

•

•

•

•

•

•



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Longitudinal Modes of Aircraft

PHUGOID or Long Period

Short Period Oscillation (SPO)



22


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Phogoid Trajectory

Short Period Longitudinal Oscillation(Δαvariation)



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Summary of Longitudinal Approximations

Phugoid Short Period

Frequency

Damping



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One Degree of Freedom Pitching MotionAircraft with one degree of freedom executes pitch oscillations

about its CG and y axis. For this we haveΔθ = Δα and Δα= Δθ = Δq

External Pitching Moment ΔM = I yy Δθ

External pitching moment ΔM of aerodynamic origin on aircraftwe know depends on ( Δα, Δ α, Δq, Δδe). For 1- DOF pitchingmotion or pure pitch oscillations, we retain this dependency on(Δα, Δ α ,Δq, Δδe) and write down

ΔM = ( ∂M/ ∂α) Δα+ (∂M/ ∂α) Δα+ (∂M/ ∂q ) Δq + ( ∂M/ ∂δe)Δδe

Now with Δq= Δαwe can write down the equation for 1 DOFpitching asI yy Δα= (∂M/ ∂α) Δα+ (∂M/ ∂α) Δα+(∂M/ ∂q) Δα+ (∂M/ ∂δe)Δδe

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●

●

●

●●

●

●

●

●

●

● ●



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One Degree of Freedom Pitching Motion (contd)Thus we have the equation for 1 DOF pitching motion as

I yy Δα= (∂M/ ∂α) Δα+[( ∂M/ ∂α) + ( ∂M/ ∂q)] Δα+ (∂M/ ∂δe)Δδe

Using subscript notation we getΔα= M α Δα+ {M α + M q}Δα+ M δe Δδe

For Free Response of aircraft in PURE pitching motion, thesecond order system is

Δα - {M α + M q}Δα - M α Δα = 0

From this we have2 ζωn = - {M α + M q} and ωn² = - M α

For statically stable aircraft M α < 0 and we have frequency anddamping as ω

n= √(- M

α) and ζ = - {M

α+ M

q} /2√(- M

α)

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Pitch Response for Step ElevatorWe had written earlier one DOF approximation for pitching

motion with elevator input asI yy Δα= (∂M/ ∂α) Δα+[( ∂M/ ∂α) + ( ∂M/ ∂q)] Δα+ (∂M/ ∂δe)Δδe

Using subscript notation and rearranging terms we haveΔα - {M

α+ M

q}Δα - M

αΔα = M

δeΔδ

e

Solution of above equation is given byΔα= ΔαTrim {1+[e(- ζωnt)/ √(1-ζ2)]sin ( √(1-ζ2) ωnt + φ)}

where ΔαTrim = - (M δeΔδe)/M α

φ = tan -1 [-√(1-ζ2)/ ζ2]ωn = √(- M α) andζ = - {M α + M q} /2√(- M α)

Above response is shown plotted for a range of damping

parameter : ζ < 1, ζ = 1 and ζ >1

●● ● ●

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●



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ωn t

Pitch Response for Step Elevator*

Δα/ΔαTrim

ζ< 1: Subcritical Dampingα overshoots a few timesbefore attaining steady statevalue as seen in oscillatory

time history

ζ = 1 : Critically DampedAperiodic response

For ζ > 1:OverdampedAperiodic response* F igur e f rom Nelson p141 F ig 4.6.

Eqn 4.45 gives solution for step elevatorinput

ζ=0.1



AE 3002 F li ght M echanics

End ofChapter 9

Longitudinal Dynamics

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Ch 9 Longitudinal Dynamics