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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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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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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
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State-Space Longitudinal EOM
T
e
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
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00
0
0
00
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●
X A X
B u
X = A X + B u
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Longitudinal Stability DerivativesM A 3703 Fl ight Dynamics
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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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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).
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Longitudinal Modes of Aircraft
PHUGOID orLong Period
Short Period Oscillation(SPO)
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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
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
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
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●
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
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
Pitching Mom
Axial Force
Normal Force
Δα = Δw/u0; Z w = 0; Z q = 0.
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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
M A 3703 F li ght Dynamics
SPO Approximation to Long. EOM for Level Flight
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Longitudinal Modes of Aircraft
PHUGOID or Long Period
Short Period Oscillation (SPO)
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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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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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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
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AE 3002 F li ght M echanics
End ofChapter 9
Longitudinal Dynamics
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