16.333: Lecture # 7
Approximate Longitudinal Dynamics Models
• A couple more stability derivatives
• Given mode shapes found identify simpler models that capture the main re­
sponses
Fall 2004
16.333 6–1
More Stability Derivatives
• Recall from 6–2 that the derivative stability derivative terms Zẇ and
Mẇ ended up on the LHS as modifications to the normal mass and
inertia terms
– These are the apparent mass effects – some of the surrounding
displaced air is “entrained” and moves with the aircraft
– Acceleration derivatives quantify this effect
– Significant for blimps, less so for aircraft.
• Main effect: rate of change of the normal velocity ẇ causes a transient
in the downwash � from the wing that creates a change in the angle
of attack of the tail some time later – Downwash Lag effect
• If aircraft flying at U0, will take approximately Δt = lt/U0 to reach
the tail.
– Instantaneous downwash at the tail �(t) is due to the wing α at
time t − Δt.
∂�
�(t) =
α(t − Δt)
∂α
– Taylor series expansion
α(t − Δt) ≈ α(t) − α̇Δt
– Note that Δ�(t) = −Δαt. Change in the tail AOA can be com­
puted as
Δ�(t) = −
d�
d� lt
α̇Δt = − α̇ = −Δαt
dα
dα U0
Fall 2004
16.333 6–2
• For the tail, we have that the lift increment due to the change in
downwash is
d� lt
ΔCLt = CLαt Δαt = CLαt α̇
dα U0
The change in lift force is then
1
ΔLt = ρ(U02)tStΔCLt
2
• In terms of the Z­force coefficient
St
d� lt
ΔLt
St
ΔCZ = −1 2 = −η ΔCLt = −η CLαt α̇
S
S
dα U0
2 ρU0 S
¯
• We use c/(2U
0 ) to nondimensionalize time, so the appropriate stabil­
ity coefficient form is (note use Cz to be general, but we are looking
at ΔCz from before):
�
�
�
�
∂CZ
2U0 ∂CZ
CZα̇ =
=
˙ c/2U0) 0
∂ (α¯
c̄
∂α̇ 0
d�
2U0 St lt
CLαt
= −η
c̄ S U0
dα
d�
= −2ηVH CLαt
dα
• The pitching moment due to the lift increment is
ΔMcg = −ltΔLt
1
2
2 ρ(U0 )t St ΔCLt
→ΔCMcg = −lt
1
2
2 ρU0 Sc̄
= −ηVH ΔCLt = −ηVH CLαt α̇
d� lt
dα U0
Fall 2004
16.333 6–3
• So that
�
CMα̇ =
∂CM
˙ c/2U0)
∂ (α¯
�
�
∂CM
∂α̇ 0
d� lt 2U0
= −ηVH CLαt
dα U0 c̄
d� lt
= −2ηVH CLαt
dα c̄
lt
≡ CZα̇
c̄
=
0
2U0
c̄
�
• Similarly, pitching motion of the aircraft changes the AOA of the tail.
Nose pitch up at rate q, increases apparent downwards velocity of tail
by qlt, changing the AOA by
Δαt =
qlt
U0
which changes the lift at the tail (and the moment about the cg).
• Following same analysis as above: Lift increment
ΔLt = CLαt
qlt 1
ρ(U02)tSt
U0 2
ΔLt
qlt
St
=
−η
C
Lαt
2 )S
S
U0
ρ(U
0
2
ΔCZ = − 1
�
CZq ≡
∂CZ
∂(qc̄/2U0)
�
0
2U0
=
c̄
�
∂CZ
∂q
• Can also show that
CMq = CZq
�
lt
c̄
2U0 lt St
CL
c̄ U0 S αt
= −2ηVH CLαt
= −η
0
Fall 2004
16.333 6–4
Approximate Aircraft Dynamic Models
• It is often good to develop simpler models of the full set of aircraft
dynamics.
– Provides insights on the role of the aerodynamic parameters on
the frequency and damping of the two modes.
– Useful for the control design work as well
• Basic approach is to recognize that the modes have very separate sets
of states that participate in the response.
– Short Period – primarily θ and w in the same phase.
The u and q response is very small.
– Phugoid – primarily θ and u, and θ lags by about 90◦.
The w and q response is very small.
• Full equations from before:
⎡ ⎤ ⎡ X
X
⎣
u̇
ẇ
q̇
θ̇
m
Zu
m−Zw˙
[Mu +Zu Γ]
Iyy
m
Zw
m−Zẇ
[Mw +Zw Γ]
Iyy
0
−g cos Θ0
Zq +mU0
m−Zẇ
[Mq +(Zq +mU0 )Γ]
Iyy
−mg sin Θ0
m−Zw
˙
Θ0 Γ
− mg sin
Iyy
0
0
1
0
u
⎦= ⎣
w
⎤�
⎦
u
w
q
θ
�
�
+
ΔX c
ΔZ c
ΔM c
0
�
Fall 2004
16.333 6–5
• For the Short Period approximation,
1. Since u ≈ 0 in this mode, then u̇ ≈ 0 and can eliminate the
X­force equation.
⎤
⎡
Zq +mU0
−mg sin Θ0
⎡ ⎤ ⎡
⎡
⎤
⎤
Zw
c
ẇ
w
ΔZ
m−Zẇ
m−Zẇ
m−Zw˙
⎥
⎢
⎣
q̇ ⎦ =
⎢ [Mw +Zw Γ] [Mq +(Zq +mU0)Γ] − mg sin Θ0Γ ⎥ ⎣
q ⎦+
⎣
ΔM c ⎦
⎦
⎣
Iyy
Iyy
Iyy
θ
0
θ̇
1
0
0
2. Typically find that Zẇ � m and Zq � mU0. Check for 747:
–Zẇ = 1909 � m = 2.8866 × 105
–Zq = 4.5 × 105 � mU0 = 6.8 × 107
Γ=
Mẇ
Mẇ
⇒Γ≈
m
m − Zẇ
⎡
Zw
U0
m
�
�
�
�
ẇ
Mẇ
Mẇ
⎢ Mw +Z
Mq +(mU0 ) m
w m
⎣
q̇
⎦=
⎢
⎣
Iyy
Iyy
⎡
⎤
θ̇
0
1
−g sin Θ0
Θ0 Mẇ
− mg Isin
m
yy
0
⎤
⎤ ⎡
⎤
c
w
ΔZ
⎥
⎥ ⎣
q ⎦+
⎣
ΔM c ⎦
⎦
θ
0
⎡
3. Set Θ0 = 0 and remove θ from the model (it can be derived from
q)
• With these approximations, the longitudinal dynamics reduce to
ẋsp = Aspxsp + Bspδe
where δe is the elevator input, and
� �
�
�
Zw /m
U0
w
xsp =
, Asp =
−1
−1
(Mw + Mẇ Zw /m) Iyy
(Mq + Mẇ U0)
Iyy
q
�
�
Zδe /m
Bsp =
−1
Iyy
(Mδe + Mẇ Zδe /m)
Fall 2004
16.333 6–6
2
• Characteristic equation for this system: s2 + 2ζspωsps + ωsp
= 0,
where the full approximation gives:
�
�
Zw Mq Mẇ
2ζspωsp = −
+
+
U0
m
Iyy
Iyy
Z w M q U0 M w
2
ωsp
=
−
Iyy
mIyy
• Given approximate magnitude of the derivatives for a typical aircraft,
can develop a coarse approximate:
�
⎫
Mq
Mq
ζsp ≈ −
2
U0M−1w Iyy
2ζspωsp ≈ −
Iyy ⎬
→
�
U0 Mw ⎭
2
ωsp ≈ − Iyy
ωsp ≈ −UI0yyMw
• Numerical values for 747
Frequency Damping
rad/sec
Full model
0.962
0.387
0.963
0.385
Full Approximate
Coarse Approximate
0.906
0.187
Both approximations give the frequency well, but full approximation
gives a much better damping estimate
• Approximations showed that short period mode frequency is deter­
mined by Mw – measure of the aerodynamic stiffness in pitch.
– Sign of Mw negative if cg sufficient far forward – changes sign
(mode goes unstable) when cg at the stick fixed neutral point.
Follows from discussion of CMα (see 2–11)
Fall 2004
16.333 6–7
• For the Phugoid approximation, start again with:
⎡
⎤
⎡ Xu
Xw
0
−g cos Θ0
m
m
u̇
Zq +mU0
−mg sin Θ0
Zu
Zw
⎢ ⎥ ⎢
⎢
ẇ
m−Z
m−Z
m−Z
m−Zẇ
⎢ ⎥ ⎢
ẇ
⎢ ⎥ = ⎢ [Mu+ZẇuΓ] [Mw +Zẇw Γ] [Mq +(Zq +mU
0 )Γ]
Θ0 Γ
⎣ q̇
⎦
⎣
− mg sin
Iyy
Iyy
Iyy
Iyy
θ̇
0
0
1
0
⎤⎡
⎤ ⎡
c
⎤
⎥ ⎢ u ⎥ ⎢ ΔXc
⎥
⎥ ⎢ w
⎥ ⎢ ΔZ ⎥
⎥ ⎢ ⎥+
⎢
⎥ ⎣
q
⎦
⎣
ΔM c
⎥
⎦
⎦
θ
0
1. Changes to w and q are very small compared to u, so we can
– Set ẇ ≈ 0 and q̇ ≈ 0
– Set Θ0 = 0
⎤
⎡ ⎤ ⎡
⎡
⎤
⎡ Xu
⎤
Xw
0
−g
c
m
m
u̇
Zq +mU0
⎢
⎥ ⎢ u ⎥ ⎢ ΔXc
⎥
Z
Z
w
u
⎢ ⎥ ⎢
0
⎥ ⎢ w
⎥ ⎢ ΔZ ⎥
m−Zẇ
m−Zẇ
⎢ 0
⎥ ⎢ m−Zẇ
⎥ ⎢ ⎥+
⎢
⎢ ⎥ = ⎢ [Mu+ZuΓ] [Mw +Zw Γ] [Mq +(Zq +mU0)Γ]
⎥ ⎣
q
⎦
⎣
ΔM c
⎥
⎣
0
⎦
⎣
⎦
0
⎦
Iyy
Iyy
Iyy
θ
0
θ̇
0
0
1
0
2. Use what is left of the Z­equation to show that with these ap­
proximations (elevator inputs)
⎡
⎤
⎡
⎤
⎡
⎤
Z
Zq +mU0
Zw
δ
Z
e
u
�
�
m−Zẇ
m−Zẇ
m−Z
ẇ
⎢
⎥ w
⎢ m−Zẇ ⎥
⎣
⎦
=−
u−
⎣
⎣
⎦
⎦
δe
[M
+Z
Γ]
q
M
+(Z
+mU
)Γ
M
+Z
Γ
u
u
[
]
[
]
q
q
0
[Mw +Zw Γ]
δe
δe
Iyy
Iyy
Iyy
Iyy
3. Use (Zẇ � m so Γ ≈ Mm
ẇ ) and (Zq � mU0) so that:
�
� � �
Z
mU
w
0
w
�
�
Mw + Zw Mmẇ [Mq + U0Mẇ ]
q
�
�
�
�
Zδe
Zu
� u− �
�
δ
= − �
e
Mẇ
Mẇ
Mδe + Zδe m
M u + Zu m
Fall 2004
16.333 6–8
4. Solve to show that
⎡
⎤
⎡
mU0Mδe − Zδe Mq
mU0Mu − ZuMq
�
�
⎢ Zw Mq − mU0Mw
⎥
⎢ Zw Mq − mU0Mw
w
⎢
⎥
=
⎣
u +
⎢
⎣
Zδe Mw − Zw Mδe
⎦
ZuMw − Zw Mu
q
Zw Mq − mU0Mw
Zw Mq − mU0Mw
⎤
⎥
⎥ δe
⎦
5. Substitute into the reduced equations to get full approximation:
�
�
⎡
⎤
mU
M
−
Z
M
u
u
q
X
X
0
u
�
�
+ m
w Zw Mq −mU0Mw −g �
�
m
u̇
⎢
⎥ u
= ⎣
⎦
�
�
θ
θ̇
Zu Mw −Zw Mu
0
Zw Mq −mU0 Mw
⎡
�
�
⎤
Xδe
Xw mU0 Mδe −Zδe Mq
+ m Zw Mq −mU0Mw
⎥
⎢ m
+
⎣
⎦
δe
Zδe Mw −Zw Mδe
Zw Mq −mU0 Mw
6. Still a bit complicated. Typically get that
– |MuZw | � |Mw Zu| (1.4:4)
– |Mw U0m| � |Mq Zw | (1:0.13)
– |MuXw /Mw | � Xu small
7. With these approximations, the longitudinal dynamics reduce to
the coarse approximation
ẋph = Aphxph + Bphδe
where δe is the elevator input.
Fall 2004
16.333 6–9
And
⎤
Xu
�
�
⎢ m −g ⎥
u
⎥
xph =
Aph = ⎢
⎣
⎦
θ
−Zu
0
mU0
⎡ �
� �
� ⎤
Xw
Xδe − Mw Mδe
⎥
⎢
⎥
⎢
m
⎥
⎢
Bph =
⎢ �
� �
�⎥
⎥
⎢ −Z + Zw M
⎦
⎣
δe
δ
e
Mw
⎡
mU0
8. Which gives
2ζphωph = −Xu/m
gZu
2
ωph
= −
mU0
Numerical values for 747
Frequency Damping
rad/sec
Full model
0.0673
0.0489
Full Approximate
0.0670
0.0419
Coarse Approximate
0.0611
0.0561
Fall 2004
16.333 6–10
• Further insights: recall that
�
�� �
�
�� �
U0
∂Z
U0
∂L
= −
≡ −(CLu + 2CL0 )
QS
∂u 0
QS
∂u 0
M2
CL − 2CL0 ≈ −2CL0
= −
1 − M2 0
so
� �
�
�
∂Z
ρUoS
2mg
Zu ≡
=
(−2CL0 ) = −
∂u 0
2
U0
• Then
�
ωph
−gZu
=
=
mU0
√ g
= 2
U0
�
mg 2
mU02
which is exactly what Lanchester’s approximation gave Ω ≈
√
2 Ug0
• Note that
�
Xu ≡
∂X
∂u
�
�
=
0
ρUoS
2
�
(−2CD0 ) = −ρUoSCD0
and
2mg = ρUo2SCL0
so
XuU0
Xu
=− √
2mωph
2 2mg
� 2
�
1 ρUo SCD0
= √
SCL0
2 � ρUo2�
1 CD0
= √
2 CL0
ζph = −
so the damping ratio of the approximate phugoid mode is inversely
proportional to the lift to drag ratio.
−2
10
−250
−200
−150
−100
−50
0
50
10
−2
10
0
10
1
2
10
10
3
4
10
Freq (rad/sec)
−1
10
Freq (rad/sec)
−1
10
0
10
0
Transfer function from elevator to flight variables
γ
de
de
|Gude|
arg Gude
|G |
arg Gγ
10
−2
−2
10
−350
−300
−250
−200
−150
−100
−50
0
10
−1
10
0
10
1
10
10
2
10
3
4
10
−1
10
Freq (rad/sec)
−1
10
Freq (rad/sec)
0
10
0
10
Fall 2004
16.333 6–11
Freq Comparison from elevator (Phugoid Model) – B747 at M=0.8. Blue– Full model, Black– Full approximate
model, Magenta– Coarse approximate model
−2
10
0
−2
10
50
100
150
200
250
300
10
−1
0
10
10
1
2
10
Freq (rad/sec)
−1
10
Freq (rad/sec)
−1
10
0
10
0
Transfer function from elevator to flight variables
γ
de
de
|Gα |
de
arg Gude
|G |
arg Gγ
10
−2
−2
10
−350
−300
−250
−200
−150
−100
−50
0
10
−1
10
0
10
1
10
2
10
−1
10
Freq (rad/sec)
−1
10
Freq (rad/sec)
0
10
0
10
Fall 2004
16.333 6–12
Freq Comparison from elevator (Short Period Model) – B747 at M=0.8. Blue– Full model, Magenta– Approx­
imate model
Fall 2004
16.333 6–13
Summary
• Approximate longitudinal models are fairly accurate
• Indicate that the aircraft responses are mainly determined by these
stability derivatives:
Property
Stability derivative
Damping of the short period
Mq
Frequency of the short period
Mw
Damping of the Phugoid
Xu
Frequency of the Phugoid
Zu
Fall 2004
16.333 6–14
• Given a change in α, expect changes in u as well. These will both
impact the lift and drag of the aircraft, requiring that we re­trim
throttle setting to maintain whatever aspects of the flight condition
might have changed (other than the ones we wanted to change). We
have:
�
� �
��
�
ΔL
Lu Lα
u
=
ΔD
Du D α
Δα
But to maintain L = W , want ΔL = 0, so u = − LLα
u Δα
�
�
Lα
Giving ΔD = − Lu Du + Dα Δα
2CL0
CL → Dα = QSCDα
πeAR α
→ Lα = QSCLα
QS
(4 − 16)
Du =
(2CD0 )
U0
QS
Lu =
(2CL0 )
(4 − 17)
U0
�
�
�
�
2CD0
CLα
ΔD = QS −
+ CDα Δα
2CL0 /U0
U0
�
�
2
2CL0
QS
−CD0 +
CLα Δα
=
CL0
πeAR
CDα =
(T0 + ΔT ) − (D0 + ΔD) −ΔD
=
L
+
Δ
L
L0
0
�
�
2CL0 CLα Δα
CD0
−
=
CL0
CL0 πeAR
tan Δγ =
For 747 (Reid 165 and Nelson 416), AR = 7.14, so πeAR ≈ 18,
CL0 = 0.654 CD0 = 0.043, CLα = 5.5, for a Δα = −0.0185rad
(6–7) Δγ = −0.0006rad. This is the opposite sign to the linear
simulation results, but they are both very small numbers.