Spacecraft and Aircraft Dynamics
Matthew M.Peet
Illinois Institute of Technology
Lecture 11:Longitudinal Dynamics
Aircraft Dynamics
Lecture 11
In this Lecture we will cover:
Longitudinal Dynamics:
•
Finding dimensional coeﬃcients from nondimensional coeﬃcients
•
Eigenvalue Analysis
•
Approximate modal behavior
short period mode
phugoid mode
M.Peet
Lecture 11:
2/25
Review:Longitudinal Dynamics
Combined Terms
Δ˙u +Δθg cos θ
0
= X
u
Δu +X
w
Δw +X
δ
e
δ
e
+X
δ
T
δ
T
Δ ˙w +Δθg sin θ
0
−u
0
Δ
˙
θ = Z
u
Δu +Z
w
Δw +Z
˙w
Δ ˙w +Z
q
Δ
˙
θ +Z
δ
e
δ
e
+Z
δ
T
δ
T
Δ
¨
θ = M
u
Δu +M
w
Δw +M
˙w
Δ ˙w +M
q
Δ
˙
θ +M
δ
e
δ
e
+M
δ
T
δ
T
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Lecture 11:
3/25
Force Coeﬃcients
Force/Moment Coeﬃcients can be found in Table 3.5 of Nelson
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Lecture 11:
4/25
Nondimensional Force Coeﬃcients
Nondimensional Force/Moment Coeﬃcients can be found in Table 3.3 of Nelson
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Lecture 11:
5/25
StateSpace
From the homework,we have a statespace representation of form
Δ˙u
Δ ˙w
Δ
˙
θ
Δ˙q
=
A
u
w
θ
q
+
B
δ
e
δ
T
Where we get A and B from X
u
,Z
u
,etc.
Recall:
•
Eigenvalues of A deﬁne stability of ˙x = Ax.
•
A is 4 ×4,so A has 4 eigenvalues.
•
Stable if eigenvalues all have negative real part.
M.Peet
Lecture 11:
6/25
Natural Motion
We mentioned that A is
•
Stable if eigenvalues all have negative real part.
Now we say more:Eigenvalues have the form
λ = λ
R
±λ
I
ı
If we have a pair of complex eigenvalues,then we have two more concepts:
1.
Natural Frequency:
ω
n
=
λ
2
R
+λ
2
I
2.
Damping Ratio:
d = −
λ
R
ω
n
M.Peet
Lecture 11:
7/25
Natural Frequency
Natural frequency is how fast the the motion oscillates.
Closely related is the
Deﬁnition 1.
The Period is the time take to
complete one oscillation
τ =
2π
ω
n
M.Peet
Lecture 11:
8/25
Damping Ratio
Damping ratio is how much amplitude decays per oscillation.
•
Even if d is large,may decay slowly is ω
n
is small
Closely related is
Deﬁnition 2.
The HalfLife is the time taken for the
amplitude to decay by half.
γ =
.693
λ
R

M.Peet
Lecture 11:
9/25
StateSpace
Example:Uncontrolled Motion
C172:V
0
= 132kt,5000ft.
Δ˙u
Δ ˙w
Δ˙q
Δ
˙
θ
=
−.0442 18.7 0 −32.2
−.0013 −2.18.97 0
.0024 −23.8 −6.08 0
0 0 1 0
u
w
q
θ
M.Peet
Lecture 11:
10/25
StateSpace
Example:Uncontrolled Motion
Using the Matlab command [u,V] = eigs(A),we ﬁnd the eigenvalues as
Phugoid (LongPeriod) Mode
λ
1,2
= −.0209 ±.18ı
and Eigenvectors
ν
1,2
=
−.1717
−.0748
.9131
−.1038
±
.2826
.1685
0
.1103
ı
ShortPeriod Mode
λ
3,4
= −4.13 ±4.39ı
and Eigenvectors
ν
3,4
=
1
−.0002
.001
−.0008
±
0
.0000001
.0000011
.0055
ı
Notice that this is hard to interpret.Lets scale u and q by equilibrium values.
M.Peet
Lecture 11:
11/25
StateSpace
Example:Uncontrolled Motion
After scaling the state by the equilibrium values,we ﬁnd the eigenvalues
unchanged (Why?) as
Phugoid (LongPeriod) Mode
λ
1,2
= −.0209 ±.18ı
but clearer Eigenvectors
ν
1,2
=
−.629
.0218
−.0016
.138
±
.0213
.0007
.0001
.765
ı
Natural Frequency:ω
n
=.181rad/s
Damping Ratio:d =.115
Period:τ = 34.7s
HalfLife:γ = 33.16s
Motion dominated by variables u and θ.
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Lecture 11:
12/25
Modal Illustration
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Lecture 11:
13/25
StateSpace
Example:Long Period Mode
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Lecture 11:
14/25
StateSpace
Example:Uncontrolled Motion
Using the Matlab command [u,V] = eigs(A),we ﬁnd the eigenvalues as
ShortPeriod Mode
λ
3,4
= −4.13 ±4.39ı
and Eigenvectors
ν
3,4
=
−.0049
−.655
−.396
−.006
±
.004
.409
.495
.0423
ı
Natural Frequency:ω
n
= 6.03rad/s
Damping Ratio:d =.685
Period:τ = 1.04s
HalfLife:γ =.167s
Motion dominated by variables w and q.
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Lecture 11:
15/25
Modal Illustration
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Lecture 11:
16/25
StateSpace
Example:Short Period Mode
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Lecture 11:
17/25
StateSpace
Modal Approximations
Now that we know that longitudinal dynamics have two modes:
•
Short Period Mode
•
Phugoid Mode (LongPeriod Mode)
Short Period Mode:
•
ﬁx u = 0 and w = 0.
•
study variation in θ and q.
•
Similar to Static Longitudinal Stability
Long Period Mode:
•
ﬁx q = 0 and w = 0.
•
study variation in θ and u.
Now we develop some simpliﬁed expressions to study these modes.
M.Peet
Lecture 11:
18/25
Short Period Approximation
For the short period mode,we have the following dynamics:
˙w
u
0
˙q
=
Z
α
u
0
1
M
α
+
M
˙α
Z
α
u
0
M
q
+M
˙α
w
u
0
q
+
Z
δ
e
u
0
M
δ
e
+
M
˙α
+Z
δ
e
u
0
δ
e
= A
sp
θ
q
+B
sp
δ
e
•
To understand stability,we need the eigenvalues of A
sp
.
•
Eigenvalues are solutions of det(λI −A
sp
) = 0.
Thus we want to solve
det(λI −A
sp
) = λ
2
−(M
q
+
Z
α
u
0
+M
˙α
)λ +(
Z
α
M
q
u
0
−M
α
) = 0
We use the quadratic formula (Lecture 1):
λ
3,4
=
1
2
(M
q
+M
˙α
+
Z
α
u
0
) ±
1
2
(M
q
+M
˙α
+
Z
α
u
0
)
2
−4(m
q
Z
α
u
0
−M
α
)
M.Peet
Lecture 11:
19/25
Short Period Approximation
Frequency and Damping Ratio
λ
3,4
=
1
2
(M
q
+M
˙α
+
Z
α
u
0
) ±
1
2
(M
q
+M
˙α
+
Z
α
u
0
)
2
−4(m
q
Z
α
u
0
−M
α
)
This leads to the Approximation Equations:
•
Natural Frequency:
ω
sp
= M
q
Z
α
u
0
•
Damping Ratio:
d
sp
= −
1
2
M
q
+M
˙α
+
Z
α
u0
ω
sp
M.Peet
Lecture 11:
20/25
Short Period Mode Approximations
Example:C127
Approximate Natural Frequency:
ω
sp
=
−481 ∗ −431
219
+27.7 = 6.10rad/s
True Natural Frequency:
ω
sp
= 6.03rad/s
Approximate Damping Ratio:
d
sp
= −
4.32 −2.20 −1.81
2 ∗ 6.10
=.683rad/s
True Damping Ratio:
d
sp
=.685rad/s
So,generally good agreement.
M.Peet
Lecture 11:
21/25
Long Period Approximation
Long period motion considers only motion in u and θ.
˙u
˙
θ
=
X
u
−g
−
Z
u
u
0
0
u
θ
This time,we must solve the simple expression
det(λI −A) = λ
2
−X
u
λ −
Z
u
u
0
g = 0
Using the quadratic formula,we get
λ
1,2
=
X
u
±
X
2
u
+4
Z
u
u
0
g
2
M.Peet
Lecture 11:
22/25
Long Period Approximation
λ
1,2
=
X
u
±
X
2
u
+4
Z
u
u
0
g
2
This leads to the Approximation Equations:
•
Natural Frequency:
ω
lp
=
−
Z
u
g
u
0
•
Damping Ratio:
d
lp
= −
X
u
2ω
lp
M.Peet
Lecture 11:
23/25
Long Period Mode Approximations
Example:C127
Approximate Natural Frequency:
ω
lp
=.181rad/s
True Natural Frequency:
ω
lp
=.208rad/s
Approximate Damping Ratio:
d
lp
=.115rad/s
True Damping Ratio:
d
lp
=.106rad/s
So,generally good,but not as good.Why?
M.Peet
Lecture 11:
24/25
Conclusion
In this lecture,we covered:
•
How to ﬁnd and interpret the eigenvalues and eigenvectors of a statespace
matrix
Natural Frequency
Damping Ratio
•
How to identify
Long Period Eigenvlaues/Motion
Short Period Eigenvalues/Motion
•
Modal Approximations
Phugoid and ShortPeriod Modes
Formulas for natural frequency
Formulae for damping ratio
M.Peet
Lecture 11:
25/25
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