Spacecraft and Aircraft Dynamics

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Nov 16, 2013 (3 years and 4 months ago)

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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 non-dimensional 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
M.Peet
Lecture 11:
3/25
Force Coefficients
Force/Moment Coefficients can be found in Table 3.5 of Nelson
M.Peet
Lecture 11:
4/25
Nondimensional Force Coefficients
Nondimensional Force/Moment Coefficients can be found in Table 3.3 of Nelson
M.Peet
Lecture 11:
5/25
State-Space
From the homework,we have a state-space 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

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 =

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 Half-Life is the time taken for the
amplitude to decay by half.
γ =
.693

R
|
M.Peet
Lecture 11:
9/25
State-Space
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
State-Space
Example:Uncontrolled Motion
Using the Matlab command [u,V] = eigs(A),we find the eigenvalues as
Phugoid (Long-Period) Mode
λ
1,2
= −.0209 ±.18ı
and Eigenvectors
ν
1,2
=




−.1717
−.0748
.9131
−.1038




±




.2826
.1685
0
.1103




ı
Short-Period 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
State-Space
Example:Uncontrolled Motion
After scaling the state by the equilibrium values,we find the eigenvalues
unchanged (Why?) as
Phugoid (Long-Period) 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
Half-Life:γ = 33.16s
Motion dominated by variables u and θ.
M.Peet
Lecture 11:
12/25
Modal Illustration
M.Peet
Lecture 11:
13/25
State-Space
Example:Long Period Mode
M.Peet
Lecture 11:
14/25
State-Space
Example:Uncontrolled Motion
Using the Matlab command [u,V] = eigs(A),we find the eigenvalues as
Short-Period 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
Half-Life:γ =.167s
Motion dominated by variables w and q.
M.Peet
Lecture 11:
15/25
Modal Illustration
M.Peet
Lecture 11:
16/25
State-Space
Example:Short Period Mode
M.Peet
Lecture 11:
17/25
State-Space
Modal Approximations
Now that we know that longitudinal dynamics have two modes:

Short Period Mode

Phugoid Mode (Long-Period 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

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 state-space
matrix
￿
Natural Frequency
￿
Damping Ratio

How to identify
￿
Long Period Eigenvlaues/Motion
￿
Short Period Eigenvalues/Motion

Modal Approximations
￿
Phugoid and Short-Period Modes
￿
Formulas for natural frequency
￿
Formulae for damping ratio
M.Peet
Lecture 11:
25/25