Download - 441 Lecture 11
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Spacecraft and Aircraft Dynamics
Matthew M. PeetIllinois Institute of Technology
Lecture 11: Longitudinal Dynamics
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Aircraft DynamicsLecture 11
In this Lecture we will cover:
Longitudinal Dynamics:• Finding dimensional coefficients from non-dimensional coefficients• Eigenvalue Analysis• Approximate modal behavior
short period modephugoid mode
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Review: Longitudinal Dynamics
Combined Terms
∆ u̇ + ∆ θg cos θ0 = X u ∆ u + X w ∆ w + X δ e δ e + X δ T δ T
∆ ẇ + ∆ θg sin θ0 − u 0 ∆ θ̇ = Z u ∆ u + Z w ∆ w + Z ẇ ∆ ẇ + Z q ∆ θ̇ + Z δ e δ e + Z δ T δ T ∆ θ̈ = M u ∆ u + M w ∆ w + M ẇ ∆ ẇ + M q ∆ θ̇ + M δ e δ e + M δ T δ T
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Force CoefficientsForce/Moment Coefficients can be found in Table 3.5 of Nelson
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Nondimensional Force CoefficientsNondimensional Force/Moment Coefficients can be found in Table 3.3 of Nelson
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State-Space
From the homework, we have a state-space representation of form
∆ u̇∆ ẇ∆ θ̇
∆ q̇
= A
uwθ
q
+ B δ eδ T
Where we get A and B from X u , Z u , etc.
Recall:• Eigenvalues of A dene stability of ẋ = Ax .• A is 4 × 4, so A has 4 eigenvalues.• Stable if eigenvalues all have negative real part.
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Natural Motion
We mentioned that A is• Stable if eigenvalues all have negative real part.
Now we say more: Eigenvalues have the form
λ = λR ± λ I ı
If we have a pair of complex eigenvalues, then we have two more concepts:1. Natural Frequency:
ωn = λ 2R + λ2I 2. Damping Ratio:
d = −λRωn
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Natural Frequency
Natural frequency is how fast the the motion oscillates.
Closely related is theDenition 1.The Period is the time take tocomplete one oscillation
τ = 2πωn
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Damping Ratio
Damping ratio is how much amplitude decays per oscillation.• Even if d is large, may decay slowly is ωn is small
Closely related is
Denition 2.The Half-Life is the time taken for theamplitude to decay by half.
γ = .693|λR |
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State-SpaceExample: Uncontrolled Motion
C172: V 0 = 132 kt , 5000f t .
∆ u̇∆ ẇ∆ q̇ ∆ θ̇
=− .0442 18.7 0 − 32.2− .0013 − 2.18 .97 0.0024 − 23.8 − 6.08 0
0 0 1 0
uwq θ
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State-SpaceExample: Uncontrolled Motion
Using the Matlab command [u,V] = eigs(A) , we nd the eigenvalues as
Phugoid (Long-Period) Modeλ 1 , 2 = − .0209 ± .18ı
and Eigenvectors
ν 1 ,2 =
− .1717
− .0748.9131− .1038
±
.2826
.16850
.1103ı
Short-Period Modeλ 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
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State-SpaceExample: Uncontrolled Motion
After scaling the state by the equilibrium values, we nd the eigenvaluesunchanged (Why?) asPhugoid (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: ωn = .181rad/sDamping Ratio: d = .115Period: τ = 34 .7sHalf-Life: γ = 33 .16sMotion dominated by variables u and θ.
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Modal Illustration
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State-SpaceExample: Long Period Mode
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State-SpaceExample: Uncontrolled Motion
Using the Matlab command [u,V] = eigs(A) , we nd the eigenvalues asShort-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: ωn = 6 .03rad/s
Damping Ratio: d = .685Period: τ = 1 .04sHalf-Life: γ = .167sMotion dominated by variables w and q .
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Modal Illustration
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State-SpaceExample: Short Period Mode
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State-SpaceModal Approximations
Now that we know that longitudinal dynamics have two modes:• Short Period Mode• Phugoid Mode (Long-Period Mode)
Short Period Mode:• x u = 0 and w = 0 .• study variation in θ and q .• Similar to Static Longitudinal Stability
Long Period Mode:• x q = 0 and w = 0 .• study variation in θ and u.
Now we develop some simplied expressions to study these modes.
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h d
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Short Period Approximation
For the short period mode, we have the following dynamics:
ẇ
u 0q̇ =
Z α
u 0 1M α + M α̇ Z αu 0 M q + M ̇α
w
u 0q +
Z δ e
u 0M δ e +
M α̇ + Z δ eu 0
δ e
= A spθq + Bsp δ e
• To understand stability, we need the eigenvalues of Asp .• Eigenvalues are solutions of det( λI − Asp ) = 0 .
Thus we want to solve
det( λI − Asp ) = λ2 − (M q + Z α
u0+ M ̇α )λ + (
Z α M q
u0− M α ) = 0
We use the quadratic formula (Lecture 1):
λ 3 , 4 = 12
(M q + M ̇α + Z αu0
) ± 12 (M q + M α̇ + Z αu0 )2 − 4(m q Z αu0 − M α )
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Sh P i d A i i
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Short Period ApproximationFrequency and Damping Ratio
λ 3 , 4 = 12
(M q + M ̇α + Z αu0
) ± 12 (M q + M α̇ + Z αu0 )2 − 4(m q Z αu0 − M α )
This leads to the Approximation Equations :• Natural Frequency:
ωsp = M qZ αu0
• Damping Ratio:
dsp = − 12M q + M α̇ + Z αu 0
ωsp
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Sh P i d M d A i i
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Short Period Mode ApproximationsExample: C127
Approximate Natural Frequency:
ωsp = − 481∗ −431219 + 27 .7 = 6 .10rad/sTrue Natural Frequency:
ωsp = 6 .03rad/s
Approximate Damping Ratio:
dsp = −4.32 − 2.20 − 1.81
2∗6.10 = .683rad/s
True Damping Ratio:dsp = .685rad/s
So, generally good agreement.
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L P i d A i ti
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Long Period Approximation
Long period motion considers only motion in u and θ.u̇θ̇ =
X u − g− Z uu 0 0
uθ
This time, we must solve the simple expression
det( λI − A) = λ 2 − X u λ − Z uu0
g = 0
Using the quadratic formula, we get
λ 1 , 2 = X u ± X
2u + 4
Z uu 0 g
2
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L g P i d A i ti
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Long Period Approximation
λ 1 , 2 =X u ± X 2u + 4 Z uu 0 g2
This leads to the Approximation Equations :• Natural Frequency:
ωlp = − Z u gu0• Damping Ratio:
dlp = − X u2ωlp
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Long Period Mode Approximations
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Long Period Mode ApproximationsExample: C127
Approximate Natural Frequency:ωlp = .181rad/s
True Natural Frequency:ωlp = .208rad/s
Approximate Damping Ratio:
dlp = .115rad/s
True Damping Ratio:dlp = .106rad/s
So, generally good, but not as good. Why?
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Conclusion
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Conclusion
In this lecture, we covered:• How to nd and interpret the eigenvalues and eigenvectors of a state-space
matrixNatural FrequencyDamping Ratio
• How to identifyLong Period Eigenvlaues/MotionShort Period Eigenvalues/Motion
• Modal ApproximationsPhugoid and Short-Period Modes
Formulas for natural frequencyFormulae for damping ratio
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