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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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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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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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.


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 )


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


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.


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


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


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?


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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441 Lecture 11