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16.333 Handout #1 Prof. J. P. How Sept 14, 2004

Due: Sept 28, 2004

16.333 Homework Assignment #1

1. Consider a glider flying in a vertical plane at an angle γ to the horizontal. As we discussed in class, for a glider γ < 0. In a steady glide, the force balance is given by:

D + W sin γ = 0 (1)

L −W cos γ = 0 (2)

The kinematic equations for the system give us that the distance covered along the ground satisfies x = V cos γ and h = V sin γ.

For small γ we showed that the flight velocity that gives the flattest glide is given by

2W K Vfg = 4

ρS CD0

where CD = CD0 + KC2 L.

(a) Show that, for a given height, the distance covered with respect to the ground satisfies

dx 1 = = −E R = EΔh

dh γ ⇒

where E = CL/CD and R is the range (assume a constant angle of attack so that E is constant during the glide). What speed does this suggest we should glide at to maximize the range?

(b) Now consider the sink rate hs = −h. Show that

DV hs ≈ −V γ ≈ =

2W CD 3/2W ρS CL

This suggests that the sink rate is a minimum when CD is a minimum. Show 3/2CL

that this corresponds to a flight speed of

2W K Vms = 4 ≈ 0.76Vfg

ρS 3CD0

(Hint: find the CL that minimizes CD 3/2 ).

CL

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Discussion: The conclusion from this analysis is that to maximize range you fly at the speed that minimizes drag but slightly slower than that to minimize sink rate. Sailplane pilots tend to fly at Vms when in “lift” mode, i.e. find an upward gust and then accelerate to Vf g when the updraft dies and they what to cover the most ground looking for another “lift”.

2. Consider the “weathervane” shown in the figure which allows us to explore some of the basics associated with aircraft motion. The aerodynamic surface (tail) acts like a wing, with a lift force proportional to the angle of attack α. The drag of this wing can safely be ignored.

Figure 1: Test Configuration for Problem #2.

˙(a) When the arm is moving with rate Ψ, find the effective angle of attack for the tail.

(b) Assuming small angles, linearize and find the differential equation that governs the dynamics of this system. Assume that the damping is added at the support post by a torsional spring and is proportional to Ψ. The damping is less than the critical level.

(c) What are the roots of this system when: (i) the weathervane is as shown, and (ii) the vane is turned into the wind so that the tail is at the front?

(d) Qualitatively discuss the response of the weathervane to a 20 degree change in the wind direction (U0).

(e) Develop a simulink simulation of the full nonlinear equations and compare the response of the nonlinear and linear systems.

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3. (Adapted from Etkin and Reid, 2.5) The following data apply to a 1/25th scale wind tunnel model of a transport airplane. The full­scale mass of the aircraft is 22,680kg. Assume that the aerodynamic data can be applied to the full­scale vehicle. For level unaccelerated flight at V=239 knots (123 m/s) of the full­scale aircraft, under the assumption that propulsion effects can be ignored,

(a) Find the limits on the tail angle it and CG position h imposed by the conditions Cm0 and Cmα < 0.

(b) For trimmed flight with δe = 0, plot it vs. h for the aircraft and indicate where this line meets the boundaries in part (a).

Geometric data

• wing area S = 0.139m2 (so need to scale this are up by 252 – lengths scale by 25)

• wing mean aerodynamic chord c = 15.61cm

• lt = 38.84cm

• Tail area St = 0.0342m2

Aerodynamic Data (does not need to scale)

• CLw = 0.077/deg (convert to radians)

• CLt = 0.064/deg

• �0 = 0.72◦

• d�0/dα = 0.30

= −0.018• Cmacw

• h¯ = 0.25 (notes page 2–4) n

• ρ = 1.225kg/m3

4. Stability

(a) Draw a picture similar to those on 2–1 of the notes for a system that is statically stable, but not dynamically stable.

(b) Consider a second order system such as a mass vibrating on a spring/damper.

(i) The condition for static stability is that there be a restoring force when the system is displaced from equilibrium. What condition does that impose on the spring constant k? What does that imply about the roots of the second order system?

(ii) What does the condition of dynamic stability imply about the damper con­stant c? What does that condition imply about the roots of the system?

(iii) Do these conditions support the claim that static stability is a necessary but not sufficient condition for dynamic stability?

5. Download the Aerosim blockset discussed in the first class. Modify navion demo1 so that you can plot the altitude as a function of time.

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16.333 Handout #2 Prof. J. P. How Sept 28, 2004

Due: Oct 12, 2004

16.333 Homework Assignment #2

Please include all code used to solve these problems.

1. Date of Lindbergh’s famous flight? When was the first solo flight across the Pacific? 2. Sketch­plot versus time the Euler angles for the various maneuvers starting at Φ =

Θ = Ψ = 0 at t = 0.: (a) Half an inside loop followed by a snap roll at the top (Immelmann turn). (b) outside loop (c) coordinated turn (constant vehicle speed)

3. Consider a jet transport at 40,000 ft with the data given below: (a) Find the short period and phugoid roots. Be careful with the dimensionalization

of these coefficients (see Reid, page 119, 207). Compare the phugoid frequencies with Lanchester’s approximation.

(b) Plot the phugoid response and the short period time response (i.e., find initial conditions that excite each of these modes separately).

(c) Graph the equivalent response plots in Figs. 6–8, 6–10, 6–12, and 6–13. Are the results similar?

Basic aircraft data Mach M=0.62 Nominal speed U0 = 600 [ft/s] Density ρ = 5.85 × 10−4 [slug/ft3] Tail distance lt/c = 2.89 Wing area S=2400 [ft2] Wing chord c=20.2 [ft] Mass m=5800 [slug] Moment of inertia IY Y = 2.62 × 106 [slug ft2] Body pitch angle Θ0 = 0◦

Basic Aerodynamic data Non − dimensional Dimensional CXu = −0.088

→ Xu = SQ CXuU0

→ CXα = 0.392 Xw = SQ CXαU0

→ CZu = −1.48 ?CZα = −4.46

→ Zw = SQ CZαU0 � �→

cCZ = −1.13 Zw = SQ CZ α → U0 2U0 α

cCZq = −3.94 → Zq = SQ 2U0

CZq

CMu = 0 Mu = SQc CMuU0

CMα = −0.619 →

Mw = SQc CMαU0→

CM = −3.27 ?α →

cCMq = −11.4 → Mq = SQc 2U0

CMq

CZδe = −0.246 → Zδe = SQCZδe

CMδe = −0.710 → Mδe = SQcCMδe

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4. For the general aviation Navion, find the trim angle of attack (αtrim) and elevator deflection (δetrim ) for the flight condition discussed in class. Note that from Nelson page 400 we have that CLδe = 0.355 and CMδe = −0.923. How do these trim values change if the tail incidence angle is changed to −3◦?

5. An aircraft has instruments that continuously record P , Q, R. Assuming that the initial flight conditions are known, develop a Simulink block diagram that can be used to calculate Θ, Φ, and Ψ for future time. Plot the angles as a function of time that result when P (t) = 0.01 rad/sec, Q(t) = −0.01 rad/sec, R(t) = 0.01 rad/sec starting from zero initial conditions.

6. The equations of motion for a spinning missile can be approximated as follows:

• Center­of­mass moves in a straight line through inertial space (Vcm is constant). • There is a constant rolling velocity P0. • The quantities Θ, Ψ, Q, and R are always small enough to permit linearization. • The X­force equation and the rolling­moment equation may be assumed automat­

ically satisfied by proper deflections of the control surfaces. • The pitching moment, M , and yaw­moment N consist mainly of moments, pro­

portional to W and V respectively which tend to restore the X­axis parallelism with the flight direction.

Under these approximations, show how Q and R can be eliminated from the remaining four equations of motion.

(a) With U approximately constant, show how the last two equations become:

¨ W + (ωy /P0)2 − I1 W + (1 + I1)V = 0

¨ −(1 + I2)W + V + (ωz /P0)2 − I2 V = 0

˙ dwhere ( ) = dτ , and τ = P0t, I1 = (Izz − Ixx)/Iyy > 0, I2 = (Iyy − Ixx)/Izz > 0,

Ixz ≈ 0, and ωy is the natural frequency of the pitching oscillation in the absence of rolling (with the restoring moment, M ), and ωz is the natural frequency of the yawing oscillation in the absence of rolling (with the restoring moment, N ).

(b) Find the characteristic equation for this system, and identify the ranges of the parameters I1, I2, ωy /P0, and ωz /P0 where the spinning systems is stable or unstable.

7. Discuss the main points in the paper by McRuer. In particular, answer the following questions:

(a) What was the biggest change in aeronautics from 1956 onwards? (b) Who was Byron, and what did he contribute? (c) What was the primary reason for the gap in the tools used by the theoretician

and the tinkerers? And what role did the YB49 serve in reducing this gap? (This gap is still very large!)

(d) Why was the flight of Robert C. Lee so remarkable?

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16.333 Handout #3 Prof. J. P. How Oct 12, 2004

Due: Oct 26, 2004

16.333 Homework Assignment #3

Please include all code used to solve these problems.

1. What is the wing surface area of the “Peacemaker” B­36?

2. For the F­4C aircraft we get the data on the following page from the tables. There are three flight conditions. For each flight condition:

(a) Find the phugoid and short period frequencies.

(b) Comment on how these mode frequencies change with the flight condition, and how do the numbers compare with the B747 analyzed in class?

(c) Find the Spiral, Dutch roll, and Roll modes frequencies. Comment on how these mode frequencies change with these flight conditions. How do the numbers com­pare with the B747 analyzed in class?

3. The longitudinal model approximations are thought to be significantly better than those developed for the lateral dynamics.

(a) The Dutch roll approximate model is obtained by looking at sideslip and yawing motions, neglecting the rolling motion. Show that the resulting model is of the form: � � � � � � � �

v Yv/m −Uo v Yδr /m= + δr r (I � + Nv/I� (I � + Nr/I

� r (I � + Nδr /I�

zxLv zz) zxLr zz) zxLδr zz)

For the B747, examine this conjecture by plotting the following transfer functions for the actual and approximate models given below:

• Gvδr (s) ­ actual and Dutch roll approximate models

• Grδr (s) ­ actual and Dutch roll approximate models

(b) Compare the accuracy of the approximate model in the frequency range near the mode that it approximates. Do your results support the conjecture given above? How well does this model approximate the actual dynamics in the other (higher and lower) frequencies ranges?

(c) An approximate model for the spiral mode is obtained by looking at changes in the bank and heading angles. Sideslip is usually small (can ignore the side

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force equation), but cannot ignore β completely. Not much roll motion, so set p = p = φ = θ0 = 0. The result is of the form:

r + λr r = 0

What is λr , and how well does this model agree with the full dynamics?

(d) An approximate roll model is given by the equation

p + λpp = 0

What is λp, and how well does this model agree with the full dynamics?

4. Using classical techniques (PID or Lead/Lag), design a pitch attitude autopilot for the B747 Jet using pitch angle and/or rate feedback. Use the short period approximation of vehicle dynamics. The goal is to put the short period roots in the vicinity of wsp = 4 rad/sec and ζsp = 0.4. Assume there is an actuator servo that can be modeled as a first order lag with a time constant of 0.1 sec and has a DC gain of 1.

(a) Show how you arrived at your design (show a root locus or Bode plot).

(b) Plot the time response to a step θc command.

(c) Check your pole locations on the full set of longitudinal dynamics. Is the response stable?

(d) Use the short period model and design the controller using state space techniques. Develop a full state feedback controller that puts the regulator poles where re­quired. Compare the time response to a step θc command to your classical con­troller.

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5. The attached figures show the planform of the monocoupe that we have just started flying autonomously. Here are some key additional parameters:

• U0 = 20m/s

• Mass=9.55Kg,

η = 0.9•

• it ≈ 0, iw ≈ 0

• Tail section is (NACA 0009), so that Clα ≈ 6.25/rad

• CD min ≈ 0.017,

• Can assume CG at 1/4 chord point of the wing

• CLαw = 4.4/rad.

Given this information, estimate the four main longitudinal derivatives Xu, Zu, Mw , and Mq and use them to predict the frequency and damping of the Phugoid and short period modes.

Some basic questions:

(a) What is your estimate of the trim angle of attack. Recall that:

St d�0CLαT

= CLαw + η CLαt

1 −S dα

(b) What are CL0 and CD0 ?

(c) What is Cmcg ?

= Cm0 + Cmα αCmcg

Cm0 = Cmacw + ηVH CLαt

(�0 + iw − it)

d� Cmα = CLαw

(h − hn) − ηVH CLαt 1 −

dα

(d) Can you estimate the derivatives needed to form the more accurate estimates of the frequencies and damping (e.g., using the full approximations)?

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F − 4C S = 530 ft2 , b = 38.67ft c = 16 ft m = 1210slugs

Condition 1 2 3

U0 ft/sec h ft

1.4520e+3 4.5000e+4

9.5200e+2 1.5000e+4

5.8400e+2 3.5000e+4

Ixx 2.5040e+4 2.4970e+4 2.7360e+4 Iyy 1.2219e+5 1.2219e+5 1.2219e+5 Izz 1.3973e+5 1.3980e+5 1.3741e+5 Ixz ­3.0330e+3 1.1750e+3 ­1.6432e+4 Θ0 0 0 0 Xu ­8.7120e+0 ­2.6015e+1 ­2.1296e+1 Xw ­2.4983e+1 ­7.5371e+0 ­5.0244e+1 Zu ­1.4520e+1 ­1.7424e+2 ­1.4036e+2 Zw ­5.9725e+2 ­1.4022e+3 ­3.3607e+2 Z w ­4.3250e­1 ­2.5420e+0 ­1.2245e+0 Zq ­2.7104e+3 ­7.2600e+3 ­2.2022e+3 Mu 3.0548e+2 ­5.3764e+2 ­1.2219e+1 Mw ­2.4354e+3 ­2.1820e+3 ­4.0319e+2 M w ­1.0267e+1 ­5.8656e+1 ­3.0129e+1 Mq ­5.9629e+4 ­1.2133e+5 ­3.7512e+4 Yv ­1.4275e+2 ­2.6018e+2 ­6.8477e+1 Lv ­1.7206e+2 ­7.3048e+2 ­3.8660e+2 Lp ­2.4614e+4 ­5.6532e+4 ­1.8796e+4 Lr 7.9878e+3 1.6930e+4 7.6882e+3 Nv 9.7965e+2 1.7578e+3 5.0705e+2 Np ­1.1178e+3 1.1184e+3 ­8.2446e+2 Nr ­4.4294e+4 ­7.5632e+4 ­2.0474e+4 Xde 0 0 0 Zde ­8.5511e+4 ­1.2947e+5 ­2.5386e+4 Mde ­1.9550e+6 ­3.0548e+6 ­5.9873e+5 Ydr 1.7364e+4 3.2368e+4 7.9860e+3 Ldr 4.0064e+4 1.3199e+5 ­9.4666e+3 Ndr ­2.9008e+5 ­7.9211e+5 ­1.9279e+5 Yda ­3.4969e+3 ­5.7487e+3 ­1.0672e+3 Lda 1.7022e+5 4.3648e+5 1.1609e+5 Nda 3.0042e+4 6.2491e+4 ­1.7039e+4

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2.451

0.246

0.222

0.3870.076

0.606

0.146

0.108

0.638

0.4320.489

0.111 0.097

X

Y

Wing Area = 0.9420 M2

Stab Area = 0.0628 M2

Elevator Area = 0.0418 M2

¼ Scale Monocoupe 90A

Units = SI

Fin Area = 0.0270 M2

Rudder Area = 0.0335 M2

¼ Scale Monocoupe 90A

Units = SI

0.345 0.894

0.159

0.146

0.311

0.1300.8940.129

0.149

0.116

X

Z

¼ Scale Monocoupe 90A

Units = SI

2.451

0.610

0.084

0.149

Z

Y

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Figure 1: Monocoupe Planform

16.333 Handout #4 Prof. J. P. How Oct 26, 2004

Due: Nov 16, 2004

16.333 Homework Assignment #4

Please include all code used to solve these problems.

1. Given the following model of the aerosonde vehicle extracted from the (trim condition, level flight at 23 m/s at sea level)

152.2s + 913.1 Guδe (s) =

Δ(s) −25.72s2 − 2.439s − 3.85

Gαδe (s) = Δ(s)

2−24.18s3 − 99.13s − 9.672s Gqδe (s) =

Δ(s) −24.18s2 − 99.13s − 9.672

Gθδe (s) = Δ(s)

1.236s3 + 10.46s2 + 131.4s + 38.5 Guδa (s) =

t Δ(s) −1.192s2 + 0.3228s − 0.1366

Gαδa (s) = t Δ(s)

−0.7646s3 + 3.058s Gqδa (s) =

t Δ(s) −0.7646s2 + 3.058

Gθδa (s) = t Δ(s)

whereΔ(s) = s 4 + 8.28s 3 + 105.1s 2 + 14.22s + 24.29

3and δa = H(s)δtc, with H(s) =

s+3 to capture the engine lag. The elevators on this t

aircraft are fast enough that they can be ignored.

Part of the system model available on­line is shown in the figure. u and α are available as the first and third outputs of VelW, q is available as the 5th element of States, and θ is available as the second element of Euler. The elevator is the second control input, the throttle (δt

c) is the fourth. Note that a very simple roll loop has been added to facilitate analysis of the longitudinal dynamics.

• Compare the OL dynamics to the 747 ­ any surprises?

• Given these dynamics, design an altitude autopilot

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• Use classical, multi­loop closure techniques as discussed in class

• Use Full state feedback. Discuss you rationale for where to locate the regulator poles.

• Use output feedback with u, θ, and α measurements. Discuss you rationale for where to locate the estimator poles.

• Use the simplified block diagram available on the class web page, implement the controller in simulink. Compare the nonlinear simulation response with your predictions ­ any surprises?

2. Continue Question #4 of HW3, but this time use the short period model and design the controller using state space techniques. As part of this design,

(a) Develop a full state feedback controller that puts the regulator poles where re­quired.

(b) Then develop a closed­loop estimator for the system assuming that you can mea­sure the pitch angle θ. Choose estimator pole locations that have the same imag­inary part as the regulator poles, but a real part that is 3–4 times larger (in magnitude).

(c) Put the regulator and estimator together to form the compensator Gc(s) that maps y = θ to u = δe (recall that actually u = −Gcy). To implement this design, perform the same trick that we did in the notes, and use u = Gce, where e = θc −θ. You should now be able to perform closed­loop simulations of the response of the system to a step in θc.

(d) Check your pole locations on the full set of longitudinal dynamics. Is the response stable?

(e) Compare the frequency response of this state space controller and the controller that you designed in HW3.

3. Using the same approach given in class, design a heading autopilot for the F­4C (i.e. one that can track a given Ψd) using the dynamics in condition 2. Assume that the actuator servo dynamics has the transfer function Hs(s) = 20/(s + 20).

(a) This design will consist of a yaw damper, a roll controller, and a feedback on the heading ψ.

(b) Simulate the response to an interesting Ψ sequence (like a sequence of 45 deg turns) and comment on the performance. Include a limiter on the desired bank angle of ±15 degs.

4. Write a brief summary of the paper by Vincenti that was handed out. In particular, be sure state his main point and whether or not you agree with it.

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Figure 1: Simplified block diagram for the Aerosonde3