Final Exam

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1


Final Exam

AerE355

Fall 2011 (Take
-
Home)


Name _____________________________________


***DUE by 2pm Monday @ the front desk of the AerE main office ***


PROBLEM 1 (15
pts)

The lateral mode dynamics are described by:










































































r
a
r
p
v
r
p
v
r
p
v
r
r
r
a
a
N
L
u
Y
N
L
r
p
N
N
N
L
L
L
u
g
u
Y
u
Y
u
Y
r
p












0
/
0
0
0
0
1
0
0
0
/
cos
1
/
/
/
0
0
0
0
0
0






(1.1)


Equation (1.1) can be expressed in compact notation as:
u
B
x
A
x






.

(a) (8pts)
Give the explicit definition of

the response variable

x

, and define in aerodynamics term
s

what each of its
elements is.

Solution
:


tr
x





(b) (4pts)
Give the explicit definition of the input
u

, and define in aerodynamics terms what each of its elements is.

Solution
:


tr
u






(c
) (3
pt
s)

The complex
-
valued scalar


i
s


is said to be an
eigenvalue

of
A

if for some
v

, we have the equality
v
s
v
A



.
The homogeneous equation associated with (1.1) is
)
(
)
(
t
x
A
t
x




. Assume a
comp
limentary
solution of the
form
o
st
c
x
e
t
x



)
(
, Insert this solution into the homogeneous equation. Then use the result to explain why the scalar,
s,

must be an eigenvalue of
A,
and
o
x


must b
e its corresponding eigenvector,

v

.

Solution
:


2



PROBLEM 2 (15
pts)

Consider a general aviation plane whose lateral dynamics described by (1.1) include:
























0
0
1
0
0
76
.
35
.
49
.
4
0
19
.
2
40
.
8
02
.
16
18
.
1
0
25
.
A
.

(a)
(3
pts)

Use
Matlab
to f
ind the eigenvalu
e
s of the 4
-
D system (1.1).

Solution
:


(b)(3
pts)
You should have obtained two real eigenvalues and a pair of complex eigenvalues. Hence, there are three
modes. Use the eigenvalues to argue whether or not the system is stable.

Argument
:



(c)(6
pts)
For eac
h mode, obtain numerical values for the following dynamic parameters, as appropriate:

(i) time constant

, (ii) damped natural frequency, (iii) damping ratio.

Solution
:

mode #1:


mode #2:



mode #3:



(d)(3pts)
Each of the three lateral modes has a name. Give the name for each mode.

Answer
:

mode #1

.
mode #2


mode #3


3


PROBLEM 3 (3
5
pts)
The longitudina
l dynamics of a plane are described by:




e
q
w
u
w
u
w
u
e
e
e
M
Z
X
q
w
u
M
M
M
u
Z
Z
g
X
X
q
w
u


































































0
0
1
0
0
0
0
0
0




.

(3.1)

The phugoid mode has the approximate 2
-
D dynamics described by:


e
u
u
u
Z
X
u
u
Z
g
X
u
e
e






































0
0
/
0
/


.

(3.2)

Suppo
se that for a given plane, we have:


































0
1
0
0
0
3
04
.
0
0
180
2
4
.
2
.
32
0
04
.
04
.
0
1
0
0
0
0
0
0
q
w
u
w
u
w
u
M
M
M
u
Z
Z
g
X
X

(3.3)

(a)(10
pts)
Compute the eigenvalues of the systems (3.1) and (3.2). Then compute the % error of the approximation (3.2)
to in relation to (i) the

4
settling time, and (ii) the damped natural period
d
d
T


/
2

. [Note: define a % error to be
%
100
]
/*
)
*
[(*
4
4
2


D
D
D
.]

Answer
:

(b)
(5pts)

Consider the initial condition


0
3
/
0
10
0


tr
x

. use

the

Matlab

command sequence below

to obtain

plots of

the initial condition responses
,
)
(
t
u


and
)
(
t


.

x0= [10; 0;pi/3
;0];

dt = 5*tau/1000;

tvec = 0:dt:5*tau;

C=eye(4);

sys4 = ss(A,[],C,[]);

G

= initial(sys4,x0,tvec);

figure(1)

subplot(2,1,1), plot(tvec
,G
(:,1),
'k'
,
'LineWidth'
,2)

title(
'u(t) phugoid response'
)

grid

subplot(2,1,2), plot(tvec,G
(:,4),
'k'
,
'LineWidth'
,2)

title(
'theta(t) phugoid response'
)

xlabel(
'Time (sec)'
)

grid


Plots
:
4


(c)(5pts)
Modify the above code to obtain plots of
)
(
t
u


and
)
(
t



associated with the 2
-
D system (3.2). Overlay these
pl
ots on those in (b). In the event that a response obtained using the 2
-
D approximation is poor, offer an explanation.

Answer
:


(d)(5pts)
Recall that for matrices
A

and
B
,

if both of the following two conditions hold, then
B

is the inverse of
A
:

(C1)
I
AB

, and (C2)
I
BA

. Consider the matrices







d
b
c
a
A

and
)
/(
bc
ad
a
b
c
d
B










.

Prove that
B

is the inverse of
A

(i.
e.
1


A
B
).


Solution
: (C1):


(C2):


(e)(5
pts)
Use the formula in (d
)

in relation

to (3.2) to obtain the Laplace domain expression for


tr
s
s
U
)
(
)
(



for
the initial condition given (b) [Assume no forcing functio
n.]

Solution
:




(f)(5pts)
For

your expression for


tr
s
s
U
)
(
)
(



use a table of Laplace transforms to obtain the expressions for


tr
t
t
u
)
(
)
(



.

Identify the transform used in your table and reference the table.


Solution
:







5


PROBLEM 4 (20pts)
The approximate 2
-
D phugoid and short period models are:

e
u
u
u
Z
X
u
u
Z
g
X
u
e
e






































0
0
/
0
/




&

e
q
e
e
M
u
Z
q
M
M
u
Z
q






































0
0
/
1
/



Each of these systems is a 1
-
input/2
-
output system. Hence each has two transfer functions.

(a)(5pts)
Use the result

in (d
) of PROBLEM 3 to obtain the transfer functions for the phugoid model.

Solution
:


(b
)(5pts)
Use the resul
t in (d
) of PROBLEM 3 to obtain the transfer functions for the short period model.

Solution



(c)(5
pts)
Supp
ose that the transfer function
0483
.
04
.
0
002
.
01
.
)
(
)
(
)
(
2









s
s
s
s
s
U
s
H
e
u


.
Use the ‘bode’

command to obtain a
plot

of the fr
equency response function (FRF)
)
(

i
H
u
.

Solution
:


(d)(5pts)
Suppose that the FRF in (c) is:

-80
-60
-40
-20
0
20
Magnitude (dB)
10
-2
10
-1
10
0
10
1
10
2
90
180
270
360
Phase (deg)
Bode Diagram
Frequency (rad/sec)

Use the magnitude plot to obtain estimates of (i) the system
static ga
in

(in dB) and (ii) the damped natural frequency.

Solution
:
dB
g
s
_____


and
s
rad
d
/
_____





6



PROBLEM 5 (20pts)
DC motors have traditionally been used to activate various control surfaces of an aircraft. Suppose
that a given moto
r has the transfer function











volt
s
s
V
s
s
G
sec
deg/
1
5
)
(
)
(
)
(

, where
)
(
&
)
(
s
s
V


are the Laplace
transforms of
)
(
&
)
(
t
t
v

, respectively.

Since
)
(
)
(
t
t




, it follows
that (ignoring i.c.
s)
)
(
)
(
s
s
s



. Hence,











volt
s
s
s
V
s
s
G
deg
)
1
(
5
)
(
)
(
)
(

.

(a)(5pts)
Use the
Matlab

command sequence ‘tf’ and ‘step’ to obtain the response of the angular position system,
)
(
s
G

to a unit step voltage.

Solution
:





(b)(5pts)
In (a), you should have found that a 1
-
volt step inp
ut to the motor resulted in


)
(
t

. This makes sense,
because this input will result in constant
speed.
Hence, as a positioning system, a DC motor is
unstable.

Consider the
following block diagram of a feedback control system that is desig
ned to make the motor a
stable
positioning system:




Show that the closed loop system transfer function,
)
(
)
(
)
(
s
s
s
W
r




is equal to
)
(
)
(
1
)
(
)
(
s
G
s
G
s
G
s
G
c
c



.

Solution
:


(c)(5pts)
Consider a
proportional controller
,
p
c
K
s
G

)
(
. For th
is controller, obtain the expression for
)
(
s
W
. Your
expression should be simplified to be a ratio of only two polynomials.

Solution
:.


(d)(5pts)
You should have found that
)
(
s
W
is a 2
nd

order system. Find the value of
p
K
so that the damping ratio

2
/
1



(i.e. the system is ‘optimally’ damped).

Solution
:

)
(
s
G
c
)
(
s
G

)
(
t
r

)
(
t
e

)
(
t


)
(
t
v
7


APPENDIX
Matlab
code

for PROBLEM 3