# I. Modal Analysis

Urban and Civil

Nov 16, 2013 (4 years and 5 months ago)

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UAV Design and Build

Assignment 4

I. Modal Analysis

eig_lon =

-
13.5925 + 9.0398i

-
13.5925
-

9.0398i

-
0.1975 + 0.4735i

-
0.1975
-

0.4735i

eig_lat =

0

-
12.5370

-
0.0171

-
1.6845 + 4.6478i

-
1.6845
-

4.6478i

1.

From
the eigenvalues we see that the longitudinal modes are two decaying sinusoidal poles and
that the lateral/directional modes are two decaying modes and 1 decaying sinusoidal modes.

2.

This is consistent with aerodynamic theory.

The two decaying sinusoidal pole
s for the longitudinal case are the Short Period and Phugoid.

The two exponential decaying modes for the lateral case are the Roll and Spiral modes while the
sinusoidal decaying mode is the Dutch Roll Mode
.

3.

Since all the poles are in the imaginary left hal
f plane, all modes decay and the aircraft is free
-
response stable.

II. Simulation of the AC Response

1.

Shown on next page

2.

Inputs: Elevator and Thrust commands

Outputs: X
-
Velocity, Descent Rate, Pitch, and Pitch rate (Longitudinal Dynamics)

Inputs: Rudder
and Aileron commands

Outputs: Yaw, Roll, Horizontal Velocity
, Yaw rate, Roll rate

3. and 4.

First Test was an elevator doublet with a magnitude of 1° over 1 second
.

By exciting the system with a doublet on the elevator command we excite the phugoid mo
de this can be
seen in the pitch plunge response of the horizontal velocity. The doublet is clearly mapped to the pitch
rate output and decent speed output. There is a slight phugoid oscillation in decent speed but the gain is
very low.

0
1
2
3
4
5
6
7
8
9
10
-0.2
0
0.2
u (m/s)
Longitudinal Response
0
1
2
3
4
5
6
7
8
9
10
-0.2
0
0.2
w (m/s)
0
1
2
3
4
5
6
7
8
9
10
-10
0
10
q (

/s)
0
1
2
3
4
5
6
7
8
9
10
-4
-2
0
2
time (sec)

(

)
Second Test

was
a
aileron
doublet with a magnitude
of 1° over 1 second

The
aileron input excites both the dutch roll and spiral modes. The dutch roll mode can be seen in
the damped oscillations on p and r while the spiral mode with its large time constant can be seen in

the offset of the roll angle. The roll mode is very quick and visually cannot be separated from dutch
roll.

0
1
2
3
4
5
6
7
8
9
10
-0.5
0
0.5
v (m/s)
Lateral Response
0
1
2
3
4
5
6
7
8
9
10
-20
0
20
p (

/s)
0
1
2
3
4
5
6
7
8
9
10
-5
0
5
r (

/s)
0
1
2
3
4
5
6
7
8
9
10
-5
0
5

(

)
0
1
2
3
4
5
6
7
8
9
10
-2
0
2
time (sec)

(

)
The Final test was a step response on the rudder of 1°.

The step on the rudder excites the dutch roll mode which can be seen in the initial os
cillations. The
large coupling between the longitudinal inputs can been seen here as the roll angle experiences a large
deviation from trip from rudder input.

5. The oscillatory responses generated in our simulation closely match those generated in Dorob
antu et
al the magnitudes will be scaled differntly. Our steps are of lesser magnitude and do not include
corrective input for divergence.

III Transfer Function and Frequency Response

1.

Transfer functions of the longitudinal from elevator to

U(s)

-
0.36 s^3 + 39 s^2 + 369.3 s + 1.46e04

---------------------------------------------

s^4 + 27.58 s^3 + 277.5 s^2 + 112.4 s + 70.14

0
1
2
3
4
5
6
7
8
9
10
-0.5
0
0.5
v (m/s)
Lateral Response
0
1
2
3
4
5
6
7
8
9
10
-5
0
5
p (

/s)
0
1
2
3
4
5
6
7
8
9
10
-20
-10
0
r (

/s)
0
1
2
3
4
5
6
7
8
9
10
-50
0
50

(

)
0
1
2
3
4
5
6
7
8
9
10
-100
-50
0
time (sec)

(

)

W(s)

-
3.62 s^3
-

2431 s^2
-

939 s
-

1355

---------------------------------------------

s^4 + 27.58 s^
3 + 277.5 s^2 + 112.4 s + 70.14

q(s)

-
141.6 s^3
-

1542 s^2
-

651.7 s

---------------------------------------------

s^4 + 27.58 s^3 + 277.5 s^2 + 112.4 s + 70.14

theta(s)

-
141.6 s^2
-

1542 s
-

651.7

----------------------------
-----------------

s^4 + 27.58 s^3 + 277.5 s^2 + 112.4 s + 70.14

Transfer functions of the lateral from aileron to

V(s)

-
2.06 s^4
-

192.1 s^3
-

3309 s^2
-

3440 s

-------------------------------------------------

s^5 + 15.92 s^4 + 66.95 s^3 +

307.5 s^2 + 5.225 s

P(s)

-
138.3 s^4
-

449.7 s^3
-

2927 s^2 + 42.19 s

-------------------------------------------------

s^5 + 15.92 s^4 + 66.95 s^3 + 307.5 s^2 + 5.225 s

R(s)

5.329 s^4 + 97.47 s^3
-

36.62 s^2
-

1406 s

----------
---------------------------------------

s^5 + 15.92 s^4 + 66.95 s^3 + 307.5 s^2 + 5.225 s

Psi(s)

-
138.1 s^3
-

446.8 s^2
-

2929 s
-

2.475e
-
16

-------------------------------------------------

s^5 + 15.92 s^4 + 66.95 s^3 + 307.5 s^2 + 5.225 s

Phi(s)

5.329 s^3 + 97.47 s^2
-

36.62 s
-

1406

-------------------------------------------------

s^5 + 15.92 s^4 + 66.95 s^3 + 307.5 s^2 + 5.225 s

The transfer functions of the lateral from rudder to

V(s)

2.98 s^4 + 524 s^3 + 6059 s^2
-

930 s
-

1.008e
-
13

-------------------------------------------------

s^5 + 15.92 s^4 + 66.95 s^3 + 307.5 s^2 + 5.225 s

P(S)

2.405 s^4
-

91.99 s^3
-

888 s^2 + 12.97 s + 9.081e
-
15

-----------------------------------------------------

s^5 + 15.9
2 s^4 + 66.95 s^3 + 307.5 s^2 + 5.225 s

R(S)

-
26.21 s^4
-

341.7 s^3
-

187.3 s^2
-

432.3 s

-------------------------------------------------

s^5 + 15.92 s^4 + 66.95 s^3 + 307.5 s^2 + 5.225 s

Psi(S)

1.618 s^3
-

102.2 s^2
-

893.6 s + 4.148e
-
15

-------------------------------------------------

s^5 + 15.92 s^4 + 66.95 s^3 + 307.5 s^2 + 5.225 s

Phi(S)

-
26.21 s^3
-

341.7 s^2
-

187.3 s
-

432.3

--------------------------------------------
-----

s^5 + 15.92 s^4 + 66.95 s^3 + 307.5 s^2 + 5.225 s

3.

The characteristic equation is the denominator of the above equations.

4.

The roots are equal to the eigen values

The roots of the longitudinal mode

-
13.5925 + 9.0398i

-
13.5925
-

9.0398i

-
0.19
75 + 0.4735i

-
0.1975
-

0.4735i

The roots of the lateral mode

0

-
12.5370

-
1.6845 + 4.6478i

-
1.6845
-

4.6478i

-
0.0171

IV. Ultrastick Nonlinear Simulation

Elevator doublet from sim

Aileron doublet from sim

0
5
10
15
16.85
16.9
16.95
17
17.05
17.1
Linear Velocity to Doublet Sequence [elevator,rudder,aileron]
Time (sec)
u (m/sec)

0
5
10
15
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Time (sec)
v (m/sec)
0
5
10
15
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Time (sec)
w (m/sec)
0
5
10
15
-4
-3.5
-3
-2.5
-2
Angular Velocity to Doublet Sequence [elevator,rudder,aileron]
Time (sec)
p (deg/sec)
0
5
10
15
4
6
8
10
12
14
16
18
Time (sec)
q (deg/sec)
0
5
10
15
15.5
16
16.5
17
17.5
18
Time (sec)
r (deg/sec)
NonLinear
Linear

The nonlinear sim produced by the group is a simplified version of this code and therefore captures less
of the dynamics of the air craft.

0
5
10
15
16.8
16.85
16.9
16.95
17
Linear Velocity to Doublet Sequence [elevator,rudder,aileron]
Time (sec)
u (m/sec)

0
5
10
15
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Time (sec)
v (m/sec)
0
5
10
15
1.04
1.06
1.08
1.1
1.12
Time (sec)
w (m/sec)
0
5
10
15
-15
-10
-5
0
5
10
Angular Velocity to Doublet Sequence [elevator,rudder,aileron]
Time (sec)
p (deg/sec)
0
5
10
15
9
9.5
10
10.5
11
Time (sec)
q (deg/sec)
0
5
10
15
12
13
14
15
16
17
18
19
Time (sec)
r (deg/sec)
NonLinear
Linear