Modeling
Q
UESTION
16
Linearize the differential equation
(
29
(
29
(
29
(
29
3
2
cos
0
x t
x t
x t
x t
+
+

=
&&
&
.
The two states are
(
29
(
29
1
x
t
x t
=
(1)
(
29
(
29
2
x
t
x t
=
&
(2)
The two first order nonlinear differential equations are
(
29
(
29
(
29
1
1
2
f
t
x
t
x
t
=
=
&
(3)
(
29
(
29
(
29
(
29
(
29
2
2
2
1
1
3
2
cos
f
t
x
t
x
t
x
t
x
t
=
= 

+
&
(4)
At equilibrium, the derivatives are equal to zero and equation (3) becomes
2
2
0
0
x
x
=
→
=
(5)
At equilibrium equation (4) becomes
(
29
(
29
2
1
1
1
1
0
3
2
cos
2
cos
0
x
x
x
x
x
= 

+
→

=
(6)
Using graphical techniques or the bisection method to solve equation (6),
1
0.4502
x
=
. The first
incremental state and its first derivative with respect to time are
(
29
(
29
(
29
(
29
1
1
1
1
1
ˆ
ˆ
x
t
x
t
x
x
t
x
t
=

→
=
&
&
(7)
The second incremental state and its first derivative with respect to time are
(
29
(
29
(
29
(
29
2
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 Spring '11
 LANDERS
 Derivative, X1

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