Biomedical Engineering Reference
In-Depth Information
>>
syms D
>>
¼
A
[-1.1 0 0 0 0;1.1 -.91 0 0 0;0.9 -15 .5 30 .4;0 0 15 -31 0;0 0.5 0 -.45];
>>
det(D*eye(5)-A)
ans
¼
^
þ
^
þ
^
þ
^
þ
D
5
(1224*D
4)/25
(293191*D
3)/2000
(787421*D
2)/5000
þ
(2656059*D)/40000
301301/40000
>>
eig(A)
ans
¼
0.1748
0.9392
45.8360
0.9100
1.1000
The reconstructed differential equation for
q
3
is
q
3
þ
q
3
þ
_____
q
3
þ
48
:
92
____
q
3
þ
146
:
6
__
q
3
þ
157
:
5
66
:
4
7
:
53
q
3
¼
0
From the roots, the response is written as
q
3
¼
B
1
e
0:18
t
þ
B
2
e
0:94
t
þ
B
3
e
45:84
t
þ
B
4
e
0:91
t
þ
B
5
e
1:1
t
ð
7
:
123
Þ
which is the same form as Eq. (7.122). To calculate
B
1
through
B
5
, we use the initial conditions,
q
1
(0)
¼
25,
q
2
(0)
¼
0,
q
3
(0)
¼
0,
q
4
(0)
¼
0, and
q
5
(0)
¼
0, to find, after considerable effort, that
q
3
ð
q
3
ð
q
3
ð
0
Þ¼
0,
0
Þ¼
0,
0
Þ¼
24
:
75,
__
q
3
ð
0
Þ¼
433
:
3752, and
____
q
1
ð
0
Þ¼
17, 935.
Using the initial conditions and Eq. (7.123), we have
q
3
ð
0
Þ¼
0
¼
B
1
þ
B
2
þ
B
3
þ
B
4
þ
B
5
q
3
ð
0
Þ¼
0
¼
0
:
18
B
1
0
:
94
B
2
45
:
84
B
3
0
:
91
B
4
1
:
1
B
5
2
2
2
2
2
q
3
ð
0
Þ¼
24
:
75
¼ð
0
:
18
Þ
B
1
þð
0
:
94
Þ
B
2
þð
45
:
84
Þ
B
3
þð
0
:
91
Þ
B
4
þð
1
:
1
Þ
B
5
3
3
3
3
3
__
q
3
ð
0
Þ¼
433
:
3725
¼ð
0
:
18
Þ
B
1
þð
0
:
94
Þ
B
2
þð
45
:
84
Þ
B
3
þð
0
:
91
Þ
B
4
þð
1
:
1
Þ
B
5
4
4
4
4
4
__
q
3
ð
0
Þ¼
17, 935
¼ð
0
:
18
Þ
B
1
þð
0
:
94
Þ
B
2
þð
45
:
84
Þ
B
3
þð
0
:
91
Þ
B
4
þð
1
:
1
Þ
B
5
To solve for the unknown constants, we evaluate the unknown coefficients using MatLab:
2
4
3
5
2
4
3
5
2
4
3
5
1
1
1
1
1
B
1
B
2
B
3
B
4
B
0
0
24
0
:
18
0
:
94
45
:
84
0
:
91
1
:
1
0
:
031
0
:
88
2101
0
:
83
1
:
21
¼
:
75
0
:
005
0
:
83
96, 298
:
75
1
:
33
433
3725
17, 934
:
0
:
0009
0
:
78
4, 4139, 948
0
:
69
1
:
46
5
Continued
