Biomedical Engineering Reference
In-Depth Information
To make the solution more readable, matrix
A
is written as
2
3
a
11
a
12
a
13
4
5
A
¼
a
21
a
22
a
23
a
31
a
32
a
33
Solving Eq. (7.79) gives
1
Þ
1
F
Q
¼
D
ð
I
A
¼
adj
ð
D
I
A
Þ
F
det
ð
D
I
A
Þ
ð
7
:
80
Þ
or
det
ð
D
I
A
Þ
Q
¼
adj
ð
D
I
A
Þ
F
and using MATLAB, we have
>>
syms D q1 q2 q3 a11 a12 a13 a21 a22 a23 a31 a32 a33
>>
¼
A
[a11 a12 a13; a21 a22 a23; a31 a32 a33];
>>
det(D*eye(3)-A)
ans
¼
^
^
^
þ
D
3-D
2*a33-a22*D
2
D*a22*a33-D*a23*a32-
^
þ
þ
þ
a11*D
2
a11*D*a33
a11*a22*D-a11*a22*a33
a11*a23*a32-
þ
a21*a12*D
a21*a12*a33-a21*a13*a32-a31*a12*a23-
þ
a31*a13*D
a31*a13*a22
>>
¼
adj
det(D*eye(3)-A)*inv(D*eye(3)-A)
¼
adj
^
þ
þ
[D
2-D*a33-a22*D
a22*a33-a23*a32, a12*D-a12*a33
a13*a32,
a12*a23
a13*D-a13*a22]
[ a21*D-a21*a33
þ
þ
a23*a31, D
^
2-D*a33-a11*D
þ
a11*a33-a13*a31,
a23*D-a23*a11
þ
a13*a21]
[ a21*a32
þ
a31*D-a31*a22, a32*D-a32*a11
þ
a12*a31, D
^
2-a22*D-
a11*D
þ
a11*a22-a12*a21]
Substituting the values from MATLAB into Eq. (7.80) gives
Q
3
a
ð
þ
a
þ
a
Þ
D
2
þ
a
ð
þ
a
þ
a
a
a
a
Þ
D
D
a
a
a
a
a
a
11
22
33
11
22
22
33
33
11
12
21
13
31
23
32
¼
a
a
a
þ
a
a
a
þ
a
a
a
þ
a
a
a
a
a
a
a
a
a
11
22
33
11
23
32
22
13
31
33
12
21
12
23
31
13
32
21
2
4
3
5
2
D
a
ð
þ
a
Þ
D
þ
a
a
a
a
a
D
a
a
þ
a
a
a
D
þ
a
a
a
a
33
22
22
33
23
32
12
12
33
13
32
13
12
23
13
22
2
ð
Þ
D
þ
a
F
a
D
a
a
þ
a
a
D
a
þ
a
a
a
a
a
D
a
a
þ
a
a
21
21
33
23
31
33
11
11
33
13
31
23
23
11
13
21
2
a
D
þ
a
a
a
a
a
D
a
a
þ
a
a
D
a
ð
þ
a
Þ
D
þ
a
a
a
a
31
21
32
31
22
32
32
11
12
31
22
11
11
22
12
21
ð
7
:
81
Þ
Returning to the time domain gives the following independent differential equations:
__
q
1
a
ð
þ
a
þ
a
Þ
q
1
þ
a
ð
þ
a
þ
a
a
a
a
Þ
q
1
a
a
a
a
a
a
11
22
33
11
22
22
33
33
11
12
21
13
31
23
32
þ
a
ð
a
þ
a
þ
a
a
a
Þ
q
a
a
a
a
a
a
a
a
a
a
a
a
11
23
32
11
22
33
22
13
31
33
12
21
12
23
31
13
32
21
1
¼
f
Þ
f
1
þ
a
12
f
2
a
a
ð
þ
a
ð
a
a
a
Þ
f
þ
a
ð
a
a
a
Þ
f
ð
7
:
82
Þ
1
33
22
22
33
23
32
1
12
33
13
32
2
13
f
3
þ
a
þ
a
ð
a
a
a
Þ
f
12
23
13
22
3
