Information Technology Reference
In-Depth Information
The first and fifth equations correspond to unknown reactions. The remaining equations
appear as
D
22
D
23
D
24
D
26
θ
a
w
b
θ
b
θ
c
=
D
32
D
33
D
34
D
36
0
(8)
D
42
D
43
D
44
D
46
D
62
D
63
D
64
D
66
The characteristic equation is obtained from the determinant of the coefficients of (8),
i.e.,
D
22
D
23
D
24
D
26
=
D
32
D
33
D
34
D
36
∇=
0
(9)
D
42
D
43
D
44
D
46
D
62
D
63
D
64
D
66
This relationship can be obtained directly from (5).
From (6), i.e.,
D
ij
2
M
ij
,
=
K
ij
−
ω
10
6
2
D
22
=
3
.
9999
×
−
1
.
7752
ω
10
5
2
D
32
=
D
23
=
1
.
49996
×
+
0
.
1442
ω
10
4
2
D
33
=
1
.
4999
×
−
0
.
0865
ω
10
6
2
D
42
=
D
24
=
1
.
9999
×
+
1
.
3314
ω
D
43
=
D
34
=
0
(10)
10
6
2
D
44
=
7
.
9998
×
+
3
.
5505
ω
D
62
=
D
26
=
0
10
5
2
D
36
=
D
63
=−
1
.
49996
×
−
0
.
1442
ω
D
64
=
D
46
=
D
42
=
D
66
D
22
Substitution of (10) into (9) and use of factorization leads to two equations:
4
10
5
2
10
10
ω
−
41
.
0093
×
ω
+
13
.
3967
×
=
0
and
4
10
6
2
10
12
ω
−
14
.
1631
×
ω
+
8
.
7036
×
=
0
(11)
The roots of these equations are
=
ω
1
2
ω
=
181
.
47 rad
/
sec
f
1
π
=
28
.
88 cycles
/
sec
(
Hz
)
1
ω
=
802
.
37 rad
/
sec
f
2
=
127
.
70 cycles
/
sec
2
or
(12)
ω
=
2016
.
93 rad
/
sec
f
3
=
321
.
00 cycles
/
sec
3
ω
4
=
.
/
=
.
/
3676
86 rad
sec
f
4
585
19 cycles
sec
These can be compared to the exact natural frequencies (Example 10.8) of
ω
=
180
.
74
,
ω
=
1
2
.
ω
3
=
.
ω
=
.
.
722
96
,
1626
66
,
and
2891
84
4
Search WWH ::
Custom Search