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