Civil Engineering Reference
In-Depth Information
Extracting the data from Section 3.7, the stiffness matrix for the active degrees of free-
dom is
7
.
5
0
0
0
−
3
.
75
0
0
0
0
3
.
75
0
−
3
.
75
0
0
0
0
0
0
10
.
15
0
−
1
.
325
1
.
325
−
3
.
75
0
0
−
3
.
75
0
6
.
4
1
.
325
−
1
.
325
0
0
10
5
lb/in.
[
K
a
]
=
−
3
.
75
0
−
1
.
325
1
.
325
5
.
075
−
1
.
325
0
0
0
0
1
.
325
−
1
.
325
−
1
.
325
5
.
075
0
−
3
.
75
0
0
−
3
.
75
0
0
0
3
.
75
0
0
0
0
0
0
−
3
.
75
0
3
.
75
The finite element model for the truss exhibits 8 degrees of freedom; hence, the charac-
teristic determinant
2
[
M
]
+
[
K
]
|=
0
|−
yields, theoretically, eight natural frequencies of oscillation and eight corresponding
modal shapes (modal amplitude vectors). For this example, the natural modes were com-
puted using the student edition of the ANSYS program [9], with the results shown in
Table 10.1. The corresponding modal amplitude vectors (normalized to the mass matrix
as discussed relative to orthogonality) are shown in Table 10.2.
The frequencies are observed to be quite large in magnitude. The fundamental fre-
quency, about 122 cycles/sec is beyond the general comprehension of the human eye-
brain interface (30 Hz is the accepted cutoff based on computer graphics research [10]).
The high frequencies are not uncommon in such structures. The data used in this example
correspond approximately to the material properties of aluminum; a
light
material with
good stiffness relative to weight. Recalling the basic relation
=
k
/
m
, high natural
frequencies should be expected.
The mode shapes provide an indication of the geometric nature of the natural modes.
As such, the numbers in Table 10.2 are not at all indicative of amplitude values; instead,
Table 10.1
Natural Modes
Frequency
Mode
Rad/sec
Hz
1
767.1
122.1
2
2082.3
331.4
3
2958.7
470.9
4
4504.8
716.9
5
6790.9
1080.8
6
7975.9
1269.4
7
8664.5
1379.0
8
8977.4
1428.8
