Civil Engineering Reference
In-Depth Information
a
b
8
12
A10L
A11T
10
6
8
4
6
4
2
2
0
0
1
2
3
4
5
6
1
2
3
4
5
6
Frequency [Hz]
Frequency [Hz]
180
180
A11T
90
90
0
0
-90
-90
A10L
-180
-180
1
2
3
4
5
6
1
2
3
4
5
6
Frequency [Hz]
Frequency [Hz]
Fig. 1.4
Example comparisons between measured (
continuous line
) and synthesized (
dashed line
) point
inertance for Dogna bridge:
(
a
) accelerometer A11T; (
b
) accelerometer A10L
Table 1.1
Experimental Modal Analysis results: mean value of natural frequencies
p
r
and damping ratios
ξ
r
, with their
maximum deviations. T
¼
Torsional; B
¼
Bending; RB
¼
(almost) rigid-body motion
Mode order
r
-
Description
-
Natural frequency
p
r
[Hz]
Damping ratio
ξ
r
[%]
1
1st B
2.022
0.001
0.88
0.03
2
RB Transversal
3.053
0.003
2.88
0.05
3
2nd B
3.180
0.002
0.89
0.05
4
RB Longitudinal
3.605
0.002
4.33
0.07
5
RB Torsional
4.831
0.011
3.93
0.13
6
3rd B
6.887
0.046
Not available
7
1st T
6.934
0.015
1.15
0.20
8
2nd T
7.995
0.005
0.88
0.10
9
4th B
9.107
0.020
1.78
0.44
10
Coupled B-T
12.910
0.025
1.66
0.20
11
Coupled B-T
14.228
0.081
0.66
0.12
12
Coupled B-T
14.433
0.100
0.77
0.27
repeat several curve-fitting runs by varying both the modal order and the frequency interval. Figure
1.4
shows a comparison
between some measured and synthesized point inertance function of the bridge.
Vibration modes have resonance frequencies well separated, with the exception of the pairs of modes (2,3) and (6,7). The
identification of Modes 2 and 3 was made easy by the fact that their modal shape corresponds to vibrations with prevailing
amplitudes in the horizontal plane and along the vertical direction, respectively. Modes 6 and 7 are both primarily vertical
modes and, in order to identify them, it is found convenient to analyze the FRFs obtained as the half-difference and the half-
sum of the FRFs measured in points belonging to the same cross-section of the deck in order to separate the flexural and
torsional contribution on these modes. Table
1.1
summarizes the results of the EMA.
Deviations of the natural frequency values from their average value can be considered negligible for modes at low
frequency and, generally, differences increase with the mode order. The estimation of the damping ratios is equally good,
even if, as expected, differences are important for some modes, particularly for higher order modes. Moreover, mode shapes
