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(1998). In this study, we also extended the 2D model from the bent in the intact stage to that hi the
severe-damage stage by releasing boundary restriction at the ground end of pile 15 in Fig. 6.
Table 1 shows the first ten undamped natural frequencies of the bent in its intact and severe-damage
stages, in which the first to the third natural frequencies in vertical directions and their reduction
percentage in its severe damage are also indicated. To simulate the damping effects, Rayleigh
damping was used in modeling, i.e., [C] = a 0 [M] + a : [K] where M, C and K are the matrices of mass,
damping and stiffness, 0 0 and a } are the constant coefficients. We used the same force in Fig. 2a to
excite the 2D model hi Fig. 6 and obtained vibration response at nodes 53 and 37 in Fig. 6 that
correspond to the same locations in sensors 15 and 13 in Fig. 1, respectively.
Fig. 7a shows the Hilbert amplitude spectrum of the response at node 37 of the bent in intact stage with
no damping, i.e., a 0 = a } = d = 0. The response energy focuses at frequency band with its center
primarily around 12.0, 19.7, and 69.2 Hz that are respectively the fust three natural frequencies in
vertical direction in Table 1. This can be further clarified in marginal amplitude spectrum in Fig. 7b.
Without damping, the bent is fully excited at the natural frequencies. Therefore, the vibration energy
at natural frequencies is typically higher than the excitation-inherited energy at chirp frequency.
Consequently, the energy inherited from the excitation at chirp frequency is not clearly shown in Fig.
7a, except the strong portion with frequency band from 40-70 Hz at 3.5-5 s. With the damping, the
relath e intensity of the energy from the structure and excitation should be changed, which can be seen
from the Hilbert amplitude spectra in Figs. 7c,d with a Q = a : = d =0.00198 and 0.05305, respectively.
Comparing Fig. 7c with Figs. 3a and noticing the fundamental natural frequency of the bent being
11.98 Hz, we can conclude that the excitation-inherited energy in Fig. 7c is associated with chirp
frequency and other frequencies such as high-frequency band in 10-30 Hz during 0.3-0.8 s, and that the
vibration energy is associated with frequency band around 10-13 Hz during 1-1.6 s. The damped
fundamental natural frequency in vertical direction is identified in Fig. 7c around 11.5 Hz, smaller than
the undamped one with 11.98 Hz in Table 1 which is consistent with the vibration theories. Due to the
large damping used and strong excitation at chirp frequency, the vibration energy at natural
frequencies are greatly suppressed and the excitation energy is inherited, in comparison with the
undamping case (see Figs. 7a,c). This phenomenon can be further clarified by Fig. 7d with a large
damping. Fig. 7d shows that the vibration energy at 10 Hz primarily at 1.4 s. Since the damping here
is extremely high, the damped fundamental natural frequency is greatly downshifted from undamped
one at 11.98 Hz to the current one at 10 Hz. It should be noted here that it is with the damping
d=0,05305 that the ANSYS model in Fig. 6 generated time history responses that are similar to those
in recordings such as Fig. 2b. This implies that real damping of bent 12 in the vertical direction is very
high.
Depicted in Figs. 8a,b are respectively the Hilbert amplitude spectra of the response at nodes 37 and
53 of the bent in severe-damage stage with a Q =a l =d =003979. Fig. 8b shows the vibration energy
concentrates at 7 Hz from 0.5 to 1.2 s, i.e., the fundamental natural frequency of severe-damage bent
in Table 1. In contrast, the dominant vibration energy in Fig. 8a is at around 12 Hz between 0.6-1 Hz
and minor at 7 Hz at 1+ s. The difference of dominant response energy at frequency 7 Hz in Figs. 8a,b,
i.e., damage-signature from recordings, helps identify qualitatively the damage column and severity.