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structural response energy with frequency up to 15 Hz during 1-2 s. The structural response energy
below roughly 12 Hz (light lines or dots) is much less than that above 12 Hz (dark lines or dots). The
response energy is generated likely by the force with the chirp frequency during 0.4-1 s and with the
frequency band in 5-15 Hz at around 1+ s in Fig. 3a. If the fundamental natural frequency falls in the
driving frequency range of 5-15 Hz, the response energy at the fundamental natural frequency should
be much stronger than that at frequency lower, but not necessarily stronger than that at frequency
higher since there exist second and higher natural frequencies and mixed frequencies between driving
and natural frequencies. Fig. 4a suggests that the fundamental natural frequency is around 12 Hz.
Fig. 4b shows the Hilbert amplitude spectrum of vibration response at sensor 15 with the bent in
minor-damage stage. Besides the energy inherited from the excitation force such as at chirpy
frequency, the response energy during 1-2 s has lowered its dominant frequency to 5 Hz at about 1.4 s.
The observed 5 Hz in the minor-damage stage could be related to the mixed natural frequencies of the
bent and pile, since the vibration at sensor 15 should reflect the dynamic characteristics of both the
whole bent and the local member. The bent with excavated soil, i.e., the bent in the minor-damage
stage, reduces the stiffness of the pile and thus the fundamental natural frequencies of the bent and
pile. This is the signature of local damage in the recordings, which is not sensitively picked up by the
Fourier data analysis.
Such an explanation can be further strengthened from Fig. 4c, which shows the Hilbert amplitude
spectrum of vibration response at sensor 15 with the bent in severe-damage stage. As seen in the
figure, the dominant frequency corresponding to the structural energy has decreased to 3 Hz at 1.7 s.
As a comparison for damage-signature recognition from recordings that are collected at sensor 15 close
to the damage location, Figs. 5a-c show the Hilbert amplitude spectra of vibration data at sensor 13
away from the damage location for the bent in intact, minor- and severe-damage stages respectively.
Fig. 5 a shows that the dominant frequency to structural vibration energy is around 13 Hz at 2 s, slightly
higher than 12 Hz identified from Fig. 4a, which could be caused by noise and various uncertainties
introduced when recorded at sensors 13 and 15. Figs. 5b and 5c show the dominant structural
vibration energy with lowest frequency of 10 Hz at 2 and 2.2 s respectively. Since sensor 13 is
removed from the damage location, the vibration characteristics at sensor 13 will retain primarily the
dynamic features of whole bent, which likely do not change significantly due to the local damage at the
column with sensor 15. Accordingly, it is understood for the small change observed in the identified
frequency associated with the dominant structural vibration energy for the bent in three-different stages
in Figs. 5a-c.
The above analysis suggests that signature of structural vibration in terms of natural frequencies of the
whole bent and/or local members can be well distinguished from the driving frequency content by the
HHT analysis of recordings, which is sensitive to the local damage if the recordings are collected near
the damage location.
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MODEL-BASED VALIDATION
To verify the observation and assertion in the last chapter and thus to improve our understanding of the
HHT-based damage characterization, we performed the HHT analysis of simulated vibration data with
ANSYS-based two-dimensional (2D) finite-element-method (FEM) models for the bent 12 hi its intact
and severe-damage stages. Fig. 6 shows the ANSYS-based 2D FEM model for the bent 12 in intact
stage, which was built on design specifications and pertinent information by Sanayei and Santini