## 3 Detection of Damage Occurrence
### Frequency Domain Methods  
Frequency domain responses are obtained from time series responses by non-parametric estimation and signal processing techniques, relying on the application of the Fourier transformation. The transformation can drastically reduce the volume of data and compensate the loss of information by averaging the effects of random noise [25]. One of the most common methods is based on the frequency response function (FRF) that is a ratio between the structure response such as displacement, velocity, or acceleration in the frequency domain to an applied force that is a known harmonic force generally. The FRFs are used as the data source to estimate modal parameters through curve fitting methods. The commonly selected features are the shift in the resonance and anti-resonances or changes affecting the amplitudes. By applying FRF data directly in damage detection problems, [PERSON] et al. [61] compared the method based on FRF curvatures versus the method based on modal shape curvatures, and illustrated with a lumped-mass system and a real bridge with experimental data [62]. The main drawback is that this FRF curvature method works better for a frequency range before the first resonance or anti-resonance, whichever comes first. The FRF curvature method has been successfully studied in the simulation and experiment [63; 64]. Meanwhile, the direct use of FRF shapes has been studied by many researchers. For example, [PERSON] et al. [65] made use of the residual imaginary parts of FRF shapes to locate the structural damage and the abnormality caused by damage in a cantilever beam. [PERSON] et al. [66] presented a damage detection technique based on both the real and imaginary parts of measured FRF shape. [PERSON] et al. [67] evaluated the use of FRFs with the help of the Particle Swarm Optimization technique. [PERSON] et al. [68] presented the interpolation damage detection method using FRF measurements. [PERSON] and [PERSON] [69] proposed the signal anomaly index to express the amount of changes in the shape of FRFs or strain frequency response function (SFRF). [PERSON] and [PERSON] [70] compared the FRF-based approach extracted from dynamic measurements of a truss structure and the flexibility-based approach extracted from the modal data, and showed that the FRF-based approach can be utilized more explicitly than the flexibility-based approach. The FRF based method is very promising because the FRF data can be directly obtained without any further data extraction and processing. The main advantage of using the FRF shape in damage detection is that it contains higher frequency information, while the mode shape discussed later constructs a truncated frequency space. The FRF requires the simultaneous measurement of both the excitation loads and the structural responses. When the input force is unknown, the computation of the FRF is substituted by the power spectral density or the transmissibility functions. The transmissibility that is calculated from the FRFs of a system is a transformation matrix between two sets of output responses in the frequency domain. The motivation of using transmissibility for damage detection relies on the fact that the transmissibility is a local quantity, suggesting a higher sensitivity than the modal parameters to detect structure changes [71]. For example, [PERSON] et al. [72] detected damage using the novelty detection algorithm based on the measured transmissibility. [PERSON] et al. [73] studied the output-only modal identification using transmissibility under variable load conditions. [PERSON] et al. [74; 75; 76] used transmissibility to identify modal parameters from output-only transmissibility measurements and multivariable transmissibility measurements. [PERSON] and [PERSON] [77] reviewed the state-of-the-art of using transmissibility functions for damage localization of mechanical systems. [PERSON] et al. [78] studied the feasibility of detecting bridge damage using transmissibility of the vehicle response in a vehicle-bridge coupled system. [PERSON] et al.