2.1.1. Frequencies and Mode Shapes

Frequencies and mode shapes are the most frequently used modal parameters in SHM methods. When a structure is damaged, its frequencies usually drop accordingly. In particular, the lower frequencies drop slightly while higher frequencies drop a little bit significantly [16]. In addition, temperature change also affects the natural frequency of the structure. The importance of temperature effects for a damage detection method based on relative frequency shift of several weak-axis bending vibration modes of beam-like structures was investigated by Gillich et al. [17]. However, frequencies alone are not usually adopted to identify the local damage since they are global indicator and they do not contain location information. Therefore, they are usually used together with mode shapes, which contain location information and more sensitive to local damages. The extracted mode

shapes during monitoring were compared to those measured at the undamaged stage, and the damage indices of MAC and COMAC were proposed to locate and evaluate the local damages:

$$\text{MAC}\left(\boldsymbol{\varrho}\_{i}^{\text{u}}, \boldsymbol{\varrho}\_{i}^{\text{d}}\right) = \frac{\left[\left(\boldsymbol{\varrho}\_{i}^{\text{u}}\right)^{\text{T}} \boldsymbol{\varrho}\_{i}^{\text{d}}\right]^{2}}{\left[\left(\boldsymbol{\varrho}\_{i}^{\text{u}}\right)^{\text{T}} \boldsymbol{\varrho}\_{i}^{\text{u}}\right]\left[\left(\boldsymbol{\varrho}\_{i}^{\text{d}}\right)^{\text{T}} \boldsymbol{\varrho}\_{i}^{\text{d}}\right]} \tag{1}$$

$$\text{COMPAC}\left(\boldsymbol{\varrho}\_{i}^{\text{u}}\left(\boldsymbol{x}\_{j}\right), \boldsymbol{\varrho}\_{i}^{\text{d}}\left(\boldsymbol{x}\_{j}\right)\right) = \frac{\left(\boldsymbol{\varrho}\_{i}^{\text{u}}\left(\boldsymbol{x}\_{j}\right)\boldsymbol{\varrho}\_{i}^{\text{d}}\left(\boldsymbol{x}\_{j}\right)\right)^{2}}{\left(\boldsymbol{\varrho}\_{i}^{\text{u}}\left(\boldsymbol{x}\_{j}\right)\right)^{2}\left(\boldsymbol{\varrho}\_{i}^{\text{d}}\left(\boldsymbol{x}\_{j}\right)\right)^{2}}\tag{2}$$

where *ϕ*<sup>u</sup> *i* and *ϕ*d *i* are the *i*th undamaged and damaged mode shape, *xj* is the coordinate of *j*th point. It is observed that when *ϕ*d *i* exactly matches *ϕ*<sup>u</sup> *i* , the MAC value should be 1, hence a MAC value close to 1 indicates that the structure is still in good condition, but a MAC value greatly less than 1 means that the structure is damaged. Compared to MAC, COMAC has location information, the COMAC value at *xj* close to 1 indicates that the structure is still intact at *xj* and the COMAC value at *xj* greatly less than 1 means that the structure has damage at *xj*.

A lot of research [18–24] has been conducted to identify local damages by using both frequencies and mode shapes since they contain both global and local information of structures, and some improvements have also been proposed so that they can be applied successfully in practice. One direction to improve is to construct the baseline of structure mode shapes more accurately. Finite element (FE) model updating has been widely used for this purpose [25–27]. Conventional FE model updating was constructed for regenerating of baseline of frequencies and mode shapes. The frequencies and mode shapes obtained by FE model were compared to those measured by monitoring system to check the existence of local damages in the building. Then the stiffness matrix (usually, the mass matrix is not included) can be updated so that the updated frequencies and mode shapes can match the measured ones. Finally, the location and severity (stiffness loss) can be obtained by the updated FE model. In fact, the FE model updating can be generalized as a constrained optimization problem:

$$\min\_{\mathbf{x}\_k} \left\| \sum\_{i} w\_i (\lambda\_{FE,i}(\mathbf{x}\_k) - \lambda\_i) \right\|\_2^2 \\ \text{s.t.} \ x\_{lk} \le x\_k \le x\_{uk} \tag{3}$$

where *<sup>λ</sup>FE*,*<sup>i</sup>*(*xk*) is the *i*th frequency or mode shape obtained by *FE* model using design parameters *xk*, *λi* is the measured *i*th frequency or mode shape, *wi* in the range of 0 to 1 is the weight factor, *xlk* and *xuk* are the upper and lower bounds on the *k*th design variable. There are several standard procedures to solve this kind of constrained optimization problem.

To reduce iteration times and increase computation efficiency, substructure techniques have been developed [28–35]. It divided the whole structures into several small substructures, each of which was treated independently. Then the substructures were assembled to regenerate the global structure by imposing interface constraints. Weng et al. [28] proposed a new iterative substructuring method, which can accurately obtain the eigen-solutions and eigen-sensitivities of structures. Li et al. [30] proposed a sub-structure damage identification method based on frequency domain dynamic response reconstruction, and verified it numerically and experimentally. Papadimitriou et al. [31] proposed the component mode synthesis technology, which can effectively re-analyze in the generalized coordinate space of the accurate component model calculated by using the reference finite element model and the characteristic interface mode. The substructure techniques are usually more effective than conventional FE model updating method since substructure is more sensitive to local damage. The FE model updating methods including substructure techniques are considered as a typical inverse problem in mathematics, where restraint and optimization algorithm are very important.

Constraint is important since FE model updating is generally ill conditioned due to less measurements than unknown parameters to be determined. Hence, the target function to be optimized should include an additional term leading to a convex error function, and the selection of regularization parameters should be determined by specific structures and experience. The Tikhonov regularization is frequently adopted [36–38], and it is proven effective for a lot of practical scenarios, but the identified size of damage is usually larger than expect.

In addition to conventional optimization algorithms, several advanced optimization algorithms proposed for artificial intelligence and pattern recognition have been adopted in vibration-based SHM approaches, such as genetic algorithm [39–43], artificial neural network [44,45], and particle swarm optimization [46,47] and Artificial bee colony algorithm [48,49], etc. Unlike conventional optimization techniques which require established model to optimize parameters, these advanced ones are model-independent. This is actually very helpful in vibration-based SHM system since the measured and monitored data are usually insufficient and contains significant uncertainties, which brings grea<sup>t</sup> difficulties in convergence when identifying parameters by conventional optimization methods. However, it should also be admitted that these advanced approaches have their own drawbacks, for example, the computation load of genetic algorithm is very high since it is a global optimizer. In fact, for different types of structures, different optimization techniques should be considered due to various degree-of-freedoms; unfortunately, there is no common sense on how to select the optimization algorithms based on the type of structures.

Recently, machine learning methods become more popular due to the quick development of artificial intelligence [50–58]. It can definitely help to improve the reconstruction of structural model, but the model is a data-driven model rather than the physics-based model in FE model updating methods. Generally, the machine learning methods contains three steps, data acquisition, feature extraction, and feature classification, which are also the most important steps in pattern recognition. Frequencies and mode shapes are usually obtained during data acquisition and pre-processing as input of these algorithms. Feature extraction may depend on "model", which means that the features of undamaged "model" and damaged "model" should be labeled artificially during the training process. Then the algorithms are trained by the labeled data to generate the classifier. This is also known as supervised learning, and artificial neural network, convolutional neural network, and supported vector machine, etc., are the most typical ones. Actually, these methods are quite useful in real vibration-based SHM systems since it does not require regeneration of physics-based models of structures; therefore, it has grea<sup>t</sup> potential in the future application. However, it should be noted that there is a huge amount of data to be labeled during the training process, which costs a lot of manpower.

Bayesian methods [59–63] have been proposed and developed to reduce the influence of measurement noise and model errors on identifying local damages, since deterministic methods may fail when the change of frequencies and mode shapes due to damage is concealed by measurement noise or model errors. Bayesian methods use prior information from experiments and experience to construct the posterior probability of uncertainties and identified and evaluated damages accordingly. They are typical probabilistic methods, and they can even help on ill-conditioned inverse problems since they introduce a regularization term by using the probability distributions of uncertainties. However, it is noteworthy that prior information is very important in Bayesian methods, if the prior information is not accurate enough, the damage identification may fail even though the measurement is noiseless and model is perfect. In fact, the Bayesian probability of parameters θ under a given structure response **R** is as follows:

$$p(\boldsymbol{\theta} \mid \mathbf{R}) = \frac{p(\mathbf{R} \mid \boldsymbol{\theta}) p(\boldsymbol{\theta})}{p(\mathbf{R})} \tag{4}$$

where *p*(**R**|θ) is the posterior joint probability distribution of the structure response under the condition of θ, *p*(θ) is the prior probability distribution of θ, *p*(**R**) is a standardized constant, and *p*(θ|**R**) is the posterior joint probability distribution of θ under the condition of **R**. It should also be noted that the integral value of *p*(θ|**R**) equals to 1.

In practice, the sampling data for generating prior distribution is usually sparse which makes the task difficult; therefore, sparse Bayesian learning [64–69] has been proposed to construct parameterized prior which can accurately construct the prior distribution based on sparse data. Several investigations have been conducted to show the feasibility of Bayesian methods on SHM by using frequencies and mode shapes, however most of which used lab-scale experiments and numerical simulations. Further studies are expected to show the applicability of Bayesian methods on real structures.

The other direction to improve is elimination of the dependency of the baseline or the undamaged model of structures. An assumption was proposed for this purpose: the mode shapes and mode shape curvatures are smooth and no sudden change with respect to location can be found for undamaged structures [70–73]. Once the sudden change of mode shapes or mode shape curvatures is observed, it is believed that the local damage occurs there. It is also proven by some studies that the mode shape curvatures are more sensitive than mode shapes on local damages, especially for early-stage damages. However, the extraction of mode shapes from accelerations or displacements is usually polluted by noise, and the mode shape curvatures obtained by central difference on mode shapes are even less accurate, resulting that the sudden change of mode shape curvatures due to local damages are covered by numerical error. Hence, how to improve the accuracy of mode shape curvature during monitoring should be investigated. On the other hand, the machine learning methods may also be independent on "model", which means that the features from undamaged "model" and damaged "model" do not need to be completely labeled, and the algorithms themselves can identify can classify the features. This is what is called "semi-supervised learning" [58,74] and "unsupervised learning" [75,76]. It is attractive but the identification accuracy needs to improve significantly, otherwise the false alarm will be issued unexpectedly and frequently.
