Damage detection in structures using frequency response functions ensemble with extended cosine based indicator

Although extensive investigations exist for structural damage detection, which often solely considers the stiffness reduction, or specific damage type like concrete spalling, this study aims to address a general novelty detection procedure for detecting the novelty, or so-called outlier from structural dynamic analysis with applying cosine based indicator in frequency response functions. This study also addresses a short review for the cosine indicator and modal assurance criterion, and their interrelation as well. The eight degree-of-freedom system is considered for the verification, and proves well performance in novelty detection.


Introduction
In-service structures undergo various forms of occasions that can affect their dynamic responses, such as temperature, bolt loosening, concrete spalling and so on. These changes in comparison with the initial structural state cannot be always considered as damages, but might be better called as novelty, or socalled outlier. The detection of these novelties makes structural health monitoring (SHM) of high significance, and also make it arduous since various factors can induce novelty. The accurate detection of damage rather than environmental change shall be afterwards challenging. In contrast to defining a damage-caused stiffness reduction, this study considers all response changes as novelty.
Over the past few years, various inspection techniques have been developed, such as vibration based, impedance based, probability based, and so on. The techniques can involve magnetic particle testing, acoustic testing, guided wave, new sensors like fiber bragg gratting (FBG) and so on. For largescale structures, vibration based techniques still occupy a large portion of applications due to their simple conduction, and effective work [1]. Vibration based techniques can be summarized into two categories: experimental modal analysis (EMA) and operational modal analysis (OMA) [2]. EMA usually applies to the structures in laboratory testing, and measures as much data as possible especially for the excitation and response, later to use frequency response functions (FRFs) and modal parameters such as modal damping, modal frequencies, mode shapes and their extensions like modal strain energy (MSE) in identifying the potential structural damages, or more generally, novelties. OMA behaves as an alternative for EMA by avoiding the measurement of excitation, and tries to achieve the same functions as EMA, which implies further applications for real engineering structures. For OMA, transmissibility, as reference free technique, raised for decades ago, has been applied in several directions, such as damage detection [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], damage quantification [8,10,19], and so on. Review for transmissibility are available in [2,10,12]. Even alternatives can be found for novelty detection in terms of specific damage, like weight change can apply for spalling. It would be desirable to develop a strategy for general detection of novelty. This study tries to use extended cosine based indicator in the detection of novelty. Remaining work is organized as follows: section 2 gives the background of cosine indicator and its relation with modal assurance criterion (MAC) [19], and further applies in the FRFs to extract indicators for novelties; section 3 uses a eight degree-of-freedom system as verification, and section 4 addresses the concluding remarks.

Cosine indicator and modal assurance criterion (MAC)
Consine and MAC have been widely used separately, and recent advancement combined these two indicators, and addressed their kernel agreeing well, and detailed discussion can refer to [19]. For cosine indicator, which can be expressed as: (1) where X, Y mean two vectors, () T means the transpose of ().
Then, MAC can be illustrated as: (2) This relation unveils the interrelation between cosine indicator and MAC, and also explains why MAC can be applied as an objective function [19].

Frequency response functions
For a single loaded linear elastic structural system, the damped vibration can be illustrated with the second order differential equation: where M, C, and K denote the mass, damaping, and stiffness matrices of the structure, respectively. f(t) represents all the possible time dependent loading. x(t) means the responses of the structure.
Then, the FRF can also be expressed as: (4) where Xi, and Fr refer to the frequency spectrum of response xi and excitation fr. FRF can also expressed as the accumulation of discrete modes, parameters are hidden in the FRFs, and therefore possible to extract these modal parameters from FRFs through some algorithms, like expectation maximum algorithm.

Novelty indicator
From the discussion above, novelty indicators can be defined as the extension of cosine indicator [19], where () u , () d denote the value under undamaged and damaged states, respectively. This cosine extension indicator has also been applied in transmissibility for damage detection [19], and it proves to be effective in detecting damages, and can measure the relative damage.

Experimental verification
An eight degree-of-freedom system shown in Figure 1 [20] is taken into consideration for unveiling the feasibility of the proposed approach in detecting the novelty. Detailed information for this experiment can be found in [16]. Bolts, springs, translating masses form the system. Different types of damage are simulated, linear and non-linear. Excitation of the system is driven by the voltage supplied. In this study, only the 14% damage at location 5 is considered, and the excitation is random, and changing voltage with 3V, 4V, 5V and 6V. Each response is recorded with an average of 40 measurements, and hanning window is used [20]. Further information about this experimental data analysis can also refer to [21].

Results analysis
Results are computed and discussed in this section as follows in order to draw out the feasibility of the proposed novelty detection procedure. Figure 2 demonstrates the FRF (9, 2) for the structural system with and without damage under 6V excitation, and one can clearly find the natural frequency shift leftward, which implies the stiffness decrease. Certainly herein, it is possible to use the frequency change to estimate the stiffness reduction, since the change in the stiffness matrix leads to a corresponding change in the eigenvalue.  Figure 2. FRF(9, 2) for the structure with and without damage under 6V excitation. Figure 3 shows the FRF (9, 2) for the structure without damage under 6V, 5V, 4V, and 3V excitation. From this figure, clear differences exist between these FRFs in the light of different voltage excitation. Similar phenomenon can be found in Figure 4 for the case with 14% damage. In theory, FRFs should be stable without change in the existing damage and nonlinearity. Herein, these difference can attribute to small nonlinearities, operational errors, or systematic errors. In general, these factors are all summarized as novelty in this study. Further investigation is indispensable in unveiling the kernel.   , and CI (Q=4) for undamaged and damaged scenarios, under 6V, 5V, 4V and 3V excitation. All novelties are successfully detected, like damaged scenarios for these three CIs, and as Q increases from 1 to 2, and 4, the CI performances become better as the difference of CI between undamaged and damaged scenarios increases. For the novelty induced between different voltage excitations, as the excitation voltage decreases from 6V, 5V, 4V, to 3V, the CIs increase. For different FRFs, the CIs show small variations without changing the general trend.

Conclusions
This study applies the cosine extended indicator in FRFs for detecting the novelties. Compared with an experimental investigation on an eight degree-of-freedom structure, this study demonstrates that the proposed novelty detection procedure performs well in identifying the novelties, regardless of the type of damages. The sensitivity of CI depends on the Q, which relies heavily on the engineering experience.