# Beyond Correlation: Mutual Information to Detect Damage in Nonlinear Systems
## 2 Damage Detection Using Transmissibility Coherence  
Measuring input excitation to actual civil engineering structures is notoriously difficult, if not impossible. This difficulty has motivated the development of \"output-only\" system identification methods that are based only on the response of the structure. The extraction of modal parameters under ambient/operational loading is known as operational modal analysis [22], and it has several variants that operate in the time, frequency, or wavelet domains [23; 24; 25]. Although the input excitation is not measured in operational modal analysis, certain assumptions (e.g., stationary, uncorrelated, and white noise with uniform spatial distribution) are made regarding the nature of the ambient or operational excitations. Research in wind- and human-induced load modeling [26; 27; 28; 29; 30; 31] has raised questions about the validity of these assumptions. Transmissibility-based operational modal analysis [32] is an appealing way to reduce the effect of colored excitation on the estimated modal quantities. Transmissibility coherence, defined as the magnitude-squared coherence between a pair of response signals, has shown some promise as a damage-sensitive feature when used with data from numerical and experimental simulations of a simple shear frame [33]. A loss of transmissibility coherence can indicate damage. [PERSON] et al. [33] define two damage indices based on the transmissibility coherence and show that both indices can successfully detect nonlinear behavior induced in an initially linear system. However, these indices are not sensitive to mass or stiffness changes when the structure remains linear. Following [PERSON] et al. [33], transmissibility coherence and related quantities are calculated as follows. Suppose that a harmonic force is applied at a given point \(l\) in a  
Figure 1: Flowchart for detecting damage using the MI-based metric. multi-degree-of-freedom system, and \(x_{i}(t)\) and \(x_{j}(t)\) are the measured responses at point \(i\) and a reference point \(j\), respectively. The transmissibility between point \(i\) and reference point \(j\) is defined as follows:  
\[T(i,j)=\frac{X_{i}(\omega)}{X_{j}(\omega)}, \tag{1}\]  
where \(X_{i}(\omega)\) and \(X_{j}(\omega)\) are the complex amplitudes of the system responses, and \(\omega\) is the frequency. Transmissibility can also be expressed in terms of the frequency response function \((H)\) as follows:  
\[T(i,j)=\frac{X_{i}(\omega)}{\frac{F_{l}(\omega)}{X_{j}(\omega)}}=\frac{H_{il} (\omega)}{H_{jl}(\omega)}. \tag{2}\]  
As this definition requires knowledge of the excitation, a method to estimate transmissibility directly from two responses is desired. This leads to the concept of transmissibility coherence, which measures the similarity between two responses. Transmissibility coherence is computed as follows:  
\[TC_{ij}(\omega)=\frac{|S_{ij}(\omega)|^{2}}{S_{ij}(\omega)S_{jj}(\omega)}, \tag{3}\]  
where \(S_{i,j}(\omega)\) and \(S_{j,j}(\omega)\) are the auto-spectrum of \(x_{i}(t)\) and \(x_{j}(t)\), respectively, and \(S_{ij}(\omega)\) is the cross-spectrum. Transmissibility coherence can be used as a damage indicator based on the premise that damage induces a loss of coherence compared to the undamaged state. Specifically, [PERSON] et al. (2013) define two damage indices based on transmissibility coherence. The first index, called Accumulation of Transmissibility Coherence (ATC) is defined as  
\[ATC(i,j)=\int_{f_{\min}}^{f_{\max}}TC_{ij}(\,f)\,df, \tag{4}\]  
where \(f_{\min}\) and \(f_{\max}\) define the frequency interval of interest. A similar metric was previously defined by [PERSON] et al. (2014). The second index is called Transmissibility Modal Assurance Criterion (TMAC) and is calculated using the Modal Assurance Criterion (2015) between the transmissibility coherence in the undamaged and damaged states of the structure, each expressed as a column vector in the frequency domain.