# Drone-Based Vibration Monitoring and Assessment of Structures
## 3 System Development
### Method of Damage Identification  
The Dynamic Signature Response (DSR) of a system is a characteristic response of the system and was first introduced by [PERSON] et al. [23] as the B-Spline Impulse Response Function (BIRF) for soil-structure interaction analysis. The BIRF is a time-varying function that is defined between any two points in a mechanical system. It represents the time history of the response of one of the two points due to a unit amplitude B-spline excitation applied to the other point [23; 24; 27; 28]. The B-spline functions are piecewise smooth polynomials of order k and belong to a family of base functions used in data interpolation and approximation. They are defined with respect to a time knot sequence \(t_{n},n=1,2,\ldots\), by the recursive formula,  
\[B_{n}^{k}(t)= \begin{cases}1,&\text{if }t_{n}<t<t_{n+1}\\ 0,&\text{elsewhere}\end{cases} \qquad k=1, \tag{1}\]  
\[B_{n}^{k}(t)=\left(\frac{t-t_{n}}{t_{n+k-1}-t_{n}}\right)B_{n}^{k-1}(t)+ \left(\frac{t_{n+k}-t}{t_{n+k}-t_{n+1}}\right)B_{n+1}^{k-1}(t),\ \ k>1. \tag{2}\]  
The \(B_{n}^{k}\)-spline polynomials are of degree \(k-1\) and have \(k-2\) continuous derivatives. While the BIRFs are computed efficiently in numerical models of physical systems [23; 28; 29], it is almost impossible to measure the BIRFs directly in physical systems due to the difficulty of accurately reproducing the B-spline loading on the physical system. The BIRF between two points of a mechanical system, however, can be extracted from vibration tests that acquire the time history record of a response, \(R\), and excitation, \(f\), at the two points. For example, the fourth-order-based BIRF is computed in a discrete form at a time step \(N\) as:  
\[BIRF_{N}=\left(R_{N}-\sum_{n=1}^{N-2}BIRF_{n}f_{N-n+2}+BIRF_{N-1}(f_{3}-f_{1}) \right)/(f_{2}+2f_{1}), \tag{3}\]  
where subscripts indicate the time step at which the corresponding functions are evaluated. Details of the method along with verification and validation studies were presented in [PERSON] et al. [30; 31]. Structural assessment procedures through dynamic testing focus on identifying changes in the dynamic properties of the system through modal analysis and/or model updating techniques. In Equation (3), the BIRFs are independent of any external excitations and are considered the DSRs that capture the current state of the dynamic system. Any changes in the current state of the structure are captured as changes in its DSRs. Therefore, if the response time history, R, and the corresponding excitation, f, are known through vibration test measurements conducted before and after a damage-inducing event, the difference in the corresponding DSR is an indication of damage [30; 31]. The computation of the BIRF shown in Equation (3) represents in essence a discrete deconvolution of the response signal, and the BIRF can be perceived as an equivalent time domain transfer function of the system that captures all frequencies in the range defined by the sampling time interval. It is noted and had been demonstrated that the analysis of only a short duration segment of the recorded response is adequate for change detection through \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Electropermanent Magnet \#1** & **Electropermanent Magnet \#2** & **Sensor Package** & **UAS** \\ \hline Unmagnetized & Magnetized & Attached to UAS & At delivery \\ Magnetized & Unmagnetized & Attached to structure & Package deployed \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Operation status for Electropermanent Magnets #1 and #2 for each sensor package and UAS position. the DSR algorithm, compared to mainstream methods that require, in most cases, the entire response record from multiple sensors. However, the noise inherent to the time history response records may have to be removed through filtering before the DSR is extracted.