## 2 Methodology for Estimation of Damping on Bridges Based on Dynamic Tests under Railway Traffic
### Prony Method Analysis for Damping Estimation  
The Prony method analysis directly estimates the parameters of the free vibration data, \(y(t)\), by fitting, in the time domain, a sum of damped sinusoids to evenly spaced sample values of the form given by [34]:  
\[\hat{y}(t)=\sum_{i=1}^{N}A_{i}\cdot e^{-2\pi f_{i}\cdot\hat{x}_{i}\cdot t}\cdot \cos(2\pi f_{i}\cdot t+\phi_{i}) \tag{1}\]  
where \(\hat{y}(t)\) is the estimate of the free vibration data \(y(t)\) consisting of \(L\) samples evenly spaced; \(y(t)=[y(t_{1}),y(t_{2}),\ldots,y(t_{k})]\) in which \(k=0,\ 1,\ 2,\ 3,\ \ldots,\ L-1\); \(N\) is the total number of damped exponential sinusoids; \(A_{i}\), \(f_{i}\), \(\xi_{i}\), and \(\phi_{i}\) are the amplitude, frequency, damping coefficient, and phase angle, respectively, of the \(i\)th sinusoid. The methodology to perform the Prony method analysis is based on the MATLAB's Signal Processing Toolbox [46] function called prony. This function uses the Prony method analysis for an Infinite Impulse Response (IIR) digital filter for time-domain responses [47] to model a signal using a specified number of poles and zeros for a given data sequence and numerator and denominator orders. This methodology uses a variation of the covariance method of autoregressive modelling to find the denominator coefficients \(a\) and then to find the numerator coefficients \(b\) for which the resulting filter's impulse response matches exactly the first \(L-1\) samples of \(y(t_{k})\)[46]. The coefficients can be found using the transfer function of an IIR that is given by [47]:  
\[H(z)=\frac{B(z)}{A(z)}=\frac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\ldots+b_{q}z^{-q}}{ 1+a_{1}z^{-1}+a_{2}z^{-2}+\ldots+a_{p}z^{-p}}=\frac{\sum_{k=0}^{k=q}b_{k}z^{-k} }{\sum_{l=1}^{l=p}a_{l}z^{-l}} \tag{2}\]  
where \(q\) and \(p\) are the numerator and denominator orders of the transfer function, respectively; \(b_{k}\) and \(a_{k}\) are the numerator and denominator coefficients of the transfer function, respectively; and \(H(z)\) is the Z-transform of the impulse response \(h(n)\) given by  
\[H(z)=\sum_{n=-\infty}^{\infty}h(n)z^{-n} \tag{3}\]  
where \(z\) is a complex number and \(n\) is the samples of the impulse response function. More details about the calculation of the coefficients of the transfer function and about IIR digital filters can be found in Parks and Burrus [47]. The estimation of the amplitude, frequency, damping and phase angle of each sinusoid is obtained using the residuez function from the MATLAB Signal Processing Toolbox [46]. This function converts transfer functions to the partial fraction expansion form using the Z-transform, thus obtaining the residues \(r\), the poles \(p\), and the direct terms \(k\). These terms are obtained considering the coefficients \(a\) and \(b\) from the transfer functions shown above. Each transfer function has a corresponding partial fraction expansion representation, expressed as [46]:  
\[\frac{b(z)}{a(z)}=\frac{r(1)}{1-p(1)z^{-1}}+\ldots+\frac{r(n)}{1-p(n)z^{-1}}+ k(1)+k(2)z^{-1}+\ldots+k(m-n+1)z^{-(m-n)} \tag{4}\]  
where \(n\) and \(m\) are the numerator and denominator orders of the transfer function, respectively. The partial fraction expansion arises in signal processing as one method of finding the inverse Z-transform of a transfer function. With the residues and poles, \(r\) and \(p\), respectively, it is possible to calculate the parameters of each complex exponential sinusoid estimate [48]. These two parameters are in the form of a complex value \(|a+bi|\).