# Analysis of Dynamic Parameters of a Railway Bridge
## 5 Results of the Analysis of the Vibration Accelerations Model  
Eight arrays of data of measurements of vibration accelerations were received using accelerometer 8344. Signals were captured at the following time intervals: \(\tau_{\Delta}=0.0039\) s and \(\tau_{\Delta}=0.00097\) s 64 s and 16 s. \(n=16,387\) vibration signal acceleration values were fitted in each vector of the array. The expression of each measurement vector is a random function for random measurement errors, when, by bringing its expression close to the shape of a stationary function, trend components, which were calculated in application of the method of the least squares, were eliminated. Values of the quantization interval of the normalized covariance functions range from 1 to \(n/2=8000\). Estimate \(K^{\prime}_{\varphi}(\tau)\) of the normalized auto-covariance function \(K^{\prime}_{\varphi}(\tau)\) was calculated for each vector of vibrations and graphical expressions of 8 normalized auto-covariance functions were received. Estimates of normalized inter-covariance functions \(K^{\prime}_{\varphi}(\tau)\) were calculated in the same manner according to 8 vectors for both vibrations, and 28 graphical expressions of them were received according to their respective combinations. Figure 14 presents a graphical image of the summarized (spatial) correlation matrix of the array of 8 vectors of vibration accelerations. The correlation matrix expression gets a block shape of 8 pyramids, where correlation coefficient values are shown in colour spectrum shades. Colour projections of pyramids are illustrated in the horizontal plane. Figure 11: Normalized inter-covariance function of the 3 th and 4 th vibration accelerations vectors. Figure 10: Normalized inter-covariance function of the 2 nd and 5 th vibration accelerations vectors. Figure 9: Normalized inter-covariance function of the 1 st and 6 th vibration accelerations vectors. Figure 14: Graphical image of summarized (spatial) correlation matrix of the array of 8 vectors of vibration accelerations. Figure 12: Normalized inter-covariance function of the 5 th and 6 th vibration accelerations vectors. Figure 13: Normalized inter-covariance function of the 7 th and 8 th vibration accelerations vectors. Analysis of Figures 6-13 made it possible to determine the correlation values between the examined vectors (from Table 1). The results shown in Figure 6 revealed (1 st vibration accelerations vector) that the correlation coefficient value varies \(\pm 0.4\) where \(k\to 0\); accordingly, at \(k\to 1000\), the correlation coefficient value changes \(\pm 0.2\) and with increasing \(k\to 8000\), the correlation coefficient value is \(r\to 0\). The results of Figure 7 showed (7 th vibration accelerations vector) that the correlation coefficient value changes from \(+0.53\) to \(-0.44\)\(k\to 0\); accordingly, with increasing \(k\to 8000\), the correlation coefficient value is \(k\to\pm 0.2\). The assessment of the results of Figure 8 revealed (1 st and 2 nd vibration accelerations vectors) that the correlation coefficient value ranges from \(+0.21\) to \(-0.24\) at \(k\to 0\), and accordingly, at \(k\to 1000\) the correlation coefficient value varies \(\pm 0.16\) and with increasing \(k\to 8000\), the value is \(r\to 0\). An assessment of the results of Figure 9 revealed (1 st and 6 th vibration accelerations vectors) that the correlation coefficient value changes \(\pm 0.07\) at \(k\to 0\), and with increasing \(k\to 8000\), the correlation coefficient value \(r\to\pm 0.01\). The assessment of the results of Figure 10 showed (2 nd and 5 th vibration accelerations vectors) that the correlation coefficient value ranges from \(+0.06\) to \(-0.05\), when \(k\to 0\), and with respectively increasing \(k\to 8000\), the correlation coefficient value \(r\to\pm 0.005\).