# Analysis of Dynamic Parameters of a Railway Bridge
## 2 Modelling Vibration Parameters  
Accelerometers were placed on beams and the vibrations of the beams were measured. Sensors were positioned on the upper plane (i.e., the top view). The sensor layout was selected to display dynamic variations at all significant points. Signals received during measurements were processed using Bruel & Kjaer software (Sampling frequency 400 Hz). The weight of the moving locomotive was 119 t. The tests were conducted under the same environmental conditions, namely, the temperature was 30 \({}^{\circ}\)C and humidity was at 70%. The array of vibration accelerations consists of 8 vectors. Data of vectors 1-4 capture accelerations of the rear steady point 1 of the railway bridge beam at time intervals \(\tau_{k}\to 0.0039\) s during the movement of the locomotive (vectors 1 and 2) and train (vectors 3 and 4). Vectors 1 and 3 show changes in vertical acceleration, while vectors 2 and 4 show changes in longitudinal acceleration. The data of vectors 5-8 show the change of the vertical accelerations of the mid-points of beams 1 and 2 of the railway bridge at time intervals \(\tau_{k}\to 0.00097\) s during the movement of a locomotive (vectors 5 and 6) and a train (vectors 7 and 8) [11]. The study presents a means to calculate of the most reliable trend values of the vibration vector which uses the least squares method. The trend of the vibrations vector is assumed to be a discrete dimension with a fixed value. The use of the least squares method partially eliminates random vibration errors. When processing large-scale measurement data, the method of the least squares also provides asymptotically effective values of the calculated parameters when the distribution of the measurement data is not normal [12]. The array of the measurement of vibration accelerations consists of 8 vectors \(\varphi\) (columns). Each vector is understood as a random function with random measurement errors. In application of the method of least squares, the most reliable trend value, \(\overline{\varphi}\), also called the weighted average, is calculated for each vector \(\varphi\)[13]. Assuming that the vector trend value differs according to the harmonic law, when the forecasted wave length corresponds to the length of the vibration acceleration vector, the parametric equation of a single vector value \(\varphi_{i}\) is expressed as:  
\[\epsilon_{i}=\varphi_{i}-a_{i}\overline{\varphi}, \tag{1}\]  
where \(\epsilon_{i}\) is the random acceleration error, \(\varphi_{i}\) is the acceleration value, and \(\overline{\varphi}\) is the acceleration vector trend. Coefficient \(a_{i}\) is expressed as:  
\[a_{i}=cos\Delta_{i}, \tag{2}\]  
where \(\Delta_{i}=\Delta\cdot i\), \(\Delta=2\pi/n\) is the value of the unit of measure, rad, \(i=1,\ 2,\ \ldots,\ n\). Equation (1) in the expression of matrices is expressed as:  
\[\epsilon=\varphi-A\overline{\varphi}, \tag{3}\]  
where \(\epsilon\) is the vector of random errors, \(\varphi=\left(\varphi_{1},\ \varphi_{2},\ldots,\varphi_{n}\right)^{T}\) is the vector of vibrations accelerations, and \(A=\left(a_{1},\ a_{2},\ldots,a_{n}\right)^{T}\) is the matrix of coefficients of parametric equations \(\left(n\times 1\right)\)[14]. The most reliable trend value of the vibration acceleration vector \(\varphi\) is calculated using the condition of the method of the least squares:  
\[\Phi=\epsilon^{T}P\epsilon=min, \tag{4}\]  
where \(P\) is the diagonal matrix \(\left(n\times n\right)\) of weight \(p_{i}\) of vibration accelerations. Weights of single acceleration values \(\varphi_{i}\) are calculated according to the formula:  
\[p_{i}=\frac{\sigma_{0}^{2}}{\sigma_{\varphi_{i}}^{2}}, \tag{5}\]  
where \(\sigma_{0}\) is the standard deviation of the measurement result \(\varphi_{0}\), the weight of which is assumed to be equal to \(1\ \boldsymbol{p_{0}}=1\). Thus, the value \(\sigma_{0}\) is chosen freely because it has no impact on the calculation results. The measurement result \(\varphi_{0}\) is selected for weights \(p_{i}\) to be close to 1 (to reduce the scope of calculations).