## 2 Measurement of Damping Parameters
### Damping Parameter Identification Test
The vibration response was obtained by the shaker sweep excitation method, and the half-power bandwidth method was used to identify the damping parameter. The frequency response test results of the Fe-12 Cr-3A1 damping alloy specimen 1 at measurement points 1 and 2 are shown in Figures 8 and 9, respectively, where \(f_{n}\) is the resonance frequency of the corresponding order of the system and \(f_{1}\) and \(f_{2}\) are the frequencies of the half-power points, respectively. Comparing the resonance frequencies derived from the modal test and the swept frequency test, as shown in Table 1, the relative error is about 5%, indicating the test results have good accuracy. The damping ratios corresponding to the two orders of the resonance frequency are obtained and averaged according to Equation (2). Figure 8: The frequency response of measuring point 1: (a) first order, (b) second order. Figure 3.0 The measurement points of the sweep response experiment: (a) Measurement point 1 and (b) Measurement point 2. Figure 3.1 The frequency response of measuring point 2: (a) first order, (b) second order. The same test method was used to determine the damping ratios of specimen 2 and specimen 3 and the damping ratio of the Q235 carbon steel specimens, and the test results are summarised in Figure 3.2 The comparison results of the test are shown in Figure 3.3 \"The large deviation in the test results for the Fe-12 Cr-3 Al specimen 2 can be attributed to the homogeneity of the specimen material, the tightness of the specimen mounting, and the quality of the sensor mounting. It should be especially pointed out that the high damping characteristics of Fe-12 Cr-3 Al are realised by its own crystal structure, and a poor crystal structure will greatly affect its damping characteristics, which may lead to a large deviation of the damping ratio, so the homogeneity of the specimen material itself is the main influence on the test error. In this paper, the influence of other factors is reduced by testing several times and then taking the average value as the final reference value. According to the average value of the three specimens' test results, the first-order and second-order damping ratios of Fe-12 Cr-3 Al alloy are 2.75 and 1.42 times higher than those of Q235 carbon steel, respectively. It can be determined that the overall damping characteristic of the Fe-12 Cr-3 Al alloy is significantly better than that of Q235 carbon steel.\" When the excitation amplitude is fixed, the damping parameter of the Fe-12 Cr-3 Al alloy decreases with increasing frequency. The results obtained from the test provide a real and reliable database for the subsequent finite element calculation and vibration reduction analysis. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model Order** & **Measurement** & **Modal** & **Sweep** & **Relative** & **Damping** & **Average** \\  & **Points** & **Test** & **Test** & **Error** & **Ratio** & **Average** \\ \hline \multirow{2}{*}{First order} & Point 1 & 168.8 Hz & 162.5 Hz & 3.73\% & 0.02215 & \\  & Point 2 & 168.8 Hz & 162.0 Hz & 4.03\% & 0.02253 & \\ \hline Second & Point 1 & 1194.0 Hz & 1135 Hz & 4.94\% & 0.00484 & \\ order & Point 2 & 1194.0 Hz & 1134 Hz & 5.02\% & 0.00485 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The resonance frequency and damping ratio of Fe-12 Cr-3 Al. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model Order**} & \multirow{2}{*}{**Material**} & \multicolumn{4}{c}{**Damping Ratio**} \\ \cline{3-6}  & & **Specimen 1** & **Specimen 2** & **Specimen 3** & **Average** \\ \hline \multirow{2}{*}{First order} & Fe-12 Cr-3 Al & 0.02234 & 0.01278 & 0.01848 & 0.01787 \\  & Q235 & 0.00643 & 0.00655 & 0.00653 & 0.00650 \\ \hline \multirow{2}{*}{Second order} & Fe-12 Cr-3 Al & 0.00484 & 0.00517 & 0.00420 & 0.00474 \\  & Q235 & 0.00353 & 0.00334 & 0.00314 & 0.00334 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The damping ratio test results of Fe-12 Cr-3 Al and Q235.