## 4 Experimental Validation
### Experimental Results
#### 4.3.2 Modal Frequencies and Loss Factors of Sandwich Specimens  
The integral method proposed by [PERSON] (2000), which can be used to process linear and nonlinear response data, is employed to calculate modal loss factors by the narrow band experimental data. Tested modal frequencies from the second to the sixth modes of the short asymmetrical sandwich specimen and modal frequencies of its base beams are listed in Table 1. Tested modal frequencies from  
Figure 9: Normalized frequency responses of sandwich beam specimens by narrow band excitation: (**a**) short asymmetrical specimen (**b**) long asymmetrical specimen and (**c**) long symmetrical specimen.  
the second to the seventh modes of the long asymmetrical and symmetrical sandwich beam specimens and modal frequencies of their base beams are listed in Tables 2 and 3 respectively. Tested structure loss factors of three beam specimens are diagramed in Figure 10. As listed in the tables, the modal frequencies over 1 kHz are obtained by the frequency resolution of 1 Hz to reduce the FFT average time, meanwhile higher frequencies resolutions are set at lower modal frequencies.  
As illustrated in Figure 10, the modal loss factors of the short asymmetrical beam are 0.076, 0.044, 0.030, 0.022, 0.020, accordingly. The tested modal loss factors of the long asymmetrical sandwich specimen are 0.113, 0.081, 0.054, 0.046, 0.036, 0.034, accordingly. The tested modal loss factors of the long symmetrical sandwich specimen are 0.077, 0.046, 0.036, 0.025, 0.0234, 0.0209, accordingly. The modal structure loss factors of three beam specimens are mostly in the range of 0.01-0.1 which is easy to be determined from the response at resonance. It can also be observed that the structure loss factors all decrease with frequency from the second mode.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Beam Type**} & \multirow{2}{*}{**Materials**} & \multirow{2}{*}{**Thickness (mm)**} & \multicolumn{6}{c}{**Modal Frequency (Hz)**} \\ \cline{3-8}  & & & **2** & **3** & **4** & **5** & **6** \\ \hline Base 1 & Steel & 0.90 & 143.55 & 402.80 & 792.30 & 1311.0 & 1957.0 \\ Base 2 & Steel & 1.02 & 164.50 & 459.50 & 900.30 & 1488.0 & 2221.0 \\ Sandwich & — & 2.42 & 161.31 & 434.50 & 842.50 & 1388.0 & 2068.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Tested modal frequencies of the short asymmetrical sandwich beam specimen.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Beam Type**} & \multirow{2}{*}{**Materials**} & \multirow{2}{*}{**Thickness (mm)**} & \multicolumn{6}{c}{**Modal Frequency (Hz)**} \\ \cline{3-8}  & & & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline Base 1 & Steel & 0.97 & 87.63 & 246.06 & 482.80 & 798.50 & 1193.0 & 1668.0 \\ Base 2 & Aluminium & 0.90 & 76.13 & 219.00 & 432.00 & 717.60 & 1078.5 & 1512.0 \\ Sandwich & — & 2.37 & 92.94 & 245.00 & 469.00 & 768.50 & 1145.0 & 1597.0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Tested modal frequencies of the long symmetrical sandwich beam specimen.  
Figure 10: Test results of structure loss factors of three sandwich beam specimens.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Beam Type**} & \multirow{2}{*}{**Materials**} & \multirow{2}{*}{**Thickness (mm)**} & \multicolumn{6}{c}{**Modal Frequency (Hz)**} \\ \cline{3-8}  & & & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline Base 1 & Steel & 0.97 & 87.63 & 246.06 & 482.80 & 798.50 & 1193.0 & 1668.0 \\ Base 2 & Aluminium & 0.90 & 76.13 & 219.00 & 432.00 & 717.60 & 1078.5 & 1512.0 \\ Sandwich & — & 2.37 & 92.94 & 245.00 & 469.00 & 768.50 & 1145.0 & 1597.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Tested modal frequencies of the long asymmetrical sandwich beam specimen.