## 3 Numerical Simulations
### The Sparse Measurement Point Arrangement Based on the Theory of ARM  
A. The Results of CMESM  
To illustrate the good convergence of ARM with the increase of order, the attenuation curves of ARM eigenvalues with the increase of the ARM order at different excitation frequencies are shown in Figure 7a. According to Section 2.3, a set of orthogonal basis \(\mathbf{R}\) is presented to give another expression of the equivalent source strengths. To verify that CMESM could enhance the sparsity of the equivalent source strengths, the normalized modal expansion coefficients with CMESM and ESM are shown in Figure 7b. In the simulation, the quantity of equivalent sources is 168. Meanwhile, 135 measurement points are uniformly distributed on the surface of the cylindrical shell in Section 3.1 (axial: 9; circumferential: 15). It could be seen in Figure 7a that the eigenvalues of ARMs decrease rapidly with the increase of ARM order. At the higher excitation frequency, the eigenvalues decrease more slowly. Still, the big losses of ARM eigenvalues indicate that only the first few orders of ARMs contribute to the sound field. In Figure 7b, it could be seen that, compared with ESM, the modal expansion coefficients for CMESM are much sparser. Thus, though equivalent source strengths for the structure may not be sparse enough, the modal expansion coefficients of ARMs could be quite sparse, which allows sparse vibration measurement. Before the sound prediction, the sparsity of the modal expansion coefficient vector should be guaranteed to keep the prediction result accurate. Repeat the simulations in Section 3.1. The quantity of measurement points is reduced to 90 (axial: 6, circumferential: 15), which is far less than in Section 3.1. Define \(\gamma\) as the condition number of the sensing matrix, \(\gamma=\text{cond}(\mathbf{\Psi})\). The variation of \(\gamma\) versus the ARM order is shown in Figure 8. The prediction results are shown in Figure 9. Several values of  
Figure 7: (**a**). Attenuation curves of ARM eigenvalues with the increase of the ARM order at different frequencies (unit: dB); (**b**) comparison of the normalized modal expansion coefficient sparsity between ESM and CMESM at the frequency of 100 Hz. \(\gamma\) at different orders are given in Table 3. According to the study of [PERSON] [30], it is essential to reduce the value of \(\gamma\) for computing ARMs more stably. Otherwise, the poorly conditioned sensing matrix will give rise to large prediction errors. In Figure 8 and Table 3, it could be seen that, with the increase of ARM order, \(\gamma\) increases rapidly. After reaching the maximum, \(\gamma\) begins to decrease. Meanwhile, the values of \(\gamma\) at high orders are still much higher than those for the lower orders. Therefore, it is not appropriate to fix the ARM truncation number as high as possible. If \(\gamma\) is extravagantly large, the sensing matrix needs regularization [32]. In addition, it is not difficult to find that \(\gamma\) reaches the maximum when \(L1=L\). As shown in Figure 9, it could be seen that, with uniform measurement point arrangement, the sound prediction results are not satisfying. Though the prediction error of sound  
Figure 8: The variation of \(\gamma\) with the increase of ARM order when 90 uniform-arranged measurement points are used. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Case\textbackslash{Order}** & **10** & **30** & **50** & **70** & **90** & **110** & **130** & **168** \\ \hline Free Field & 2.28 & 4.55 & 569.70 & 1.17 \(\times\) 10\({}^{3}\) & 1.17 \(\times\) 10\({}^{5}\) & 5.08 \(\times\) 10\({}^{3}\) & 2.51 \(\times\) 10\({}^{3}\) & 1.59 \(\times\) 10\({}^{3}\) \\ Half-Space & 2.52 & 5.12 & 210.89 & 480.98 & 1.06 \(\times\) 10\({}^{4}\) & 1.17 \(\times\) 10\({}^{3}\) & 894.51 & 721.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Different acoustic radiation mode (ARM) truncation numbers in compressed modal equivalent source method (CMESM) for 90 measurement points.