# Analysis of the Effect of Temperature on the Sound Transmission Loss of a Curved Plate
## 3 Model Validation  
The finite element method was used to calculate the modal frequencies and STL of the curved plate and verify the accuracy of the model. Two different temperature conditions were used, assuming that the temperatures on the incident and radiating sides of the curved plate are \(T_{1}\) = 0 \({}^{\circ}\)C, \(T_{2}\) = 0 \({}^{\circ}\)C and \(T_{1}\) = \(-\)10 \({}^{\circ}\)C, \(T_{2}\) = 0 \({}^{\circ}\)C, respectively. The material parameters, dimensions, and thermodynamic parameters of the curved plate and the parameters of the air medium are listed in Table 1[33]. Because the temperature range considered in this study was not large, the negligible variation in the material parameters was not considered. The modal frequency was solved using ANSYS, and the mode with thermal stress was used for the calculation. A simply supported boundary was set as the boundary condition. The modal calculation result file \"rst\" of ANSYS was imported into the acoustic calculation software ACTRAN to calculate the STL. The simulation model in ACTRAN is shown in Figure 2. On the radiation side, a three-dimensional acoustic cavity mesh was created to simulate the air domain of acoustic wave propagation. Moreover, the outside of the cavity was given infinite element boundary properties to absorb acoustic waves and calculate the radiated sound power. According to the 1/6[34] wavelength principle, the maximum length of the acoustic cavity unit was 20 mm, which corresponded to the calculation range of 2000 Hz. The calculation frequency range for the STL was set to 20-1000 Hz, the calculation step was 5 Hz, and it took 10 h 17 min for the computer to perform one calculation. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Symbols and Values** \\ \hline Opening angle & \(\theta_{0}\) = 60\({}^{\circ}\) \\ Radius of curvature & \(R_{0}\) = 1 m \\ Axial length & \(L_{0}\) = 1.5 m \\ Thickness & \(h\) = 0.01 m \\ Young’s modulus & \(E\) = 71 GPa \\ Poisson’s ratio & \(\
u\) = 0.33 \\ Density & \(\rho\) = 2700 kg/m\({}^{3}\) \\ Specific heat & \(C\) = 880 J (kg\(\cdot\)K) \\ Thermal expansion coefficient & \(\kappa\) = 236 W/(m\(\cdot\)K) \\ Coefficient of thermal conductivity & \(\alpha\) = 2.3 \(\times\) 10\({}^{-5}\) m/(m\(\cdot\)K) \\ Air density & \(\rho\) = 1.225 kg/m\({}^{3}\) \\ Speed of sound & \(c_{0}\) = 340 m/s \\ \hline \end{tabular}
\end{table} 
Table 1: Structural dimensions and material parameters of the curved plate. The theoretical calculations were performed using the modal superposition method, which satisfied a sufficient number of terms and ensured the convergence of the numerical predictions. The existing results show that if a term is found and the level solution converges within the target frequency, then the level solution of this term converges below the target frequency [35]. In this calculation, \(f\) = 10,000 Hz was chosen to check for convergence. For the following analysis, the angle of incidence was set to \(0^{\circ}\). In the traversal calculation term, the values of \(m\) and \(n\) were always kept equal. The results are shown in Figure 3. When the number of terms is 20, the result of the series stabilizes, and the sum result of the series can converge as the number of terms continues to increase. This shows that the convergence of the theoretical results can be guaranteed when both \(m\) and \(n\) take values greater than 20. To simplify the calculation and ensure convergence, in this study, we set \(m=n=50\), and the partial terms in Equation (24) were summed separately for 2500 terms. The calculation quantities and computation times were coherent. The results of the theoretical calculation and simulation of the modal frequency of the curved plate are shown in Figure 4. The theoretical and simulation results of the STL are  
Figure 3: Level convergence verification. Figure 2: Simulation model for STL. shown in Figure 5. In particular, the acoustic wave incidence was random.