## 3 Simulation Results and Empirical Results Verifications
### Simulation Results
The results for \(Bl(\mathbf{x})\), \(K_{m}(\mathbf{x})\), \(L_{e}(\mathbf{x})\), and \(R_{m}(\mathbf{x})\) indicated high consistency between the predicted and exact curves that were obtained using the method proposed in this study. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameters (Unit)**} & \multicolumn{2}{c}{**Values**} \\ \cline{2-4}  & **Case1** & **Case2** & **Case3** \\ \hline \(M_{m}\) (kg) & \(0.049\times 10^{-3}\) & \(0.003\) & \(0.065\times 10^{-6}\) \\ \(R_{e}\) (Ohm) (fixed) & \(4.560\) & \(4.560\) & \(4.560\) \\ \(Bl(\mathbf{x})\) (N/A) & \(0.391\exp\left(\frac{-(x-0.017)^{2}}{0.319^{2}}\right)\) & \(0.813\exp\left(\frac{-(x-0.015)^{2}}{1.975}\right)\) & \(0.0271\exp\left(\frac{-(x-0.005)^{2}}{0.064^{2}}\right)\) \\ \(R_{m}(\mathbf{x})\) (kg/s) & \(0.079\exp\left(\frac{(x-0.012)^{2}}{0.224^{2}}\right)\) & \(6.652\exp\left(\frac{(x-0.011)^{2}}{2.014^{2}}\right)\) & \(0.222\exp\left(\frac{(x-0.004-1)^{2}}{0.067^{2}}\right)\) \\ \(L_{e}(\mathbf{x})\) (H) & \(0.010\times 10^{-3}\exp\left(\frac{-(x-0.017)^{2}}{0.57^{2}}\right)\) & \(0.009\times 10^{-2}\exp\left(\frac{-(x-0.015)^{2}}{1.418^{2}}\right)\) & \(0.003\times 10^{-3}\exp\left(\frac{-(x-0.005\times 10^{-1})^{2}}{0.047^{2}}\right)\) \\ \(K_{m}(\mathbf{x})\) (N/m) & \(1010.218\exp\left(\frac{(x-0.010)^{2}}{0.87^{2}}\right)\) & \(4501.962\exp\left(\frac{(x-0.009)^{2}}{7.887^{2}}\right)\) & \(300.0654\exp\left(\frac{(x-0.003\times 10^{-1})^{2}}{0.262^{2}}\right)\) \\ \hline \hline \end{tabular} Note: \(R_{e}\) which is measured by electrical meter.  
\end{table} 
Table 2: Initial guesses of linear and nonlinear parameters. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{**Case**} & \multirow{3}{*}{**Iteration Numbers**} & \multirow{3}{*}{**Minimal Value of J**} & \multicolumn{6}{c}{**Average Error for Each Parameters**} \\ \cline{3-8}  & & & & & \(\sum\limits_{\mathbf{x}=-\mathbf{x}_{sim}}^{\mathbf{X}_{sim}}\left(1\text{\emph{ Exact}}-\emph{predict}1\right)\) (\%)** & \\ \cline{3-8}  & & & & \(M_{m}\) & \(Bl(\mathbf{x})\) & \(R_{m}(\mathbf{x})\) & \(L_{e}(\mathbf{x})\) & \(K_{m}(\mathbf{x})\) \\ \hline
1 & 51 & \(7.437\times 10^{-9}\) & \(4.616\times 10^{-3}\) & \(6.851\times 10^{-3}\) & \(7.878\times 10^{-3}\) & \(1.300\times 10^{-2}\) & \(2.594\times 10^{-3}\) \\
2 & 143 & \(5.231\times 10^{-11}\) & \(9.202\times 10^{-4}\) & \(1.035\times 10^{-3}\) & \(8.259\times 10^{-4}\) & \(2.765\times 10^{-2}\) & \(7.447\times 10^{-4}\) \\
3 & 130 & \(8.175\times 10^{-8}\) & \(1.966\times 10^{-3}\) & \(4.954\times 10^{-3}\) & \(2.195\times 10^{-2}\) & \(9.085\times 10^{-4}\) & \(1.105\times 10^{-2}\) \\ \hline \hline \end{tabular} Note: Mm is the linear parameter which do not effect by displacement.  
\end{table} 
Table 3: Convergence results of linear and nonlinear parameters. As a result, in Figure 3a-d, nonlinear parameters can be accurately determined in simulation if the error is not assumed. In the practical measurement, the effect factors, such as the measurement environment, microphone, front and rear stage amplifier, A/D and D/A converter, and nonlinear distortion of the transducer itself may produce inevitable noise, which would cause measurement errors in the measured displacement from the vibration of the transducer membrane. To approximate the practical measurement in the simulation study, the equation for simulating the condition with standard deviation \(\sigma\) of the practical measurement is defined, as follows:  
\[\varpi=\frac{1}{\sigma\sqrt{2\pi}}\int_{0}^{\mathbf{x}_{sim}(t)}e^{-0S[(S- \mathbf{x}_{max}(t))/\sigma]^{2}_{t}}d\xi \tag{25}\]  
where \(\varpi(=0\sim 1)\) is a random number which is substituted into Equation (25) to obtain \(\mathbf{x}_{sim}(t)\), which has a slight deviation from \(\mathbf{x}_{mean}(t)\). The different values of \(\sigma\) may be considered to simulate the potential range of measurement error. Figure 4 shows the comparison of the displacement values with different standard deviations (\(\sigma\) = 0.01, 0.05, 0.11, 0.15).