# Nonlinear Suppression of a Dual-Tube Coriolis Mass Flowmeter Based on Synchronization Effect
## 3 Numerical Solution and Analysis
### Symmetric Additional Mass
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Additional-**} & \multicolumn{8}{c}{**Frequencies (Hz)**} & \multicolumn{8}{c}{**Amplitudes (dB)**} \\ \cline{2-13} \multicolumn{1}{c}{**Mass Position**} & \multicolumn{3}{c}{**30 g**} & **50 g** & **70 g** & **100 g** & **200 g** & **30 g** & **50 g** & **70 g** & **100 g** & **200 g** \\ \hline \multirow{3}{*}{_I/4_} & \(\omega_{1}\) & 132.57 & 131.76 & 131.40 & 130.88 & 129.20 & A\({}_{1}\) & \(-\)52.62 & \(-\)27.75 & \(-\)34.36 & \(-\)34.77 & \(-\)28.82 \\  & \(2\omega_{1}\) & 265.21 & - & - & - & - & A\({}_{2}\) & \(-\)124.95 & - & - & - & - \\  & \(3\omega_{1}\) & 397.78 & 395.14 & 394.12 & 392.58 & 387.74 & A\({}_{3}\) & \(-\)125.80 & \(-\)104.52 & \(-\)109.48 & \(-\)107.90 & \(-\)106.79 \\ \hline \multirow{3}{*}{_I/2_} & \(\omega_{1}\) & 126.64 & 123.12 & 119.82 & 115.43 & 103.49 & A\({}_{1}\) & \(-\)27.96 & \(-\)34.73 & \(-\)25.68 & \(-\)24.84 & \(-\)25.21 \\  & \(2\omega_{1}\) & 253.27 & 369.07 & 359.47 & 346.14 & 310.55 & \(-\)211.97 & \(-\)111.41 & \(-\)100.41 & \(-\)98.39 & \(-\)105.53 \\  & \(3\omega_{1}\) & 379.91 & - & - & 577.00 & 517.60 & \(\lambda_{3}\) & \(-\)106.00 & - & - & \(-\)98.99 & \(-\)107.75 \\ \hline \multirow{3}{*}{_I_\(\times\)3/4} & \(\omega_{1}\) & 113.67 & 104.74 & 97.71 & 89.28 & 71.85 & A\({}_{1}\) & \(-\)31.80 & \(-\)34.12 & \(-\)32.73 & \(-\)31.81 & \(-\)27.63 \\  & \(2\omega_{1}\) & 227.34 & - & - & - & - & - & A\({}_{2}\) & \(-\)133.98 & - & - & - & - \\  & \(3\omega_{1}\) & 341.02 & 314.28 & 293.04 & 267.99 & - & A\({}_{3}\) & \(-\)111.89 & \(-\)112.65 & \(-\)113.15 & \(-\)108.63 & - \\  & \(4\omega_{1}\) & 454.69 & - & - & - & - & A\({}_{4}\) & \(-\)139.24 & - & - & - & - \\  & \(5\omega_{1}\) & 568.36 & 523.75 & 488.38 & 446.56 & - & A\({}_{5}\) & \(-\)112.00 & \(-\)113.98 & \(-\)113.08 & \(-\)108.53 & - \\ \hline \multirow{3}{*}{_I_} & \(\omega_{1}\) & 97.85 & 85.77 & 77.20 & 68.19 & 51.78 & A\({}_{1}\) & \(-\)31.78 & \(-\)26.24 & \(-\)31.22 & \(-\)28.02 & \(-\)31.22 \\  & \(2\omega_{1}\) & 195.04 & 257.30 & 231.74 & 204.57 & - & A\({}_{2}\) & \(-\)109.85 & \(-\)97.07 & \(-\)105.29 & \(-\)97.89 & - \\  & \(3\omega_{1}\) & 293.55 & 428.91 & 386.13 & 341.02 & 258.91 & A\({}_{3}\) & \(-\)109.07 & \(-\)101.73 & \(-\)106.45 & \(-\)101.23 & \(-\)106.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Frequencies and amplitudes. Figure 5 and Table 2 show that the greater the distance from the base, the better the suppression effect on nonlinear frequency doubling, but the more sensitive to the change in additional mass. At a small mass, low-frequency doubling was obviously suppressed, but high-frequency doubling was not suppressed. At this time, the entire system had a high natural frequency, so high-frequency doubling had a small effect on the measuring accuracy in a low-frequency range. However, the measuring accuracy decreased in a high-frequency range; at a larger mass, the low-frequency doubling was basically eliminated. However, with the decrease in the overall natural frequency of the system, the high-frequency doubling appeared prematurely in a low-frequency range. In that case, the CMF working frequency range was determined by the high-frequency range. Therefore, there were two types of nonlinear frequency doubling suppressions, namely nonlinear low-frequency doubling suppression and nonlinear high-frequency doubling suppression. By rationally designing the additional masses and relative positions, the working performance of the CMF within a certain frequency range can be improved.