# Nonlinear Suppression of a Dual-Tube Coriolis Mass Flowmeter Based on Synchronization Effect
## 4 Experiment and Analysis
### CMF with Asymmetric Additional Mass  
With the help of the experimental platform, the effect of the asymmetric additional mass on the frequency doubling phenomenon was studied. For the dual-tube CMF used in the experiments, four beams were connected to the base. The asymmetric additional mass can be divided into three types by placement positions: symmetric additional masses (pipe 1 and 3, 2 and 4) were placed only on double beams in the same U tube; symmetric additional masses (pipe 1 and 2, 3 and 4) were placed on the same side of different U tubes, and an additional mass was placed on a beam. In the latter two cases, the CMF  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{**Frequencies (Hz)**} \\ \hline \multirow{4}{*}{Channel 1} & \multirow{4}{*}{\ \begin{tabular}{c} Additional- \\ mass position \\ \end{tabular}  } & \multicolumn{4}{c}{1 g} & \multicolumn{4}{c}{3 g} & \multicolumn{4}{c}{5 g} \\ \cline{2-13}  & & \(\omega_{1}\) & \(2\omega_{1}\) & \(3\omega_{1}\) & \(4\omega_{1}\) & \(\omega_{1}\) & \(2\omega_{1}\) & \(3\omega_{1}\) & \(4\omega_{1}\) & \(\omega_{1}\) & \(2\omega_{1}\) & \(3\omega_{1}\) & \(4\omega_{1}\) \\ \cline{2-13}  & \(l/2\) & \(133.29\) & \(266.57\) & \(399.55\) & - & \(133.19\) & \(266.42\) & \(399.61\) & - & \(133.13\) & \(266.32\) & \(399.55\) & - \\  & \(l\times 3/4\) & \(133.10\) & \(266.16\) & \(399.26\) & - & \(132.81\) & \(265.66\) & \(398.47\) & \(531.32\) & \(132.56\) & \(265.15\) & \(397.72\) & - \\  & \(l\) & \(132.85\) & \(265.69\) & \(398.54\) & - & \(132.15\) & \(264.33\) & \(396.48\) & - & \(131.46\) & \(262.94\) & \(394.43\) & \(525.89\) \\  & \(\mathrm{Ap}\) & \(132.91\) & \(265.85\) & - & - & \(132.34\) & \(264.49\) & \(396.80\) & - & \(131.65\) & \(263.32\) & \(395.00\) & - \\ \hline \multirow{4}{*}{Channel 2} & \multirow{4}{*}{ \begin{tabular}{c} \(l/2\) \\ \(l\times 3/4\) \\ \(l\) \\ \(l\) \\ \end{tabular}  } & \multicolumn{4}{c}{\(1\) g} & \multicolumn{4}{c}{\(3\) g} & \multicolumn{4}{c}{5 g} \\ \cline{2-13}  & & \(\omega_{1}\) & \(2\omega_{1}\) & \(3\omega_{1}\) & \(4\omega_{1}\) & \(\omega_{1}\) & \(2\omega_{1}\) & \(3\omega_{1}\) & \(4\omega_{1}\) & \(\omega_{1}\) & \(2\omega_{1}\) & \(3\omega_{1}\) & \(4\omega_{1}\) \\ \cline{2-13}  & \(l/2\) &may produce a second-order-like motion in the first order, or the CMF was under gradual oscillation. Only the first asymmetric case is discussed here. Figure 11 shows the vibration spectrograms of the beams and base only with additional mass a\({}_{1}=1\) g. From top to bottom, the vibrations at the bottoms of four beams and the base are given, where (a) and (c), (b) and (d) indicate the two beams in the same tube. It can be seen that double beams in the same U tube experience the same vibration, and the vibration of a single beam cannot be obtained at the detection point of the CMF itself. To simplify the picture information, only the vibration information of channels 1, 2, and 7 must be summarised. The vibration spectrograms of the three masses at four positions are shown in Figure 12, where (a), (b), and (c) correspond to the three channels 1, 2, and 7 respectively. Each channel comprises four pictures, corresponding to four positions \(l/2\), \(l\times 3/4\), \(l\), and the actuation position(Aq), each of which contains three additional masses, namely 1, 3, and 5. Their corresponding frequency and amplitude are given in Table 7.  
Figure 12 shows that the suppression effect of the asymmetric additional mass on the nonlinear frequency doubling phenomenon was basically the same as that of the symmetric additional mass. The larger the additional mass and the closer to the base, the better the suppression effect on the nonlinear frequency doubling phenomenon. However, in conjunction with Table 7, for the asymmetric additional mass on the beams, the larger the additional mass, and the farther from the base, the better the suppression effect on the frequency doubling amplitude.