# Nonlinear Suppression of a Dual-Tube Coriolis Mass Flowmeter Based on Synchronization Effect
## 3 Numerical Solution and Analysis
### Asymmetric Additional Mass  
In Equation (20), the additional mass of two beams on the base and its position are independent variables. Accordingly, the additional mass on only one beam can be studied, and its effect on the coupling system can be observed. In the CMF manufacturing process, some asymmetries are caused by comprehensive errors, such as mold inaccuracy and machine tool machining errors. Those asymmetries occur mainly in the measuring tube itself and in dual measuring tubes, which in the latter include mainly mass asymmetry. If the CMF has been used for a long time, the measuring tube and the pipeline become corroded or worn, which also contributes to the mass asymmetry of dual measuring tubes. Therefore, research on asymmetric additional mass can facilitate actual error analysis and provide a basis for the nondestructive testing of CMF measuring tubes.  
It can be seen from Equation (20) that the same results are obtained if the magnitude and position of an additional mass are equally changed at any of the magnitudes, which is verified by the calculation results, and only beam 1 is discussed here, given that \(m_{\text{o2}}=0\); \(\xi_{2}=0\). Because the masses of double beams are different, two first-order frequencies definitely exist, whereas the CMF is a closed-loop system and outputs only one frequency. Here, for the convenience of discussion, the external force is ignored; i.e., double beams freely vibrate, and the rest of the parameters are kept the same as they are in Section 3.1. The results are summarised in Figure 6.  
Figure 6 shows the spectrograms at four positions, \(l/4\), \(l/2\), \(l\times 3/4\), and \(l\), when additional mass is 30 g (a to d), 70 g (e to h) and 200 g (i to l). It can be seen from left to right in each row that, as the additional mass on a single beam is moved away from the base, the difference in the first-order natural frequency of the two separate beams gradually increased. In other words, the additional mass had a greater effect on the working frequency of the system. The additional mass had some suppression effect on the high-order frequency only when it was away from the base, but errors in low-order frequency were introduced. Note that, as shown in Figure 6g,h,k,l, the first two orders of frequency of the system; i.e., the first-order natural frequency of two beams, produced the beat frequency phenomenon. That phenomenon was related to the magnitude of the additional mass, but the asymmetric additional mass became one of the factors leading to nonlinear frequency doubling for coupled vibration. Also, as shown in Figure 6h,k,l, with the increase in additional mass, the high-order frequency of the system (such as \(\Omega_{3}\) and \(\Omega_{4}\)) was a multiple of the natural frequency of the beam without additional mass, which demonstrates again that the frequency is transferred between beams through the base and further confirms the [PERSON] theory.  
In conclusion, the asymmetric additional mass hindered the suppression of frequency doubling signals, but more burrs were found in the spectrum curve. However, the asymmetric additional mass was analyzed to determine the health of the CMF tubes. Further analysis with reference to Figure 6 provides a basis for the calibration of instruments in the actual industrial field.