# Nonlinear Suppression of a Dual-Tube Coriolis Mass Flowmeter Based on Synchronization Effect
## 2 Nonlinear Analysis on Coupled Vibration of Dual-Tube Model with Additional Mass  
Frequency is the most fundamental characteristic value in characterizing variations in structural properties, and the frequency of a real engineering structure is the most accessible modal parameter with high measurement accuracy [23,24]. It has been determined through previous theoretical analysis [22] that the nonlinearity invoked by the coupling effect cannot be fundamentally eliminated and is suppressed by changing the wall thickness and the mass and spacing of the measuring tube, among other structural parameters. However, once the mass of the measuring tube changes, the working frequency of the measuring tube inevitably changes. That affects the CMF performance such as measuring range and measuring accuracy and incurs higher manufacturing costs; if the CMF structure size changes, its installation conditions and working environment are affected, bringing inconvenient in use. The additional-mass method is the most commonly used method for nondestructive testing of beam structures. The effects on the key structural parameters of a CMF, such as its mass and size, can be minimized by the additional-mass method, and some correlation exists between additional mass and structural modal parameters. Therefore, a very small additional mass relative to the entire mass has little effect on the original working frequency and measuring range of the CMF. By changing the magnitude and distribution of the additional mass, the law of nonlinearity suppression can be quickly obtained, and the best suppression position can be determined, thus greatly improving the CMF measuring accuracy. Here, the CMF can be simplified to the abstract model shown in Figure 3, where \(z_{1},z_{2}\), and \(z\) are the vibration displacement of the beams in its own coordinate system and system displacement; \(m_{1},m_{2}\), and \(m\) are the masses of the beams and base, respectively; \(F_{1},F_{2}\) are the driving force of the exciting coil on the middle point of the flow tube are the equal harmonic driving force in an opposite direction; \(EI\) is the flexural rigidity of  
Figure 2: Structure diagram of a U-tube Coriolis mass flowmeter. cantilever beam \(i\), and \(\xi_{i}\) indicates the position coordinates of the additional mass \(m_{oi}\) of the coupling system on the beams. The coupled vibration model of the dual-tube model with additional mass was built, and the generalized coordinates were defined as \(q=\left(q_{1i},q_{2i},\ldots,q_{ji},z\right)\). \(z_{i}(x,t)\) is the transverse vibration displacement function of cantilever beam \(i\). According to the modal superposition method, \(z_{i}(x,t)\) can be depicted as the superposition form of each order of modal, and single beam \(i\) has the following form:  
\[z_{i}(x,t)\approx\sum_{j=1}^{n}\phi_{ji}(x)q_{ji}(t) \tag{1}\]  
where \(\phi_{ji}(x)\) is the \(j^{\text{th}}\) order vibration mode of beam \(i\) with additional mass, and \(q_{ji}(t)\) is a generalized coordinate and only time-dependent. \(j\) is the \(j^{\text{th}}\) order modal. Here, the first three orders were chosen for analysis.