# Nonlinear Suppression of a Dual-Tube Coriolis Mass Flowmeter Based on Synchronization Effect
## 1 Introduction  
Resonator sensors are currently one of the best-performing types of sensors worldwide. A traditional resonator sensor is a linear system, but nonlinear interference components exist in the actual output signal. Taking a resonant Coriolis mass flowmeter (CMF) as an example, the excitation signal from the sensor is a single sinusoidal signal, and the detection signal of the sensor exists in the form of a nonlinear signal characterized by a frequency doubling signal [1, 2]. To trace the nonlinear phenomenon of a CME, many studies have analyzed the fluid properties [3], fluid-structure interaction [4, 5], key parameters [6], and installation effects [7; 8] of measuring tubes, nonideal boundary conditions [9; 10; 11], defects resulting from structural inhomogeneity [12; 13], etc. At the same time, this phenomenon has also been observed in microfluidics; [PERSON] et al. [14; 15] took advantage of the optical monitoring of slug flow in microchannels to characterize flow nonlinearity with rich and specific experiments. It is possible to distinguish clearly the air and water passages in the optical signal. The observation shows that there appear to be some frequency doubling signals in the spectrum. However, the fundamental reasons for the nonlinear frequency doubling phenomenon of the absolute perfect CMF under zero flow conditions are rarely reported. Zero flow includes several situations: the fluid in the pipe is low-velocity fluid or a micro flow, the fluid fills the pipe and moves along the pipe at a uniform speed, or the fluid in the pipe is a gas. Synchronization generally takes place in two or more coupled and nonlinear autonomous oscillation systems of similar size. Synchronization is a fundamental theme in nonlinear phenomena and is a currently popular topic of research in the control field [16]. The earliest scientific description of synchronization dates back to 1665. The Dutch physicist [PERSON] found that two different pendulum clocks hanging on the wall swung in opposite directions, and gave the reason for the synchronous swing as being the coupled beams that the two clocks hung from interacted with each other so that the two clocks became fully synchronized after swinging for a period of time. This experimental device is shown in Figure 1[17; 18]. Following in [PERSON] footsteps, experimental research was begun on vibration-coupled pendulums, which were later replaced by modern rhythm clocks and mechanical metromomes. [PERSON] et al. fully replicated [PERSON] pendulum experiment, explored the effect of nonidentical pendulums, and explained synchronous behaviours under different coupling modes [19]. [PERSON] et al. investigated the in-phase and antiphase synchronization phenomena of two coupled mechanical metromomes under random initial conditions [20]. [PERSON] analyzed the synchronization phenomenon of two metromome systems hanging on a cart, based on a description of the function method [21]. In summary, the synchronization problem of two or more same-frequency vibration sources in a system can be explained and solved based on the synchronization phenomenon. The system model is a typical nonlinear model, and the synchronization phenomenon induces nonlinear vibration coupling of the system. Figure 2 is a structure diagram of a typical dual-tube CMF; its sensitive unit is a pair of completely symmetrical measuring tubes. When the measuring tubes are empty, a first-order bending vibration is invoked under the driving force of the exciting coil, and the measuring tube is in first-order resonance. The vibration of this U tube can be deemed as a circular motion around a fixed axis from a certain perspective. In reality, two symmetric sensitive units synchronously vibrate in a reverse direction under the action of the exciting unit, which is regarded as a typical antiphase synchronous motion. Based on synchronization theory and through numerical analysis, it has been confirmed that  
Figure 1: Diagram of [PERSON] experimental device for pendulum synchronization.