Modal simulation and experiment analysis of coriolis mass flowmeter

: This article describes the simulation and experimental determination of the oscillation modes of a Coriolis Mass Flowmeter (CMF). The authors use the SolidWorks software to create a suitable model which is a double U-type of CMF. Based on SolidWorks software, the authors established the double-U tube model for CMF, adopted ANSYS-Workbench simulation software to make the modal analysis, and the exciting mode and Coriolis force mode of CMF are obtained by simulation. According to the results of modal analysis, the authors made the harmonic response analysis of the Flowmeter, got the displacement amplitude of the U-shaped tube excited by different frequency loads under the exciting force, and the maximum displacement amplitude at the resonant frequency was determined. Next, the authors use a laser vibrometer to perform an experimental modal test on the flowmeter. The result of the simulation of the finite element model was verified by modal test of the flowmeter. Provides an effective reference for CMF modelling and simulation in the future.


Introduction
Coriolis Mass Flowmeter (CMF) [1] is used to measure the mass flow of liquid and gas. The basic measurement principle is based on a flow pipe excited to vibrate in a fundamental mode, usually called excitation mode, at the according resonance frequency. The amplitude of the motion is thereby very small, typically a fraction of a millimetre. Mass flow through the pipe then represents a relative movement to the pipe, on which the Coriolis force acts, resulting in a superimposed vibration of another mode, called a Coriolis mode [2]. The amplitude of the Coriolis mode is hereby proportional to mass flow through the pipe and usually two or more magnitudes smaller as the excitation mode and can be measured through position sensors as a phase shift of the excitation mode.
The sensitive unit is a U-shaped elastic measuring tube, which is a key component of the CMF and has a great influence on the sensitivity and natural frequency of the sensor. Generally, in order to improve the precision of the flowmeter from the sensitive mechanism and facilitate the structural optimisation design of the sensor, it is necessary to grasp the influence of various structural parameters of the U-shaped tube on the performance of the mass flowmeter.
The modal analysis technique is used to determine the vibration characteristics (i.e. natural frequencies and mode shapes) of linear elastic structures. It is the most fundamental of all dynamic analysis types [3]. Harmonic response analysis are used to determine the steady-state response of a linear structure to loads that vary sinusoidally (harmonically) with time, thus enabling you to verify whether or not your designs will successfully overcome resonance, fatigue, and other harmful effects of forced vibrations. To understand the dynamic of mechanical system is of great importance for the creation and improvement of new design as well as for solving the problems associated with the mechanical vibrations of existing structures [4,5]. An effective tool for studying the dynamic properties of the system is the modelling of the dynamic behaviour of structures using the finite element method [6,7]. Verification by modal test results is important for designing a finite element model.

Modal analysis theory
In a linear system, the actual structural system can be discretised into vibrations with n degrees of freedom. The corresponding ones will have n physical coordinates to describe its parametric model [8]. The dynamic equations are: In the case of un-damped free vibration, the solution of the natural frequency and vibration mode of the structure can be solved to the problem of the eigenvalues and eigenvectors in the above (1). The linear equation of motion for free, un-damped vibration is: where the [M] and [K] matrices are constant. Assume harmonic motion: Substituting {x} and {x¨} in the governing equation gives an eigenvalue equation: This equality is satisfied if: • This is an eigenvalue problem which may be solved for up to n roots ω 1 2 , ω 2 2 , …, ω n

Harmonic response analysis theory
In a structural system, any sustained cyclic load will produce a sustained cyclic or harmonic response. Harmonic analysis results are used to determine the steady-state response of a linear structure to loads that vary sinusoidally (harmonically) with time, thus enabling you to verify whether or not your designs will successfully overcome resonance, fatigue, and other harmful effects of forced vibrations. The purpose of the harmonic response analysis is to determine the steady-state response of a known linear structure subjected to a load that changes with harmonics over time. This is a commonly used method of structural dynamics analysis, also called frequency response analysis or sweep analysis. All loads and displacement vary sinusoidally at the same known frequency (although not necessarily in phase): where F 0 is amplitude, φ 0 is phase angle. Assumptions and Restrictions: • All loads and displacements vary sinusoidally at the same known frequency • All loads and displacements, both input and output, are assumed to occur at the same frequency • Calculated displacements are complex if: i. damping is specified ii. applied load is complex

Modal simulation analysis:
Here, SolidWorks software was used to construct the model for the U-shaped elastic tube of CMF. The three-dimensional model is shown in Fig. 1. It is imported into the ANSYS-Workbench software for modal simulation analysis.
In the simulation experiment, we adopted the double-U tube model for CMF which was added with fixed distance plate, without considering the influence of fluid, neglecting the additional mass and external support etc. The tube material was set to 316L stainless steel, the elastic modulus was 206 GPa, the Poisson ratio was 0.3, and the density was 7800 kg/m 3 . We used a zerodisplacement constraint to simulate the boundary conditions of the fixed end of the measuring tube. The grid division was generated into hexahedron mesh automatically by automatic method, which was shown as in Fig. 2. The modal frequencies of the first six orders were obtained by modal analysis under boundary conditions, as shown in Table 1.
CMFs work in two modes: the excitation mode and the Coriolis mode. The basic measurement principle is based on a flow pipe excited to vibrate in a fundamental mode, usually called excitation mode, at the according resonance frequency. Mass flow through the pipe then represents a relative movement to the pipe, on which the Coriolis force acts, resulting in a superimposed vibration of another mode, called a Coriolis mode. Corresponding to the second and sixth modes of the Ansys simulation results, respectively, as shown in Fig. 3a and b. In addition, the excitation unit and the detection unit installed on the U-shaped pipe are made of magnetic steel and coils, and their equivalent mass has a great influence on the working frequency of the measuring pipe. Therefore, the U-tube adopts an additional mass point method to simulate the natural frequency variation measured by the excitation unit and the detection unit. Here, we only discuss the working frequency of the excitation state of the measuring tube, i.e. the second-order natural frequency of the simulation result, as shown in Table 2 and Fig. 4.    Table 2 and Fig. 4, it can be concluded the natural frequency of the measuring tube decreases with the increase of the additional mass. When the additional mass is 35 g, the natural frequency is close to the exciting force of the flowmeter the natural frequency is close to the working frequency under the excitation force of the flowmeter.

Harmonic response analysis:
Assuming that the two ends of the U-shaped tube are rigidly fixed, and a simple harmonic exciting force is applied to the inner side of the tube at the middle position of the two tubes. The exciting force is: The amplitude of excitation force F 0 = 450N, the initial phase φ 0 = 0 rad, and the excitation frequency is the frequency of the corresponding order obtained by modal analysis. We only discuss the excitation mode of the flowmeter here. According to the results of modal analysis and actual working conditions, the frequency range was set to 125∼150 Hz and the solution intervals was 25. The frequency response function curve obtained by solving in the Z-axis direction is shown in Fig. 5. Fig. 5 shows that with the increase of the excitation frequency, the U-shaped tube reaches the maximum amplitude at 134 Hz, and its maximum value is 0.1343 mm.

Laser vibration measurement technology
The Laser Vibrometer (LV) basically uses the Doppler principle to measure velocity at a point where its coherent laser beam is directed to. The reflected laser light is compared with the incident light in an interferometer to give the Doppler-shifted wavelength. This shifted wavelength provides information of surface velocity in the direction of the incident laser beam [9].
LVs significantly extend measurement capabilities with respect to traditional vibration sensors [10]. LV techniques offer the possibility of performing quick and flexible non-intrusive vibration measurements at many points over a structure; the data quality and quantity allows to use them for modal analysis and model updating [11]. It has the accuracy and sensitivity that the general sensor cannot reach and plays an important role in the measurement of weak vibrations or minor changes in the target.

Laser vibration test
There is a Germany Polytec laser vibrometer in our laboratory. The physical quantity measured by this instrument is displacement, and the vibration with an amplitude of 0.1 nm can be measured, and the vibration response signal of the mass flow meter U-shaped tube can be accurately collected.
The flowmeter was placed on the isolation table and the boundary conditions are elastic support. Place 17 test points on one U-tube, as shown in Figs. 6 and 7. Based on the fundamental principle of modal analysis, applying continuous excitation on the U-tube and collecting the responding signals from a single point. Since the excitation coil of the flow meter was in the middle of the two tubes, the exciting point should be at this position. The LV was set to sample at the rate of 1 kHz. The LV measures the response signal of each test point three times, and transmitted it to the controller by the laser head, collected it by the data acquisition card, connects the PC, and recorded the response signal with Labview. When the laser beam was continuously scanned and vibrated on the vibrating U-shaped tube, the vibration signal of the corresponding test point was converted into a corresponding digital voltage signal. The voltage signal was a vibration signal of the test point at different positions and different moments. In order to obtain the corresponding mode shape under the natural frequency vibration, a series of continuous voltage signals were collected and imported into MATLAB. Fig. 8 is the frequency response diagram of the sampled signal after FFT, and Fig. 9 is the displacement amplitude of each test point obtained by fitting.
It can be concluded that the first-order natural frequency measured by the LV is 133.1 Hz, and the maximum displacement amplitude is 0.1335 mm from Figs. 8 and 9, which is basically consistent with the results of the finite element simulation. Here gives the comparison of the simulation and experimental frequency and amplitude of the tube in Table 3.
Due to the objective difference between the finite element model and the actual structure in the material physical and mechanical properties, damping characteristics, size and shape of the structure, constraints etc. the simulation results and measured results are slightly different in the modal frequency and mode shape.

Conclusions
Here, the vibration characteristics of the mass flowmeter were analysed based on the finite element simulation software. The first six natural frequencies and mode shapes obtained; only considered the vibration mode of the flowmeter, that is, the second mode of the simulation result. The U-tube was analysed with a harmonic response analysis in the frequency range of 125-150 Hz. The frequency response and the maximum displacement amplitude at a frequency of 134 Hz were obtained. Next, we used the LV to do a modal experiment, comparing test mode result with simulation mode analysis data, the finite element model is verified. The results obtained from the verified finite element model allow a sufficiently high reliability in assessing the changes of flowmeter frequencies.