## 1 Introduction  
As a representative fluid-transporting system, fluid-conveying pipes play an essential role in water conservancy and hydropower, the petrochemical industry, aerospace, and municipal drainage. The change of fluid pressure and flow velocity will cause pipe vibration and deformation. The vibration and deformation of pipe will in turn affect the internal fluid flow, which is named fluid-structure interaction (FSI). For a fluid-conveying pipe system, flow-induced vibration response is mainly a result of flow velocity pulsation and mechanical vibration caused by the pipeline pump during operation [1]. The flow velocity pulsation is caused by the interstitial pressure of the pipeline pump, and it is the main reason for pipe vibration. The mechanical vibration is caused by the vibration of the pipeline pump itself, which is attached on the pipe to affect the pipe vibration. Therefore, development of a reasonable FSI model can guide the analysis of flow-induced vibration and also help in actual pipes. With the rapid development of technology and the continuous improvement of calculation methods, more and more researchers use simulation software to analyze the vibration, stress, and strain of fluid-conveying pipes. Several researchers [2; 3; 4; 5; 6] investigated the influence of the FSI on radial forces, pressure distribution, equivalent stress, and rotor deflection in a pipeline pump. [PERSON] [7] studied the influence of pulsation frequency on the vibration characteristics by simulating a hydraulic flexible pipe. Simulation results showedthat the vibration amplitude and frequency of the pipe increased as pulsation frequency increased, causing the natural frequency and the support stiffness of the pipe to decrease. [PERSON] [8] simulated an L-shaped hydraulic bellow to study the vibration characteristics of the pipe when inlet pressure is a continuous pulse. The results showed that the L-shaped hydraulic bellow expanded and stretched because of fluid pulsation, which caused the pipe to vibrate violently. [PERSON] et al. [9] investigated the effects of flow velocity on the vibration responses of flexible marine pipes. They found that the internal flow had little influence on vibration frequency and amplitude of flexible pipe when its velocity was relatively small. [PERSON] et al. [10] simulated a fluid-conveying pipe and took into consideration friction coupling and Poisson coupling. It was found that the stress wave in the pipe wall and the pressure wave in the water were generated simultaneously. In addition to the above-mentioned theories and simulation results, some researchers attempted to measure the pressure fluctuation of the internal fluid using piezoelectric sensors. [PERSON] et al. [11] used a flexible aluminum nitride (AIN) film sensor to measure pressure fluctuation in pipe systems and vibration of pipe walls. [PERSON] et al. [12] developed a flexible lead zirconate titanate (PZT) diaphragm sensor and investigated its performance in measuring fluctuating pressures in the low-frequency range. The experimental results showed that the sensor was highly sensitive to low and quasi-static pressure load. [PERSON] et al. [13] reported that a polyvinylidene fluoride (PVDF) sensor located on a flexible side branch of a fluid-filled pipe can effectively monitor pressure in the main pipe. [PERSON] et al. [14] validated through experiments that PVDF patches are able to effectively monitor and detect leakage in metal pipes. The authors also presented a numerical model that includes FSI, which estimates the vibrations and dynamic strains on a fluid-carrying pipe as well as the impact of leaks [15]. Although these researchers have used different piezoelectric sensors to measure pressure fluctuation in pipe, no measurement principles have been given to clearly describe the relationship between detected signals of sensors and internal fluid pulsation. Recently, we carried out a study on the indirect measurement of the pressure fluctuation of internal fluid using piezoelectric PVDF sensors.