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The vibration characteristics of two piezoelectric cantilevered plates in air and water are investigated in this study. The two-piezoelectric-plate system can provide different vibration characteristics compared to the previous studies in which only one cantilever plate is used because the anti-phase motion can be evoked in this newly designed system. First, the vibration characteristics of the plates coupled with a bounded compressible inviscid fluid are derived by theoretical analysis. The governing equation of the piezoelectric plate and the superposition method are used to analyze the vibration displacement of the piezoelectric plate, and the acoustic wave equation is used to derive the fluid equation. The frequency eigenfunction of the coupled system is derived by the boundary conditions of hydrostatic equilibrium. After solving the characteristics equation, the resonant frequencies and corresponding mode shapes for assumptions about both compressible and incompressible fluids and the velocity and pressure fields of the fluid can be obtained. Next, finite element analysis (FEA) is used to verify the theoretical results. It is shown from this study that the theoretical predictions of resonant frequencies and the corresponding mode shapes have a good agreement with those of the FEA for water and compressible fluid assumption of air. The unique high pressure value in the outer or middle layer in an air-coupled condition does not occur in water because water’s compressibility is relatively smaller than that of air. The influence of the depth and properties of the fluid on the behavior of the coupled solid–fluid is also investigated. For anti-phase motion, the frequency variation with change in the depth of the middle layer is similar to that of the outer layer for both air- and water-coupled conditions. For in-phase motion, the variation in frequency with the change in the depth of the middle layer is negligible. The particular modes in fluid compounded by modes in vacuum are analyzed on the distortion of mode shapes at higher resonant modes. These results can provide complete information on the vibration characteristics of a solid–fluid coupled system and will contribute to the evolution of hydroacoustic devices in which two piezoelectric plates are used.

It is known that a propagating wave at a certain spatial point can be decomposed into plane waves propagating at different angles. In this work, by designing a gradient index phononic crystal lens (GRIN PCL) with transverse-continuous leaf-like unit cells, we theoretically and experimentally show that the spatial-domain propagating waves in finite periodic structures can be linked to their time-domain vibration behaviors. The full-field instantaneous focusing behaviors of Lamb waves in the proposed leaf-like GRIN PCL give an example of the wave-vibration linkage in finite periodic structures while allowing a certain complexity. The conclusion in this paper can help one skip iterative time-consuming finite element analysis (e.g., time-stepping solutions) to avoid possible numerical instabilities occurred in calculating transient wave field on practical finite metamaterials or phononic crystals having unit cells with complicated configurations.

The objective of this study was to explore from a theoretical perspective the relationship between dry mode shapes and the distortions associated with wet mode shapes. This study investigated the free vibration characteristics of a rectangular thin plate partially coupled with fluid. A powerful full-field experimental technique referred to as AF-ESPI was used to measure vibration modes and the corresponding resonant frequencies. The derivation of wet mode shapes from dry mode shapes proved effective in analyzing problems for plate-fluid interactions. Our analysis revealed that every wet mode shape is composed of multiple dry mode shapes with corresponding contribution coefficients. In the proposed analytic approach, the superposition method is used to deal with theoretical aspects of thin plate vibration behavior, and velocity potential is used to express the dynamic behavior of an incompressible flow field in a finite container. The accuracy of the results obtained using the proposed theoretical approach was verified via comparison with numerical results obtained using finite element method (FEM) and experimental measurements. A detailed discussion pertaining to the relationship between wet and dry mode shapes is also presented.

This study combined theoretical, experimental, and numerical analysis to investigate the vibration characteristics of a thin rectangular plate positioned horizontally at the bottom of a rectangular container filled with liquid. Flow field pressure was derived using an equation governing the behavior of incompressible fluids. Analytic solutions to vibrations in a thin plate in air served as the fundamental function of the thin plate coupled with liquid. We then used liquid pressure, and the out-of-plane deflection of the thin plate for the construction of frequency response functions for the analysis of vibration characteristics in the liquid-plate coupling system. Two experimental methods were employed to measure the vibration characteristics of the thin plate immersed in water. The first involved using sensors of polyvinylidene difluoride (PVDF) to measure transient signals of fluid-plate system subjected an impact at the thin plate. These were then converted to the frequency domain in order to obtain the resonant frequencies of the fluid-plate coupling system. The second method was amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), which was used to measure the dynamic characteristics of the thin plate in the flow field. This method was paired with the image processing techniques, temporal speckle pattern interferometry (TSPI) and temporal standard deviation (TSTD), to obtain clear mode shapes of the thin plate and resonant frequencies. Comparison of the results from theoretical analysis, finite element method, and experimental measurements confirmed the accuracy of our theoretical analysis, which was superior to the conventional approach based on beam mode shape functions. The experimental methods proposed in this study can be used to measure the resonant frequencies of underwater thin plates, and clear mode shapes can be obtained using AF-ESPI. Our results indicate that the resonant frequencies of thin plates underwater are lower than those in air. To a limited extent, increasing the depth of water enhanced the reduction of resonant frequencies.

In this paper, a theoretical method is developed to investigate the effect of a finite flow field on free vibration of a rectangular thin plate fully immersed in an incompressible and inviscid liquid. Rayleigh–Ritz method is applied to obtaining resonant frequencies and mode shapes of the wet plate, and different boundary conditions of a rectangular plate are discussed in this study, including fully clamped, free, and cantilever plates. The Galerkin method is utilized to deal with continuity conditions on the interface between plate and fluid and the beam method is applied to constructing mode shapes of a rectangular plate. Numerical results of the finite element method (FEM) are used to compare with the theoretical analysis on resonant frequencies and mode shapes and it shows the high accuracy of the proposed solution. In order to provide a convenient and accurate method to deal with the vibration characteristics of an immersed plate, the convergence of the Galerkin method is discussed and a modified computation method is presented. Finally, the effect of fluid's boundary on the resonating plate and the relation between the flow field and plate are investigated.

This paper presents a mathematical derivation of the vibration characteristics of an elastic thin plate placed at the bottom of a three dimensional rectangular container filled with compressible inviscid fluid. A set of beam functions is used as the admissible functions of the plate in a fluid–plate system, and the motion of the fluid induced by the deformation of the plate is obtained from a three-dimensional acoustic equation. Pressure from the fluid over the fluid–plate interface is integrated to form a virtual mass matrix. The frequency equation of the fluid–plate system is derived by combining mass, stiffness, and the virtual mass matrix. Solving the frequency equation makes it possible to obtain the dynamic characteristic of the fluid–plate system, such as resonant frequencies, corresponding mode shapes, and velocity of the fluid. Numerical calculations were performed for plates coupled with fluids with various degrees of compressibility to illustrate the difference between compressible and incompressible fluids in a fluid–plate system. The proposed method could be used to predict resonant frequencies and mode shapes with accuracy compared to that of incompressible fluid theory (IFT). The proposed method can be used to analyze cases involving high value of sound velocity, such as incompressible fluids. When the sound velocity approaches infinity, the results obtained for compressible fluids are similar to those of incompressible fluids. We also examined the influence of fluid compressibility on vibration characteristics in which a decrease in sound velocity was shown to correspond to a decrease in resonant frequency. Additional modes, not observed in incompressible fluids, were obtained in cases of low sound velocity, particularly at higher resonant frequencies. Fluid velocity plots clearly reveal that the additional resonant modes can be attributed to the compressible behavior of the fluid.

This study employed theoretical analysis and experimental measurements in an exploration of the transient behavior of a cantilever plate subjected to a dynamic impact force. The solution of transient displacement is established as a product of the time function and mode shapes for the cantilever plate. The superposition method was used to obtain the mode shapes and resonant frequencies for free vibrations, and the orthogonality of the mode shapes was used to determine the time function. We demonstrate that the results of transient behaviors including transient displacement, strain, velocity and acceleration of the cantilever plate can be explicitly presented in closed forms. In the experiment, the dynamic external force was produced by the impact of a steel ball on the cantilever plate, and a pair of Polyvinylidene fluoride (PVDF) sensors are used to measure the force history of the impact loading. Fotonic sensor (FS) was also used to obtain the time signals of transient displacement, and PVDF film sensors were used to determine transient strain in the cantilever plate. Our results obtained in the theoretical analysis are highly consistent with experimental measurement and FEM simulations in the time domain. This is a clear demonstration of the effectiveness of pairing theoretical analysis with impact loading histories measured using PVDF sensors in the representation of transient wave propagation behavior in a cantilever plate. Based on the proposed theoretical solution, we established the relationship between transient behaviors and vibrational characteristics that allow us to get more information of the physical phenomenon of the transient response. In transient signal, the contribution of each mode is related to the position of impact point and observation point. The position of exciting force and observation near the nodal lines of particular mode shapes can reduce the contribution of those modes in the transient waveform. We also investigated the influence on the transient response of a cantilever plate subjected to various impact loading duration, and indicate that the amplitude of transient displacement after the release of external force is depended on the impact force duration. Setting the force released at the time of period for specific mode, which provide the maximum contribution, can suppress the amplitude of oscillation in subsequent transient displacement.