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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.

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 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.