Finite Plate Research Articles

Vibrational damping properties of finite microperforated plates

Microperforated plates (MPP) are traditionally used to advantageously absorb acoustic waves in sound control technologies. However, less is known concerning the additional structural damping they can induce. The latter is activated through exchanges in the viscous and thermal boundary layers near the fluid-structure interface of the microperforations. MPPs therefore offer an alternative to, or can be used together with, viscoelastic materials, commonly implemented to damp vibrations at medium and high frequencies. In this work, the structural damping capabilities of MPP are investigated. To this end, the damping performance of a finite size MPP is explored analytically through an alternative form of the Biot model, classically devoted to porous plates, and considering energy dissipation through viscous friction mechanisms. Analytical results are compared to experimental measurements of structural damping factors on various MPP samples. The model is validated and confirms the damping effect added by the microperforations in the low frequency range. A sensitivity analysis on the perforation rate and perforation diameter provides a condition for which additional damping is maximized.

Resonance mode analysis of finite plate strip with acoustic black holes: the gap between bandgap and attenuation band

Acoustic Black Hole (ABH) lattice structures show promise for achieving broadband bandgap in lightweight design. Existing ABH lattice research usually assumes that the bandgaps of infinite periodic structure are roughly the same as the attenuation band with strong energy attenuation in the finite counterpart. This work is concerned with comparison of the real attenuation bandwidth of finite periodic ABH structures and the bandgap of the infinite counterpart. The plate strips consisting of different numbers of periodic ABH elements are considered. It is observed that the bandgap-predicted attenuation band is split into two narrow attenuation bands, so that the periodic ABHs fails to ensure a broadband and continuous attenuation in the finite scenarios. Results show that there are resonance modes of finite plate strips with ABHs falling into the bandgap due to the boundary reflection and would result in high transmission peaks within the bandgap-predicted attenuation band. This unexpected phenomenon suggests the gap between the attenuation bandwidth of finite periodic structure and the bandgap of the corresponding infinite one. Analysis shows that the resonance mode of finite plate strips, which reduces the bandgap-predicted attenuation bandwidth, can be tuned by geometrically disturbing the periodicity and symmetry of the finite periodic configuration.

Strategic distribution of resonators' parameters for broadband vibration mitigation

Locally resonant metamaterials have recently come to the fore as novel lightweight and compact Noise, Vibrations and Harshness (NVH) solutions. They can be designed/obtained by assembling nominally equal resonant elements, which are tuned to the desired frequency range, onto a host structure in a sub-wavelength manner. This combination leads to tunable frequency ranges, known as stop bands i.e. frequency regions where free wave propagation is not allowed. Nonetheless, these stop bands are typically only effective in a limited frequency region. In order to broaden the frequency range of noise and vibration reduction, an optimization procedure could be applied in order to obtain the ideal distribution of resonance frequencies and masses to achieve a minimal sound radiation over a certain frequency region. However, this approach typically requires a considerable computational time. For this reason, this paper proposes a set of guidelines that define a strategy for broadband NVH reduction without the use of optimization. The proposed strategy is applied to a finite plate with a predefined added grid for the resonator positions. The performance of the strategy is numerically analyzed by assessing the vibration response of the plate for multiple excitation locations and within different frequency ranges. The obtained responses are then compared to the response of the plate with an optimized grid of resonators. This study shows that the proposed strategy leads to a reasonable vibration reduction with similar levels with respect to the response obtained from optimization.

Analysis on fracture mechanics theory of roof cutting instability mechanism with large mining height face in shallow coal seam

The roof in a fully-mechanized face of a shallow coal seam with large mining height is prone to form a combined cantilever—articulated rock beam structure. When the support resistance is insufficient, the articulated rock beam will sink. This will make the combined cantilever beam rotate and fracture. It is then easy to induce the sliding and instability of the articulated rock beam, which results in large-scale roof cutting and the support crushing. Taking the combined cantilever beam structure as the main research object, and considering the mining damaged characteristics of cantilever beam rock stratum, the rock beam was regarded as a finite plate model with an edge crack of arbitrary dip angle. In addition, a fracture mechanics model controlled by a set of structural planes was established, the instability conditions of rock beam and the main control factors were analyzed, and the method of determining the support resistance were discussed. The results show that the cantilever beam rotates and fractures. This causes a chain reaction of the rock beam that leads to fracture. The combined cantilever beam then loses stability with the increase of the length of the crack length and the crack dip angle, and therefore it is easier to penetrate the cantilever beam and cause roof instability. The necessary condition for rock beam instability was crack activation, and the sufficient condition was the airfoil branch crack propagate through the rock beam. The influence degree of each parameter on the support resistance was thus determined: crack length a > crack dip angle β > rock thickness h > weighting interval l. The theoretical analysis results were proven to be reasonable by an in situ monitoring example of no. 22,310 working face in the Daliuta coal mine, China. On this basis, the reasonable value of support resistance was obtained. The conclusions of this research provide a new method for researching the roof instability mechanism. They are also conducive to the green and sustainable development of mines.

A Verification and Validation Study on a Loosely Two-Way Coupled Hydroelastic Model of Wedge Water Entry

_ The interaction between the structural response and hydrodynamic loading (hydroelasticity) must be considered for design and operation purposes of high-speed planing craft made of composites that are prone to frequent water impact (slamming). A computational approach was proposed to study the hydroelastic slamming of a flexible wedge. The computational approach is a loose two-way coupling between a Wagner-based hydrodynamic solution and a linear finite element plate model. Verification and validation (V&V) was performed on this coupled model. It was found that the model overpredicts rigid-body/spray root kinematics by <15% and hydrodynamic loading/ structural response by <26%. Introduction One of the primary constraints on the operational envelope of high-speed craft is slamming (water impact). Slamming occurs between the hull body and the water surface when a portion/whole of the craft exits the water and then reenters at high-enough velocity (Lloyd 1989; Faltinsen 2005). The frequent water impacts, which work like “water hammers,” along with their induced acceleration pose great jeopardy on hull structures as well as crew and instrument on-board (Yamamoto et al. 1985; Ensign et al. 2000; Hirdaris et al. 2014). With the growing use of lightweight materials, the interaction between the structural deformation and the hydrodynamic loading (hydroelasticity) becomes more prevalent. The current design criteria of high-speed craft are based on empirical procedures with no regard to hydroelasticity due to the lack of understanding of this complex phenomenon (DNV 2013; ABS 2016). Therefore, a better comprehension of hydroelastic slamming is the first step to designing more high-performance craft (Fu et al. 2014; Judge et al. 2020).

Theoretical Predictions of Perturbed Magnetic Flux Density Components Due to Narrow Flaws in ECT

This work proposes a semianalytical technique based on the boundary element method (BEM) to predict the signals due to flaws (cracks) that are typically encountered in eddy current (ECT) nondestructive testing (NDT). Two case studies, for two different narrow opening notches in a finite thickness aluminium plate, were performed by experimental measurements to corroborate the theoretical model. The signal which has been modelled is the perturbation in the magnetic flux density (MFD) components <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B_x</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B_y</i> , and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B_z</i> due to narrow opening flaws. The proposed analytical model uses the magnetic vector potential (MVP), arising from electric dipole sources, to determine the voltage induced in a hypothetical small circular loop located around the evaluation point. Based on the linear relationship between the voltage induced and the magnetic flux in the surface enclosed by the hypothetical loop, a linearity constant that transforms the induced voltage to the MFD component has been computed. Appropriate vector potential Green’s functions have been used to transform the electric dipole sources to the magnetic vector potential components to determine the voltage induced in the hypothetical loop.

Vibration analysis of a composite elastic metamaterial plate based on a semi-analytical method

A semi-analytical vibration analyzing process is presented for a kind of finite composite elastic metamaterial plate by combining the effective medium theory and the first-order shear deformation sandwich plate theory, and the natural and steady-state vibration characteristics of such complex composite structure are investigated by means of equivalent analysis. The effectiveness of the semi-analytical analyzing process is demonstrated and some extraordinary vibration behaviors of such composite structure are revealed, such as there is no natural frequency in the bandgap, the modal density consistently increases around the bandgap's starting frequency, and the modes of the appropriate orders before and after the bandgap are consistent at the macroscopic level. The semi-analytical analyzing process and the special vibration behaviors may provide useful reference for relevant investigation and design of composite elastic metamaterial plate structures in the field of vibration control and isolation.

Analysis of stress intensity factor of a finite plate with centre crack containing different hole shapes

Analysis of stress intensity factor of a finite plate with centre crack containing different hole shapes

Vibroacoustic behavior of a rectangular composite plate in thermal environment

This paper is a study of the vibroacoustic behavior of orthotropic laminated composite rectangular plate under a sound wave excitation in thermal environments. An improved analytical procedure has been developed that allows for an efficient solution of the finite composite plate sound transmission problem. A symmetrically orthotropic laminated composite plate is considered. The plate is modeled with classic thin-plate theory and is assumed to be clamped on all four sides. The incident acoustic pressure is modeled as a harmonic plane wave impinging on the plate at an arbitrary angle. The sound transmission loss is calculated from the ratio of incident to transmitted acoustic powers.

Plate capacitor problem as a benchmark case for verifying the finite element implementation

In this work, parallel plate capacitors are numerically simulated by solving weak forms within the framework of the finite element method. Two different domains are studied. We study the infinite parallel plate capacitor problem and verify the implementation by deriving analytical solutions with a single layer and multiple layers between two plates. Furthermore, we study the finite parallel plate capacitor problem and verify it by Love’s potential equation and Xiang’s capacitance equation. Moreover, the fringing effect is considered and extended to problems with multiple dielectric layers, such a solution is not possible by means of the existing analytical solutions. Besides, we realize the possibility of choosing different boundary conditions (electric potential boundary conditions and charge density boundary conditions) by changing the weak form. Finally, a transient solution that includes dielectric loss and calculates the quality factor of a capacitor is presented, which may be used in capacitor design. Convergence and consistency of results are demonstrated by comparing the results between analytical and numerical solutions and also the results from different boundary conditions.

A plastic wave model for the perforation of a finite metal plate struck by a flat-ended projectile

A theoretical study is presented herein on the perforation of a finite metal plate struck by a flat-nosed projectile at normal incidence. Projectile deformations and plug formation are analyzed using plastic wave theory. Three modes of perforation (namely, perforation by a rigid projectile, perforation by a deformable projectile and perforation by an erosive projectile) are studied theoretically. Furthermore, a two-stage adiabatic shear model is introduced to describe the instability of adiabatic shear during the perforation process. Various equations are established and the critical conditions for the transition among the three modes are determined. It transpires that the present model predictions are in good agreement with the available experimental data for Weldox 460E steel plates struck transversely by flat-ended projectiles in terms of ballistic limit, residual velocity, transition temperature, projectile residual diameter, projectile residual length and plug thickness.

Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate

This paper investigates the complex band diagram of flexural waves in the phononic plate with semi-infinite heavy fluid loading. The system under examination is a square plate lattice with two-dimensional periodicity immersed in a fluid domain with infinite height. The numerical models based on the wave field transformation and the Galerkin method combined with the finite element discretization technique are developed to investigate the real and imaginary parts of the dispersion relation of flexural waves propagating through the phononic plate incorporating the fluid-loading effects. A perfect agreement is found between the location and width of stop bands from the real band diagram and the attenuation diagram, which supports the validity of the numerical models. Moreover, the complex band diagram is verified by the transverse vibration transmittance of the finite phononic plate. The results demonstrate that the external fluid loading is able to adjust the location, bandwidth, and decaying level as well as affect the degree of attenuation anisotropy of the complete and directional band gaps.

Attenuation band splitting in a finite plate strip with two-dimensional acoustic black holes

Acoustic black holes (ABH) lattice is an effective way to realize the bandgaps (BGs) through lightweight design. The predicted BGs using infinitely large periodic structures are commonly believed to enable corresponding attenuation bands (ABs) even the structure only contains a limited number of unit cells. This paper reports an unusual splitting phenomenon in the ABs of a finite plate strip with two-dimensional (2D) ABHs. A wide AB predicted by BG is split into two ABs in the finite strip separated by a high resonance peak, which challenges the aforementioned common belief. Results show that the increasing number of periodic unit cells has a negligible effect on the frequency and amplitude of the resonance peak. Its underlying mechanism is clarified by investigating the relationship between the BGs and ABs, obtained from infinite and finite periodic plate strips with the same 2D ABHs respectively. Complex wavenumber analysis results show the existence of two flexural wave modes in the considered frequency range due to the finite width of the strip. Two types of BGs, in which one satisfies the Bragg condition, are found when both wave modes fail to propagate. Analysis of structural modes further uncover the physical reason for the emerging AB splitting: a certain natural frequency of the boundary portion of the 2D ABHs strip falling into the theoretical BG. Due to the boundary reflection and energy concentration, this structural portion can loosely be regarded as a clamped-free plate strip. It is demonstrated that the resonance peak splitting the AB becomes tunable via disrupting the structural details of finite ABH strips. In particular, extending the terminal end of a finite ABH strip allows the change in boundary portion, which is conducive to eliminating the splitting peak. The reported study is helpful for the design of finite periodic structures with broadband attenuation bandgap.

Voltage-controlled topological interface states for bending waves in soft dielectric phononic crystal plates

The operating frequency range of passive topological phononic crystals is generally fixed and narrow, limiting their practical applications. To overcome this difficulty, here we design and investigate a one-dimensional soft dielectric phononic crystal (PC) plate system with actively tunable topological interface states via the mechanical and electric loads. We use nonlinear electroelasticity theory and linearized incremental theory to derive the governing equations. First we determine the nonlinear static response of the soft dielectric PC plate subjected to a combination of axial force and electric voltage. Then we study the motion of superimposed incremental bending waves. By adopting the Spectral Element Method, we obtain the dispersion relation for the infinite PC plate and the transmission coefficient for the finite PC plate waveguide. Numerical results show that the low-frequency topological interface state exists at the interface of the finite phononic plate waveguide with two topologically different elements. By simply adjusting the axial force or the electric voltage, an increase or decrease in the frequency of the topological interface state can be realized. Furthermore, applying the electric voltage separately on different elements of the PC plate waveguide is a flexible and smart method to tune the topological interface state in a wide range. These results provide guidance for designing soft smart wave devices with low-frequency tunable topological interface states.

Theoretical and experimental investigation on the low-frequency vibro-acoustic characteristics of a finite locally resonant plate

This study investigates the low-frequency vibro-acoustic characteristics of a finite locally resonant (LR) plate. A dynamic model of the finite LR plate consisting of periodic arrays of beam-like resonators attached to a thin aluminum plate with simply supported boundary conditions is established, and the average vibration response and radiated efficiency are theoretically determined by using modal-superposition and harmonic-balance methods. In addition, the study investigates the influence of the parameters and number of additional resonators on the vibro-acoustic performance of the finite LR plate. Finally, a vibration experiment of a finite plate with 8 × 10 uniformly distributed beam-like resonators validates the theoretical analysis results. The numerical and experimental results show that the finite LR plate has a low-frequency bandgap that can suppress the vibration and radiated noise of the structure, and the bandgap position is close to the resonance frequency of resonators. The position and performance of bandgaps can be influenced by changing the parameters and number of resonators. The experimental results show a bandgap ranging from 370 to 425 Hz, which is consistent with the theoretical prediction. The finite LR plates proposed in this study can find potential applications in the attenuation of low-frequency vibration and noise.

Using the extended finite element method to integrate the level-set method to simulate the stress concentration factor at the circular holes near the material boundary of a functionally-graded material plate

Several numerical methods have developed into indispensable and effective tools for solving science-technology problems. The finite element method has particularly become a popular and effective technology, among others—a method that cannot be separated from technical programming MATLAB codes. In order to simulate the discontinuity boundary, enrichment functions have been added to the displacement field expression of the finite element method to produce the extended finite element method.This research introduced the application of MATLAB codes combined with extended finite element method based on the level-set method to simulate the stress concentration factor at circular holes near the material boundary. Both cases were compared with the following analytical results: the stress intensity factors of the finite functionally-graded material isotropic plate had a crack with an average error of 3.37%; the stress concentration factor of the isotropic plate had a circular hole near the material boundary with an error of 4.19%. In particular, the extended finite element method had significant errors compared to the Exact and experimental methods.

Efficient Model Assisted Probability of Detection Estimations in Eddy Current NDT with ACA-SVD Based Forward Solver.

Model assisted probability of detection (MAPoD) is crucial for quantifying the inspection capability of a nondestructive testing (NDT) system which uses the coil or probe to sense the size and location of the cracks. Unfortunately, it may be computationally intensive for the simulation models. To improve the efficiency of the MAPoD, in this article, an efficient 3D eddy current nondestructive evaluation (ECNDE) forward solver is proposed to make estimations for PoD study. It is the first time that singular value decomposition (SVD) is used as the recompression technique to improve the overall performance of the adaptive cross approximation (ACA) algorithm-based boundary element method (BEM) ECNDE forward solver for implementation of PoD. Both the robustness and efficiency of the proposed solver are demonstrated and testified by comparing the predicted impedance variations of the coil with analytical, semi-analytical and experimental benchmarks. Calculation of PoD curves assisted by the proposed simulation model is performed on a finite thickness plate with a rectangular surface flaw. The features, which are the maximum impedance variations of the coil for various flaw lengths, are obtained entirely by the proposed model with selection of the liftoff distance as the uncertain parameter in a Gaussian distribution. The results show that the proposed ACA-SVD based BEM fast ECNDE forward solver is an excellent simulation model to make estimations for MAPoD study.

Phased array guided wave propagation in curved plates

For the monitoring of plate structures, phased array guided wave has great application prospects to effectively determine damage location and size. Locating damage via phased array focusing is a challenging task, especially in curved plates. Hence, a systematic understanding of the behavior of the phased array guided wave propagation in curved plates is needed. In this study, a phased array focusing method was proposed for the damage localization of curved plates. Different damage scenarios and phased array focusing schemes are investigated in the finite element plate models with different radii. A high-speed signal generation and acquisition system including PZT actuators and sensors is set up. Experimental tests are carried out on a curved aluminum plate with embedded piezoelectric elements to verify the finite element models. Parametric studies are then performed to evaluate the effect of different radii and damage scenarios on the behavior of phased array guided wave propagation in curved plates. Numerical simulations and experimental results show that the proposed PA method has the capabilities of wave focusing and damage identification in curved plates. The damage localization method using phased array guided wave has the potential to be applied to the real-time monitoring of structures with curved plates.

A novel locally resonant metastructure with soft-material rings for broadband and low frequency vibration attenuation

Metastructure has been discovered as a new and effective means of vibration suppression in recent studies. Based on the basic principle of the locally resonant (LR) metastructure, a novel two-dimensional LR metastruture is proposed in this paper. The host plate is punched out several concentric holes and filled with soft materials, with the resonator installed in the center. The band structure of an infinite structure is calculated by using Gaussian expansion method as well as the finite element method. To further evaluate the vibration attenuation performance, the vibration transmission of a finite plate is obtained by both numerical and experimental methods. Formation mechanism of band gaps (BGs) is revealed by analyzing eigenmodes at boundary frequencies. The significance of the novel design is clarified from the mechanism level, about the characteristics of generating broad band gaps in low frequencies. The influence of both geometric parameters and material damping properties on BGs are discussed, including examples of a non-periodic plate and a periodic damped plate. Because of its features and simple design, it can be expected to be widely applied in engineering.

The finite deformation approximate solution and pressure–impulse​ diagram of an elastic–brittle rectangular glass plate under pressure loading

Glass windows and curtain walls are increasingly used in modern buildings. To judge whether a glass panel fails under a certain blast, the pressure–impulse (P–I) diagram is necessary. In contrast to the commonly adopted P–I diagrams given by the single degree of freedom theory or the high-cost finite element (FE) simulations, a P–I diagram is given in this study by employing a proposed approximate analytical solution. The theoretical solution of the finite deformation rectangular plate under pressure loading is based on the traditional von Kármán equations and the uniform stress assumption. Through comparisons with the corresponding FE simulations, experimental data, and internationally acknowledged standards, the solution is confirmed to be both accurate and efficient. Furthermore, the P–I diagram given is also confirmed to be both reliable and efficient when compared with the experimental data.