Vibration Of Plates Research Articles

Two-dimensional orthotropic plate problems in a thermal environment: Refined crack modelling

Three two-dimensional (2D) refined models are devised for the analysis of orthotropic thin plates with part-through surface and internal cracks under in-plane forces due to temperature differences. The refined models are used to formulate the governing equations for cracked orthotropic thin plates in a thermal environment. Subsequently, the refined models and their ensuing modified forms can be directly applied to all-over and finite-length cracks, respectively. Using an interface-fitted numerical scheme (i.e., matched interface and boundary technique), the resulting equations of motion for the buckling and free vibration of cracked orthotropic thin plates are solved. The effects of various crack types and dimensions on the critical buckling temperatures and vibration frequencies of orthotropic thin plates are investigated. The present models are also validated by comparing their numerical results with the line-spring model and three-dimensional finite-element solutions. Illustrative examples, including boron fiber/epoxy and glass-fibre-reinforced composite materials, are considered to show the superiority of the proposed models over the conventional one, especially in terms of understanding the effect of in-plane forces arising from temperature differences at the cracked location.

Free Vibration Analysis of Trapezoidal Bi-Stable Laminates Supported at the Elastic Midpoint of the Median Line

This study investigates the natural vibration of trapezoidal bi-stable laminates (TBL) with elastic supports at the midpoints of the median lines. Configuration of the midplane of the TBL is expressed by a polynomial with 17 parameters. Then, the first order shear deformation theory, curing temperature, and nonlinear strain displacement relations combining energy principles are applied to obtain the bi-stable shapes numerically. Three translational springs and two rotational springs are added at the midpoint of the median line in the trapezoidal bi-stable laminate to acquire elastic point supports. And, by varying the stiffness of the springs, arbitrary elastic point support boundary conditions can be achieved. Chebyshev polynomials are applied to characterize the mode shape function of the TBL. The vibration mode functions of the TBL are mapped to a square area under the new coordinate system by using the coordinate mapping method. Furthermore, the effects of geometry, layup sequence, and spring stiffness on the natural vibrations of the TBL are analyzed, which provides a reference for research in this field. The innovation and highlights lie in the following: (1) the natural frequencies and modes of trapezoidal bi-stable plates are solved; (2) arbitrary elastic support is achieved by a set of artificial springs; (3) the influences of spring stiffness, layer sequence, and trapezoidal base angle on the natural vibration of a trapezoidal bi-stable plate are studied.

Electro-mechanical vibro-acoustic characteristics of submerged functionally graded piezoelectric plates with general boundary conditions

The present paper focuses on studying the underwater electro-mechanical vibro-acoustic characteristics of functionally graded piezoelectric material (FGPM) plates with general boundary conditions. For this purpose, the collocation points method is firstly developed to handle vibro-acoustic coupling, which can directly calculate the virtual work of secondary sound pressure via an easy double integral, without pre-simplifying the complex quadruple integral. The proposed FGPM plate model is established in the framework of first-order shear deformation theory (FSDT) and second type constitutive equations of piezoelectric materials, and the macroscopically inhomogeneous electro-mechanical properties are considered by means of Voigt model. The multi-physical fields including electric, mechanical and acoustic fields are uniformly extended in Chebyshev polynomials. Meanwhile, penalty function method is introduced to satisfy the arbitrary mechanical and electric constraints. All energy formulations are processed by Rayleigh-Ritz procedure and a standard governing equation is finally derived. Based on the proposed unified model, the convergence and accuracy for solving vibration and acoustic radiation of FGPM plates are validated firstly. Additionally, the influence of some key parameters on underwater vibro-acoustic characteristics of FGPM plates is fully examined. The results of this paper may provide a reference for future application of FGPM-based smart structures.

Geometric eccentricity effect on thickness-shear vibration of an elliptical flexoelectric crystal plate

In this paper, a thickness-shear vibration model for a two-dimensional finite flexoelectric crystal plate with slowly varying thickness and elliptical cross section is presented. The influences of eccentricity and flexoelectricity on the fundamental frequencies of thickness-shear vibration and electric potential distributions under short circuit boundary condition are investigated. The proposed model uses the first order McLaughlin series to approximate the elliptical function and the first derivative of elliptical function is ignored since the plate thickness varies slowly. For the finite plate, the symmetric mode in length and antisymmetric mode in thickness are adopted for the displacement function. In addition, only the shear strain gradient through the thickness is used in the mathematical model for the thin and long flexoelectric crystal plate. By implementing the variational principle, the governing equations are obtained and further solved by the Galerkin method. The obtained frequencies of thickness-shear mode are expressed in terms of eccentricity and length-to-thickness ratio. The results show that the eccentricity has great influence on the nondimensional fundamental frequencies. A small reduction in eccentricity results in significant increment of the nondimensional frequency implying that it is possible to increase the fundamental frequency by changing the cross section of the elliptical flexoelectric plate. The flexoelectric effect is also found to affect the nondimensional frequency and electric potential distribution significantly. Thus, for accurate design of nano/micro scale high precision and high frequency flexoelectric/piezoelectric devices, flexoelectricity must be taken into consideration.

A new finite-element procedure for vibration analysis of FGP sandwich plates resting on Kerr foundation

Abstract This article provides a new finite-element procedure based on Reddy’s third-order shear deformation plate theory (TSDT) to establish the motion equation of functionally graded porous (FGP) sandwich plates resting on Kerr foundation (KF). Although Reddy’s TSDT is attractive, it cannot be exploited as expected in finite-element analysis due to the difficulties in satisfying the zero shear stress boundary condition. In this study, the proposed element has four nodes, each with seven degrees of freedom (DOF). The performance of this element is confirmed by conducting various examples, showing its accuracy and range of applications. Then, some studies are performed to present the effects of input parameters on the vibration of FGP sandwich plates resting on KF.

Thermoelastic nonlinear vibration and dynamical response of geometrically imperfect carbon nanotube-reinforced composite plates on elastic foundations including tangential edge constraints

The combined influences of initial geometrical imperfection, elasticity of in-plane constraints of edges, elastic foundations and elevated temperature on the nonlinear vibration and dynamical response of carbon nanotube-reinforced composite rectangular plates are investigated in this paper. Carbon nanotubes (CNTs) are reinforced into matrix according to functionally graded distributions. The properties of CNTs and matrix are assumed to be temperature dependent and effective properties of nanocomposite are determined using an extended rule of mixture. Governing equations in terms of deflection and stress function are established within the framework of thin plate theory including von Kármán nonlinearity, geometric imperfection and interactive pressure from elastic foundation. Analytical solutions are assumed for simply supported plates and Galerkin method is applied to result in nonlinear time differential equation. This nonlinear equation is solved using fourth-order Runge–Kutta scheme. Parametric studies are executed to examine numerous influences on the natural frequency, nonlinear to linear frequency ratio and nonlinear dynamical response of CNT-reinforced composite plates. The results reveal that increase in imperfection size results in increase and decrease in natural frequencies and the amplitude of forced vibration, respectively. In contrast, it is found that the elevated temperature reduces the natural frequencies and enhances the amplitude of forced vibration.

A subwavelength ventilated structure for efficient broadband sound insulation

Multifunctional structures such as ventilated sound barriers, have become the focus of recent research on the noise reduction and environmental comfort. However, its large size and complex inner structure hinder its potential applications. Novel structures with side-branch sectorial Helmholtz resonators and double-layered perforated slit plates enlightened by macro-perforated plates to enhance the soundproof performance and facilitate natural ventilation are proposed and experimentally validated. Compared with simple muffler ducts, the combinations with slit plates provide a smoother transmission loss (TL) curve with a broad and continuous TL band. We also study the influences of the independent parts and interactive effects of the assembly on the sound field, including the frequency migration and plate vibration. The proposed sub-wavelength structures with a thickness of 15 mm can obtain TL values up to 25 dB with a broad bandwidth from 930 Hz to 1600 Hz. Moreover, soundproof walls can be fabricated by using these structures with plenty of ventilated slits to freely exchange air and heat. This ventilation sound barrier is suitable for acoustic landscape buildings as it covers the main frequency spectrum of a human equal loudness contour.

Nonlinear active control of thermally induced pyro-coupled vibrations in porous-agglomerated CNT core sandwich plate with magneto-piezo-elastic facings

In this article, the damped nonlinear transient response of a smart sandwich plate (SSP) comprising of agglomerated CNT-reinforced porous nanocomposite core with multifunctional magneto-piezo-elastic (MPE) facesheets, subjected to the thermal environment, is numerically investigated. The synergistic influence of agglomeration, porosity and pyro-coupling on vibration control is studied for the first time under the finite element framework. The attenuation of the vibrations is caused by active constrained layer damping (ACLD) treatment. The kinematics of the plate is based on the layer-wise shear deformation theory and von-Karman’s nonlinearity. The viscoelastic properties of the ACLD patch and CNT agglomeration of the core are mathematically modelled using Golla–Hughes–McTavish and Eshelby–Mori–Tanaka methods, respectively. A comprehensive examination of the inter-related effects of different agglomeration states, porosity distributions and thermal loading profiles has been performed. The new insights on controlling pyro-coupled induced vibrations of smart sandwich plates by supplying control voltage directly to the MPE facesheets without ACLD treatment have been discussed thoroughly. The numerical analysis confirms the significant effects of pyro-coupling associated with active vibration control response of SSP.

Acoustic properties of piezoelectric cubic crystals

AbstractThis paper is offered as a complementary adjunct to the many treatments of the electronic and photonic properties of cubic III–V and II–VI compounds appearing in the literature. These crystals typically exhibit piezoelectricity, due to the molecular dissymmetry, thereby allowing the inclusion of classical mechanical/acoustic features along with the quantum. We discuss the history of this modality and then illustrate its use by applying it to an electro‐elastic problem that has the estimable virtues of having an exact solution, along with wide practical applicability: determination of the piezocoupling values governing the excitation of thickness vibrations in thin cubic films or plates of arbitrary crystallographic orientation by electric fields directed either along, or lateral to, the thickness. Explicit results are given for orientations along the great‐circle paths connecting the principal directions [100], [110], and [111]. The formalism is then applied to GaAs as an example; it is further demonstrated that various results, such as the orientational variations of piezocoupling factors, are generally applicable to other members of the III–V and II–VI families by scaling. Ancillary aspects, such as errors due to misorientations, nonlinearities, and equivalent circuit representations, are described and discussed. This work is dedicated to Gerald W. Farnell (1925–2015), Prof. Emeritus, McGill University, Montréal, Canada.

Dynamic modelling and chaos control for a thin plate oscillator using Bubnov–Galerkin integral method

The utilization of thin plate systems based on acoustic vibration holds significant importance in micro-nano manipulation and the exploration of nonlinear science. This paper focuses on the analysis of an actual thin plate system driven by acoustic wave signals. By combining the mechanical analysis of thin plate microelements with the Bubnov–Galerkin integral method, the governing equation for the forced vibration of a square thin plate is derived. Notably, the reaction force of the thin plate vibration system is defined as f = α|w|, resembling Hooke’s law. The energy function and energy level curve of the system are also analyzed. Subsequently, the amplitude–frequency response function of the thin plate oscillator is solved using the harmonic balance method. Through numerical simulations, the amplitude–frequency curves are analyzed for different vibration modes under the influence of various parameters. Furthermore, the paper demonstrates the occurrence of conservative chaotic motions in the thin plate oscillator using theoretical and numerical methods. Dynamics maps illustrating the system’s states are presented to reveal the evolution laws of the system. By exploring the effects of force fields and system energy, the underlying mechanism of chaos is interpreted. Additionally, the phenomenon of chaos in the oscillator can be controlled through the method of velocity and displacement states feedback, which holds significance for engineering applications.

A Moving Kriging Interpolation Meshless for Bending and Free Vibration Analysis of the Stiffened FGM Plates in Thermal Environment

This paper adopts the Moving Kriging (MK) interpolation meshless method to analyze the static and dynamic behaviors of stiffened functionally graded material (FGM) plate in thermal environment based on the physical neutral surface. The ribbed FGM plate is regarded as a composite structure of a FGM plate and ribs. The displacement transformation relationship between stiffeners and FGM plates is obtained through the displacement compatible conditions and MK interpolation. The meshfree model for ribbed FGM plate is obtained by superimposing the total energy of the FGM plate and the stiffeners based on the first-order shear deformation theory (FSDT) and physical neutral surface. The nonlinear temperature field along thickness direction is introduced into the meshless model of stiffened FGM plate. The equations governing the bending and free vibration of the ribbed FGM plate in thermal environment are obtained according to the principle of Minimum Potential Energy and Hamilton’s Principle. Thereafter, several ribbed FGM plate examples in different temperatures and with different locations of ribs are calculated. The results are compared with those given by the ABAQUS and literature. The results show that the effectiveness and accuracy of the proposed method in analyzing the ribbed FGM plate in thermal environment.

Effect of Graphene Reinforcement on Free Vibration and Material Properties of the FG-GPLRC Porous Cantilever Torsional Plate

This paper researches the free vibrations and corresponding material parameters of a functional gradient graphene platelets-reinforced composite (FG-GPLRC) cantilever torsional plates with both pore and graphene variations in the transverse direction. Utilizing the feature of closed-cell cellular solids, the mixture rule, and the modified Halpin–Tsai model, the material parameters of the composite are determined for different volume fractions of component materials. Then, using the classical plate theory (CLPT), the Rayleigh–Ritz technique and polynomials, the dynamic equation that can be used to obtain the free vibration mode shapes and frequencies of the rotating cantilever torsional plate is given. Comparison studies with the previous calculation results from available literature and the finite element (FE) models of cantilever plates are conducted, and the correctness of the present theoretical formulation and numerical calculation is verified. Finally, the effects of graphene platelet (GPL) distribution, porosity distribution (PD), GPL content, rotational speed, and average geometric size of GPL on free vibrations of the system are studied in depth.

Impact protection enhancement by negative mass meta-honeycombs with local resonance plates

Impact and blast waves are difficult to isolate entirely because of their tremendous energy and wide-band characteristics. In this study, the meta-honeycombs are designed by adding the local resonance plates (LRP) to the honeycombs to improve the impact resistance. The negative mass property and the directional bandgap property of the LRP are investigated to form wider bandgaps to predict the impact resistance of the meta-honeycombs. For one thing, in the corresponding frequency range of the equivalent negative mass, the LRP and the impact wave vibrate towards each other, which can resist wave propagation. For another, when the frequency of impact energy is concentrated in the directional bandgap, the energy is trapped in the vibrations of the LRP plate, accelerating the attenuation of the impact wave. It is verified that the three meta-honeycombs, whose cores are re-entrant, hexagonal, and semi-re-entrant honeycombs, can effectively protect the stability of the structure against impact and blast waves through experiments and simulations. Further, the mechanical properties of the meta-honeycombs are optimized by coupling the bandgap generated by the LRP containing different resonators. The universality of the design method lays the foundation for engineering application.

Vibration and sound radiation of an acoustic black hole plate immersed in heavy fluid

Vibrational acoustic black holes (ABHs) have shown great promise for reducing structural vibrations and sound radiation in light fluids. However, it is still unknown whether the acoustic black hole (ABH) effect can be materialized in heavy fluids. This paper discusses this issue by developing a semi-analytical model on a simply supported ABH plate that vibrates and radiates sound into water. The proposed model is validated by finite element models and used to investigate the vibration and sound radiation properties of the ABH plate in different frequency ranges. The results show that the ABH effect can be systematically manifested in heavy fluids, as reflected by a significant increase in structural damping and a decrease in vibration and sound radiation. Numerical analysis of the radiation damping and mass loading effects shows that the radiation damping has little effect on the vibration reduction of the water-loaded plate. However, the mass loading effect mitigates the low-frequency drawback of conventional ABH structures in air, resulting in a broadband reduction in structural vibration and sound radiation from the water-loaded ABH plate.

Investigation on Vibration Characteristics of Thin-Walled Steel Structures under Shock Waves.

Thin-walled steel structures, prized for their lightweight properties, material efficiency, and excellent mechanical characteristics, find wide-ranging applications in ships, aircraft, and vehicles. Given their typical role in various types of equipment, it is crucial to investigate the response of thin-walled structures to shock waves for the design and development of innovative equipment. In this study, a shock tube was employed to generate shock waves, and a rectangular steel plate with dimensions of 2400.0 mm × 1200.0 mm × 4.0 mm (length × width × thickness) was designed for conducting research on transient shock vibration. The steel plate was mounted on an adjustable bracket capable of moving vertically. Accelerometers were installed on the transverse and longitudinal symmetric axes of the steel plate. Transient shock loading was achieved at nine discrete positions on a steel plate by adjusting the horizontal position of the shock tube and the vertical position of the adjustable bracket. For each test, vibration data of eight different test positions were obtained. The wavelet transform (WT) and the improved ensemble empirical mode decomposition (EEMD) methods were introduced to perform a time-frequency analysis on the vibration of the steel plate. The results indicated that the EEMD method effectively alleviated the modal aliasing in the vibration response decomposition of thin-walled structures, as well as the incompletely continuous frequency domain issue in WT. Moreover, the duration of vibration at different frequencies and the variation of amplitude size with time under various shock conditions were determined for thin-walled structures. These findings offer valuable insights for the design and development of vehicles with enhanced resistance to shock wave loading.

Free vibrations of rectangular plates

The paper investigates free vibrations of rectangular metal plates. The finite element method and analytical calculation are particularly used to determine the vibration frequency. The analytical calculation is based on the equation of motion of a thin rectangular plate. The asymptotic method is applied to determine the dynamic edge effect. As a result, the free vibration frequency is determined for the rectangular metal plate. The finite element analysis is performed in Lira and SolidWorks software packages. For this, a solid plate model with sensors is created to measure the free vibration frequency; the effective mass participation factor was determined.The plate vibration tests were conducted to confirm the results of analytical calculations. The method of smooth sinusoidal vibrations is used. Spectrum graphs of the plate vibrations are suggested based on the vibration acceleration of sensors. Errors are identified in the free vibration frequencies depending on the applied method. The paper does not consider frequencies with the low effective mass participation factor.

Nonlinear 2D-DQ volume-preservative global–local dynamic analysis of composite sandwich plates with soft hyperelastic cores and viscoelastic Winkler-Pasternak foundations

Nonlinear large amplitude vibration and dynamic deformations of sandwich plates with composite face sheets and hyperelastic cores that encounter large thickness changes during the vibration are investigated here for the first time. Moreover, a new global–local zigzag hyperelastic plate theory that considers the incompressibility and large transverse deformability of the core, and employs transverse normal strains whose order is higher than that of the in-plane strains is proposed. The plate is resting on a viscoelastic Winkler-Pasternak substrate, exposed to various spatial and time distributions of loads, and experiencing a diversity of edge conditions. The governing equations of motion are obtained by using Hamilton’s principle, von Kármán’s assumptions, and the series expansion of the volume-preserving strain energy density function. A 2D differential quadrature (2D-DQ) technique is employed for the spatial discretization of a structure with both constitutive and kinematic nonlinearities, for the first time. The time-dependent combination of the nonlinear coupled motion equations and boundary conditions is treated by an iterative/updating Runge-Kutta time-marching technique and the effects of the thickness and aspect ratio of the plate, viscoelastic and shear parameters of the foundation, types of the edge conditions, constitutive parameters of the composite materials, fiber orientation angle, and magnitude and distribution pattern of the loads on the resulting dynamic lateral deflections are studied. Results reveal that while the vibration frequencies and amplitudes of the face sheets are quite different, the damping is more remarkable when stiffer cores are utilized. Moreover, the effects of the higher vibration modes are more apparent in looser edge conditions, smaller fiber angles, and special excitation frequencies.

A von Kármán-type model for two-layer laminated glass plates, with applications to buckling and free vibration under in-plane edge loads

This article presents a layerwise geometrically nonlinear two-dimensional model for the undamped dynamical response of thin two-layer plates with partial shear interaction. The model is tailored for the analysis of laminated glass plates of the type most commonly used in building structures (two glass layers bonded by a polymeric interlayer). The geometrical nonlinearities are introduced via the so-called von Kármán strains. The process of dimensional reduction is based on Podio-Guidugli’s method of internal constraints. An unconventional choice of generalised displacements highlights several direct similarities with the single-layer plate theories of von Kármán and Mindlin.The model is used to study the flexural buckling and free vibration of two-layer plates under in-plane edge loads. Analytical solutions are obtained for rectangular plates with (i) all edges simply supported without shear restraint and (ii) a two-parameter system of uniform in-plane normal edge loads. These solutions degenerate into those of single-layer von Kármán plates in the two limiting cases of zero and full shear interaction. A conforming rectangular finite element is formulated and shown to perform effectively. When using uniform meshes, the observed asymptotic rate of convergence for the finite element approximation of the lowest buckling load and vibration frequency is quadratic.

Sandwiched Plate Vibration Analysis with Open and Closed Lattice Cell Core

This work attempts to replace the sandwich core's traditional shape and material with a cellular pattern, where the cells have a regular shape, distribution, and size. The contribution of this paper is to design two structures, one open-celled and the other closed, and to evaluate the performance of sandwich plates with lattice cell core as it is used for many industrial applications, particularly in automobile engineering. The new theoretical formulations are constructed for two structures to find the free vibration characteristics. The results of the new design are compared with the traditional shape. Derivation of equations to predict mechanical properties based on relative density with the chosen shapes, specific vibration equation of three-layer sandwich plate, and substitution by equation using excel sheet. Results are promising, and the effectiveness of cellular pattern theoretical analysis estimation. Limitations and error rates for the mechanical properties come through the empirical equations, and their ratio to the relative density values are higher depending on the behavior of the core material. Findings reveal, with open cell decrease in modulus of elasticity by (PLA: -90.4%) and (TPU: -90.4%), increases natural frequency by (PLA: 44.5%) and (TPU: 46.4%), as for closed-cell decreases in the modulus of elasticity by (PLA: -66.9%) and (TPU: -64.4%), increases natural frequency by (PLA: 36%) and (TPU: 37.7%). Converting a solid substance or replacing a foam form with a cellular pattern is one way to better performance and save weight through the selected cell pattern in absorbing the energy of the vibration wave.

Higher-Order Finite Element Vibration Analysis of Circular Raft on Winkler Foundation

On a Winkler foundation, solid circular plate vibration is examined using a higher-order finite element in polar co-ordinate system. The present formulation has developed a Mat-lab code to handle any boundary conditions. Validation of the code is carried out after the convergence studies. The results are compared to other researchers and show excellent conformity. Furthermore, a parametric analysis gave the first 10 natural frequency characteristics in tabular and graphical form. The authors conclude that the present formulation is straightforward, behaves exceptionally well for thin solid circular plates on elastic foundations with reasonable convergence rate and accuracy, and requires less computational effort, resources, and time.