Kirchhoff Thin Plate Research Articles

Dynamic responses of a plate on layered transversely isotropic poroelastic subgrade under moving thermal and mechanical loads

In this paper, the dynamic responses of a plate on layered transversely isotropic (TI) poroelastic subgrade under moving thermo-mechanical loads are investigated. Based on the thermodynamics, poroelastic mechanics and the Kirchhoff's thin plate theory, the governing equations of the plate and layered TI saturated subgrade are derived in the wavenumber domain. Then, by introducing the flexibility coefficient of the layered TI poroelastic subgrade and the plate-subgrade interface condition, the solution of the plate-subgrade dynamic interaction is obtained through the extended precise integration method. After the presented method is validated, parametric studies are carried out to confirm the influences of the thermal load, plate-subgrade modulus ratio, plate thickness, and the speed of moving load. Parametric analysis shows that the effect of the thermal load cannot be ignored when considering the plate-subgrade dynamic interaction. In addition, the plate-subgrade modulus ratio, plate thickness, and the speed of moving load have a significant effect on the plate-subgrade dynamic interaction.

Semi-analytical approach to study wave interaction with a pair of obliquely submerged heterogeneous flexible structures

ABSTRACT Analyzing inclined flexible plates of varying thickness is crucial for optimising and attaining precise control over wave reflection and transmission. This is particularly significant in the context of breakwater construction. In contrast to horizontal breakwaters, inclined barriers can penetrate multiple layers of fluid with varying particle velocities, facilitating interactions and causing wave breaking, which results in the dissipation of wave energy. Moreover, compared to vertical structures, inclined ones demonstrate more effective wave attenuation. In the context of the present study, we investigate the interaction of water waves with a pair of obliquely submerged flexible thin plates characterised by non-uniform thickness. In order to model this complex physical scenario, we employ the principles of linear water wave theory in conjunction with Kirchhoff's thin plate theory. By employing repeated integration and Green's integral theorem, we reduce the boundary value problem to a system of coupled integral equations. Through appropriate approximations, we solve this system and obtain numerical values for various hydrodynamic quantities. This model provides comprehensive results for both horizontal and vertical plates, making it highly versatile. We examine the impact of varying thickness and the inclination of the two flexible plates to understand their roles in the wave scattering process and the associated physical parameters. The results obtained from this study shed light on the intricate dynamics involved in wave-structure interactions and can inform the design and optimisation of breakwater systems.

Theoretical modeling and vibration analysis of composite laminated wing-box structures of hydrogen-electric aircraft under hygrothermal environment

For the sake of achieving the goal of ultra-low or even zero carbon emissions during flight, hydrogen-electric aircraft has been receiving increasing emphasis. Nevertheless, the hydrogen-electric aircraft owns an all-composite fuselage configuration and is subjected hygrothermal environment, which have a significant effect on the aircraft dynamics, especially for wing structure. In this study, an improved orthogonal polynomial solution based on the Rayleigh-Ritz method is developed for the dynamic analysis of composite laminated wing-box structures (CLWBSs) of hydrogen-electric aircraft with humid and hot environment and elastic boundary. In this solution approach, the potential energy, kinetic energy, elastic potential energy and hygrothermal potential energy of CLWBSs are derived from the framework of Kirchhoff thin plate hypothesis. A set of artificial springs is imported to the free edges of CLWBSs to simulate the elastic boundary and connection between two plates. The convergence, accuracy and computational efficiency of the proposed formulation are validated by experiment and finite element analysis. The effect of the coupling parameters and the elastic boundaries due to flexible composite fuselage in hygrothermal environment created by the hydrogen fuel cells are systematically investigated which have been confirmed by theoretical research, finite element analysis and experiment. The related findings indicate that the present approach provided an effective formulation for the vibration analysis and optimization of flexible composite wing-box structures. It can be applied in the field of aerospace and navigation industry at the same time.

A mesoscale computational approach to predict ABD matrix of thin woven composites

The ABD matrix is a fundamental method to characterize the overall stiffness behavior of laminated composite structures. Although classical laminate theory has been widely used, it has limitations in predicting the ABD matrix for woven composites. To address this issue, this paper presents a mesoscale homogenization approach aimed at computing the ABD matrix for thin woven composites accurately. The mesoscale representative volume element (RVE) of the woven composite is generated using TexGen and imposed with periodic boundary conditions to enforce the Kirchhoff thin plate assumption. The ABD matrix is computed by conducting six separate finite element simulations, each representing one simple in-plane or out-of-plane deformation in a specified direction. Moreover, to facilitate the implementation of the method, an open-source plugin tool was developed within ABAQUS CAE, automating the ABD matrix calculation for various types of woven composites including 2D weave, 3D weave, and multiaxial. The accuracy of the proposed method was validated through benchmark calculations against existing literature results.

An analytical model for the analysis of vibration and energy flow in a clamped stiffened plate using integral transform technique

Based on Kirchhoff thin plate and Mindlin thick plate theories, the vibration and energy flow characteristics of clamped stiffened plate are studied by using the analytical model constructed by finite integral transform method. The results show that the energy flow characteristics of the stiffened plate at the beam/plate coupling interface depend on the position of the rib in the vibration modes of the plate. The effects of shear deformation and rotatory inertia on the energy flow across the beam/plate coupling interface of the stiffened plate are further investigated. It is found that the inclusion of rotatory inertia of the beam and plate in the model only affects the energy flow component controlled by the moment coupling but not that controlled by the shear force coupling. Whilst the inclusion of the shear deformation of the beam and plate mainly causes a decreased amplitude of the energy flow for the mode group where the beam is located away from both the nodal and antinodal lines of modes, in addition to the shear deformation of the plate which also leads to an increased amplitude of the energy flow component controlled by the shear force coupling for the mode group where the beam locates at the antinodal line of modes. The understanding of energy flow characteristics of the stiffened plate at the beam/plate interface is essential to effectively control the noise and vibration problems of structures such as transformer tanks and machine covers.

Computational and Theoretical Investigation of Acoustical and Vibrational Properties of Rigid Thin Material

A computational and theoretical investigation of acoustical and vibrational properties of rigid thin fiberglass material was carried out for different boundary conditions. Fiberglass materials could be applied in industries varying from the aircraft and automotive sectors to the built environment and construction sectors. Plate vibration and acoustic radiation were applied to predict the deflection of the thin fiberglass material and sound radiation efficiency at different locations on its surface, while a study-controlled equation of motion known as the Kirchhoff thin plate theory was applied for a COMSOL simulation of the thin material to determine the deflection of the plate and to obtain stress distribution, velocity contour, displacement, and acoustic pressure at the first resonance of the material. The results of this paper show that thin fiberglass material could be applied to sandwich building elements to form panels for reducing airborne noise and to lessen the sound transmission of structural borne noise, to cover noise barriers to make them more sustainable and weather resistant, to dampen the vibration of machines, and to reduce the structural vibration of buildings.

Planar piezoelectric metamaterials: Sound transmission and applicable frequency range in oblique incidence

Frequency limit and applicable frequency range of planar piezoelectric metamaterials connected to external circuits have not been well defined in estimating the sound transmission loss. This article extends the classical transfer matrix method for use in evaluating the sound transmission of thin-plate piezoelectric metamaterials in oblique incidence. Using the Kirchhoff thin plate theory, a modified transfer matrix method that takes factors of external circuits into consideration is developed for attenuation and control of acoustic waves. Several vibro-acoustic analytical models are compared, including the Kirchhoff thin plate theory, the Reissner–Mindlin thick plate theory and the theory of wave propagation in elastic solids. These theories are used to determine the dispersion relation, coincidence and transition frequency of thin and thick plate theories in analysing piezoelectric acoustic metamaterials, alongside a validation using the finite element method. Then acoustic properties of piezoelectric plates connected to passive external circuits are studied parametrically with both dimensional and dimensionless variables based on an equivalent Kirchhoff plate approximation. The findings show that external electrical impedance alone can be used to adjust the resonance frequency over a broad range and thereby control sound transmission loss. This provides considerable flexibility in modifying the acoustic properties of the piezoelectric metamaterial in comparison to traditional, fixed-structure metamaterials. The study indicates a straightforward and powerful analytical approach for the optimization of acoustic insulation using thin-plate piezoelectric metamaterials.

Homogenization and analysis of planar piezoelectric metamaterials: A comprehensive study

This article presents a comprehensive investigation into the homogenization and analysis of planar piezoelectric metamaterials. The classical transfer matrix method is extended by using homogenization methods to evaluate the sound transmission of piezoelectric metamaterials in both normal and oblique incidences. This has been validated numerically using the finite element method and experimentally in the solid medium propagation in normal incidence. Several vibro-acoustic analytical models are compared in oblique incidence, including the Kirchhoff thin plate theory, the Reissner-Mindlin thick plate theory, and the theory of wave propagation in elastic solids. These theories are used to determine the dispersion relation, coincidence, and transition frequency of thin and thick plate theories in analyzing piezoelectric metamaterials. Additionally, acoustic properties of piezoelectric layers connected to external circuits are parametrically studied, exploring both dimensional and dimensionless variables. Results indicate that significant control over the resonance frequency and sound transmission can be enabled by adjusting the external electrical impedance, flat-layer structures, and piezoelectric materials. This demonstrates excellent tunability and compactness of planar piezoelectric metamaterials for space-sensitive applications. The study indicates a straightforward and powerful analytical approach for the optimization of acoustic insulation using planar piezoelectric metamaterials.

Symplectic analytical solution for forced vibration of a multilayer plate system

The classical laminate and lattice sandwich plate structure can be simplified into a multilayer plate system, wherein the plate components of the system are continuously joined along the transverse direction by elastic layers and can have different combinations of boundary conditions. A symplectic analytical wave propagation approach is developed for the forced vibration of a system of multiple elastically connected thin plates considering the Kirchhoff thin plate theory. The proposed method overcomes the limitation of the traditional analytical method, wherein the exact vibration field function only exists for a system with all edges of the plate components simply supported. First, the coupled partial differential equations governing the vibration of the multi-plate system are decoupled using a technique based on matrix theory; for decoupled equations, a general “vibration” state is innovatively introduced into the symplectic dual system. Next, the general “vibration” state can be analytically described in symplectic space by solving the symplectic eigenproblem and utilizing wave propagation theory. Finally, by using these analytical wave shapes and satisfying the physical boundary conditions of the system, the forced responses can be analytically calculated. In the numerical examples, the forced transverse vibrations of the double- and three-plate systems are investigated, and the cases with various combinations of boundary conditions are considered. The effectiveness of the present method is validated by comparing the present results with the results from the literature and those calculated using the finite element method. The influence of the elastic layer stiffness and number of plate components on the vibration is also investigated.

The MLS based numerical manifold method for bending analysis of thin plates on elastic foundations

Compared with the finite element method, H2-regularity in the Galerkin based approximation to the Kirchhoff thin plate model can be easily realized using either the moving least squares (MLS) or the generalized moving least squares (GMLS), which take the Lagrange form and the Hermite form, respectively. Coupling (G)MLS with the numerical manifold method (NMM) can greatly improve numerical properties of NMM in the treatment of plates of complicated shape, thereby denoted by MLS-NMM and GMLS-NMM. In the (G)MLS-NMM, the mathematical cover is composed of simply connected and partially overlapped mathematical patches that are the influence domains of (G)MLS nodes. GMLS-NMM appears to better fit to the Kirchhoff plate because it is equipped with rotation angle degrees of freedom. Through numerical tests and theoretical analysis in solving problems of thin plates on elastic foundations, however, this study shows that MLS-NMM is much more advantageous over GMLS-NMM from the aspects of both accuracy and memory usage.

Dynamic Response Analysis of Orthotropic Saturated Subgrade-Pavement Slab

Based on the Kirchhoff thin plate theory, a three-dimensional spatial mechanical model of the subgrade pavement with an infinite elastic plate on the orthotropic saturated foundation in the rectangular coordinate system is established, and the differential equations under a moving harmonic load are derived. The partial differential equations are solved by using coordinate transform and Fourier transform, and the analytical solutions of the dynamic responses of the plate and foundation are obtained. A three-dimensional spatial finite element model of subgrade-pavement is established by ABAQUS finite element software. The correctness of the method is verified by comparing the theoretical solution with the finite element solution. The influence of foundation parameters on plate displacement and vertical normal stress of soil is further studied. The results show that considering the orthogonal anisotropy of saturated soil, the dynamic response of subgrade-pavement slab interaction can be described more accurately.

Theoretical Analysis of Free Vibration and Transient Response of Rectangular Plate–Cavity System Under Impact Loading

AbstractThis paper presents a theoretical method to solve the free vibration and transient responses of a rectangular plate–cavity system. The spectral collocation method was used to solve the resonant frequencies and corresponding mode shapes of rectangular plates based on Kirchhoff thin plate and Mindlin–Reissner thick plate theories. A linear velocity potential function was employed to model the fluid pressure applied to the plate surface. Unlike in previous studies, it was not assumed that the wet-mode shapes were the same as the dry-mode ones. Rather, the wet modes were assumed to be the superposition of the dry modes; then, the resonant frequencies and corresponding mode shapes of the wet modes could be obtained by solving the equations of the coupled system by exploiting the orthogonality of dry modes. Using dry modes’ orthogonality and superposition of the wet modes, the transient responses of the rectangular plate–cavity system under impact loading can be solved. A method for estimating the resonant frequencies of the coupled system is proposed based on parametric studies to determine the influence of the fluid properties and plate materials on resonant frequencies. As a result, the resonant frequencies and transient responses obtained from the proposed theoretical methods are in excellent agreement with those obtained from finite element analysis.

Equivalent Solution Method for the Analytical Transverse Modal Shape of Hollow Slab Bridges

Hollow slab bridges are the most widely used form of small- and medium-span bridges. The existing research on the dynamic characteristics of hollow slab bridges is mostly based on numerical models, but there is a lack of theoretical analyses of their dynamic characteristics. In this paper, the relationship between the dynamic characteristic parameters and structural parameters of a hollow slab bridge is explored theoretically. Firstly, the solid model of a hollow slab bridge was established, and a modal analysis was carried out on it as a reference. Then, an orthotropic plate was used as an equivalent dynamic analysis model, and the analytical form of the transverse modal shape was deduced based on Kirchhoff thin plate theory. Furthermore, one hinge joint was considered as being equivalent to the elastic support boundary, and the local structure and the equivalent elastic support boundary were used to reflect the transverse modal shape of the original structure. The analysis shows that the influence of hinge joints on the transverse modal shape is mainly reflected in the transmission of bending deformation. Through comparison and verification, the results show that the analytical expression of the transverse modal shape can well describe the low-order transverse modal shape of a hollow slab bridge.

Metamaterial based piezoelectric acoustic energy harvesting: Electromechanical coupled modeling and experimental validation

Piezoelectric energy harvesting technology utilizing acoustic metamaterials investigated by experiment and simulation improves the sound energy density and conversion efficiency. A fully coupled electromechanical model in both the mechanical and electric domains is crucial for accurate prediction and optimization of acoustic metamaterial based on sound energy harvesting. Based on the Kirchhoff thin plate theory, constitutive laws of isotropic aluminum substrate and transversely isotropic piezoelectric bimorph, the electromechanical coupled governing equation in the mechanical domain is deduced for a two-dimensional acoustic metamaterial based piezoelectric energy harvester. Each piezoelectric layer is represented as a current source in series with its internal capacitance for derivation of the electromechanical coupled governing equation in the electrical domain via Gauss’s law and Kirchhoff’s laws. With generalized boundary conditions, modal analysis is performed using the Galerkin method and Kelvin-Kirchhoff condition. The general transient response and frequency response functions are obtained under the excitation of sound waves and the reaction forces of the silicone rubbers. The time-history, power spectrum and phase portrait of the measured harvested voltage at 90 fixed frequencies agree well with the model predictions. The theoretical maximal harvested power is 3.09 mW, with the optimal resistance of 28180 Ω and inductance of 0.28H connected in parallel. Near the resonant frequency, the utilization rate of converting sound energy into electrical power is maximized. The proposed electromechanical coupled model can be used as a basis for further topology design and optimization and underlying physics for experimental study.

Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

AbstractIn this paper, based on the extended Melnikov method the homoclinic analysis for chaotic vibration of a double layered viscoelastic nanoplates (DLNPs) embedded in Winkler–Pasternak elastic foundation and under in‐plane excited biaxial loads using the Eringen's nonlocal elasticity theory has been investigated. The governing nonlinear coupled partial differential equations (PDEs) of motion for DLNP considering viscoelastic Kelvin–Voigt model in the Kirchhoff's thin plate theory in conjunction with the von‐Karman geometrical nonlinear strain–displacement relations have been derived. Then, applying the Galerkin method the governing nonlocal coupled PDEs of motion of DLNP have been converted into the nonlinear coupled ordinary differential equations (ODEs) using mode summation technique and discretized in terms of time. The criteria for the homoclinic transverse intersection for synchronous and asynchronous types of buckling are studied to predict the homoclinic orbits and chaotic motions applying the generalized Melnikov method. Besides, the critical buckling load has been determined for asynchronous and synchronous types of buckling. The obtained results are validated with comparing them with the results presented in the literature for a special case. Then, the effect of changes of different parameters including nonlocal small‐scale parameter, foundation coefficients, biaxial load ratio, geometric aspect ratio on the critical buckling load and ratio of nonlocal to local natural frequency have been studied. The critical threshold surface for the synchronous and asynchronous manifolds to predict transversely condition for the chaotic motion are discussed on the parametric space.

Dynamics of plates resting on layered transversely isotropic poroelastic media under moving loads

This paper investigates the dynamic interaction between a plate and multi-layered transversely isotropic (TI) hysteretically damped poroelastic media under a moving rectangular load. Following Kirchhoff's thin plate theory, the plate deflection is derived utilizing the Fourier series expansion and the integral transform technique. The governing equation for the layered TI hysteretically damped poroelastic media is formulated through the standard hysteretic damping model and Biot's theory. The solution for the plate-soil dynamic interaction in the wavenumber domain considering the plate-soil interface condition is obtained by the extended precise integration method (PIM) for the layered TI saturated porous media. This solution can be further converted into the spatial domain by the inverse transform technique. We have verified the efficiency and accuracy of the proposed method. Several numerical examples are presented to investigate the influences of the plate thickness, plate modulus ratio, surface layer, speed of the moving load, and hysteretic damping of the solid skeleton on the dynamic behaviors of the plate on layered TI poroelastic media.

Singular elastic field induced by a rigid line adhering to a micro/nanoscale plate during bending

In this work, the elastic analysis of a micro/nanoscale plate with a stiffer ribbon adhering to it is studied. The stiffer ribbon is understood as a rigid line. The problem is solved by superposition and the singular elastic field induced by a rigid line is concerned when the plate is during bending. The singular elastic field of a micro/nanoscale elastic thin plate with surface elasticity is determined for a parabolic rigid line. The theoretical analysis is based on a mixed boundary value problem that is solved through the Fourier integral transform technique. According to the classical Kirchhoff thin plate theory incorporating surface elasticity, some basic equations are established. The mixed boundary value problem is reduced to a singular integral equation of the first kind with Cauchy kernel. Explicit expressions for the moments and effective shear forces as well as the surface and bulk stress components in any position of the whole nanoplate are obtained. The singularity coefficients of the moment and effective shear force at the tip of the rigid line are determined in closed form. From the obtained results, we find that the normal stress components for bulk and surface phases and bending moments at the rigid line tips have a usual inverse square-root singularity, but the effective shear forces behave like r−3/2, r being the distance from the tip of the rigid line or admit an r−3/2 singularity. The numerical results are displayed graphically and show that the singularity coefficients are heavily dependent on the surface and bulk material constants.

An Electromechanical Model for Clamped-Edge Bimorph Disk Type Piezoelectric Transformer Utilizing Kirchhoff Thin Plate Theory.

In this paper, an analytical solution for a clamped-edge bimorph disk-type piezoelectric transformer with Kirchhoff thin plate theory is proposed. The electromechanical equations for transient motions are first derived, and coupled expressions for mechanical response and voltage output are obtained. For the case of excitation around the first resonant frequency, the resulting equations are further simplified. There are analytical solutions for a mechanical response, voltage, current, and power outputs. According to the analytical model, the output voltage is affected by the inner radius of the input and output electrodes, the radius of the piezoelectric transformer (PT), and the thickness ratio between the lead zirconate titanate (PZT) layer and the substrate. When the inner radius of the input electrode approaches zero (electrode becomes circular shape), it achieves maximum output voltage at the first resonance frequency excitation. On the contrary, when the inner radius of the output electrode approaches zero, the output voltage reaches its minimum value. Voltage ratios remain constant as the disk radius changes, and the first resonance frequency is inversely proportional to the square of the disk radius. The voltage ratio is fixed even with the miniaturization of the PT.

Frequency Comb Free Vibration Behavior of a Single-Span Plate Pumped by Low-Speed Moving Inertial Loads

Structures carrying moving loads present rich vibration phenomena and colorful spectrum characteristics. This study analytically investigates the moving-inertial-load-pumped frequency comb free vibration behavior of a single-span rectangular linear elastic Kirchhoff thin plate in the low-speed region. Multiple-time-scales expansion is made and the asymptotic solution describing the non-resonance modal oscillation is derived. Case studies on a single-span rectangular isotropic plate with the clamped opposite boundaries dominated by the first two transverse flexural modes are presented. Numerical computation based on finite element method structure modeling and response numerical integration verifies the modal frequency comb characteristics as predicted by the proposed theoretical model. The results also present the weak modal interaction for the studied single-span structure in the observed light-mass low moving speed region. The outcomes of this study will benefit the related structural dynamic system design and intervention.

The dynamic responses of an infinite plate resting on a poroelastic layered half‐space soil medium with imperfect interface to a moving load

AbstractThe present study deals with the dynamic responses of an infinite plate resting on a poroelastic layered half‐space soil medium with imperfect interfaces to a moving load. Kirchhoff thin plate theory is employed to analyze the plate, while Biot's fully dynamic poroelastic theory is used to characterize the poroelastic half‐space. Applying the Fourier transform and vector functions, the completely coupled system of equations of motion for the layered system are reduced to a system of decoupled ordinary differential equations which could be solved semi‐analytically. The propagation relationships for the layered structure with imperfect interfaces are derived by the novel and efficient dual variable and position (DVP) method. In so doing, the calculated expansion coefficients in the transformed domain are utilized to calculate the unknowns in the physical domain by integration at each field point. Numerical analyses validate the obtained results by comparing with the existing work, and soil vertical displacements, accelerations and pore water pressures induced by moving load are calculated. Moreover, the effects of the different factors are investigated. Computed result shows apparent influences of load velocity, intrinsic permeability and imperfect bonded condition of the soil medium on the dynamic responses of the whole system.