Mindlin Plate Research Articles

Constrained layer damping for mitigating vibration of a rotating disk-drum coupled structure

Rotating disk-drum coupled structures are commonly employed in large rotating machines. They are prone to have vibration issues caused by their decreasing thickness to meet lightweight demands. Constrained layer damping (CLD) is a widely used technique to reduce vibrations, but no studies have yet investigated its application on rotating coupled structures. In this paper, a universal model is proposed and experiments are conducted to investigate the influence of CLD on a rotating cylindrical shell ended by an annular disk. This approach considers the rotating effects, and formulates the equations of motion using the Rayleigh-Ritz method with the help of the Sanders shell theory and Mindlin plate theory. Different sets of artificial springs are applied to model the coupling and boundary conditions. The proposed approach is validated by comparing the numerical results with those obtained from modal and rotor experiments. Then the modal parameters and vibration responses are evaluated for different rotational speeds, and the influences of the location and thickness ratio of the CLD patch and the coupling stiffness on the dynamic characteristics of the system are discussed.

Damping optimization of viscoelastic thin structures, application and analysis

In this work, damping properties of bending viscoelastic thin structures are enhanced by topology optimization. Homogeneous linear viscoelastic plates are optimized and compared when modeled by either the Kirchhoff–Love or Reissner–Mindlin plate theories as well as by the bulk 3D viscoelastic constitutive equations. Mechanical equations are numerically solved by the finite element method and designs are represented by the level-set approach. High performance computing techniques allow to solve the transient viscoelastic problem for very thin 3D meshes, enabling a wider range of applications. The considered isotropic material is characterized by a generalized Maxwell model accounting for the viscoelasticity of both Young modulus and Poisson’s ratio. Numerical results show considerable design differences according to the chosen mechanical model, and highlights a counter-intuitive section shrinking phenomenon discussed at length. The final numerical example extends the problem to an actual shoe sole application, performing its damping optimization in an industrial context.

Analytical modeling on the vibration response of a beam‐stiffened Mindlin thick plate with free boundary conditions

AbstractThis paper presents a new analytical solution for the vibration response of a beam‐stiffened Mindlin plate having a completely free boundary condition by utilizing a finite cosine integral transform. In the solution, the unknown coupling force and moments at the beam/plate interface and the unknown modal constants from the integral transform are determined by the continuity and compatibility conditions at the interface as well as the boundary conditions. It provides an easily implemented tool for exploring complex edge value problems for a class of higher‐order partial differential equations represented by fully free‐stiffened Mindlin thick plates. The validity of the model is evaluated by comparing the calculated free and forced vibration responses of the beam‐stiffened plate with those calculated using a beam‐stiffened thin plate and those from finite element analysis.

Stochastic Finite Element Analysis of Plate Structures Considering Spatial Parameter Random Fields

For plate structures, their random parameters can be regarded as a two-dimensional random field in the plane. To solve the plate theory considering a two-dimensional random field, an efficient strategy for the stochastic finite element method was adopted. Firstly, the stochastic finite element method was used to establish the plate structural model, in which the random field characteristics of the parameter were considered, and the mathematical expression of its random field was obtained through the Karhunen–Loève expansion; secondly, the point estimate method was applied to calculate the statistics of random structures. The computational efficiency can be significantly improved through the reference point selection strategy. The accuracy and efficiency of the calculation strategy were verified, and the influences of correlation length and coefficient of variation of the parameter on the random response of plate structures under different plate types (including Kirchhoff plate and Mindlin plate) and boundary conditions (including simply supported and clamped supported) were discussed. The proposed method can provide some help in solving static problems of plate structures.

Parametric study for modified couple stress theory on postbuckling of size-dependent FG saturated porous mindlin microplates

Using modified couple stress and Mindlin plate theory, the present study is a parametric study for revealing the effect of length scale parameter on postbuckling of saturated poroelastic rectangular microplates with pore distribution being functionally graded across thickness. The microplate postbuckling is a nonlinear eigenvalue problem with coupled partial differential equations of equilibrium. In order to find the mode shapes, the eigenfunctions have been approximated by some complete polynomial series expansions in accordance with the conventional Ritz approximation technique. After discretizing the governing equations throughout the microplate grid points, the assembled difference equations construct a nonlinear eigenvalue problem with coupled algebraic equations. The nontrivial coefficients of the expansions were calculated such that the eigenfunctions extremize the microplate potential energy for satisfying the equilibrium homogeneous equations. Variation of the potential energy have been derived with respect to the expansions coefficients from the vantage of using Maple software. By gradual reduction in the initial end shortening of the movable supports and solving the assembled difference equations via Newton-Raphson iterations, the critical postbuckling displacement imposed to the movable supports are found along with the nontrivial coefficients regarding eigenfunctions. The results of the parametric study have revealed how geometric parameters, pore distribution, the Skempton’s coefficient, porosity, and different Supports modify the influence of length scale parameter on the microplate postbuckling.

An Improved Cell-Based Smoothed Discrete Shear Gap Method (CS-DSG3) for Static and Dynamic Analyses of Reissner–Mindlin Plates

The efficiency and accuracy of the cell-based smoothed discrete shear gap (CS-DSG3) are improved in simplifying the application process of the cell-based strain smoothing technique and redefining the stabilized parameter of Stenberg’s stabilization method, respectively. In the original CS-DSG3, both the smoothed bending strain and the smoothed shear strain are derived by the cell-based strain smoothing technique. This study proves that the smoothed bending strain matrix in the original CS-DSG3 is equal to the unsmoothed bending strain matrix. Theoretically, abandoning the bending strain smoothing operation can improve the efficiency while maintaining the accuracy of the original CS-DSG3. The introduction of Stenberg’s stabilization method can boost the accuracy of shear strain in the original CS-DSG3, whereas the relationships of the stabilized parameter in Stenberg’s stabilization method with the mesh density, the thickness–span ratio, and the boundary condition are still unclear. In this study, the range of the optimal stabilized parameter for different analyses and boundary conditions is determined through the static and dynamic analysis of the plate. The new stabilized parameter is applied to static analyses, free vibration analyses, and frequency response analyses for verification and comparison, which demonstrates that it can significantly improve the accuracy of the CS-DSG3.

Semi-analytical solution for interfacial debonding of high-speed railway ballastless track under thermal loading using a quasi-dynamic method

Interfacial debonding of multi-layered ballastless track systems has proved to be one of the most common diseases in the high-speed railway during long-term operations. This paper focuses on the theoretical modeling of track temperature field and consequent bond-slip behavior between track layers under thermal loading. By establishing heat conduction differential equations, whose boundary conditions are combined with complicated geographical and meteorological factors, the temperature field of a multi-layered track structure is obtained via the integral transform technique. The reliability of the heat conduction model is well-validated by an on-site experiment in a high-speed railway line. Then, we originally extend an explicit integration algorithm for dynamics analysis to solve nonlinear statics problems with small deformation by proposing a quasi-dynamic method (QDM). Equilibrium equations of a track slab subjected to quasi-dynamic equivalent thermal loads and mixed-mode interfacial cohesive forces are constructed based on the Mindlin plate in-plane and flexural vibration equations. The application of the QDM to solve interface responses is verified by comparing results obtained from finite element (FE) software. Finally, numerical discussions are conducted to reveal the nonlinear distribution characteristics of temperature field, evolution process of the interface damage and failure modes under time-varying thermal loads, which provide a better understanding of the interfacial debonding phenomenon encountered in ballastless track systems.

A compound strip method based on the Mindlin plate theory for static and buckling analyses of thin-walled members with transverse stiffeners

The compound strip method origins from the finite strip method (FSM) aimed to analyze thin-walled members with additional parts such as bolts and stiffeners. Traditionally, FSM mainly solves the buckling problem of prismatic members with longitudinally uniform sections. However, existing FSM could not take into account the stiffeners’ effect in the present form. The paper proposes a compound strip method based on the Mindlin plate theory (MCSM), in which members are modeled by finite strips while stiffeners are by MITC4 shell elements. This combination can effectively connect the stiffeners with the flange/web of the members. Meanwhile, the contribution from the thickness of stiffeners to buckling strengths could be considered accurately. The validity of MCSM is demonstrated through the comparison of the deformation and capacity from static and buckling analyses with solutions of the finite element method (FEM). The developed MCSM is then used to explore the impact of the number and thickness of stiffeners on the members. Besides, the discussion on how sectional characteristics affect stiffeners in improving the buckling loads is carried out. To ensure the calculation in MCSM is accurate and efficient, a study on meshing schemes is also presented.

A variational multiscale stabilized finite element formulation for Reissner–Mindlin plates and Timoshenko beams

A variational multiscale stabilized finite element formulation for Reissner–Mindlin plates and Timoshenko beams

A Discontinuous Petrov–Galerkin Method for Reissner–Mindlin Plates

We present a discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model. Our method is based on a variational formulation that utilizes a Helmholtz decomposition of the shear force. It produces approximations of the primitive variables and the bending moments. For any canonical selection of boundary conditions the method converges quasi-optimally. In the case of hard-clamped convex plates, we prove that the lowest-order scheme is locking free. Several numerical experiments confirm our results.

Equivalent plate dynamic modeling of space periodic truss structures

Antenna structures used in spaceflight always contain nonlinear joints distributed along their cross section; hence, an equivalent beam model (EBM) that uses a point to describe the cross section is not ideally suited to depicting connection nonlinearity in the structure. To address this difficulty, a novel orthotropic equivalent plate model (EPM) of a space periodic truss structure is proposed. Firstly, utilizing the Mindlin plate hypothesis, strain in the periodic truss element is expanded into a binary high-order Taylor series, and corrections are introduced to normal strain. Secondly, employing the principle of energy equivalence, the equivalent elastic and inertial matrices of the EPM are calculated for in-plane and out-of-plane vibration. Finally, the static condensation method is utilized to reduce the dimensions of the equivalent model to establish an orthotropic equivalent Mindlin plate dynamic model, and its inherent characteristics are analyzed. The simulations show that when in-plane shear strain is expanded to the third order, the EPM can more accurately represent torsional vibration of the periodic truss structure. By introducing normal strain correction coefficients in the longitudinal direction, lateral bending vibrations of the periodic truss structure can be simulated more precisely. Compared with the EBM, the EPM can more accurately depict mechanical characteristics along the cross section. The EPM broadens the scope of continuum equivalent modeling of periodic trusses, and is more useful in engineering practice. In addition, the frequencies and mode shapes of the analytical EPM are consistent with a model using the finite element method, which supports the accuracy and effectiveness of the EPM. The analytical EPM provides a convenient way to study the nonlinear connections and vibration control of antenna structures with multiple joints along their cross section.

Underwater sound radiation from a Mindlin plate with an acoustic black hole

Acoustic black holes (ABHs) have been shown very efficient to reduce sound radiation in the air. However, their underwater performance is not clear. In this paper, we consider a Mindlin plate with an embedded circular ABH indentation. The Gaussian expansion method is used to compute the rotations and bending displacement at the mid-surface, with boundary conditions imposed via the nullspace method. The interaction between the solid and fluid is accounted for by the work done by the sound pressure. The theoretical method is validated against FEM and hybrid BEM-FEM simulations. Results show that the ABH is very effective for suppressing the underwater sound power between the cut-on frequency and the critical frequency. Yet at higher frequencies, the advantage of the ABH is not significant. The underlying physics is revealed using modal loss factor (MLF) that is obtained by solving the eigenvalue problem with the Newton iteration method. It is found that beyond the critical frequency, the radiation damping is very high, because of energy transmission from the plate to the water. Observation on the non-negative intensity (NNI) demonstrates that at higher frequencies the radiation area moves from the plate corners to the force point.

Dynamic Analysis of Mindlin Plate Subjected to a Moving Mass-Spring-Damper System Using the Moving Element Method

The moving element method (MEM) is applied in this paper to study the dynamic behavior of the Mindlin plate subjected to a moving mass-spring-damper system. Reissner- Mindlin plate theory is utilized to study the plate structure. The plate is modeled by the Isoparametric quadrilateral nine-node element. The governing equation of motion is built into a coordinate system which moves with the moving mass-spring-damper system and is based on the principle of virtual work. The MEM is proposed to solve the governing equation of motion of the plate resting on a Pasternak foundation under the moving mass-spring-damper system including roughness of plate surface which is a dynamic source. Numerical results are investigated and verified by comparing with the published results. The effects of different parameters, such as spring stiffness, damping coefficient and velocity of the moving load system, plate thickness, foundation parameters and roughness of the plate surface on the dynamic response of the Mindlin plate are investigated.

Deep-Learning-Based Acoustic Metamaterial Design for Attenuating Structure-Borne Noise in Auditory Frequency Bands

In engineering acoustics, the propagation of elastic flexural waves in plate and shell structures is a common transmission path of vibrations and structure-borne noises. Phononic metamaterials with a frequency band gap can effectively block elastic waves in certain frequency ranges, but often require a tedious trial-and-error design process. In recent years, deep neural networks (DNNs) have shown competence in solving various inverse problems. This study proposes a deep-learning-based workflow for phononic plate metamaterial design. The Mindlin plate formulation was used to expedite the forward calculations, and the neural network was trained for inverse design. We showed that, with only 360 sets of data for training and testing, the neural network attained a 2% error in achieving the target band gap, by optimizing five design parameters. The designed metamaterial plate showed a -1 dB/mm omnidirectional attenuation for flexural waves around 3 kHz.

Design and bandgap optimization of multi-scale composite origami-inspired metamaterials

This paper seeks to present the design and bandgap optimization of multi-scale composite origami-inspired metamaterials (MCOMs) reinforced with grapheme platelets (GPLs) and carbon fibers. The MCOMs are constructed with a series of hybrid origami single-configuration that spatially reconfigure within prescribing valley and mountain lines. A semi-analytical periodic spectral plate method for the wave analysis of two-dimensional (2D) periodic origami-inspired metamaterials is proposed based on the Mindlin plate theory, higher-order spectral elements, and the Bloch theorem. Various significant parameters, including the weight fraction of GPLs, the volume fraction of carbon fibers, and the fiber orientations, are considered to investigate wave propagation in MCOMs. The results indicate that complete bandgaps (BGs) can be realized in MCOMs with three different spatial configurations, and distributions of nano-reinforced materials cause extraordinary changes in band structures. Subsequently, given that the BG at low frequencies is more attractive in practical engineering applications, the main objective of this paper is to maximize the sum of bandwidths of the first and second BGs employing various significant parameters. The optimization framework is suggested by incorporating the semi-analytical method in an optimization scheme based on particle swarm optimization (PSO). The design objective of MCOMs with three different spatial configurations generated from the proposed method achieves 185.6%, 10.2%, and 61.5% improvement compared with that obtained from the initial design, respectively, which proves that the proposed novel MCOMs and non-uniform distributions of nano-reinforced materials are quite effective for broadband wave attenuation. This work can provide a new thought for designing novel composites and metamaterials.

Estimation of plate parameters from vertical displacement data using a family of plate models

We develop a method for estimation of parameters of an elastic plate resting on a Winkler-type elastic foundation solely from data on the vertical displacements of the plate. The method allows one to estimate components of the external body force density field, plate thickness, elastic foundation stiffness parameters, horizontal displacements of the plate, and stresses. The key idea of the method is that multiple plate models are used simultaneously, namely the proposed reduced three-dimensional (R3D) plate model, the Mindlin plate model, and the thin plate model. The three plate models form a hierarchy of elastic plate models based on assumptions imposed on stresses, with the R3D plate model being the most generalized model and the thin plate model being the most constrained one. The hierarchical relationship among the plate models allows one to incorporate prior information into the estimation technique. The applicability of the proposed estimation method is illustrated by a numerical example. For more information see https://ejde.math.txstate.edu/Volumes/2023/15/abstr.html

Effect of crack characteristics on the vibration behavior of post-buckled functionally graded plates

This study investigates the free vibration analysis of a cracked post-buckled functionally graded plate under a uniaxial compressive load. The crack is assumed to be open-edged and modeled using a linear, rotational spring. The material is distributed through its thickness gradually. The nonlinear differential equations of motion are derived using the Mindlin plate theory for an initially imperfect plate. First, the nonlinear static differential equations are converted to algebraic equations using the differential quadrature method and solved by an arc-length strategy. The solution of the post-buckling equations gives the equilibrium state of the vibration. Next, the results were implemented into the motion equations to derive the vibrational differential equations. Then, the solution of the standard linearized eigenvalue problem, using the differential quadrature method, results in the plate's natural frequencies and mode shapes. The study compares the effects of various plate parameters on its vibrational behavior before and after buckling. It can be an excellent reference to consider the effect of crack characteristics on vibrations of the functionally graded plates before and after buckling. The presented results agree well with those available in the literature and those computed from the finite element method.

A novel method for sub-wavelength focusing of flexural waves

A novel method for achieving sub-wavelength focusing of reflected flexural waves in a single-phase plate is proposed in this paper. The original plane lens is divided into two adjacent lenses, and modulating the internal angle between them, more reflected wave components from broader directions can arrive at the focusing position, which leads to the sub-wavelength focusing phenomenon with the focusing size smaller than half wavelength. Firstly, Mindlin plate theory is adopted to solve the reflected wave, and lenses with different width values are designed based on phase modulation for wave focusing. After that, the sub-wavelength focusing is exemplified numerically and experimentally, with the reason explained systematically. Remarkably, the focusing size perpendicular to wave propagation direction locates (0.4λ, 0.5λ), which is controlled by the internal angle. It is demonstrated that the subwavelength focusing is general because it is still suitable when the lenses are designed via the thickness variation or other phase distribution. Additionally, the lenses exhibit broadband property, which can work in a certain frequency region centered the designed frequency. Not limited by flexural wave in plates, the design methodology in this paper can also lay the foundation for wave control in electromagnetics and acoustics.

Finite element analysis of bending waves in Mindlin plates with Perfectly Matched Layers

It is important to determine the boundary conditions of walls and floors precisely when simulating the building acoustics. For a certain room, the extension of the spans can be considered as infinite edges. The Perfect Matched Layer(PML) is an artificial absorbing domain for the wave propagations and is widely used in finite element analysis to simulate the acoustical free field conditions right now. In this paper, an effective PML technique for the plate structure will be presented. The PML formulation will be derived based on the Mindlin plate theory and the implementation method will be introduced. This technique will be validated through the numerical experiments. The accuracy and limits of the presented technique will be discussed based on the numerical results compared with the analytical results. The results show that the presented PML technique is effective and accurate to simulate the plate as an infinite large plate. It is expected to implement the technique in the further research of the structure borne sound, such as floor impact sounds.

An improved analytic model for designing the polymer-composite stepped-plate transducer using the modified Mindlin plate theory

A stepped-plate transducer (SPT) uses an extensive radiating plate to produce highly-directional ultrasound beams. In this paper, we present an improved analytical model for designing the polymer-composite stepped-plate transducer (PCSPT). The polymer-composite features the lightweight and flexible properties, and there can be little change in the resonant frequency and mode shape when the steps are attached. With the outstanding merit, it is feasible to construct SPTs with polymer-composite steps without taking the steps into consideration. The modified Mindlin plate theory (MMPT) is applied to improve the accuracy in the equivalent circuit model (ECM) that is used to predict the high-frequency vibratory responses. Our analytical model can be used to design well-tuned SPTs to achieve the desired dynamic responses such as resonant frequencies, mode shape and bandwidth for various high-power ultrasonic applications. We use several numerical design examples to illustrate that the design of the transducer can be accomplished without analyzing the sophisticated stepped-plate’s behavior. We also perform a series of experiments to verify that the PCSPT is capable of functioning as a high-power ultrasonic transducer.