Spectral Element Matrix Research Articles

Bandgap characteristics of periodic mindlin plates under arbitrary boundary conditions via the Spectral Element Method

This paper employs the Spectral Element Method (SEM) to investigate the bandgap characteristics in periodic plates under arbitrary boundary conditions. The spectral element model of Mindlin plate is derived via force-displacement relations. The convergence of the SEM results for a single plate structure is validated by comparing them with the Finite Element Method (FEM) results at distinct mesh resolutions, demonstrating that SEM maintains high computational accuracy even with fewer elements. Like FEM, the spectral element matrix is assembled to establish the spectral element model of the periodic plate. The influences of boundary conditions, substructures, material properties, and geometric dimensions on the bandgap characteristics in the periodic plate are explored.

Bandgap analysis of partial-interaction composite beams periodically attached vibration absorbers

Elastic metamaterials with local resonant components exhibit unique bandgap behavior that can be utilized to control the vibration and noise of mechanical structures. This study aims to introduce the bandgap characteristics into the vibroacoustic control of partial-interaction composite beams (PICBs) widely adopted in civil engineering, aviation, and marine fields. Vibration absorbers are arranged periodically on the beam to generate a bandgap in PICBs, thereby transforming a conventional PICB into a metamaterial beam. An analytical approach is proposed to establish the spectral element (SE) matrix of a metamaterial beam based on the Timoshenko–Ehrenfest beam theory. Specifically, the governing equations of motion for PICBs follow Hamilton's principle. This model considers the effects of shear slip between two-layer sub-beams, shear deformations, and rotary inertia. The SE matrix of PICBs is then constructed and applied to obtain the SE matrix of the metamaterial beam by assembling the SE matrix of the vibration absorbers. Subsequently, the wave finite element (WFE) method is employed to predict the dispersion curve of the infinite metamaterial beam and the dynamic response of the finite metamaterial beam. In a benchmark example, the complex band structure and transmission indicated by the proposed approach are validated compared to those calculated by evaluation formulas of the literature and the finite element method. Lastly, the bandgap behavior for the vibration absorber parameters, lattice constants, and shear connector stiffness is investigated based on the defined dimensionless parameters. These results are valuable for optimizing vibration and noise reduction in PICBs equipped with attached vibration absorbers.

Frequency dependent effective modulus of square grid lattice using spectral element method

A spectral element method-based formulation is developed to obtain the frequency-dependent effective Young’s and shear modulus of a two-dimensional square grid lattice. The methodology presented in this work is a simplified approach to obtain effective parameters. The lattice is assumed to be constituted of Euler–Bernoulli beam elements which are arranged in a specific order to obtain the said geometry. Spectral element formulation of the beam element considering axial and flexural deformation is presented, which in due course is employed to obtain the global spectral element matrix of the unit cell. A detailed procedure to obtain the frequency-dependent stiffness matrix of the unit cell is presented on which proper boundary conditions are applied using specific transformation matrices for each case. The frequency-dependent properties are also calculated using the software package COMSOL Multiphysics to validate the results. Frequency responses of the unit cell are obtained using the proposed methodology as well as the software package, and the results are compared. It is observed that the proposed methodology predicts the peaks at nearly the same frequency range as obtained through the finite element solution obtained in COMSOL Multiphysics. Finally, the results are presented which show the frequency-dependent effective parameters for the unit cell. The methodology presented here opens up a new way to calculate the effective properties of two-dimensional lattice structures and with some modifications can also be extended to three-dimensional lattice structures.

The exact spectral element modeling and vibration analysis of the acoustic black hole double-beam system

In this paper, we present a study of vibration characteristics of the double-beam structures combined with the concept of the acoustic black hole (ABH), which is an effective technique for vibration and noise control. In the proposed double-beam structure, the ABH beam as the vibration damper is attached to the primary uniform beam by a range of translational and rotational springs. We formulate the closed-form spectral element matrix of the double-beam structure and calculate the natural frequencies and mode shapes of the system. The results demonstrate the ABH effect of increasing the modal damping ratios. We also study the power flow and mechanical intensity of the system to offer physical insight into the vibration suppression mechanism of the ABH. A detailed parametric analysis of both translational and rotational springs is carried out. The investigation contributes to the exact dynamic modelling and analysis of the double-beam system containing ABH elements. Furthermore, the proposed ABH double-beam structure shows great potential for vibration control and energy harvesting.

Flexural Wave Propagation in Rigid Elastic Combined Metabeam

Abstract In this paper, flexural wave propagation, attenuation, and reflection through finite number of rigid elastic combined metabeam (RECM) elements sandwiched between two Euler Bernoulli beams has been studied, implementing the spectral element, inverse Fourier transform, and transfer matrix method. Spectral element has been formulated for the unit representative cell of RECM employing the rigid body dynamics. Governing dimensionless parameters are identified. Furthermore, the sensitivity analysis has been carried out to comprehend the influence of non-dimensional parameters, such as mass ratio, length ratio, and rotary inertia ratio on the attenuation profile. Rotary inertia of rigid body produces local resonance (LR) band, which may abridge the gap between the two Bragg scattering (BS) bands and results in an ultra-wide stop band for the specific combination of governing non-dimensional parameters. A total of 164% normalized attenuation band is possible to obtain in RECM. Natural frequencies for the finite RECM have also been evaluated from the global spectral element matrix and observed that some natural frequencies lie in the attenuation band. Therefore, the level of attenuation near that natural frequencies is significantly less and cannot be identified from the dispersion diagram of the infinite RECM.

Development of spectral element method for free vibration of axially-loaded functionally-graded beams using the first-order shear deformation theory

In this study, the spectral element method to assess free vibration of axially-loaded functionally-graded beams according to the first-order shear deformation theory is developed. In this approach, the underlying equations of motion and related boundary conditions were determined by using Hamilton's principle. Analytical solutions are presented for simply-simply, clamped-clamped, clamped-free, and clamped-simply supported FG beams. The general solutions corresponding to governing equations determined for axially loaded FG beams are presented in the spectral-element matrix. A comparison is performed to validate the proposed formulation and solution between the obtained results with the existing solutions provided in previous studies. Finally, the parametric study is developed to investigate the impacts of slenderness ratio, end supports, gradient parameter, and axial load value on the free vibration characteristics in the axially-loaded FG beams. A comparison of the results of the developed theory and the results of the existing solutions shows that the proposed solution developed based on the spectral element method is the appropriate accuracy and efficiency in determining the dimensionless natural frequencies parameter. Based on the findings, all four parameters are effective in the free vibration of axially-loaded FG beams.

Exact frequency-domain spectral element model for the transverse vibration of a reangular Kirchhoff plate

As an extension of the previous work on membranes by Kim and Lee (2020), an exact frequency-domain spectral element model is proposed for the transverse vibrations of a rectangular Kirchhoff plate, based on the following procedure. First, in the frequency-domain, the general solution of a finite rectangular plate element is derived in the spectral form in the spatial domain, after removing all the trigonometric terms that vanish at the four boundary edges. Second, as an important theoretical improvement, the boundary functions that represent the boundary conditions at the four boundary edges are also expressed in the spectral forms in the spatial domain. Next, by using the projection method based on the orthogonality of trigonometric functions, the spatial-domain spectral coefficients of the general solution are related to those of the boundary functions. Finally, from the force-displacement relationships, the spectral element matrix (or dynamic stiffness matrix) is formulated for a finite rectangular plate element. As both the general solution and boundary functions are expressed in the spectral forms in both the temporal and spatial domains, a fast Fourier transform (FFT) algorithm can be applied to efficiently simulate the vibration responses and waves in a plate. The accuracy and computational efficiency of the proposed SEM are evaluated by comparison with the exact theory, modal analysis method, and the commercial finite-element-analysis software ANSYS.

Elastic wave and vibration bandgaps in planar square metamaterial-based lattice structures

In this work, a new type of planar square lattice structures for the attenuation of elastic wave propagation is proposed and designed. To obtain a high vibration attenuation in low frequency ranges, the internal resonance mechanism of the acoustic metamaterials (AMs) is introduced into the continuous square lattice system with a cross-type core consisting of two segments with a radius jump discontinuity in each unit-cell. The wave propagation and vibration bandgap properties of the AM-based lattice structures are studied by using a spectral element method (SEM) which can provide accurate dynamic characteristics of a lattice structure due to its analytical spectral element matrix. The band structures calculated show that the phononic bandgaps induced by the local resonance and destructive interference co-exist in the considered systems. The effects of the structural and material parameters on the bandgaps are investigated in detail by the frequency response analysis of a spectral element model subjected to a dynamic excitation. Finally, in order to obtain lower and wider frequency bandgaps, the mechanism to optimize the unit-cell of the AM-based lattice structures is illustrated and a composite lattice model is suggested.

Exact spectral element model for rectangular membranes subjected to transverse vibrations

This study proposes an exact frequency-domain spectral element model for a rectangular membrane subjected to arbitrary boundary conditions. The frequency-domain general solution of a finite rectangular membrane element is first assumed in the trigonometric Fourier series with the spatial coordinates x and y. After all the trigonometric Fourier series terms that vanish at the four boundary edges are eliminated, the general solution is expressed in the spectral (i.e., discrete Fourier transform) forms with respect to the spatial coordinates x and y. By using the projection method, the spatial-domain spectral coefficients of the general solution are related to the spatial cosine/sine series coefficients of the functions specified for the geometric and natural boundary conditions. Lastly, the spectral element matrix (i.e., dynamic stiffness matrix) is formulated from the force–displacement relationships. Owing to the spectral representation of the general solution in both temporal and spatial coordinates, the well-known fast Fourier transform algorithm can be applied for efficiently obtaining the dynamic responses and wave propagations in a membrane, in both the time and spatial domains. The high accuracy of the proposed spectral element model for the rectangular membranes is verified via comparison with existing solution techniques, such as the exact theory, modal analysis method, and finite element method.

State-vector equation method for the frequency domain spectral element modeling of non-uniform one-dimensional structures

This paper presents a state-vector equation method (SVEM)-based spectral element method (SEM) to formulate frequency domain spectral element models for non-uniform 1-D structures. In the SVEM-based SEM, the frequency domain homogeneous governing differential equations for non-uniform 1-D structures are first transformed into state-vector equations. The system matrix and state-vector are represented with power series. These power series representations are then substituted into the state-vector equation, and the differential transformation method is used to efficiently derive a recurrence formula for the coefficients of the power series solution of the state-vector equation. By computing these coefficients from the recurrence formula, the transfer function that relates the state-vector at an arbitrary position to that at the initial position of a finite element is derived. The transfer matrix is then obtained from the transfer function. Finally, the spectral element matrix or exact dynamic stiffness matrix is obtained from the transfer matrix. The accuracy and computational efficiency of the proposed SVEM-based SEM are verified in due course through comparisons with other solution methods for non-uniform axial rods, Bernoulli−Euler beams, and Timoshenko beams with different spatial variations.

Axial-bending coupled vibration analysis of an axially-loaded stepped multi-layered beam with arbitrary boundary conditions

This paper presents an analytical method for solving both the free and forced vibration of an axially loaded stepped laminated beam with general layers and arbitrary boundary conditions based on the first-order shear deformation theory (FSDT). The beam is divided into appropriate segments and the spectral element matrix is employed to formulate the solutions for each segment. The transfer matrix method is adopted to analytically combine the solutions and boundary conditions of each segment. The solutions to the free and forced vibration of the beam are then obtained by solving a 6 × 6 matrix equation. The results obtained by the proposed method show good agreement with those by the finite element method (FEM). Parametric studies are presented to study the effect of beam parameters on the dynamic characteristics of the beams.

Dynamic response of circular and annular circular plates using spectral element method

This paper presents development of the spectral element method (SEM) for analysis of circular and annular circular plates vibration under impact load. A novel formulation is proposed in frequency domain for conducting spectral element matrix. Several numerical examples are presented in order to demonstrate performance of the presented method compared to other numerical methods in literature. The presented formulation was applied in order to analyze vibration of annular circular plates with various thicknesses and various internal-to-external radius ratios. In addition, annular circular plate displacements subjected to an impact load were calculated using the presented SEM.

A spectral element model for thermal effect on vibration and buckling of laminated beams based on trigonometric shear deformation theory

A spectral element model is presented for analyzing thermal effect on free vibration and buckling behaviors of generally laminated composite beams subjected to uniform temperature distribution. Coupled differential equations of motion for the generally layered composite beams are derived on the basis of trigonometric shear deformation theory, in which temperature-dependent material properties, temperature change and Poisson effect are incorporated in the mathematical formulation. A spectral element matrix is formulated from the exact analytical solutions of the governing differential equations of the composite beams in free vibration. The spectral element method together with the Wittrick–Williams algorithm is employed to obtain the natural frequencies and buckling temperature change of the generally layered composite beams with various boundary conditions. Comparison of the natural frequencies and/or buckling temperature changes with the ANSYS solutions and/or corresponding results in literature validates the derived spectral element formulation. The effects of temperature change, Poisson effect, boundary condition, material anisotropy and thermal expansion coefficient on the natural frequencies and/or buckling temperature changes of the composite beams are investigated.

Spectral finite element analysis of in-plane free vibration of laminated composite shallow arches

A spectral finite element model is developed for predicting the free vibration characteristics of cross-ply laminated shallow arches vibrating in the in-plane. The exponential shear deformation theory is used to define the arches’ dynamic behavior. The governing differential equations of laminated shallow arches are derived using Hamilton’s principle. The spectral element matrix is formulated using the dynamic shape functions based on the exact analytical solutions of the homogeneous governing equations. The derived spectral element matrix is then used in conjunction with the Wittrick–Williams algorithm to calculate the natural frequencies and mode shapes of the cross-ply laminated shallow arches. Effects of stacking sequence, material orthotropy ratio, curvature ratio and length to thickness ratio, as well as end condition, on the natural frequencies of the laminated arches are thoroughly investigated.

Comparison of various shear deformation theories for free vibration of laminated composite beams with general lay-ups

This study deals with the numerical evaluations of various higher-order shear deformation beam theories. Free vibration analysis of generally laminated composite beams with arbitrary boundary conditions is performed by using the spectral finite element method. A unified type of assumed displacement field is considered and the influences of shear deformation, rotary inertia and Poisson effect are included in the formulation. The spectral element matrix is derived by rigorous use of the analytical solutions of the governing differential equations of the laminated beam in free vibration. The application of the spectral element matrix to study the free vibration characteristics of the laminated beams is demonstrated by adopting the Wittrick–Williams algorithm. Numerical results for the natural frequencies of the illustrative examples, which are found by various higher-order shear deformation beam models, are compared with each other and with the available solutions in the literature.

Building spectral element dynamic matrices using finite element models of waveguide slices and elastodynamic equations

Structural spectral elements are formulated using the analytical solution of the applicable elastodynamic equations and, therefore, mesh refinement is not needed to analyze high frequency behavior provided the elastodynamic equations used remain valid. However, for modeling complex structures, standard spectral elements require long and cumbersome analytical formulation. In this work, a method to build spectral finite elements from a finite element model of a slice of a structural waveguide (a structure with one dimension much larger than the other two) is proposed. First, the transfer matrix of the structural waveguide is obtained from the finite element model of a thin slice. Then, the wavenumbers and wave propagation modes are obtained from the transfer matrix and used to build the spectral element matrix. These spectral elements can be used to model homogeneous waveguides with constant cross section over long spans without the need of refining the finite element mesh along the waveguide. As an illustrating example, spectral elements are derived for straight uniform rods and beams and used to calculate the forced response in the longitudinal and transverse directions. Results obtained with the spectral element formulation are shown to agree well with results obtained with a finite element model of the whole beam. The proposed approach can be used to generate spectral elements of waveguides of arbitrary cross section and, potentially, of arbitrary order.

Building Spectral Element Dynamic Matrices Using Finite Element Models of Waveguide Slices and Elastodynamic Equations

Structural spectral elements are formulated using the analytical solution of the applicable elastodynamic equations and, therefore, mesh refinement is not needed to analyze high frequency behavior provided the elastodynamic equations used remain valid. However, for modeling complex structures, standard spectral elements require long and cumbersome analytical formulation. In this work, a method to build spectral finite elements from a finite element model of a slice of a structural waveguide (a structure with one dimension much larger than the other two) is proposed. First, the transfer matrix of the structural waveguide is obtained from the finite element model of a thin slice. Then, the wavenumbers and wave propagation modes are obtained from the transfer matrix and used to build the spectral element matrix. These spectral elements can be used to model homogeneous waveguides with constant cross section over long spans without the need of refining the finite element mesh along the waveguide. As an illustrating example, spectral elements are derived for straight uniform rods and beams and used to calculate the forced response in the longitudinal and transverse directions. Results obtained with the spectral element formulation are shown to agree well with results obtained with a finite element model of the whole beam. The proposed approach can be used to generate spectral elements of waveguides of arbitrary cross section and, potentially, of arbitrary order.

Free vibration of axially loaded composite beams with general boundary conditions using hyperbolic shear deformation theory

The natural frequencies, mode shapes and buckling loads of the laminated composite beams subject to concentrated axial forces are investigated on the basis of hyperbolic shear deformation theory. In the analysis the effects of axial force, Poisson effect, shear deformation, rotary inertia and coupling between axial and transverse deformations are taken into account. The spectral finite element method is adopted with particular reference to the Wittrick–Williams algorithm when investigating the free vibration of axially loaded composite beams. The exact spectral element matrix is established by directly solving the governing differential equations of the laminated beams in free vibration. Two illustrative examples are worked out to show the influences of axial force and boundary condition on the free vibration and buckling behaviors of the laminated composite beams. Comparison with the available solutions in the literature demonstrates the accuracy and reliability of the proposed formulation.

Spectral element analysis of the axial-bending-shear coupled vibrations of composite Timoshenko beams

This article presents a spectral element model for the axially loaded axial-bending-shear coupled vibrations of composite laminated beams, which are represented by the Timoshenko beam models based on the first-order shear deformation theory. The variation approach is used to formulate the frequency-dependent spectral element matrix (often called exact dynamic stiffness matrix) for the present spectral element model. As the spectral element matrix is formulated from exact wave solutions satisfying the frequency domain governing equations of motion transformed by the use of discrete Fourier transform theory, the present spectral element model cannot only provide extremely accurate solutions with using only a minimum number of degrees of freedom but also contribute to improving the computation efficiency. The high accuracy of the present spectral element model is numerically verified by comparing its solutions with exact analytical solutions available from references as well as with the solutions obtained by conventional finite element method. The effects of the axial loading and damping are also numerically investigated. For the numerical verification, the finite element model is also provided for the axially loaded axial-bending-shear coupled composite laminated Timoshenko beams.

ICONE19-43440 SPECTRAL ELEMENT MODEL FOR THE AXIAL-BENDING-SHEAR COUPLED VIBRATION OF A COMPOSITE TIMOSHENKO BEAM

In this paper, a spectral element model is developed for the axial-bending-shear coupled vibration of an axially loaded composite laminated beam which is represented by the Timoshenko beam model based on the first-order shear deformation theory (FSDT). The spectral element matrix is formulated from exact wave solutions which satisfy the frequency-domain governing equations transformed by the use of discrete Fourier transform theory, the present spectral element model can not only provide extremely accurate solutions with using only a minimum number of degrees of freedom but also contribute to improving the computation efficiency. The high accuracy of the present spectral element model is numerically verified by comparing its solutions with exact analytical solutions available from references as well as with the solutions obtained by conventional finite element method.