Element For Free Vibration Analysis Research Articles

An accurate and efficient 4-noded quadrilateral plate element for free vibration analysis of laminated composite plates using a refined third-order shear deformation plate theory

An accurate, efficient and reliable four-noded quadrilateral plate element is developed for the free vibration analysis of laminated composite plates. The element formulation is based on a refined third-order shear deformation plate theory (TSDT) and the quasi-conforming element technique. The equations of motion and the variational consistent mass matrix are derived through Hamilton’s principle. The resulting composite plate element is free from shear locking. Furthermore, both its element stiffness and mass matrices are computed explicitly. The convergence and accuracy of the present element are evaluated by the standard static and dynamic examples of isotropic plates. The accuracy of the computed natural frequencies of laminated composite plates is verified by the examples with various lamination schemes, elastic modulus ratios, aspect ratios and boundary conditions and by the comparison with other results. The effect of the associated boundary conditions in different TSDTs is also evaluated through numerical results. The influence of the higher-order and coupling mass matrices on the frequencies of higher-mode vibration is conducted too. The numerical results clearly demonstrate that the present plate element is not only very efficient but also capable of yielding accurate natural frequencies of both the fundamental and the higher-mode vibrations of laminated composite plates.

An efficient nine-node quadrilateral element for free vibration analysis of deep doubly curved soft-core sandwich shells

An efficient nine-node quadrilateral element for free vibration analysis of deep doubly curved soft-core sandwich shells

Adaptive finite element analysis of free vibration of elastic membranes via element energy projection technique

Purpose A general strategy is developed for adaptive finite element (FE) analysis of free vibration of elastic membranes based on the element energy projection (EEP) technique. Design/methodology/approach By linearizing the free vibration problem of elastic membranes into a series of linear equivalent problems, reliable a posteriori point-wise error estimator is constructed via EEP super-convergent technique. Hierarchical local mesh refinement is incorporated to better deal with tough problems. Findings Several classical examples were analyzed, confirming the effectiveness of the EEP-based error estimation and overall adaptive procedure equipped with a local mesh refinement scheme. The computational results show that the adaptively-generated meshes reasonably catch the difficulties inherent in the problems and the procedure yields both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm. Originality/value By reasonable linearization, the linear-problem-based EEP technique is successfully transferred to two-dimensional eigenproblems with local mesh refinement incorporated to effectively and flexibly deal with singularity problems. The corresponding adaptive strategy can produce both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm and thus can be expected to apply to other types of eigenproblems.

An Edge-Based Smoothed Finite Element for Free Vibration Analysis of Functionally Graded Porous (FGP) Plates on Elastic Foundation Taking into Mass (EFTIM)

In this paper, free vibration analysis of the functionally graded porous (FGP) plates on the elastic foundation taking into mass (EFTIM) is presented. The fundamental equations of the FGP plate are derived using Hamilton’s principle. The mixed interpolation of the tensorial components (MITC) approach and the edge-based smoothed finite element method (ES-FEM) is employed to avoid the shear locking as well as to improve the accuracy for the triangular element. The EFTIM is a foundation model based on the two-parameter Winkler–Pasternak model but added a mass parameter of foundation. Materials of the plate are FGP with a power-law distribution and maximum porosity distributions in the forms of cosine functions. Some numerical examples are examined to demonstrate the accuracy and reliability of the proposed method in comparison with those available in the literature.

Free Vibration of FGSW Plates Partially Supported by Pasternak Foundation Based on Refined Shear Deformation Theories

A refined third-order shear deformation theory (RTSDT), in which the transverse displacement is split into bending and shear parts, is employed to formulate a four-node quadrilateral finite element for free vibration analysis of functionally graded sandwich (FGSW) plates partially supported by a Pasternak foundation. An element based on the refined first-order shear deformation theory (RFSDT) which requires a shear correction factor is also derived for comparison purpose. The plates consist of a fully ceramic core and two functionally graded skin layers with material properties varying in the thickness direction by a power gradation law. The Mori–Tanaka scheme is employed to evaluate the effective moduli. The elements are derived using Lagrangian and Hermitian polynomials to interpolate the in-plane and transverse displacements, respectively. The numerical result reveals that the frequencies obtained by the RTSDT element are slightly higher than the ones using the RFSDT element. It is also shown that the foundation supporting area plays an important role on the vibration of the plates, and the effect of the material distribution on the frequencies is dependent on this parameter. A parametric study is carried out to highlight the effects of the material inhomogeneity, the foundation stiffness parameters, and the foundation supporting area on the frequencies and vibration modes. The influence of the layer thickness and aspect ratios on the frequencies is also examined and highlighted.

Free vibration of isotropic plates with various cutout configurations using finite elements and design of experiments

This work represents a finite element analysis of free vibration of isotropic plates with different cutout shapes, areas, locations and aspect ratios. Modal analysis was carried out using the ANSYS APDL software to evaluate the fundamental frequencies. ANSYS model was validated in the first stage and showed good agreement with the selected literature works. Furthermore, the selected cutout parameters were investigated to assess which parameter is more effective on the frequency. It is found that these parameters was overlapping, therefore Design of Experiments was conducted. Results revealed that the aspect ratio of the cutouts is the most significant factor on the fundamental frequency.

An efficient matrix tridiagonalization method for 3D finite element analysis of free vibration

In order to avoid mechanical resonance, a vibrating structure needs to be designed such that its working frequency interval is sufficiently far from its natural frequencies. Natural frequencies of a mechanical system are obtained from free vibration analysis commonly done by the finite element method. One important step in the analysis is matrix tridiagonalization commonly performed by the block Lanczos method. However, the classical block Lanczos method suffers numerical instability due to the loss of matrix orthogonality. In this paper, we demonstrate the implementation of the block Lanczos method with an orthogonality fixing scheme for 3D free vibration problems. The solution accuracy and computational time of this method are compared with those of the classical block Lanczos method and the Householder method. The results show that the block Lanczos method with the orthogonality fixing scheme employed in this work can effectively avoid the numerical instability due to the loss of matrix orthogonality. Furthermore, the block Lanczos method with the orthogonality fixing scheme provides solution accuracy as good as the Householder method while using significantly less computational time.

Finite element analysis of free vibration of beams with composite coats

This paper presents a finite element model to investigate the analysis of free vibration of beams with composite coats. We used two sandwich beam models; the core is made from an isotropic material, the steel as heavy material, for first model, and the foam as light material, employed in the second model. The faces are made from glass/epoxy composite material for the two models. The natural frequency and mode shapes of the sandwich beam are controlled by choosing the proper fiber orientation, the laminate thickness and the boundary conditions. The effects of these parameters are examined for the two models with different boundary conditions imposed on the beam. Good agreements were achieved between the finite element method and analytical solutions. DOI: http://dx.doi.org/10.5755/j01.mech.21.4.9849

Multiple crack identification in Euler beams by means of B-spline wavelet

In this paper, an effective method for identification of multiple cracks is presented based on discrete wavelet transform in a cracked beam. First, a compactly supported semi-orthogonal B-spline wavelet on interval (BSWI) is employed to construct Euler beam-bending element for free vibration analysis of cracked beams. Next, the construction of general order one-dimensional B-spline wavelets is presented and applied for damage identification in a cantilever beam modeled by wavelet-based elements. Also, principles of an appropriate wavelet selection are presented. Natural vibration modes of a cantilever beam with three cracks are analyzed using one-dimensional fourth-order B-spline wavelet. The results illustrate that BSWI elements can be used in determining the un-cracked and cracked beam natural frequencies with a high accuracy and efficiency. Moreover, the applicability of the presented method in crack identification is studied by numerical examples under several situations, such as in the presence of random noises, and the efficiency of B-spline wavelets in damage prognosis is compared with other types of wavelet functions. The obtained results show the effectiveness of B-spline wavelets in modeling of the damaged beam and identifying multiple crack locations in a free baseline scheme.

Finite element analysis of free vibration of the delaminated composite plate with variable kinematic multilayered plate elements

Composite laminates are prone to delamination. Implementation of delamination in the Carrera Unified Formulation frame work using nine noded quadrilateral MITC9 element is discussed in this article. MITC9 element is devoid of shear locking and membrane locking. Delaminated as well as healthy structure is analyzed for free mode vibration. The results from the present work are compared with the available experimental or/and research article or/and the three dimensional finite element simulations. The effect of different kinds and different percentages of area of delamination on the first three natural frequencies of the structure is discussed. The presence of open-mode delamination mode shape for large delaminations within the first three natural frequencies is discussed. Also, the switching of places between the second bending mode, with that of the first torsional mode frequency is discussed. Results obtained from different ordered theories are compared in the presence of delamination. Advantage of layerwise theories as compared to equivalent single layer theories for very large delaminations is stated. The effect of different kinds of delamination and their effect on the second bending and first torsional mode shape is discussed.

A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells

An exact free vibration analysis of doubly-curved laminated composite shallow shells has been carried out by combining the dynamic stiffness method (DSM) and a higher order shear deformation theory (HSDT). In essence, the HSDT has been exploited to develop first the dynamic stiffness (DS) element matrix and then the global DS matrix of composite cylindrical and spherical shallow shell structures by assembling the individual DS elements. As an essential prerequisite, Hamilton’s principle is used to derive the governing differential equations and the related natural boundary conditions. The equations are solved symbolically in an exact sense and the DS matrix is formulated by imposing the natural boundary conditions in algebraic form. The Wittrick–Williams algorithm is used as a solution technique to compute the eigenvalues of the overall DS matrix. The effect of several parameters such as boundary conditions, orthotropic ratio, length-to-thickness ratio, radius-to-length ratio and stacking sequence on the natural frequencies and mode shapes is investigated in details. Results are compared with those available in the literature. Finally some concluding remarks are drawn.

적층된 복합 및 샌드위치 판 구조의 자유진동 해석을 위한 EAS 고체 유한요소

This study deals with an enhanced assumed strain (EAS) three-dimensional element for free vibration analysis of laminated composite and sandwich structures. The three-dimensional finite element (FE) formulation based on the EAS method for composite structures shows excellence from the standpoints of computational efficiency, especially for distorted element shapes. Using the EAS FE formulation developed for this study, the effects of side-to-thickness ratios, aspect ratios and ply orientations on the natural frequency are studied and compared with the available elasticity solutions and other plate theories. The numerical results obtained are in good agreement with those reported by other investigators. The new approach works well for the numerical experiments tested, especially for complex structures such as sandwich plates with laminated composite faces.

A Dynamic Stiffness Element for Free Vibration Analysis of Delaminated Layered Beams

A dynamic stiffness element for flexural vibration analysis of delaminated multilayer beams is developed and subsequently used to investigate the natural frequencies and modes of two-layer beam configurations. Using the Euler-Bernoulli bending beam theory, the governing differential equations are exploited and representative, frequency-dependent, field variables are chosen based on the closed form solution to these equations. The boundary conditions are then imposed to formulate the dynamic stiffness matrix (DSM), which relates harmonically varying loads to harmonically varying displacements at the beam ends. The bending vibration of an illustrative example problem, characterized by delamination zone of variable length, is investigated. Two computer codes, based on the conventional Finite Element Method (FEM) and the analytical solutions reported in the literature, are also developed and used for comparison. The intact and defective beam natural frequencies and modes obtained from the proposed DSM method are presented along with the FEM and analytical results and those available in the literature.

C0-continuous isoparametric Timoshenko beam element for rotating

This paper presents a C0-continuous isoparametric finite element for free vibration analysis of a rotating, tapered Timoshenko beam mounted on the periphery of a rotating rigid hub, at a setting angle with the plane of rotation. The finite element has three nodes and two degrees of freedom per node and employs modified shape functions for rotational displacement associated with the shear strain energy to avoid shear locking. To obtain a finite element equation of the generalized eigenvalue problem, Hamilton’s principle is applied to kinetic and potential energy expressions of a rotating Timoshenko beam with non-zero setting angle. The numerical solutions for various situations including variations of rotational speed, taper ratio, slenderness ratio, hub radius and setting angle are compared with other numerical results available in the literature whenever possible. The results show that the new 3-noded isoparametric element yields accurate results when compared to other numerical ones.

An accurate higher-order theory and C0 finite element for free vibration analysis of laminated composite and sandwich plates

Abstract At present, it is difficult to accurately predict natural frequencies of sandwich plates with soft core by using the C 0 plate bending elements. Thus, the C 1 plate bending elements have to be employed to predict accurately dynamic response of such structures. This paper proposes an accurate higher-order C 0 theory which is very different from other published higher-order theory satisfying the interlaminar stress continuity, as the first derivative of transverse displacement has been taken out from the in-plane displacement fields of the present theory. Therefore, the C 0 interpolation functions is only required during its finite element implementation. Based on the Hamilton’s principle and Navier’s technique, analytical solutions to the natural frequency analysis of simply-supported laminated plates have been presented. To further extend the ranges of application of the proposed theory, an eight-node C 0 continuous isoparametric element is used to model the proposed theory. Numerical results show the present C 0 finite element can accurately predict the natural frequencies of sandwich plate with soft core, whereas other global higher-order theories are unsuitable for free vibration analysis of such soft-core structures.

A dynamic stiffness element for free vibration analysis of composite beams and its application to aircraft wings

The dynamic stiffness matrix of a composite beam that exhibits both geometric and material coupling between bending and torsional motions is developed and subsequently used to investigate its free vibration characteristics. The formulation is based on Hamilton’s principle leading to the governing differential equations of motion in free vibration, which are solved in closed analytical form for harmonic oscillation. By applying the boundary conditions the frequency dependent dynamic stiffness matrix that relates the amplitudes of loads to those of responses is then derived. Finally the Wittrick–Williams algorithm is applied to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of an illustrative example. The results are discussed and some conclusions are drawn. The theory can be applied for modal analysis of high aspect ratio composite wings and can be further extended to aeroelastic studies.

A new finite element for free vibration analysis of stiffened composite plates

A new 6-noded stiffened triangular plate element for the analysis of stiffened composite plates based on Mindlins deformation plate theory has been developed. The stiffened plate element is a combination of basic triangular element and bar component. The element can accommodate any number of arbitrarily oriented stiffeners and obviates the use of mesh lines along the stiffeners. Free vibration analyses of stiffened laminated plates have been carried out with this element and the results are compared with those published. The finite element results show very good matching with the experimental ones.

Hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.

WITHDRAWN: A dynamic stiffness element for free vibration analysis of composite beams and its application to aircraft wings

The Publisher regrets that this article is an accidental duplication of an article that has already been published in Compos. Struct., doi:10.1016/j.compstruc.2007.04.027. The duplicate article has therefore been withdrawn.

A refined higher order finite element for asymmetric composite beams

A refined 2-node, 4 DOF/node beam element is derived based on higher order shear deformation theory for axial–flexural-shear coupled deformation in asymmetrically stacked laminated composite beams. The element has an exact shape function matrix, which is derived by satisfying the static part of the governing equations of motion, where a general ply-stacking is considered. This makes the element super-convergent in static analysis. The numerical results are validated by considering uniformly distributed load with various boundary conditions. Subsequently, the efficiency of the element for free vibration analysis is studied. Numerical examples showing the nature of interlaminar stresses for various ply-stacking configurations and boundary conditions are demonstrated. Also, parametric studies are performed to study the effect of coupling and stiffness on the natural frequencies. Based on these studies, number of observations are drawn.