Formulation For Free Vibration Analysis Research Articles

Dynamic stiffness formulation for free vibration analysis of rotating cross-ply laminated combined elliptical-cylindrical-conical shell

Dynamic stiffness formulation for free vibration analysis of rotating cross-ply laminated combined elliptical-cylindrical-conical shell

Dynamic stiffness formulation for vibration analysis of an open cylindrical shell and its coupling structures based on a generalized superposition method

• A dynamic stiffness formulation is provided for open cylindrical shells. • The dynamic stiffness formulation is derived by combining a generalized superposition method with the projection method. • Compared with available ones, the derivation process of the dynamic stiffness matrix is easier in the present formulation and its dimension is also largely reduced. • The present formulation is appropriate for various boundary conditions. • The present formulation is extended to the vibration analysis of coupled shell structures. • Parametric studies are performed on the vibration of coupled shell structures. This paper presents a dynamic stiffness formulation for free vibration analysis of open cylindrical shells and their coupling structures. Based on the Flügge's thin shell theory, the dynamic stiffness matrix of an open cylindrical shell is derived by employing the projection method and a newly developed superposition method, which considers the entire domain of the system as a whole part instead of dividing it into small subdomains in the traditional Gorman's superposition method. By doing this, the derivation process of the dynamic stiffness formulation is much easier and the analytical model has a significantly lower order than the traditional dynamic stiffness formulation. The presented dynamic stiffness formulation is then extended to the vibration analysis of coupled shell structures. To achieve this goal, the coupled shell structures are divided into several sub-shells, and the global dynamic stiffness matrices are assembled according to the geometrical coupled conditions between them. Several typical coupled shell structures are taken as examples and their vibration characteristics are studied by the present formulation. The convergence and accuracy of the present formulation are verified by comparing present results with those obtained by literature and the finite element method (FEM). Parameter studies for the vibration analysis of the coupled shell structures are presented, indicating that the circumference angles, length-to-radius ratios, and thickness-to-radius ratios have significant influences on the dynamic characteristics of the coupled shell structures.

Free vibration analysis of laminated rectangular plates with varying thickness using Legendre-radial point interpolation method

The purpose of this paper is to investigate the free vibration characteristics of laminated rectangular plate with varying thickness (LRPVT) using the meshfree strong form method. The theoretical formulation for free vibration analysis is based on the first order shear deformation theory (FSDT), and the governing equations for the LRPVT are derived using Hamilton's principle. In order to approximate the displacement function in the governing equations, a novel Legendre-radial point interpolation method (LRPIM) shape function using combined basis of thin plate spline (TPS) radial functions and Legendre polynomials is presented. The boundary conditions are generalized by the introduction of an artificial spring technique, and the type of boundary conditions is selected according to the spring stiffness values. The accuracy and reliability of the LRPIM shape function are verified by comparing with the results of the literatures and finite element software ABAQUS. The free vibration characteristics including natural frequencies and mode shapes of the LRPVT with various geometry and boundary conditions are presented through numerical examples.

Three-dimensional free vibration analysis of thick laminated combination shell using a meshfree approach

In this paper, a novel meshfree approach is proposed for the three-dimensional free vibration analysis of laminated composite combination shells. The three-dimensional theory of elasticity is applied to the formulation for free vibration analysis of thick laminated elliptical–cylindrical and conical–cylindrical combination shells, and the field functions are approximated by a novel meshfree Tchebychev-point interpolation method shape function using Tchebychev polynomials as the basis. The governing equation and boundary conditions for the individual layers of the substructures of the laminated combination shell are derived, and the equations of the whole system are obtained by combining them using a continuous condition. The boundary and continuous conditions are generalized by the introduction of an artificial spring technique, and the type of boundary conditions is selected according to the spring stiffness. The accuracy and reliability of the proposed method are verified by comparing the results of the proposed method with those of literature and finite element program ABAQUS. The free vibration characteristics including natural frequencies and mode shapes of thick combination shells with various geometries and boundary conditions are presented through numerical examples.

Free vibration analysis of laminated closed conical, cylindrical shells and annular plates with a hole using a meshfree method

In this paper, a novel meshfree approach is proposed for the free vibration analysis of laminated closed conical, cylindrical shells and annular plate with a hole. The theoretical formulation for free vibration analysis is based on the first order shear deformation theory (FSDT) and the field functions are approximated by a Tchebychev-radial point interpolation method (TRIPM) shape function using various radial functions augmented with Tchebychev polynomials as the basis. A laminated closed shell with a hole is decomposed into five open shells without holes, and continuous conditions of displacement are applied at the interfaces between the open shells. The governing equations and boundary conditions for the laminated open shells without hole are derived using Hamilton’s principle. The boundary and continuous conditions are generalized by the introduction of an artificial spring technique, and the type of the conditions is selected according to the spring stiffness values. A comparison with the results of the literatures and finite element software ABAQUS is conducted to verify the accuracy and reliability of the proposed method. The investigation of the vibration characteristics of the laminated closed conical, cylindrical shells and annular plates with a hole according to various radial functions, geometric dimensions and boundary conditions is presented through numerical examples.

A rotation-free quadrature element formulation for free vibration analysis of thin sectorial plates with arbitrary boundary supports

Due to corner stress singularities, free vibration analysis of thin sectorial plates with arbitrary boundary supports including a free vertex is a rather challenging problem. In this paper, a novel rotation-free quadrature element formulation based on Kirchhoff plate theory is presented to solve the problem. Different from all existing quadrature thin plate element formulations, rotational degrees of freedom are not used. Instead, a boundary point is modeled by two nodes separated by a very small distance and thus the C1 continuity is easily enforced. Explicit formulations with arbitrary number of nodes and nodes of any type are worked out to ease the p-refinement. Numerical results demonstrate that the proposed formulation is simple, straightforward and has excellent performance for solving thin sectorial plate with arbitrary combinations of boundary supports. Some tabulated data are believed new which may serve as a reference.

A meshfree method for free vibration analysis of ply drop-off laminated rectangular plates

The purpose of this paper is to develop a more effective meshfree shape function for free vibration analysis of ply drop-off laminated rectangular plates. The proposed meshfree Tchebychev-radial point interpolation method (TRIPM) shape function uses the Tchebychev polynomials and multi-quadrics (MQ) radial functions as the basis. The first order shear deformation theory (FSDT) is applied to the formulation for free vibration analysis of ply drop-off laminated rectangular plate, and the field functions are approximated by TRIPM shape function. The governing equations and boundary conditions for the substructures of ply drop-off laminated plate are derived, and the equations of the whole system are obtained by combining them using a continuous condition. The boundary and continuous conditions are generalized by the introduction of an artificial spring technique, and the type of boundary conditions is selected according to the spring stiffness. The accuracy and reliability of the proposed method are verified by comparing the results of the literature and the finite element program ABAQUS. The free vibration characteristics including natural frequencies and mode shapes of ply drop-off laminated rectangular plates with various geometrical dimensions and boundary conditions are presented through numerical examples.

Comparisons of matched interface and boundary (MIB) method and its interpolation formulation for free vibration analysis of stepped beams and plates

This work conducts a comparison study of matched interface and boundary (MIB) method and its interpolation formulation for free vibration analysis of stepped beams and plates. Two types of schemes are used to describe the relations between interfaces and grid points in one-dimensional and two-dimensional domains. In the first scheme, the interfaces are located on grid points, while, in the second scheme, the interfaces lie between grid points. Detailed procedures of MIB and the interpolation formulation of MIB (IMIB) are presented in these two schemes. Based on the in-depth understanding of MIB, a new algorithm is also developed to deal with cross derivatives of interface conditions of plates with interfaces lying between grid points to improve the accuracy. Various examples of stepped beams and plates are chosen to illustrate the performance of MIB and IMIB. This research work reveals that on the whole, MIB and IMIB are equivalent in the solutions of eigenvalues of beams and plates. However, some differences can also be observed for very large computational bandwidths. Some important conclusions are drawn at the end of this study.

An Efficient Three Nodded Finite Element Formulation for Free Vibration Analysis of Sandwich Arches with Graded Metallic Cellular Core

An efficient finite element model based on three nodded element has been developed for the vibration analysis of sandwich arches with graded metallic cellular (GMC) core. The present formulation is based on the higher-order shear deformation theory and orthogonal curvilinear coordinate axes. The arch consists of two isotropic face sheets and a GMC core layer. The internal pores in the core layer follow the different types of distributions. The material properties of the GMC core layer of the sandwich arches vary in the thickness direction as a function in terms of porosity coefficient and mass density. Three types of porosity distributions have been considered to accomplish the vibration responses of sandwich arches. The present formulation is validated with limited results available in the literature. Few new results are computed and the effects of different influencing parameters such as porosity coefficient [Formula: see text], porosity distribution type, the thickness-to-length ratio [Formula: see text], boundary conditions and opening angle [Formula: see text] on the free vibration characteristics of sandwich arches with the GMC core are observed. The present finite element model gives better convergence and more accurate results than a conventional two nodded element-based finite element model.

Free Vibration Analysis of Bidirectional Functionally Graded Conical/Cylindrical Shells and Annular Plates on Nonlinear Elastic Foundations, Based on a Unified Differential Transform Analytical Formulation

In the present research, a unified formulation for free vibration analysis of the bidirectional functionally graded conical and cylindrical shells and annular plates on elastic foundations is developed. To cover more individual cases and optimally tailored material properties, the material properties are assumed to vary in both the meridian/radial and transverse directions. The shell/plate is assumed to be supported by a non-uniform Winkler-type elastic foundation in addition to the edge constraints. Therefore, the considered problem contains some complexities that have not been considered together in the available researches. The proposed unified formulation is derived based on the principle of minimum total potential energy and solved using a differential transform analytical method whose center is located at the outer edge of the shell or plate; so that the resulting semi-analytical solution can be employed not only for truncated conical shells and annular plates, but also for complete conical shells and circular plates. Accuracy of results of the proposed unified formulation is verified by comparing the results with those of the three-dimensional theory of elasticity extracted from the ABAQUS finite element analysis code. A variety of the edge condition combinations are considered in the results section. A comprehensive parametric study including assessment of influences of the material properties indices, thickness to radius ratio, stiffness distribution of the elastic foundation, and various boundary conditions, is accomplished. Results reveal that influence of the meridian variations of the material properties on the natural frequencies is more remarkable than that of the transverse gradation.

A global Ritz formulation for the free vibration analysis of hybrid metal-composite thick trapezoidal plates

A general variational formulation for free vibration analysis of hybrid (metalcomposite) plates with a trapezoidal platform is presented in this paper. The plate is composed of two distinguished parts in the span direction, where the inboard section is assumed to be made of an isotropic metal, and the outboard section is from a laminated composite material. Kinematics, corresponding to the First-order Shear Deformation plate Theory (FSDT), is used to take into account the non-classic e ects of transverse shear deformation and rotational inertia for both sections in the analysis. The developed approach is based on the global Ritz method, where the transverse de flection and two rotations of the plate are independently approximated by simple and Legendre polynomials. The algorithm allows one to obtain an approximate analytical solution for the hybrid plate with di erent geometric aspect ratios, numbers of layers, staking sequences and metal to composite ratios. The results of the present work for special cases, including pure metal and pure composite plates, are in good agreement with previous works.

A layerwise finite element formulation for free vibration analysis of functionally graded sandwich shells

This paper presents a layerwise finite element formulation for dynamic analysis of functionally graded material (FGM) sandwich shell in thermal and non-thermal environments. The layerwise theory used in this work is based on the first-order shear deformation theory (FSDT) for each layer satisfying the displacement continuity at layer interface. Two configurations of FGM sandwich shells are considered, the first with FGM facesheet and homogenous core, and the second having homogenous facesheets and FGM core. Effective material properties of the FGM are estimated according to two micromechanical models, namely, Voigt’s rule of mixture (ROM) and Mori–Tanaka (MT) scheme. For the shells in thermal environment, a nonlinear temperature distribution in thickness direction is considered and the elastic properties are assumed to be temperature dependent. The results obtained from the proposed formulation are validated with those available in the literature. Natural frequencies obtained from Sanders’, Love’s and Donnell’s shell theories for different geometric and boundary conditions are compared to assess the performance of different shell theories for FGM sandwich shells under non-thermal environment. The effects of volume fraction index, core thickness and temperature gradient on natural frequencies of FGM sandwich shells are investigated. The present formulation is simple, accurate and computationally efficient.

A layerwise p-version finite element formulation for free vibration analysis of thick composite laminates with curvilinear fibres

Herein, modes of vibration of a novel type of composite laminates, manufactured by tow-placement machines and which are called Variable Stiffness Composite Laminates (VSCL), are of interest. Layerwise theory is chosen, because it leads to an accurate prediction of displacements through the thickness of laminates, even when these are thick. Furthermore, it naturally has the capability of modelling the bending-membrane coupling that occurs on free vibration of VSCL plates with unsymmetric stacking sequences. The continuity of displacements at layer interfaces is imposed and in-plane displacements vary in a zig-zag fashion along the thickness. The accuracy of the model is confirmed by testing Constant Stiffness Composite Laminates (CSCL) analysed in the literature by others, who employed either Equivalent Single Layer (ESL) or layerwise theories. Abaqus commercial finite element software is as well employed in order to test the present approach. Finally, published natural frequencies of VSCLs, which were studied by an ESL theory, are compared with frequencies resulting from the present layerwise theory. The verification of the proposed model is followed by an investigation on the effect of curvilinear fibre orientation parameters on natural frequencies of thin to thick composite plates, with symmetric and unsymmetric lay-ups.

Modeling and free vibration behavior of rotating composite thin-walled closed-section beams with SMA fibers

Smart structure with active materials embedded in a rotating composite thin-walled beam is a class of typical structure which is using in study of vibration control of helicopter blades and wind turbine blades. The dynamic behavior investigation of these structures has significance in theory and practice. However, so far dynamic study on the above-mentioned structures is limited only the rotating composite beams with piezoelectric actuation. The free vibration of the rotating composite thin-walled beams with shape memory alloy(SMA) fiber actuation is studied. SMA fiber actuators are embedded into the walls of the composite beam. The equations of motion are derived based on Hamilton's principle and the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin's method. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. Numerical results of natural frequency are obtained for two configuration composite beams. It is shown that natural frequencies of the composite thin-walled beam decrease as SMA fiber volume and initial strain increase and the decrease in natural frequency becomes more significant as SMA fiber volume increases. The actuation performance of SMA fibers is found to be closely related to the rotational speeds and ply-angle. In addition, the effect of the pitch angle appears to be more significant for the lower-bending mode ones. Finally, in all cases, the precone angle appears to have marginal effect on free vibration frequencies. The developed model can be capable of describing natural vibration behaviors of rotating composite thin-walled beam with active SMA fiber actuation. The present work extends the previous analysis done for modeling passive rotating composite thin-walled beam.

High accuracy mixed finite element-Ritz formulation for free vibration analysis of plates with general boundary conditions

A high order accurate mixed finite element-Ritz method is introduced and developed to study the vibration problem of plates with general boundary conditions. The finite element method (FEM) with higher order interpolation functions is first used to discretize the spatial partial derivatives with respect to a co-ordinate direction of the plate. The Ritz method is then employed to analogize the resulting system of ordinary differential equations. A novel technique is also presented to exactly satisfy the mixed natural boundary conditions. The proposed method is applied here to solve some benchmark vibration problems of plates including isotropic and anisotropic rectangular plates, variable thickness rectangular plates, multi-span rectangular plates, and skew plates. Comparisons with existing numerical and analytical solutions show that the proposed mixed method can yield highly accurate results for vibration problem of plates involving free edges, free corners and irregular boundaries using a small number of finite elements and Ritz terms.

Free vibration multiquadric boundary elements applied to plane elasticity

This paper presents a new boundary element formulation for free vibration analysis of two-dimensional elastic structures. The dual reciprocity method (DRM) is introduced using the multiquadric radial basis functions (MQ). The required particular solution kernels for displacement and traction are derived and smoothed using simple mathematical trick; hence the limiting values of these kernels are computed. The eigen-problem of displacement is formulated and solved to obtain the required frequencies of different vibration modes. Finally, several numerical problems are studied to demonstrate the validity and accuracy of the developed formulation. The results are compared to those obtained from other interpolation functions, and finite element analysis to demonstrate the validity and superiority of the present formulation.

Free vibration analysis of horizontally curved steel I-girder bridges

Presented herein is a finite element formulation for free vibration analysis of horizontally curved steel I-girder bridges. Stiffness as well as mass matrices of the curved and the straight beam elements is formulated. Each node of both of them possesses seven degrees of freedom including the warping degree of freedom. The curved beam element is derived based on the Kang and Yoo's thin-walled curved beam theory in 1994. A computer program is developed to carry out free vibration analyses of the various bridges. Comparing with the frequencies using the general purpose program ABAQUS, the validity of the presented numerical formulation is shown. The numerical formulation is extensively applied to investigate free vibration characteristics of the bridges considering effects of the initial curvature, boundary condition, modeling method, and degrees of freedom of cross frame. Invaluable information which help practicing engineers better understand the vibration characteristics is provided.

Incompatible finite dynamic element formulation for free vibration analysis by hybrid stress formulation

Based on Hellinger–Reissner functional and with the inclusion both of displacements and stresses, a hybrid stress formulation of dynamic finite element method is derived in this paper. In this formulation, the displacements are separated into zeroth displacement modes and high-order incompatible dynamic dis placement modes which are related to the dynamic characteristics. The rationality, feasibility and effec tiveness of the present method are also numerically justified. Copyright © 2000 John Wiley & Sons, Ltd.

A finite element formulation for free vibration analysis of shells of general shape

The free, undamped vibration analysis of a shell of general shape is formulated with a curved isoparametric trapezoidal finite element, including thickness shear deformations and without neglecting z R in comparison with unity and rotatory inertia effects with consistent mass matrix. The shell element with eight nodes and 40 d.f. is applicable for both thin and moderately thick shell analysis. The accuracy and efficiency of this proposed finite element are illustrated by examples and the results are compared with those obtained by other analytical and numerical methods. The comparisons show that the method yields very good results with a relatively small number of elements.

Stability of Thin-Walled Open Beams Under Nonconservative Loads

ABSTRACT A finite element formulation for free-vibration analysis of straight prismatic beams of general thin-walled open cross-section, under conservative and nonconservative loads, is presented. The formulation is used to calculate the flutter load for a number of beam problems and is verified by comparison with pre-existing numerical solutions.