Free Vibration Analysis Of Structures Research Articles

Strain-based plane element for fracture mechanics’ problems

A triangular cracked plane element is proposed for application in fracture mechanics’ problems. The new element is based on the strain formulation and the compliance concept. To formulate the non-cracked element, the assumed strain formulation is used. Due to strain formulation, the derived non-cracked element is free from shear parasitic error. Moreover, it demonstrates very low sensitivity to the mesh distortions. In comparison to the classical displacement-based elements, the proposed element leads to more accurate responses for the coarse meshes. To incorporate crack effects in the element, the additional flexibility caused by the crack is computed based on the principles of the linear elastic fracture mechanics. Because of acceptable accuracy of the proposed element in free-vibration analysis of structures with stable open cracks, it is applicable for vibration-based crack detection. Various numerical examples evaluate accuracy and efficiency of the cracked and non-cracked formulations.

Free vibration analysis of elastic structures submerged in an infinite or semi-infinite fluid domain by means of a coupled FE–BE solver

The vibration behavior of thin elastic structures can be noticeably influenced by the surrounding water, which represents a kind of heavy fluid. Since the feedback of the acoustic pressure onto the structure cannot be neglected in this case, a strong coupled scheme between the structural and fluid domains is usually required. In this work, a coupled finite element and boundary element (FE–BE) solver is developed for the free vibration analysis of structures submerged in an infinite fluid domain or a semi-infinite fluid domain with a free water surface. The structure is modeled by the finite element method (FEM). The compressibility of the fluid is taken into account, and hence the Helmholtz equation serves as the governing equation of the fluid domain. The boundary element method (BEM) is employed to model the fluid domain, and a boundary integral formulation with a half-space fundamental solution is used to satisfy the Dirichlet boundary condition on the free water surface exactly. The resulting nonlinear eigenvalue problem (NEVP) is converted into a small linear one by using a contour integral method. Adequate modifications are suggested to improve the efficiency of the contour integral method and avoid missing the eigenfrequencies of interest. The Burton–Miller method is used to filter out the fictitious eigenfrequencies of the boundary integral formulations. Numerical examples are given to demonstrate the accuracy and applicability of the developed eigensolver, and also show that the fluid-loading effect strongly depends on both the water depth and the mode shapes.

Analiza slobodnih ravanskih oscilacija struktura sastavljenih od krutih tela i elastičnih grednih segmenata

This article presents free vibration analysis of structures composed of rigid bodies connected with elastic beam segments. It is assumed that the mass centers of rigid bodies are not located on the neutral axes of undeformed elastic beam segments as well as rigid bodies perform planar motion in the same plane and their mass centers are located in that plane. For determination of natural frequencies of the system, modification of the conventional continuous-mass transfer matrix method has been performed. The elastic beam segments are treated as Euler-Bernoulli beams. Numerical example is presented.

Static and Free Vibration Analysis of Structures Composed of Functionally Graded Material with Random Material Properties

Functionally graded materials (FGMs) have mechanical properties that vary continuously from one phase to another within a confined volume. In general, these materials exhibit certain amount of scatter in their properties due to different factors. The dispersion in the response values of a structure is due to the scatter in the values of material properties and applied external load. For design purposes, it is essential to know the potential variations in the structural response due to the system material or external randomness. In the present work, free vibration and static analysis on FGM structures with material randomness are considered.

A fuzzy finite element algorithm based on response surface method for free vibration analysis of structure

This paper introduces an improved response surface-based fuzzy finite element analysis of structural dynamics. The free vibration of structure is established using superposition method, so that fuzzy displacement responses can be presented as functions of fuzzy mode shapes and fuzzy natural frequencies. Instead of direct determination of these fuzzy quantities by modal analysis which will involve the calculation of the whole finite element model, the paper proposes a felicitous approach to design the response surface as surrogate model for the problem. In the design of the surrogate model, complete quadratic polynomials are selected with all fuzzy variables are transformed to standardized fuzzy variables. This methodology allows accurate determination of the fuzzy dynamic outputs, which is the major issue in response surface based techniques. The effectiveness of the proposed fuzzy finite element algorithm is illustrated through a numerical analysis of a linear two-storey shear frame structure.

Establishment of structural similitude for elastic models and validation of scaling laws

Scaled down models are widely used for experimental investigations on huge structures due to the limitation in the capacities of testing facilities, and, moreover, the experimentation on scaled models is less expensive. Also small scale models are often built with materials dissimilar to those of the prototype. However, only few studies have been carried out on the similitude laws for the prototype structures with small scale models built with a different material. In the present study, an attempt has been made to develop a scaling law for models to carry out the free vibration analysis of structures, based on the similitude requirement. The established scaling laws are validated analytically using simple test problems. The similitude relationship between the prototype and the model is validated with the aid of a case study using finite element analysis software.

Dynamic stiffness method for exact inplane free vibration analysis of plates and plate assemblies

The inplane free vibration behaviour of plates is investigated using the dynamic stiffness method. Some distinctive modes which went unnoticed in earlier investigations using the dynamic stiffness method have been addressed by revisiting the problem and focusing on the special set of missing solutions. Results are validated extensively both by published results as well as by numerical studies using NASTRAN and ABAQUS. The accuracy of the finite element method for inplane free vibration analysis is assessed and critically examined through the provision of benchmark solutions. Some representative modes that are missed by well-established dynamic-stiffness-based computer programs are presented. The inplane dynamic stiffness matrix presented is of great importance when combined with the out of plane matrix in order to obtain the closed-form solution for free vibration analysis of structures with complex geometries.

Efficient buckling and free vibration analysis of cyclically repeated space truss structures

Eigenproblems play a key role in the stability and free vibration analysis of structures. In large structural models, the solutions of these problems need a considerable computational effort. There are special types of structures whose special properties can be utilized to achieve solutions in a much simpler way. In this paper, an efficient method is presented for buckling and free vibration analysis of cyclically repeated space truss type structures. First, for a three dimensional truss element, stiffness, geometric stiffness and mass matrices are expressed in cylindrical coordinate system and this leads to the formation of a special pattern for the related matrices of entire of such structures. Second, using this pattern where some concepts of Kronecker product, initial generalized eigenproblems are decomposed into some subproblems with smaller dimensions and their solutions can easily be obtained. Finally, the efficiency of the present method is illustrated through some examples.

A stiffness equation transfer method for natural frequencies of structures

A stiffness equation transfer method is proposed for obtaining vibration frequencies of structures. This method is an extension of the finite element-transfer matrix (FE-TM) method. In the present method, the transfer of state vectors from left to right in the ordinary FE-TM method is changed into the transfer of stiffness equations of every section from left to right. This method reduces the propagation of round-off errors produced in the ordinary transfer matrix method. Furthermore, the drawback that the number of degrees of freedom on the left boundary must be the same as that on the right boundary in the ordinary FE-TM method, is now avoided. Besides, this method finds out the values of the frequency by Newton–Raphson iteration method, so no plotting of the value of the determinant versus assumed frequency is necessary. An IFETM—W program based on this method for use on an IBM PC586 microcomputer is developed. Finally, numerical examples are presented to demonstrate the accuracy as well as the potential of the proposed method for free vibration analysis of structures.

Free vibration analysis of a multiple-layered structure by the transfer influence coefficient method 5th report, considerations of a few points for a doulble-layered structure.

The authors apply the transfer influence coefficient method to the in-plane flexural free vibration analysis of a structure with double layers regarded as a discrete system with lumped mass, lumped inertia moment, and massless linear and rotational springs. They describe an effective method of reducing the computation time for the structure in which the lengths of layers are remarkably different from each other, and a systematic and generalized treatment of the releases and rolls in each layer at which the displacements are discontinuous. The results of the comparatively simple numerical computational examples on a personal computer also demonstrate the validity of the present algorithm for the particular structure, that is, the numerical high accuracy, the high speed and the flexibility for programming of the present method, compared with the transfer matrix method.

Free vibration analysis of plates, bridges and axisymmetric shells using a thick finite strip method

A unified approach for the vibration analysis of curved or straight prismatic plates and bridges and axisymmetric shells using a finite strip method based in Reissner—Mindlin shell theory is presented. Details of obtaining all relevant strip matrices and vectors are given. It is also shown how the use of the simple linear two node strip with reduced integration leads to direct explicit forms of all relevant matrices. Examples of application which show the accuracy of the linear strip for free vibration analysis of structures are presented.

Vibration analysis of quasi-symmetric structures

An efficient computational procedure is presented for the free vibration analysis of structures with unsymmetric geometry. The procedure is based on approximating the unsymmetric vibrational response of the structure by a linear combination of a few symmetric and antisymmetric modes (global approximation vectors), each obtained using approximately half the degrees of freedom of the original model. The three key elements of the procedure are: (a) use of mixed finite element models having independent shape functions for the internal forces (stress resultants) and generalized displacements, with the internal forces allowed to be discontinuous at interelement boundaries, (b) operator splitting, or additive decomposition of the different arrays in the governing finite element equations to delineate the contributions to the symmetric and antisymmetric response vectors, and (c) use of a reduction method through successive application of the finite element method and the classical Bubnov-Galerkin technique. The finite element method is first used to generate a few symmetric and antisymmetric global approximation response vectors. Then, the classical Bubnov-Galerkin technique is used to substantially reduce the size of the eigenvalue problem. An initial set of global approximation vectors is selected to be a few symmetric and antisymmetric eigenvectors, and their various-order derivatives with respect to a tracing parameter identifying all the correction terms to the symmetric (and antisymmetric) eigenvectors. A modified (improved) set of approximation vectors is obtained by using the inverse iteration procedure. The effectiveness of the proposed procedure is demonstrated by means of a numerical example.

A Numerical Method for Free Vibration Analysis of Structures with Small Design Changes : 2nd Report: Analysis of Degenerate or Nearly Degenerate Cases by a Subspace Technique

This paper presents a numerical method for free vibration analysis of structures with small design changes or perturbations. The proposed method is an extension of the iteration scheme introduced in the first report of this paper. A subspace technique is employed to deal with variation of degenerate or nearly degenerate eigenvalues of the unperturbed problems. The algorithm is designed such that it can be easily implemented in large-scale finite element computations. To see the effectiveness and fundamental properties of the proposed method, some numerical results are given for lateral vibration problems of a cantilever beam, a circular plate, and a parallelogramic membrane. It is shown that the present approach can actually analyze the separation of degenerate eigenvalues. Furthermore, it can improve the convergence behavior of the iteration process when it is compared with the method in the first report.

A Numerical Method for Free Vibration Analysis of Structures with Small Design Changes

Small design changes are usually repeated at design phase of structures, and it appears to be of practical significance to develop effective numerical methods for analysis of free vibration problems of structures with small perturbations. This paper presents a numerical method for perturbation analysis of matrix eigenvalue problems, where the eigenpairs necessary to be considered are assumed to be non-degenerate in unperturbed states. The proposed method is based on an iteration scheme combined with the use of generalized inverses of singular matrices. The scheme is designed to be easily implemented on large-scale finite element programs. To see effectiveness and fundamental properties of the proposed method, some numerical results are given for lateral vibrations of cantilever beams and circular plates.

A Numerical Method for Free Vibration Analysis of Structures with Small Design Changes

実際の設計では,基準となる構造物に対し各種の小さな設計変更を繰返すことが多い。本論文では,自由振動問題について,基準となる構造物の振動特性を利用して,設計変更による振動特性の変化を求める計算法を提出する。本法では,反復法と一般化逆マトリックスを利用して固有対の変化を計算するが,有限要素プログラムとの整合性に配慮した。さらに,はり,円板の横振動について数値例を与え,本法の特性,有効性を明らかにした。

Development of a unified numerical procedure for free vibration analysis of structures

AbstractThis paper presents the details of a unified numerical algorithm and the associated computer program developed for the efficient determination of natural frequencies and modes of free vibration of structures. Both spinning and nonspinning structures with or without viscous and/or structural damping may be solved by the current procedure; in a addition, the program is capable of solving static problems with multiple load cases as well as the quadratic matric eigenproblem associated with a finite dynamic element structural discretization. A special symmetric matrix decomposition scheme has been adopted for matrix triangularization, which renders the program rather efficient and economical. Also, a novel bisection scheme is described that further accelerates the solution convergence rate, particularly for the case of repeated roots.The associated computer program adopts and out‐of‐core solution strategy, thereby enabling effective solutions to be achieved for large, complex, practical structures. A complete listing of the program written in FORTRAN V, for the UNIVAC 1100/82 computer, along with the source deck is available for ready use.

On a finite dynamic element method for free vibration analysis of structures

This paper explores the concept of finite dynamic elements involving higher order dynamic correction terms in the associated stiffness and mass matrices. Such matrices are then developed for a rectangular prestressed membrane element. Next, efficient analysis techniques for the eigenproblem solution of the resulting quadratic matrix equations are described in detail. These are followed by suitable numerical examples which indicate that employment of such dynamic elements in conjunction with an efficient quadratic matric solution technique will result in a most significant economy in the free vibration analysis of structures.

Solution of quadratic matrix equations for free vibration analysis of structures

AbstractAn efficient digital computer procedure and the related numerical algorithm are presented herein for the solution of quadratic matrix equations associated with free vibration analysis of structures. Such a procedure enables accurate and economical analysis of natural frequencies and associated modes of discretized structures.The numerically stable algorithm is based on the Sturm sequence method, which fully exploits the banded form of associated stiffness and mass matrices. The related computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be substantially more accurate and economical than other existing procedures of such analysis. Numerical examples are presented for two structures: a cantilever beam and a semicircular arch.