Transcendental Eigenvalue Problem Research Articles

An efficient optimization method for transcendental eigenvalue problems based on mode count constraints and heuristic algorithm

An efficient optimization method for transcendental eigenvalue problems based on mode count constraints and heuristic algorithm

Free vibration analysis of composite beams and laminated reinforced panels by refined dynamic stiffness method and CUF-based component-wise theory

This paper presents a comprehensive analysis of the free vibration behavior of composite beams and laminated reinforced panels. Employing high-order theories with displacement variables only, the investigation combines dynamic stiffness method (DSM) and the Carrera unified formulation (CUF). The 3D displacement field can be expanded in the framework of CUF as any order of generic unknown variables over the cross-section, in the case of beam theories. Specifically, Lagrange expansions (LE) of cross-sectional displacement variables are considered, enabling the refinement modeling of complex cross-sections with different layers (layer-wise) and components (component-wise). The governing differential equations and natural boundary conditions are derived using the principle of virtual displacements. Subsequently, an exact dynamic stiffness matrix is developed by establishing a relationship between the amplitudes of harmonically varying loads and the corresponding responses. The Wittrick–Williams algorithm is employed to solve the transcendental eigenvalue problem resulting from this approach. The Lagrange-based CUF(LE)-DSM outperforms other polynomials (such as Taylor)-based CUF in analyzing composite structures, enabling detailed analysis with various geometries, lamination schemes and boundary conditions. It delivers accurate results in reinforced panel analysis, requiring only 13% of the DOFs compared to 3D FE models, thereby reducing computational costs significantly, as confirmed by comparative studies and validations.

A two-dimensional wave and Rayleigh–Ritz method for complex dispersion in periodic arrays of circular damped acoustic black holes

Two-dimensional arrays of circular acoustic black holes (ABHs) on plates are known to exhibit many surprising properties such as bandgap formation, scattering, topological edge states and negative refraction and bi-refraction, to name a few. These effects can be deduced from the analysis of dispersion relations. However, to date, most of the work on periodic ABH arrays has only dealt with real dispersion curves that do not adequately reproduce their behavior. The role of the damping layer and intrinsic losses, which is essential to ABHs, is ignored. This can be accounted for by complex dispersion. In this paper, we extend the wave and Rayleigh–Ritz method (WRRM) previously developed for periodic ABHs in beams to address complex dispersion in two-dimensional periodic ABH arrays. This poses several challenges. On the one hand, in the WRRM the solution of the equations of motion is expanded in terms of the basis of the nullspace determined by the periodic boundary conditions. Although this only involves point connections for beams, in the two-dimensional case continuous line constraints are present and care must be taken to find the appropriate nullspace. To obtain the complex dispersion surface and complex equi-frequency contours of two-dimensional periodic ABH arrays we then need to set the eigenvalue problem in the k(ω) approach. For a fixed angular frequency ω∈R and a direction of propagation, θ, whose tangent is the quotient of the unknown wavenumber components in the y and x directions, we derive polynomial and transcendental eigenvalue problems for the complex wavenumber k∈ℂ×ℂ as a function of θ. The complex dispersion surface and equi-frequency contours of the periodic ABH arrays can be obtained by solving these eigenvalue problems with an iteration method. The two-dimensional WRRM is validated by comparison with finite element simulations. Finally, we apply it to calculate the complex dispersion of an infinite periodic strip and of an infinite periodic array of circular ABHs. In the latter case, collimation effects are observed. The transmissibility of finite ABH strips and arrays is also presented for comparison.

Spectral model and experimental validation of hysteretic and aerodynamic damping in dynamic analysis of overhead transmission conductor

Abstract The paper treats a new approach of dynamic analysis of a conductor cable of an overhead transmission line under the theoretical background of spectral element method (SEM). The methodology relies on the analytical solution of the displacement wave equation in the frequency domain; moreover, it both enhances the accuracy of model predictions and reduces the computational efforts when compared to a finite element (FE) model. Two numerical models based on SEM are built for transmission lines taking into account hysteretic and aerodynamics damping and whose analyses consider dispersion diagrams and frequency response functions (FRFs). As SEM leads to a transcendental eigenvalue problem, to obtain the natural frequencies of the conductor, it is used the Wittrick–Williams algorithm. The results presented in the paper show the sensitivity of the response conductor changes according to the tensile load and damping parameters. Finally, the numerical models have compared with the analytical solution of the cable and experimental tests performed at CEPEL’s (Electric Power Research Center) laboratory with the overhead conductor Grosbeak cable. The results show outstanding accuracy in both experimental measurements and analytical analysis aspects.

Dynamic finite element formulations for moderately thick plate vibrations based on the modified Mindlin theory

In this paper, the modified Mindlin theory is used for the construction of the dynamic stiffness matrix, the flexibility matrix, and the transfer matrix of a thick plate simply supported at two opposite edges. The modified Mindlin theory operates with bending deflection as the basic variable for the determination of the total (bending+shear) deflection and the angles of rotation. It is shown that the appropriate application of the constructed matrices to various boundary conditions leads to a determined formulation of the eigenvalue problem. As a result, the problem can be treated using ordinary algorithms for linear eigenvalue problems. Therefore the application of the relatively complex Wittrick-Williams algorithm, developed for the transcendental eigenvalue problems with the unusual forms of non-zero determinant is avoided. Using this technique, dynamic finite elements can be obtained in a simpler form than that based on the application of the conventional Mindlin theory. All phenomena related to the dynamic finite element application are investigated in the case of an axial bar vibration in a transparent analytical way. Furthermore, the application of the developed thick plate elements is illustrated through several numerical examples. Examples include using single elements as well as an assembly of elements. In the examples of single elements transcendental eigenfunctions are derived. Also a dynamic beam finite element is considered as a special case of the plate finite strip which exhibits no variation in the transverse direction.

Distributed parallel computing of the recursive eigenvalue search in the context of transfer matrix method for multibody systems

The modeling and solving a transcendental eigenvalue problem are important issues in the transfer matrix method for linear multibody systems. Based on the recursive eigenvalue search algorithm for transfer matrix method for linear multibody system, the distributed parallel approach for assembling overall transfer matrix and searching eigenvalues is proposed. This is achieved based on Message Parallel Interface. The influence of the CPU core number as well as the distributed network environment on the final computational time is analyzed through numerical examples of both a non-uniform beam and a multiple launch rocket system. The results indicate that the computational time is significantly reduced by the proposed parallel computing method, so that the computational efficiency on optimization and design of complex multibody systems can be improved.

Directionality of wave propagation and attenuation in plates with resonant shunting arrays

This article proposes an analytical method to evaluate the propagation constant in arbitrary direction in the plate with shunting arrays, which can circumvent the complicated transcendental eigenvalue problem. Based on this method, the directionality of waves propagating or decaying in the plate with arrays of resonant shunts is examined, including directionality of attenuation constant, location of band gaps, and group velocity in pass band. Moreover, the theoretical results are verified by simulation in well-accepted commercial finite element software.

Coupled flexural–torsional dynamic stiffness matrix of an elastically supported axially loaded beam with shear resistant in‐fill

SummaryThe dynamic stiffness matrix of an axially loaded elastically supported uniform beam with doubly asymmetric cross‐section that exhibits coupling between flexural and torsional motions is developed and subsequently used to investigate its free vibration characteristics. The beam comprises a thin‐walled outer section that encloses, and works compositely with, a core of shear resistant in‐fill material. The outer layer provides flexure, warping and Saint–Venant rigidity, while the inner layer provides both Saint–Venant and shear rigidity. A three‐parameter Winkler model is used to describe the distributed elastic support. Hamilton's principle is used to derive the partial differential equations governing the free vibration of the beam, together with the associated natural boundary conditions. This gives rise to three coupled equations that are subsequently combined into a single, 12th order, ordinary differential equation. Throughout the process, the uniform distribution of mass in the member is accounted for exactly and thus necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick–Williams algorithm, which enables the required natural frequencies to be converged upon to any required accuracy with the certain knowledge that none have been missed. Finally, in order to verify the accuracy of the presented theory, the numerical solutions are given and compared with the results that are available in the literature and finite element solutions using abaqus software. Copyright © 2014 John Wiley & Sons, Ltd.

Free vibration analysis of beams and frames with multiple cracks for damage detection

The problem of calculating the natural frequencies of beams with multiple cracks and frames with cracked beams is studied. The natural frequencies are obtained using a new method in which a rotational spring model is used to represent the cracks. For beams, dynamic stiffness matrices of order 4 are obtained in a recursive manner, according to the number of cracks, by applying partial Gaussian elimination. The Wittrick–Williams algorithm is used to compute the natural frequencies in the resulting transcendental eigenvalue problem. Published numerical examples for cracked beams are used for validation. The global dynamic stiffness matrix of a frame with multiply cracked members is then assembled. A published two bay frame example is used to evaluate the new method. The effect of changing the location of a crack in a two bay two storey frame is studied numerically, giving insight into the inverse problem of damage detection.

Refined dynamic stiffness elements applied to free vibration analysis of generally laminated composite beams with arbitrary boundary conditions

The Carrera Unified Formulation (CUF) is used in this paper to develop higher-order beam theories for composite laminates. In the framework of the CUF, the three-dimensional displacement field is approximated as a truncated Taylor-type expansion series of the generalized displacements, which lie on the beam axis. The truncation of the series determines the theory order N, which is a free parameter of the formulation. The strong form of the principle of virtual displacements is used to obtain the equations of motion and the natural boundary conditions for the laminated beam in free vibration. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The Wittrick–Williams algorithm is applied to solve the transcendental eigenvalue problem resulting from the present approach. Composite beams with arbitrary boundary conditions, geometries and lamination schemes can be analysed with the proposed method. Numerical investigations are carried out and the results are compared with reference solutions from the literature and with solutions from commercial finite elements codes.

Wave propagation and attenuation in plates with periodic arrays of shunted piezo-patches

Periodic arrays of shunted piezoelectric patches are employed to control the wave propagation in a thin plate. The performance is characterized through the application of finite element method and Bloch theorem. This article proposes an effective approach to gain the dispersion properties of the periodically shunted plate in any directions from the solutions of transcendental eigenvalue problems. The results show that resistive shunts can tune the location and attenuation constants of the Bragg gap, while the internal resonances of resonant shunting system split the dispersion curves and form a locally resonant band gap. Moreover, the Bragg gap is directional, whose width varies enormously with directions. However, the locally resonant gap almost keeps the same in different directions, i.e., the gap is complete. Also, the influences of different shunting parameters to the band gaps are investigated in detail.

A method to solve the nonlinear eigenvalue problem of Timoshenko plane frames with rigid offsets and end releases

The study reported here makes use of the dynamic stiffness matrix, DSM, for Timoshenko beams, hence considering rotatory inertia and shear deflection, to obtain the exact solution of the associated free vibration problem. The natural frequencies and vibration modes in the presence of end releases and rigid offsets are all exactly obtained for different combinations of boundary conditions and in plane arrangements of members. This is possible by solving the resulted nonlinear and transcendental eigenvalue problem using two general methods, one based on the Wittrick–Williams algorithm adapted to handle end release and rigid offset, and the other on the power deflation secant method. The latter, offered here as a novelty, is proven to be an efficient technique judging from the examples given, using both algorithms and the finite element method. It is concluded that the power deflation method is a powerful tool to solve the nonlinear eigenvalue problem, leading to exact solutions of the free vibration of complex skeleton-like structures.

On the bi-planar motion of a Timoshenko beam with shear resistant in-fill

An exact dynamic stiffness matrix is developed to describe the motion of a uniform beam in which the flexural and torsional motion is coupled in three dimensions by its doubly asymmetric cross-section. The beam comprises a thin walled outer section that encloses, and works compositely with, a core of shear resistant in-fill material. The outer layer can be of open or closed section and flexure is modeled according to Timoshenko theory. In addition, it provides warping, shear and Saint-Venant rigidity, while the core material provides Saint-Venant and shear rigidity. The partial differential equations governing the free vibration of the beam, together with the associated natural boundary conditions, are derived using Hamilton's principle. This gives rise to five coupled equations that are subsequently combined into a single, twelfth order, ordinary differential equation. Throughout the process, the uniform distribution of mass in the member is accounted for exactly and thus necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick–Williams algorithm, which enables the required natural frequencies to be converged upon to any required accuracy with the certain knowledge that none have been missed. Finally, examples are given to confirm the accuracy of the theory presented.

Sensitivity Analysis of the Eigenvalue Problem for General Dynamic Systems with Application to Bridge Deck Flutter

The mathematical models governing the dynamics of various engineering systems, such as airplane wings and bridge decks subjected to aerodynamic forces, mechanical and civil structures interacting with fluid or soil, or systems with time delays, yield transcendental eigenvalue problems. In this work, a general transcendental eigenvalue problem is first formulated and a biorthogonality relationship between eigenvectors is derived. Then, the sensitivities of eigenvalues and eigenvectors with respect to a system parameter are obtained. The method is employed to analyze in detail a transcendental eigenvalue problem arising in the analysis of a bridge deck subjected to aerodynamic forces. The sensitivities of eigenvalues and eigenvectors are successfully used to improve the performance of an iterative method used for solving the eigenvalue problem.

Exact dynamic stiffness matrix for a thin-walled beam-column of doubly asymmetric cross-section

Bernoulli-Euler beam theory is used to develop an exact dynamic stiffness matrix for the flexural-torsional coupled motion of a three-dimensional, axially loaded, thin-walled beam of doubly asymmetric cross-section. This is achieved through solution of the differential equations governing the motion of the beam including warping stiffness. The uniform distribution of mass in the member is also accounted for exactly, thus necessitating the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick-Williams algorithm. Finally, examples are given to confirm the accuracy of the theory presented, together with an assessment of the effects of axial load and loading eccentricity.

Power secant method applied to natural frequency extraction of Timoshenko beam structures

This work deals with an improved plane frame formulation whose exact dynamic stiffness matrix (DSM) presents, uniquely, null determinant for the natural frequencies. In comparison with the classical DSM, the formulation herein presented has some major advantages: local mode shapes are preserved in the formulation so that, for any positive frequency, the DSM will never be ill-conditioned; in the absence of poles, it is possible to employ the secant method in order to have a more computationally efficient eigenvalue extraction procedure. Applying the procedure to the more general case of Timoshenko beams, we introduce a new technique, named "power deflation", that makes the secant method suitable for the transcendental nonlinear eigenvalue problems based on the improved DSM. In order to avoid overflow occurrences that can hinder the secant method iterations, limiting frequencies are formulated, with scaling also applied to the eigenvalue problem. Comparisons with results available in the literature demonstrate the strength of the proposed method. Computational efficiency is compared with solutions obtained both by FEM and by the Wittrick-Williams algorithm.

Coupled lateral–torsional frequencies of asymmetric, three-dimensional structures comprising shear-wall and core assemblies with stepwise variable cross-section

A simple and accurate model for asymmetric, three-dimensional wall-core structures is developed that enables any desired natural frequency to be determined by a method which guarantees that no natural frequencies can be missed. The model assumes that the primary walls and cores run in two orthogonal directions and that their properties may vary in a stepwise fashion at one or more storey levels. A vectorial approach is used to generate the governing differential equations for coupled flexural-torsional motion that are finally incorporated into an exact dynamic stiffness matrix (exact finite element) that can model any uniform segment of the structure. A model of the original structure can then be assembled in the usual way. Since the mass of each segment is assumed to be uniformly distributed, it is necessary to solve a transcendental eigenvalue problem, which is accomplished using the Wittrick–Williams algorithm. When the structure can be represented realistically by a uniform cantilever, solutions can be found easily, by hand. A parametric study comprising five, three-dimensional, asymmetric wall-core structures is given to compare the accuracy of the current approach with that of a full finite element analysis.

Buckling of functionally graded and elastically restrained non-uniform columns

Columns with non-uniform distribution of geometrical or material parameters i.e. functionally graded material distribution, varying cross-sectional area and flexural stiffness provide an economical solution to carry the desired higher compressive loads in engineering structures. In this paper, a low-dimensional mathematical model is presented, which is capable of computing the buckling loads of uniform and non-uniform functionally graded columns in the axial direction. The columns with spatial variation of flexural stiffness, due to material grading and/or non-uniform shape, are approximated by an equivalent column with piecewise constant geometrical and material properties. Such a formulation leads to certain transcendental eigenvalue problems where the matrix elements are transcendental functions. This model is further extended in analyzing some uniform and non-uniform elastically restrained or braced axially graded columns with equal or unequal spans. The mathematical modeling, closed-form transcendental functions and numerical solution technique are described and several examples of estimating buckling loads for various boundary configurations are presented. Some of the results are also validated against available solutions, representing the convergence, effectiveness, accuracy and versatility of the proposed modeling and numerical method. Formulation of such low-dimensional eigenvalue problems can also be extended for analyzing, designing and optimizing the static and dynamic behavior of structural components that are made of functionally graded materials.

A method for finding repeated roots in transcendental eigenvalue problems

The eigenvalues of a gyroscopic system depend on a parameter, and the system may become unstable when eigenvalues collide. For continuous systems, the eigenvalue problem is represented by a matrix with transcendental elements. A method is developed to determine the value of the parameter that causes eigenvalues to collide. The method is implemented using a certain formula for the determination of the derivative of a matrix determinant without the use of symbolic expansion of the determinant. This formula simplifies the computational task considerably. Applications of the method are demonstrated by problems involving buckling by circulatory forces and oscillations of a gyroscopic system.

Vibration analysis of doubly asymmetric, three-dimensional structures comprising wall and frame assemblies with variable cross-section

A global analysis approach to modelling doubly asymmetric, three-dimensional, multi-bay, multi-storey, wall–frame structures is presented in a form that enables the lower numbered natural frequencies to be determined approximately with the certain knowledge that none have been missed. It is assumed that the primary walls and frames of the original structure run in two orthogonal directions and that their properties may vary in a step-wise fashion at one or more storey levels. The structure therefore divides naturally into uniform segments between changes of section properties. A typical segment is then replaced by an equivalent shear–flexure–torsion coupled beam whose governing differential equations are formulated using a continuum approach and posed in the form of a dynamic member stiffness matrix. The original structure can then be re-modelled as a sophisticated stepped cantilever in the usual way. Since the mass of each segment is assumed to be uniformly distributed, it is necessary to solve a transcendental eigenvalue problem, which is accomplished using the Wittrick–Williams algorithm. A parametric study on a series of wall–frame structures of varying height with different plan configurations is given to compare the accuracy of the current approach with datum results from fully converged finite element analyses.