Classical Rayleigh-Ritz Method Research Articles

Compressive local buckling of integrated fluted-core sandwich composite panels

Fluted-core sandwich structures, consisting of fiber-reinforced plastics, are frequently utilized in the aerospace industry owing to their excellent strength-to-weight ratio. However, the thin-walled structures are prone to local buckling, which reduces their load-bearing capacity and eventually leads to material failure. The current study presents analytical models for predicting the local buckling of fluted-core sandwich panels subjected to uniform compressive load. The proposed method is based on a discrete plate analysis approach, which considers the face-sheets between two webs as rotationally-restrained plates, wherein the rotational restraint stiffness is determined based on the geometric configuration and material properties. Herein, two boundary conditions, i.e., simply-supported and clamped, along the loaded edges were studied. Several deflection functions were proposed based on the classical Rayleigh–Ritz method, satisfying the boundary conditions. The experimental tests and finite element simulations were carried out to analyze the local buckling behavior, rendering excellent consistency and demonstrating the reliability of the proposed method. The effects of geometric parameters on the local buckling behavior were systematically studied and the results revealed that the current analytical models with high computational efficiency are reliable, exhibiting promise of eventual success in the optimization of preliminary engineering designs.

Nonlinear structural analysis of a flexible multibody system using the classical Rayleigh–Ritz method

A new formulation based on the classical Rayleigh–Ritz method (CRRM) is proposed in this study to conduct a geometrically nonlinear analysis for flexible multibody structures (FMSs). The proposed formulation can employ various shape functions for global discretization. This feature renders the proposed formulation straightforward and suitable for computer programming. The convergence characteristics of various shape functions for the proposed formulation were first examined with some numerical examples. Then we investigated the efficiency of the proposed formulation for obtaining converged solutions of the displacement, reaction force, maximum stress, and the location where the maximum stress occurs by comparing the degrees of freedom (DOFs) used for the proposed formulation and FEM formulation. The comparative study showed that the proposed formulation can be used to solve geometrically nonlinear FMS problems much more efficiently than the FEM formulation with the same level of accuracy.

Stiffness matrix for the analysis and design of partial-interaction composite beams

Compared to the classical Rayleigh-Ritz method and other analytical solutions, finite element (FE) method is more efficient and capable in calculating the deformations and stress states of partial-interaction composite beams, as well as manipulating the material and geometrical parameters for better engineering designs. Stiffness matrix of composite beams considering the interlayer slips is derived based on the kinematic assumptions of the Timoshenko’s beam theory by taking into account of the transverse shear deformations. A detailed derivation is elaborated to obtain the local stiffness matrix for a composite beam element, while the higher-order interpolation functions are adopted for the displacement fields (deflection, rotation, and interlayer slip). Then a finite element program is developed by assembling the local stiffness matrices and applying corresponding equivalent nodal stresses. Several numerical results are presented and compared against the analytical solutions available in the literature to demonstrate the accuracy of the proposed FE stiffness matrix. Finally, a design procedure by connecting particle swarm optimization technique with the present FE analysis is created to reduce the deformations of simply supported composite beams while the quantity of shear connectors remains the same, to prove the superior simulation capacity and efficiency of the derived FE stiffness matrix with other techniques. Compared to the analytical methods, the proposed finite element is more convenient and applicable in the analysis of partial-interaction composite beams under more complicated loading and boundary conditions. In the meantime, the explicitly expressed local stiffness matrix can be easily implemented into other commercial software packages asa subroutine for both professional and personal engineering designs and calculations.

Free Vibrations of Uniform Timoshenko Beams on Pasternak Foundation Using Coupled Displacement Field Method

Abstract Complex structures used in various engineering applications are made up of simple structural members like beams, plates and shells. The fundamental frequency is absolutely essential in determining the response of these structural elements subjected to the dynamic loads. However, for short beams, one has to consider the effect of shear deformation and rotary inertia in order to evaluate their fundamental linear frequencies. In this paper, the authors developed a Coupled Displacement Field method where the number of undetermined coefficients 2n existing in the classical Rayleigh-Ritz method are reduced to n, which significantly simplifies the procedure to obtain the analytical solution. This is accomplished by using a coupling equation derived from the static equilibrium of the shear flexible structural element. In this paper, the free vibration behaviour in terms of slenderness ratio and foundation parameters have been derived for the most practically used shear flexible uniform Timoshenko Hinged-Hinged, Clamped-Clamped beams resting on Pasternak foundation. The findings obtained by the present Coupled Displacement Field Method are compared with the existing literature wherever possible and the agreement is good.

A general substructuring synthesis method for the analysis of complex vibro-acoustic interaction systems

A general substructuring synthesis method is proposed for analyzing structural-acoustic interaction problems in engineering applications. The approach is performed by systematically partitioning the entire system into a series of distinct components according to geometries and boundaries. The motion of substructures and fluid subdomains is then discretized and represented by the classical Rayleigh-Ritz method, while the perturbed Lagrangian method is employed to connect the disjoint substructures and subdomains by imposing approximate compatibility conditions. The established framework enables one to easily model the interior acoustics of combined fluid structural systems with geometric complexities and to effectively analyze vibro-acoustic behaviors with sufficient accuracy at relatively high frequencies. Several numerical examples are carried out to demonstrate the reliability and applicability of the present method, which illustrate a significant computational advantage as compared to classical Finite Element procedures.

Closed-form analysis of the buckling loads of composite laminates under uniaxial compressive load explicitly accounting for bending–twisting-coupling

This paper presents a method for the determination of the buckling loads of symmetrically laminated composite plates under uniaxial compressive load with explicit consideration of bending–twisting-coupling (BTC). The plates under consideration are assumed to be thin and balanced with a symmetric stacking sequence and consist of layers with ply orientations of {0°/±45°/90°}, but otherwise arbitrary lamination schemes. The characteristic laminate quantities are described in a non-dimensional notation in order to capture all combinations of material and stacking sequences that are physically possible. Plates with rectangular planforms in the framework of Classical Laminate Theory (CLT) under several different sets of boundary conditions are considered, and the buckling problem of such composite laminated plates is solved in an approximate manner by using the classical Rayleigh–Ritz method. Several combinations of the boundary conditions ‘simply supported’ and ‘clamped’ are analyzed, and the obtained results are compared with finite-element simulations for verification purposes. Based on the computed data, two different design equations for sufficiently long plates are developed which allow for a rapid and closed-form analytical assessment of the buckling behavior of symmetrically layered composite plates under explicit consideration of bending–twisting coupling. It is found that the closed-form analytical analysis equations are simple in their application and yet still work with very satisfying accuracy which makes them very suitable for use in practical engineering work.

A free vibration analysis of toroidal composite shells in free space

The main aim of this paper is to present a dynamic analysis of toroidal shell structures in free space. The simplified thin shell theory, based on the typical small deflection assumption, but enriched by the transverse shear terms, was used. The constituting material is not isotropic but a multilayer composite angle-ply laminate, with non-uniform thickness.A numerical procedure based on the classical Rayleigh–Ritz method, was utilized. The dynamic independent variables were expressed in terms of a double Fourier series expansion as function of the azimuth and internal meridian circular angle. This allowed to determine easily the strain and kinetic energy expressions, and consequently the stiffness and mass matrices.Then, the numerical problem was simplified to the classical generalized eigenvalue problem relation by differentiating the whole energetic functional, as in the Rayleigh–Ritz method. The solutions were determined by using an appropriate algorithm. The high frequency vibration modes were considered and their peculiarities with respect to the low frequency modes were pointed out. The dependence of the obtained results on the laminate winding angle was also considered.Comparisons with the other authors results were necessary to validate the utilized numerical approach.

Procedure for Optimization of Truss Space Structureswith Fixed Minimum Vibration Frequency

The considered structure, with an attached rigid body at the external free end, is typical of satellite appendages. The same value of the lowest vibration frequency of the reference truss structure, with uniform thickness over all the component beams, has been prescribed to prevent possible resonances between satellite and appendages. A minimum value of the thickness over each component beam of the analyzed truss has been established a priori. Focusing attention on the dynamic behavior, a numerical procedure (based on the classical Rayleigh–Ritz method but combined with the finite element method) has been used. Polynomial power series expansions have been used both for the dynamic variables (displacements and rotations) and for the thickness axial distribution over each component beam of the truss structure. The classical Euler–Lagrange optimality criterion has been used to find the requested solution of each optimization operation, which consists of the search for stationary conditions of the Lagrangian functional. A series of repeated optimization operations have been performed to arrive at the thickness optimizedprofilewithineachstructuralcomponent,withthepreestablishedminimumvalueofthebeams’thickness. It has also been possible to evaluate the weight reduction.

Vibration analysis of multiple-stepped beams with the composite element model

A new approach to analyze the free and forced vibrations of beams with multiple cross-section steps is proposed using a composite element method in this paper. The results are compared to receptance function method and classical Rayleigh–Ritz method and finite element results. The natural frequencies obtained from the composite element method are found to be in close agreement with other methods mentioned above and finite element method. The forced vibration responses of the stepped beam are also calculated from the composite element method and compared with those obtained from the finite element method. Time histories from both methods are found to match very well. This indicates the correctness of the proposed method for vibration analyses of the stepped beam. The proposed method can be extended easily to deal with beams consisting of an arbitrary number of nonuniform segments.

Substructure synthesis method for buckling analysis in frames

An everlasting objective of Structural Stability and Structural Dynamics is the development of approximate methods that yield highly accurate results (critical loads, and natural frequencies and modes) and that involve low numbers of degrees of freedom. The Rayleigh–Ritz–Meirovitch substructure synthesis method (RRMSSM) has proved to be more efficient than the finite element method (FEM) in dynamic analysis of structures that can be modeled as assemblages of simple substructures, and the question is if it would be also more efficient in stability analysis of these structures. The RRMSSM is the classical Rayleigh–Ritz method with Meirovitch’s introduction of quasicomparison functions [1] and extension to multibody systems [2]. This Rayleigh-Ritz methodology with both developments was presented by Meirovitch and Kwak [3]. Lately, it was extended to multiply supported or framed structures [4, 5]. In these investigations, the

Determination of the steady state response of viscoelastically corner point-supported rectangular orthotropic plates

Vibration of orthotropic rectangular plates having viscoelastic point supports at the corners is analyzed. The Lagrange’s equations is used to examine the free vibration characteristics and steady state response to a sinusoidally varying force affecting at the center of a viscoelastically point-supported orthotropic elastic plate of rectangular shape. In the study, for applying the Lagrange’s equations, the trial function denoting the deflection of the plate is expressed in the polynomial form: This procedure may be considered as the classical Rayleigh–Ritz method. By using the Lagrange’s equations, the problem is reduced to the solution of a system of algebraic equations. The influence of the mechanical properties, and of the damping of the supports to the mode shapes and to the steady state response of the viscoelastically point-supported rectangular plates is investigated numerically for the concentrated load at center for various values of the mechanical properties characterizing the anisotropy of the plate material and for various damping and stiffness of the supports. The results are given for the frequencies and mode shapes of the first three symmetrical modes. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with other solutions based on the Kirchhoff–Love plate theory.

Reproducing kernel particle method for free and forced vibration analysis

A reproducing kernel particle method (RKPM) is presented to analyze the natural frequencies of Euler–Bernoulli beams as well as Kirchhoff plates. In addition, RKPM is also used to predict the forced vibration responses of buried pipelines due to longitudinal travelling waves. Two different approaches, Lagrange multipliers as well as transformation method , are employed to enforce essential boundary conditions. Based on the reproducing kernel approximation, the domain of interest is discretized by a set of particles without the employment of a structured mesh, which constitutes an advantage over the finite element method. Meanwhile, RKPM also exhibits advantages over the classical Rayleigh–Ritz method and its counterparts. Numerical results presented here demonstrate the effectiveness of this novel approach for both free and forced vibration analysis.

ON LOWER AND UPPER BOUNDS FOR ENERGY EIGENVALUES OF QUANTUM DOTS

The problem of precise estimation of the low-lying discrete energy spectrum of quantum dots was addressed rigorously by means of the methods of upper and lower bounds for eigenvalues of linear Hermitian operators in Hilbert space. It was shown that the method of intermediate problems for eigenvalues and the classical Rayleigh–Ritz method may be fused into effective tool of analytical and numerical investigation of the energy spectrum of quantum dots thus enabling and facilitating calculation of eigenvalues with controlled or prescribed precision. As an example, the energy spectrum problem for a simple two-electron quantum dot with harmonic-oscillator spherical confinement potential was thoroughly analyzed. Precise numerical calculations corroborated that in the absence of external magnetic field the ground state of the dot is the spin singlet for quantum dots of arbitrary characteristic size, so that no spontaneous singlet-triplet "phase transition" can occur.

VIBRATIONS OF FLUID-FILLED HERMETIC CANS

Free, conservative vibrations of a hermetic can, simply supported at the base, are studied. The can is composed by a circular cylindrical shell and two identical circular plates connected to the shell at its ends. The artificial spring method, which is an extension of the classical Rayleigh–Ritz method, is used to solve the system by using substructuring. The can is studied empty and filled with an inviscid and incompressible fluid. Fluid volume conservation is applied. The interaction between the plates and the shell via the fluid is considered, and exact expressions for the fluid velocity potential are used. The effect of flexibility of joints between plates and shell is investigated. Results for a fluid-filled, simply supported shell closed by rigid ends are also obtained and compared to the classical open-end shell.

A new approach for numerical modal analysis using the element‐free method

A novel approach in analysing the vibration of elastic body systems is presented and discussed in this paper. This method uses a meshless spatial approximation based only on nodes, which constitutes an advantage over the finite element method. As it is applicable to an elastic body with arbitrary shape, it also has advantages over the classical Rayleigh–Ritz method and its extensions. The paper is organized as follows: in Section 2, Hamilton's principle is used for obtaining the equations of motion for a three-dimensonal simply connected elastic body. In Section 3, the development of weighted base functions for the moving least- squares interpolant is described. In Section 4, the mass and stiffness matrices are derived using the previous results. In Section 5, a system synthesis description is given, in which we also develop the approach used for integrating the boundary conditions. Finally in Section 6, the theoretical results are validated by case studies chosen to highlight various features of the approach, including optimization of the shape function parameters. The particle method has already been used for structural dynamics (Liu et al., International Journal for Numerical Methods in Engineering 1995;38:1655–1679), but for free vibration of beams and plates, to the authors knowledge, this paper is the first application of the element-free method to modal analysis. Copyright © 1999 John Wiley & Sons, Ltd.

A new approach for numerical modal analysis using the element-free method

A novel approach in analysing the vibration of elastic body systems is presented and discussed in this paper. This method uses a meshless spatial approximation based only on nodes, which constitutes an advantage over the finite element method. As it is applicable to an elastic body with arbitrary shape, it also has advantages over the classical Rayleigh-Ritz method and its extensions. The paper is organized as follows: in Section 2, Hamilton's principle is used for obtaining the equations of motion for a three-dimensonal simply connected elastic body. In Section 3, the development of weighted base functions for the moving least-squares interpolant is described. In Section 4, the mass and stiffness matrices are derived using the previous results. In Section 5, a system synthesis description is given, in which we also develop the approach used for integrating the boundary conditions. Finally in Section 6, the theoretical results are validated by case studies chosen to highlight various features of the approach, including optimization of the shape function parameters. The particle method has already been used for structural dynamics (Liu et al., International Journal for Numerical Methods in Engineering 1995;38:1655-1679), but for free vibration of beams and plates, to the authors knowledge, this paper is the first application of the element-free method to modal analysis. Copyright (C) 1999 John Wiley & Sons, Ltd.

VIBRATIONS AND ELASTIC STABILITY OF THIN CIRCULAR PLATES WITH VARIABLE PROFILE

Circular plates of thicknesses varying to the functional relation h o[1 + γ( r/ a) n ], where nis a positive integer, are studied in the present paper. Uniform, elastic boundary restraints are considered and the first two natural frequency coefficients corresponding to axisymmetric modes and the buckling, in plane pressure are determined using simple polynomial co-ordinate functions which identically satisfy the boundary conditions and the classical Rayleigh-Ritz method. An independent solution is also obtained by means of the differential quadrature technique.

Hybrid equations of motion for flexible multibody systems using quasicoordinates

A variety of engineering systems, such as automobiles, aircraft, rotorcraft, robots, spacecraft, etc., can be modeled as flexible multibody systems. The individual flexible bodies are in general characterized by distributed parameters. In most earlier investigations they were approximated by some spatial discretization procedure, such as the classical Rayleigh-Ritz method or the finite element method. This paper presents a mathematical formulation for distributed-parameter multibody systems consisting of a set of hybrid (ordinary and partial) differential equations of motion in terms of quasi-coordinates. Moreover, the equations for the elastic motions include rotatory inertia and shear deformation effects. The hybrid set is cast in state form, thus making it suitable for control design.

Rayleigh-Ritz based substructure synthesis for flexible multibody systems

This paper is concerned with the modeling of flexible multibody systems by a Rayleigh-Ritz based substructure synthesis method, so that certain advantages can be accrued by using the variational approach to derive the eigenvalue problem. As with the classical Rayleigh-Ritz method, if the admissible functions used to represent the motion of the substructures are not chosen properly, convergence can suffer. This paper presents a new substructure synthesis method with superior convergence characteristics achieved by representing the motion by means of a recently developed class of functions, namely, the class of quasi-compari son functions. This improved convergence is shown to be related to improved approximation of both the differential equations and the natural boundary conditions. The theory is demonstrated by means of a numerical example. I. Introduction T HIS investigation is concerned with the modeling of flexible multibody systems. Many structures, such as fixedwing aircraft, helicopters, flexible spacecraft, flexible robots, etc., can be modeled as assemblages of interacting flexible bodies. A method proposed by Hurty1'2 in the early 1960s, and known as component-mode synthesis, consists of modeling the motion of the individual substructures, referred to as components, by means of He uses three types of modes to represent the motion of a substructure: rigid-body modes, constraint modes, and normal modes. To define constraint modes, Hurty divides the constraints into two classes, statically determinate and redundant. The constraint modes are equal in number to the number of redundant constraints and are defined by producing a unit displacement on each redundant constraint in turn, with all other constraints fixed. The normal modes represent the modes of vibration of the components with all constraints fixed. An approach by Craig and Hampton3 differs from that of Hurty 1'2 mainly in the selection of the component modes. Indeed, Craig and Hampton suggest two types of component modes: boundary modes providing for displacements and rotations at points along the substructure boundaries, and related to the constraint modes of Hurty, and substructure modes corresponding to completely restrained boundaries. The substructures modeled individually in Refs. 1-3 are made to act together as a single structure by eliminating the redundant generalized coordinates arising from the fact that displacements at points common to two adjacent substructures are included twice in the overall problem formulation, once for each substructure. The elimination process is based on the use of constraint equations resulting from the enforcement of compatibility conditions, which amounts to saying that the displacement of a boundary point shared by two substructures is the same. In another approach to the problem, proposed by

Analysis of closed arbitrary dielectric waveguides using a modified Rayleigh-Ritz technique

To avoid the meshing difficulties of the finite-element method, the classical Rayleigh-Ritz method is combined with an additional optimization to analyze closed arbitrary dielectric waveguides. The method is easily implemented in compact code and is user-friendly. The method and its rationale are developed, and numerical examples are presented to demonstrate its accuracy in propagation constant and nonperturbational loss calculations. The method is shown to be rapidly convergent and extremely stable.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>