Classical Ritz Method Research Articles

Variational Solution and Numerical Simulation of Bimodular Functionally Graded Thin Circular Plates under Large Deformation

In this study, the variational method and numerical simulation technique were used to solve the problem of bimodular functionally graded thin plates under large deformation. During the application of the variational method, the functional was established on the elastic strain energy of the plate while the variation in the functional was realized by changing undetermined coefficients in the functional. As a result, the classical Ritz method was adopted to obtain the important relationship between load and maximum deflection that is of great concern in engineering design. At the same time, the numerical simulation technique was also utilized by applying the software ABAQUS6.14.4, in which the bimodular effect and functionally graded properties of the materials were simulated by subareas in tension and compression, as well as the layering along the direction of plate thickness, respectively. This study indicates that the numerical simulation results agree with those from the variational solution, by comparing the maximum deflection of the plate, which verifies the validity of the variational solution obtained. The results presented in this study are helpful for the refined analysis and optimization design of flexible structures, which are composed of bimodular functionally graded materials, while the structure is under large deformation.

Application of the Variational Method to the Large Deformation Problem of Thin Cylindrical Shells with Different Moduli in Tension and Compression

In this study, the variational method concerning displacement components is applied to solve the large deformation problem of a thin cylindrical shell with its four sides fully fixed and under uniformly distributed loads, in which the material that constitutes the shell has a bimodular effect, in comparison to traditional materials, that is, the material will present different moduli of elasticity when it is in tension and compression. For the purpose of the use of the displacement variational method, the physical equations on the bimodular material model and the geometrical equation under large deformation are derived first. Thereafter, the total strain potential energy is expressed in terms of the displacement component, thus bringing the possibilities for the classical Ritz method. Finally, the relationship between load and central deflection is obtained, which is validated with the numerical simulation, and the jumping phenomenon of thin cylindrical shell with a bimodular effect is analyzed. The results indicate that the bimodular effect will change the stiffness of the shell, thus resulting in the corresponding change in the deformation magnitude. When the shell is relatively thin, the bimodular effect will influence the occurrence of the jumping phenomenon of the cylindrical shell.

Design Scalability Study of the Γ-Shaped Piezoelectric Harvester Based on Generalized Classical Ritz Method and Optimization

This paper studies the design scalability of a Γ-shaped piezoelectric energy harvester (ΓEH) using the generalized classical Ritz method (GCRM) and differential evolution algorithm. The generalized classical Ritz method (GCRM) is the advanced version of the classical Ritz method (CRM) that can handle a multibody system by assembling its equations of motion interconnected by the constraint equations. In this study, the GCRM is extended for analysis of the piezoelectric energy harvesters with material and/or orientation discontinuity between members. The electromechanical equations of motion are derived for the PE harvester using GCRM, and the accuracy of the numerical simulation is experimentally validated by comparing frequency response functions for voltage and power output. Then the GCRM is used in the power maximization design study that considers four different total masses—15 g, 30 g, 45 g, 60 g—to understand design scalability. The optimized ΓEH has the maximum normalized power density of 23.1 × 103 kg·s·m−3 which is the highest among the reviewed PE harvesters. We discuss how the design parameters need to be determined at different harvester scales.

Generalized classical Ritz method for modeling geometrically nonlinear flexible multibody systems having a general topology

In the field of structural dynamics, the classical Ritz method is famous for its computational efficiency and employed for modeling numerous types of structures. However, applications of the method have been limited to structures having simple configuration since the method specializes in modeling simple structures that consist of a single or a few structural elements. To overcome the limitation of the method, we developed a generalized version of the method to conduct geometrically nonlinear analyses of flexible multibody systems having a general topology. To examine the accuracy, efficiency, and stability of the proposed method, we solved five numerical examples and compared the results obtained with the proposed method with those obtained with the finite element method. The comparison study illustrated the better performance of the proposed method in computational efficiency and numerical stability.

Postbuckling of a uniformly compressed simply supported plate with free in-plane translating edges

The postbuckling of a Kirchhoff isotropic simply supported plate is considered in detail. The in-plane displacements on the edges of the plate are not constrained. The solution is obtained using the principle of the total potential energy stationarity. The expression for energy is written in the three versions: in terms of the Biot strains, the Cauchy-Green strains, and the strains corresponding to Fuppl-von Karman plate theory. Some approximate solution is constructed by the classical Ritz method. The basis functions are taken in the form of Legendre polynomials and their linear combinations. The obtained diagram of equilibrium states is rather similar to the classical diagrams of compressed shells. We show the failure of Foppl-von Karman theory under large deflections. Using the Biot strains and the Cauchy-Green strains leads to the discrepancy between the results of at most 5 %. We demonstrate the high accuracy and convergence of the approximate solution.

Effect of Ritz Vectors on Random Seismic Response of Cantilever Beam

To understand the effect of Ritz vectors on the random seismic response of non-uniform cantilever beam, the following works are performed: Firstly, the equation of random seismic response of non-uniform cantilever beam based on Hamilton theory is presented. The solution is derived using the combined classical Ritz and pseudo-excitation method. Secondly, by considering the principle of virtual displacement, the pertinent function can be obtained from the characteristics of pseudo-excitation method. The mode of analysis is transformed from structural stationary random to structural harmonic response. Finally, the results of the analysis of the cantilever beam samples of different Ritz vectors are presented. The functional extremum corresponding to increasing Ritz vectors is compared. It can be concluded based on the result findings that the Ritz vectors, which satisfy both the geometrical and the natural boundary conditions, do not necessarily lead to accurate results. It is, therefore, recommended that the extreme value of the function for the whole system should be considered as an evaluation criterion for the Ritz vectors.

Dynamics of cable truss via polynomial shape functions

A model for the in-plane dynamic behaviour of a biconcave cable structure, subject to large static deformations and potentially slack harnesses is proposed, based on polynomial shape functions, in line with the classical Ritz method. The model provides a semi-analytical approach to the calculation of natural frequencies and modal shapes of the structure. The proposed formulation leads to an eigenvalue problem, based on a reduced number of degrees of freedom compared with equivalent FEM solutions, providing the basis for fast and accurate sensitivity analysis. The behaviour of the deformed structure is analysed in detail to understand the non-linear effects of non-symmetric mass and load distribution and slack harnesses on natural frequencies and corresponding modal shapes. Results confirm the relevance of the non-linear effects, due to the statically loaded configuration, on the linear vibrations of the structure, in particular evidencing the influence of the slackening of harnesses on modal shapes. Results are compared to analytical models, where available (single sagged cable), and to FEM solutions (for cable trusses with non-uniform mass and load distribution and potentially slack harnesses), providing good agreement.

Effect of cracks on nonlinear flexural vibration of rotating Timoshenko functionally graded material beam having large amplitude motion

This study investigates the stiffening effect due to rotation on the nonlinear vibrational characteristics for cracked Timoshenko beam for the first time. Fixed end of the beam is attached to a rotating hub. Functionally graded material is taken into consideration, in which the properties vary as a continuous function along the depth of the beam. An elastic mass-less rotational spring is assumed in the place of crack, which splits the beam into two different parts. The point on the neutral axis at the fixed end is assumed to be the center of rotation of the beam. Centrifugal force is considered to act towards the spanwise direction and along the neutral axis. An additional displacement due to rotation of the beam along with the centrifugal force is incorporated with the energy formulation. Timoshenko beam theory and classical Ritz method is employed to derive the governing equations. In order to solve the nonlinear governing equations, direct substitution iterative technique is used. Effects of various parameters such as rotating speeds, radius of hub, depth of crack, location of crack, and different functionally graded material properties on linear and nonlinear vibration characteristics are studied. Validity of the present methodology is assured by comparing the results with some of the results from the existing literatures.

Parametric stability of symmetrically laminated composite super-elliptical plates

Static and parametric stability of thin symmetrically laminated composite super-elliptical plates resting on Winkler-type foundation and subjected to uniform in-plane harmonic loads, under clamped, simply supported and free boundary conditions, are investigated based on the classical laminated plate theory. The governing equations are obtained from a variational approach and then the classical Ritz method is used to reduce the problem into a set of coupled Mathieu–Hill equations. Hsu’s technique is utilized to determine the dynamic instability regions of principal and combination resonance frequencies. Extensive numerical data are provided to examine the effects of plate aspect ratio, super-ellipticity power, foundation stiffness parameter, stacking sequence, and fiber orientation on the vibrational, static, and parametric stability characteristics of symmetrically laminated super-elliptical composite plates. Furthermore, three-dimensional buckling mode shapes are illustrated. The accuracy of formulation is checked by performing convergence studies and the validity of results is established by comparison with the existing results in the literature, exact results obtained from the analytical approach, and as well as from FEM results.

On the Curse of Dimensionality in the Ritz Method

It is shown that the classical Ritz method of the calculus of variations suffers from the "curse of dimensionality," i.e., an exponential growth, as a function of the number of variables, of the dimension a linear subspace needs in order to achieve a desired relative improvement in the accuracy of approximation of the optimal solution value. The proof is constructive and is obtained by exhibiting a family of infinite-dimensional optimization problems for which this happens, namely those with quadratic functional and spherical constraint. The results provide a theoretical motivation for the search of alternative solution methods, such as the so-called "extended Ritz method," to deal with the curse of dimensionality.

Direct Method of Analysis of an Isotropic Rectangular Plate Using Characteristic Orthogonal Polynomials

This work evaluates the static analysis of an isotropic rectangular plate with various boundary conditions. The direct variation method according to Ritz is used to obtain the total potential energy of the plate by employing the static elastic theory of plate. The shape function of fourth order for the plate under uniformly distributed load for various boundary conditions were formulated using the characteristic orthogonal polynomials (COP), which is also known as product of orthogonal strips of the plate along the two axes. Analyses of results obtained were compared with exact results presented in the monograph of Timoshenko and Woinowsky – Krieger. The previous work adopted shape function in the form of trigonometric series to solve the fourth order governing differential equation of the plate. This work is helpful for obtaining the shape functions not only for simply supported plates but also for rectangular plates with various kinds of support conditions, where the exact method presents intricate and complex solution. Finally, comparison has been done for the coefficients of deflection and moments at various aspect ratios of the plate for different support conditions. The results obtained by the use of the COP in the classical Ritz method are in close agreement with the results obtained from exact solutions from classical method. http://dx.doi.org/10.4314/njt.v34i2.3

Random seismic response analysis of jacket structure with Timoshenko's beam theory

Today many complex three-dimensional (3D) models, typically finite element method (FEM) models, have been used in the analysis of jacket platforms. However, these comprehensive models are not readily adopted in engineering practice, especially during the preliminary design stage. The main reasons are the large computing resources required and the lack of detailed information for setting up the FEM model. In this paper, a simplified response analysis method is proposed to make preliminary estimate of the random seismic responses on jacket platforms. Based on Timoshenko's beam theory, the equation of bending motion was proposed and solved by using the classical Ritz method combined with the pseudo-excitation method. A computer code for computing random seismic responses was also developed and verified. A six-legged jacket platform was used as an example, whereby three types of Ritz functions were used to illustrate the validity of the model. The results of this study were compared with those obtained from a well-established complex 3D FEM model. The findings of the comparison showed that the accuracy of the simplified model was better than 95%. It was concluded that the proposed simplified method was effective, useful and practical in seismic design of platforms.

A simplified computational method for random seismic responses of a jacket platform

A simplified method for modeling Jacket Offshore Platform was developed and numerical examples were presented to demonstrate the validity of the proposed method. In this method, a jacket platform was described as a cantilever beam subjected to an axial compressive load acting at the top-end centre of the jacket. The random seismic responses of the model were derived by using both the pseudo-excitation method and the classical Ritz method. The authors found and demonstrated in this paper that this proposed method may be used to calculate the random responses accurately and efficiently. In particular, the responses of a non-uniform beam, in the form of an auto-PSD function, cross-PSD or higher spectral moments could be solved directly without the need to determine the normal modes. The writers included an example in which the solution was obtained using the proposed method with a selected Ritz function. The results were compared with that obtained using the conventional finite element method (FEM). The findings of the study showed that the proposed method was effective, practical and useful in the seismic design of platforms.

Influence of boundary stress singularities on the vibration of clamped and simply-supported sectorial plates with arbitrary radial edge conditions

This paper extends previous studies made for sectorial plates having re-entrant (i.e., interior) corners causing stress singularities, to provide accurate frequencies when the circular edge is either clamped or simply-supported. An extensive review of the literature is also given herein spanning nearly the past two decades explaining the free vibration characteristics of sectorial plates. In this work, the classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include: (1) mathematically complete algebraic–trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained and (2) corner functions which account for the bending moment singularities at the re-entrant vertex corner of the radial edges having arbitrary edge conditions. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of non-dimensional frequencies for sectorial plates having either a clamped or hinged circumferential edge and various combinations of clamped, hinged, and free conditions on the radial edges. Accurate (to at least four significant figure) frequencies and normalized contours of the transverse vibratory displacement are presented for the spectra of sector angles [90°, 180° (semi-circular), 270°, 300°, 330°, 350°, 355°, 360° (complete circular)] causing a re-entrant vertex corner of the radial edges. For sector angles of 360°, a clamped–clamped, clamped–hinged, clamped–free, hinged–free or free–free radial crack ensues. One general observation is the substantial reduction in the first six frequencies as the sector angle increases for all plates, except in the first two modes of plates having free–free radial edges.

A simplified calculation method for non-stationary random seismic response of jacket platform

Jacket platform was simulated by non-uniform cantilever beam subjected to axial loading. Based on the Hamilton theory, the equation of bending motion was developed and solved by the classical Ritz method combined with the pseudo-excitation method (PEM) for non-stationary random response with non-classical damping. Usually, random response of this continuous structure is obtained by orthogonality of modes and some normal modes of the structure are needed, causing inconvenience in the analysis of the non-uniform beam whose normal modes are not easy to be obtained. However, if the PEM is extended to calculate random response by combining it with the classical Ritz method, the responses of non-uniform beam, such as auto-power spectral density (PSD) function, cross-PSD and higher spectral moments can be solved directly avoiding the calculation of normal modes. The numerical results show that the present method is effective and useful in aseismic design of platforms.

Free vibrations of anisotropic rectangular plates with holes and attached masses

Fil: Rossit, Carlos Adolfo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria; Argentina

Approximate study of the free vibrations of a cantilever anisotropic plate carrying a concentrated mass

This study is concerned with the vibration analysis of a cantilevered rectangular anisotropic plate when a concentrated mass is rigidly attached to its center point. Based on the classical theory of anisotropic plates, the Ritz method is employed to perform the analysis. The deflection of the plate is approximated by a set of beam functions in each principal coordinate direction. The influence of the mass magnitude on the natural frequencies and modal shapes of vibration is studied for a boron-epoxy plate and also in the case of a generic anisotropic material. The classical Ritz method with beam functions as the spatial approximation proved to be a suitable procedure to solve a problem of this analytical complexity.

Vibrations of Mindlin Sectorial Plates Using the Ritz Method Considering Stress Singularities

This paper reports very accurate vibration frequencies of moderately thick sectorial plates with various boundary conditions and vertex angles (α=90°, 180°, 270°, 300°, 330°, and 355°) based on Mindlin plate theory, and provides the nodal patterns of their vibration modes for the first time in the published literature. Most of the extensive frequencies presented are exact to the four digits shown. The classical Ritz method is employed, using corner functions and algebraic trigonometric functions as the admissible functions. Because the corner functions properly describe the singularity behaviors of moments and shear forces in the vicinity of the vertex of a sectorial plate, they substantially enhance the convergence and accuracy of the numerical results, which is shown by convergence studies.

New A Priori FEM Error Estimates for Eigenvalues

We analyze the Ritz--Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, what is apparently the first truly a priori error estimates that show the levels of the eigenvalue errors depending on approximability of eigenfunctions in the corresponding eigenspace. These estimates reflect a known phenomenon that different eigenfunctions in the corresponding eigenspace may have different approximabilities, thus resulting in different levels of errors for the approximate eigenvalues. For clustered eigenvalues, we derive eigenvalue error bounds that do not depend on the width of the cluster. Our results are readily applicable to the classical Ritz method for compact symmetric integral operators and to finite element method eigenvalue approximation for symmetric positive definite differential operators.

Error Estimates for Approximate Optimization by the Extended Ritz Method

An alternative to the classical Ritz method for approximate optimization is investigated. In the extended Ritz method, sets of admissible solutions are approximated by their intersections with sets of linear combinations of all n-tuples of functions from a given basis. This alternative scheme, called variable-basis approximation, includes functions computable by trigonometric polynomials with free frequencies, free-node splines, neural networks, and other nonlinear approximating families. Estimates of rates of approximate optimization by the extended Ritz method are derived. Upper bounds on rates of convergence of suboptimal solutions to the optimal one are expressed in terms of the degree n of variable-basis functions, the modulus of continuity of the functional to be minimized, the modulus of Tikhonov well-posedness of the problem, and certain norms tailored to the type of basis. The results are applied to convex best approximation and to kernel methods in machine learning.