Papers by Keyword: Ritz Method

Abstract: This paper describes an approach extended from Ritz method to analyze the free vibration of thin isotropic annular plates in good accuracy, and presents comprehensive lists of natural frequencies of the plate for all possible sets of classical boundary conditions. Analytical process is developed to introduce the boundary index that allows to accommodate any sets of free, simple supported and clamped edges along inner and outer boundary of the plate. Convergence and comparison studies are made to demonstrate numerical accuracy in the frequency parameters. Results are summarized for nine sets of boundary conditions and six different ratios of (inner radius)/(outer radius), and are intended to serve for uses of design data and comparison in relevant future papers.

Abstract: An analytical method is presented for free vibration of a symmetrically laminated rectangular plate with point masses, and experimental modal analysis is conducted to compare both sets of the frequency data. The problem is solved by an extending Ritz method to include kinetic energy caused by added point masses under any sets of edge conditions, and a frequency equation is derived by minimizing the energy functional. In numerical computation, the accuracy of the solution is studied by convergence test and comparison with the existing result in the specific case. Then, the experimental modal analysis is applied to measure the natural frequencies and mode shapes. The two sets of results are compared, and the validity of both theoretical and experimental approaches is established.

Abstract: The size-dependent bending and static stability characteristics of nanobeams made of bi-directional functionally graded materials (2D-FGMs) under different boundary conditions are comprehensively investigated. Based on the modified couple stress theory and surface elasticity theory, the size-dependent model is formulated for 2D-FG Euler-Bernoulli beam. The material properties of the beam smoothly change along both the axial and thickness directions according to power-law distribution. The continuous spatial variations of the single material length scale parameter and the three surface constants are incorporated to describe the effects of microstructure and surface energy, respectively. This model accounts for the axial and transverse displacements, the exact position of the physical neutral plane, and Poisson’s effect. To obtain the static response of the present model, Ritz method is employed by approximating the axial and transverse displacements in terms of polynomial forms. Different boundary conditions, i.e., Simply-simply (S-S), Clamped-clamped (C-C), Clamped-simply (C-S), and Clamped-free (C-F), are considered and satisfied by adding auxiliary functions to the displacement functions. Numerical results with various cases of boundary conditions are performed with an insight to explore the effects of gradient indices in thickness and length directions, surface energy, material length scale parameter, slenderness ratio, and thickness on the static deflection and buckling responses of 2D-FG nanobeams. Results disclose that, the material properties, the surface energy, and microstructure effects have a significant effect on the bending, and buckling responses of 2D-FG nanobeams. Hence, this study can be helpful in the design and optimization of 2D-FG nanobeams in bending and buckling responses.

Abstract: To design vibration control system for flexible structures their mathematical model should be reduced. In the paper we consider the influence of the model reduction on the dynamics of the real closed-loop system. A simply cantilever beam is an object of consideration since we are able to formulate the exact analytical model of such structure. As a result of reduction the model with low frequency resonances is usually separated from the high frequency dynamics because high frequency part of the model is naturally strong damped. In order to estimate dynamical system for control purposes in the paper we applied a few orthogonal methods such as: modal, Rayleigh-Ritz and Schur decompositions. As it is shown all methods well calculate resonances frequencies but generate different anti-resonances frequencies. From control strategy in point of view of the flexible structures these anti-resonances have significantly influence on the stability and dynamics of the closed-loop systems.

Abstract: A Rayleigh-Ritz approach for the analysis of buckling and post-buckling behavior of cracked composite stiffened plates is presented. The structure is modeled as the assembly of plate elements modeled by the first order shear deformation theory and taking geometric nonlinearities into account through the von Karman’s theory assumptions. Continuity along the plate elements connected edges and the enforcement of rigid and elastic restraints of the plate boundaries are obtained by using penalty techniques, which also allow to straightforwardly implement efficient crack modeling strategies. General symmetric and unsymmetric stacking sequences are considered and numerical procedures have been developed and used to validate the present solution by comparison with FEA results. Original results are presented for post-buckling solution of multilayered stiffened plates with through-the-thickness cracks, showing the effects of large displacements on the cracked plate post-buckling behavior.

Abstract: The Ritz method is used for the buckling analysis of stiffened plates under compressive load. Suggested approximations of displacements allow taking into account the features of system wave formation. The results are compared with data, received by finite element method.

Abstract: The work considers using a multilayer elastomeric package for the vibroisolation of engineering constructions under the action of periodic vibration. The multilayer elastomeric package is located between the protected object and a vibrating base; it consists of alternating thin metallic and elastomeric layers jointed by vulcanization or gluing. The paper discusses flat rubber-metal elements of circular shapes with reinforcing steel layers and describes kinematic excitation directed flatwise. The analytical expression of the characteristics of the compression stiffness of the plane multilayered elastomeric structure is derived on the basis of the variational principle, and metallic plates-layers are assumed to be perfectly rigid. An analytical solution was confirmed by experimental data. The fitted equation for “force-displacement” was derived and used in the equation of motion on of the protected object.

Abstract: This paper presents an investigation on the free vibration of rectangular nanocomposite plates reinforced by aligned single-walled carbon nanotubes (SWCNTs). The CNT reinforcement may be uniformly distributed (UD) or functionally graded (FG) over the thickness direction of a plate. The material properties of the CNT composite are determined through a micromechanical model. The eigenvalue equation governing the plate vibration problem is derived by the p-Ritz method through minimizing the virtual strain and kinetic energies of a CNT composite plate. The influences of CNT distribution and reinforcing angle, plate thickness ratio, aspect ratio and support conditions on the vibration behaviour of the plates are discussed.

Abstract: The elastic collapse of long cylinders under combined external pressure and axial force was investigated using analytical approach. Long cylindrical pipelines laid on the seabed are subjected to external pressure, initial defect will cause the local collapse of the pipelines. Due to the change of subsea environments and construction conditions, circular pipelines are subjected not only to the hydrostatic pressure, but also to forces of other forms, such as axial tension or compression, so on and so forth. This paper studies the local collapse and the morphological characteristics of a circular pipelines subjected to hydrostatic pressure and axial force. Governing equations based on Karman-Donnell`s shell theory are derived and are solved using Ritz method.

Abstract: Functionally graded materials (FGMs) are a new kind of composite materials which have a smooth variation of material properties along one or more directions. At each interface, the material is chosen according to specific applications and environment loadings. This paper presents some solutions to study the free vibration of FGM plates made of ceramic and metal. The formulation used is based on Reddys higher order shear deformation plate theory. Material properties are taken to be temperature-dependent, and vary continuously through the thickness direction according to a power law distribution (P-FGM). The plate is assumed to be initially stressed by temperature rise through the thickness. The energy functional of the system is obtained by using energy principles. Free vibration frequencies are then obtained by using a set of characteristic orthogonal polynomials and by applying Ritz method.