An Improved Fourier–Ritz Method for Analyzing In-Plane Free Vibration of Sectorial Plates

2017 ◽  
Vol 84 (9) ◽  
Author(s):  
Siyuan Bao ◽  
Shuodao Wang ◽  
Bo Wang
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Displacement Function ◽  
Accurate Solution ◽  
Previous Method ◽  
Fourier Series Expansion ◽  
Arbitrary Boundary ◽  
Fourier Cosine ◽  
Radial Variable

A modified Fourier–Ritz approach is developed in this study to analyze the free in-plane vibration of orthotropic annular sector plates with general boundary conditions. In this approach, two auxiliary sine functions are added to the standard Fourier cosine series to obtain a robust function set. The introduction of a logarithmic radial variable simplifies the expressions of total energy and the Lagrangian function. The improved Fourier expansion based on the new variable eliminates all the potential discontinuities of the original displacement function and its derivatives in the entire domain and effectively improves the convergence of the results. The radial and circumferential displacements are formulated with the modified Fourier series expansion, and the arbitrary boundary conditions are simulated by the artificial boundary spring technique. The number of terms in the truncated Fourier series and the appropriate value of the boundary spring retraining stiffness are discussed. The developed Ritz procedure is used to obtain accurate solution with adequately smooth displacement field in the entire solution domain. Numerical examples involving plates with various boundary conditions demonstrate the robustness, precision, and versatility of this method. The method developed here is found to be computationally economic compared with the previous method that does not adopt the logarithmic radial variable.

Static and Dynamic Analysis of Annular Sector Plates Subjected to Arbitrary Boundary Conditions

2014 ◽  
Vol 624 ◽  
pp. 240-244
Author(s):  
Kai Peng Zhang ◽  
Cheng Yang ◽  
Han Wu
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Current Method ◽  
Displacement Function ◽  
Current Approach ◽  
Arbitrary Boundary ◽  
Static And Dynamic Analysis ◽  
Arbitrary Boundary Conditions ◽  
Annular Sector

In this investigation, an improved Fourier series method (IFSM) is employed to predict the static and dynamic characteristics of annular sector plates with arbitrary boundary conditions. Regardless of boundary supports, the displacement function is invariantly expressed as a modified two-dimensional Fourier series containing sine and cosine function. It is capable of dealing with the possible discontinuities at elastic boundary edges. The unknown Fourier coefficients are treated as generalized coordinates, and determined using Rayleigh-Ritz method. Unlike most of the existing solution techniques, the current approach can be universally applied to a variety of edge restraints including all classical cases and their combinations. The accuracy and reliability of the current method are fully illustrated through all the numerical examples.

scholarly journals A Unified Spectro-Geometric-Ritz Method for Vibration Analysis of Open and Closed Shells with Arbitrary Boundary Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-30 ◽  
Author(s):  
Dongyan Shi ◽  
Yunke Zhao ◽  
Qingshan Wang ◽  
Xiaoyan Teng ◽  
Fuzhen Pang
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Analysis Model ◽  
Arbitrary Boundary ◽  
Fourier Cosine ◽  
Auxiliary Functions ◽  
Arbitrary Boundary Conditions ◽  
Closed Shells

This paper presents free vibration analysis of open and closed shells with arbitrary boundary conditions using a spectro-geometric-Ritz method. In this method, regardless of the boundary conditions, each of the displacement components of open and closed shells is represented simultaneously as a standard Fourier cosine series and several auxiliary functions. The auxiliary functions are introduced to accelerate the convergence of the series expansion and eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries. The boundary conditions are modeled using the spring stiffness technique. All the expansion coefficients are treated equally and independently as the generalized coordinates and determined using Rayleigh-Ritz method. By using this method, a unified vibration analysis model for the open and closed shells with arbitrary boundary conditions can be established without the need of changing either the equations of motion or the expression of the displacement components. The reliability and accuracy of the proposed method are validated with the FEM results and those from the literature.

A Modified Fourier Solution for Free Damped Vibration Analysis of Sandwich Viscoelastic-Core Conical Shells and Annular Plates with Arbitrary Restraints

2016 ◽  
Vol 08 (08) ◽  
pp. 1650094 ◽  
Author(s):  
Chuanmeng Yang ◽  
Guoyong Jin ◽  
Weijian Xu ◽  
Zhigang Liu
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Current Method ◽  
Fourier Series Expansion ◽  
Algebraic Equations ◽  
Arbitrary Boundary ◽  
Conical Shells ◽  
Annular Plates ◽  
Damping Characteristics ◽  
Loss Factors

In this paper, arbitrary boundary conditions including classical and elastic ones are considered in analyzing the vibration and damping characteristics of the sandwich conical shells and annular plates using a simple and efficient modified Fourier solution. The displacement field is expressed as the linear combination of a standard Fourier series and several supplementary terms. The addition of these terms make the Fourier series expansion applicable to any boundary conditions, and the Fourier series expansions improved drastically regarding its accuracy and convergence. Instead of adopting conventional differentiation procedure, a Rayleigh–Ritz technique based on the energy function is conducted which leads to a set of algebraic equations. Then natural frequencies and loss factors can be obtained by solving the algebraic equations. Accuracy and reliability of the current method are checked by comparing the present results with the existing solutions. Influences of some vital parameters on the free vibration and damping performance of sandwich shells and plates are discussed. The detailed effect of restraints from different directions on the frequencies and loss factors is investigated. So, the method can provide a guide to design sandwich structures with desired vibration characteristic and well damping performance by reasonably adjusting the boundary condition. Some new numerical results are presented for future validation of various approximate/numerical methods.

Moving Force-Induced Vibration of a Rotating Beam with Elastic Boundary Conditions

2015 ◽  
Vol 15 (01) ◽  
pp. 1450035 ◽  
Author(s):  
Binglin Lv ◽  
Wanyou Li ◽  
Huajiang Ouyang
Keyword(s):  
Boundary Conditions ◽  
Displacement Function ◽  
Mode Shapes ◽  
Work Piece ◽  
Rotating Beam ◽  
Arbitrary Boundary ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Analytical Technique ◽  
Rayleigh Beam

In this paper, an analytical technique, the so-called Fourier Spectral method (FSM), is extended to the vibration analysis of a rotating Rayleigh beam considering the gyroscopic effect. The model presented can have arbitrary boundary conditions specified in terms of elastic constraints in the translations and rotations or even in terms of attached lumped masses and inertias. Each displacement function is universally expressed as a linear combination of a standard Fourier cosine series and several supplementary functions introduced to ensure and accelerate the convergence of the series expansion. Lagrange's equation is established for all the unknown Fourier coefficients viewed as a set of independent generalized coordinates. A numerical model is constructed for the rotating beam. First, a numerical example considering simply supported boundary conditions at both ends is calculated and the results are compared with those of a published paper to show the accuracy and convergence of the proposed model. Then, the method is applied to one real work piece structure with elastically supported boundary conditions updated from the modal experiment results including both the frequencies and mode shapes using the method of least squares. Several numerical examples of the updated model are studied to show the effects of some parameters on the dynamic characteristics of the work piece subjected to moving loads at different constant velocities.

Dynamic Analysis of Circular Cylindrical Shells With General Boundary Conditions Using Modified Fourier Series Method

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Lu Dai ◽  
Tiejun Yang ◽  
W. L. Li ◽  
Jingtao Du ◽  
Guoyong Jin
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Current Method ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Arbitrary Boundary ◽  
Circular Cylindrical Shells

Dynamic behavior of cylindrical shell structures is an important research topic since they have been extensively used in practical engineering applications. However, the dynamic analysis of circular cylindrical shells with general boundary conditions is rarely studied in the literature probably because of a lack of viable analytical or numerical techniques. In addition, the use of existing solution procedures, which are often only customized for a specific set of different boundary conditions, can easily be inundated by the variety of possible boundary conditions encountered in practice. For instance, even only considering the classical (homogeneous) boundary conditions, one will have a total of 136 different combinations. In this investigation, the flexural and in-plane displacements are generally sought, regardless of boundary conditions, as a simple Fourier series supplemented by several closed-form functions. As a result, a unified analytical method is generally developed for the vibration analysis of circular cylindrical shells with arbitrary boundary conditions including all the classical ones. The Rayleigh-Ritz method is employed to find the displacement solutions. Several examples are given to demonstrate the accuracy and convergence of the current solutions. The modal characteristics and vibration responses of elastically supported shells are discussed for various restraining stiffnesses and configurations. Although the stiffness distributions are here considered to be uniform along the circumferences, the current method can be readily extended to cylindrical shells with nonuniform elastic restraints.

An Exact Fourier Series Method for the Vibration Analysis of Multispan Beam Systems

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Wen L. Li ◽  
Hongan Xu
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Series Expansion ◽  
Vibration Analysis ◽  
Series Solution ◽  
Series Representation ◽  
Displacement Function ◽  
Fourier Series Expansion ◽  
Series Method ◽  
Fourier Series Method

An exact Fourier series method is developed for the vibration analysis of multispan beam systems. In this method, the displacement on each beam is expressed as a Fourier series expansion plus an auxiliary closed-form function such as polynomials. The auxiliary function is used to deal with all the possible discontinuities, at the end points, with the original displacement function and its derivatives when they are periodically extended over the entire x-axis as implied by a Fourier series representation. As a result, not only is it always possible to expand the beam displacements into Fourier series under any boundary conditions, but also the series solution will be substantially improved in terms of its accuracy and convergence. Mathematically, the current Fourier series expansion represents an exact solution to a class of beam problems in the sense that both the governing equations and the boundary/coupling conditions are simultaneously satisfied to any specified degree of accuracy. In the multispan beam system model, any two adjacent beams are generally connected together via a pair of linear and rotational springs, allowing a better modeling of many real-world joints. Each beam in the system can also be independently and elastically restrained at its ends so that all boundary conditions including the classical homogeneous boundary conditions at the end and intermediate supports can be universally dealt with by simply varying the stiffnesses of the restraining springs accordingly, which does not involve any modification of basis functions, formulations, or solution procedures. The excellent accuracy and convergence of this series solution is demonstrated through numerical examples.

Free vibration analysis of three-layer thin cylindrical shell with variable thickness two-dimensional FGM middle layer under arbitrary boundary conditions

2021 ◽  
pp. 109963622110204
Author(s):  
Xue-Yang Miao ◽  
Chao-Feng Li ◽  
Yu-Lin Jiang ◽  
Zi-Xuan Zhang
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Admissible Function ◽  
Free Vibration Analysis ◽  
Middle Layer ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions

In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.

scholarly journals A Modified Fourier–Ritz Formulation for Vibration Analysis of Arbitrarily Restrained Rectangular Plate with Cutouts

2018 ◽  
Vol 2018 ◽  
pp. 1-22
Author(s):  
Yiming Liu ◽  
Zhuang Lin ◽  
Hu Ding ◽  
Guoyong Jin ◽  
Sensen Yan
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Flexural Vibration ◽  
Vibration Characteristics ◽  
Geometric Parameters ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions ◽  
Stiffness Constant ◽  
Proper Values

A modified Fourier–Ritz method is developed for the flexural and in-plane vibration analysis of plates with two rectangular cutouts with arbitrary boundary conditions, aiming to provide a unified solving process for cases that the plate has various locations or sizes of cutout, and different kinds of boundary conditions. Under the current framework, modifying the position of the cutout or the boundary conditions of the plate is just as changing the geometric parameters of the plate, and there is no need to change the solution procedures. The arbitrary boundary conditions can be obtained by setting the stiffness constant of the boundary springs which are fixed uniformly along the edges of the plate at proper values. The strain and kinetic energy functions of a plate with rectangular cutout are derived in detail. The convergence and accuracy of the present method are demonstrated by comparing the present results with those obtained from the FEM software. In this paper, free in-plane and flexural vibration characteristics of the plate with rectangular cutout under general boundary conditions are studied. From the results, it can be found that the geometric parameters and positions of the cutout and the boundary conditions of the plate will obviously influence the natural vibration characteristics of the structures.

scholarly journals Optimal multistage relaxation with automatic parameter selection

2021 ◽  
Author(s):  
Alvin Wong
Keyword(s):  
Fourier Series ◽  
Relaxation Method ◽  
Accurate Solution ◽  
Convergence Time ◽  
Fourier Series Expansion ◽  
End User ◽  
Local Flow ◽  
Flow Problems ◽  
User Expertise ◽  
Complex Fourier Series

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

scholarly journals Optimal multistage relaxation with automatic parameter selection

2021 ◽  
Author(s):  
Alvin Wong
Keyword(s):  
Fourier Series ◽  
Relaxation Method ◽  
Accurate Solution ◽  
Convergence Time ◽  
Fourier Series Expansion ◽  
End User ◽  
Local Flow ◽  
Flow Problems ◽  
User Expertise ◽  
Complex Fourier Series

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

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