Vibration of circular plate with multiple eccentric circular perforations by the Rayleigh-Ritz method

Khodabakhsh Saeedi,Alfin Leo,Rama B. Bhat,Ion Stiharu 대한기계학회 2012 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.26 No.5

"The free vibration of a circular plate with multiple perforations is analyzed by using the Rayleigh-Ritz method. Admissible functions are assumed to be separable functions of radial and tangential coordinates. Trigonometric functions are assumed in the circumferential direction. The radial shape functions are the boundary characteristic orthogonal polynomials generated following the Gram-Schmidt recurrence scheme. The assumed functions are used to estimate the kinetic and the potential energies of the plate depending on the number and the position of the perforations. The eigenvalues, representing the dimensionless natural frequencies, are compared with the results obtained using Bessel functions, where the exact solution is available. Moreover, the eigenvectors, which are the unknown coefficients of the Rayleigh-Ritz method, are used to present the mode shapes of the plate. To validate the analytical results of the plates with multiple perforations, experimental investigations are also performed. Two unique case studies that are not addressed in the existing literature are considered. The results of the Rayleigh-Ritz method are found to be in good agreement with those from the experiments. Although the method presented can be employed in the vibration analysis of plates with different boundary conditions and shapes of the perforations, circular perforations that are free on the edges are studied in this paper. The results are presented in terms of dimensionless frequencies and mode shapes."

Block-partitioned Rayleigh–Ritz method for efficient eigenpair reanalysis of large-scale finite element models

정연호,부승환,Yim Solomon C 한국CDE학회 2023 Journal of computational design and engineering Vol.10 No.3

In this manuscript, we propose a new effective method for eigenpair reanalysis of large-scale finite element (FE) models. Our method utilizes the matrix block-partitioning algorithm in the Rayleigh–Ritz approach and expresses the Ritz basis matrix using thousands of block matrices of very small size. To avoid significant computational costs from the projection procedure, we derive a new formulation that uses tiny block computations instead of global matrix computations. Additionally, we present an algorithm that recognizes which blocks are changed in the modified FE model to achieve computational cost savings when computing new eigenpairs. Through selective updating for the recognized blocks, we can effectively construct the new Ritz basis matrix and the new reduced mass and stiffness matrices corresponding to the modified FE model. To demonstrate the performance of our proposed method, we solve several practical engineering problems and compare the results with those of the combined approximation method, the most well-known eigenpair reanalysis method, and ARPACK, an eigenvalue solver embedded in many numerical programs.

Natural mode analysis of a telescopic boom system with multiple sections using the Rayleigh-Ritz method

구본용,장세명,강승현 대한기계학회 2018 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.32 No.4

A truck crane, a type of construction machine, contains a boom consisting of several strips with different cross-sectional data. A set of the overall natural frequencies of each mode should be obtained for a boom body with multiple sections to analyze vibrational characteristics. The proposed tool for analysis is Rayleigh-Ritz, which is an energy method that employs the calculus of variations. We develop a general computer code for a set of sub-beam structures with a symbolic expression in MATLAB and compare the result with that of a numerical analysis using the finite element method (FEM) with COMSOL, a commercial code. Results of the boom model show that the first-mode natural frequency shows an error of 4.6 % between the Rayleigh-Ritz method and 3D FEM. The solution reaches stable convergence after increasing the terms of base functions, and the present application produces reasonable results.

ON THE COMPUTATION OF EIGENVALUE BOUNDS OF ANHARMONIC OSCILLATOR USING AN INTERMEDIATE PROBLEM METHOD

Lee, Gyou-Bong,Lee, Ok-Ran 한국전산응용수학회 2002 The Korean journal of computational & applied math Vol.9 No.1

We apply an Intermediate Problem Method to compute eigenvalues of an anharmonic oscillator. The method produces lower bounds to the eigenvalues while the Rayleigh-Ritz method yields upper bounds. We show the convergence rate of the Intermediate Problem Method is the same as the rate of the Rayleigh-Ritz method.

Needle Deflection during Insertion into Soft Tissue Based on Virtual Spring Model

Haiyan Du,Yongde Zhang,Jingang Jiang,Yanjiang Zhao 보안공학연구지원센터 2015 International Journal of Multimedia and Ubiquitous Vol.10 No.1

Needle insertion is the most common procedure of minimally invasive interventions. During insertion into a soft tissue, the needle with bevel tip can deflect due to the asymmetric forces acting on the tip of the needle. In this paper, a mechanics-based model is developed to predict the needle deflection. In the model, the needle is considered as a cantilever beam supported by a series of nonlinear springs each of which has stiffness different from each other. The value of stiffness can be calculated by cutting force acting on the needle tip. Based on the model and the analysis of cutting force and friction force, Rayleigh-Ritz method is used to estimate the amount of needle deflection. Experiment shows that the simulation model can accurately predict the deflection of the bevel-tipped needle.

Optimal locations of point supports in laminated rectangular plates for maximum fundamental frequency

Wang, C.M.,Xiang, Y.,Kitipornchai, S. Techno-Press 1997 Structural Engineering and Mechanics, An Int'l Jou Vol.5 No.6

This paper investigates the optimal locations of internal point supports in a symmetric crossply laminated rectangular plate for maximum fundamental frequency of vibration. The method used for solving this optimization problem involves the Rayleigh-Ritz method for the vibration analysis and the simplex method of Nelder and Mead for the iterative search of the optimum support locations. Being a continuum method, the Rayleigh-Ritz method allows easy handling of the changing point support locations during the optimization search. Rectangular plates of various boundary conditions, aspect ratios, composed of different numbers of layers, and with one, two and three internal point supports are analysed. The interesting results on the optimal locations of the point supports showed that (a) there are multiple solutions; (b) the locations are dependent on both the plate aspect ratios and the number of layers (c) the fundamental frequency may be raised significantly with appropriate positioning of the point supports.

Study on three-dimensional elastic deformation characteristics of flexspline in harmonic drive based on Rayleigh-Ritz method with accelerated convergence strategy

Linfeng Qiu,Manyi Chen,Gang Song 대한기계학회 2023 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.37 No.7

The 3D deformation mechanism of flexspline (FS) is the foundation of harmonic drive. In this paper, a novel methodology based on Rayleigh-Ritz method (RRM) is proposed to solve the three-dimensional elastic deformation of FS. First, the analytical model of RRM for 3D elastic deformation is formulated. A three-dimensional displacement trial function is designed according to the structure and constraints of FS. The RRM improves the accuracy of the solution by increasing the number of terms of the trial function. According to the completeness of the trial function, the real elastic deformation state can be approached arbitrarily in theory. The convergence efficiency of the actual iteration process is greatly affected by the iteration strategy. Therefore , an accelerated convergence strategy is proposed to overcome the drawback of poor convergence rate in the general iterative process. The elastic deformation characteristics of FS are studied by the new method, and a sensitivity analysis of structural parameters is performed. The results demonstrate that the new method is highly accurate and efficient. The radial deflection is slightly arch-shaped along the axial direction, reaching the maximum value near the axial position of the displacement constraint. The tangential deflection is 0 on the symmetry plane, and the value is smaller when it is farther away from the symmetry plane, which reflects the tendency of the gear tooth deflecting towards the major shaft section of the wave generator. The descending order of sensitivity of structural parameters affecting taper deformation is cylinder length (Pl), radii difference (Pd), cylinder thickness (Pt), hole radius (Ph), fillet radius (Pf) and gear rim width (Pw). The descending order of sensitivity of structural parameters affecting tooth skewness is Pl, Pd, Ph, Pf, Pw and Pt.

Rayleigh-Ritz법에 의한 단순지지된 타원형판의 고유진동수

강재훈,이은택,장경호 중앙대학교 건설산업기술연구소 2000 건설산업기술연구소 논문집 Vol.1 No.-

The problem of free vibration of a simply-supported elliptical plate has no published results in the literature. The present work uses the Rayleigh-Ritz technique with a three-term deflection function to obtain accurate fundamental frequencies. Tabulated results are given for the complete spectrum of Poisson's ratios and aspect ratios.

Rayleigh-Ritz법을 이용한 샌드위치 패널의 진동 및 소음방사 특성 분석

김동규(Kim, Dong-Kyu),김재현(Kim, Jae-Hyun),전진용(Jeon, Jin-Yong),박준홍(Park, Jun-Hong) 한국소음진동공학회 2011 한국소음진동공학회 논문집 Vol.21 No.5

The purpose of this study is to analyze the vibration and sound generation characteristics of the sandwich panel. Two thick panels were assumed to be separated by a compliant viscoelastic core. The transverse vibration induced by an external impact was analyzed using the Rayleigh-Ritz method. For applying arbitrary boundary condition of the panels, the edges were assumed to be supported by the translational and rotational springs. The beam functions were used as the trial functions. The effect of the boundary condition and viscoelastic core on the resulting vibration characteristics was investigated. The radiated sound power was analyzed using the proposed numerical model and the Rayleigh integral. The dynamic properties of the core and the mass-stiffness-mass resonance frequency had significant influence on the impact sound.

Study of non-relativistic energy and fine structure splitting using a Rayleigh–Ritz method for a high-angular-momentum state

Liu Xin,Zhang Jingchao 한국물리학회 2022 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.80 No.3

Based on the Rayleigh–Ritz variation method, the non-relativistic energies of highly excited 1s2nl (l = g, h) states of a lithiumlike isoelectronic sequence have been studied. We also investigated the fine structure splitting of the 1s2nl (l = g, h) states, which includes spin–orbital effects, spin–other orbital effects, quantum electrodynamics (QED) corrections and higher order relativistic corrections. The fine structure splitting values of the 1s25g state are compared with those in the existing literature. At present, no relevant data on 1s2nl (l = g, h) (n ≥ 6) have been found. Consequently, the calculation results in this paper should provide a valuable reference. We also obtained the regularities of the fine structure splitting with the nuclear charge number Z and the main quantum number n and found a physical law that the fine structures of the 1s2nl (l = g, h) states is proportional to the fourth power of the effective nuclear charge Zeff.