Oscillator Basis in the J-Matrix Method: Convergence of Expansions, Asymptotics of Expansion Coefficients and Boundary Conditions

2008 ◽  
pp. 49-66 ◽  
Author(s):  
S.Yu. Igashov
Keyword(s):  
Boundary Conditions ◽  
Matrix Method ◽  
Oscillator Basis ◽  
Expansion Coefficients

Flexural Vibration Response of Beams With General Boundary Conditions: A Transfer Matrix Approach

1991 ◽  
Author(s):  
T. Önsay
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vibration Response ◽  
General Boundary ◽  
Phase Variable ◽  
General Boundary Conditions ◽  
Transfer Matrices ◽  
Analytical Results

Abstract The wave-mode representation is utilized to obtain a more efficient form to the conventional transfer matrix method for bending vibrations of beams. The proposed improvement is based on a phase-variable canonical state representation of the equation governing the time-harmonic flexural vibrations of a beam. Transfer matrices are obtained for external forces, step-change of beam properties, intermediate supports and for boundaries. The transfer matrices are utilized to obtain the vibration response of a point-excited single-span beam with general boundary conditions. The general characteristic equation and the transfer mobility of a single-span beam are determined. The application of the analytical results are demonstrated on physical structures with different boundary conditions. A hybrid model is developed to incorporate measured impedance of nonideal boundaries into the transfer matrix method. The analytical results are found to be in excellent agreement with experimental measurements.

Vibration Analysis of Drug Delivery CNTs Using Transfer Matrix Method

2011 ◽  
Author(s):  
Anooshiravan Farshidianfar ◽  
Ali A. Ghassabi ◽  
Mohammad H. Farshidianfar ◽  
Mohammad Hoseinzadeh
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Equation Of Motion ◽  
Beam Model ◽  
Multi Wall Carbon Nanotubes ◽  
Resonance Frequencies ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

The free vibration and instability of fluid-conveying multi-wall carbon nanotubes (MWCNTs) are studied based on an Euler-Bernoulli beam model. A theory based on the transfer matrix method (TMM) is presented. The validity of the theory was confirmed for MWCNTs with different boundary conditions. The effects of the fluid flow velocity were studied on MWCNTs with simply-supported and clamped boundary conditions. Furthermore, the effects of the CNTs’ thickness, radius and length were investigated on resonance frequencies. The CNT was found to posses certain frequency behaviors at different geometries. The effect of the damping corriolis term was studied in the equation of motion. Finally, a useful simplification is introduced in the equation of motion.

Application of the Transfer Matrix Method to the Analysis of Hydro-Mechanical Vibration of NPP Piping

2013 ◽  
Author(s):  
Igor Orynyak ◽  
Sergii Radchenko ◽  
Iaroslav Dubyk
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Nuclear Power ◽  
Mechanical Vibration ◽  
Piping System ◽  
Harmonic Vibrations ◽  
Water Power ◽  
Poisson Contraction

The transfer matrix method (TMM) was used for description of harmonic vibrations of piping with transported medium. Apart from 12 well-known mechanical parameters which characterize the state of piping system in each cross section two additional parameters that characterize the vibration of the medium, namely its translation and pressure pulsations were considered. The solution of these equations, which take into account the Poisson contraction of the pipe wall, in the form suitable for the transfer matrix method application was derived. The biggest uncertainty in the analytical modeling is to adopt the boundary conditions for above mentioned 2 parameters for the considered piping section. To solve this problem of identification of the most probable induced frequency we developed the technique of choosing such boundary conditions at which the maximum of energy is confined within the considered piping section. The validity of the approach was tested on some analytical examples. This method was used to analyze the forced vibration of the second circuit loop of unit 1 Zaporizhia Nuclear Power Plant (ZNPP) with VVER-1000 (from Russian: Vodo-Vodyanoi Energetichesky Reactor; Water-Water Power Reactor) arising from turbulent eddies in the flow of steam. Natural frequencies and forms of mechanical, hydrodynamic, and related hydro-mechanical vibration were found, a number of recommendations were given to reduce the vibration levels.

Structural dynamics of the foldable stairs

2012 ◽  
Vol 226 (10) ◽  
pp. 2549-2572 ◽  
Author(s):  
Jing-Shan Zhao ◽  
Jian-Yi Wang ◽  
Fu-Lei Chu ◽  
Zhi-Jing Feng ◽  
Jian S Dai
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Structural Dynamics ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Huge Number ◽  
Traditional Methods ◽  
Revolute Joints ◽  
Dynamics Method

This article proposes a structural dynamics method for foldable stairs based on transfer matrix. The stairs are made up of a number of identical scissor-like elements which are supposed to be Euler–Bernoulli beams. The dynamics of each segment beam between every two adjacent revolute joints can be precisely expressed by the transfer matrix of the segment with the variables of boundary conditions of the joints. Therefore, the structural dynamics of the whole stairs is built using the least number of variables compared with the traditional methods. In addition, this method avoids the problem of the traditional transfer-matrix method that the number of variables greatly increases when there are a huge number of cross-joints within a structure.

Numerical Solution of Bending Problem for Elliptical Plate Using Differentiation Matrix Method Based on Barycentric Lagrange Interpolation

2014 ◽  
Vol 638-640 ◽  
pp. 1720-1724 ◽  
Author(s):  
Zhao Qing Wang ◽  
Jian Jiang ◽  
Bing Tao Tang ◽  
Wei Zheng
Keyword(s):  
Boundary Conditions ◽  
Governing Equation ◽  
Matrix Method ◽  
Lagrange Interpolation ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Uniform Load ◽  
Tensor Form ◽  
Elliptical Plate ◽  
Differentiation Matrix

A differentiation matrix method based on barycentric Lagrange interpolation for numerical analysis of bending problem for elliptical plate is presented. Embedded the elliptical domain into a rectangular, the barycentric Lagrange interpolation in tensor form is used to approximate unknown function. The governing equation of bending plate is discretized by the differentiation matrix derived from barycentric Lagrange interpolation to form a system of algebraic equations. The boundary conditions on curved boundary are directly discretized using barycentric Lagrange interpolation. Combining discrete algebraic equations of governing equation and boundary conditions to form an over-constraints system of equations, the numerical solutions on rectangular can be obtained by solving it. Then, the numerical solutions on elliptical domain are obtained by interpolating the data on rectangular. Numerical results of elliptical plate with uniform load illustrate the effectiveness and accuracy of the proposed method.

Dynamic Analysis of T-Shaped Plate with General Boundary Conditions

2013 ◽  
Vol 572 ◽  
pp. 509-512
Author(s):  
Dong Yan Shi ◽  
Qing Shan Wang ◽  
Xian Jie Shi
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Double Fourier Series ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Arbitrary Boundary ◽  
Out Of Plane ◽  
Expansion Coefficients ◽  
Good Agreement

In this investigation, an analytical method is proposed for the dynamic analysis of T-shaped plates with general boundary conditions. Four types of springs are uniformly distributed along each edge, which are used to simulate the transverse shearing forces, bending moments, in-plane longitudinal forces and in-plane shearing forces, respectively. Arbitrary boundary conditions can be readily realized by setting the stiffness of the four types restraining springs. The interactions between the plates have been taken into account at the junction with four types of coupling springs. The in-plane and out-of-plane displacements are invariantly expressed, regardless of boundary conditions, as a new form of double Fourier series with a drastically improved convergence as compared with the traditional Fourier series. The expansion coefficients are considered as the generalized coordinates, and determined using the Rayleigh-Ritz technique. Numerical examples are presented to validate the accuracy and reliability of the proposed method. A good agreement is observed between the current results and FEA results. The present method can be directly extended to more complicated structures with any number of plates.

A Fast New Algorithm for Solving a Nonlinear Beam Equation under Nonlinear Boundary Conditions

2017 ◽  
Vol 72 (5) ◽  
pp. 397-400 ◽  
Author(s):  
Chein-Shan Liu ◽  
Botong Li
Keyword(s):  
Boundary Conditions ◽  
Numerical Solution ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
Nonlinear Boundary Conditions ◽  
Beam Equation ◽  
Test Functions ◽  
Nonlinear Beam ◽  
Expansion Coefficients ◽  
Sinusoidal Functions

AbstractFor the problem of a nonlinear beam equation under nonlinear boundary conditions of moments, a fast iterative method is developed by transforming the ordinary differential equation into an integral one. The sinusoidal functions are used subtly as test functions as well as the bases of numerical solution in the calculation. Due to the orthogonality of the sinusoidal functions, the expansion coefficients of numerical solution in closed form can be found easily. Hence, the iterative scheme converges very fast to find numerical solutions with high accuracy.

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