Simulations of string vibrations with boundary conditions of third kind using the functional transformation method

2005 ◽  
Vol 118 (3) ◽  
pp. 1763-1775 ◽  
Author(s):  
L. Trautmann ◽  
S. Petrausch ◽  
M. Bauer
Keyword(s):  
Boundary Conditions ◽  
Transformation Method ◽  
Functional Transformation ◽  
String Vibrations

scholarly journals Differential Quadrature and Differential Transformation Methods in Buckling Analysis of Nanobeams

2019 ◽  
Vol 6 (1) ◽  
pp. 68-76 ◽  
Author(s):  
Subrat Kumar Jena ◽  
S. Chakraverty
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Transformation Method ◽  
Nonlocal Theory ◽  
Buckling Analysis ◽  
Differential Quadrature ◽  
Load Parameter ◽  
Computationally Efficient ◽  
Differential Transformation

AbstractIn this paper, two computationally efficient techniques viz. Differential Quadrature Method (DQM) and Differential Transformation Method (DTM) have been used for buckling analysis of Euler-Bernoulli nanobeam incorporation with the nonlocal theory of Eringen. Complete procedures of both the methods along with their mathematical formulations are discussed, and MATLAB codes have been developed for both the methods to handle the boundary conditions. Various classical boundary conditions such as SS, CS, and CC have been considered for investigation. A comparative study for the convergence of DQM and DTM approaches are carried out, and the obtained results are also illustrated to demonstrate the effects of the nonlocal parameter, aspect ratio (L/h) and the boundary condition on the critical buckling load parameter.

Implementing of Displacement Boundary Conditions of Element-Free Galerkin Method and its Applications Used in Fracture Mechanics

2012 ◽  
Vol 629 ◽  
pp. 606-610
Author(s):  
Gang Cheng ◽  
Wei Dong Wang ◽  
Dun Fu Zhang
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Reproducing Kernel ◽  
Particle Method ◽  
Transformation Method ◽  
Computational Method ◽  
Reproducing Kernel Particle Method ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Element Free

The main draw back of the Moving Least Squares (MLS) approximate used in element free Galerkin method (EFGM) is its lack the property of the delta function. To alleviate difficulties in the treatment of essential boundary conditions in EFGM, the local transformation method and the boundary singular weight method, which are used in the reproducing kernel particle method, is combined with the element free Galerkin method. The computational method is given to analyze the stress intensity factors and the numerical simulation of crack propagation of two-dimentional problems of the elastic fracture analysis. The application examples reveal the effectiveness and feasibility of the present methods.

scholarly journals Analysis of beam-column elements on non-homogeneous soil using the differential transformation method

2021 ◽  
Author(s):  
Juan Sebastián Carvajal-Muñoz ◽  
Carlos Alberto Vega-Posada ◽  
Julio César Saldarriaga-Molina
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Mathematical Formulation ◽  
Transformation Method ◽  
Transverse Load ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Pasternak Elastic Foundation ◽  
Wide Range ◽  
Homogeneous Soil

This paper describes an analytical approach to conduct an analysis of beam-column elements with generalized end-boundary conditions on a homogeneous or non-homogeneous Pasternak elastic foundation. The mathematical formulation utilized herein is that presented by the senior author in a recent work. The differential equation (DE) governing the behavior of the beam-column element is solved using the differential transformation method (DTM). The DTM offers practical advantages over other conventional approaches when solving the proposed structural model. The proposed formulation provides the flexibility to account for i) combined lateral and axial load at the ends of the element, ii) homogeneous or non-homogeneous soil, iii) Pasternak elastic foundation, and iv) an external arbitrary transverse load acting on the element. The effects of various slenderness ratios, pile-soil stiffness ratios, and classical and semirigid boundary conditions can be easily studied with the proposed formulation. Examples are presented to validate the accuracy of the model and its applicability over a wide range of analyses.

Exact Response of a Translating String With Arbitrarily Varying Length Under General Excitation

2008 ◽  
Vol 75 (3) ◽  
Author(s):  
W. D. Zhu ◽  
N. A. Zheng
Keyword(s):  
Boundary Conditions ◽  
Initial Conditions ◽  
Control Volume ◽  
Parametric Instability ◽  
Transformation Method ◽  
Forced Response ◽  
Rate Of Change ◽  
Solution Methods ◽  
Vibrating String ◽  
Varying Length

The exact response of a translating string with constant tension and arbitrarily varying length is determined under general initial conditions and external excitation. The governing equation is transformed to a standard hyperbolic equation using characteristic transformation. The domain of interest for the transformed equation is divided into groups of subdomains according to the properties of wave propagation. d’Alembert’s solution for any point in the zeroth subdomain group is obtained by using the initial conditions. The solution is extended to the whole domain of interest by using the boundary conditions, and a recursive mapping is found for the solution in the second and higher groups of subdomains. The least upper bound of the displacement of the freely vibrating string is obtained for an arbitrary movement profile. The forced response of the string with nonhomogeneous boundary conditions is obtained using a transformation method and the direct wave method. A new method is used to derive the rate of change of the vibratory energy of the translating string from the system viewpoint. Three different approaches are used to derive and interpret the rate of change of the vibratory energy of the string within a control volume, and the energy growth mechanism of the string during retraction is elucidated. The solution methods are applied to a moving elevator cable with variable length. An interesting parametric instability phenomenon in a translating string with sinusoidally varying length is discovered.

The problem of boundary control of string vibrationsby displacement of the left end when the right end is fixedwith the given values of the deflection functionat intermediate times

2020 ◽  
pp. 131-146
Author(s):  
Vanya R. Barseghyan ◽  
Svetlana V. Solodusha
Keyword(s):  
Boundary Conditions ◽  
Control Problem ◽  
Boundary Control ◽  
Constructive Method ◽  
Boundary Control Problem ◽  
String Vibrations ◽  
Classical Boundary ◽  
The Right ◽  
The Given

We consider the boundary control problem for the homogeneous string vibrationequation with given the classical boundary (initial and final) conditions and with given valuesof the deflection function at intermediate times. The control is performed by displacementof the left end of the string when the right end is fixed. The problem is reduced to thecontrol problem with zero boundary conditions. We propose the constructive method forconstructing the boundary control of the process of string vibrations with given values ofthe deflection function at intermediate times.We present the results of numerical experimentsand the corresponding graphs confirm the validity of the results.

Free Vibration Analysis of Rotating Tapered Bresse-Rayleigh Beams Using the Differential Transformation Method

2009 ◽  
Author(s):  
Dominic R. Jackson ◽  
S. Olutunde Oyadiji
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Slenderness Ratio ◽  
Free Vibration Analysis ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Algebraic Equations ◽  
Differential Transformation ◽  
Rayleigh Beam

The free vibration characteristics of a rotating tapered Rayleigh beam is analysed in this study. First, the strain-displacement relationship for the rotating beam is formulated and used to derive the kinetic and strain energies in explicit analytical form. Second, Hamilton’s variational principle is used to derive the governing differential equation of motion and the associated boundary conditions. Third, the Differential Transformation Method (DTM) is applied to reduce the governing differential equations of motion and the boundary conditions to a set of algebraic equations from which the frequency equation is derived. Next, a numerical algorithm implemented in the software package Mathematica is used to compute the natural frequencies of vibration for a few paired combinations of clamped, pinned and free end conditions of the beam. Also, the variation of the natural frequencies of vibration with respect to variations in the rotational speed, hub radius, taper ratio and the slenderness ratio is studied. The results obtained from the Bresse-Rayleigh theory are compared with results obtained from the Bernoulli-Euler and Timoshenko theories to demonstrate the accuracy and relevance of their application.

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