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

2007 ◽  
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 sub-domains according to the properties of wave propagation. The d’Alembert’s solution for any point in the zeroth sub-domain 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 sub-domains. 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 non-homogeneous 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 the 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.

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.

Forced response of translating media with variable length and tension: Application to high-speed elevators

2005 ◽  
Vol 219 (1) ◽  
pp. 35-53 ◽  
Author(s):  
W D Zhu ◽  
Y Chen
Keyword(s):  
High Speed ◽  
Initial Conditions ◽  
Control Volume ◽  
String Model ◽  
Forced Response ◽  
Rate Of Change ◽  
Variable Length ◽  
Lateral Response ◽  
Discretization Schemes ◽  
Fixed Boundaries

The lateral response of vertically translating media with variable length and tension, subjected to general initial conditions and external excitation, are determined. The translating media are modelled as a taut string and tensioned beams with pinned and fixed boundaries. In each model a rigid body is attached to the lower end of the medium and has a prescribed displacement along the horizontal direction. The rate of change in the energy of the translating medium is analysed from the control volume and system viewpoints. The models are used to predict the forced response of a moving cable in a high-speed elevator. Three spatial discretization schemes are used to calculate the response and shown to yield the same results. The convergence of the solution for each model is investigated. The approximate solution for the string model with constant tension is compared with its exact solution from the wave method.

Forced Response of Translating Media With Variable Length and Tension: Application to High-Speed Elevators

2004 ◽  
Author(s):  
W. D. Zhu ◽  
Y. Chen
Keyword(s):  
High Speed ◽  
Initial Conditions ◽  
Control Volume ◽  
String Model ◽  
Forced Response ◽  
Rate Of Change ◽  
Variable Length ◽  
Lateral Response ◽  
Discretization Schemes ◽  
Fixed Boundaries

The lateral response of vertically translating media with variable length and tension, subjected to general initial conditions and external excitation, are determined. The translating media are modeled as a taut string and tensioned beams with pinned and fixed boundaries. In each model a rigid body is attached to the lower end of the medium and has a prescribed displacement along the horizontal direction. The rate of change of the energy of the translating medium is analyzed from the control volume and system viewpoints. The models are used to predict the forced response of a moving cable in a high-speed elevator. Three spatial discretization schemes are used to calculate the response and shown to yield the same results. The convergence of the solution for each model is investigated. The approximate solution for the string model with constant tension is compared with its exact solution from the wave method.

Forced Response of Translating Media With Variable Length and Tension and a Mass-Spring Termination

2005 ◽  
Author(s):  
Yan Chen ◽  
W. D. Zhu
Keyword(s):  
High Speed ◽  
Initial Conditions ◽  
Resonance Condition ◽  
Control Volume ◽  
Forced Response ◽  
Variable Length ◽  
Lateral Response ◽  
Suspension Stiffness ◽  
Rail Irregularity ◽  
Mass Spring

The lateral response of vertically translating media with variable length and tension and a mass-spring termination is determined for general initial conditions and external excitation. The translating media are modeled as a taut string and tensioned beams with pinned and fixed boundary at the upper end. In each case a spring-mass sub-system is attached at the lower end of the translating medium. The rates of change of the energies in each case are analyzed and interpreted from the control volume and system viewpoints. The models are used to predict the forced response of cable-car systems in high-speed elevators due to building sway, pulley eccentricity, and guide-rail irregularity. The optimal design of the suspension stiffness of the car is investigated. A resonance condition y(x,t) that leads to dramatically increasing vibratory energy of the cable-car system is identified.

scholarly journals Life is determined by its environment

2016 ◽  
Vol 15 (4) ◽  
pp. 345-350 ◽  
Author(s):  
John S. Torday ◽  
William B. Miller
Keyword(s):  
Boundary Conditions ◽  
First Principles ◽  
Evolutionary Biology ◽  
Initial Conditions ◽  
Cellular Communication ◽  
Evolutionary Development ◽  
Molecular Approach ◽  
Unicellular Organisms ◽  
Varying Length ◽  
Life On Earth

AbstractA well-developed theory of evolutionary biology requires understanding of the origins of life on Earth. However, the initial conditions (ontology) and causal (epistemology) bases on which physiology proceeded have more recently been called into question, given the teleologic nature of Darwinian evolutionary thinking. When evolutionary development is focused on cellular communication, a distinctly different perspective unfolds. The cellular communicative-molecular approach affords a logical progression for the evolutionary narrative based on the basic physiologic properties of the cell.Critical to this appraisal is recognition of the cell as a fundamental reiterative unit of reciprocating communication that receives information from and reacts to epiphenomena to solve problems. Following the course of vertebrate physiology from its unicellular origins instead of its overt phenotypic appearances and functional associations provides a robust, predictive picture for the means by which complex physiology evolved from unicellular organisms. With this foreknowledge of physiologic principles, we can determine the fundamentals of Physiology based on cellular first principles using a logical, predictable method. Thus, evolutionary creativity on our planet can be viewed as a paradoxical product of boundary conditions that permit homeostatic moments of varying length and amplitude that can productively absorb a variety of epigenetic impacts to meet environmental challenges.

scholarly journals Solution of Second-Order IVP and BVP of Matrix Differential Models Using Matrix DTM

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Reza Abazari ◽  
Adem Kılıcman
Keyword(s):  
Boundary Conditions ◽  
Matrix Models ◽  
Boundary Value Problems ◽  
Initial Value Problems ◽  
Present Method ◽  
Initial Conditions ◽  
Transformation Method ◽  
Second Order ◽  
Initial Value ◽  
Matrix Differential

We introduce a matrix form of differential transformation method (DTM) and apply for nonlinear second-order initial value problems (IVPs) and boundary value problems (BVPs) of matrix models which are given by and subject to initial conditions and boundary conditions , where . Also the convergence of present method is established. Several illustrative examples are given to demonstrate the effectiveness of the present method.

scholarly journals Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems

Pharmaceutics ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 140 ◽  
Author(s):  
Constantin Mircioiu ◽  
Victor Voicu ◽  
Valentina Anuta ◽  
Andra Tudose ◽  
Christian Celia ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Moving Boundary ◽  
Initial Conditions ◽  
Integral Transformation ◽  
Release Kinetics ◽  
Separation Of Variables ◽  
Transformation Method ◽  
Mathematical Methods ◽  
Important Place ◽  
First Choice

Embedding of active substances in supramolecular systems has as the main goal to ensure the controlled release of the active ingredients. Whatever the final architecture or entrapment mechanism, modeling of release is challenging due to the moving boundary conditions and complex initial conditions. Despite huge diversity of formulations, diffusion phenomena are involved in practically all release processes. The approach in this paper starts, therefore, from mathematical methods for solving the diffusion equation in initial and boundary conditions, which are further connected with phenomenological conditions, simplified and idealized in order to lead to problems which can be analytically solved. Consequently, the release models are classified starting from the geometry of diffusion domain, initial conditions, and conditions on frontiers. Taking into account that practically all solutions of the models use the separation of variables method and integral transformation method, two specific applications of these methods are included. This paper suggests that “good modeling practice” of release kinetics consists essentially of identifying the most appropriate mathematical conditions corresponding to implied physicochemical phenomena. However, in most of the cases, models can be written but analytical solutions for these models cannot be obtained. Consequently, empiric models remain the first choice, and they receive an important place in the review.

DYNAMICS OF THERMAL LOSSES BY CONVECTION AND RADIATION OF THE SPHERICAL HEAT ACCUMULATOR OF SOLAR PLANTS

2020 ◽  
Vol 4 (41) ◽  
pp. 57-62
Author(s):  
SHAVKAT KLYCHEV ◽  
◽  
BAKHRAMOV SAGDULLA ◽  
VALERIY KHARCHENKO ◽  
VLADIMIR PANCHENKO ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Heat Loss ◽  
Initial Conditions ◽  
Water System ◽  
Heat Losses ◽  
Research Purpose ◽  
Heat Accumulator ◽  
Water Heaters ◽  
Intrinsic Radiation ◽  
Over Time

There are needed energy (heat) accumulators to increase the efficiency of solar installations, including solar collectors (water heaters, air heaters, dryers). One of the tasks of designing heat accumulators is to ensure its minimal heat loss. The article considers the problem of determining the distribution of temperatures and heat losses by convection and radiation of the heat insulation-accumulating body (water) system for a ball heat accumulator under symmetric boundary conditions. The problem is solved numerically according to the program developed on the basis of the proposed «gap method». (Research purpose) The research purpose is in determining heat losses by convection and radiation of a two-layer ball heat accumulator with symmetric boundary conditions. (Materials and methods) Authors used the Fourier heat equation for spherical bodies. The article presents the determined boundary and initial conditions for bodies and their surfaces. (Results and discussion) The thickness of the insulation and the volume of the heat accumulator affect the dynamics and values of heat loss. The effect of increasing the thickness of the thermal insulation decreases with increasing its thickness, starting with a certain volume of the heat accumulator or with R > 0.3 meters, the heat losses change almost linearly over time. The dynamics of heat loss decreases with increasing shelf life, but the losses remain large. (Conclusions) Authors have developed a method and program for numerical calculation of heat loss and temperature over time in a spherical two-layer heat accumulator with symmetric boundary conditions, taking into account both falling and intrinsic radiation. The proposed method allows to unify the boundary conditions between contacting bodies.

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.

scholarly journals Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ji Lin ◽  
Yuhui Zhang ◽  
Chein-Shan Liu
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Initial Conditions ◽  
Boundary Value ◽  
Accurate Solution ◽  
Point Boundary ◽  
Third Order ◽  
Boundary Shape ◽  
Free Function ◽  
New Algorithms

AbstractFor nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

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