scholarly journals A Force Method Model for Dynamic Analysis of Flat-Sag Cable Structures

2009 ◽  
Vol 16 (6) ◽  
pp. 623-635 ◽  
Author(s):  
Xing Ma ◽  
John W. Butterworth
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Travelling Wave ◽  
Governing Equations ◽  
Compatibility Equation ◽  
Partial Differential ◽  
Force Method ◽  
Infinite Degree ◽  
Cable Systems ◽  
Method Model

A new force method is proposed for analysing the dynamic behaviour of oscillating cables with small sags. The accepted dynamic model of such cables reduces to a partial differential equation (the equation of motion) and an integral equation (the compatibility equation). In the paper, D’Alembert’s travelling wave solution is applied to the partial differential equation (PDE). Substituting the solution into the compatibility and boundary conditions, the governing equation is obtained in terms of the dynamic tension increment. This equation has been named the force method dynamic equation (FMDE). In this way the infinite-degree-of-freedom dynamic system is effectively simplified to a system with only one unknown. Explicit solutions for both single-span and multi-span cable systems are derived. The natural frequencies obtained from the FMDE are shown to be identical to those deduced using the conventional displacement method (DM). Nonlinear governing equations are developed by considering the effect of quadratic and cubic displacement terms. Finally, two examples are presented to illustrate the accuracy of the proposed force method for single and multi-span cable systems subjected to harmonic forces.

A Nonlinear Force Method Model for Dynamic Analysis of Cable Trusses

2011 ◽  
Vol 250-253 ◽  
pp. 2281-2284 ◽  
Author(s):  
Xing Ma
Keyword(s):  
Governing Equation ◽  
Travelling Wave ◽  
Governing Equations ◽  
Dynamic Tension ◽  
Force Method ◽  
Support Reaction ◽  
Infinite Degree ◽  
Nonlinear Force ◽  
Reaction Forces ◽  
Method Model

A nonlinear force method model is proposed to study the dynamic behavior of cable trusses. In the paper, travelling wave method is employed to solve the governing equation of motion. After support reaction forces are considered as excitations, cable trusses are extended to infinitude, and D’Alembert solution to the partial differential equation (PDE) is achieved. Substituting the solution into compatibility condition and boundary conditions, the governing equation expressed by dynamic tension is derived, which is named force method dynamic equation (FMDE). Hence the dynamic system of infinite-degree-of-freedom is simplified as a system with only one unknown without any loss of precision. Nonlinear governing equations are developed through considering the effect of quadratic terms of displacements. At last, an example is given to verify the force method model presented in the paper.

scholarly journals EVOLUTION OF MODULATIONAL INSTABILITY IN TRAVELLING WAVE SOLUTION OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATION

2020 ◽  
Vol 5 (1) ◽  
pp. 1-7
Author(s):  
Ram Dayal Pankaj ◽  
Arun Kumar ◽  
Chandrawati Sindhi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Solution ◽  
Modulational Instability ◽  
Nonlinear Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Partial Differential ◽  
Jacobian Elliptic Functions

The Ritz variational method has been applied to the nonlinear partial differential equation to construct a model for travelling wave solution. The spatially periodic trial function was chosen in the form of combination of Jacobian Elliptic functions, with the dependence of its parameters

scholarly journals Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves

2016 ◽  
Author(s):  
Stuart T. Johnston ◽  
Ruth E. Baker ◽  
D.L. Sean McElwain ◽  
Matthew J. Simpson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Diffusion ◽  
Cell Biology ◽  
Travelling Waves ◽  
Field Approximation ◽  
Travelling Wave ◽  
Mean Field Approximation ◽  
Partial Differential ◽  
Birth Death

AbstractInvasion processes are ubiquitous throughout cell biology and ecology. During invasion, individuals can become isolated from the bulk population and behave differently. We present a discrete, exclusion-based description of the birth, death and movement of individuals. The model distinguishes between individuals that are part of, or are isolated from, the bulk population by imposing different rates of birth, death and movement. This enables the simulation of various co-operative or competitive mechanisms, where there is either a positive or negative benefit associated with being part of the bulk population, respectively. The mean-field approximation of the discrete process gives rise to 22 different classes of partial differential equation, which can include Allee kinetics and nonlinear diffusion. Here we examine the ability of each class of partial differential equation to support travelling wave solutions and interpret the long time behaviour in terms of the individual-level parameters. For the first time we show that the strong Allee effect and nonlinear diffusion can result in shock-fronted travelling waves. We also demonstrate how differences in group and individual motility rates can influence the persistence of a population and provide conditions for the successful invasion of a population.

scholarly journals A Force Method Model for Static Analysis of Transmission Line System Subjected to in-Plane and Out-of-Plane Loadings

2011 ◽  
Vol 368-373 ◽  
pp. 3535-3538
Author(s):  
Xing Ma ◽  
Julie E. Mills ◽  
Graham Brown ◽  
Zhu Ge Yan
Keyword(s):  
Transmission Line ◽  
Static Response ◽  
Cable Model ◽  
Equilibrium Equations ◽  
Compatibility Equation ◽  
Line System ◽  
Force Method ◽  
Out Of Plane ◽  
Infinite Degree ◽  
Method Model

In this paper, transmission line systems are modeled as multi-span cable structures. A force method model is proposed for analysing the static response of the multi-span cables with small sags. The accepted cable model reduces to two groups of differential equations (the equilibrium equations in y, z directions) and an integral equation (the compatibility equation). Substituting the differential equation solutions into the compatibility condition, the governing equation is obtained in terms of the tension component in chord direction. This equation has been named the force method equation (FME). In this way the infinite-degree-of-freedom dynamic system is effectively simplified to a system with only one unknown. Finally, one example is presented to illustrate the application of the proposed force method.

scholarly journals Nonlinear Frequencies for Transverse Free Oscillations of a Transporting Tensioned Beam

2011 ◽  
Vol 2-3 ◽  
pp. 807-816
Author(s):  
Hu Ding ◽  
Li Qun Chen ◽  
Hai Yan Jiang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Natural Frequencies ◽  
Nonlinear Oscillations ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equation ◽  
Nonlinear Coefficient ◽  
Governing Equations ◽  
Partial Differential ◽  
Transverse Oscillations

This study focuses on the natural frequencies of nonlinear free transverse oscillations of transporting tensioned beams via the standard fast Fourier transform (FFT). The transverse oscillations of transporting tensioned beams can be governed by a nonlinear partial-differential equation or a nonlinear integro-partial-differential equation. Numerical schemes are respectively presented for the two governing equations via the differential quadrature method. For each nonlinear equation, the nonlinear oscillations frequencies are investigated via FFT with the time responses histories. The numerical results depict the tendencies of the frequencies of nonlinear free transverse oscillations of transporting tensioned beams with the changing oscillations amplitude, transporting speed, the nonlinear coefficient and the flexural stiffness.

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