An exact, closed-form solution for equilibrium of trveling, sagged, elastic cables under uniformly distributed loading

IntroctionThquilibrium of sagged, elastic cable s under its own weight loading is needed in tiledesign of cable structures. In 1851 RO'htsLll first investigated a uniform, inextensible suspendedchain hanging freely under its own weight. Ro.th[Zj considered a model of a heterogeneouschain hanging in the form of a cycloid, and application of this model to the uniform chainreduced to the Rohrs model when the sag ratio is small. However, the chain was modeled asinextensible. Pugsley['] develope…     

Traveling waves for a diffusive predator–prey system with general functional response

By combining the simplified shooting method with a sandwich method, the existence and nonexistence of two types of traveling wave solutions for a class of diffusive predator–prey systems with general functional response are investigated. These two types of point-to-point traveling waves include the connections of the zero equilibrium to the positive equilibrium and the boundary equilibrium to the positive equilibrium. Furthermore, we also give a discussion about the parameter threshold value whether any of the traveling waves approaches the positive equilibrium monotonically or has exponentially damped oscillations about the positive equilibrium. Some applications with different functional response functions are given to illustrate the application of our results.     

Nonlinear Analysis and Optimum Design of Guyed Masts

In this paper the nonlinear analysis and design optimization of guyed masts is addressed. The mast is modeled as a 3D truss and is supported by catenary cable elements that have nonlinear elastic behavior. For nonlinear static analysis, an innovative procedure is proposed that divides the structure into linear and nonlinear parts and analyzes them separately. The proposed method satisfies the equilibrium and compatibility by establishing and solution of a set of nonlinear equations. The optimization problem employs the sizes of members, initial cable tensions and the positions of anchor on the ground and tie level of cables on the mast as design variables. To facilitate the optimization solution, a compatible sensitivity analysis procedure is proposed. Sensitivities of objective function, displacement and strength constraints in the mast and cables, subjected to a variety of load combinations including dead, wind and ice loads are calculated. Numerical examples are provided to show the nonlinear analysis procedure and the applicability of the algorithm to optimum design of practical guyed masts.     

Multiplicity of supercritical fronts for reaction–diffusion equations in cylinders

We study multiplicity of the supercritical traveling front solutions for scalar reaction–diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions connecting an unstable equilibrium to the closest stable equilibrium for all speeds exceeding a critical value. We show that these are, in fact, the only traveling front solutions in the considered problems for sufficiently large speeds. In addition, we show that other traveling fronts connecting to the unstable equilibrium may exist in a certain range of the wave speed. These results are obtained with the help of a variational characterization of such solutions.     

Exact traveling wave solutions and bifurcations in a nonlinear elastic rod equation

The exact parametric representations of the traveling wave solutions for a nonlinear elastic rod equation are considered. By using the method of planar dynamical systems, in different parameter regions, the phase portraits of the corresponding traveling wave system are given. Exact explicit kink wave solutions, periodic wave solutions and some unbounded wave solutions are obtained.     

A new formalism for the dynamic modelling of cables

This article proposes a new formalism for the dynamic modelling of cables that can even be applied when they are submitted to cross flow of water or air. An important application is the case of umbilical cables used in remotely operated vehicles. The primary basis for the formulation is to assume that the continuous flexibility is represented by a discrete approach, consisting of rigid links connected by elastic joints, allowing movement in three dimensions. Each elastic joint allows three independent movements, called elevation, azimuth and torsion (twist). A significant contribution of the proposed formalism is the development of a compact equation that allows obtaining the Lagrangian of the system directly and automatically, regardless of the number of links chosen to form a chain of rigid bodies connected by flexible joints to represent the continuous flexibility of the cable. This formulation allows the construction of an algorithm for obtaining the equations of the dynamic model of flexible cables.     

Bounded traveling waves of the Burgers-Huxley equation

In order to investigate bounded traveling waves of the Burgers-Huxley equation, bifurcations of codimension 1 and 2 are discussed for its traveling wave system. By reduction to center manifolds and normal forms we give conditions for the appearance of homoclinic solutions, heteroclinic solutions and periodic solutions, which correspondingly give conditions of existence for solitary waves, kink waves and periodic waves, three basic types of bounded traveling waves. Furthermore, their evolutions are discussed to investigate the existence of other types of bounded traveling waves, such as the oscillatory traveling waves corresponding to connections between an equilibrium and a periodic orbit and the oscillatory kink waves corresponding to connections of saddle-focus.     

Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added. For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point. We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set. Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point. This gives us a traveling wave of the viscous–capillary compressible Euler equations.     

Boundedness and persistence of traveling wave solutions for a non-cooperative lattice-diffusion system with time delay

In this paper we study traveling wave solutions of a non-cooperative lattice-diffusion system with time delay, which includes predator–prey models and disease-transmission models. Minimal wave speed of traveling wave solutions is given. Schauder’s fixed-point theorem is applied to show the existence of semi-traveling wave solutions. The boundness and persistence of traveling wave solutions are overcome by using rescaling method and Laplace transform, where the application of Laplace transform to persistence is very novel and creative. The traveling wave solutions for some specific models are shown to connect to a positive equilibrium by using Lyapunov function and LaSalle’s invariance principle.     

Solitary wave in a nonlinear elastic structural element of large deflection

General form nonlinear governing equations for the wave traveling in a nonlinear elastic structural element of large deflection are derived in the present research. An asymptotic solution of solitary wave in the elastic element is derived and investigated by means of a modified complete approximate method. Numerical computations for the solution are carried out. Characteristics of the solitary wave are investigated with various system parameters and initial conditions. Shapes and the propagation of the nonlinear elastic wave are also illustrated with figures. Based on the theoretical and numerical analyses of the research, quantitative conclusions are obtained for the wave motion of the elastic structural element.     

Oscillatory waves in reaction–diffusion equations with nonlocal delay and crossing-monostability

This paper is concerned with the existence of oscillatory waves in reaction–diffusion equations with nonlocal delay and crossing-monostability, which include many population models, and two main results are presented. In the first one, we establish the existence of non-monotone traveling waves from the trivial solution to the positive equilibrium. The approach is based on the construction of two associated auxiliary reaction–diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using traveling fronts of the auxiliary equations. In the second one, we obtain the existence of periodic waves around the positive equilibrium by using Hopf bifurcation theorem.     

Generation of surface waves through friction: FEM investigation

Sliding friction between two bodies can generate elastic vibration. This study uses a finite-element model comprising an elastic body sliding against a flat rigid surface with constant coefficient of friction. For the elastic body a structured topography is taken into account. The model shows traveling surface waves, which depend on the asperities of the sliding surface. It can be shown that the surface structure and its inertia are the cause for elastic waves in the contact region. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An SIS epidemic model with diffusion

The aim of this paper is to study the dynamics of an SIS epidemic model with diffusion. We first study the well-posedness of the model. And then, by using linearization method and constructing suitable Lyapunov function, we establish the local and global stability of the disease-free equilibrium and the endemic equilibrium, respectively. Furthermore, in view of Schauder fixed point theorem, we show that the model admits traveling wave solutions connecting the disease-free equilibrium and the endemic equilibrium when R_0 1 and c c~*. And also, by virtue of the two-sided Laplace transform, we prove that the model has no traveling wave solution connecting the two equilibria when R_0 1 and c ∈ [0, c~*).     

Traveling Waves in Elastic Rods with Arbitrary Curvature and Torsion

The dynamic Kirchhoff equations, describing a thin elastic rod of infinite length, are considered in connection with the study of the conformations of polymeric chains. A?novel special traveling wave solution that can be interpreted as a conformational soliton propagating at constant speed is obtained, featuring arbitrary non-constant curvature and torsion of the rod, in the simple case of constant cross-section, homogeneous density and elastic isotropy. This traveling wave corresponds to a specific constraint on the twist-to-bend ratio of the constant stiffness parameters, which in turn appears to be compatible with the experimental evidence for the mechanical properties of real polymeric chains. Due to such a constraint, the square of the velocity of the solitary wave is directly proportional to the bending stiffness and inversely proportional to the density and to the principal momentum of inertia of the rod. Several applications to the study of conformational changes in polymeric chains are given.     

Extinction and periodic oscillations in an age-structured population model in a patchy environment

We consider an age-structured single-species population model in a patch environment consisting of infinitely many patches. Previous work shows that if the nonlinear birth rate is sufficiently large and the maturation time is small, then the model exhibits the usual transition from the trivial equilibrium to the positive (spatially homogeneous) equilibrium represented by a traveling wavefront. Here we show that (i) if the birth rate is so small that a patch alone cannot sustain a positive equilibrium then the whole population in the patchy environment will become extinct, and (ii) if the birth rate is large enough that each patch can sustain a positive equilibrium and if the maturation time is moderate then the model exhibits nonlinear oscillations characterized by the occurrence of multiple periodic traveling waves.     

具有全局交互作用的时滞周期格微分系统的front-like整体解

研究了一类一维空间周期格上的具有时滞和全局交互作用的微分系统的front⁃like整体解.通过建立适当的比较原理,并融合不同方向的波前解与连接稳定态和不稳定态的空间周期解,构造了front⁃like整体解并证明了一些定性性质.与波前解相比,front⁃like整体解能够展示出新的动力学行为.     

Minimal wave speed and spread speed of competing pioneer and climax species

In this article, for a diffusive population model describing interaction of pioneer-climax species, we explore the issues of spreading speed, linear determinacy and traveling wave fronts. Applying the theory developed by Weinberger et al. [J. Math. Biol. 2002;45:183–218], we identify some ranges of model parameters within which, the model is shown to have a single spreading speed which is linearly determinate and coincides with the corresponding minimal speed for the traveling wave fronts connecting two relevant equilibria, one being a boundary equilibrium and the other being a coexistence equilibrium.     

Attenuating Resonance Modes in a Cylindrical Waveguide Placed in an Elastic Medium

Interference attenuating waves traveling in a cylindrical elastic waveguide, placed in an elastic medium, are considered. The group velocity of these waves is intermediate between that of the P wave and that of the S wave; the phase velocity equals that of the P wave. The frequency of the waves is almost constant and is determined by the requirement of constructive interference. The dispersion and attenuation of these waves are described. Bibliography: 3 titles.     

Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response

A class of reaction-diffusion equations with time delay and nonlocal response is considered. Assuming that the corresponding reaction equations have heteroclinic orbits connecting an equilibrium point and a periodic solution, we show the existence of traveling wave solutions of large wave speed joining an equilibrium point and a periodic solution for reaction-diffusion equations. Our approach is based on a transformation of the differential equations to integral equations in a Banach space and the rigorous analysis of the property for a corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by the application of Liapunov-Schmidt method and the Implicit Function Theorem.