Methods of Analysis of Discrete - Continuous Systems Vibrations and Their Application to Railway Transport Problems

In the paper some aspects of the interaction between the current collector and the overhead equipment have been presented. A model of the system that consists of a discrete subsystem modelling the pantograph and a continuous subsystem modelling the contact wire has been introduced. The dynamic interaction between the discrete oscillator of two degrees of freedom and a continuous string has been studied. At the contact point of the pantograph and the wire a varying component of force appears. It is the source of waves that propagate along the contact wire. There are also two standing sources of wave generation, i.e. varying forces at the points of catenary supports. General results have been illustrated by numerical examples in which the effect of wave propagation is confirmed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effect of stiffness on the transverse and longitudinal motions of a musical string: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The effect of stiffness on the propagation of longitudinal and transverse waves and vibrations in prestretched strings is considered. The contribution of the longitudinal and transverse components to the dynamic load is of the same order. The longitudinal vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena are possible. Low stiffness, which is characteristic for musical strings, leads to a small change in the frequencies of the whole spectrum of transverse and longitudinal vibrations, but to a considerable change in the shape of the string at strike and mounting points and on the transverse wave front.     

Asymptotic behavior of the correlation coefficients of transverse momenta in the model with string fusion

In the framework of the model with fusion of quark–gluon strings on the transverse lattice, we find the asymptotic behavior of the correlation coefficients between observables in separated rapidity intervals with a high string density in a realistic case with an inhomogeneous distribution of strings in the impact parameter plane. We calculate the asymptotic forms for three types of correlations: between the average transverse momenta of particles with rapidity in these intervals, between the average transverse momentum of particles in one rapidity interval and the multiplicity of particles in another, and also between the multiplicities of charged particles in these intervals. We show that the previously found independence of the asymptotic form of the correlation coefficient between the average transverse momenta from the variance in the number of particles produced in string fragmentation holds only in the case of a uniform distribution of strings in the transverse plane. We also show that the found general expressions for the long-range correlation coefficients in the particular case with a uniform distribution of strings in the transverse plane become the formulas previously obtained by another method applicable only in this simple case.     

Waves and vibrations in strings with windings: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The equations of the propagation of transverse, twisting and longitudinal waves and vibrations are obtained, taking into account their interactions in musical strings with windings. Their solutions are obtained. The occurrence of transverse and twisting motions leads to the appearance of longitudinal motions, while the transverse and longitudinal components play the role of inducing forces for the twisting components. The contributions of the transverse, twisting and longitudinal components to the dynamic loading of the string are of the same order. The longitudinal-twisting vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena between the individual modes of these vibrations are possible.     

Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients

In this paper, we study the Riesz basis property and the problem of stabilization of two vibrating strings connected by a point mass with variable physical coefficients under a boundary feedback control acts at one extreme point and Dirichlet boundary condition on the other end. It is shown that the system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. By a detailed spectral analysis, it is proved that this hybrid system is asymptotically stable but not exponentially stable.     

Vibration of a Complex Strings System under a Moving Force

We consider vibrations of an elastically connected double-spring complex system subjected to moving concentrated forces. This system is treated as two strings connected by an elastic layer. The motion of the system is described by two partial differential equations. It is shown that some solutions of the system response can be obtained not only as series but also in closed forms. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new Jacobi rational–Gauss collocation method for numerical solution of generalized pantograph equations

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, a new spectral collocation method is applied to solve the generalized pantograph equation with variable coefficients on a semi-infinite domain. This method is based on Jacobi rational functions and Gauss quadrature integration. The Jacobi rational-Gauss method reduces solving the generalized pantograph equation to a system of algebraic equations. Reasonable numerical results are obtained by selecting few Jacobi rational–Gauss collocation points. The proposed Jacobi rational–Gauss method is favorably compared with other methods. Numerical results demonstrate its accuracy, efficiency, and versatility on the half-line.     

Dynamic analysis and experiment investigation of a cracked dual-disc bearing-rotor system based on orbit morphological characteristics

The paper is aimed to examine dynamic behaviors of a dual-disc bearing-rotor system in multi-fault state, and the crack detection based on the orbit morphological characteristics and vibration responses is proposed. Dynamic response and vibration signal analysis are two significant studies in rotor system. Most researchers have simulated the nonlinear dynamics and analyzed the fault signal using various methods separately. However, the fault feature from vibration signal is tightly connected with the dynamic mechanism in the rotor system, especially in rotor system with coupling multi-fault. In the paper, the dynamic model of the dual-disc bearing-rotor system is established, which takes into account the effects of crack, rub-impact and nonlinear oil-film forces. The vibration responses and the effect of crack on dual-disc rotor system with multi faults are investigated. The existence of crack and the coupling effect of multi faults enrich dynamic behavior of the dual-disc bearing-rotor system, and the response near the 1/2 subcritical speed provides a criterion for crack detection. Experiment investigation is attempted for the first time, which is based on the changes of crack depth and rotation speed for multi-fault dual-disc rotor system. The analysis of the dynamic response and the orbit morphological characteristics from experiment can effectively detect the crack information.     

Ultrasonic wave propagation in a polymerizing medium

It is shown that by monitoring the propagation velocities of longitudinal and shear ultrasonic waves in a polymerizing system it is possible to keep a continuous track on the conversion of the monomer, estimate the effect of various technological factors — sensitizing additives, dose rate, absorbed dose — and trace the behavior of the dynamic moduli of elasticity and the dynamic Poisson's ratio of the polymeric material. Relations between the wavelength and the transverse dimension of the polystyrene specimens that satisfy the conditions for determining the speed of ultrasound in an infinite medium and in a rod are obtained.     

Control and exponential stabilization for the equation of an axially moving viscoelastic strip

The aim through this work is to suppress the transverse vibrations of an axially moving viscoelastic strip. A controller mechanism (dynamic actuator) is attached at the right boundary to control the undesirable vibrations. The moving strip is modeled as a moving beam pulled at a constant speed through 2 eyelets. The left eyelet is fixed in the sense that there is no transverse displacement (see Figure 1 ). The mathematical model of this system consists of an integro‐partial differential equation describing the dynamic of the strip and an integro‐differential equation describing the dynamic of the actuator. The multiplier method is used to design a boundary control law ensuring an exponential stabilization result.     

Elastic waves in regions with multiple interfaces

The purpose of this article is to illustrate how the study of partial differential equations (PDEs) can be made more accessible to undergraduates through the use of applications and computers. The analysis initially concentrates on travelling waves on strings and their reflections from boundaries. Then wave behaviour on strings with multiple interfaces, where incident waves are repeatedly reflected and transmitted, is investigated. Finally, the results are extended to waves in three-dimensional bodies in which longitudinal and transverse waves occur simultaneously. The effectiveness of a computer algebra system (CAS) for visualizing and understanding solutions is demonstrated. In particular, snapshots and sample code for Mathematica animations are provided to demonstrate the physical nature of the solutions. Although intended for PDE students, the contents of this paper are accessible to undergraduates who have studied ordinary differential equations (ODEs) and would be appealing to anyone with a natural curiousity about waves.     

The modeling and dynamic analysis of two jointed beams with clearance

The dynamic behaviors of two beams connected by a joint with clearance are investigated. A new equivalent joint model using the transverse and torsion spring systems is developed to describe the dynamics of two jointed beams with clearance. Based on the finite element method (FEM), the equations of motion of the two jointed beams with clearance are established using Hamilton's principle. A modified numerical solution method is presented based on the Newmark integration method to solve the equations with the nonlinearity due to clearance. The contributions of the clearance size and stiffness on the amplitude-frequency response are discussed. The effects of the clearance on the vibration transfer between the two connected beams and the impact force of the joint are investigated.     

Stability results for impulsive pantograph equations

By employing the Lyapunov functions and Razumikhin technique, some stability results are obtained for pantograph equations with impulses. Our results reveal the fact that certain impulses may make an unstable system stable and that the stability of pantograph equations may also be inherited by impulsive pantograph ones under appropriate impulsive perturbations.     

On the structure of run-maximal strings

We present a combinatorial structure consisting of a special cover of a string by squares. We characterize the covering property of run-maximal strings, i.e. strings achieving the maximal number of runs. The covering property leads to a compression scheme which is particularly efficient for run-maximal strings. It also yields a significant speed improvement in the computer search for good run-maximal string candidates. The implementation of the search and preliminary computational results are discussed.     

Failure detection and reconstruction in switched nonlinear systems

This work presents a novel algorithm for the invertibility of switched nonlinear systems. An invertible switched system is one for which it is possible to reconstruct the entire unknown inputs and the associated evolving mode. In our approach, the dynamic of the inverse systems is given by means of a projection which decomposes the state space into two transverse foliations: one is tangent to the unknown inputs and the other is transverse. Illustrative examples and a missile application are given and discussed in this paper.     

轴向变速运动粘弹性弦线横向振动的复模态Galerkin方法

在考虑初始张力和轴向速度简谐涨落的情况下,利用含预应力三维变形体的运动方程,建立了轴向变速运动弦线横向振动的非线性控制方程,材料的粘弹性行为由Kelvin模型描述．利用匀速运动线性弦线的模态函数构造了变速运动非线性弦线复模态Galerkin方法的基底函数,并借助构造出来的基底函数研究了复模态Galerkin方法在轴向变速运动粘弹性弦线非线性振动分析中的应用．数值结果表明,复模态Galerkin方法相比实模态Galerkin方法对变系数陀螺系统有较高的收敛速度．     

Numerical Treatment of Catenary Systems

Simulation is of great importance in the development of railway cable systems. Mathematical models and numerical methods for the computation of both static equilibria and dynamic oscillations of railroad catenaries are being derived. These cable systems form a complex network of string and beam elements and lead to coupled partial differential equations in space and time where constraints and corresponding Lagrange multipliers express the interaction between carrier, contact wire, and pantograph head. For computing static equilibria, two different algorithms are presented, while the dynamic case is treated by a finite element method in space, combined with stabilized time integration of the resulting differential algebraic system. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Operational matrices of Chebyshev cardinal functions and their application for solving delay differential equations arising in electrodynamics with error estimation

In this paper, a new and effective direct method to determine the numerical solution of pantograph equation, pantograph equation with neutral term and Multiple-delay Volterra integral equation with large domain is proposed. The pantograph equation is a delay differential equation which arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probability, mechanics and electrodynamics. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration, product and delay of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a pantograph equation can be transformed to a system of algebraic equations. An efficient error estimation for the Chebyshev cardinal method is also introduced. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Interference between the transverse and longitudinal vibrations in musical strings

Two problems of the vibrations of strings are considered using the approach described previously in [1]: the vibrations of the string of a plucked musical instrument, drawn out at one of the points and at rest at the initial instant of time (Problem 1), and the vibrations of the string of a keyboard musical instrument, the points of which are given an initial velocity at the initial instant of time by a hammer of small width (Problem 2). It is established that forced longitudinal oscillations of the string occur at frequencies of the transverse vibrations, the condition for possible resonance of the longitudinal vibrations is derived, and the nature of the vibrations at the point where the string is fastened due to elasticity and the related shift in the frequency of transverse vibrations is established.