The steady vibrations and resistance of a railway track to the uniform motion of an unbalanced wheel

A model of a railway track, in the form of an infinite Timoshenko beam resting on equally spaced massive visco-elastic supports, is considered. Steady vertical vibrations of the track due to a harmonic force moving along it at a constant velocity are investigated. The vertical displacement of the track is represented in a moving system of coordinates by a generalized Fourier series. The steady vertical vibrations of a massive rigid wheel rolling along the track at a constant velocity and loaded by a vertical harmonic force are investigated. The track-wheel interaction force is expressed as a generalized Fourier series whose coefficients are determined using an equality relating the vertical displacements of the wheel and the track. Vibrations of the wheel due to centrifugal force and periodic changes in the track parameters are considered. Parametric vibrations of a wheel moving at a constant velocity under a static load due to periodic variation in the stiffness of the track are investigated. The force with which the track resists the uniform motion of an unbalanced wheel is computed.     

The dynamics of a caterpillar drive

The steady-state vertical oscillations of an infinite horizontal periodic chain of identical uniform solid rods joined by hinges and supported by a uniform viscoelastic base are studied. The oscillations are excited by a vertical complex harmonic force that moves along the chain at a constant speed. They are assumed to be steady oscillations in the following sense: a shift along the chain by a distance equal to the length of a rod causes the complex vertical deflection of the chain to be delayed for a time, during which the complex harmonic force is displaced by this distance. The problem is solved using a Fourier transform under the assumption that each point of the chain is in a quiescent state long before the approach of the harmonic force, and returns to the same state under the action of the viscous forces of the base after the harmonic force has withdrawn to infinity.     

Diffraction of a monochromatic plane shear wave on a reinforced cylindrical cavity in an elastic half-space

This article examines a boundary-value problem concerning the diffraction of a monochromatic plane shear wave on a reinforced cylindrical cavity in an elastic half-space. It is assumed that longitudinal shear stresses are absent and that the normal displacements over the entire boundary are specified. Through the use of a special form of the Lamé representation in cylindrical coordinates, the problem is reduced to the determination of scalar functions which satisfy the Helmholtz equation. The coefficients of the Fourier expansions of these functions in the angular coordinate are written as the sum of Fourier and Weber integrals. The densities of these integrals are determined exactly. A specific example is examined.Translated from Dinamicheskie Sistemy, No. 5 pp. 42–49, 1986.     

On a particular form of stabilized capillary-gravitational waves of finite amplitude at the surface of fluid over an undulating bed : PMM vol. 37, n6, 1973, pp. 1020–1034

The problem of stabilized plane capillary-gravitational waves of finite amplitude at the surface of a stream of perfect incompressible fluid flowing over an undulating bed and subjected to pressure periodically distributed along the surface and defined by some infinite trigonometric series is considered. The intersection of the bed with a vertical plane is assumed to be a periodic curve, called the bed line, defined by some infinite trigonometric series. The problem is rigorously formulated and reduced to the solution of a system of nonlinear integral and transcendental equations. The solution is constructed in the form of series in powers of a small dimensionless parameter to which amplitudes of the first harmonics of the bed line and of the surface pressure wave are proportional. An approximate equation is derived for the wave profile.The particular case is considered, when the length of the bed line wave arc is equal to the length of the stabilized free wave line corresponding to the specified flow velocity over a horizontal flat bed and constant pressure along the surface. In such case the parameter of the integral equation is equal to one of the eigenvalues of the kernel of that equation and the solution is constructed in the form of series in powers of the cube root of the small parameter mentioned above.A similar problem but for constant pressure along the surface was considered by the author in [1, 2] and in his paper presented at the 13-th International Congress on Theoretical and Applied Mechanics (Moscow, 1972 [3]).Another similar problem of capillary-gravitational waves over an undulating bed was considered in [4], where besides the topological proof of the existence and uniqueness of solution the algorithm for constructing the latter is given, but the calculation of approximations is only outlined and the mechanical meaning of solution is not investigated in depth.Unlike in [4] the equation of the bed line and the expression for pressure at the surface are specified here in a form which makes it possible to express any approximations in the form of finite sums, and an analysis of the fundamental system of nonlinear integral and transcendental equations by the LiapunovSchmidt analytical methods and their developments is presented.     

A model of a reinforced tyre with a rod protector

A tyre design consisting of a steel-cord-reinforced rigid bond with sides connected to the wheel disc and a protector(tread) in contact with the road is examined. The tread is in the form of a set of rods connected by one end to the band, with the other end either free or in contact with the road. The rod end in contact with the road is acted upon by a force applied from the road, represented by a force normal to the road plane and a shear force due to dry friction. If the modulus of the shear force does not exceed the magnitude of the normal force multiplied by the dry friction coefficient, there is no slip at the contact point. In the opposite case, the rod end will be displaced along the road by an amount sufficient to distribute the normal and shear forces. The dynamics of longitudinal and transverse strains of the rods in contact with the road is analysed using the motion separation method in the quasi-static approximation. The behaviour of the tread rods as a function of the vertical displacement of the wheel centre is investigated, the contact area is found and the conditions are determined under which the contact area is divided into parts in which either slip of the rod ends occurs or does not occur, depending on the magnitude of the longitudinal displacement of the wheel centre or its turning relative to the horizontal axis. An analogue of a continuous model of a rod-like tread is considered, and the magnitudes of the forces and moments are found as a function of the wheel disc displacements. The equations of wheel rolling are obtained, and the conditions under which steady motions exist are found.     

On the transition mode characterizing the triggering of a vibrator in the subsonic boundary layer on a plate

The problem of the development of two-dimensional linear perturbations in a boundary layer, generated by the triggering of a vibrator, is considered. Fourier transformations in the longitudinal coordinate and a Laplace transform in time are used to construct the solution. The inverse transforms are evaluated for large values of the characteristic time t and all values of the longitudinal coordinate x. Domains located downstream of the vibrator are studied in the first of which the perturbations will have the form of Tollmien-Schlichting waves that go over into a wave packet in the second domain. The identity in the structure of the wave packets, which are orthonormalized to the maximum amplitude for this packet for different frequencies of vibrator oscillation is noted.     

Interaction of a Solitary Wave with a Porous Elliptical Cylinder

In this paper the diffraction of a plane first order solitary wave by a vertical permeable elliptical cylinder is investigated and the corresponding wave loading is computed. The analytical model presented here is employs Boussinesq equations to describe the diffraction. The solution procedure is based on the perturbation theory, where a perturbation parameter defined in terms of the surface geometry of the cylinder is used. The flow through the porous cylinder is assumed to obey Darcy's law. The effects of porosity, relative wave length and the incident wave angle are discussed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

参数激励圆柱形容器中的非线性Faraday波

在柱坐标系下,通过奇异摄动理论的多尺度展开法求解势流方程,研究了垂直强迫激励圆柱形容器中的单一模式水表面驻波模式。假设流体是无粘、不可压且运动是无旋的,在忽略了表面张力的影响下,用两变量时间展开法得到一个具有立方项以及底部驱动项影响的非线性振幅方程。对上述方程进行了数值计算,计算的结果显示了在不同驱动振幅和驱动频率下,会激发不同自由水表面驻波模式,从等高线的图像来看,和以往的实验结果相当吻合。     

Non-linear oscillations of a satellite at 1:1:1 resonance

The motion of a satellite about its centre of mass in a central Newtonian gravitational field is investigated. The satellite is considered to be a dynamically symmetrical rigid body. It is assumed that the ratio of the polar and equatorial principal central moments of inertia of the satellite is 4/3, or close to this. The orbit of the centre of mass is elliptic, and the orbit eccentricity is assumed to be small. In the limit case, when the orbit of the centre of mass is circular, a steady motion exists (corresponding to relative equilibrium of the satellite in the orbital system of coordinates) in which the axis of dynamic symmetry is directed along the velocity vector of the centre of mass of the satellite; here, the frequencies of the small linear oscillations of the axis of symmetry are equal or close to one another. But in an elliptic orbit of small eccentricity, multiple 1:1:1 resonance occurs in this case, as the oscillation frequencies mentioned are equal or close to the frequency of motion of the centre of mass of the satellite in orbit. The non-linear problem of the existence, bifurcations and stability of periodic motions of the satellite with a period equal to the rotation period of its centre of mass in orbit is investigated.     

The motion of a railway wheelset

The rolling of a railway wheelset along rails without slipping is investigated taking the creep hypothesis into account. The wheelset is represented by two cones that have a common base, and the rails are represented by two circular cylinders with parallel axes. The kinematic characteristics of the unperturbed rolling motion of the wheelset, which occurs when the centre of mass moves along a straight line, and of the perturbed motion, which occurs when the centre of mass of the wheelset describes a sinusoidal trajectory, are determined. The constraint reactions are found for the motions investigated up to small second-order values of the perturbed variables. When the elastic properties of the material in the contact area are taken into account, the creep hypothesis is used, averaging over the fast variables is employed, and the value of the critical speed, above which the rectilinear rolling of the wheelset becomes unstable, is found using averaged equations. In the latter case a periodic mode with two time intervals when the wheel flanges come into contact with the rails is investigated. The reaction force, the work of the dry friction force, and the moment of the active forces needed to maintain the periodic mode are found at the flange/rail contact point within the dry friction model. The boundaries of the stability regions, the parameters of the periodic mode and the moment of the resistance forces as functions of the problem parameters are determined from the formulae obtained by analytical methods.     

Rolling Contact Problem with the Generalized Coulomb Friction

This paper deals with the numerical solution of the wheel - rail rolling contact problems. The unilateral dynamic contact problem between a rigid wheel and a viscoelastic rail lying on a rigid foundation is considered. The contact with the generalized Coulomb friction law occurs at a portion of the boundary of the contacting bodies. The Coulomb friction model where the friction coefficient is assumed to be Lipschitz continuous function of the sliding velocity is assumed. Moreover Archard's law of wear in the contact zone is assumed. This contact problem is governed by the evolutionary variational inequality of the second order. Finite difference and finite element methods are used to discretize this dynamic contact problem. Numerical examples are provided. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Calculation of pressure in the linear problem of vibrator in a supersonic boundary layer : PMM vol. 43, no. 6, 1979, pp. 1014–1028

The supersonic flow over a body consisting of a triangular oscillating plate the vibrator— mounted between two flat plates is investigated. The body is assumed to be thermally insulated, and the vibrator dimensions and the oscillation frequencies to be such that the flow can be defined by equations of a boundary layer with self-induced pressure [1 — 5]. The oscillation amplitude is assumed small so that these equations can be linearized. The Fourier transform of the longitudinal coordinate is used for solution derivation. The inverse Fourier transform is obtained by numerical methods. It is shown that the perturbations of flow parameters induced by the vibrator are damped upstream and downstream in accordance with an exponential law.     

Wave parameter identification problem for ocean test structure data,part 1, continuous formulation

This paper deals with the solution of the wave parameter identification problem for ocean test structure data. A continuous formulation is assumed. An ocean test structure is considered, and wave elevation and velocities are assumed to be measured with a number of sensors. Within the frame of linear wave theory, a Fourier series model is chosen for the wave elevation and velocities. Then, the following problem is posed: Find the amplitudes of the various wave components of specified frequency and direction, so that the assumed model of wave elevation and velocities provides the best fit to the measured data. Here, the term best fit is employed in the least-square sense over a given time interval.At each time instant, the wave representation involves three indexes (frequency, direction, instrument); hence, three-dimensional arrays are required. This formal difficulty can be avoided by switching to an alternative representation involving only two indexes (frequency-direction, instrument); hence, standard vector-matrix notation can be used. Within this frame, optimality conditions are derived for the amplitudes of the assumed wave model.Numerical results are presented. The effect of various system parameters (number of frequencies, number of directions, sampling time, number of sensors, and location of sensors) is investigated in connection with global or strong accuracy, local or weak accuracy, integral accuracy, and condition number of the system matrix.From the numerical experiments, it appears that the identification problem has a unique solution if the number of directions is smaller than or equal to the number of sensors; it has an infinite number of solutions otherwise. In the case where a unique solution exists, the condition number of the system matrix increases as the size of the system increases, and this has a detrimental effect on the accuracy. However, the accuracy can be improved by proper selection of the sampling time and by proper choice of the number and location of the sensors.This work was supported by Exxon Production Research Company, Houston, Texas. This paper is based partly on Ref. 1–4.     

Diffraction of Plane Electromagnetic Waves on a Small Obstacle Inside a Layer

The problem on the diffraction of the electromagnetic plane wave on a small obstacle in a layer is investigated. The obstacle is assumed to be a circular cylinder and its radius is small in comparison with the length of the incident wave. It is proved that the small obstacle radiates as a linear source. Its intensity is proportional to the area of a cross-section and the jumps of the dielectric and magnetic constants on the interfaces. Bibliography: 4 titles.     

Control of the rolling of a wheel by changing its imbalance

The motion of a wheel (rolling without slipping) in a vertical (longitudinal) plane is considered. The load, which is modelled as a point mass, can be moved within the wheel in a hollow which is positioned along its diameter. By moving the load from one end of the hollow to the other in a specific way, it is possible to manage the rolling of the wheel. An algorithm is constructed for control of the position of the load using which the load is periodically moved from one end of the hollow to the opposite end while the wheel rolls (non-uniformly) over the surface.     

Effect of Hall currents on interaction of pulsatile and peristaltic transport induced flows of a particle-fluid suspension

The interaction of purely periodic mean flow with a peristaltic induced flow is investigated within the framework of a two-dimensional analogue. The mathematical model considers a viscous incompressible fluid under the effect of transverse magnetic field, taking into account the effect of Hall currents for a magneto-fluid with suspended particles between infinite parallel walls on which a sinusoidal traveling wave is imposed. A perturbation solution to the complete set of Navier-Stokes equations is found for the case in which the frequency of the traveling wave and that of the imposed pressure gradient are equal. The ratio of the traveling wave amplitude to channel width is assumed to be small. For this case a first order steady flow is found to exist, as contrasted to a second order effect in the absence of the imposed periodic pressure gradient. The effect of Hall parameter, Hartmann number and the various parameters included in the problem are discussed numerically.     

The motion of a ball on a grassy lawn

The motion of a ball of constant spherical shape on a grassy lawn represented by a set of deformable rods with one end rigidly attached to a plane, while the other end is free and can contact the ball, is investigated. The contact area on the ball surface is represented by a spherical sector, whose dimensions and location on the ball surface depend on the position of the centre of the ball and its orientation. The projection of the contact area onto the plane is a circle, whose centre and radius vary as the centre of the ball moves. When the rods come into contact with the ball, they experience longitudinal and flexural strains, as well as impact actions on the boundary of the contact area. The translational–rotational motion of a ball on a lawn of rods without friction, as well as motion with viscous friction at the points of contact of the ends of the rods with the ball, is considered.     

车灯光源的优化设计

通过建立直角坐标系,利用线、面方程得出以下结论:1.根据由光源发射的辐射能均匀分布在以点光源为中心的球形波阵面上,把所求的总功率之比转化为以线光源一点F′为球心,|F′B|为半径(B为灯口边缘最上的顶点)的球面被灯口所截得的两个球冠面积之比,经计算得直射光总功率反射光总功率≈0.690683.2.直射光的亮区是由圆滑动而成.3.光源放在车灯焦点处效果最佳.     

半无穷长圆管内的低雷诺数入口流

1969年Lew及Fung[1]计算了圆管内的低雷诸数入口流.1982年Dagan等人[2]得到了有限长圆柱形孔道内蠕动流的级数解.[1]中所得的数值解实质上代表有限长圆管内的低雷诺数入口流,因为一般解中的富氏积分已用富氏级数代替.本文直接计算富氏积分,更精确地求出了真正的半无穷长圆管内Stokes入口流的速度分布,压力分布以及流函数,与此对应的入口段长度为圆管半径的1.2倍,接近于Lew及Fung得到的结果1.3倍.此外,本文还研究了配置法的收敛性,证明了此法在入口流问题中具有很好的收敛性,因此可以在其他类似的问题中采用.     

Scattering of Plane Elastic Waves on a Small Obstacle Inside a Layer

The problem on the diffraction of a plane elastic wave with horizontal polarization (SH wave) on a small obstacle placed in a layer is investigated. The layer is situated on a half-space with stress-free boundary. The obstacle is assumed to be a circular cylinder with radius small in comparison with the length of the incident wave. The polarization of the incident wave is parallel to the axis of the cylinder. It is proved that the small inhomogeneity radiates as a linear source such that the intensity of the radiation is proportional to the area of the cross-section of the obstacle and the jump of the squared transverse velocities in the layer and in the obstacle. Bibliography: 5 titles.