Infinite Euler-Bernoulli Beam Research Articles

Dynamic response of high-speed rails due to heavy braking

The dynamic response of a high-speed rail experiencing heavy braking is investigated using the moving element method. Possible sliding of train wheels over the rails as the train decelerates is accounted for. The train is modelled as a 14-DOF system comprising a car body, bogies and wheel sets interconnected by spring-damping units. The railway track is modelled as an infinite Euler–Bernoulli beam resting on a two-parameter elastic-damped foundation. A convected coordinate system attached to the moving train is employed in the formulation of the governing equations. The effects of braking torque, coefficient of static friction between wheels and rail, initial train speed and the severity of railhead roughness (track irregularity) on the dynamic response of the high-speed rail, including the occurrence of the ‘jumping wheel’ phenomenon, are examined. The phenomenon describes the momentary loss of contact between the wheel and track. A combination of high braking torque, large static friction coefficient, high initial train speed and severe track condition promotes larger dynamic effects.

Experimental validation of wavelet based solution for dynamic response of railway track subjected to a moving train

New approaches allowing effective analysis of railway structures dynamic behaviour are needed for appropriate modelling and understanding of phenomena associated with train transportation. The literature highlights the fact that nonlinear assumptions are of importance in dynamic analysis of railway tracks. This paper presents wavelet based semi-analytical solution for the infinite Euler–Bernoulli beam resting on a nonlinear foundation and subjected to a set of moving forces, being representation of railway track with moving train, along with its preliminary experimental validation. It is shown that this model, although very simplified, with an assumption of viscous damping of foundation, can be considered as a good enough approximation of realistic structures behaviour. The steady-state response of the beam is obtained by applying the Galilean co-ordinate system and the Adomian׳s decomposition method combined with coiflet based approximation, leading to analytical estimation of transverse displacements. The applied approach, using parameters taken from real measurements carried out on the Polish Railways network for fast train Pendolino EMU-250, shows ability of the proposed method to analyse parametrically dynamic systems associated with transportation. The obtained results are in accordance with measurement data in wide range of physical parameters, which can be treated as a validation of the developed wavelet based approach. The conducted investigation is supplemented by several numerical examples.

Modeling on energy harvesting from a railway system using piezoelectric transducers

Theoretical models of piezoelectric energy harvesting from railway systems using patch-type and stack-type piezoelectric transducers are studied. An infinite Euler–Bernoulli beam on a Winkler foundation subjected to moving multi-loads is adopted to describe the dynamic behavior of railway track. The voltage and electric power of piezoelectric transducers installed at the bottom of a steel rail are derived analytically. Comparisons with earlier works and experimental results are given, indicating that the present solutions are reliable. Additionally, a parametric study is conducted to discuss the effects of axle loads, running velocity and load resistors on the solutions. The numerical results show that patch-type and stack-type piezoelectric transducers can harvest the available energy from track vibration to supply power for a wireless sensor network node and can also serve as sensors to monitor basic train information, such as the running velocity, the location and the axle load. The present investigations provide a theoretical guide in the design of piezoelectric patch and stack energy harvesters used in railway systems, which can serve as power sources for distributed wireless sensor networks in remote areas. The research results also demonstrate the potential of piezoelectric patches and stack harvesters in designing self-powered wireless sensor networks used in railway systems to ensure train operation safety.

Dynamic Response of Railway track using two parameter model

In the present analysis, an infinite Euler–Bernoulli beam of constant cross-section resting on an elastic foundation is considered. The beam and foundation are assumed to be homogeneous and isotropic. The foundation is modeled using two parameters with damping. The beam is subjected to a constant point load moving with a constant speed along the beam. An effort has been made to find the solution of the governing differential equation analytically. It gives beam deflection under moving load in closed form for damped case. At subcritical speed the absolute value of the deflection of the beam increases with increase of the load velocity. Peak maximum deflection appears when the load travels at the critical speed.

Theoretical analysis of piezoelectric energy harvesting from traffic induced deformation of pavements

The problem of energy harvesting using piezoelectric transducers for pavement system applications is formulated with a focus on moving vehicle excitations. The pavement behavior is described by an infinite Bernoulli–Euler beam subjected to a moving line load and resting on a Winkler foundation. A closed-form dynamic response of the pavement is determined by a Fourier transform and the residue theorem. The voltage and power outputs of the piezoelectric harvester embedded in the pavements are then obtained by the direct piezoelectric effect. A comprehensive parametric study is conducted to show the effect of damping, the Winkler modulus, and the velocity of moving vehicles on the voltage and power output of the piezoelectric harvester. It is found that the output increases sharply when the velocity of the vehicle is close to the so-called critical velocity.

Analytical Solution for an Infinite Euler-Bernoulli Beam on a Viscoelastic Foundation Subjected to Arbitrary Dynamic Loads

AbstractAn analytical solution for the dynamic response of an infinite beam resting on a viscoelastic foundation and subjected to arbitrary dynamic loads is developed in this paper. Fourier and Laplace transforms are utilized to simplify the governing equation of the beam to an algebraic equation, so that the solution can be conveniently obtained in the frequency domain. The convolution theorem is employed to convert the solution into the time domain. Final solutions of beam responses investigated are deflection, velocity, acceleration, bending moment, and shear force. The validation of the proposed solution is verified by considering the solutions of several special dynamic loads and comparing the degraded solution to the known results. Further complicated dynamic loads, such as impulsive loads and time-lag loads, are also discussed and analytical solutions are presented. These relationships can be an effective tool for practitioners.

Modelling the Floating Ladder Track Response to a Moving Load by an Infinite Bernoulli-Euler Beam on Periodic Flexible Supports

Abstract.An infinite Bernoulli-Euler beam (representing the “combined rail” consisting of the rail and longitudinal sleeper) mounted on periodic flexible point supports (representing the railpads) has already proven to be a suitable mathematical model for the floating ladder track (FLT), to define its natural vibrations and its forced response due to a moving load. Adopting deliberately conservative parameters for the existing FLT design, we present further results for the response to a steadily (uniformly) moving load when the periodic supports are assumed to be elastic, and then introduce the mass and viscous damping of the periodic supports. Typical support damping significantly moderates the resulting steady deflexion at any load speed, and in particular substantially reduces the magnitude of the resonant response at the critical speed. The linear mathematical analysis is then extended to include the inertia of the load that otherwise moves uniformly along the beam, generating overstability at supercritical speeds – i.e. at load speeds notably above the critical speed predicted for the resonant response when the load inertia is neglected. Neither the resonance nor the overstability should prevent the safe implementation of the FLT design in modern high speed rail systems.

Symplectic analysis of vertical random vibration for coupled vehicle–track systems

A computational model for random vibration analysis of vehicle–track systems is proposed and solutions use the pseudo excitation method (PEM) and the symplectic method. The vehicle is modelled as a mass, spring and damping system with 10 degrees of freedom (dofs) which consist of vertical and pitching motion for the vehicle body and its two bogies and vertical motion for the four wheelsets. The track is treated as an infinite Bernoulli–Euler beam connected to sleepers and hence to ballast and is regarded as a periodic structure. Linear springs couple the vehicle and the track. Hence, the coupled vehicle–track system has only 26 dofs. A fixed excitation model is used, i.e. the vehicle does not move along the track but instead the track irregularity profile moves backwards at the vehicle velocity. This irregularity is assumed to be a stationary random process. Random vibration theory is used to obtain the response power spectral densities (PSDs), by using PEM to transform this random multiexcitation problem into a deterministic harmonic excitation one and then applying symplectic solution methodology. Numerical results for an example include verification of the proposed method by comparing with finite element method (FEM) results; comparison between the present model and the traditional rigid track model and; discussion of the influences of track damping and vehicle velocity.

Vibration and stability of axial loaded beams on elastic foundation under moving harmonic loads

The vibration and stability of an infinite Bernoulli–Euler beam resting on a Winkler-type elastic foundation have been investigated when the system is subjected to a static axial force and a moving load with either constant or harmonic amplitude variations. A distributed load with a constant advance velocity and damping of a linear hysteretic nature for the foundation were considered. Formulations were developed in the transformed field domains of time and moving space, and a Fourier transform was used to obtain the steady-state response to a moving harmonic load and the response to a moving load of constant amplitude. Analyses were performed: (1) to investigate the effects of various parameters, such as the load velocity, load frequency, and damping, on the deflected shape, maximum displacement, and critical values of the velocity, frequency, and axial force, and (2) to examine how the axial force affects the vibration and stability of the system. Expressions to predict the critical (resonance) velocity, critical frequency, and axial buckling force were proposed.

Natural flexural vibrations of a continuous beam on discrete elastic supports

It has been suggested that modern rail systems might exploit so-called floating tracks, to minimise traffic vibration and noise. This paper discusses the transverse deflexion of an infinite Bernoulli–Euler beam mounted on discrete elastic supports, a model considered suitable to explore low-frequency vibrations and associated resonances in such systems. The dynamics is governed by the eigenvalues and eigenvectors of a transfer matrix, which relates the deflexion of any beam span to the deflexions of its neighbours. Important “extensive” contributions, rather than “spatially damped” modes, occur whenever the transfer matrix has one or more eigenvalues of modulus 1. Responses such as the so-called “pinned–pinned resonance” occur when these eigenvalues of modulus 1 are real (i.e., the eigenvalues are ∓1); and further modes corresponding to two complex conjugate eigenvalues coalescing into ∓1 arise at other wavelengths, when the supports are elastic—i.e., in addition to the resonant modes identified in many earlier analyses assuming fixed supports. There is no average energy flux from span to span for any mode defined by a real eigenvector, and we infer that zero-energy transfer between spans is a characteristic of the resonant response of the system to a stationary vibrating source located on some particular span.

Vibrations of a track caused by variation of the foundation stiffness

The paper deals with the stochastic analysis of a single-degree-of-freedom vehicle moving at constant velocity along a simple track structure with randomly varying support stiffness. The track is modelled as an infinite Bernoulli–Euler beam resting on a Kelvin foundation, which has been modified by the introduction of a shear layer. The vertical spring stiffness in the support is assumed to be a stochastic homogeneous field consisting of a small random variation around a deterministic mean value. First, the equations of motion for the vehicle and beam are formulated in a moving frame of reference following the vehicle. Next, a first-order perturbation method is proposed to establish the relationship between the variation of the spring stiffness and the responses of the vehicle mass and the beam. Numerical examples are given for various parameters of the track. The response spectra obtained from the perturbation analysis are compared with the numerical solution, in which finite elements with transparent boundary conditions are used. The circumstances, under which the first-order perturbation approach provides satisfactory results, are discussed.

Vehicle Moving Along an Infinite Beam With Random Surface Irregularities on a Kelvin Foundation

The paper deals with the stochastic analysis of a single-degree-of-freedom vehicle moving at a constant velocity along an infinite Bernoulli-Euler beam with surface irregularities supported by a Kelvin foundation. Both the Bernoulli-Euler beam and the Kelvin foundation are assumed to be constant and deterministic. This also applies to the mass, spring stiffness, and damping coefficient of the vehicle. At first the equations of motion for the vehicle and beam are formulated in a coordinate system following the vehicle. The frequency response functions for the displacement of the vehicle and beam are determined for harmonically varying surface irregularities. Next, the surface irregularities are modeled as a random process. The variance response of the mass of the vehicle as well as the displacement variance of the beam under the oscillator are determined in terms of the autospectrum of the surface irregularities.

CONTROL OF FLEXURAL WAVES ON A BEAM USING A TUNABLE VIBRATION NEUTRALISER

This paper describes an analytical and experimental investigation into the use of a tunable vibration neutraliser to control the transmission of flexural propagating waves on an infinite Euler-Bernoulli beam. The paper investigates the way in which the physical properties of the neutraliser affect the attentuation of an incident flexural wave, and the frequency at which the neutraliser should be tuned to in order to achieve the maximum attentuation of this wave. Expressions are derived for the attentuation, the tuned frequency and the bandwidth of the device. A simple control parameter is also proposed that uses two signals measured with accelerometers on the beam at the base of the neutraliser. This facilitates a compact control device. The theoretical predictions are validated by a simple experiment.

On Continuous Subsystem Modelling in the Dynamic Interaction Problem of a Train-Track-System

SUMMARY The paper deals with the dynamics of a periodic structure by means of which such a mechanical system as a railway track can be modelled. The structure considered consists of an infinite Bernoulli-Euler beam resting on periodically spaced spring-damper-mass elements. At first, the case of free wave propagation in a pure elastic structure is considered. Then the steady-state system dynamic response to a moving harmonic force is studied. In both cases a procerure is used which bases on Floquet's theorem. The analysis of dispersion relations makes it possible to determine passing and stopping bands and the eigenforms corresponding to cut-off frequencies. A simple way of calculating critical velocities and frequencies of the load is proposed ana a physical explanation for the increase of wave amplitudes is given.

The motion of a carriage with constant velocity along a beam of infinite length resting on a base with two elastic characteristics

The vertical oscillations of an infinite Bernoulli-Euler beam resting on a viscous inertial base with two elastic characteristics under the action of a deforming carriage which is continuously moving at a constant velocity is considered. The carriage, consisting of a system of rigid bodies with viscoelastic couplings between them, makes contact with the beam via viscoelastic springs at a finite number of points which allows the small vertical oscillations of the elements of the carriage to be described by a system of ordinary differential equations with constant coefficients. A technique is proposed for obtaining the asymptotic forms of the solution at long times. The asymptotic forms of the solutions are presented and they are analysed for certain types of loads.

Mean-square response of an infinite Bernoulli-Euler beam to nonstationary random excitation

This paper presents expressions for the mean-square response of an infinite length Bernoulli-Euler beam, including linear viscous damping and a Winkler foundation, subjected to weakly nonstationary random excitations. The method used is the time domain approach, incorporating impulse response functions and the convolution integral. Results are compared to published solutions for the mean-square response of a finite simply supported beam analyzed using the modal approach. The comparison indicates that the infinite length beam response rise and decay, and time to reach maximum, differs significantly from that of the finite length beam.

Dynamic Response of an Infinite Beam

Both the exact and an approximate solution for the dynamic response of an infinite Bernoulli-Euler beam under an instantaneously applied, concentrated load are presented in this paper. The exact solution is obtained by means of complex Fourier transforms. The approximate solution is obtained by assuming the dynamic response has the form of a deflected infinite beam on an elastic foundation, with wavelength a function of time. This assumption is motivated by the similarity between the dynamic response problem and the problem of an infinite beam on an elastic foundation. A governing equation for the wavelength in the assumed response is derived by application of the principle of conservation of energy, and solved by straightforward methods. A comparison of the two solutions shows good agreement near the point of loading. Results applicable to pipe whip problems are presented.

On the dynamic response of an infinite Bernoulli-Euler beam

This paper presents a theory of the transient Bernoulli-Euler beam problem on an elastic foundation which takes into account the effects of axial load and linear damping. An analytical solution of the steady state and the transient components has been obtained due to physically realistic load distributions. With a view to extend its practical applicability, the characteristic features of the solution are explored. Several limiting situations are investigated as special cases. It is shown that the steady state vibration can be achieved as the limit of the solution of the transient problem.

On the pressure under an infinite beam resting on an elastic half-space

The equivalent stiffness of a saturated poroelastic halfspace that supports an infinite Euler-Bernoulli beam under a moving point load has been investigated in this paper. The smooth contact condition is assumed for the interface of beam and halfspace, however, with an improved continuity condition, i.e. the continuities of contact normal stress and contact vertical displacements are imposed across the width of the beam. The equivalent stiffness of the halfspace has been evaluated using a contour integration approach such that contributions of the Rayleigh pole and the dispersion branches can be explicitly taken into account. Influences of the continuity condition and the permeability on the equivalent stiffness have been studied. It is found that the imaginary part of the equivalent stiffness is nonzero since the viscous coupling between two phases of the saturated halfspace has been considered. This observation fundamentally differentiates the present paper to the work by Shi and Selvadurai (2016) where an infinite permeability has been assumed for the halfspace, i.e. null viscous coupling.