Random Vibration Analysis Of Linear Continuous Systems Subjected To Moving Random Loads

With trains are becoming increasingly faster and heavier,the environmental vibration induced by moving loads has recently received special attention.Trains travelling at speed present moving loads that are known to excite large amplitude and wide frequency spectrum vibration in the track,in which the vibration can propagate over long distances.Such vibrations can enter buildings through their foundations and affect the operation of sensitive equipment and human comfort.At present,in most moving loads induced environmental vibration problems,the loads are assumed to be constant or harmonic.Actually the moving loads in train-track domain are random in nature,due in part to rail irregularities.The random characteristics of train induced vibration analysis of the track-foundation,to be a random vibration problem of a continuous system induced by moving random excitations.Generally speaking.the main difficulties in the continuous structure vibration problems induced by moving loads can be concluded as:1)Non-stationary random vibration problem.Due to the moving characteristic of the load,even if the load is a stationary random process,the response at any location is a non-stationary random process,which causes the process to obtain the probability and statistical properties of the response very difficult.2)Solution of high-order partial differential equations.On the one hand the dynamic equations of the continuous structure are high-order partial differential equations,on the other hand the random characteristic of the moving load need to be considered,the two obstacles cause the solution to be a formidable task.3)Discussion on the resonance mechanism of continuum.The resonance of continuum due to moving random load is rely on the load’s velocity and frequency characteristics,which brings a richer physical significance.Because of the complexity of the problem,so far the current research results are rare and effective research methods have not yet formed.Due to the aforementioned situation,the initial motivation of this paper is to study the essential mechanism of the problem.In order to study the propagation law of the vibration wave and the mechanism of resonance as well as the characteristics of the response varied with parameters,a stochastic analysis method on the problem of continuous structures induced by moving random load is presented,which is constructed on the dynamic theory of elasticity and random vibration theory.The main works are presented as follows:1.To study the vibration of an infinite long beam rested on a Kelvin foundation under a moving random load,a random vibration analysis method for structural response is proposed by using the Green function method and Fourier transform.Using this method,the analytical solution of the evolving power spectrum of the system can be obtained,which is simple and is convenient to analyse the resonant mechanism and vibration characteristic of the system in the frequency domain,as well as to observe the non-stationary property through the expression.Compared with the Monte Carlo method,the proposed method is highly efficiency.The new concept of the critical velocity of the system induced by moving random load is proposed,furthermore,the resonance mechanism of the system induced by deterministic load,harmonic load and random load is discussed,respectively.The response characteristics caused by the Doppler effect are analysed,furthermore,the effects of the random load’s velocity,structural damping and stiffness characteristics to the response of non-stationary random vibrations are also analysed.2.A general method is proposed to study the stochastic dynamic response of an elastic body,i.e.,the PEM is introduced to deal with the problem of an elastic half-space subjected to a moving random load in frequency domain.According to the field theory,the displacement is decomposed into a non-rotating field and a non-diverging field.The governing equations are transformed into frequency-wavenumber domain,thus the high-order partial differential equations are changed into ordinary differential equations.One solution is to obtain the Green’s function of the system and to construct the pseudo excitation form,then through the general Duhamel’s integral one can obtain the pseudo response of the system.Another solution is to transform the pseudo excitation into frequency-wavenumber domain,then substitute the transformed excitation into the governing equations,thus the obtained response can be changed into physical domain.For the integral form solution,the singularity of the integral is avoided by introducing a material damping,and an adaptive algorithm is adopted to solve the oscillation of the integrand.An important conclusion is proposed that the ground resonance induced by a moving random load depends on the moving velocity and the fluctuation spectrum of the load.The wave propagation differences induced by deterministic and stochastic loads are compared and explained.3.Based on Green’s function,a semi-analytical method is proposed to deal with the problem of continuous structure subjected to multi-point excitations.For fully coherent multi-point excitation,the pseudo excitation form is constructed according to the PEM.The random responses of the system are derived by the Green function method,which can be used in any linear continuous structures.For the half-space,the works are focused on the characteristics including the PSD’s distribution on frequency,the influence of velocity to the PSD,wave propagation of elastic wave in the medium,the influence of wave effect to the response.For the infinitely long beam,the influence of wave effect to the response on various velocities is studied.Numerical results show that when the load velocity is high,the adjacent load will have influence to the neighboring response,so the wave effect must be considered.4.In order to study the dynamic response of a beam-soil structure under a moving random load,a hybrid method based on the PEM,and wavelet theory is proposed.The governing equations of the beam and the soil are converted into frequency-wavenumber domain,and the equations are decoupled using the boundary conditions of the beam and soil at the interface,hence the analytical expression of the power spectrum in the integral form is obtained.The high oscillation of the integral kernel caused by the viscous damping makes the numerical calculation very difficult.The numerical instability is eliminated by the wavelet method,and the numerical results show that the wavelet approach is efficient and accuracy.The influence of viscous damping and shear modulus of the soil on critical and time delay is studied.It is first proposed that the critical velocity of the beam-soil structure under the moving random load is closely related to the compressive wave velocity of the soil.