An iterative method for non-stationary response analysis of non-linear random systems

A method for calculating the non-stationary response of non-linear systems subjected to random excitations is formulated. The time-dependent equivalent linear system is considered and an iterative procedure for evaluating the non-stationary mean-square responses is developed. Several examples are presented and applicability of the technique is illustrated.     

Non-stationary response of a beam to a moving random force

The non-stationary random vibration of a beam is investigated. The beam is subjected to a random force with constant mean value which is moving with constant speed along the beam. The statistical characteristics of the first and second order for the deflection and bending moment of the beam are computed by using the correlation method. The numerical results of the coefficient of variation of the deflection at beam span mid-point are given for five basic types of convariances of the force (white noise, constant, exponential cosine, exponential, and cosine wave). The effect of the speed of the movement of the force along the beam as well as the effect of the beam damping is investigated in detail. It is concluded that the resulting beam vibration turns out to be a non-stationary process even though the motion considered is that of a stationary random force.     

The use of moment equations for calculating the mean square response of a linear system to non-stationary random excitation

The moment equations approach is used to calculate the mean square response of a linear system to non-stationary random excitation which is expressed as a product of a deterministic envelope function and a Gaussian stationary non-white noise. The moment equations are derived by performing single integrations in the time domain and are solved numerically by digital computer. Numerical examples are given for the response of single and two degree-of-freedom systems which are excited by noise with an exponentially decaying harmonic correlation function. It is shown that an overshoot, in the sense that the transient response exceeds its stationary value, may occur even in the case of an exponential envelope function, but that the response does not exhibit overshoot when the natural frequency of the system is almost coincident with the dominant frequency of the input.     

Dynamic response of a pre-stressed cylindrical shell to a moving load

The work presented in this paper is concerned with the response of a pre-stressed, finite, thin circular cylindrical shell under a moving local load with a constant velocity. An analysis is carried out by a dynamic method, and the solutions which are bounded even at the critical velocity are obtained. The effects of the initial stresses on the dynamic responses of the displacement and the stresses are examined in connection with the velocity of the load.     

Dynamic response of a poroelastic stratum to moving oscillating load

The dynamic response of a poroelastic stratum subjected to moving load is studied. The governing dynamic equations for poroelastic medium are solved by using Fourier transform. The general solutions for the stresses and displacements in the transformed domain are established. Based on the general solutions, with the consideration of boundary conditions, the final expressions of stresses and displacements in physical domain are put forward for the three-dimensional single-layer medium. Some numerical solutions for the stresses, displacements and pore fluid pressure are presented and reveal that the response of a poroelastic stratum varies obviously with the moving velocity.     

Stationariness provided by filtration to a periodic non-stationary random process

The concept of generalized spectral density is used for evaluating the variance of the response of a simple linear oscillator excited by periodic non-stationary random functions. The generalized Wiener-Khintchine relations are slightly reshaped, so that the generalized spectral density of the excitation function could be expressed as the sum of Dirac delta functions, whose coefficients could be given in the form of closed analytical expressions. The results show that, apart from a few exceptions, the response of a lightly damped system is practically stationary. It is noted in general that filtration brings about stationarization.     

Vibration of a beam due to a random stream of moving forces with random velocity

The problem of dynamic response of a beam to the passage of a train of concentrated forces with random amplitudes and velocities is considered. Force arrivals at the beam are assumed to constitute the point stochastic process of events. Thus, the excitation process is an idealization of vehicular traffic loads on a bridge. An analytical technique is developed to determine the response of the beam. Explicit expressions for the expected value and the variance of the beam deflection are provided. As an example, the response of a beam to a stationary stream of forces is determined for some practical situations, and discussed.     

Dynamic response of a prestressed,orthotropic thick plate strip to a moving line load

This paper is a study of the steady state response of an orthotropic plate strip to a moving line load. The plate is of infinite length and subjected to initial in-plane stresses parallel and perpendicular to the edges. The solution is obtained on the basis of a thick plate theory which takes into account the effects of shear deformation and rotatory inertia. The critical speed of the load which brings about a resonance effect in the system is determined. Further, the bending moment in the plate is calculated for several values of the load speed and the initial stress parameters and shown graphically as a function of the space variable moving with the load.     

Steady state response of an infinite beam on a viscoelastic foundation with moving distributed mass and load

Compared with the moving concentrated load model, it is more realistic and proper to use the moving distributed mass and load model to simulate the dynamics of a train moving along a railway track. In the problem of a moving concentrated load, there is only one critical velocity, which divides the load moving velocity into two categories: subcritical and supercritical. The locus of a concentrated load demarcates the space into two parts: the waves in these two domains are called the front and rear waves,respectively. In comparison, in the problem of moving distributed mass and load, there are two critical velocities, which results in three categories of the distributed mass moving velocity. Due to the presence of the distributed mass and load, the space is divided into three domains, in which three different waves exist. Much richer and different variation patterns of wave shapes arise in the problem of the moving distributed mass and load. The mechanisms responsible for these variation patterns are systematically studied. A semi-analytical solution to the steady-state is also obtained, which recovers that of the classical problem of a moving concentrated load when the length of the distributed mass and load approaches zero.     

A restoration procedure for (non-stationary) signals from moving sources

Moving sources, such as trains, cars and airplanes, provide non-stationary sound and vibration signals in situations where the receiver is not moving with the source. For non-stationary signals there are strong limitations on the use of computerized analysis techniques based on Fourier transformation. For instance, it is not possible to compute reliably either power spectral density functions or coherence functions. A procedure has been developed, and is discussed in this paper, that restores non-stationary signals into stationary ones, thus enabling one to apply the analysis techniques mentioned above to moving source data with a reliable outcome.     

Dynamic response of an annular disk to a moving concentrated,in-plane edge load

In-plane dynamic behaviour of a thin annular disk with a clamped inner boundary is analyzed. The frequencies of free in-plane vibration with a free outer boundary are first evaluated, by using Lamé potentials, for various radius ratios ranging from 0.2 to 0.8. The steady state dynamic stresses induced by a concentrated load moving at a constant angular speed at the outer boundary are then evaluated through a Galilean transformation. Results are presented for a radius ratio of 0.5.     

Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load

The present paper investigates the convergence of the Galerkin method for the dynamic response of an elastic beam resting on a nonlinear foundation with viscous damping subjected to a moving concentrated load. It also studies the effect of different boundary conditions and span length on the convergence and dynamic response. A train–track or vehicle–pavement system is modeled as a force moving along a finite length Euler–Bernoulli beam on a nonlinear foundation. Nonlinear foundation is assumed to be cubic. The Galerkin method is utilized in order to discretize the nonlinear partial differential governing equation of the forced vibration. The dynamic response of the beam is obtained via the fourth-order Runge–Kutta method. Three types of the conventional boundary conditions are investigated. The railway tracks on stiff soil foundation running the train and the asphalt pavement on soft soil foundation moving the vehicle are treated as examples. The dependence of the convergence of the Galerkin method on boundary conditions, span length and other system parameters are studied.     

Intensity fluctuations in a moving random medium

We consider the intensity fluctuations arising when a point source of radiation moves in a randomly inhomogeneous scattering medium. The medium itself can also move with a velocity whose component normal to the direction of propagation can have an arbitrary distribution. We derive an expression for the space–time autocorrelation function of the intensity fluctuations transverse to the direction of propagation. The result is analysed for some particular cases and it is shown how the resulting information can be useful in examining the behaviour of random media in situations of practical interest.     

Intensity fluctuations in a moving random medium

We consider the intensity fluctuations arising when a point source of radiation moves in a randomly inhomogeneous scattering medium. The medium itself can also move with a velocity whose component normal to the direction of propagation can have an arbitrary distribution. We derive an expression for the space-time autocorrelation function of the intensity fluctuations transverse to the direction of propagation. The result is analysed for some particular cases and it is shown how the resulting information can be useful in examining the behaviour of random media in situations of practical interest.