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 random response of a finite cable

Numerous problems of current concern involve the designs of aerodynamic systems which either travel at high speeds or contain structural elements which are excited by moving pressure fluctuations. In a number of recent papers responses of dynamic systems to random excitation have been considered. The appropriate theory for calculating the mean square response of linear systems to both stationary and non-stationary random excitation is well known [1–7]. In this paper, the mean square response of a finite cable to non-stationary random excitation is considered. The non-stationary random excitation is of the form s(t) = e(t)α(t), where e(t) is a well defined envelope function and α (t) is the Guassian, narrow band, stationary part of the excitation which has zero mean. Both the unit step and rectangular step functions are used for the envelope function, and both white noise and noise with an exponentially decaying harmonic correlation function are used to prescribe the statistical property of the excitation. The results obtained are shown to be a complete expression for the mean square response when checked for accuracy by reduction to expressions previously obtained by Lyon [4]. It is felt that these results will aid the design of both linear and two-dimensional aerodynamic systems excited by random pressure fluctuations.     

The non-stationary freefield response for a moving load with a random amplitude

This paper presents a stochastic solution procedure for the calculation of the non-stationary freefield response due to a moving load with a random amplitude. In this case, a non-stationary autocorrelation function and a time-dependent spectral density are required to characterize the response at a fixed point in the freefield. The non-stationary solution is derived from the solution in the case of a moving load with a deterministic amplitude. It is shown how the deterministic solution can be calculated in an efficient way by means of integral transformation methods if the problem geometry exhibits a translational invariance in the direction of the moving load. A key ingredient is the transfer function between the source and the receiver that represents the fundamental response in the freefield due to an impulse load at a fixed location. The solution in the case of a moving load with a random amplitude is formulated in terms of the double forward Fourier transform of the non-stationary autocorrelation function. The solution procedure is illustrated with an example where the non-stationary autocorrelation function and the time-dependent standard deviation of the freefield response are computed for a moving harmonic load with a random phase shift. The results are compared with the response in the deterministic case.     

Approximate,time domain,non-stationary analysis of stochastically excited,non-linear systems with particular reference to the motion of vehicles on rough ground

An approximate state-space method for obtaining the time varying mean and covariance of non-linear systems excited by non-stationary random processes is presented. In particular the class of non-stationarity associated with the motion of a vehicle on rough ground (i.e., the process is “frequency modulated” as a result of the vehicle's variable velocity) is of interest. The method is based on a technique of modelling the input process as a “shaping filter” in the spatial domain which may be linked to the vehicle dynamic equations through the velocity function. The non-linear problem is overcome by using the technique of statistical linearization. An example is briefly discussed.     

Time-dependent variance and covariance of responses of structures to non-stationary random excitations

In this paper techniques for the analysis of non-stationary random responses of linear structures, discretized by the finite element method so that they can be analyzed as multi-degree of freedom systems, subjected to non-stationary random excitation are developed. The non-stationary random excitation is represented as a product of (a) an exponentially decaying function and a white noise process, and (b) a modulating function in the form of an exponential envelope and a white noise process. Closed form expressions for the time-dependent variance and covariance of response of structures are presented. Application of these expressions is made for the analysis of non-stationary random responses of a physical model of a class of mast antenna structures subjected to base excitation. It is concluded that (a) the coupling terms do have a definite influence on the response; the magnitude of the influence is proportional to the amount of damping in the structure and proximity of the modes excited; (b) the non-stationary random excitations considered are general in that the modulating functions are not necessarily identical, and therefore the influence of various modulating functions of the excitations applied to different locations of the structure on responses can be examined quantivatively; and (c) for a given damping parameter the magnitudes of the modulating function parameters cannot be chosen arbitrarily though the shapes of normalized modulating functions can be selected to best fit the excitation realizations.     

Non-stationary responses of a non-symmetric non-linear system subjected to a wide class of random excitation

The non-stationary variance and mean responses of a single-degree-of-freedom mechanical system with non-symmetric non-linearities subjected to a wide class of random excitation are investigated. For non-white excitation a modified equivalent linearization technique is proposed, in which the equivalent linear system is subjected to the equivalent forcing function with shifted mean. The results obtained by the present method are compared with the corresponding digital simulation results. The general form is provided for a multi-degree-of-freedom non-linear system with non-symmetric non-linearity.     

The covariance response of linear systems to locally stationary random excitation

A simple method is given for calculating the covariance response of linear, time-invariant systems to random excitation processes which are locally stationary, or approximately so. As an illustration, the method is used to estimate the response of an idealized model of a ten-storey building to non-stationary ground acceleration; the accuracy of the estimated response is assessed by a comparison with the results of a less approximate, but lengthier, general calculation method, previously published.     

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.     

Transient vibration phenomena in deep mine hoisting cables. Part 1: Mathematical model

The classical moving co-ordinate frame approach and Hamilton's principle are employed to derive a distributed-parameter mathematical model to investigate the dynamic behaviour of deep mine hoisting cables. This model describes the coupled lateral-longitudinal dynamic response of the cables in terms of non-linear partial differential equations that accommodate the non-stationary nature of the system. Subsequently, the Rayleigh-Ritz procedure is applied to formulate a discrete mathematical model. Consequently, a system of non-linear non-stationary coupled second order ordinary differential equations arises to govern the temporal behaviour of the cable system. This discrete model with quadratic and cubic non-linear terms describes the modal interactions between lateral oscillations of the catenary cable and longitudinal oscillations of the vertical rope. It is shown that the response of the catenary-vertical rope system may feature a number of resonance phenomena, including external, parametric and autoparametric resonances. The parameters of a typical deep mine winder are used to identify the depth locations of the resonance regions during the ascending cycles with various winding velocities.     

Statistical errors for non-linear system measurements involving square-law operations

This paper develops normalized random error formulas for special bispectra estimates and associated frequency response function estimates in finite memory square-law systems. Error formulas are also derived for output spectrum estimates from these non-linear systems and for associated non-linear coherence functions. These formulas are useful to evaluate such measured non-linear results as well as to design experimental programs.     

First passage time of nonlinear ship rolling in narrow band non-stationary random seas

The generalized extended stochastic central difference (GESCD) method is applied to study the response statistics and first passage time of nonlinear ship rolling in narrow band stationary and non-stationary random seas. The GESCD method is based on a combination of the extended stochastic central difference method with a statistical linearization technique, modified adaptive time scheme, and time coordinate transformation. The extended stochastic central difference method is, however, an extension of the stochastic central difference method for the determination of the recursive mean square or covariance of responses of systems under narrow band stationary and non-stationary random disturbances. Approximate first passage probabilities of nonlinear systems based on the modified mean rate of various crossings proposed earlier by the first author were determined. It is concluded that the GESCD method is very accurate, simple and efficient to apply compared with Monte Carlo simulation. The proposed method is applicable to cases with large nonlinearities and intensive random excitations. The approximate first passage probabilities of the nonlinear system determined by the proposed approach are very accurate as they are in excellent agreement with those evaluated by the Monte Carlo simulation. It is believed that the model considered in this paper is a closer representation to reality than those reported earlier in the literature.     

STATISTICAL SEISMIC RESPONSE ANALYSIS AND RELIABILITY DESIGN OF NONLINEAR STRUCTURE SYSTEM

Industrial structure systems may have non-linearity, and are also sometimes exposed to the danger of earthquake. In the design of such system, these factors should be accounted for from the viewpoint of reliability. This paper proposes a method to analyze seismic response and reliability design of a complex non-linear structure system under random excitation. The actual random excitation is represented to the corresponding Gaussian process for the statistical analysis. Then, the non-linear system is subjected to this random process. The non-linear structure system is modelled by substructure synthesis method (SSM) procedure. The non-linear equations are expanded sequentially. Then, the perturbed equations are solved in probabilistic method. Several statistical properties of a random process that are of interest in random vibration applications are reviewed in accordance with the non-linear stochastic problem. The system performance condition in the design of system is that responses caused by random excitation be limited within safe bounds. Thus, the reliability of the system is considered according to the crossing theory. Comparing with the results of the numerical simulation proved the efficiency of the proposed method.     

Non-linear dynamics and geometrical properties of localized wave structures

The present state of investigations of geometric properties of spatially localized configurations of electromagnetic wave fields with TE and TM polarizations is reviewed. The spatial structures of these field configurations are determined by their discrete amplitude spectra. The possibilities of controlled formation of localized field structure with different symmetries are analysed. The non-linear dynamics of such distributions, depending upon their initial geometric properties, are considered for series of realistic models of non-linear media. The utilization of adequate analytical methods, including a geometric analysis in special space, a generalized variational approach, an inverse scattering method for non-stationary processes, is illustrated. The new tendencies in non-linear dynamics of short polarized signals in directional systems are outlined.     

Recurrence quantification analysis of the logistic equation with transients

Recurrence quantification analysis (RQA) detects state changes in drifting dynamical systems without necessitating any a priori constraining mathematical assumptions. Study of the logistic equation with transients posits that RQA may be ideal for analyzing complex biological systems whose equations are unknown and whose dynamics are characteristically non-linear and non-stationary.     

Non-stationary characteristics of instability in a single-mode Nd:YVO 4 laser with fiber feedback

Random chaotic burst generation was experimentally observed in a single-mode microchip Nd:YVO₄ laser with fiber feedback. As the feedback strength was increased, a transition from stable relaxation oscillation state to unstable random chaotic burst state appeared. Furthermore, the non-stationary characteristic of probability association was experimentally identified at the transition of the two states while similar characteristics were reported only by numerical simulations of simple dynamical systems. This implies the general feature of non-stationary property of the dynamic switching between two states at transition. The observed chaotic burst generation and non-stationary nature were reproduced numerically based on the Lang-Kobayashi model. Received 28 March 2001 and Received in final form 5 June 2001     

Dynamics of a class of processes with Smoluchowski noises

One-dimensional evolution equations with a linear random parameter, which is a colored-noise stochastic process, are analyzed. Exact analytical expression for the probability distribution of the considered processes is given explicitly. The existence of stationary states and critical properties of the systems are considered. An analytical example is studied. It is shown that there exist a class of one-dimensional non-stationary markovian processes for which the one-dimensional distribution is the same as for the processes of interest.     

Stationary response of multi-degree-of-freedom vibro-impact systems to Poisson white noises

The stationary response of multi-degree-of-freedom (MDOF) vibro-impact (VI) systems to random pulse trains is studied. The system is formulated as a stochastically excited and dissipated Hamiltonian system. The constraints are modeled as non-linear springs according to the Hertz contact law. The random pulse trains are modeled as Poisson white noises. The approximate stationary probability density function (PDF) for the response of MDOF dissipated Hamiltonian systems to Poisson white noises is obtained by solving the fourth-order generalized Fokker-Planck-Kolmogorov (FPK) equation using perturbation approach. As examples, two-degree-of-freedom (2DOF) VI systems under external and parametric Poisson white noise excitations, respectively, are investigated. The validity of the proposed approach is confirmed by using the results obtained from Monte Carlo simulation. It is shown that the non-Gaussian behaviour depends on the product of the mean arrival rate of the impulses and the relaxation time of the oscillator.     

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.     

Random vibration of a beam impacting stops

The random vibration of a beam impacting a spring-like stop is discussed. The mean square response and the frequency of impacts are obtained by an equivalent linearization. Reasonable agreement is obtained between these results and the results for an equivalent non-linear single-degree-of-freedom system.