RECONSTRUCTION OF GAUSSIAN RANDOM FUNCTIONS FROM SPECTRUM DATA

It is known that a stationary random process is represented as a superposition of harmonic oscillations with real frequencies and uncorrelated amplitudes. In the study of nonstationary processes, it is natural to have increasing or declining oscillationсs. This raises the problem of constructing algorithms that would allow constructing broad classes of nonstationary processes from elementary nonstationary random processes. A natural generalization of the concept of the spectrum of a nonstationary random process is the transition from the real spectrum in the case of stationary to a complex or infinite multiple spectrum in the nonstationary case. There is also the problem of describing within the correlation theory of random processes in which the spectrum has no analogues in the case of stationary random processes, namely, the spectrum point is real, but it has infinite multiplicity for the operator image of the corresponding operator, and when the spectrum itself is complex. Reconstruction of the complex spectrum of a nonstationary random function is a very important problem in both theoretical and applied aspects. In the paper the procedure of reconstruction of random process, sequence, field from a spectrum for Gaussian random functions is developed. Compared to the stationary case, there are wider possibilities, for example, the construction of a nonstationary random process with a real spectrum, which has infinite multiplicity and which can be distributed over the entire finite segment of the real axis. The presence of such a spectrum leads, in contrast to the case of a stationary random process, to the appearance of new components in the spectral decomposition of random functions that correspond to the internal states of «strings», i.e. generated by solutions of systems of equations in partial derivatives of hyperbolic type. The paper deals with various cases of the spectrum of a non-self-adjoint operator , namely, the case of a discrete spectrum and the case of a continuous spectrum, which is located on a finite segment of the real axis, which is the range of values of the real non-decreasing function a(x). The cases a(x)=0, a(x)=a0, a(x)=x and a(x) is a piecewise constant function are studied. The authors consider the recovery of nonstationary sequences for different cases of the spectrum of a non-self-adjoint operator promising since spectral decompositions are a superposition of discrete or continuous internal states of oscillators with complex frequencies and uncorrelated amplitudes and therefore have deep physical meaning.

Reliability analysis of geogrid material with random nonlinear viscoelastic characteristics

Introduction. The behavior in the course of a time of geogrid material with random nonlinear viscoelastic characteristics under tension is analysed. Parameters of viscoelasticity are represented in form of Gaussian random vector. The components of this vector are taken from experimental data. Aim of the research. The objective of this research is the analysis of influence of different factors (value of applied load and the application of load in the form of random value instead of dead one, number of realizations, change of given level of strain) on providing of needed service life of geogrid material with given reliability level. Here reliability is interpreted as function of probability of non-failure. The first crossing of some given level by random strain is considered as a failure. The strain value corresponding to yield limit of geogrid material is accepted as the given level of longitudinal strain. Methods. The realizations of Gaussian random vector of viscoelastic parameters of material with given correlation matrix were imitated by means of linear transformation method. Results. It is demonstrated that longitudinal strain is Gaussian nonstationary random process which stochastic analysis can be made on base of 10 000 realizations. The dependencies on time of mathematical expectation and standard deviation of random longitudinal strain as well as function of probability of non-failure are found. Conclusion. It is shown that durability estimation found on base of the deterministic problem solution is overestimated in comparison with stochastic problem solution if the condition of given service life providing with some reliability level is set up.

Wear reliability of spur gear based on the cross-analysis method of a nonstationary random process

Tooth wear is one of the main reasons that lead to gear failure. The amount of wear is nonlinearly related to temperature, lubrication, load, and various random factors of materials, with obvious randomness and slow time-varying characteristics. Wear is a nonstationary random process, which has no accurate mathematical model or accurate reliability estimation method. This article proposes a reliability model of spur gears which works under a nonstationary random process that exceeds the limit, and the time-varying wear reliability is studied based on the level crossing analysis method. The wear at tooth root is revised in the calculation under the nonstationary random process, and the reliability curves are obtained afterwards. An experiment is carried out on the spur gear meshing test rig, and the reliability model and wear performance are verified and analyzed. Results obtained with the proposed tooth surface wear reliability model match well with the experimental results. Therefore, this model is applicable for situations under a nonstationary random process. The new method makes contribution to the assessment of gear running status and is of great significance in the prediction of wear life under a nonstationary random process.

Ocean Eddy Dynamics in a Coupled Ocean–Atmosphere Model*

The role of mesoscale oceanic eddies is analyzed in a quasigeostrophic coupled ocean–atmosphere model operating at a large Reynolds number. The model dynamics are characterized by decadal variability that involves nonlinear adjustment of the ocean to coherent north–south shifts of the atmosphere. The oceanic eddy effects are diagnosed by the dynamical decomposition method adapted for nonstationary external forcing. The main effects of the eddies are an enhancement of the oceanic eastward jet separating the subpolar and subtropical gyres and a weakening of the gyres. The flow-enhancing effect is due to nonlinear rectification driven by fluctuations of the eddy forcing. This is a nonlocal process involving generation of the eddies by the flow instabilities in the western boundary current and the upstream part of the eastward jet. The eddies are advected by the mean current to the east, where they backscatter into the rectified enhancement of the eastward jet. The gyre-weakening effect, which is due to the time-mean buoyancy component of the eddy forcing, is a result of the baroclinic instability of the westward return currents. The diagnosed eddy forcing is parameterized in a non-eddy-resolving ocean model, as a nonstationary random process, in which the corresponding parameters are derived from the control coupled simulation. The key parameter of the random process—its variance—is related to the large-scale flow baroclinicity index. It is shown that the coupled model with the non-eddy-resolving ocean component and the parameterized eddies correctly simulates climatology and low-frequency variability of the control eddy-resolving coupled solution.

Information Theoretic Equivalent Bandwidths of Random Processes and Their Applications

Summary Objectives : Since most of the biomedical signals, such as electroencephalogram (EEG), electromyogram (EMG) and phonocardiogram (PCG), are nonstationary random processes, the time-frequency analysis has recently been extensively applied to those signals in order to achieve precise characterization and classification. In this paper, we have first defined a new class of information theoretic equivalent bandwidths (EBWs) of stationary random processes, then instantaneous EBWs (IEBWs) using nonnegative time-frequenc distributions have been defined in order to track the change of the EBW of a nonstationary random process. Methods : The new class of EBWs which includes spectral flatness measure (SFM) for stationary random processes is defined by using generalized Burg entropy. Generalized Burg entropy is derived from the relation between Rényi entropy and Rényi information divergence of order α. In order to track the change of EBWs of a nonstationary random process, the IEBWs are defined on the nonnegative time-frequency distributions, which are constructed by the Copula theory. Results : We evaluate the IEBWs for a first order stationary auto-regressive (AR) process and three types of time-varying AR processes. The results show that the IEBWs proposed here properly represent a signal bandwidth. In practical application to PCGs, the proposed method was successful in extracting the information that the bandwidth of the innocent systolic murmur was much smaller than that of the abnormal systolic murmur. Conclusions : We have defined new information theoretic EBWs and have proposed a novel method to track the change of the IEBWs. Some computer simulation showed effectiveness of the methods. Applying the IEBWs to PCGs, we could extract some features of a systolic murmur.

Characterization and Simulation of Gunfire with Karhunen-Loeve Expansion

Gunfire is used as an example to illustrate how the Karhunen-Loeve (K-L) expansion can be used to characterize and simulate nonstationary random events. This paper will develop a method to describe the nonstationary random process in terms of a K-L expansion. The gunfire record is broken up into a sequence of transient waveforms, each representing the response to the firing of a single round. First, the mean is estimated and subtracted from each waveform. The mean is an estimate of the deterministic part of the gunfire. The autocovariance matrix is estimated from the matrix of these single-round gunfire records. Each column is a realization of the firing of a single round. The gunfire is characterized with the K-L expansion of the autocovariance matrix. The gunfire is simulated by generating realizations of records of a single-round firing from the expansion and the mean waveform. The individual realizations are then assembled into a realization of a time history of many rounds firing. The method is straightforward and easy to implement, and produces a simulated record very much like the original measured gunfire record.

Reduction of First Excursion Probability Due to Plastic Deformation and Absorbed Energy

When the system is subjected to excess seismic loading, spring element has hysteresis loop characteristic caused by plastic deformation. Energy is dissipated by hysteresis loop characteristic. Elastic-plastic damper which utilizes energy absorption is practically used. On the other hand, response is nonstationary random process because seismic loading is nonstationary random process. In such a case, reliability of the system should be evaluated in probabilistic manner. Some failure modes are observed. First excursion failure is one of the most important failure modes. First excursion probability, that is, probability of occurrence of first excursion failure, also represents the characteristic of random process. In this paper, the first excursion probability and absorbed energy by hysteresis loop characteristic are obtained from theoretical method. This method is based on equivalent linearization method. Bilinear hysteresis loop characteristic is used as force-deformation relation. It is concluded that the first excursion probability decreases with the increase of absorbed energy.

Simplified Calculation Method of Seismic Response Energy of Mechanical System

When the mechanical systems are subjected to seismic excitation, the responses are nonstationary random process, since seismic excitation has nonstationary characteristics. Mean square response is a representative value of the statistical properties of the response. Seismic response energy which is integral of mean square response is used to evaluate absorbed energy and cumulative damage of mechanical system. Theoretical method for obtaining nonstationary mean square response of the secondary system is very complicated and time consuming. Thus, some approximate methods are sometimes used. In this paper, an approximate method for calculation method of seismic response energy using statistical properties of stationary response is proposed. As input excitation, nonstationary white noise is used. The input is defined by product of stationary white noise and envelope function. Mean square response of absolute acceleration of the mechanical system, relative velocity of the mechanical system to the ground and relative displacement are obtained. Some numerical results are shown. It is found that the proposed method gives exact values of seismic response energy independent of the damping ratio, mass ratio and the natural period.