CORRELATION-SPECTRAL ANALYSIS OF THE TOPOGRAPHY OF ENGINEERING SURFACES AT THE NANOSCALE LEVEL

Исследована нанотопография некоторых типичных технических поверхностей и экспериментально определены характеристики профиля наношероховатости как случайного процесса - автокорреляционная функция и спектральная плотность. Показано, что для исследованных поверхностей их профилограммы могут рассматриваться как реализации случайного стационарного нормального эргодического процесса. Проведена визуальная проверка нормальности процесса сравнением экспериментальных значений ординат профиля с теоретическими значениями, подчиняющимися нормальному распределению, а также сравнением полигона частот с теоретической функцией плотности вероятности нормального распределения. Количественное подтверждение нормальности процесса выполнено с применением критерия согласия Колмогорова. Показано, что на уровне значимости p = 0,05 гипотеза о нормальности случайного процесса (профиля наношероховатости поверхности) не противоречит экспериментальным результатам. Определены интервалы корреляции рассмотренных процессов. Вид автокорреляционных функций и величины интервалов корреляции говорят о случайном характере профиля поверхности: на интервале, равном одному - двум средним значениям шага неровностей профиля его ординаты становятся практически некоррелированными. Графики спектральных плотностей свидетельствуют о том, что профиль поверхности можно рассматривать как широкополосный случайный шум с преобладанием низкочастотных составляющих. The nanotopography of some typical technical surfaces is investigated and the characteristics of the nanoroughness profile as a random process are experimentally determined - the autocorrelation function and spectral density. It is shown that for the investigated surfaces, their profilograms can be considered as realizations of a random stationary normal ergodic process. A visual check of the process normality was carried out by comparing the experimental values of the profile ordinates with theoretical values obeying the normal distribution, as well as by comparing the frequency polygon with the theoretical probability density function of the normal distribution. Quantitative confirmation of the process normality was carried out using the Kolmogorov goodness-of-fit test. It is shown that at the significance level p = 0,05, the hypothesis about the normality of a random process (surface nanoroughness profile) does not contradict the experimental results. The correlation intervals of the considered processes are determined. The form of the autocorrelation functions and the values of the correlation intervals indicate the random nature of the surface profile: in the interval equal to one or two average values of the step of the irregularities of the profile, its ordinates become practically uncorrelated. Spectral density plots indicate that the surface profile can be considered as a wide-band random noise with a predominance of low-frequency components.

Stochastic model of the gearbox pair vibration

The model of vibration signal of gearbox pair in the form of periodically correlated non-stationary random process is considered. It is shown that hidden periodicities in biperiodic correlated random process mean and covariance function, characterizing the vibrations of gearbox pair can be detected using the component and least square methods. Seven particular cases of the bi-rhythmic hidden periodicity for different modulation modes are analyzed.

ON THE APPLICATION OF THE EXTREME VALUE THEORY IN SHIP’S STRENGTH CALCULATIONS

A procedure is proposed for application of the extreme value theory (EVT) approach considering not only the maximal value of the corresponding random variable but also its probability of exceedance. It substantially reduces the probability of exceedance of any given limit value used in the case when traditional EVT is applied. Examples are provided to illustrate its application when records of the random process are available.

On the local time of a recurrent random walk on ℤ²

We prove a functional limit theorem for the number of visits by a planar random walk on Z 2 \mathbb {Z}^2 with zero mean and finite second moment to the points of a fixed finite set P ⊂ Z 2 P\subset \mathbb {Z}^2 . The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.

On the Range Assignment in Wireless Sensor Networks for Minimizing the Coverage-Connectivity Cost

This article deals with reliable and unreliable mobile sensors having identical sensing radius r , communication radius R , provided that r ≤ R and initially randomly deployed on the plane by dropping them from an aircraft according to general random process. The sensors have to move from their initial random positions to the final destinations to provide greedy path k 1 -coverage simultaneously with k 2 -connectivity. In particular, we are interested in assigning the sensing radius r and communication radius R to minimize the time required and the energy consumption of transportation cost for sensors to provide the desired k 1 -coverage with k 2 -connectivity. We prove that for both of these optimization problems, the optimal solution is to assign the sensing radius equal to r = k 1 || E [S]||/2 and the communication radius R = k 2 || E [S]||/2, where || E [S]|| is the characteristic of general random process according to which the sensors are deployed. When r < k 1 || E [S]||/2 or R < k 2 || E [S]||/ 2, and sensors are reliable, we discover and explain the sharp increase in the time required and the energy consumption in transportation cost to ensure the desired k 1 -coverage with k 2 -connectivity.

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.

A structured scaffold underlies activity in the hippocampus

Place cells are believed to organize memory across space and time, inspiring the idea of the cognitive map. Yet unlike the structured activity in the associated grid and head-direction cells, they remain an enigma: their responses have been difficult to predict and are complex enough to be statistically well-described by a random process. Here we report one step toward the ultimate goal of understanding place cells well enough to predict their fields. Within a theoretical framework in which place fields are derived as a conjunction of external cues with internal grid cell inputs, we predict that even apparently random place cell responses should reflect the structure of their grid inputs and that this structure can be unmasked if probed in sufficiently large neural populations and large environments. To test the theory, we design experiments in long, locally featureless spaces to demonstrate that structured scaffolds undergird place cell responses. Our findings, together with other theoretical and experimental results, suggest that place cells build memories of external inputs by attaching them to a largely prespecified grid scaffold.

Evaluating the Dynamic Response of the Bridge-Vehicle System considering Random Road Roughness Based on the Moment Method

The bridge-vehicle interaction (BVI) system vibration is caused by the vehicles passing through the bridge. The road roughness has a great impact on the system vibration. In this regard, poor road roughness is known to affect the comfort of the vehicle crossing the bridge and aggravate the fatigue damage of the bridge. Road roughness is usually regarded as a random process in numerical calculation. To fully consider the influence of road roughness randomness on the response of the BVI system, a random BVI model was established. Thereafter, the random process of road roughness was expressed by Karhunen–Loeve expansion (KLE), after which the moment method was used to calculate the maximum probability value of the BVI system response. The proposed method has higher accuracy and efficiency than the Monte Carlo simulation (MCS) calculation method. Subsequently, the influences of vehicle speed, roughness grade, and bridge span on the impact factor (IMF) were analyzed. The results show that the road roughness grade has a greater impact on the bridge IMF than the bridge span and vehicle speed.