scholarly journals Analysis of Stationary Random Vibrating Systems Using Smooth Decomposition

2013 ◽  
Vol 20 (3) ◽  
pp. 493-502 ◽  
Author(s):  
Sergio Bellizzi ◽  
Rubens Sampaio
Keyword(s):  
Random Process ◽  
Covariance Matrix ◽  
Random Fields ◽  
Decomposition Method ◽  
Time Derivative ◽  
Random Processes ◽  
Stationary Random Processes ◽  
Proper Orthogonal ◽  
Free Beam ◽  
Vibrating Systems

A modified Karhunen-Loève Decomposition/Proper Orthogonal Decomposition method, named Smooth Decomposition (SD) (also named smooth Karhunen-Loève decomposition), was recently introduced to analyze stationary random signal. It is based on a generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated time-derivative random process. The SD appears to be an interesting tool in terms of modal analysis. In this paper, the SD will be described in case of stationary random processes and extended also to stationary random fields. The main properties will be discussed and illustrated on a randomly excited clamped-free beam.

Modal Analysis With Smooth Karhunen-Loe`ve Decomposition

2009 ◽  
Author(s):  
S. Bellizzi ◽  
Rubens Sampaio
Keyword(s):  
Random Field ◽  
Modal Analysis ◽  
Covariance Matrix ◽  
Random Fields ◽  
Time Derivative ◽  
Normal Modes ◽  
Orthogonal Decomposition ◽  
White Noise Excitation ◽  
Smooth Orthogonal Decomposition ◽  
Vibrating Systems

In this paper, the Smooth Orthogonal Decomposition is formulated in term of a Smooth Karhunen-Loe`ve Decomposition (SKLD) to analyze random fields. The SKLD is obtained solving a generalized eigenproblem defined from the covariance matrix of the random field and the covariance matrix of the associated time derivative random field. The main properties of the SKLD are described and compared to the classical Karhunen-Loe`ve decomposition. The SKLD is then applied to the responses of randomly excited vibrating systems with a view to performing modal analysis. The associated SKLD characteristics are interpreted in case of linear vibrating systems subjected to white noise excitation in terms of normal modes. Discrete and continuous mechanical systems are considered in this study.

Information Theoretic Equivalent Bandwidths of Random Processes and Their Applications

2007 ◽  
Vol 46 (02) ◽  
pp. 110-116 ◽  
Author(s):  
S. Kikkawa ◽  
H. Yoshida
Keyword(s):  
Random Process ◽  
Random Processes ◽  
Systolic Murmur ◽  
Nonstationary Random Process ◽  
Frequency Distributions ◽  
Copula Theory ◽  
Information Theoretic ◽  
Time Frequency ◽  
New Class ◽  
Stationary Random Processes

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.

scholarly journals Fourier Expansions with Polynomial Terms for Random Processes

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Zhihua Zhang
Keyword(s):  
Random Process ◽  
Best Approximation ◽  
Decomposition Method ◽  
Fourier Expansion ◽  
Boundary Effect ◽  
Random Processes ◽  
Approximation Order ◽  
Obvious Advantage ◽  
The Best Approximation ◽  
Fourier Expansions

Based on calculus of random processes, we present a kind of Fourier expansions with simple polynomial terms via our decomposition method of random processes. Using our method, the expectations and variances of the corresponding coefficients decay fast and partial sum approximations attain the best approximation order. Moreover, since we remove boundary effect in our decomposition of random process, these coefficients can discover the instinct frequency information of this random process. Therefore, our method has an obvious advantage over traditional Fourier expansion. These results are also new for deterministic functions.

scholarly journals Ergodicity test of the eddy correlation method

2014 ◽  
Vol 14 (12) ◽  
pp. 18207-18254 ◽  
Author(s):  
J. Chen ◽  
Y. Hu ◽  
Y. Yu ◽  
S. Lü
Keyword(s):  
Boundary Layer ◽  
Random Process ◽  
Spatial Scale ◽  
Atmospheric Turbulence ◽  
Turbulent Flux ◽  
Similarity Theory ◽  
Random Processes ◽  
Eddy Correlation ◽  
Wide Sense ◽  
Stationary Random Processes

Abstract. The turbulent flux observation in the near-surface layer is a scientific issue which researchers in the fields of atmospheric science, ecology, geography science, etc. are commonly interested in. For eddy correlation measurement in the atmospheric surface layer, the ergodicity of turbulence is a basic assumption of the Monin–Obukhov (M–O) similarity theory, which is confined to steady turbulent flow and homogenous surface; this conflicts with turbulent flow under the conditions of complex terrain and unsteady, long observational period, which the study of modern turbulent flux tends to focus on. In this paper, two sets of data from the Nagqu Station of Plateau Climate and Environment (NaPlaCE) and the cooperative atmosphere–surface exchange study 1999 (CASE99) were used to analyze and verify the ergodicity of turbulence measured by the eddy covariance system. Through verification by observational data, the vortex of atmospheric turbulence, which is smaller than the scale of the atmospheric boundary layer (i.e., its spatial scale is less than 1000 m and temporal scale is shorter than 10 min) can effectively meet the conditions of the average ergodic theorem, and belong to a wide sense stationary random processes. Meanwhile, the vortex, of which the spatial scale is larger than the scale of the boundary layer, cannot meet the conditions of the average ergodic theorem, and thus it involves non-ergodic stationary random processes. Therefore, if the finite time average is used to substitute for the ensemble average to calculate the average random variable of the atmospheric turbulence, then the stationary random process of the vortex, of which spatial scale was less than 1000 m and thus below the scale of the boundary layer, was possibly captured. However, the non-ergodic random process of the vortex, of which the spatial scale was larger than that of the boundary layer, could not be completely captured. Consequently, when the finite time average was used to substitute for the ensemble average, a large rate of error would occur with use of the eddy correction method due to losing the low frequency component information of the larger vortex. When the multi-station observation was compared with the single-station observation, the wide sense of stationary random process originating from the multi-station observation expanded from a vortex which was about 1000 m smaller than a boundary layer scale to the turbulent vortex, which was larger than the boundary layer scale of 2000 m. Therefore, the calculation of the turbulence average or variance and turbulent flux could effectively meet the ergodic assumption, and the results would be approximate to the actual values. Regardless of vertical velocity and temperature, if the ergodic stationary random processes could be met, then the variance of the vortexes in the different temporal scales could follow M–O similarity theory; in the case of the non-ergodic random process, its vortex variance deviated from the M–O similarity relations. The exploration of ergodicity in the atmospheric turbulence measurements is doubtlessly helpful to understanding the issues in atmospheric turbulent flux observation, and provides a theoretical basis for overcoming related difficulties.

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