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.

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.

Enhanced Proper Orthogonal Decomposition for the Modal Analysis of Homogeneous Structures

2002 ◽  
Vol 8 (1) ◽  
pp. 19-40 ◽  
Author(s):  
S. Han ◽  
B. F. Feeny
Keyword(s):  
Finite Element ◽  
Proper Orthogonal Decomposition ◽  
Modal Analysis ◽  
Normal Modes ◽  
Orthogonality Condition ◽  
Orthogonal Decomposition ◽  
Analysis Tool ◽  
Homogeneous Structures ◽  
Proper Orthogonal ◽  
Modal Coordinates

Proper orthogonal decomposition (POD) is studied in an effort to increase its applicability as a modal analysis tool. A modification is proposed to make better use of spatial resolution and to accommodate arbitrary spacing in the discretization. The theory for this modification is rooted in the discrete approximation of the integral orthogonality condition for continuous normal modes. The modified POD is applied to a finite element beam and an experimental beam sensed with accelerometers, and the resulting proper orthogonal modes (POMs) are compared to the theoretical modes of the beam. The POMs are used as a basis for decomposing the signal ensemble into proper modal coordinates. The proper modal coordinates are used to evaluate the POMs and to match modes with modal frequencies and damping.

A New Method for Vibration Mode Analysis

2005 ◽  
Author(s):  
David Chelidze ◽  
Wenliang Zhou
Keyword(s):  
Modal Analysis ◽  
A Priori ◽  
Normal Modes ◽  
Orthogonal Decomposition ◽  
Mode Analysis ◽  
Continuous Systems ◽  
Analysis Technique ◽  
Priori Information ◽  
Mathematical Justification ◽  
Proper Orthogonal

In this paper, a new modal analysis method based on a novel multivariate data analysis technique called smooth orthogonal decomposition (SOD) is proposed. The development of the SOD and its main properties are described. The mathematical justification for the application in modal analysis is also provided. The proper orthogonal decomposition (POD) and its application in modal analysis are provided for comparison. Numerical simulations of discrete and continuous systems are used in this comparison. It is demonstrated that the SOD-based analysis overcomes main deficiencies of the POD. The SOD identifies linear normal modes and corresponding frequencies without requiring any a priori information about the distribution of mass in the system.

Modal Identification of damped vibrating systems by iterative smooth orthogonal decomposition method

2020 ◽  
pp. 136943322096844
Author(s):  
Zhixaing Hu ◽  
Jun Li ◽  
Lunhai Zhi ◽  
Xiao Huang
Keyword(s):  
Decomposition Method ◽  
Experimental Studies ◽  
Covariance Matrices ◽  
Transformation Matrix ◽  
Orthogonal Decomposition ◽  
Modal Identification ◽  
Iteration Step ◽  
Coupled Modes ◽  
Smooth Orthogonal Decomposition ◽  
Vibrating Systems

The smooth orthogonal decomposition method (SOD) is an efficient algorithm that can be used to extract modal matrix and frequencies of lightly damped vibrating systems. It uses the covariance matrices of output-only displacement and velocity responses to form a generalized eigenvalues problem (EVP). The mode shape vectors are estimated by the eigenvectors of the EVP. It is stated in this work that the accuracy of the SOD method is mainly affected by the correlation characteristic of modal coordinate responses. For the damped vibration systems, biases will be contained in the results of using the SOD. Therefore, an iterative smooth orthogonal decomposition (ISOD) method is proposed to identify modal parameters of the damped system from the covariance matrices of the displacement, velocity, and acceleration responses. The modal matrix given by the SOD method is updated in each iteration step using a transformation matrix. The transformation matrix can be efficiently computed using a set of analytical formulations. Meanwhile, natural frequencies and damping ratios are obtained by using a simple search method. The performance of the proposed ISOD method is verified by numerical and experimental studies. The results demonstrate that, by considering the correlation of modal responses, the ISOD method can be used to extract accurately the modal information of vibration systems with coupled modes.

Extended Smooth Orthogonal Decomposition for Modal Analysis

2018 ◽  
Vol 140 (4) ◽  
Author(s):  
Zhi-Xiang Hu ◽  
Xiao Huang ◽  
Yixian Wang ◽  
Feiyu Wang
Keyword(s):  
Modal Analysis ◽  
Degrees Of Freedom ◽  
Eigenvalue Problems ◽  
Search Algorithm ◽  
Damping Ratio ◽  
Orthogonal Decomposition ◽  
Vibration System ◽  
Output Only ◽  
Damped Systems ◽  
Smooth Orthogonal Decomposition

The smooth orthogonal decomposition (SOD) is an output-only modal analysis method, which has simple structure and gives good results for undamped or lightly damped vibration systems. In the present study, the SOD method is extended to incorporate various measurements that contain the displacement, the velocity, the acceleration, and even the jerk (derivation of the acceleration). Several generalized eigenvalue problems (EVPs) are put forward considering different measurement combinations, and it is proved that all these EVPs can reduce to the eigenvalue problems of the undamped vibration system. These different methods are called extended smooth orthogonal decomposition (ESOD) methods in this paper. For the damped vibration system, the frequencies obtained by different ESOD methods are different from each other. Thus, a cost function is defined and a search algorithm is proposed to find the optimal frequency and damping ratio that can explain these differences. Although the search algorithm is derived for the single-degree-of-freedom (SDOF) vibration systems, it is effective for the multi-degrees-of-freedom (MDOF) vibration system after assuming that the smooth orthogonal coordinates (SOCs) computed by the ESOD methods are approximate to the modal coordinate responses. In order to verify the ESOD methods and the search algorithm, simulations are carried out and the results indicate that all ESOD methods reach correct results for undamped vibration systems and the search algorithm can give accurate frequency and damping ratio for damped systems. In addition, the effects of measurement noises are considered and the results show that the proposed method has anti-noise property to some extent.

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