Advances in Acoustics and Vibration

Volume 2016 (2016), Article ID 3837971, 7 pages

http://dx.doi.org/10.1155/2016/3837971

## Comparison of the Time Domain Windows Specified in the ISO 18431 Standards Used to Estimate Modal Parameters in Steel Plates

^{1}Escuela de Ingeniería Electrónica, Universitaria de Investigación (UDI), Grupo de Investigación en Procesamiento de Señales (GPS), Bucaramanga, Colombia^{2}Universidad Pontificia Bolivariana, Bucaramanga, Colombia

Received 29 April 2016; Revised 14 July 2016; Accepted 3 August 2016

Academic Editor: Massimo Viscardi

Copyright © 2016 Jhonatan Camacho-Navarro et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The procedures used to estimate structural modal parameters as natural frequency, damping ratios, and mode shapes are generally based on frequency methods. However, methods of time-frequency analysis are highly sensible to the parameters used to calculate the discrete Fourier transform: windowing, resolution, and preprocessing. Thus, the uncertainty of the modal parameters is increased if a proper parameter selection is not considered. In this work, the influence of three different time domain windows functions (Hanning, flat-top, and rectangular) used to estimate modal parameters are discussed in the framework of ISO 18431 standard. Experimental results are conducted over an AISI 1020 steel plate, which is excited by means of a hammer element. Vibration response is acquired by using acceleration records according to the ISO 7626-5 reference guides. The results are compared with a theoretical method and it is obtained that the flat-top window is the best function for experimental modal analysis.

#### 1. Introduction

Physical behavior of complex engineering systems can be studied through prediction and simulation analysis by means of specialized software [1, 2]. Thus, it is possible to check abnormal performance with the help of monitoring methods based on simulation tools. In particular, the analysis of structural dynamics can be addressed by determining modal parameters defined by mode shapes, natural frequencies, and damping ratios. In this sense, by means of Operational Modal Analysis it is possible to conduct nondestructive testing, fatigue analysis, and issues concerned with field in structural analysis [3]. These tasks also involve updated finite element models used to predict the dynamic behavior of the structure reliability [4]. For instance, an application of modal analysis is the assessment of highway-bridge by dynamic testing and finite-element model updating [5]. Also, damage detection has been carried out in reinforced concrete beams by using modal flexibility residuals [6].

Because of the importance of modal analysis in the field of structural analysis, well established procedures to obtain proper estimations of modal parameters are required. Although there exists a huge documentation about methods used for modal analysis [7, 8], their practical implementation is still difficult because there are many parameters involved in the procedure, which must be decided by engineers of different areas and knowledge. In this sense, the 7626 standard ISO specifies a guideline of methodological steps to conduct experiments in order to obtain the frequency response measurement and the 18431 ISO describes the procedures for time-frequency analysis of vibrational records. Thus, by considering the recommendations of the ISO standards, the further estimation of modal model parameters by means of well-known modal analysis methods (as, e.g., peak-picking or least squares among others) is facilitated.

In this paper, a practical implementation of the abovementioned ISO standards with a special emphasis on computing the natural frequency values is demonstrated. Thus, the influence of using three domain window functions (*Hanning, flat-top, and rectangular*) to estimate natural frequencies is discussed. Experimental results are conducted over an AISI 1020 steel plate, which is excited by means of a hammer element.

#### 2. Theoretical Framework

The procedure used in this paper to estimate the natural frequencies of a modal model is based on the analysis of measurements from Frequency response functions (FRFs). In this sense, the extraction of relevant frequency information is performed by applying spectrum estimation techniques to structural vibrational records. Thus, FRFs are approximated by the cross-power spectral density (PSD) between the vibrational responses.

In this section, the conceptual issues involved in the estimation of structural natural frequencies by means of FRFs are detailed. Also, fundamentals of methods used to estimate the frequency decomposition are presented, focusing on given details about the parameters with great influence for its implementation.

##### 2.1. Frequency Response Functions (FRFs)

Dynamical model of structures is constructed on physical knowledge and fundamental laws of motion according to [9]where , , and are the stiffness, damping, and mass matrices, respectively. Likewise, and denote the forcing vector and displacement. Assuming that is a delta-correlated exciting force and using properties demonstrated in the Natural Excitation Technique (NExT), it is possible to write the law motion of (1) in the form of [9] where is a vector of correlation functions for all positive lags , between the displacement vector and a reference signal, which must be uncorrelated with respect to excitation signal. Thus, (2) establishes that correlation function between acceleration records and a reference signal can be treated as free vibration data, which allows determining modal characteristics of the structures.

Moreover to law motion and NExT equations used in methodologies for modal parameter identification, the FRF relationship in structures must be specified. The FRF for one* n-degree* of freedom system represented by (1) or (2) can be written in the form of partial fractions as is expressed in (3) (classical pole/residue) [10, 11].where is the output PSD matrix, is the total number of modes of interest, is the pole of the mode, is the modal damping (constant decay), is the damped natural frequency, and the superscript denotes complex conjugate. The typical PSD curve for the FRF determined by (3) is depicted in Figure 1.