Advances in Materials Science and Engineering

Volume 2017 (2017), Article ID 5821835, 9 pages

https://doi.org/10.1155/2017/5821835

## Comparison of Damage Detection Methods Depending on FRFs within Specified Frequency Ranges

^{1}Department of Architectural Engineering, Chung-Ang University, Seoul, Republic of Korea^{2}Department of Architectural Engineering, Kangwon National University, Chuncheon, Republic of Korea

Correspondence should be addressed to Hee-Chang Eun

Received 8 April 2017; Accepted 20 July 2017; Published 29 August 2017

Academic Editor: Xiao-Jian Gao

Copyright © 2017 Yong-Su Kim and Hee-Chang Eun. 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

Structural damage can be detected using frequency response function (FRF) measured by an impact and the corresponding responses. The change in the mechanical properties of dynamic system for damage detection can seldom be estimated using FRF data extracted from a very limited frequency range. Proper orthogonal modes (POMs) from the FRFs extracted in given frequency ranges and their modified forms can be utilized as damage indices to detect damage. The POM-based damage detection methods must be sensitive to the selected FRFs. This work compares the effectiveness of the damage detection approaches taking the POMs estimated by the FRFs within five different frequency ranges including resonance frequency and antiresonance frequency. It is shown from a numerical example that the POMs extracted from the FRFs within antiresonance frequency ranges provide more explicit information on the damage locations than the ones within resonance frequency ranges.

#### 1. Introduction

Structural damage is detected based on the variation in dynamic responses due to the local change of physical properties at damage region. There have been many attempts to provide the accurate damage detection methods related to structural health monitoring. Many indices like mode shapes, mode shape curvature, frequency response function (FRF) curvature, modal strain energy, and so forth to evaluate the structural performance have been utilized [1, 2].

One of the measurement data types is FRF data set. The FRF can be estimated by impact hammer test and the energy is propagated from the impact point. The FRF data measured from experimental work are utilized to estimate the characteristics of dynamic system. It indicates that the measured FRF data set can be used as an index to detect damage. Phani and Woodhouse [3] proposed the FRF curvature method based on only the measured data without the need for any modal identification. Lee and Kim [4] proposed a structural damage detection method to determine both location and magnitude of damage from perturbation equations of FRF data. Fritzen [5] evaluated a FRF-based model updating algorithm using experimentally collected data and presented a protocol for measurement selection and a regularization technique. Rahmatalla et al. [6] presented a feasible method for structural vibration-based health monitoring to reduce the dimension of the initial FRF data and to employ artificial neural network. Sampaio et al. [7] provided a structural damage identification technique using changes of the FRF to be related to the changes of the stiffness and mass through damage sensitivity equations. Wang et al. [8] developed a methodology by coupled response monitoring through the emergence of peaks on a FRF plot to determine the damage existence. Garcia-Palencia et al. [9] exhibited that the curvatures of FRF at frequencies other than natural frequencies can be used for identifying both the existence of damage and the location of damage. Bandara et al. [10] presented a damage identification method using the changes in the distribution of the compliance of the structure due to damage. Bandara et al. [11] investigated the advantages and limitations to use vibration-based damage detection methods from the measurements of mode shapes and FRFs. Nozarian and Esfandiari [12] identified parameter matrices using FRF data measured at specific positions and constraints and provided a damage detection method from their variations.

Proper orthogonal decomposition (POD) analysis captures most of the kinetic energy in the least number of modes possible. Its application is similar to that of Fourier analysis, except that it normally requires far less modes to represent the system within a desired level. POMs extracted from the FRF data in a prescribed frequency range are utilized as an index to recognize the existence of damage. The POD method has been widely applied in various fields of engineering and science. Ramanamurthy et al. [13] related the POM to normal modes of vibration in systems for lightly modally damped systems. They exhibited that the POMs represent the principal axes of inertia formed by the distribution of data on the modal coordinate curve. Using statistical process control technique, Mondal et al. [14] investigated the applications of principal component analysis for damage diagnosis. Choi et al. [15] provided a damage detection algorithm based on the POD technique to filter out the influence of operational/environmental variation in the dynamic response. De Medeiros et al. [16] applied a POD statistical method to the field of damage detection for continuous static monitoring systems. Feeny and Kappagantu [17] presented a damage detection method based on fuzzy -means clustering algorithm and measured FRF data reduced by principal component analysis. De Boe and Golinval [18] developed a POMs-based damage identification approach using acceleration dynamic responses of shear-type buildings. Shane and Jha [19] presented a neural networks-based damage detection method using FRF data. The method reduces the dimension of the initial FRF data and employs an artificial neural network for the actual damage.

The parameter matrices as well as damage region of dynamic systems can be predicted using FRF data sets within a specific frequency range rather than at a specific frequency. The FRF data sets are transformed to the POM to represent the principal axes of inertia formed by the distribution of data on the modal coordinate curve. The POM can be changed depending on the FRF data sets extracted from full sets of FRFs. Beginning with the FRFs measured from the dynamic finite element model, this work investigates the sensitivity of the damage detection method using the POMs to be estimated from the FRFs corresponding to five different frequency ranges including resonance frequency and antiresonance frequency. A numerical example compares the sensitivity of the damage detection method depending on the extracted FRFs.

#### 2. FRF and POM

The dynamic behavior of a structure, which is assumed to be linear and approximately discretized for* n* DOFs, can be described by the equations of motion:where** M**,** K**, and** C** denote the analytical mass, stiffness, and damping matrices, respectively, , and is the load excitation vector.

Frequency response function is defined as the ratio of the complex spectrum of the response to the complex spectrum of the excitation:where and denote the complex spectrum of the response and the complex spectrum of the excitation, respectively. The magnitude of the FRF denotes the ratio of .

For the case of a displacement response at station and a disturbing force at station , the numerical frequency response can be constructed aswhere denotes the th element of the vector and and denote the circular natural frequency and the damping ratio, respectively, for the th mode, .

The FRF data, , contaminated by external noise rarely provide the accurate information on the dynamic system. In this study, they can be described by adding a series of random numbers to the calculated FRF response data expressed by where denotes the relative magnitude of the error, and is a random number variant in the range .

The measured FRFs can be reduced by the POD and are transformed to the POD to extract extremal data set. The POD technique is effective method because basis elements are formed in an optimal way. The POD basis collects snapshots. The FRFs of a system are generated by forcing the system. expresses an FRF on a finite number of points in space and a finite frequency interval . The FRFs at dofs are sampled frequencies and the data are arranged in a snapshot matrix :

Let the autocovariance matrixbe defined aswhere is the number of the FRF data sets extracted. Matrix** C** is a Hermitian positive semidefinite matrix to possess a complete set of orthogonal eigenvectors with corresponding nonnegative real eigenvalues. Solving the eigenvalue problem of (6) at the core of the POD method, (6) satisfieswhere the eigenvalues reflect the energies in different POMs and are arranged in descending order as follows:where the eigenvalues are the POVs, and the corresponding eigenvector of the extreme value problem is associated with a POV . The greatest POV is the optimal vector to characterize the snapshots.

The POMs may be used as a basis for the decomposition of . The POM associated with the greatest POV is the optimal vector. If the eigenvalues are normalized to unit magnitude, then they represent the relative energy captured by the corresponding POM. The eigenvalue reflects the relative kinetic energy associated with the corresponding mode. The energy is defined as the sum of the POVs. The POMs are written aswhere denotes the FRF matrix in (5) and represents the eigenvector corresponding to the eigenvalue . The POMs are arranged as follows:

The slope of the POM data corresponding to two adjacent nodes from Figure 1 can be defined aswhere is the distance between two adjacent nodes. The sum of the squared slope of the POM at two adjacent nodes is defined as