Power mode shapes for early damage detection in linear structures
Introduction
Non-destructive damage detection has been widely studied over the past decades [1], [2], [3]. Damage detection based on modal parameters and the corresponding derived indices, such as modal curvature and flexibility, can be found in a wide range of literature [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Modal parameters can be extracted by using different techniques [13], [14], [15], [16], [17] and generally the spectrum estimation of signals is the first step in the entire procedure. The procedure is also expected to detect any damage based on some non-modal parameters, which avoids modal parameter extraction. The study of non-modal parameters from known power spectra or spectral densities, which may be regarded as a parallel branch to conventional modal parameter extraction in frequency domain, is worthy of investigation. This kind of non-modal parameter is based on the statistical properties of objective signals, such as means, variances and root mean squares (RMS), and by these means the statistical methodology is introduced into the damage detection area. Using statistical analysis is expected to reduce measurement noise influence to some extent, especially for random noise. The application potential of such research can be seen but so far little research has been carried out. Therefore this study attempts to propose damage localization indices based on the statistical properties calculated using the power spectral densities (PSDs) of vibration signals. Meanwhile, the proposed indices are also expected to locate minor damage or defects.
Related to the work developed in this paper, Das and Dey [18] studied the random responses of two numerical locally damaged beams. The response PSDs of response characteristics within a cut-off frequency range were computed for damage detection on a simply supported beam and a five-span continuous beam under clipped white noise and acoustic jet noise, respectively. It was found that the displacement PSD is sensitive to the damage especially when the damage approaches the center of the simply supported beam. Cacciola et al. [19] investigated the vibration response of a numerical nonlinear cantilever beam using the Monte Carlo method to evaluate the higher order statistical properties in time domain. It was observed that the skewness coefficient of the rotational degrees of freedom is highly sensitive to the nonlinear behavior of the structure and thus can be used for damage detection purposes. In order to locate the single damage in a simply supported beam, an index combining mode shapes and RMS values estimated in the narrow bandwidths near the frequency resonances was proposed by Liberatore and Carman [20]. It was shown that in both the numerical and the experimental analysis, distinct peaks appeared at the damage locations in the index value plot along the beam length. A minor defect could be the presence of the symmetric peaks in some scenarios in which the damage might not be uniquely identified. As an alternative method to Ref. [20], Fang et al. [21] gave a damage localization index constructed on the standard deviations of structural response PSDs. The index was verified against a numerically simulated beam and an experimental reinforced concrete beam and the results demonstrated that this index appeared to be sensitive to the damage location especially for the cases studied in which noise was not present. Another recent approach using the cross correlation function amplitude vectors (CorVs) of the measured responses was described by Yang et al. [22]. The normalized CorVs were used to construct a damage location index, which was consequently used to detect any fasteners becoming loose in an aircraft panel. The authors concluded that this non-model-based approach can be applied to structural health monitoring (SHM) with steady ambient excitations.
In this study, a new concept of power mode shape (PMS), as an alternative to the conventional mode shape concept, is proposed based on the statistical analysis of vibration signals. The PMS utilizes the RMS values calculated directly from the PSDs of objective signals and is found to have a similar shape to the conventional mode shape. Meanwhile, referring to the concepts of modal curvature and flexibility, PMS is used to derive two extended parameters power mode shape curvature (PMSC) and power flexibility (PF), which are subsequently used for establishing the desired damage localization indices. Indices based on the absolute differences in PMSC and PF between damaged and undamaged structures are used to localize the damage in different structures including a single-span beam, a mechanical system and a bridge taking account of the noise effect. On the other hand, it should be noted that this is a piece of initial research to investigate the feasibility of such a statistics-based method and thus a further validation on more engineering structures is a prerequisite before any practical application.
Section snippets
Power spectral density analysis
Provided a linear structure is excited at point r by a stationary and ergodic random force having zero mean and the corresponding response at point i is represented as , then the autocorrelation function of agrees with its mean square value (MSV), , when the time lag :The last term of this equation represents the average power of the output signal, which demonstrates its relation with the MSV. Since and its root vary with
Damage localization indices based on PMSs
By extending the concepts of modal curvature and flexibility to our case, two new damage localization indices have been defined based on PMSs.
Description
A numerical simulation on a single-span beam was firstly adopted to examine the performance of the damage indices proposed in Section 3. A schematic diagram of this beam with its geometric dimensions and material properties is shown in Fig. 1. The beam was assumed to be lightly damped with a constant damping ratio of 0.2%. For analysis purposes the beam was divided into 20 two-dimensional beam elements and fixed ends were considered.
The damage was numerically simulated by introducing stiffness
Problem description
An experimentally tested 8-dof mechanical system [23] (Fig. 5) was also used for the validation of the proposed damage indices. This system consists of eight translating masses connected by seven coil springs where the masses are able to slide along a highly polished steel rod. In the undamaged state, all the masses are identical with the same geometric dimensions and weight of 419.4 g except the 559.3 g weight attached to the force transducer. All the springs share the same material properties
Problem description
The I-40 bridge over the Rio Grande in New Mexico (USA) was dynamically tested by Farrar et al. [24] and its dataset was analyzed in the past by many researchers using different damage detection methods. With the same purpose, an FE benchmarked model of the bridge has been used in this work to evaluate the feasibility and reliability of the proposed method under different damage scenarios.
The I-40 bridge consists of twin spans made up of a concrete deck supported by two welded-steel plate
Conclusions
Two damage localization indices based on a novel concept of power mode shapes are proposed in this paper. Instead of the conventional modal parameters, the statistical properties of random signals and bandwidth-localized energy concept are used for their formulation. The proposed indices are designed for linear and lightly damped structures and were found to be capable of locating minor damage in different numerical and experimental cases. The ability to detect minor damage qualifies the
Acknowledgments
This research is supported by the Ministry of Education and Science of Spain (Project: BIA2007-67790). The authors gratefully acknowledge the work that was done by the Los Alamos National Lab for the data used in this study. The authors also express their deepest gratitude to the reviewers for their invaluable comments and suggestions which greatly helped in improving the quality of the paper.
References (26)
- et al.
Assessment of vibration-based identification techniques damage
Journal of Sound and Vibration
(2006) - et al.
Development in vibration-based structural damage detection technique
Mechanical Systems and Signal Processing
(2007) - et al.
Damage detection from changes in curvature mode shapes
Journal of Sound and Vibration
(1991) - et al.
Damage detection in bridges using accurate modal parameters
Finite Elements in Analysis and Design
(2004) - et al.
Damage detection in structures using changes in flexibility
Journal of Sound and Vibration
(1994) - et al.
Experimental verification of flexibility difference method for locating damage in structures
Journal of Sound and Vibration
(1995) - et al.
Damage assessment using mode shape sensitivities
Mechanical Systems and Signal Processing
(2003) - et al.
A flexibility-based continuum damage identification approach
Journal of Sound and Vibration
(2005) - et al.
An evolutionary multiobjective framework for structural damage localization and quantification
Engineering Structures
(2007) - et al.
A multistage FE updating procedure for damage identification in large-scale structures based on multiobjective evolutionary optimization
Mechanical Systems and Signal Processing
(2008)
Spectral and modal parameter estimation from output-only measurements
Mechanical Systems and Signal Processing
System identification from ambient vibration measurements on a bridge
Journal of Sound and Vibration
Reference-based stochastic subspace identification for output—only modal analysis
Mechanical Systems and Signal Processing
Cited by (70)
A novel sensitivity-based finite element model updating and damage detection using time domain response
2022, Journal of Sound and VibrationCivil structure condition assessment by a two-stage FE model update based on neural network enhanced power mode shapes and an adaptive roaming damage method
2020, Engineering StructuresCitation Excerpt :RDM provides a more direct and accurate means of locating damage and assessing its severity However, a new concept of PSD-derived power mode shapes was proposed for use in damage detection by Fang and Perera [26]. Damage indicative features using the curvature and the flexibility of these PMSs were used to determine likely areas of damage.
Deterministically generated negative selection algorithm for damage detection in civil engineering systems
2019, Engineering StructuresStatistical modeling for fast Fourier transform coefficients of operational vibration measurements with non-Gaussianity using complex-valued t distribution
2019, Mechanical Systems and Signal ProcessingDamage localization of beam structures using mode shape extracted from moving vehicle response
2018, Measurement: Journal of the International Measurement ConfederationDetection of damaged supports under railway track based on frequency shift
2017, Journal of Sound and VibrationCitation Excerpt :Various vibrational characteristics have been investigated and among them, mode shape curvature [3], strain energy [4], flexible matrix [5] and flexibility curvature [6] are very famous, based on which many advanced damage detection methods and indices are developed. Fang and Ricardo [7] proposed a concept of power mode shape which is equivalent to the square of mode shape to detect damages in linear structures. Grande and Imbimbo [8] presented a fusion approach to detect multiple damage locations based on changes of flexibility.