Latest revision as of 20:29, 20 August 2019

Abstract: Dynamic characterization of structures from field measurements is useful for different purposes (e.g. retrofit validation, model updating, structural health monitoring, etc.). The identification of high spatial density mode shapes has been recently a challenge tackled using mobile sensors. These sensors travel over the structure and continuously acquire vibration data that is used to identify modal coordinates with a higher spatial density than can generally be obtained using a limited number of stationary sensors. The recorded signal from a mobile sensor is non-stationary, thus, it has significant variations in its spectral content over time, requiring a suitable processing to extract the properties not only for the time but also for the frequency domain. In this paper, Cohen’s class Time-Frequency Distributions (TFD) are proposed for the output-only dynamics identification of structures based on non-stationary signals recorded with mobile sensors. Identification is achieved through cross-time-frequency estimators using Smoothed Pseudo-Wigner-Ville (SPWVD) distribution. Results from numerical simulations using a simply supported beam subject to ambient vibration are shown and the sensibility of the proposed identification to the presence of measurement noise is evaluated. Numerical results show that use of the cross-time-frequency estimators is effective in extracting modal properties of the structures and filtering noise.

Keywords: Modal Identification, Mobile Sensors, Non-Stationary Signals, Time-Frequency Distributions

1. Introduction

Structural Health Monitoring (SHM) Systems usually play a crucial duty in early failure or damage detection for construction, maintenance or catastrophe management. Civil infrastructure and many other sectors can greatly benefit from systems that monitor behaviour under normal operating conditions, without the need of interrupting regular day-to-day activity (Brownjohn, 2007; Karbhari et al., 2009).

The future trend in instrumentation for SHM is the use of smart mobile sensing along with wireless intelligent networks, as they not only provide similar information to conventional cabled systems, but can also be installed at a much lower cost and process data autonomously using their embedded microprocessors and software (Lynch, 2006; Spencer et al., 2004). Modal shapes are traditionally identified with stationary sensors fixed at locations that provide representative structural responses, however, these schemes contain restricted spatial information (Figure 1). If more spatial and temporal data is obtained from a vibrating structure, the structure’s response can be evaluated more accurately, and hence the advantage of implementing the structure with as many stationary sensors as possible during the acquisition of data. However, despite the improved accuracy in estimations, these or dense sensor array approaches can be impractical for many reasons including the cost of sensors and setup time, memory constrains, network reliability, power requirements, and physical limitations due to structure geometries (Magalhães et al., 2008, 2009; Matarazzo T. J. and Pakzad, S. N. 2013). Mobile sensing is a novel alternative in Structural Health Monitoring (SHM). The objective is to solve failings from fixed sensor array setups. Few sensors are usually used to collect dense spatial information in mobile sensing setups. Mobile sensing provides several advantages over schemes using stationary sensors, but the main advantage is that a single mobile sensor can be used to record signals continuously along the structure. Modal coordinates are identified with a higher spatial density than using stationary sensors because the mobile sensor continuously records vibration data as it travels along trajectories on the structure (Figure 2). Mobile sensing is not only easier to implement, but also more cost effective when compared to dense stationary sensor arrays (Marulanda et al., 2016; Marulanda, J., 2014; Matarazzo T. J. and Pakzad S. N., 2013).

Recent implementation of mobile sensor networks has been diverse, although limited, and is still under development. Zhu et al. (2010, 2012) proposed a sensing device. Such device will record data at selected nodes, but will not do so while the sensor is moving. Sibley et al. (2002) and Dantu et al. (2005) implemented Robomote, miniature low cost robots for mobile sensors networks. Partial modal identification include the use of a moving vehicle to identify frequencies of a single span bridge using frequency domain techniques (Cerda F. et al., 2012; Lin and Yang, 2005). Marulanda et al. (2016) and Marulanda (2014) developed a novel technique that uses two sensors, one mobile and the other stationary, in which the mobile sensor records continuously, and, in a subsequent procedure, performs the modal identification through spectrograms. Matarazzo et al. (2014, 2016) proposed a method of collecting data through continuous mobile detection in the presence of missing time and space observations. As the corresponding matrices have incompatible data records, they developed and successfully implemented an algorithm, called STRIDE, to cover for such “empty spaces”.

The use of mobile sensors, typically accelerometers, require an analysis of non-stationary response signals. Time-Frequency analysis has been proposed by several authors to analyse non-stationary responses in structural identification and in the assessment of mechanical damage. Staszewski et al. (1994) and Rioul et al. (1991) have devoted several papers to modal identification using the Short-Time Fourier Transform (STFT) and Wavelet Transform (WT). The STFT transforms the signal in a two-dimensional time-frequency plane, assuming that the signal is stationary when viewed through a limited extension window. The Fourier Transform of the windowed signal leads to a time-frequency distribution (Gabor, 1946; Arango, 2009; Oppenheim, 2010). The STFT is linear and therefore provides poor energy information about the signal. Quadratic Time-Frequency Distributions (TFDs), on the other hand, allow the interpretation of representations from a much richer energy point of view (Hlawatsch and Boudreaux-Bartels, 1992; Loughlin et al., 1993). Marginal properties express such interpretation. However, in view of the Uncertainty Principle, such properties are not sufficient to identify an energy density at each point on the time-frequency plane. (Ceravolo, 2009). A spectrogram, or quadratic representation of the amplitude of the STFT, is a three dimensional representation of the time-averaged Fourier Transform over adjacent time segments of a random process (Bendat and Piersol, 2011). It does not fulfil marginal properties and violates the Linearity Principle, generating crossed or interference terms. These spurious terms are restricted to those regions of the time-frequency domain where the auto-terms overlap. In the specific case of the spectrogram, for two sufficiently separated components in the time-frequency plane, its interference terms are practically nil (Ceravolo, 2009). Spectrograms can be used to obtain modal coordinates from the signals recorded from a mobile and a stationary sensor. These signals are divided into quasi-stationary segments and then the auto-spectral density and cross-spectral density functions for segments of both records are calculated. Finally a modal identification procedure is performed (Marulanda et al., 2016; Marulanda, J, 2014).

In this paper, the non-stationary signals analysis recorded from sensors is performed by using Cohen’s class TFDs. The adoption of a quadratic representation helps in overcoming the limitations from time-frequency resolution, as these transformations, the energetic and the correlative, are not based on the segmentation of signals. (Cohen, 1995). Invariance to time and frequency shifts characterize the Cohen’s Class Distributions (Hlawatsch and Auger, 2013). The system response is recorded in the time-frequency plane as the evolution of the spectral components corresponding to the energy of individual vibration modes whenever the Cohen’s Class Distributions is in use (Bonato et al., 1998, 1999; Bonato et al., 2000; Bonato P. et al., 1997; Cohen, 1995; Hammond and White, 1996; Hlawatsch and Boudreaux-Bartels, 1992). The proposed technique assumes that the frequency range of the system input spans the vibration modes. As the input spectrum increases, so does the identification accuracy. The instantaneous amplitude ratios directly determine the modal amplitude ratios for the time-frequency representations of signals recorded from a mobile sensor and a stationary sensor. Additionally, the phase of the cross-time- frequency representation between the two signals estimate the phase relationships. The estimators retain their dependence on time since they are derived directly from the two-dimensional functions of frequency and time variables. modal responses in linear time invariant systems show a constant relationship between the amplitude and the phase, and, therefore, the identified mode shapes are characterized by their stability over time (Bonato et al., 1998).

2. Theoretical Background

Cohen’s Class Distributions

Cohen’s class distributions display a clear invariance of their representations for time and frequency shifts, a desirable property when correlating signal characteristics to phenomena occurring in the physical system generating the signal. The following equation describes all the Cohen’s class distributions (Hlawatsch and Auger, 2013):

where ${\textstyle s\left(t\right)}$ is the signal, ${\textstyle {s}^{\ast }\left(t\right)}$ is its complex conjugate, ${\textstyle t}$ is time, ${\textstyle f}$ is frequency and ${\textstyle \tau }$ is the time delay, ${\textstyle \omega =}$${\displaystyle 2\pi f}$ is the circular frequency; ${\textstyle \Phi (\theta ,\tau )}$ is the kernel of the distribution and ${\textstyle D\left(t,f\right)}$ is the Cohen Class TFD. The cross-time-frequency distribution between two signals ${\textstyle s\left(t\right)}$ and ${\textstyle y\left(t\right)}$ can be defined as follows:

The Wigner–Ville distribution (WVD) is the basic TFD and is a quadratic form that measures a local time-frequency energy. This distribution is obtained from equation (1) using ${\textstyle {\Phi }_{WV}\left(\theta ,\tau \right)=}$${\displaystyle 1}$. Although the WVD satisfies a several important mathematical properties (Bonato et al., 2001; Flandrin, 1984; Wang and McFadden, 1993), it is not very suitable for applications with multiple components signals since the bi-linearity of the transform causes the appearance of interference terms . These are spurious terms when using the time-frequency characteristics of the signal for modal identification. The smoothed pseudo-Wigner-Ville transform (SPWVD) results from ${\textstyle {\Phi }_{SPWV}\left(\theta ,\tau \right)=}$${\displaystyle \pi {e}^{-{T}^{2}{\pi }^{2}{\theta }^{2}}{e}^{-{B}^{2}{\pi }^{2}{t}^{2}}}$, where ${\textstyle B}$ and ${\textstyle T}$ are the standard deviations in time and frequency, respectively. With the exception of real value, time-shift-invariance, and frequency invariance, the application of the smoothed function causes the loss of most of the mathematical properties of the WVD, (Hlawatsch and Boudreaux-Bartels, 1992). The foregoing independence between windows makes the SPWVD very versatile for reducing cross—terms in the WVD. This property is useful for modal identification.The Choi-Williams Distribution (CWD) (Choi and Williams, 1989; Papandreou and Boudreaux-Bertels, 1993) is defined by ${\textstyle {\Phi }_{CW}\left(\theta ,\tau \right)=}$${\displaystyle {e}^{\frac {-{\theta }^{2}{\tau }^{2}}{\sigma }}}$. This kernel is defined in the ambiguity plane ${\textstyle \left(\theta ,\tau \right)}$ . If ${\textstyle \sigma }$ is very large, the kernel vanishes, therefore leading to the WVD, otherwise, the kernel application will have a filtering effect. For small values ​​of ${\textstyle \sigma }$ , the effect of the multiplication is to preserve the ambiguity function close the origin of the plane ${\textstyle \left(\theta ,\tau \right)}$ , therefore, this kernel satisfies the minimization property (Cohen, 1995). In multicomponent signals the authentic terms are usually close to the ambiguity plane axis, and the interference terms are dispersed away. The resulting characteristic function reduces the interference terms without significantly affecting the signal components (Bonato et al., 1997). Therefore, although the CWD violates the time and frequency support properties, it can be shown that the introduced error is generally negligible (Lippmann, 1989; Papandreou and Boudreaux-Bertels, 1993), so the ${\textstyle \sigma }$ value should be adapted to each individual case, since the location of the interference terms and the signal components change with respect to the characteristics of the signal (Bonato et al., 1997). In this paper, SPWVD distribution is used. In this case, by parameterizing the windows, as they attenuate the interference terms without significantly modifying the signals, good modal identification results are obtained.

Modal Identification Using Mobile Sensors

The proposed methodology requires a mobile sensor that acquires data continuously along the structure to be identified and simultaneous acquisition by a stationary sensor. Under ambient excitation using a zero mean Gaussian white noise process, the outputs of the system are also zero mean Gaussian processes, exciting the fundamental frequencies (Bendat and Piersol, 2011). Since the recorded signal from the mobile sensor is non-stationary, a procedure based on the Cohen Class Distributions (i.e. SPWVD) is proposed. These distributions are invariable to time and frequency shifts, which is required for proper system identification.

Assuming that the structure to be identified is the one-dimensional beam shown in Figure 3 has ${\textstyle n}$discrete degrees of freedom (d.o.f.), wich each d.o.f. corresponds to a segment of the signal from the mobile sensor (Figure 4).

Let ${\textstyle {s}_{i}\left(t\right)}$ be the response (displacement) in the ${\textstyle i}$th position of the system, ${\textstyle {d}^{\left(k\right)}(t)}$ the response (displacement) associated with the ${\textstyle k}$th vibration mode, and ${\textstyle {\varnothing }_{i}^{\left(k\right)}}$ a term of the normalized eigenvectors matrix, then ${\textstyle {s}_{i}\left(t\right)}$ can be in decouple form as,

The term of the sum in equation (3) extends to the vibration modes that are detected in the ${\textstyle i}$th position corresponding to a discretized d.o.f. The structure under ambient excitation is equipped with a mobile sensor with constant velocity that continuously records the acceleration and, simultaneously, with a stationary sensor also recording the acceleration vibrations. For each ${\textstyle ith}$d.o.f., the individual modal components appear in the form of an energy peak in the time-frequency plane that can be written in complex form as (Bonato et al., 2000):

where ${\textstyle {\tilde {d}}^{\left(k\right)}(t)=}$${\displaystyle {A}^{\left(k\right)}(t){e}^{j{\varphi }^{\left(k\right)}(t)}}$ is a baseband signal related to the ${\textstyle kth}$ mode; ${\textstyle {A}^{\left(k\right)}\left(t\right)}$ and ${\textstyle {\varphi }^{\left(k\right)}(t)}$ are the amplitude and phase modulation waveforms respectively; ${\textstyle {f}^{(k)}}$ is the natural frequency; ${\textstyle {\varnothing }_{i}^{\left(k\right)}}$ is the amplitude of the ${\textstyle kth}$ mode shape in the ${\textstyle ith}$ position. The complex form of equation (4) represents an analytical signal and, therefore, the spectrum of ${\textstyle {d}^{(k)}~\left(t\right)}$ is single-sided. The WVD will transform the signal into: