Estimación de las frecuencias naturales de estructuras civiles a través del algoritmo clasificación múltiple de señales y la transformada rápida de Fourier

“Natural frequencies estimation of civil structures throughout the Multiple Signal Classification algorithm and the Fast Fourier Transform”

“Estimación de las frecuencias naturales de estructuras civiles a través del algoritmo clasificación múltiple de señales y la transformada rápida de Fourier”

Abstract

The modal parameters identification, in particular natural frequencies, is a challenging task since the signals used are noise-corrupted, making the selection of the best-suited technique a complicated decision-making process in order to estimate them with a high accuracy. In this paper, the natural frequencies estimation of civil structures throughout the Fast Fourier Transform (FFT) and the Multiple Signal Classification (MUSIC) algorithm is presented. In order to assess the algorithms performance under ideal and noisy environments, a synthetic signal with different levels of noise is analyzed. The accuracy of the proposed algorithms is also demonstrated using the simulated response of a 4-story 2 × 2 bay 3D frame structure-based finite element model. The obtained results show that the Multiple Signal Classification technique presents a high accuracy in comparison with the Fast Fourier Transform for natural frequencies estimation of civil structures.

Keywords: Civil Structure, FFT, MUSIC, Natural Frequencies, Signal processing.

Resumen

La identificación de los parámetros modales, en particular de las frecuencias naturales de las estructuras civiles, es una tarea que requiere algoritmos de procesamiento de señales con características específicas, ya que las señales a procesar generalmente están contaminadas con un alto nivel de ruido. Esta característica hace que la selección de la técnica apropiada para la estimación de las frecuencias naturales sea un proceso arduo, ya que la estimación de las mismas debe ser con una gran exactitud. En este trabajo, la estimación de las frecuencias naturales de estructuras civiles a través de la transformada rápida de Fourier (FFT) y el algoritmo clasificación múltiple de señales (MUSIC) es presentado. Con el objeto de verificar el desempeño de ambos algoritmos, una señal sintética con diferentes niveles de ruido es analizada. Posteriormente, la señal estimada a partir de un modelo de elemento finito de una estructura de cuatro pisos es procesada. Los resultados obtenidos indican que las frecuencias obtenidas con el algoritmo de clasificación múltiple de señales es más exacta que las obtenidas con la transformada de Fourier.

Palabras Clave: Estructura civil, FFT, Frecuencia natural, MUSIC, Procesamiento de señales.

1. Introduction

The Structural Health Monitoring (SHM) technology provides technical solutions to review the condition of a wide variety of mechanical structures (Catbas et al., 2013; Ettouney and Alampalli, 2012; Hu and Ji, 2017) SHM has been used in applications from various fields such as aerospace, mechanics, automotive, and civil engineering, among others, since it allows the continuous verification of structural characteristics of the civil structure under study. SHM is focused on generating a constant monitoring of civil structures throughout the measuring of the structure physical properties such as mass, stiffness, among others, which can produce variations in the structural properties, allowing the detection damages or deterioration in order to guide its inspection and maintenance (Aktan et al., 2000; Brownjohn and Pan, 2008; Hu and Ji, 2017). In particular, the modal parameters estimation such as natural frequencies can be used to create systems capable of estimating damages into civil structures; however, their correct estimation represent challenge because the measured signals are embedded in high-level noise and present non-stationary properties (Amezquita-Sanchez and Adeli, 2014). For these reasons, a signal processing technique that allows identifying them with high accuracy is of vital importance.

In order to calculate the natural frequencies of a civil structure, the signal processing techniques used for this task have been divided into two groups: time-domain and frequency-domain techniques. Time-domain techniques require to set the number of frequencies to be found, which in most of the real-life applications is difficult to know (Perez-Ramirez et al., 2016), thus limiting their application. For this reason, frequency-domain techniques represent a better option because they do not require a fine calibration and are easy of implementing (Amezquita-Sanchez and Adeli, 2014). In this sense, the Fast Fourier Transform (FFT) has been used to estimate the modal parameters of 2- and 10-story buildings, a reinforced concrete (RC) building, and a truss structure subjected to ambient vibrations (Brincker et al., 2001; Yuen and Katafygiotis, 2005; Amezquita-Sanchez, 2012). Further, FFT has been used to detect damages in pipes (Cheraghi et al., 2005), steel bridges (Lee and Kim, 2007), a six-story steel building (Hsu et al., 2011), and highways (Hu et al., 2013). Despite the results presented by the authors of the abovementioned works, it is found that the FFT has the following limitations: (1) it cannot be used for monitoring civil structures subjected to time-varying signals and (2) its accuracy is reduced in noisy signals such as the measured in civil structures subjected to ambient vibrations. The first limitation makes FFT unable to correctly analyze signals that are variant over time; on the other hand, second limitation indicates that the noise contained in the measured signal can introduce spurious frequencies, masking the true ones. Therefore, it is necessary to explore other signal processing techniques that can work in environments such as those previously described. A promising technique is the Multiple Signal Classification (MUSIC), which has allowed notable results, which are reported in the detection of the direction of sounds arrival (Nagata et al., 2009) and damages detection in induction motors (Garcia-Perez et al., 2012). In addition, the MUSIC algorithm is known to have a greater noise immunity and an improved weak frequency detectability with respect to the FFT (Amezquita-Sanchez and Adeli, 2014). For these reasons, the use of MUSIC for the natural frequencies estimation of civil structures should be explored.

In this paper, the MUSIC algorithm is presented for the estimation of the natural frequencies of civil structures. The effectiveness of the MUSIC algorithm is evaluated through two cases. In the first case, the simulated free vibration response of a 3-degree of freedom system is used to evaluate the accuracy and noise immunity of the MUSIC algorithm for estimating the natural frequencies. In order to demonstrate the superiority of the proposed methodology, the results are compared with those obtained from the FFT method. In the second case, the proposed technique is applied to estimate the natural frequencies of a 4-story 2 × 2 bay 3D steel frame structure subjected to dynamic excitation produced by noise white. The obtained results are compared with those obtained by a finite element model provided by Dyke et al. (2001) and the ones obtained by means of the FFT method. The obtained results show that the MUSIC technique presents a high accuracy in comparison with the FFT for natural frequencies estimation of civil structures.

2. THEORETICAL FRAMEWORK.

1. Fast Fourier Transform (FFT).

The FFT algorithm is a very efficient strategy to estimate the frequencies contained in stationary signals. It obtains its results calculating the Discrete Fourier Transform (DFT). The DFT uses N² multiplications, while FFT only uses Nlog2(N) multiplications, where N is the number of samples of the signal being analyzed. The DFT is obtained as (Proakis and Manolakis, 1999):

                                                                                                                                       (1)

      Rewriting Equation (1) is obtained:

                                                                                                                                      (2)

where  This simplification allows observing that the same values of  are calculated several times while the DFT is calculated. Using the symmetry property of the twiddle factor, the calculation of DFT can be performed efficiently by expressing the Equation (2) as:

                                                                              (3)                  

Simplifying the indices within the summations and rearranging the last term, Equation (3) can be expressed as:

                                                                                         (4a)

                                                                                                                               (4b)

Consequently, a DFT of N points can be obtained as two transformed with N/2-1 samples; one of them for even samples and another for odd ones.

1. Multiple Signal Classification

The MUSIC algorithm is known for being a high-resolution method capable of identifying frequencies in signals embedded in high-level noise. To achieve the detection, the method assumes that the discretized signal in study, x[n], is represented as a sum of complex sine-wave samples m plus the noise associated with the measurement process:

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