**1. Introduction**

With the rapid growth in world energy demand, the number of offshore oil platforms keeps increasing gradually with progressively upgrading potential security hazards. Statistics show that several structural damage accidents in offshore oil platforms happen every year, and nearly half of them are caused by severe weather, such as typhoons, hurricanes, tsunami, earthquakes, etc. Furthermore, with the increasingly complex offshore environment, ships or unidentified objects may hit the platform occasionally [1]. Once an accident occurs, it will lead to heavy casualties, property loss, and environmental pollution. Hence, the monitoring of dynamic responses of offshore oil platforms is of vital importance to safety in the offshore oil industry.

Time and frequency data are essential in investigating the dynamic responses of offshore oil platforms. To obtain high-precision dynamic displacement, extracting frequency dominant information is necessary, which can also act as a principal method to analyze

**Citation:** Wang, J.; Liu, X.; Li, W.; Liu, F.; Hancock, C. Time–Frequency Extraction Model Based on Variational Mode Decomposition and Hilbert–Huang Transform for Offshore Oil Platforms Using MIMU Data. *Symmetry* **2021**, *13*, 1443. https://doi.org/10.3390/sym13081443

Academic Editors: Yang Yang, Ying Lei, Xiaolin Meng and Jun Li

Received: 11 July 2021 Accepted: 1 August 2021 Published: 6 August 2021

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dynamic response characteristics and structural symmetry. The fast Fourier transform (FFT) was traditionally applied to extract frequencies and amplitudes from monitoring datasets [2]. Nevertheless, FFT can neither extract local frequencies nor process nonstationary signals that constantly change [3]. The short-time Fourier transform (STFT) has been proposed to overcome such defects, allowing local characteristics of frequencies and amplitudes to be obtained using a moving window. STFT is also suitable for nonstationary signals as signals intercepted by the moving window can be regarded as linear [4]. However, neither FFT nor STFT can detect the relationships between time and frequency domains for time-varying signals.

To analyze frequency characteristics of nonstationary signals precisely in the time domain, the wavelet transform method was suggested because of its excellent local timefrequency properties [5]. As the wavelet transform method was of poor adaptive ability due to its fixed wavelet basis [6–8], the empirical mode decomposition (EMD) was subsequently proposed to overcome these limitations by decomposing the wavelet function adaptively [9]. Later, EMD was widely used as a time–frequency analysis method for its good performances in adaptive decomposition and nonlinear signal analysis [10,11]. However, several limitations of the EMD method have been identified, such as mode mixing (difficulty in separating modes effectively according to the time scale) and the endpoint effect (a problem of signal divergence caused by the repeated use of cubic spline interpolation) [12]. Improvements to the EMD have since been made and been applied as the ensemble empirical mode decomposition (EEMD) [13,14] and complementary ensemble empirical mode decomposition (CEEMD) [15]. However, the above methods cannot fundamentally remedy the inherent defects of EMD, primarily the phenomenon of mode mixing.

To avoid the limitations of the signal processing methods mentioned above, a variety of integration approaches can be employed, such as the EMD–wavelet and the Hilbert–Huang transform (HHT) [16,17]. The latter has been widely used to analyze dynamic responses of signals since its first application in information extraction from seismic waves [18,19]. The major advantages of HHT are as follows: (i) it is suitable for processing non-linear and nonstationary signals; (ii) it can modify the time scale adaptively; (iii) it can obtain 3D information consisting of time, frequency, and energy [20]. HHT is made up of Hilbert transform and EMD, in which EMD is the core. Accordingly, it is essential to improve the performance of EMD. Various approaches combining improved EMD and HHT are proposed, such as EEMD–HHT and CEEMD–HHT [21,22]. These approaches reduce the defects of mode mixing and endpoint effect of EMD to a certain extent.

In this study, variational mode decomposition (VMD) was first verified to avoid mode mixing and end effects [23]. Then, a combined model of VMD and Hilbert transform (VMD–HHT) was established to extract time–frequency–energy characteristics. Meanwhile, we efficiently eliminated the noise component from accelerometer data according to extracted frequency range using VMD–HHT. Thus, displacement responses were calculated using the frequency-domain integration approach (FDIA) using the accelerometer data. Additionally, torsion angles were obtained by the complementary filtering algorithm based on six degrees of freedom data from the MIMU. A series of simulation shaking-table tests were performed using accelerom in the MIMU to verify the reliability of the VMD–HHT model and displacement reconstruction method. Finally, a field test in the offshore oil platform was conducted. The results prove that VMD–HHT can notably help extract time–frequency–energy characteristics and symmetric information of offshore oil platforms wholly and accurately and improve displacement calculation by FDIA. Moreover, dynamic displacement responses and torsion angle information of the offshore oil platform can be calculated by MIMU alone.
