*5.2. Data Acquisition*

The monitoring trial of the CB4A offshore oil platform was carried out in Dongying city, Shandong Province, on 3 December 2019. The MIMU and GPS data sampling frequencies were set at 100 Hz and 5 Hz, respectively, with a 4.5 min sampling time (GPS time: 200088.0–200358.0). We planned to design an experiment for researching the dynamic response of offshore oil platforms induced by wind and waves from the beginning. However,

as CB4A offshore oil platform was a stable rigid body at the height of only 7 m, it would be difficult to produce dynamic responses caused by slight winds or waves. Unfortunately, the weather was suitable, with no strong winds and/or waves during the tests. Thus, wind and waves were replaced with ship collision events. The ship collision experiment was conducted on a calm afternoon, which would avoid the influence of wind and waves. A small ship, about 5 m long, successive hit the leg of the platform traveling from south to north at a speed of 20 km per hour. When the small ship started to hit the leg of the platform, the experimenter recorded the current Beijing standard time. The continuous collision lasted for about 4 min. Eight collisions were made, and the time interval of each impact was less than 1 min. The time of each impact was recorded manually and is shown in Table 4.

**Table 4.** Record schedule of ship collision.


*5.3. Dynamic Responses Time–Frequency Extraction*

As the X-axis installation of the inertial sensor and ship collision were both directed toward the north, which means, the X-axis was the main direction of the dynamic response. Therefore, the paper only analyzed the X-axis data for dynamic response monitoring. As shown in Figure 15, the original X-axis data of the accelerometer were collected, and changes of acceleration were probably caused by multiple ship collisions; the corresponding collision time was estimated to be at the 7th s, 59th s, 89th s, 147th s, 198th s, 225th s, 237th s, and 252nd s, respectively (the second and third collisions were not obvious to be identified directly). They were basically consistent with manual records without abnormality; hence, subsequent analyses could be carried out.

**Figure 15.** Acceleration and corresponding spectrum densities: raw signal of accelerometer (**top**) and power spectral density (**bottom**).

PSD was utilized to extract frequencies of dynamic responses of ship collisions, as represented in the bottom of Figure 15, where four peaks can be seen at 8.55 Hz, 9.23 Hz, 11.18 Hz, and 17.50 Hz separately, with the largest peak at 8.55 Hz. It could be concluded that much information was missing in the spectrum as (i) the sequence and time of collisions were unable to be identified corresponding to the four frequencies without the time-domain information in the spectrum and (ii) only four peaks were produced by eight ship collisions, which indicated the possibility of a common frequency phenomenon or the covering of high-power spectral density on the low-power ones. Therefore, PSD could not extract complete and reliable frequencies of dynamic responses for the ship collisions.

To further study the frequency characteristics of collisions in the time domain, the VMD–HHT method proposed in this paper was adopted to carry out a three-dimensional synchronous analysis of time–frequency energy. First, five IMFs were produced using selfadaptive VMD, as illustrated in Figure 16. All those functions contained dynamic responses of ship collisions, indicating the accelerometer's high accuracy, which was unaffected by high- and low-frequency noise and could be directly reconstructed without denoising. Later, following the Hilbert transform of the reconstructed acceleration data, the Hilbert time–frequency spectrum was obtained, as shown in Figure 11. It was evident that seven collisions, along with each corresponding time, frequency range, and energy (collision intensity), were displayed synchronously, except that the frequency range of the fifth collision was found to be between 3 Hz and 15 Hz, while others were between 5 Hz and 12 Hz. Referring to the qualitative analysis of energy, the third collision (89 s) was invisible, indicating that its intensity was the smallest. Among the seven visible collisions (Figure 17), the intensity of the second collision was the lowest, consistent with the analysis results of the original data; the intensity of the fifth collision was the highest, and its enlarged view is reflected in Figure 18, while the collision lasted for about 7 s, and the intensity changed from high to low. In summary, this VMD–HHT-based method could accurately extract frequency ranges of collisions in the time domain and clearly characterize the intensities of each collision.

**Figure 16.** IMF components of accelerometer-recorded acceleration based on VMD.

**Figure 17.** HHT spectrum of acceleration.

**Figure 18.** HHT spectrum of the fifth collision.

### *5.4. Reconstruction of Dynamic Displacement Based on VMD–HHT Using Accelerometer*

The frequency-domain integral equation is a powerful approach to process acceleration data using a double integral to reconstruct dynamic displacement information. According to the frequency range obtained through the VMD–HHT method, the minimum cutoff frequency was set as 1 Hz, and the maximum was set as 20 Hz. Figure 19 shows the accelerometer-derived displacement after double integration. We can observe that the displacement of the third collision (89th s) was minimal. Thus, the third collision on the platform was weak. Referring to the manual record, original data, and the HHT spectrum, the unknown displacement occurred at 180th s without any collision; the displacements of eight collisions were within 6 mm. Compared with related deformation data of bridges, high-rise buildings, and dams, deformation of the offshore oil platform collided by ships was smaller [1,9]. The reason for this is that the offshore oil platform was a stable rigid body, and the colliding ship was small when anticollision rubbers were tied to the platform pile.

**Figure 19.** Accelerometer-derived displacement by frequency-domain integration.

### *5.5. Evaluation of the Torsion Angle Based on Mahony Complementary Filter Using MIMU*

To obtain torsion angle responses caused by ship collisions, the gyroscope data of MIMU were analyzed by Mahony complementary filter. Figure 20 shows the three-axis angular rate data correctly without bias, suggesting that the Y-axis of the gyroscope was influenced by the collision most. From the above analyses, it can be concluded that X-axis was the main direction of displacement response. When the ship stroked the north of the platform (X-axis), displacement responses would be caused in the same direction, while the platform was found out to incline towards Y-axis. A visual representation of the process is given in Figure 21.

Figure 22 (bottom) shows the PSD in correspondence to angular rate output in the Y-axis of the gyroscope, which included five peaks whose corresponding frequencies were 8.57 Hz, 9.23 Hz, 11.18 Hz, 13.45 Hz, and 15.00 Hz, respectively. The main peak frequency was 8.57 Hz. Referring to Figure 2, the frequencies extracted from the X-axis of the accelerometer were 8.55 Hz, 9.23 Hz, 11.18 Hz, and 17.50 Hz independently, while the main peak frequency was 8.55 Hz. By comparing two sets of data, it is evident that

collision frequencies extracted from the X-axis of accelerometer and Y-axis of gyroscope were very similar. To be more specific, the distribution was consistent, but the amplitude corresponding to each frequency had a high coherence. To summarize, the frequencies of dynamic responses could be extracted from gyroscope data of MIMU.

**Figure 20.** Three-axis raw signals recorded by gyroscope.

**Figure 21.** Schematic of dynamic displacement and pitch caused by the ship collision.

**Figure 22.** Gyroscope-recorded signal and corresponding spectrums: raw signal at the direction of Y (**top**) and power spectral density (**bottom**).

To further explore the relationship between the accelerometer and gyroscope axis system during collisions, three-axis PSD of accelerometer and gyroscope were given, as indicated in Figures 23 and 24. Amplitudes processed by PSD through accelerometer from high to low were X-axis ≥ Z-axis ≥ Y-axis. Those of gyroscope were Y-axis ≥ Z-axis ≥ X-axis, which means the event had the most significant impact on the X-axis of the accelerometer and the Y-axis of gyroscope, followed by their Z-axes. By examining the frequency distribution, the similarity of PSD was high among X-axes and Z-axes of the accelerometer and Y-axis of the gyroscope. According to the results, when the ship struck the platform from X-direction (north direction), a displacement response in the Z-axis would be caused by the displacement response of the X-axis and the rotation of the Y-axis; the PSD of the Y-axis of the accelerometer was similar to that of Z-axis of the gyroscope (as X-axis of the gyroscope was seriously affected by the noise; thus, no comparative analysis would be performed). Hereby, two conjectures are proposed as follows: (i) Since the collision direction was not completely in the X-direction and there was a component on the Y-axis, it means that a small-angle oblique impact has occurred to cause the rotation of the Z-axis, and the horizontal distortion of the platform is illustrated in Figure 25; (ii) the X-axis of the installed inertial sensor did not coincide with the north direction, that is, the installed axis deviated from the local geographical coordinate system, resulting in the component on the Y-axis to cause a small-angle oblique collision and Z-axis' rotation.

**Figure 23.** PSD of the acceleration recorded by the accelerometer in three directions.

**Figure 24.** PSD of the angular velocity recorded by the gyroscope in three directions.

**Figure 25.** Schematic of dynamic displacement and torsion angle caused by ship collisions.

To obtain the torsion angle response data of the platform, Mahony Complementary Filtering was used to solve the torsion angle change of platforms by integrating the information of accelerometer and gyroscope. As can be seen from the torsion angle calculation results of Figure 26, the pitch angle (Y-axis rotation) changed significantly with a maximum angle of 1.2◦, followed by the roll angle (X-axis rotation) with a maximum angle of 0.15◦. The heading angle (Z-axis rotation) did not change significantly with the impact event, but it deviated about 1.1◦ from the X direction (north direction). Therefore, Conjecture 2 could be verified. In other words, the misalignment between the installation axes of MIMU and the geographical coordinate system leads to the displacement component of the Y-axis and the rotation of the Z-axis. All the above analyses show that ship collisions will cause displacement responses and make the platform tilt or even twist. In addition, the relationship between the accelerometer and the gyroscope is also revealed: The impact on the X direction caused the Y-axis to rotate, that is, the pitch angle changes. The impact direction had a component on the Y-axis to cause the X-axis rotation, which is, the roll angle changes. The Z-axis rotation was caused by a small-angle oblique impact, which was the change of heading angle.

**Figure 26.** Three-axis torsion angle by Mahony complementary filter using MIMU.
