Research on the low - frequency multiline spectrum vibration control of o ﬀ shore platforms

: With the increasing scale, complexity and diver sity of supporting equipment of o ﬀ shore platform, the low - frequency vibration of equipment such as dynamic positioning system and the main engine is di ﬃ cult to attenuate in the propagation process of the platform struc -ture, which causes a local resonance of platform, aggra vates the fatigue damage of structure and causes discomfort to the human body. Dynamic vibration absorption is widely used in the low - frequency vibration control of o ﬀ shore plat forms; however, there is little research about the multiline spectrum vibration control method in the local resonance region of platforms. In the current research, we ﬁ rst take the sti ﬀ ened plate under multipoint excitation as the research object, and the e ﬀ ectiveness of the optimal homology design method of dynamic vibration absorption is veri ﬁ ed. Subsequently, the low - frequency multiline spectrum vibra tion control method about the local resonance region of the o ﬀ shore platform is proposed. Finally, a large o ﬀ shore plat form is chosen as the research object and the measured load of the main engine is taken as the input to calculate the vibration response of the platform. The e ﬀ ect of distributed dynamic vibration absorption of the resonance area veri ﬁ es the e ﬀ ectiveness of the vibration control method presented in the article and provides a basis for the engineering application.


Introduction
In the background of global economic development, offshore platform plays an increasingly important role in the exploration and exploitation of offshore oil and gas resources. Meanwhile, with the increasing scale and complexity of the offshore platform and supporting equipment, the structural vibration of the offshore platform caused by equipment cannot be ignored [1][2][3]. The familiar vibration control methods of offshore platforms can be divided into three types: passive control, semi-active control [4] and active control [5,6]. Passive control relies on the interaction between the device and the main structure, which does not need the input of external energy. In addition, passive control has the advantage of simple structure, good economy and stable performance [7]. For this reason, it has been widely used in marine engineering.
The common passive control forms include vibration isolation, energy dissipation and vibration absorption [2]. Vibration isolation is achieved by installing vibration isolation devices between the main structure and the vibration source. Common vibration isolation methods for offshore platforms can be divided into two categories: foundation vibration isolation and structural vibration isolation [8]. Foundation vibration isolation is realized by adding vibration isolation and energy dissipation devices between the bottom of the platform deck and the jacket [9]. The structural vibration isolation is implemented by changing the local structure of the platform. For example, the support leg of the offshore platform is divided into the vibration sharing structure composed of inner and outer pipes. Numerous experimental and calculation results have shown that the vibration isolation device can effectively reduce the vibration of the offshore platform [8,10,11]. However, vibration isolation measures are usually applicable to newly constructed platforms. It is neither convenient nor economical to adopt vibration isolation measures for completed platforms.
In the method of energy dissipation, the vibration of the main structure is consumed in the deformation and reciprocating movement of energy dissipation dampers. The common energy dissipation dampers for offshore platforms can be divided into two types: velocity-and displacement-dependent dampers. Displacement-dependent dampers include friction dampers and metal yield dampers. The energy dissipation of displacement-dependent dampers [12,13] is related to displacement and has nothing to do with the structural velocity response and frequency, so it is suitable for low-frequency vibration of platforms [14]. Shape memory alloy (SMA) is a classical kind of displacement-dependent damper.
As a novel type of smart materials, SMAs exhibit unique properties such as shape memory effect (SME), corrosion resistance, fatigue resistance and high reliability [15,16]. SME was first recognized by Buehler and Wang in the alloy of nickel and titanium, which withstands large deformation of up to 10 percent with no residual strain. The SME of smart materials makes it a terrific candidate for energy dissipation appliances and dampers [17]. In the recent studies of SMAs, Choi et al. [18] researched the impacts of geometric parameters on recovering aptitude of SMA fibers, and test samples with different diameters and crimped lengths have been investigated in detail. To better understand the tribological behavior of SMAs, Levintant-Zayonts et al. [19] conducted the ball-on-plate reciprocating sliding wear tests on NiTi SMAs under different test working conditions. Dutta and Majumder [20] applied Nitinol (an alloy of Ni and Ti) SMA damper to control the vibration of structures affected by the underground blast. A steel structure with various arrangements of the dampers was analyzed and compared with conventional steel bracing. Kamarian et al. [21] compared the thermal bucking behavior of simply supported composite beams composed of SMAs and carbon nanotubes. Dynamic mechanical thermal analysis and thermomechanical analysis experiments were also conducted on the above materials. Sheikhi et al. [22] studied the static and dynamic behavior of rubber bearing composed of SMA and structural steel (SS). By means of the finite element method, the thickness ratio between SMA and SS on different dampers has also been investigated in different bridge models. Li et al. [23][24][25][26][27] proposed a new semi-analytical method to analyze the vibration behavior of typical structures composed of laminated materials and functionally graded materials.
However, energy dissipation usually requires large relative deformation and the relative deformation of damping devices in the offshore platform is not always large enough, which results in a poor damping effect. To improve the vibration control effect of SMAs in offshore platforms, Ghasemi et al. [28] combined the advantage of SMA and tuned mass damper (TMD) to control the vibration of offshore platforms by means of simplifying the platform and the dynamic vibration absorption as a multi-degree of freedom system; the effects of SMA TMD under broadband frequency excitations have been discussed in detail.
The dynamic vibration absorption is not limited by the relative deformation of the main structure and it has been a wide concern to scholars and engineers. By means of arranging the dynamic vibration absorption subsystem on the main structure and adjusting the parameters of the subsystem, the vibration of the main structure is absorbed through the movement of the dynamic vibration absorption subsystem.
At present, tuned liquid damper (TLD) [29,30] and TMD [28,[31][32][33][34] are the two main dynamic vibration absorption forms used in offshore platforms. TLD is a subsystem attached to the main structure that contains liquid. When the TLD moves with the main structure, the sloshing and viscous motion of liquid in the container will dissipate the energy of the main structure and reduce the vibration. Vandiver and Mitome [35] proposed the application of TLD to the vibration control of the offshore platform in 1979, and the vibration characteristics of the platform after placement of TLD were analyzed. Finally, the vibration reduction effect of TLD was verified. Lee et al. [36] studied the vibration reduction effect of TLD on a typical tension leg floating platform. Through numerical simulation and experiments, it is verified that the tuned TLD system can effectively reduce the dynamic response of the offshore platform. Jin et al. [37] studied the influence of TLD on the vibration response of the platform under seismic waves, and the corresponding numerical analysis was conducted through the lumped mass method and compared with the model test results subsequently. The comparison results showed that the centralized mass method can effectively simulate the effect of TLD on the platform vibration control, and the frequency and mass ratio of TLD are the main influential factors of the vibration control effect. Lotfollahi et al. [38] also analyzed the control effect of TLD on the platform vibration under seismic wave excitation, and the parameters of TLD were optimized based on the finite element analysis results.
TMD can be simplified as a subsystem composed of mass, damping and spring. Many scholars have studied the theory and the application of TMD in offshore platform vibration control. Yue et al. [39] carried out many field tests on the Bohai Bay oil platform. The test results showed that when the ice sheet passes through the pile legs of the jacket platform, the main frequency of ice-induced load excitation is close to the natural frequency of the platform, which leads to ice-induced vibration of the platform. The calculation results showed that the ice-induced vibration of the platform can be effectively reduced by placing TMD. Taflanidis et al. [40] designed TMD for the tension leg platform under random wave load based on the simulation results, and an excellent vibration absorption effect is achieved in two different directions. Chandrasekaran et al. [41] also took the tension leg platform under random wave load as the research object, and the vibration control effects of single TMD and multiple TMD were compared.
In a word, many researchers have carried out research in the application of dynamic vibration absorption on offshore platforms and achieved a number of results. Whereas most research is limited to single-order global vibration of offshore platforms, a few studies have been conducted on the vibration control of local resonance region and low-frequency multiline spectrum dynamic vibration absorption of offshore platforms.
In view of the above shortcomings, we are inspired by the following literature studies [42][43][44][45] to solve this problem. The main innovations of the current research can be summarized as follows: First, the low-frequency multiline spectrum vibration control method of offshore platforms is put forward in the current research. By arranging distributed dynamic vibration absorption, the multiline spectrum vibration of the platform can be controlled simultaneously. Second, the modal superposition method is combined with the low-frequency multiline spectrum vibration control method. Through the establishment of an equivalent numerical model of the local resonance region, the efficiency of the dynamic vibration absorption design is greatly improved. Finally, the effectiveness of the current method is verified by the calculation of a largescale offshore platform, which lays the foundation for engineering applications.

Theoretical basis
When the main structure is simplified as a single degree of freedom system, the schematic diagram of the main structure with dynamic vibration absorber can be described as in Figure 1. As displayed in Figure 1, the mass, stiffness and damping of the main structure are, respectively, m 1 , c 1 and k 1 . Meanwhile, the mass, stiffness and damping of the dynamic vibration absorber correspond to m 2 , c 2 and k 2 , respectively. When the force acting on the main structure is F e iωt 1 , the displacements of the main system and dynamic vibration absorber are x 1 and x 2 , and the coordinate origin is located at static equilibrium positions, respectively.
According to Newton's second law, the general form of the differential equation of the dynamic vibration absorber and the main structure can be expressed as follows [5,32]: (1) The special solutions of equation (1) can be set as follows: Further solution of the steady-state response amplitude of the main structure is expressed as follows: In the above equation (3), υ is the frequency ratio between the dynamic vibration absorber and the main structure, η is the frequency ratio between the external load and the main structure and μ is the mass ratio: In addition, ζ is the damping ratio of the dynamic vibration absorber, which is the ratio between damping of TMD and critical damping: In equations (7) and (8), c c is the critical damping, m 2 and k 2 are, respectively, mass and stiffness of the dynamic vibration absorber. Den Hartog [46] derived the closed formulas of the optimal frequency ratio and damping parameters and expressed as follows: The loading points displayed in Figure 2(a) are, respectively, geometric center (2.5,2.5) and (2.5,3.5) of the stiffened plate, and the assessment points are displayed in Figure 2(b).
The first three natural frequencies of the stiffened plate under clamped boundary conditions are shown in Table 1.
Under the action of a unit force at the loading points shown in Figure 2(a), the vibration displacement curve of assessment points in Figure 2(b) is shown in Figure 3.
It can be seen from Figure 3 that the peak corresponds to 19.8 Hz (the third-order natural frequency) does not appear in the frequency-response curve of assessment point 1. Analysis suggests that the reason is point 1 is located at the node of the third-order vibration mode. Compared with assessment point 1, the new peak generates at the third-order natural frequency of 19.8 Hz in the vibration curve of assessment point 2. The vibration-response curve of assessment points 3 and 4 is almost the same as assessment point 2.
As displayed in Figure 3, the maximum and secondorder resonance peaks of typical assessment points, respectively, appear at the first-order 9.3 Hz and the third-order 19.8 Hz. Therefore, the firstand third-order modes of the stiffened plate will be chosen as the control object in the subsequent design of the low-frequency multiline spectrum distributed by the dynamic vibration absorption device.  First, the equivalent modal mass [46] to be controlled at the position of the dynamic vibration absorption of the stiffened plate is solved by equation (11): In equation (11), M ik represents the equivalent mass of point k in the ith order, m Δ ik represents the additional given mass at the layout position k of the dynamic vibration absorption, Ω i represents the ith natural frequency (rad·s −1 ) of the stiffened plate, and ω ik is the ith natural frequency (rad·s −1 ) of the coupling system after the stiffened plate is attached with the given mass.
According to the calculation, the corresponding equivalent mass at the antinode of the first-order mode of the stiffened plate is 898.8 kg, and the corresponding equivalent mass at the antinode of the third-order mode is 988.2 kg. The optimal frequency ratio and damping parameters are, respectively displayed, in equations (9) and (10). When the mass ratio is set as 0.02, the parameters of the dynamic vibration absorption can be obtained by referring to the above formula.
The dynamic vibration absorption devices are set as the parameters in Table 2 and installed at the antinode of the firstand third-order vibration mode of the stiffened plate, respectively. The layout position of dynamic vibration absorption and the schematic diagram of assessment points are separately shown in Figure 4 The comparison of vibration-response curves of typical assessment points in Figure 4(b) before and after the installation of dynamic vibration absorber are demonstrated in Figure 5.  The Figure 5 displays the comparison of frequency-response curves of typical points based on the optimal homology design method. When the mass ratio of the dynamic vibration absorption device is set to 0.02, the firstand third-order resonance peak of each assessment point decreases more than 27 and 25 dB, which verifies the effective control of the low-frequency multiline spectrum of the stiffened plate. In addition, the vibration absorption effect decreases with the increase of the distance between the assessment point and the installation position of the dynamic vibration absorption.

Low-frequency multi-line spectrum vibration control method of offshore platforms
In the previous section, we analyzed the distributed dynamic vibration absorption of the stiffened plate under multipoint excitation. On the basis of this, the research on the low-frequency multiline spectrum vibration control method of platforms will be carried out in this section. The control process is shown in Figure 6. It can be seen from Figure 6 that the low-frequency multiline spectrum vibration control method of the offshore platform mainly includes the following parts: (1) Establishment of an equivalent numerical model of the local resonance area to be controlled. (2) The solution of equivalent mass and parameters of dynamic vibration absorption device. (3) Verification of dynamic vibration absorption effect. The specific process is given in the following sections.

Equivalent numerical model of the local resonance area
By means of combining the finite element cloud diagram and frequency-response curve of the assessment points, the local resonance region can be determined. Then, the free vibration of the local resonance region can be calculated. Finally, the boundary conditions of the equivalent numerical model of the local resonance region can be determined by comparing with the frequency-response curve.

Parameter design of the dynamic vibration absorber
Based on the vibration mode of the equivalent numerical model of the local resonance area, the vicinity of the Placement position of TMD antinode of the mode to be controlled is selected as the installation position of the dynamic vibration absorption device. Further, the equivalent mass of the mode to be controlled at the installation position of the dynamic vibration absorption is determined based on formula (11). In addition, the parameters of the dynamic vibration absorption device are determined by combining the optimal homology equations (9) and (10).

Verification and analysis of the vibration absorption effect
Finally, the effect of the dynamic vibration absorption device is verified on the basis of the finite element method.

Verification of the low-frequency multiline spectral vibration control of the offshore platform
A large offshore platform is chosen as the research object in this section, and the low-frequency multiline spectrum vibration of the local resonance area of the offshore platform has been controlled on the basis of the research mentioned above.

Dynamic analysis model and the load of the platform
As a complex structure, when analyzing its dynamic characteristics, it is necessary to reasonably simplify the offshore platform, equipment and flow field. In addition, it is necessary to select the appropriate size of the mesh to divide the model. During the meshing process, the longitudinal and transverse spacings of the grid shall not be greater than the strong frame spacing and longitudinal truss spacing, respectively. Taking into account the solution accuracy and efficiency, the finite element scale is finally determined to be 0.5 m. The final finite element model of the platform includes 340,904 surface elements and 140,900 beam elements.
Six main engines are arranged on the platform, with a power of 4,950 kW. The speed of the main engine is 720 rpm and the weight of each main engine is 84 t. The vibration acceleration of the main engine tested on the ship is displayed in Figure 7, which is applied vertically  to the mass point above the base corresponding to the main engines.

Platform vibration response
Based on the finite element model superposition method, the vibration response of the platform is analyzed. The partial response cloud plot of the superstructure cab at frequencies of 12 and 19 Hz are, respectively, shown in Figure 8   Assessment points #7 and #8 are both located near the antinode of the first-order vibration mode of the superstructure cab. The response of assessment points #7 at 12 Hz is greater than 19 Hz, and the response of assessment points #8 at 12 Hz and 19 Hz is similar. In addition, assessment points #9 and #10 are located near the antinode of the vibration mode corresponding to 19 Hz. Therefore, the response of the above assessment points #9 and #10 at 19 Hz is slightly greater than 12 Hz. It follows that the amplitude of the vibration-response curve of the assessment point is determined by the location of assessment points and load characteristics.

Dynamic vibration absorption design in the local resonance region
According to the flow in Figure 6, the low-frequency multiline spectrum vibration control of the superstructure cab has been conducted. Depending on the vibration response nephogram of the superstructure cab and the bulkhead partition, the local resonance region is partitioned. When the boundary condition is set as simply supported, the vibration modes of the local resonance region at corresponding frequencies are basically consistent with the overall finite element calculation results of the platform. The antinodes of 12 and 19 Hz modal shapes in the local resonance area of the superstructure cab are, respectively, selected as the installation position of the dynamic vibration absorption. According to equation (11), the equivalent mass of the 12 and 19 Hz mode antinodes are 7348.5 and 4026.7 kg, respectively. When the mass ratio is set as 0.02, the parameters of dynamic vibration absorption are as listed in Table 3.
When the dynamic vibration absorption devices are arranged at antinodes of 12 and 19 Hz modes, respectively. The cloud diagram of the vibration response before and after the dynamic vibration absorber is arranged as shown in Figure 10: It is easy to find that under the combined action of the distributed dynamic vibration absorber, the vibration response of the driving deck at 12 and 19 Hz is significantly reduced. Meanwhile, the corresponding comparison of vibration acceleration response curves of typical assessment points is listed in Figure 11. To sum up, by combining the low-frequency multiline spectrum vibration control method of the platform with the finite element method, the low-frequency multiline spectrum vibration of the local resonance region of the platform is effectively controlled. The vibration acceleration of typical assessment points at 12 and 19 Hz decreased by more than 13 dB.

Conclusion
In the current research, by means of combining the lowfrequency multiline spectrum vibration control process and modal superposition method, the low-frequency multiline spectrum distributed dynamic vibration absorption method of the platform is proposed.
First, the current research status of the vibration control methods of offshore platforms is briefly introduced. Then, we give a brief introduction to the theoretical basis of dynamic vibration absorption. Afterward, a stiffened plate under clamped boundary conditions is chosen as the research object to investigate the effect of distributed dynamic vibration absorption devices. Finally, the lowfrequency multiline spectrum vibration control process of the offshore platform is proposed. The effectiveness of the proposed method is also verified in the calculation of the platform. The conclusions of the current research are as follows: (1) The amplitude of the vibration-response curve of the assessment point is determined by the location of the assessment point and load characteristics. at signature frequencies (12 and 19 Hz) decreases by more than 13 dB under the action of distributed dynamic vibration absorption.
(3) By means of combining the low-frequency multiline spectrum vibration control method with the finite element method, the multiline spectrum vibration in the local resonance region of the offshore platform is controlled effectively and quickly.