Experimental Study and Fatigue Analysis of Vortex-Induced Vibration of Umbilical Cable Considering Internal Friction

In order to investigate the effect of internal friction of umbilical cable on its vortex-induced vibration (VIV) responses, the experimental study on VIV of bond umbilical cable (BUC) and un-bond umbilical cable (UBUC) was carried out in an experimental tank. A current generator in the laboratory simulated the uniform current, and the current velocities were observed in real time by using a Doppler Velocimeter. In addition, different sizes of top tension were applied to the umbilical cable model. The VIV responses of the umbilical cable model were measured by using Fiber Bragg grating (FBG) strain sensors. The displacement responses of umbilical cable model were reconstructed based on the experimental strain data processed by modal superposition method. In this paper, the traveling wave characteristics, the spatial-temporal distribution characteristics of frequency and fatigue damage of the BUC and UBUC under VIV are studied. The experimental results show that there are obvious differences between BUC and UBUC in the response characteristics of VIV. The UBUC appears the traveling wave sooner than BUC, but its standing wave characteristics are more obvious than those of BUC at high velocities. Compared with BUC, the spatial-temporal distribution of UBUC frequencies appears wide-band distribution sooner, but has narrower bandwidth in the “lock-in” state. The level of fatigue damage of BUC was approximately the same as that of UBUC.


Introduction
Umbilical cable is one of the most important equipment in floating production system, which is mainly used for power transmission and signal control. The umbilical cable is always affected by the current during operation. A certain velocity of current causes alternating vortexes on both sides of the umbilical cable, and the shedding of vortexes leads periodic vibrations of the umbilical cable structure. Such vibrations are called vortex-induced vibration (VIV). When the vortex shedding frequency is close to the natural frequency of the umbilical cable, the vibrations of the umbilical cable will be intensified, resulting in "lock-in" phenomenon. This phenomenon not only leads to the fatigue damage of the umbilical cable structure, but also causes damage to the internal optical fiber of umbilical cable, which can lead to communication interruption and seriously threaten the safety of the entire oil and gas system. A large number of studies have shown that the diversity of cross-section components and the nonlinear factors (such as interlayer sliding, internal friction) have different intensities of effects on the performance of marine risers and cables, including cross-section mechanical properties, overall mechanical responses and fatigue damage (Kraincanic and Kebadze, 2001;So̸ dahl et al., 2010;Vinogradov and Atatekin, 1986;Witz and Tan, 1992). Therefore, the studies of the effect of umbilical cable's internal friction on its VIV have a great significance in engineering practice.
Over the last decade, many researchers have carried out experimental studies and numerical simulation on VIV. Vandiver et al. (2005) designed an experiment for a riser with large aspect ratio in Seneca Lake and measured the VIV responses of the bare riser model and straked model that was fully covered by helical strakes. In order to improve the understanding of the physical phenomena of VIV of risers and verify the prediction model of VIV, Chaplin et al. (2005) designed the top tension riser VIV experiment in the towing tank. In this experiment, the number of modes reached the 8-th order. In addition, the phenomenon of multi-modes participating in VIV was observed. Chaplin and King (2018) carried out an experimental study on the VIV of a high-curvature steel catenary riser under steady current, and analyzed the VIV responses of the riser in the cross-flow (CF) direction in detail. Trim et al. (2005) carried out an experimental study on VIV responses of a flexible riser model whose aspect ratio is 1400 under uniform current and linear shear current, the vibration suppression effects of helical strakes with different coverage rates and different geometric shapes were further studied. Lie and Kaasen (2006) carried out a series of experiments on a 90meter top tension riser model under shear current condition. The experiment mainly measured the response modes of the riser with large aspect ratio, and elaborated the modal analysis method that is used for the analysis of experimental data. Guo et al. (2004Guo et al. ( , 2006 and Guo and Lou (2008) conducted an experimental study on the VIV of risers under the combined action of inflow, outflow and top tension in an experiment tank, and the corresponding numerical simulations were carried out. The influences of inflow on VIV of riser with top tension were studied in depth. Vandiver et al. (2009) proposed some provocative insights on previously unknown or unexplained phenomena of VIV of long flexible cylinder, such as VIV response dominated by traveling wave and stable vibration trajectory in pure traveling wave region, etc. Bourguet et al. (2011) studied the VIV of a long cylindrical tensioned beam with L/D=200 in a linear shear flow by using three-dimensional direct numerical simulation. The traveling wave, standing wave and dominant frequency of VIV are analyzed in detail. Gu et al. (2013) studied the low-mode vibration and IL-CF coupled VIV response of a vertical flexible cylinder in the towing tank, and obtained some conclusions, which are different from the previous study. In addition, a method for studying the fluid force acting on a flexible cylinder is proposed. Huera-Huarte et al. (2014) carried out VIV experiment of flexible cylinders with the mass ratios of 1.1 and 1.7, respectively in the towing tank. The results show that maximum amplitudes of the model with the lowest mass ratio reach more than 3 diameters, and the drag coefficient of it is also relatively high. Wang et al. (2015a) conducted experimental research on the VIV of steel catenary riser model under vessel motions, and the results showed that the VIV caused by pure top vessel motions has distinctive time-varying features and the vessel motion-induced VIV is closely related to the KC number and instantaneous velocity profile.
Domestic and foreign researchers have also studied the fatigue damage caused by VIV extensively. Baarholm et al. (2006) observed and analyzed the Hanøytangen experiment and obtained the following conclusions: in the low modes, the influences of cross-flow (CF) direction vibration and inline (IL) direction vibration on fatigue damage of riser are equally important, while in the high modes, the CF direction vibration plays a major role in the fatigue damage. Trim et al. (2005) studied the VIV fatigue damage of slender flexible riser model when it was in the bare state and straked state where the riser was arranged with helical strakes of different shapes and different coverage rates. Li et al. (2010) analyzed the fatigue life of top tension riser under VIV with consideration of the effect of internal flowing fluid. The influence of riser parameters such as modulus of elasticity, top tension and internal flow velocity on riser fatigue life is analyzed in detail. Gao et al. (2011) studied the fatigue damage of SCR under VIV based on the modal superposition method and S-N curve, and analyzed the relationship between the fatigue damage of SCR under VIV and current velocity. Wang et al. (2014) carried out an experimental study on the VIV fatigue damage of the steel catenary riser model under vessel motions, and found that the fatigue damage was mainly related to the amplitude and period of the top vessel motions, and pointed out the position of the high-probability fatigue damage point. Wang et al. (2015b) conducted experimental research on VIV fatigue damage of a slender flexible straight pipe in oscillating flow, and proposed a simplified calculation method for fatigue damage of the riser under oscillating flow. Sun et al. (2014) used the pseudo-excitation method to simulate the VIV responses of a deep-water riser under shear current condition, and studied the VIV fatigue life of the riser. Xue et al. (2014) proposed a prediction model of VIV fatigue damage for riser accounting for both CF and IL vibrations, the prediction results of the model are in good agreement with the experimental results, and the parameter sensitivity analysis of the fatigue damage of the model under shear current is carried out.
In the above studies, most of the experimental models are simple with uniform riser sections, while the VIV experiments of umbilical cable models with complex sections are rarely studied. Lyons et al. (2003) analyzed the VIV response of umbilical cable in detail, based on the high-quality data for VIV provided by the Foinaven Umbilical Monitoring System (FUMS). The research shows that VIV can cause the drag coefficient amplification of umbilical cable, which has an important impact on the total hydrodynamic load of umbilical cable. Lie et al. (2007) conducted a fullscale free-span VIV test of umbilical cables, and the measured data have been extensively studied using modal analysis and fatigue analysis.
The VIV experiments of umbilical cable considering nonlinear factors such as interlayer sliding and internal friction have not been reported (to the best of the authors' knowledge).
From what has been discussed above, the aim of this paper is to further improve the understanding of VIV and to obtain the effects of internal friction on the VIV responses and fatigue damage of umbilical cable. In the following parts, the experimental facility and models are introduced in detail. Secondly, the data processing method is introduced briefly, and the calculation method of fatigue damage is expounded. And then, the characteristics of traveling wave, the spatial-temporal distribution characteristics of frequency and fatigue damage of BUC and UBUC are analyzed in detail. Finally, the effects of internal friction on the responses of VIV and fatigue damage are obtained by comparing the response characteristics of BUC and UBUC.

Description of experimental facility
The experiment was carried out in the Engineering Hydraulics Laboratory of Ocean University of China.
During the experiment, the current generator simulated the uniform current with velocities of 0.1−0.7 m/s, and the Doppler Velocimeter was installed in the tank to detect the current velocities in the acquisition time step in real time. The main sketch of the experimental device is illustrated in Fig. 1. The top end and bottom end of the umbilical cable model are fixed, the upper 1.2 m is exposed to air, and the lower 0.8m is in uniform flow. The top end of the model is connected to the experimental bracket by a universal joint and a dynamometer. The dynamometer can directly measure the top tension of umbilical cable and observe the change of the top tension in real time to ensure the accuracy of the top tension. The bottom end of the umbilical cable model is directly connected with the experimental bracket through a universal joint. The clamping devices are used to fix the bracket and the experiment tank to prevent the vibration coupling between the bracket and the umbilical cable model under the action of current. After testing the mechanical properties of various types of pipes and considering the stiffness and modal requirements, transparent Plexiglass tubes and copper cables were selected as the materials of umbilical cable models.
As shown in Fig. 2, the umbilical cable model consists of an outer Plexiglass tube and an inner copper cable. Each layer of UBUC model exists independently, which allows relative sliding. The BUC model uses modified acylate adhesive to bond all layers of materials to form a whole to eliminate internal friction caused by interlayer sliding. Four FBG strain sensors are evenly arranged along the cross section of the umbilical cable model at 90° intervals, and six gauging points are arranged along each FBG strain sensor. The specific location of the gauging points along the umbilical cable model is shown in Fig. 3. The key parameters of umbilical cable models are shown in Table 1. The mass of adhesive can be neglected, and it is considered that the BUC model and UBUC model have the same unit mass in this paper. The detailed test matrix is shown in Table 2. VIV responses of BUC and UBUC under the combined action of external velocities and top tensions are measured.

Strain data analysis
During the experiment, all the data acquisitions are car-     GU Hong-lu et al. China Ocean Eng., 2020, Vol. 34, No. 2, P. 151-161 ried out after the current velocities are stable and the strains are cleared, which eliminates the influence of the initial bending strains caused by the initial drag force in the IL direction. Therefore, the strains of CF direction and IL direction obtained during the experiment only include the axial strains produced by the initial tension and the axial strains produced by the VIV. According to the data processing method of Gao et al. (2015a), the bending strains caused by VIV in CF direction and IL direction can be written as: (1) where and are the strains caused by VIV in the CF direction and the IL direction.
, , , and are the measured strains obtained from the experiment. In order to eliminate the influence of ambient noise on the measured strains, it is necessary to filter the measured strains. In this paper, threshold wavelet denoising is applied to filtering measured strains. Fig. 4 shows the strain data before and after filtering. We can see that the filtered signal not only eliminates the high-frequency signal and the lowfrequency signal, but also keeps the basic characteristics of the original signal well. After processing the measured strains, the displacement responses of the VIV of the umbilical cable can be obtained based on the modal superposition method (Trim et al., 2005).

Modal superposition method
As long as a sufficient number of sensors are arranged along the umbilical cable and the position is reasonable, the displacement response at any position can be obtained by modal superposition method.
The displacement response of umbilical cable is written in the form of superposition of different modes based on modal superposition method: is the vibration mode function; is the weighting function; z is the position along the umbilical cable.
For umbilical cable structures in this paper, the relationship between the bending strain and curvature can be formulated as: in which R is the outer radius of the umbilical cable. According to the geometric relationship, the curvature value can be approximated as the second derivative of the displacement with respect to the spatial variable: By substituting Eq. (5) into Eq. (4), Eq. (6) can be obtained: where N is the highest mode number involved in the umbilical cable vibration. Eq. (6) can be abbreviated as: Only if the number of sensors M is larger than N can Eq. (7) have a solution. Eq. (7) can be expressed as: The vibration mode function can be formulated as: Therefore, Eq.
By solving Eq. (11), the corresponding weighting function of each mode can be obtained. By substituting the weighting function into Eq. (10), the time history of displacement at each point of umbilical cable can be calculated.

Fatigue damage calculation
According to Gao et al. (2015b), based on the preliminary processed strains and the elastic modulus of the umbilical cable model, the stress of the riser can be written as: . Further, fatigue damage of umbilical cable model can be obtained: Firstly, the rain-flow counting method (Anzai and Endo, 1979) is used to obtain the stress range, the mean stress value and the number of occurrences of stress range in the stress time history of umbilical cable based on the experimental strains. Then, according to the principle of equivalent damage, the following Goodman empirical formula is used to transform the stress range of non-zero stress mean value into the stress range of zero stress mean value.
where is the equivalent zero-mean stress; is the i-th stress amplitude; is the j-th mean stress; is the ultimate strength of the material.
For a constant stress range, the number of failure stress cycles can be expressed by S−N curve as follows: S log a log a = 11.687 m = 3 where is the stress amplitude; is the interception of S−N curve in double logarithmic coordinates; m is the slope of S−N curve in double logarithmic coordinates. In reference to DNV RP C203 (DNV, 2008), this paper chooses D curve in the curve without cathodic protection in seawater environment.
, . According to Palmgren−Miner linear cumulative damage theory and referring to DNV RP C203 (DNV, 2008), the fatigue damage, which shall be satisfied, may be written as: where is the cumulative measure of fatigue damage during experiment data acquisition time; D is the cumulative measure of fatigue damage during a year; is the stress cycle counts obtained by the rain-flow counting method; is the number of fatigue failure cycles determined by the corresponding S−N curve; is the stress amplitude of umbilical cable; is the usage factor; f DFF is the design fatigue factor, according to the DNV RP F204 (DNV, 2005), f DFF =10; is the experiment data acquisition time.

Results and Discussion
The reduced velocity is defined as: where U is the current velocity; D e is the external diameter of the model; is the first natural frequency calculated by Eq. (17) L T e where is the total length of the model; is the top tension; m is the mass per unit length, including structural mass and added mass.
The reduced velocities used in this section are calculated, as shown in Table 3. Because the reduced velocities of BUC and UBUC are different, in order to compare their properties more intuitively, the current velocity is used in the analysis and discussion of this section.
The reduced velocities provide data reference for the subsequent analysis of whether the "lock-in" is reached or not.  2017). When the velocity increases to 0.6 m/s, the traveling wave behavior begins to appear, but the VIV is still dominated by standing wave behavior. With the increase of velocity to 0.7 m/s, the response envelope in CF direction appears obvious node (minima of the response envelope), but the RMS value associated with the node is far from zero, which indicates that the responses of VIV changes from standing wave dominance to traveling wave dominance. These phenomena are consistent with the findings of Gao et al. (2016). The results of Fig. 5b show that the VIV responses of UBUC displayed traveling wave behavior sooner than BUC's VIV responses. When the velocities are 0.55− 0.7 m/s, it can be seen that the maximum displacement of UBUC begins to move along the length of umbilical cable with time, indicating that traveling wave behavior begins to appear. However, standing wave behavior also appears at both ends of umbilical cable due to reflection, indicating that the completely VIV responses are the combination of standing wave behavior and traveling wave behavior. Nevertheless, it can be observed that standing wave behavior still dominates the responses of VIV when the velocity of current is 0.55 m/s, which is coinciding well with the observations of Vandiver et al. (2009) andHuera-Huarte et al. (2014).

Traveling wave analysis of VIV
With the increase of velocity, the responses of VIV begin to be dominated by traveling wave behavior. However, it can be observed that the RMS value of the displacement corresponding to the node gradually approaches zero, which means that the standing wave characteristics of the VIV responses in CF direction begin to become obvious. In addition, by comparing Fig. 5a with Fig. 5b, we can see that the standing wave behavior of UBUC is more obvious than that of BUC after traveling wave behavior occurs. Moreover, the RMS value corresponding to the node position in the response envelope of UBUC is closer to zero after the node appears in the response envelope of CF direction, which is mainly because the second modal weight of UBUC is larger than that of BUC in the responses of VIV at high velocities.
The black arrow indicates the direction of traveling wave propagation. From the change of the slope of the arrow, it can be seen that when the current velocity increases from 0.55 m/s to 0.7 m/s, the propagation speed of traveling wave increases gradually. It shows that with the increase of velocity, the power input from fluid to VIV increases continuously. Meanwhile, as the amplitude of transverse vibration increases, the spanwise propagation acceler-ates (Zhu et al., 2019). The traveling wave propagation speed of BUC is larger than that of UBUC at the same velocity, indicating that the VIV of BUC requires more power input, as traveling wave propagates along the umbilical cable from the power-in region to the power-out region (Duan et al., 2018). It can be seen that the power-in region of VIV is mainly concentrated at the bottom of umbilical cable, which is mainly because the main power input of VIV of umbilical cable comes from the impact of current. Fig. 6 shows the spatial-temporal distribution of the response frequencies of the VIV of BUC and UBUC.

Frequency responses analysis of VIV
From Fig. 6a, it can be seen that when the velocities are 0.35−0.55 m/s, VIV responses of BUC have only one dominant frequency, which shows a narrow-band distribution 4.5 < V r < 10 and is relatively concentrated over time. Especially when the velocity is 0.4 m/s, the frequency distribution is almost linear and continuous over time, which illustrates that the response frequencies are very concentrated and do not change over time, and participate in the whole process of the response of the VIV. This is mainly because the umbilical cable is in the "lock-in" state and the power is concentrated at this velocity. Because the VIV "lock-in" of the cylinder occurs in the range of (Chen and Jendrzejczyk, 1979). It can also be seen from the reduced velocity of BUC that the structure is in the "lock-in" state at this velocity. With the increase of velocity to 0.6 m/s, 0.7 m/s, the response frequencies of BUC begin to exhibit broad-band distribution and exhibit oscillatory and intermittent distribution over time. These phenomena indicate that the response frequency energy is more scattering, the response frequency components are complex, and the response frequencies change constantly during the process of VIV and intermittently participate in the process of VIV at high velocities. These phenomena observed in this paper are consistent with those in Gao et al. (2016Gao et al. ( , 2017. It can also be seen from the spatial distribution of frequencies that the distribution of frequency along the length of umbilical cable is narrow-band at low velocities and broad-band at high velocities. When the velocities are 0.35−0.55 m/s, the peak energy mainly concentrates in the middle of the umbilical cable. When the velocity increases to 0.6 m/s, the response frequency spectrum does not distribute continuously along the length of the umbilical cable and two peak regions begin to appear. The peak energy regions are mainly located at #2 and #5 gauging points. This phenomenon indicates that the second modal characteristics of the umbilical cable begin to appear, and the peak energy region coincides with the anti-nodes (maxima of the response envelope) position in the response envelope of displacement. In addition, it can also be observed that the trend of power spectral density (PSD) changes with the velocity is "increase -decrease -re-increase", which is mainly because the VIV of umbilical cable has higher power when it is near the "lock-in" region. Fig. 6b gives the response frequency characteristics of UBUC. It can be observed that the spatial-temporal distribution of the response frequency of UBUC appears broadband distribution sooner than that of BUC. From the reduced velocity and frequency response of UBUC, it can be judged that when the velocities are 0.35 m/s and 0.4 m/s, the UBUC enters the "lock-in" state. It can be seen that the bandwidth of the spatial-temporal distribution of the UBUC response frequency is narrower under the "lock-in" state than that of BUC, which indicates that the UBUC response frequency is single and more concentrated. It can also be observed that the response frequencies of UBUC have better continuity and stability over time. Besides, the UBUC appears second modal characteristic earlier, and the PSD of UBUC is higher than that of BUC with the same velocity, which indicates that the power of UBUC is higher and the response amplitude is larger in the process of VIV.

Effect of current velocity on fatigue damage
In this section, the fatigue damage of BUC and UBUC under 11-stage current velocities are calculated when the top tension is 112 N. Then the change rules of fatigue damage of BUC and UBUC in IL direction and CF direction are obtained.
It can be seen from Fig. 7 that the fatigue damage of umbilical cable in CF direction is the same as that in IL direction at low current velocities. However, with the increase of current velocity, the fatigue damage in IL direction is gradually smaller than that in CF direction. This is mainly because the vibration amplitude in CF direction is much larger than that in IL direction at the same current velocity, but the vibration frequency in both directions does not strictly follow the double relationship at high velocities during the experiment. As shown in Fig. 8, there is an obvious double relationship between the dominant response frequencies of CF direction and IL direction at low current velocities, but the vibration frequencies of both directions are equal at high current velocities. The main reason for this phenomenon is that the "Beat Phenomena" (Jong and Vandiver, 1983) at both ends of the model is enhanced at high velocities, which results in the vibration frequencies of CF direction being consistent with those of IL direction.
Compared with UBUC, the overall trend of fatigue damage of BUC is gentler. The trend of fatigue damage of umbilical cable changes with the current velocity is "increase -  decrease -re-increase" in CF and IL direction. The main reason is that the vibration amplitude increases and the power of response frequencies are more concentrated when VIV is close to the "lock-in" region. It is this strong vibration that makes the umbilical cable in a higher stress cycle level, which results in a higher degree of fatigue damage. In addition, the fatigue damages of BUC and UBUC are basically at the same level. This is mainly due to the large response amplitude and low frequency of UBUC and the small response amplitude and high frequency of BUC under the same current velocity. However, the fatigue damage of BUC is slightly higher than that of UBUC at most current velocities. The main reasons are as follows: the elastic modulus of BUC is higher than that of UBUC, and the increase of elastic modulus will lead to the increase of fatigue damage (Li et al., 2010;Xue et al., 2014); The internal friction of UBUC will play a damper role, which consumes part of the input power and reduces the fatigue damage.
The fatigue damage of UBUC is higher than that of BUC only when the external velocity is 0.35 m/s, 0.4 m/s, 0.6 m/s and 0.7 m/s. This is because UBUC enters the "lock-in" region earlier than BUC. With these velocities, UBUC is closer to its "lock-in" region and its response power is higher and more concentrated than that of BUC. It can be seen from Fig. 9−Fig. 12 that the fatigue damage of umbilical cable is sensitive to the change of top tension. This is mainly due to the fact that the fatigue damage of umbilical cable is mainly controlled by tension under low modes (Baarholm et al., 2006). The variation trend of fatigue damage over top tensions in CF direction is basically the same as that in IL direction at the same current velocity. In addition, the sensitivity of UBUC to the change of top tensions is higher than that of BUC. It is observed that the trend of fatigue damage of umbilical cable is different with the increase of top tension at different current velocities. This is mainly because the change of top tension leads to the change of "lock-in" region by changing the natural vibration frequency of umbilical cable. When the current velocity is 0.3 m/s, the fatigue damage in CF direction and IL direction decreases with the increase of top tension. The main reason is that with the increase of top tension, the natural frequency of umbilical cable increases, the response amplitude decreases, and the umbilical cables gradually move away from the first-order "lock-in" region, and then the fatigue damage decreases.

Effect of top tension on fatigue damage
When the current velocity is 0.45 m/s, the trend of fatigue damage of umbilical cables with top tension is opposite to that of 0.3 m/s. This is because when the current velocity is 0.45 m/s, both BUC and UBUC begin to break away from the first-order "lock-in" region. However, the increase    of top tension results in the backward movement of the umbilical cables' "lock-in" region. Therefore, the umbilical cables are closer to the "lock-in" state and have a higher degree of fatigue damage at this current velocity.

Fatigue damage along spanwise direction
The fatigue damage along the spanwise direction of umbilical cable model is analyzed by selecting the current velocity that produces the fatigue damage extremum. In this paper, the CF fatigue damage is larger than the IL fatigue damage, so only the CF fatigue damage is analyzed.
As shown in Fig. 13 and Fig. 14, the fatigue damage of umbilical cable has an extremum point at low velocity and two extremum points at high velocity. The number of extremum points is the same as the number of response modes of umbilical cable at this velocity. In addition, the fatigue extremum point of umbilical cable is located at #3 gauging point at low velocity, and at #2 and #5 gauging points at high velocity. It can be seen that the location of fatigue extremum points coincide with the location of maximum displacement and peak energy region. Therefore, there is no accurate fatigue extremum position along the umbilical cable spanwise direction.

Conclusions
In order to obtain the effect of internal friction on the VIV of umbilical cable, the VIV experiment of BUC and UBUC was carried out. The traveling wave characteristics, the spatial-temporal distribution characteristics of fre-quency and fatigue damage of the BUC and UBUC were studied. The main conclusions are as follows.
VIV responses of BUC and UBUC show a transition from standing wave to traveling wave with the increase of current velocity. VIV responses of UBUC displayed traveling wave behavior sooner than BUC's VIV responses and the standing wave characteristics of UBUC are more obvious than those of BUC at the same current velocity.
The traveling wave propagation speed of BUC and UBUC accelerates with the increase of current velocity. In addition, the traveling wave propagation speed of BUC is larger than that of UBUC, which means that the VIV of BUC requires more power input.
The spatial-temporal distribution of response frequencies of BUC and UBUC changed from narrow-band distribution to broad-band distribution with the increase of current velocity, and the UBUC appeared broad-band distribution sooner than BUC. The bandwidth of frequency spatialtemporal distribution of UBUC is smaller than that of BUC in "lock-in" region, that is to say, the frequency response power of UBUC is more continuous and concentrated than that of BUC.
The fatigue damage of UBUC is more sensitive to the change of velocity and top tension than that of BUC. The trend of fatigue damage of BUC and UBUC with the current velocity is "increase-decrease-re-increase" both in CF and IL direction. The fatigue damage of BUC and UBUC is basically at the same level. The fatigue damage level of BUC is in general slightly higher than that of UBUC at the same current velocity.