Study on Gyroscopic Effect of Floating Offshore Wind Turbines

Compared with bottom-fixed wind turbines, the supporting platform of a floating offshore wind turbine has a larger range of motion, so the gyroscopic effects of the system will be more obvious. In this paper, the mathematical analytic expression of the gyroscopic moment of a floating offshore wind turbine is derived firstly. Then, FAST software is utilized to perform a numerical analysis on the model of a spar-type horizontal axis floating offshore wind turbine, OC3-Hywind, so as to verify the correctness of the theoretical analytical formula and take an investigation on the characteristics of gyroscopic effect. It is found that the gyroscopic moment of the horizontal axis floating offshore wind turbine is essentially caused by the vector change of the rotating rotor, which may be due to the pitch or yaw motion of the floating platform or the yawing motion of the nacelle. When the rotor is rotating, the pitch motion of the platform mainly excites the gyroscopic moment in the rotor’s yaw direction, and the yaw motion of the platform largely excites the rotor’s gyroscopic moment in pitch direction, accordingly. The results show that the gyroscopic moment of the FOWT is roughly linearly related to the rotor’s inertia, the rotor speed, and the angular velocity of the platform motion.


Introduction
As climate change related issues worsen with environmental pollution as a result of fossil energy use, clean renewable energy is attracting more and more attention all over the world (Sethuraman, 2014). Offshore wind turbines industry has been in a position to prove its increasing maturity, cost competitiveness and efficiency in recent decades (Xu, 2016). GWEC Market Intelligence expects that more than 55 GW of new installations of offshore wind turbines each year until 2023 (GWEC, 2019).
Offshore wind turbines can be divided into two categories (Coulling et al., 2013), namely, bottom-fixed offshore wind turbines (hereafter, BFOWTs) and floating offshore wind turbines (hereafter, FOWTs). As the low-hanging fruit of shallow near-shore sites is exhausted with each passing day, there will be great technical challenges and constrain efforts to reduce costs by using BFOWTs in the deep-water area (Jeon et al., 2013). By comparison, FOWTs are capable of unlocking abundant wind resources in deeper water regions in the future (Musial et al., 2004). In addition, FOWTs, having a smaller impact on ecological surroundings, are suitable for larger wind turbines, lower installation and major repair costs by removing the need for expensive heavy-lift installation (Muliawan et al., 2013). ETI expects enormous potential for floating wind in emerging offshore wind markets in Asia, the United States, and elsewhere in Europe (ETIP, 2019).
Currently, dozens of FOWTs concepts come up all over the world. According to their stability principle, FOWTs have four fundamental types (Butterfield et al., 2005): barge (Borisade et al., 2016), semi-submersible (Chen et al., 2019), spar buoy (Jonkman, 2009) and tension leg platform (Shen et al., 2016). In recent years, FOWTs technology has attracted more and more concerns from academic and industrial fields. Prototype tests projects have been carried out for the past ten years. In 2009, the world's first full-scale FOWT demo project Hywind using a spar-type FOWT, was installed 10 km southwest of Karmøy, Norway (Skaare et al., 2011); In 2011, a semi-type FOWT, WindFloat, was placed 5 km off Portugal's coast (Aubault et al., 2012); In 2017, a barge-type FOWT DampingPool was installed 20 km off the Bay of Biscay as the first French FOWT demo project-Floatgen .
Nevertheless, there are still some engineering challenges and technical unknowns in FOWTs, which threaten FOWTs' system security. One is the gyroscopic effect of FOWTs. As is known, the gyroscopic effect is caused by the gyroscopic moment, which is defined as the principal moment of the Coriolis inertia force from a rotating body (Martin, 1975). Hence, gyroscopic effect is described as: any couple, apparently tending to incline the axis of the rotating body in a given direction, actually causes an inclination of the axis in the plane perpendicular to that given direction (Shilovski, 1924;Greenhill, 1919). In other words, gyroscopic effect is the inertia of keeping the direction of rotation of a rotator. For an example as shown in Fig. 1, a wheel rotates around its spin axis at angular speed , and its gravity makes it tend to fall down. As a result, the inclination of the axis of the rotating wheel produces a gyroscopic moment, which makes a precession of the rotating wheel around the vertical axis of the foundation. This is the socalled gyroscopic effect. Although people are still confused about the gyroscopic effect and have different interpretations of its essential reason, the gyroscopic effect has been widely applied in the modern industrial community, such as navigation on ship, altitude control in airplanes and so on.
For BFOWTs, gyroscopic effect is mainly caused by the vibration of the tower or yaw motion of the nacelle, with a revolving rotor. Wilson et al. (2008) investigated the gyroscopic moment using the measured data when the nacelle of a small bottom-fixed wind turbine was yawing in response to wind direction, and found that the moment was significantly attributed to the cyclic Coriolis acceleration of the rotating and yawing blades ; Hamdi et al. (2014) carried out an investigation on dynamic response of a horizontal axis wind turbine blade under gyroscopic moment, and found that gyroscopic effect clearly had a negative impact on the first and second mode shapes of the wind turbine blades etc.
By comparison, the floating foundation of an FOWT is restrained by mooring lines, thus the FOWT motion moves violently in operation. Hence, gyroscopic effect of an FOWT will be more obvious and should be paid more attention to. However, research regarding the gyroscopic effect of FOWTs is limited up to now. Blusseau and Patel (2012) investigated the effects of gyroscopic couples on the behaviour of a large vertical axis wind turbine mounted on a floating semi-submersible platform, and found that the gyroscopic effect introduced significant motions at the key frequency related to rotor speed. However, aerodynamic loads were not excluded in the simulation. As a result, it is hard to identify the gyroscopic moment of an FOWT from complex aerodynamic loads. Mostafa et al. (2012) carried out a scaled basin tests on a horizontal axis floating wind turbine model, and found that the gyroscopic effect depends on the angular velocity and inertia moment of the blade. However, this test model with 1/360 scale ratio was too small to achieve accurate results because of scale effects in the model test.
In this paper, gyroscopic effects of a spar-type horizontal axis FOWT, OC3-Hywind, are investigated systematically. Firstly, analytic formula of gyroscopic moment of an FOWT is derived, and the characteristics of the gyroscopic effect are concluded by using the analytic formula. In order to verify the gyroscopic moment analytic formula and investigate the characteristics of gyroscopic moment of an FOWT further, the gyroscopic effect of the OC3-Hywind FOWT is investigated with numerical results excluding the interference from aerodynamic loads. The conclusions obtained in this paper contribute to deeper understanding of the mechanism of gyroscopic effect of an FOWT, and improve control or optimization of the gyro-moment of an FOWT in the future.

Kinematics analysis
This section will adopt a simplified horizontal axis FOWT numerical model to derive the theoretical expression of the gyroscopic moment. As shown in Fig. 2, the FOWT ( ) can be divided into two parts: the rotor and the supporting foundation . For specific, includes hub, blade, rotating shaft, and includes nacelle, tower, floating support platform. Let the right-hand Cartesian frame E r (x r , y r , z r ) be the body-fixed frame of the rotor . The origin of the body-fixed frame E r is located at the centroid of the turbine, the direction of the positive axis of x r is pointing downwind along the nacelle. z r is perpendicular to the nacelle at the initial moment, but the frame E r rotates with the rotor. Let the Cartesian frame E 0 (x 0 , y 0 , z 0 ) be bodyfixed frame of the supporting foundation . The origin of this body-fixed frame E 0 is located at the intersection of the centerline of the tower and the still water surface at the start. The coordinate axis z 0 is vertically upward along the tower,  and the positive axis of x 0 is along the wave direction. Let the inertial (earth) frame be E g (x g , y g , z g ) at the initial time and coincide with the body-fixed frame E 0 (x 0 , y 0 , z 0 ) of . The origin of the body-fixed frame (E 0 ) of the above-mentioned is referred to as the "dynamic reference point of the system".

X
When the FOWT is taken as an overall model, the degrees of freedom of the supporting platform can set to be , including three degrees of freedom of translational motion (surge, sway and heave) and three degrees of freedom of rotational motion (roll, pitch and yaw) can be described by the relationship between the body-fixed frame E 0 and the global inertial frame E g , such as: Surge, sway and heave are: Roll, pitch and yaw are: where, and are the translational and rotational displacements of the overall system, respectively. The superscript " " indicates that the vector is relative to the global inertial frame E g . It can be seen that the translational displacements vector can be quantified by the sagittal distance between the origin of the body-fixed frame E 0 and the origin of the global inertial frame E g . The rotational displacements vector is quantified by the rotation angle of the body-fixed frame. The velocity and acceleration of the centroid of the whole system with respect to its body-fixed frame E 0 can be written as: where, is the radius vector from the origin of the global inertial frame E g to the centroid of the overall system. is the radius vector from the origin of the body-fixed frame E 0 to the centroid of the system C. The sign " " represents the b T b←g coordinate matrix of the vector. By introducing the form of the coordinate matrix, the vector cross product can be converted into a point multiplication. The superscript " " represents that vectors are relative to the system body-fixed frame E 0 .
is the transformation matrix (direction cosine matrix) from the global inertial frame to the body-fixed frame.
B 0 With the entire system , the Newton−Euler equations with respect to Point O are used as dynamical governing equations for the FOWT system, yielding where, is the mass and is the inertia moment of the whole FOWT. is the external force and is the external moment acting on the whole FOWT. and are the aerodynamic force and moment acting on the rotor, respectively. and are the hydrodynamic force and moment acting on the supporting platform, respectively. and are the mooring line force and moment acting on the supporting platform, respectively. and are the gravityinduced force and moment acting on the rotor, respectively.

Gyroscopic effect
In the above dynamic model, the aerodynamic load of the FOWT is directly incorperated into the dynamic equation as an external force, but the rotation of the rotor is ignored. Thus, the gyroscopic moment induced by the rotation of the rotor and the support platform is also ignored. To address this problem, the FOWT needs to be divided into two component structures: rotor and supporting platform . The frame is defined as shown in Fig. 2. It is assumed that the angular velocity of the floating platform is , and the rotor rotates at a relative angular velocity . Hence, the absolute angular velocity vector of the rotor relative to the global inertial frame is: The absolute angular acceleration of the rotor can be given by: where the sign "" denotes the first-order derivative versus time with respect to the local rotor frame E r . The second term of the above equation is the Coriolis effects between the angular motion of the floating supporting platform and the rotation of the rotor, which is also the main cause for the gyroscopic effects in an FOWT system.
At this time, the dynamic governing equation of the FOWT divided into two parts relative to the dynamic reference point O can be written as follows.
For the rotor : B 2 For the support platform : where and are the mass of the rotor and the supporting platform , respectively. and are the inertia moment of the rotor and the supporting platform as separate rigid bodies. and are the accelerations of the radius vector from the origin of the global inertial frame to the centroid of the rotor and the supporting platform , respectively. and are the radius vectors from the origin of the body-fixed frame E 0 to the centroid of the rotor and the supporting platform , respectively. and are the gravity-induced force and moment acting on the rotor , respectively. and are the gravityinduced force and moment acting on the supporting platform , respectively. and are the constraining force and moment from the supporting platform acting on the rotor , respectively. In contrast, and are the constraining force and moment acting on the supporting platform from the rotor , respectively, yielding (12) Combining Eqs. (9) and (10) and simplifying them by using Eqs. (11) and (12), eliminating constraining force and counterforce, yield Substituting Eqs. (6), (7) and (8) into Eq. (13), simplify Compared with the above-mentioned FOWT overall model in Eq. (6), there are additional terms and . It is assumed that the rotor is rotating constantly, and the relative angular acceleration of the rotor is zero ( ), thus the above additional term − can be neglected. Let: Eq. (15) is the gyroscopic moment term of the FOWT, which is added as an external force term to the dynamic model of the overall model , and the dynamic model of the FOWT with modified gyroscopic moment can be obtained as: The relative angular velocity of the rotor without con-sideration of the shaft tilt angle, yields: Ω Assuming that the floating platform moves in pitch direction ( ), the gyroscopic moment of Eq. (20) can be further simplified as: Eq. (21) indicates that the gyroscopic moment could excite the roll and yaw motion of the floating platform when the floating platform moves in pitch direction. In addition, the inertia tensor is generally small and even close to zero (depending on the selection of the coordinate system). For the rotor frame E r in this paper, the value of is small; as a result, the gyroscopic moment in roll motion is small, but the yaw motion is dominant. In other words, the combination of the rotor rotation of the FOWT and the platform pitch motion may excite a gyroscopic moment that causes the FOWT to make a significant yaw motion. The above equation can be simplified to: For the same reason, suppose that the support foundation of the FOWT is under yaw motion or the nacelle yawing to the incoming wind direction. At this time, the angular motion of the support foundation satisfies: , and the gyroscopic moment can be simplified as:

J Ryx
Consider that the inertia tensor is also usually small, that is, the roll motion excited by the rotor's gyroscopic moment is usually much smaller than the pitch motion. At this time, the combination of the rotor rotation of the rotor and the platform's yaw motion may excite a gyroscopic moment that causes the FOWT to undergo significant pitch motion.
It can be known from Eq. (15) that the gyroscopic moment of the FOWT is roughly linearly related to the rotor's inertia, the rotor speed, and the angular velocity of the platform motion. As shown in Fig. 3, the direction of the gyroscopic moment vector is determined by the rotation angular velocity vector of the rotor and the supporting platform. The right-hand theorem of vector cross product is observed. As shown in Eq. (22), the pitch motion of the supporting platform and the rotating motion of the rotor mainly stimulate the gyroscopic moment in yaw direction, but as the pitch angle of the platform changes, the space direction of the rotational angular velocity vector of the rotor is changing, so the gyroscopic moment will produce a roll component. Similarly, as shown in Eq. (24), the yaw motion of the supporting platform and the rotating motion of the rotor mainly excite the gyroscopic moment in the pitch direction. As a matter of fact, due to the gyroscopic moment, there is a complicated coupling process of pitch and yaw motion when FOWT is working. In summary, from the theoretical formula derived above, we know: (1) The FOWT has a more obvious gyroscopic effect than the bottom-fixed wind turbine because of the more violent motion of the floating supporting platform.
(2) According to the theoretical formula, the gyroscopic moment is mainly caused by the vector change of the rotating rotor. Therefore, for the FOWT, both the pitch and yaw motion of the platform may cause the system to generate the gyroscopic moment.
(3) The direction of the gyroscopic moment is orthogonal to the direction of the rotation angular velocity vector of the rotor and the direction of the angular velocity vector of the platform motions. Therefore, for an FOWT with the rotor rotating, the pitch motion of the platform will excite the gyroscopic moment in yaw direction of rotor, and the yaw motion of the platform will excite the gyroscopic moment in the pitch direction of the whole system. The floating wind turbine does not simply move in one direction, so the excited gyroscopic moment component is complex, time-varying and non-linear, as a matter of fact.
(4) The gyroscopic moment is related to the inertia of the rotor, the angular velocity of the rotor, and the angular velocity of the supporting platform. As the angular velocity of the platform is affected by the wave, the gyroscopic moment of the FOWT is affected by the incoming wave as well. Moreover, non-uniformity or mis-alignment of the wind will cause a more significant force in the yaw direc-tion, especially for a farm environment where turbines may lie in the waked flow of other turbines.

Results and discussion
According to Eq. (15) deduced by the paper, the gyroscopic moment of an FOWT is generated by the tilt of the rotating rotor as a matter of fact. In order to verify the above conclusion and carry out an investigation of characteristics of the gyroscopic effects on an FOWT, relevant numerical calculations and analyses are conducted by using the numerical tool FAST. The software FAST is developed by National Renewable Energy Laboratory (NREL) for numerical simulation of on-shore wind turbines in the early stages (Jonkman and Buhl, 2005). Later, Jonkman (2007) recoded the FAST and developed its capacity for calculating hydrodynamic loads and mooring loads for an FOWT. At present, the software FAST has been a well known code for simulating wind turbines (bottom-fixed ones and floating ones) with improved versions, e.g., FAST 7, 8 and OpenFAST. The accuracy of the code has been verified by a series of code-to-code (OC3 project (Jonkman and Musial, 2010;Duan et al., 2016) and code-to-experiment tests (OC4 project (Benitz et al., 2014;Shin et al., 2013) and OC5 project (Robertson et al., 2015(Robertson et al., , 2017Chen et al., 2018). In the following section, a spar-type horizontal axis floating wind turbine, OC3-Hywind, has been selected as the test object because of its sensitivity to the gyroscopic moment (Jonkman, 2010). Some characteristics of the OC3-Hywind are listed in Fig. 4, and more details can be found in Jonkman (2010).

Pitch motion and gyroscopic moment
FOWTs have significant pitch motion during its normal service, which will excite the gyroscopic moment in the yaw direction and a small moment component in the system's roll motion, according to Eq. (21).
In order to verify the above conclusion and investigate the gyroscopic effect during pitch motion, this section compares three load cases, namely: the rotor rotation only, the platform motions only, and the supporting platform mo- CHEN Jia-hao et al. China Ocean Eng., 2021, Vol. 35, No. 2, P. 201-214 205 tions combined with the rotor rotation. In this section, the rotor's rotation speed is set to 12.1 rpm, and the incoming wind speed is set to 0 m/s. During the test, the wind load calculation is turned off to avoid the interference, so as to observe, correlate and analyze the torque of the FOWT's gyroscopic moment. In this test, the periodic motion of the platform is forced by a regular wave load. The regular wave has a wave height of 4 m and a period of 10 s, as shown in Fig. 5. Under the action of this regular wave, the FOWT will produce obvious pitch motion, as shown in Fig. 6. The above three working conditions are shown in Table 1. The yaw moment of the wind turbine, the torque at the bottom of the tower, and the yaw motion of the platform under these three conditions are compared in Fig. 7. It is found that only the rotation of the rotor (R12.1H0) or the platform motion only (R0H4) is not enough to cause the FOWT to generate a significant yaw moment and motion. By contrast, when the rotor rotates and the platform foundation moves in pitch motion (R12.1H4), a larger yaw moment will be generated at the yaw bearing at this time. And the bottom of the tower will also be subjected to this torque, which will cause the platform yaw motion to be noticeable. It is no doubt that the dynamic responses are caused by the gyroscopic moment. In addition, the gyroscopic moment also caused a slight roll motion of the platform during the pitch of the platform, and the fluctuation has a multi-frequency component, but the amplitude is small. This is in accordance with the foregoing theory.
The power spectral density of the pitch, yaw and roll motions of the FOWT under the LC3 (the rotor rotates at 12.1 rpm and withstands the action of regular waves) can be obtained as shown in Fig. 8. It can be found that the frequency-domain response curve of the pitch motion obviously includes the wave frequency and natural frequency components of the pitch motion, and the yaw motion mainly includes the wave frequency components. In contrast, in the yaw motion, the components of the natural frequency response of the yaw and the pitch are much smaller. It shows that the yaw motion is driven by external force, which accords with the characteristics of the gyroscopic external moment. The roll motion is more complicated, and its frequency components are multi-peak as well, but its intensity is much smaller than that of the pitch or yaw motion. Peaks of the roll motion in frequency-domain in Fig. 8c are mainly at the natural frequency of roll motion (which is almost the same as the natural frequency of pitch), the wave frequency component, the natural frequency component of the yaw, and the rotor rotation frequency of 1P.
In reality, FOWTs are often subjected to waves and wind loads and produce significant pitch motions. If the rotor is rotating, it will produce a significant gyroscopic moment in the yaw direction at this moment based on the above analysis. The gyroscopic moment has the same order of magnitude as the torque induced by the wind and wave, which gives rise to fatigue damage of the yaw bearing and the tower structure.

Impact factors
According to Eq. (15), the gyroscopic moment is dependent on the inertia moment and rotating speed of the rotor and the platform motion. Relevant tests (Mostafa et al., 2012) have proved that the gyroscopic moment of an FOWT is affected by the inertia moment of the rotor. In order to verify the above-mentioned theoretical derivation and further investigate the characteristics of gyroscopic moment of a horizontal axis FOWT, the effect of rotating speed of rotor and platform pitch motion amplitude and frequency on the gyroscopic moment will be studied with numerical calculation results in this section.

Rotating speed of rotor
Set a regular wave (wave height H is 4 m and wave period T is 10 s) to force a periodic pitch motion of the FOWT and set the rotor to rotate at different speed R in terms of 0,    -hao et al. China Ocean Eng., 2021, Vol. 35, No. 2, P. 201-214 6.9, 8.5, 10, and 12.1 rpm, respectively, so as to investigate the effect of the rotating speed on the gyroscope moment of the FOWT. The calculation results in Fig. 9 show that as the rotating speed increases, the torque at the yaw bearing of the FOWT and the fluctuation amplitude of the yaw motion of the platform also increase, which is approximately linear. The rotor has the highest speed in the rated condition, so the corresponding gyroscopic moment will be more obvious at this time.

Support platform motion
This section will study the effect of platform motion on the gyroscope moment. Assume that the FOWT is subject to regular waves, resulting in regular motion of the supporting platform, as follows. η Let regular wave elevation in time series: is the amplitude of the regular wave, is the frequency of the regular wave and is the time series. At this time, the platform's position under the action of wave load can be written as: ξ The time series of velocity of the forced motion of the platform : H i (ω) , i = 1, 2, . . . , 6.
(27) By substituting Eq. (27) into Eq. (24), the gyroscopic moment expression can be written as:  It can be seen from Eq. (28) that the gyroscopic moment is affected by the wave frequency and wave amplitude , due to the motions of the platform. This is also a characteristic of the gyroscopic moment of the FOWT, which is significantly different from the onshore wind turbine. Therefore, the effects of wave frequency and wave amplitude on the gyroscopic moment of an FOWT will be studied separately in the following tests. Set the rotating speed as 12.1 rpm, set regular waves with different wave amplitude and period, force the FOWT to do pitch motion, and study the gyroscopic moment effect of the platform under different motion period and amplitude.
(1) Amplitude A m Set the period of the regular wave as 10 s and the wave amplitude as 1 m, 2 m and 4 m respectively. The rotating speed is constant at 12.1 rpm. The calculation results in Fig. 10 show that the excited pitch motion amplitude and regular wave amplitude have a linear relationship, that is, the larger the wave amplitude, the larger the pitch motion amplitude. As a result, the torque at nacelle yaw bearing and the yaw motion is also increased as the wave amplitude. All results show that the gyroscopic moment is affected by the pitch motion, which is consistent with the theoretical formula.
(2) Period Set the rotation speed of the rotor as 12.1 rpm, the regular wave height 4 m, and the wave periods T are 5 s, 10 s and 20 s respectively. It is known that the larger the period of regular wave, the smaller the corresponding frequency. As shown in Fig. 11, the yaw motion reaches a maximum when the wave period is 10 s (corresponding frequency is 0.1 Hz). This is inconsistent with the theoretical formula which indicates that the fluctuation amplitude of the gyroscopic moment and the frequency of the motion are positively correlated. In fact, the structural motion responses caused by periodic loads are related to the natural frequency of the structure. The closer the frequency of the periodic load is to the natural frequency of the structural motion, the larger the amplitude of the motion will be. Since the natural pitch period of the OC3-Hywind is 29.75 s, the corresponding natural frequency is about 0.0336 Hz, and the natural yaw first period of the OC3-Hywind is 10 s, and the corresponding natural frequency is about 0.1 Hz. The motion is within the range of this test, and the closer it is to the natural frequency, the larger the amplitude of the motion response.
(3) Wave direction The wave sets a wave direction every 15° from 0° to 90°, which are 0°, 15°, 30°, 45°, 60°, 75°, and 90°. By comparing the time history of pitch and roll motions, it is found that as the wave direction changes from 0° to 90°, the pitch amplitude decreases but the roll amplitude increases in turn. This is because the energy of the incoming wave at different directions changes with the incident angle of the incoming wave: The longitudinal component of the energy of the incoming wave gradually decreases, but its lateral component gradually increases. The comparison results are shown in Fig. 12. It is found that the moment and the motion of the yaw direction decrease with the increased incident angle, which is consistent with the pitch motion. As mentioned above, the gyroscopic moment in yaw is generated by the combination of the pitch motion of the supporting platform and the rotor rotation. In Figs. 12c and 12d, it is found that there is a phase change relationship between the yaw moment and the yaw motion at different incident angles of the incoming wave. The phase difference between the yaw moment and the yaw motion generated by waves at 0° and 90°i ncident angles is 45°.

Platform yaw motion
The above numerical calculation results show that the pitch motion of the supporting platform superimposed with the angular velocity of the rotor will excite the gyroscopic moment in the yaw direction. Eq. (24) shows that the yaw motion of an FOWT will also cause a gyroscopic moment in the pitch direction. In this section, the gyroscopic moment and dynamic response of the FOWT are studied when the rotor rotates and moves in yaw motion. In this section, the initial yaw offset of a given platform is used to allow the FOWT to perform free decay motion (yaw), and to explore the gyroscopic moment during the yaw motion of the FOWT.
In this section, three cases are set: the rotor rotates only, the platform yaw decays only, and the platform yaw decays with the rotor rotating. The rotor rotates at 12.1 rpm, and the initial yaw offset of the platform is 5°. Fig. 13 shows that only when the platform has a yaw motion and the rotor is rotating, the moment is obviously excited to cause the change of pitch motion of the overall system, which is the so-called gyroscopic moment at this time. The motion has obvious fluctuations, and the direc- tion of the fluctuations is opposite to the direction of the original pitch motion, which is consistent with Eq. (24).

Nacelle yaw motion
This section first compares the moment and dynamic response of the nacelle's yaw, whether or not the rotor rotates. And then observe under which conditions the gyroscopic moment occurs. Among them, the rotation time of the nacelle is shown in Fig. 14. The nacelle starts to rotate unidirectionally from 200 s to 300 s and the rotating speed of the rotor is 12.1 rpm.
It can be seen from Fig. 15 that only when the rotor rotates (in operation) and the nacelle's yaw occurs, a significant torque fluctuation and offset will occur at the yaw bearing. The torque is generated by the gyroscopic moment. With the forward center of gravity of the RNA (Rotor-Nacelle-Assemble) structure, the FOWT has a pitch decay fluctuation at the start. When the RNA is yawing, the for-ward tilt of the platform will be changed, so the equilibrium position of the pitch motion will be closer to the 0° position. By comparing the two cases in Fig. 15b, it can be found that only when the rotor rotates, the nacelle's yaw occurs at the same time, which will cause the platform to swing sharply and move in the opposite direction. The dynamic response is the same as the direction of the moment at the bearing (shown in Fig. 15a) , which are caused by the gyroscopic moment at this time.
Since the wind direction often changes in the actual case, the FOWT has to frequently perform yaw-to-wind motion, which will cause the yaw bearing to be subjected to the moment in the pitch direction. The tower will vibrate as a result, and the platform will also have the pitch fluctuations, which have severely affected the fatigue of the foundation structure, especially tower-bottom structure. Therefore, the effect of different yaw speed will be discussed below so as to find a way to mitigate gyroscopic moment. Set up four different nacelle yaw conditions, starting from 200 s. All the conditions start the nacelle for yaw-to-wind, which take 50 s, 100 s, 150 s, and 200 s, respectively (see Fig. 16). The entire yaw process is performed at a constant speed, therefore, the effect of different yaw speed will be discussed below.
It can be seen from Fig. 17 that the above test results show that the shorter the nacelle yaw execution time is, the larger the yaw speed will be. In addition, the larger the pitch moment to which the nacelle yaw bearing is subjected, the more obvious change in the pitch motion of the FOWT. The above test results show that reducing the speed of the yawing motion of the nacelle can reduce the gyroscopic moment in the pitch direction generated during the yaw-towind process of the FOWT.
It is known that the gyroscopic moment of an FOWT is generated when the direction of the rotation vector of the ro-   CHEN Jia-hao et al. China Ocean Eng., 2021, Vol. 35, No. 2, P. 201-214 211 tor is changed. For an FOWT, this change is often caused by the superimposition of the platform's pitch motion (including platform motion and tower vibration) or the yaw motion of the nacelle to the wind direction. The above research shows that the gyroscopic moment of the FOWT is related to the amplitude and period of angular velocity of the supporting platform and the rotation velocity of the rotor. When the rotor is rotating, the pitch motion of the platform will excite a gyroscopic moment in the yaw direction. During the pitch process, the gyroscopic moment will have a roll moment component causing slight amplitude. Similarly, the yaw of the platform or the nacelles' yaw motion will gener-ate a gyroscopic moment in the pitch direction and cause the overall system to make a pitch fluctuation as well. In summary, it can be known that the pitch and yaw of an FOWT will produce a coupling effect through the gyroscopic moment. As a matter of fact, the FOWT works in a more complex sea environment than the test cases listed in the paper. The process of actual gyro-moment is more complicated under the action of wind, wave and current loads.

Conclusions
Compared with bottom-fixed wind turbines, the supporting platform of the floating offshore wind turbine is restrained only by the mooring system. Hence, the FOWT's supporting platform moves more violently, and the resulting gyroscopic effect is more obvious as well. This paper systematically discussed the gyroscopic moment effect of the spar-type horizontal axis floating offshore wind turbine, OC3-Hywind and the following conclusions are drawn.
(1) According to the results in the paper, for a horizontal axis FOWT, the pitch and yaw motion of the platform may cause the gyroscopic moment to be generated. The direction is orthogonal to the direction of the rotation angular velocity vector of the rotor and the direction of the angular velocity vector of the supporting platform's motion. Therefore, for a rotating case, the pitch motion of the platform will stimulate the rotor's yaw direction gyroscopic moment.   In the same way, the yaw motion of the platform or rotor will excite the gyroscopic moment of the overall system in the pitch direction. In the actual service process, the FOWT does not simply move in one direction, so the excited gyroscopic moment component is more complex in fact. Although no research is conducted for a vertical axis FOWT in the paper, it can be speculated that the pitch and roll motion of the platform of a vertical axis FOWT may cause the gyroscopic moment according to the theoretical formula deduced in the paper. Compared with the horizontal axis FOWT, the vertical axis FOWT has a lower center of mass and no yaw system, and may perform better when subjected to gyroscopic moment.
(2) The gyroscopic moment is related to the angular velocity of the rotor, which will generate larger gyroscopic moment as the value increases. Therefore, during the violent motion of the platform or the yaw of the nacelle aligned on the incoming wind, it may be appropriate to consider reducing the rotor speed or reducing the yawing speed of the nacelle to restrain the induced gyroscopic moment at this time.
(3) The gyroscopic moment is related to the angular velocity of the supporting platform. In general, the larger the angular velocity of the supporting platform is, the larger the corresponding gyroscopic moment will be. Furthermore, the platform's motion is affected by loads such as waves, which also contains components such as wave frequency and the natural frequency of its own motions. Therefore, the wave height and frequency characteristics of the wave will affect the gyroscopic moment and dynamic response of the overall system as well. Therefore, seakeeping and damping optimization of the floating platform are vital in restraining gyroscopic effect.
Based on the theoretical derivation and a series of numerical tests in this paper, the principle of the gyroscopic moment characteristics of the horizontal axis FOWT is clarified. At the same time, related factors such as the magnitude and direction of the gyroscopic moment are discussed. This provides a reference for better understanding of the mechanism of the FOWT and control or optimization of the gyroscopic moment. In the future, experimental method would be used to verify the above-mentioned theoretical derivation and numerical test results, and to explore more about the dynamic responses of the gyroscope moment of an FOWT. Some techniques in the control, e.g., individual pitch control (IPC), might play a part in restraining gyroscopic effect of the floating wind turbine, which would be studied.