Internal resonances for heave, roll and pitch modes of a spar platform considering wave and vortex-induced loads in the main roll resonance

We present a study of the nonlinear coupling internal resonance for the heave roll and pitch performance of a spar platform under the wave and vortex-induced loads when the ratio of the frequencies of heave, roll and pitch are approximately 2:1:1. In consideration of varying wet surface, the three DOFs nonlinear coupled equations are established for the spar platform under the effect of the first-order wave loads in the heave and pitch, and vortexinduced loads in the roll. By utilizing the method of multi-scales when the vortex-induced frequency is close to the natural roll frequency, the first-order perturbation solution is obtained analytically and further validated by the numerical integration. Sensitivity analysis is performed to understand the influence of the damping and the internal detuning parameter. Two cases with internal resonance are shown. The first case is that no saturation phenomenon exists under small vortex-induced loads. The first order perturbation solution illustrates that only the vortex-induced frequency motion in roll and the super-harmonic frequency motion in heave are excited. The second case is that the vortex-induced loads are large enough to excite the pitch and a saturation phenomenon in the heave mode follows. The results show that there is no steady response occurrence for some cases. For these cases chaos occurs and large amplitudes response can be induced by the vortex-induced excitation.


Introduction
Spar platforms will undergo internal resonance if the ratio of the natural frequency of the heave and pitch is close to 2:1. Rho et al. (2002Rho et al. ( , 2003 studied the motion stability considering the influence of the mooring system by numerical simulations. When the spar platform exhibits an internal resonance, the mooring system cannot prevent unstable phenomenon from happening. Hong et al. (2005) compared the free attenuation characteristics and the nonlinear response characteristics of four spar platform by a model test. The results show that when the incident wave frequency is close to the natural frequency of the heave, unstable pitch response occurs. The stability of Mathieu equation with damping was used to explain this phenomenon. Neves et al. (2008) carried out a test about a cylinder model using different mooring arrangements. The result shows that when the heave inherent frequency is about double of the pitch frequency, the cylinder will have a unstable roll/pitch parametric resonance in the main heave resonance region which in-duce a substantially roll and pitch response. The response result is connected with the way of mooring. Zhao (2010) carried out a thorough research about the heave-pitch parametric resonance, coupled internal resonance, combination resonance on traditional Spar platform by using the non-linear dynamics method, numerical analysis and model test in combination. And many useful results were obtained. Shen et al. (2012) studied the nonlinear stochastic motion characteristics of a truss spar in the heave mode. It was observed that the time-domain analysis was more conservative than the frequency-domain method. Liu et al. (2014) further studied the heave-pitch nonlinear motion response of spar platform with the model experiment and numerical calculation. Yang and Xu (2015) proposed the pitch stability diagram in random waves of a spar platform. The unstable property was determined by analytical method and numerical simulation.
As a large-scale offshore platform, the deep draft cylinder structure determines the vortex-induced motions (VIM) under certain flow conditions for all types of spars. Wang (2010) studied the feature of wake field, fluid exciting force and VIM performance of a spar platform with experimental and computational fluid dynamics (CFD). The nonlinear effects of the mooring stiffness were fitted in surge and sway. A comprehensive test program was undertaken based on interactions of the current with regular and irregular waves. Zhang (2013) investigated the flow characteristics around a simplified Truss Spar with large-eddy simulation (LES). The comparisons of finite and infinite circular cylinder were discussed and the rationality of the finite circular cylinder simplification was proven by model test.
Many studies have been done on the single degree-offreedom (DOF) parametric resonance with a pitch model using the Mathieu equation or the two DOFs nonlinear coupling response of a spar platform with a heave-pitch model in a plane. However few studies have focused on the nonlinear coupling dynamic response of the heave, roll and pitch in three DOFs. In fact besides the nonlinear phenomenon in the heave, pitch and surge which are in one plane, the nonlinear unstable motion will also occur in the roll mode. From the results of the experiments, the unstable roll motion plays a dominant role in the motion of a spar platform which is much larger than the pitch mode. Most researches on the vortex-induced motions of spar platform only focus on the surge and sway mode in the horizontal plane while ignoring the pitch and roll caused by vortex. For the internal resonance of a spar platform, many studies have been done on the case in the main heave resonance. When the vortex-induced frequency is close to the roll frequency, the lock-in effect cannot be ignored. However few studies have been performed in the case that the excitation frequency is near the roll and pitch frequency.
We establish the three DOFs nonlinear coupled equations of the heave, roll and pitch considering the changes of instantaneous wet surface. With the first-order wave force in the heave and pitch, and the vortex-induced force in the roll, the method of multiple scales is utilized to solve the analytical solutions when the vortex-induced frequency approaches the natural roll frequency. The accuracy of the first-order perturbation solution is established by comparing it with a numerical solution. Two cases exist with the internal resonance phenomenon. If the vortex-induced excitation amplitude is small, there is no saturation phenomenon. Only the roll with the vortex-induced frequency and heave with super-harmonic frequency are excited. Multiple solution regions are found and the jump phenomenon exists. When the vortex-induced excitation amplitude is large enough, the pitch mode is excited with the vortex-induced frequency. In this case, there is a saturation phenomenon in the heave mode. The results also show that there is no steady state response in some cases. For these cases, large amplitude response may be caused by the vortex-induced excitation.

Three DOFs coupled nonlinear equations
There is obvious coupling relationship among the heave, roll and pitch for a spar platform. When a large amplitude heave motion occurs, the center of the buoyancy changes with the varying draft, which further causes the changes of the initial stability height and the volume of the displacement.
The notation and method in this paper follows the work by Zhao (2010). The heave, roll and pitch coupling schematic diagram is shown in Fig. 1.
The heave hydrostatic restoring force of the spar platform can be expressed by where ρ is the density of seawater; A w is the area of water plane; H g is the distance from static water surface to the center of gravity; ξ 3 , ξ 4 and ξ 5 are the heave displacements, roll and pitch angular displacement, respectively; η is the transient wave elevation.
The new hydrostatic restoring moment can be written by the changed metacentric height and displacement M The instantaneous displacement of the platform is expressed by The instantaneous metacentric height is expressed by GM Following the two element Taylor's theorem yields The higher order small terms are neglected based on the small angle assumption. The heave, roll and pitch hydrostatic restoring forces are expressed as: LI Wei et al. China Ocean Eng., 2017, Vol. 31, No. 4, P. 408-417 409 The first-order wave excitation force is considered in heave and pitch modes, and the vortex-induced force is taken into account in the roll. The three DOFs equations can be written as: where, m is the mass of spar, m 33 is the added mass of spar, I is the inertia moment for roll and pitch, I 44 is the roll added moments of the inertia and I 55 is the pitch added moments of the inertia, B i is the radiation damping, F 3 is the heave first order wave force, F 4 is the vortex-induced force, F 5 is the pitch first order wave force, Ω 1 is the wave frequency and Ω 2 is the frequency of the vortex-induced force. θ 1 , θ 2 and θ 3 are phase angles. By neglecting higher order nonlinear terms and wave elevation, Eq. (7) can be simplified intö , , Considering the symmetry of the spar platform, we can obtain a 4 =a 5 , and ω 4 =ω 5 approximately.

Steady-state solution of the three DOFs coupled equations
The multiple scales method can be utilized to solve the nonlinear equation and study the nonlinear characteristics of the offshore structures. Nayfeh et al. (1973) studied the pitch-roll nonlinear coupling response of a ship with this method. Zhao (2010) analyzed the heave-pitch nonlinear instability of a spar platform with the same method.
In order to eliminate a 3 , a 4 and a 5 from Eq. (8), new dependent variables are defined ξ By substituting Eq. (10) into Eq. (8), the equations with new variables are obtained.
The reduced forms are written as: In this section the first-order perturbation solution of Eq. (11) is obtained with the method of multiple scales. For convenience, ε is defined as a small dimensionless parameter, 0<ε<1. is defined as , i=3, 4, 5. Two time scales are introduced to obtain the first-order perturbation solution (13) The natural frequency ratio for heave, roll and pitch is close to 2:1:1. The vortex-induced frequency approaches to the natural frequency of the roll. We introduce the detuning parameters in order to describe the nearness of the frequencies to the resonance conditions Ω where σ 1 , σ 2 and σ 3 are called the detuning parameters.
In order to make the vortex-induced force term come together with the damping term in the same first order perturbation equation when Ω 2 ≈ω 4 , we assume that yields The derivatives are transformed according to Substituting Eq. (17) into Eq. (16) and equating the coefficients of equal powers of ε lead to The solutions to Eq. (19) can be written in the form where , , cc is the complex conjugate. A i is the function of T 1 and determined by satisfying the solvability conditions at the next level of the approximation. Substituting Eq. (21) into Eq. (20), we obtain where σ 1 , σ 2 , and σ 3 are the detuning factors that express the closeness of ω 4 to Ω 2 , 2ω 4 to ω 3 and 2ω 5 to ω 3 . Substituting Eq. (14) into Eqs. (22)-(24), the conditions for the elimination of secular terms can be written as: Under the polar coordinates, A i can be expressed with complex exponentials as: and are the real function of T 1 . Substituting Eq. (26) into Eq. (25) and separating the result into real and imaginary parts, we can obtain where The steady-state response is given by , and we have two possibilities: (a) α 5 is zero, while α 3 and α 4 are nonzero; and (b) α 3 , α 4 and α 5 are nonzero.
Case (a): . (32) The first-order perturbation solution for Eq. (11) is written as: In this case, the pitch mode corresponds to an uncoupled simple harmonic motion with the frequency of the wave force. In the heave mode, besides an uncoupled simple harmonic motion with the frequency of the wave force, the super-harmonic response driven by the vortex-induced excitation f 4 occurs. The heave amplitude α 3 is only dependent on the vortex-induced excitation f 4 in the roll mode. The roll mode is a simple-harmonic response coupled with the heave, and has the same frequency as the vortex-induced excitation. Consequently, in Case (a), the vortex-induced excitation in the roll induces the super-harmonic resonance only in the heave mode.
Case (b): , , and are the functions of α 5 , (39) The first-order perturbation solution for Eq. (11) is In Case (b), the super-harmonic response also occurs in the heave mode, while the amplitudes of heave α 3 , roll α 4 , and pitch α 5 are quite different from Case (a). An interesting feature of the heave is shown by Eq. (34): the amplitude of the heave mode α 3 has nothing to do with the vortex-induced excitation f 4 , its exciting force f 3 , and the heave damping μ 3 . The heave amplitude α 3 is only dependent on the pitch damping μ 5 . In the pitch mode, besides an uncoupled simple harmonic motion with the frequency of the wave force, a simple harmonic motion with the frequency of the vortex-induced force occurs. The roll mode is a simple harmonic motion coupled with the pitch, and has the same period as the vortex-induced excitation. The amplitudes of roll α 4 and pitch α 5 are only dependent on the vortex-induced excitation f 4 . Consequently, in Case (b), the vortexinduced excitation in the roll induces the internal resonance responses both in heave and pitch.
In order to analyze the stability of Eq. (27), we express Eq. (26) in terms of the Cartesian coordinates. According to Eq. (28), we can obtain

Substituting Eq. (41) into Eq. (26) we have
Substituting Eq. (42) into Eq. (27) and separating the results into real and imaginary parts, we obtaiṅ For a given solution, we calculate the eigenvalues of the Jacobi matrix (43). If all the real parts of the eigenvalues are negative, the solution is stable. If at least one eigenvalue is positive, the given solution is unstable.

Discussion on numerical results
The platform in this paper is not on a specific spar platform, which means that the basic character of the solution is useful to all spar platforms with the characteristics described in this paper. In order to validate the performance of the analytical results, a numerical integration is used to solve Eq. (11), and the results are compared with the perturbation solution.
The spar platform parameters in Hong et al. (2005) are used here to illustrate the results. The main particulars of this platform are summarized in Table 1.
In Fig. 2 the amplitude of heave α 3 , roll α 4 and pitch α 5 are plotted as a function of the amplitude of the vortex-induced excitation f 4 , when the vortex-induced frequency is near the nature roll frequency (σ 1 =0.02). The perturbation solution is validated by the numerical integration solution.
Only one stable solution exists in all regions, so jump phenomenon does not occur in this situation.
There is no saturation phenomenon when the solution is given by Eqs. (29)-(31) in Case (a) for the region of 0<f 4 <ζ.
With the increase of f 4 from zero, α 3 and α 4 increase. However, α 5 remains zero until f 4 =ζ. We can see a bifurcation phenomenon at ζ. There is a saturation phenomenon in the heave mode, when the solution is given by Eqs. (34)-(36) in Case (b) for the region of f 4 >ζ. For the first order approximation, the heave amplitude α 3 now reaches the maximum. The heave amplitude α 3 does not continuously increase. Increase in f 4 continuously produces the increase in the roll amplitude α 4 and the pitch amplitude α 5 increases from zero.
Consequently, when the vortex-induced excitation frequency is near the natural heave frequency, as the vortex-induced excitation in roll increases from zero, the nonlinear response are excited not only in roll but also in heave. All the kinetic energy is in the roll and heave mode, while the pitch is not excited. However, when the heave amplitude α 3 reaches the value of α * , the heave mode is saturated and cannot hold more kinetic energy. Increase in the vortex-induced excitation results in the increase in the kinetic energy in the roll, and the pitch mode is excited. That is the kinetic energy transfers from roll to heave, then to pitch. As a result, it is possible to produce large amplitude responses for roll and pitch. In Fig. 3, the amplitudes of heave α 3 , roll α 4 and pitch α 5 are plotted as a function of the amplitude of the excitation f 4 for three values of damping, when the vortex-induced excitation frequency is equal to the natural roll frequency. Fig. 3 shows the influence of the damping on the solution. It is seen that as μ 3 , μ 4 , and μ 5 increase, the bifurcation value ζ increases. This indicates that when the damping increases, the larger amplitude of vortex-induce excitation is required to make the heave mode reach the saturated value and excite the pitch mode. We conclude from Fig. 3 that the mechanism of energy infiltration becomes less effective as the damping increases, which is beneficial to the spar platform.
From Eq. (34), we note that the heave amplitude α 3 is independent of the values of the heave damping μ 3 . Combined with Fig. 3, when the heave mode is saturated, we find more interesting results. For a given f 4 , when the heave damping μ 3 increases from 0.0257 to 0.0411, and μ 4 , μ 5 remain unchanged (μ 4 =μ 5 =0.0204), the heave amplitude α 3 and roll amplitude α 4 stay the same. However, the pitch amplitude decreases. When μ 3 remains unchanged (μ 3 = 0.0257), and μ 4 , μ 5 increase from 0.0204 to 0.0408, the heave amplitude α 3 increases, whereas the amplitudes of roll and pitch decrease.
For Case (a), the pitch mode is not excited, so α 5 =0. In Fig. 4, the amplitudes of heave α 3 and roll α 4 are plotted as a function of σ 1 when Ω 2 ≈ω 4 . In the center region, no stable solution exists. In the region -0118<σ 1 <-0.095, and 0.095<σ 1 <0.118, there are two stable solutions. For these cases, the initial condition decides which stable solution is the real response. In other regions there is only one stable solution.
The jump phenomenon is indicated by the arrows with the change of the incident vortex-induced frequency. When the vortex-induced frequency is such that σ 1 increases from -0.15 to 0.15, α 3 and α 4 follow the curve from A to B. With the increase of σ 1 , α 3 and α 4 jump up from B to C and then follow the curve from C to D. When σ 1 increases to the right side of D, α 3 and α 4 jump down from D to E. This result is verified by the numerical method. It shows that with the change of the vortex-induced frequency, the response may increase or decrease suddenly which may cause accidence of the spar platform.
In Fig. 5, α 3 and α 4 are plotted as functions of σ 1 for different values of internal detuning factor σ 2 . We note that as σ 2 increases from zero to 0.08, the frequency response curve loses the symmetry. The maximum value of a stable solution increases with the increase of σ 2 . For a given σ 1 , the multiple solution region changes with σ 2 . In the center dip region, there is no stable solution. As σ 2 changes from zero to a certain value, the unstable center dip region makes an offset from the main roll resonance point (σ 1 =0, in Fig. 5). It indicates that for a spar platform in this situation, a large amplitude unstable response may occur in the region away from the primary resonance tuning point. Fig. 6 shows the influence of the damping. In Fig. 6, the multiple solution region changes with various values of damping. We note that when μ 3 =0.0368 and μ 4 =μ 5 =0.0321, there is only one stable solution in all regions. This indicates that the two stable solution regions disappear as the damping increases. And the heave, roll and pitch response will no longer exhibit two stable responses and the jump phenomenon.
For Case (b), the pitch amplitude is excited. In Fig. 7,   Fig. 4. Frequency-response curves when σ 2 =0 and Ω 2 ≈ω 4 , Case (a).  the amplitudes of heave α 3 and roll α 4 and pitch α 5 are plotted as a function of σ 1 with Ω 2 ≈ω 4 . Fig. 8 shows the influence of damping on the solution. The conclusions are similar to those in Fig. 3. In the center region, there is only one stable solution. In other regions, no stable solution exists. The numerical integration of Eq. (11) shows a chaos phenomenon in these regions. And the results are shown in Fig. 9. Fig. 9a shows the time histories for heave, roll and pitch, when σ 1 =0.042, σ 2 =0, Ω 1 =0.31 rad/s, f 3 =3.3×10 -5 , f 4 =7.0×10 -6 , f 5 =1.5×10 -4 . The energy exchanges in the heave, roll and pitch mode. Fig. 9b is the corresponding frequency spectrum, in which we find that the response frequency components are complicated. The Spar platform mainly vibrates with the vortex-induced frequency Ω 2 in pitch and roll, and the super-harmonic frequency 2Ω 2 in heave. The responses caused by the wave frequency Ω 1 component are very small.

Conclusions
The coupling internal resonances for heave, roll and pitch of a spar platform are studied when the vortex-induced frequency is close to the roll frequency with the frequencies in the ratio of 2:1:1. The first order perturbation solution is solved through the multi-scales method. The accuracy of the first-order perturbation solution is validated by comparing it with a numerical solution. The results are summarized as follows.
(1) There are two cases with the internal resonance. For Case (a), as the vortex-induced excitation in roll increases from zero, the roll amplitude grows, meanwhile a super-harmonic response increases in the heave mode. In this case, pitch is not excited and no saturation phenomenon exists. For Case (b), as the excitation amplitude increases further, the heave mode is saturated. The extra energy transfers from heave to pitch. The increase in the vortex-induced excitation results in the increase of the kinetic energy in roll and pitch.
(2) For Case (a), there are two stable solutions when the vortex-induced frequency is such that the internal detuning is away from the perfect resonance region. For these mul- Fig. 7. Heave, roll and pitch frequency-response curves when σ 2 =0 and Ω 2 ≈ω 4 , Case (b). tiple solution regions, the jump phenomenon is produced by varying of the vortex-induced frequency. Consequently, the amplitude of the spar platform would increase or decrease suddenly, which will possibly lead to the destruction of the mooring and riser system.
(3) For Case (b), no jump phenomenon is produced by the variation of the vortex-induced frequency. There is only one stable solution when the vortex-induced frequency is near the perfect resonance region. While for the cases the vortex-induced frequency away from the perfect resonance region, there is no stable solution. The numerical results show a chaos phenomenon in these cases, which will possibly lead to large amplitude responses for the spar platform.
(4) Without an internal resonance, the increase in the heave damping can decrease the heave amplitude. However, the heave amplitude is independent from the heave damping with an internal resonance. The heave amplitude depends on the damping in pitch and the internal detuning parameter. As the damping increases further, the response no longer exhibits the two stable motions and the jump phenomenon. The mechanism of energy infiltration becomes less effective as the damping and internal detuning increase.