Parametric Resonance Analyses for Spar Platform in Irregular Waves

The parametric instability of a spar platform in irregular waves is analyzed. Parametric resonance is a phenomenon that may occur when a mechanical system parameter varies over time. When it occurs, a spar platform will have excessive pitch motion and may capsize. Therefore, avoiding parametric resonance is an important design requirement. The traditional methodology includes only a prediction of the Mathieu stability with harmonic excitation in regular waves. However, real sea conditions are irregular, and it has been observed that parametric resonance also occurs in non-harmonic excitations. Thus, it is imperative to predict the parametric resonance of a spar platform in irregular waves. A Hill equation is derived in this work, which can be used to analyze the parametric resonance under multi-frequency excitations. The derived Hill equation for predicting the instability of a spar can include non-harmonic excitation and random phases. The stability charts for multi-frequency excitation in irregular waves are given and compared with that for single frequency excitation in regular waves. Simulations of the pitch dynamic responses are carried out to check the stability. Three-dimensional stability charts with various damping coefficients for irregular waves are also investigated. The results show that the stability property in irregular waves has notable differences compared with that in case of regular waves. In addition, using the Hill equation to obtain the stability chart is an effective method to predict the parametric instability of spar platforms. Moreover, some suggestions for designing spar platforms to avoid parametric resonance are presented, such as increasing the damping coefficient, using an appropriate RAO and increasing the metacentric height.


Introduction
Understanding parametric resonance is very important for the safety of floating structures. Parametric resonance may occur when a mechanical system parameter varies over time. For platforms and ships, this parameter is usually the metacentric height. Parametric resonance is known to occur during the roll motion of ships under certain conditions (Fossen and Nijmeijer, 2011). Parametric roll is a dangerous phenomenon for ships, and has been considered to be the reason for some accidents (Galeazzi et al., 2013). If a vessel experiences parametric rolling, accidents may occur, which can lead to high economic losses (Ginsberg, 1998;France et al., 2003). Because accidents can occur as a result of parametric resonance in irregular waves, parametric resonance should be regarded as a real danger in not only regular waves but also irregular waves. For many years, researchers have paid more attention to parametric rolling for ships in regular seas (Paulling, 1961;Dunwoody, 1989aDunwoody, , 1989bSpyrou, 2000;Francescutto, 2001;Bulian, 2006;Neves and Rodríguez, 2007;Chang, 2008;Spyrou et al., 2008). Similar to ship parametric rolling, spar platform parametric resonance in regular waves has also been observed in experimental tests (Haslum and Faltinsen, 1999;Hong et al., 2005;Neves et al., 2008). Several researchers have suggested parametric instability analyses for offshore structures in irregular waves using wave spectrums (Witz, 1995;Spyrou, 2000;Zhang et al., 2002;Radhakrishnan et al., 2007). Pettersen and Machado-Damhaug (2007) found that the parametric instability of a spar platform was triggered by irregular waves in experimental tests. Most studies of parametric instability have been based on harmonic excitation and characterized by regular waves. The parametric instability prediction methodology for a spar platform in irregular waves is still at the level of a recommendation and requires further investigation. The accurate experimental investigation of the parametric resonance of offshore structures is not easy; moreover, it is very timeconsuming (Bulian et al., 2008). Stability analyses should be extended to "realistic" seaways (Spyrou and Thompson, 2000). Therefore, the development of a methodology for predicting parametric resonance in irregular waves is urgently required for spar platforms.
Several studies have been carried out to predict spar parametric resonance in regular waves on the basis of the Mathieu equation (Haslum and Faltinsen, 1999;Zhang et al., 2002;Rho et al., 2005;Radhakrishnan et al., 2007;Zhao et al., 2010). A spar platform under real sea conditions may experience non-harmonic forces. However, the Mathieu equation was derived using harmonic excitations in regular waves. Thus, predicting the parametric instability of a spar platform in irregular waves is a challenging task. The Mathieu equation was used by several researchers to discuss the instability properties of ships and offshore structures (Yang and Xiao, 2014;Liu et al., 2015;Bhaskara Rao and Panneer Selvam, 2016;Rho et al., 2005;Radhakrishnan et al., 2007;Chang, 2008;Spyrou et al., 2008). Haslum and Faltinsen (1999) studied the instability property of spar platforms using a simplified method. Mathieu's instability phenomenon was found using both numerical simulations and model tests. Tao and Cai (2004) presented heave motion suppression of a spar platform with a heave plate. They calculated the heave damping forces by directly solving the Navier-Stokes equation. Zhang et al. (2002) drew spar instability regions on a Mathieu chart with damping changes. They pointed out that the probability of occurrence of regular waves with period above 20 s under the real sea conditions is very small. It is more important to develop stability diagrams in irregular waves with wave spectrums. Radhakrishnan et al. (2007) presented an experimental analysis of the instability of a tethered buoy in regular waves. They recommended further investigation of the instability that occurs in irregular waves. Parametric resonance has been found in the experiments for spar platforms in irregular waves (Pettersen and Machado-Damhaug, 2007). However, very few investigations have predicted the instability by developing a stability diagram on the basis of the damping effects for irregular waves with a specific spectrum (Wang and Zou, 2006).
In this paper, the parametric resonance of a spar in irregular waves is discussed using a Hill equation. It is derived from the motion equation for a spar platform based on the linear wave theory and response amplitude operator (RAO). The corresponding stability charts for a spar platform in irregular waves are developed by analyzing the equation. A comparison between Mathieu's equation for a single frequency and Hill's equation for multiple frequencies is also carried out. This work also provides design guidelines to avoid the occurrence of parametric instability in a spar platform.

Irregular wave theory
Irregular waves can be represented as the sum of har-monic wave components with random phases according to a wave spectrum. The Joint North Sea Wave Analysis Project (JONSWAP) spectrum is one of the most commonly used wave spectra, its formula is as follows: where, H s is the significant wave height, ω is the wave frequency, ω p is the peak wave frequency, γ is the peak enhancement factor, and σ is given as: The wave spectrum shows the distribution of the energy by frequency. Assume that the wave frequencies are in the range of ω 0 -ω n . Some frequencies ω 1 , ω 2 , …, ω n-1 can then be chosen using equal division of the frequency range or energy.
The wave height at one point can be expressed as follows: where, Δω i =ω i -ω i-1 , =(ω i-1 +ω i )/2, is random within the interval of (ω i-1 , ω i ), and ε i is the random phase for each element.

Dynamic responses of spar platform in irregular waves
Obtaining the heave motion of a platform on the basis of a numerical model is a complex and time-consuming process. Therefore, the RAO, which is easier to apply, is more frequently used when performing motion analyses of offshore floating structures.
The main purpose of RAO is to show the relationship between the structure's motion and the incoming unit wave. In this study, the pitch motion is assumed to be within a quite small range before parametric resonance. Thus, the nonlinear effects of pitch motion on heave motion can be ignored before parametric pitch occurs, and it can be assumed that the response of the heave motion of a spar platform to an incoming wave is a linear system. And when the parametric pitch occurs, though the nonlinear effects cannot be ignored, it is already predicted to be unstable. So, it is proper to apply the RAO of heave to simulate the heave motion in the prediction of the parametric pitch.
Both the amplitude of the wave and that of the spar motion can be represented as the superposition of sine/cosine waves with the same frequencies. For each component of the superposition of the spar motion, the amplitude and phase can be obtained by the corresponding component of the superposition of incoming wave with the same frequency and RAO.
The RAO curve for the H(T) value of the studied spar platform is shown in Fig. 1, with a JONSWAP spectrum. For a single wave frequency, the heave motion can be given by According to the linear wave theory, assuming that the response of the spar platform is a linear system, the heave motion of the spar under random waves is The heave motion amplitudes can also be obtained from the spectrum of the spar heave motion by applying RAO to the wave spectrum, as shown below: The heave spectrum in Fig. 2 shows the spectrum density of the heave motion energy with the time period. Compared with Fig. 1, it can be observed that the peak frequency of the heave motion is the peak of RAO, namely the natural frequency of heave, but not that of the wave spectrum. Therefore under the condition of irregular, the frequency is focused on the relation between natural frequencies of the heave and pitch motions.
The discretization of the spectrum is conducted by choosing a set of time periods and calculating the amplitude for each time period, as in the case of the wave spectrum. The heave motion is then given by their superposition, which can be rewritten to be the same as Eq. (5).
2.3 Instability prediction using Mathieu and Hill equations Most of the previous studies on spar platform parametric resonance analysis have assumed harmonic behavior for the time-varying restoring arm of metacentric height. The parametric resonance has been modeled as a Mathieu type equation in this assumption. However, a Mathieu equation has several limitations. First, the accuracy of the instability prediction is low. Second, a Mathieu equation is only valid for regular waves with a constant phase. The regular wave assumption is physically unrealistic, except perhaps under certain swell conditions (Witz, 1995). A stability prediction using a Mathieu equation is limited to a single harmonic excitation frequency. In this study, the Hill equation derived for spar instability prediction can include non-harmonic excitation and random phases.

Mathieu equation
One classic method for studying parametric stability is the use of a Mathieu equation. In this method, the heavepitch motion equation for a spar platform is simplified as: , (8) where, I is the moment of inertia of the pitch, M is the added mass coefficient, C is the coefficient of the linear damping, Δ is the displacement, GM is the initial value of metacentric height, ηcos(ωt) is the single-frequency heave motion, and Fcos(ωt+γ) is the force of the external excitation. One method that is used to determine whether the result is stable is to see whether the solutions of a variation equation tend to be zero. The variation equation is By using the variable substitutions and , Eq. (9) can be rewritten as: This equation can then be written as a damped Mathieu equation as follows: The substitutions are as follows: where ω 5 is the natural pitch frequency.

Hill equation
Under the condition of irregular waves, the heave motion cannot be simply described by a cosine wave. The equation of motion can then be written as: Similar to the above, the variation equation is In this study, the analyses mainly focus on the effects of the size of these parameters, thus the values of these parameters are assumed to be constant, ignoring the effects of different wave frequencies and some other aspects.
The heave motion, which is the linear superposition of a group of sine and cosine waves in irregular waves, can be written in the form of Fourier expansion, as shown below: where, ω k =2kω 0 , ω 0 is half the chosen base frequency, A k = and . The normal form of Hill equation is To deal with the heave motion of the platform, φ(τ) is taken as: where, , and , a k = .
The variation equation can then be rewritten as the Hill equation by making the following substitutions: By using Bubnov-Galerkin approach, the stability of the solution of the Hill equation can be determined (Pedersen, 1980). Though, with the determinant order larger, the obtained boundary could be more accurate, and it will cost much more time in calculation. Thus, in order to obtain a accurate boundary within acceptable time, the determinant order is recommended to be 2n+1, where n is the order of the expansion of φ(τ). In this study, n is chosen to be 15 to give a more efficient analysis (Yang and Xu, 2015).
According to the method above, to study the parametric instability by using this equation, only the natural pitch frequency and metacentric height are needed. This is very convenient to avoiding parametric instability in a preliminary design. The Hill equation can capture non-harmonic excitations under real sea conditions. The effects of damping and random phases are also considered in the stability prediction.

Stability charts in regular and irregular waves
The Mathieu equation is very general and does not capture the non-harmonic excitation property. If the excitation is not a single frequency harmonic, the system cannot be represented by a Mathieu equation. In such a case we can represent the time-varying coefficients as a Fourier expansion. The derived equation is called a Hill equation.

Stability chart in regular sea
The stability chart shows the stability region for the corresponding values of a and q according to the solution of the Mathieu equation. By changing damping c, instability regions with different damping coefficients can be obtained.
Most of the previous studies have used stability charts that do not indicate the effect of damping. Damping dramatically affects the boundaries between the stable and unstable regions. As shown in Fig. 3, the instability region is suppressed when damping increases. A two-dimensional chart is the most common method for predicting the parametric resonance of a spar platform. A major drawback of the method is that the chart does not depend on the spar characteristics. A three-dimensional chart that depends on the spar parameters would be a more practical approach for an engineering project. By applying the chart, the occurrence of parametric resonance can be predicted more conveniently in the preliminary design stage. To visualize the effect of damping on the instability regions directly, a threedimensional stability region figure in regular waves is given in Fig. 4, in which the added axis corresponds to the varying damping coefficient.
It can be seen from the above figures that increasing the damping can reduce the region of parametric instability, with a higher order resulting in a greater damping effect. The results of these charts agree with the results of previous studies.

Stability prediction for spar platform in irregular sea
The heave motion is simulated according to the wave spectrum and RAO. With the Fourier expansion, the heavepitch coupled motion equation can be rewritten as a Hill equation, and the stability chart can be given. Boundary lines in Fig. 5 with c=0 show the stability chart for a spar platform with a damping coefficient of zero. Besides, in this section the initial phases are chosen to be zero for convenience.
In Fig. 5, it can be observed that there are more instability regions, with most being much thinner. Compared with a single frequency condition, the stability regions are distributed more widely. Thus, in irregular waves, the occurrence of parametric resonance is more random.

Effects of damping coefficient
As in the case of a single wave, increasing the damping coefficient can also decrease the instability region in the stability chart. Fig. 5 shows the different instability regions with various damping coefficients.
The three-dimensional stability region for multiple frequencies is shown in Fig. 6, from which the effect of the damping coefficient can be directly observed. In Fig. 6, the two outside axes show the values of variable a and q, where the value of a shows the relationship between the natural frequency of the pitch and that of the heave, and the value of q shows the effect of the amplitude and the initial value of the metacentric height. The third axis represents the value of the damping coefficient. The first three instability regions are omitted because they are too small to be shown in the figure. Unlike the single frequency condition, where the first-order region is the least affected by the damping, under the multi-frequency condition, the effects of the damping on the regions show a slope shape. The region which corresponds to the first region under the condition of single frequency is least affected by the damping. It is supposed that the property of the instability regions under the multifrequency condition is affected most by the wave element with the largest amplitude among the set of harmonic waves. These stability charts can act as a guide in the preliminary design of a spar.

Checking for instability using stability charts
To verify the effectiveness of the charts, some typical design points are chosen to check the stability. The parametric values of the design points are applied in the equation of motion, and the time histories of the pitch are given. The results of the numerical simulations are compared with the positions of the points in the charts to see whether the charts predict the stability correctly.
The results of the simulations are listed in Fig. 7, Fig. 8 and Table 1. The results show that the stability charts can effectively distinguish the stability and instability regions.

Effects of the metacentric height and heave amplitude
The design points for a spar platform in a stability chart can be regarded as a set of points along a line that passes through the origin point with a slope of 4η/GM, which can be called the design line. When the design points fall in an instability region, the design is thought to be a failure. Therefore, if the design line has less intersection with an instability region, it is easier to avoid parametric pitch in the design.
In previous studies, it has been suggested that a smaller GM amplitude and larger initial GM value could reduce the possibility of the instability. In Fig. 9, the effect of smaller GM amplitude and larger initial GM value is that the slope of design line becomes smaller. The damping of the pitch is also considered in Fig. 9, which is shown as a decrease in the area of the instability regions.
The effects of these parameters can be easily observed in Fig. 9. With a larger initial GM, smaller amplitude for the heave motion, and larger damping coefficient, the instability is more likely to be avoided. These results agree with those of previous studies. However, for different regions, increasing the damping of the system and the metacentric height has different levels of effect. For the largest region, a larger GM can significantly decrease the area of instability, while for other thinner regions, increasing the damping works better.

Effects of random phases
In regular waves, the initial phase of wave has little effect on the responses of spar platforms. And in the motion equation, the phase can be eliminated through variable substitution. However, under the condition of irregular waves, when simulating it with multi-frequency waves, the effects of random phases cannot be dealt in this way. Thus, it is important to analyze how the random phases affect the property of parametric instability of a spar platform. Fig. 10 shows the instability regions with three groups of initial phases. For the first group, all initial phases are chosen to be zero. The initial phases in other two groups are chosen randomly based on the average distribution in the range of (-π, π].
It can be observed that, though the regions are different under different initial phases, the differences are quite small, especially for the main region. Also, for each region, with smaller value of q, the effect becomes smaller. Besides, for most regions, the ranges are wider when the initial phases are chosen to be zero.
Thus, it is acceptable to ignore the effects of random initial phases when analyzing the property of the parametric pitch of spar platform. However, for more accurate prediction in numerical simulation, it should be considered. The stability charts are more recommended to be applied for qualitative analyses.

Numerical model of spar platform
The main parameters of the classical spar platform be-  YANG He-zhen, XU Pei-ji China Ocean Eng., 2018, Vol. 32, No. 2, P. 236-244 ing investigated are listed in Table 2, where the parameters are the initial values when no heave or pitch motion occurs. The hull shape of classical spar is shown in Fig. 11. The JONSWAP wave spectrum with the peak wave frequency ω p =0.314 rad/s and peak enhancement factor λ=1.05 is used to simulate the irregular waves.
The steps for analyzing the parametric resonance of a spar platform are shown in Fig. 12. These can be divided into three main steps. First, the wave spectrum for real sea conditions is given with the chosen wave parameters. In this work, the JONSWAP spectrum is used. By applying the heave RAO of a spar platform, the spectrum of the heave motion is obtained. Second, the heave motion is simulated by using a Fourier expansion or obtained directly from the spectrum. Then, the heave motion can be discretisized into the superposition of a set of cosine waves, the periods of which are T 0 , T 0 /2, T 0 /3, … and the amplitude of each is cal-culated. Third, the values of the amplitudes are substituted into the equation of motion and the Hill equation is derived from it. The time history of the pitch motion can then be simulated by solving the motion equation with a numerical solution. The stability chart can be given by analyzing the Hill equation as shown in the previous section.

Numerical results
A prediction of the parametric resonance of a classical spar platform is carried out. The heave motion is simulated for both a single frequency and multiple frequencies. A length of time of 600 s is selected.
The simulations of the pitch motion are carried out on the basis of the motion equation. The amplitudes of the heave motion are imported into the equation and the equation is solved by using the Runge-Kutta algorithm. The results under two conditions are shown in Figs. 13 and 14. The length of the simulation time is also set to be 600 s. The pitch motion is simulated for different values of damping coefficient c and metacentric height GM.
Several conclusions can be drawn from Figs. 13 and 14. First, with the increasing damping, the pitch amplitude decreases with time, and a larger damping coefficient pro-  Fig. 11. Hull shape of classical spar. duces the faster decrease in the pitch motion. Second, a larger metacentric height GM makes the pitch amplitude smoother, and even though for some parts the amplitudes may be larger for a larger GM, the maximum amplitude of the larger GM is always smaller. Third, even when the heave motion is regular, the pitch motion may become irregular. It can be concluded from the above that increasing both the damping and the metacentric height can make the spar platform more stable. However, for a stable case, the two methods have different effects.

Conclusions
This paper presents a methodology for a parametric instability analysis of a spar platform in irregular waves. A mathematical model is derived in the form of a Hill equation. The three-dimensional stability charts of irregular waves not only display information about the stability boundaries, but the damping changes are also considered. Stability charts and simulations for different damping coefficients were also investigated. From the present work, the following conclusions can be drawn.
(1) Through a numerical simulation of the pitch motion, it can be confirmed that parametric instability may occur in irregular waves. However, its occurrence is more random compared with under a regular wave condition. The stability charts showed a wider distribution of the regions of instability under an irregular wave condition, whereas the distribution was more concentrated under a regular wave condition.
(2) By comparing the analyses using a Mathieu equation in regular waves with a Hill equation in irregular waves, it can be observed that the parametric instability properties of a spar in regular and irregular waves have obvious differences. Thus, it is necessary to predict the parametric resonance of a spar in irregular waves.
(3) Predicting the parametric instability of a spar platform by using a Hill equation and stability charts is an effective method. With a given heave motion, the method only requires the natural period of the pitch and the initial value of metacentric height GM. This makes it very convenient to use for design.
(4) Smaller metacentric height changes make it easier to avoid instability (when the damping is not zero). For spar platforms, it means a decrease of the amplitude of the heave motion. Additional heave damping can be used to decrease the peak value of RAO, such as through the use of damping plates. It is also important for the peak RAO to avoid the  YANG He-zhen, XU Pei-ji China Ocean Eng., 2018, Vol. 32, No. 2, P. 236-244 243 peak frequency of the waves.
(5) Increasing the initial value of metacentric height GM is also an effective method to reduce the possibility of parametric instability. The most effective way is to make the center of gravity lower. However, it is not easy to make a large change in GM in an engineering design, and some other parameters may also change with it.
(6) Increasing the damping of the pitch motion has a great effect under most conditions for both regular and irregular waves. In the stability charts, it is shown as a reduction in the instability areas. However, for the instability regions determined by the wave element with the largest amplitude, the effect of damping is smaller than that in other regions. Thus, it is recommended that the design point be kept away from these regions.
The methodology presented herein may help designers estimate parametric instability at the preliminary design stage for spar platforms in a more practical way. Unfortunately, it is not yet supported by experimental results. Thus, experimental tests are necessary in the future. To thoroughly study the parametric resonance of a spar platform, some investigations can be carried out in the future using a simulation model that includes the influences of the hull geometrical form and an analysis of stochastic stability in irregular waves.