Parametric instability prediction in a top-tensioned riser in irregular waves

Ocean Engineering 70 (2013) 39–50

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Parametric instability prediction in a top-tensioned riser in irregular waves Hezhen Yang a,b,n, Fei Xiao b, Peiji Xu b a b

State Key Lab of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China School of Naval Architecture. Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China

art ic l e i nf o

a b s t r a c t

Article history: Received 28 October 2012 Accepted 11 May 2013 Available online 19 June 2013

The parametric instability of a top-tensioned riser (TTR) in irregular waves was predicted based on multifrequency excitation. As the exploration of oil and gas moves into deeper water and the corresponding platforms encounter continuous wave loads, parametric resonance becomes an increasing challenge for TTR design. TTRs may suffer damage due to excessive stress or significant fatigue because of parametric resonance. Studies of the stability of long slender structures have been limited to regular waves, with an emphasis on explaining parametric resonance corresponding to simplified situations in which the related displacement or tension fluctuation is assumed to be an ideal harmonic signal. Waves are irregular in real sea conditions, therefore, the present study proposes a methodology to investigate the instability properties of TTRs in irregular waves. Hill's equation of a TTR system under multi-frequency excitation is derived based on linear-wave theory using a Pierson–Moskowitz wave spectrum. Analytic results are given for the stability limit of the related Hill's equation and they describe the stability of a TTR under irregular wave excitation. These new limits are compared to Mathieu-type stability limits. The stability diagram for Hill's equation with random phase modulation in multi-frequency excitation is obtained. To evaluate the proposed method, an example of the application of the stability diagram to TTR design is provided. Simulation results are compared under two types of wave excitation: single-frequency excitation and multi-frequency excitation, generated according to specific sea spectra. Parametric stability properties of a TTR system predicted by regular wave and irregular wave theory are different from each other. Although the transition curves both move to the upper zone of the parametric plane and the region of unstable zones shrink when the damping coefficient increases, the impact of damping is different. The results show that it is necessary to predict the parametric stability of TTR using irregular waves. Applying the stability diagram to a TTR in irregular waves shows that the introduction of extra damping to an unstable system can suppress the instability. In addition, the single-frequency excitation method predicts that the lower vibration mode is more likely to be excited, whereas the multi-frequency excitation method predicts that the higher vibration mode is more likely to fall into an unstable zone. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Riser Irregular waves Parametric resonance Instability Damping Mathieu

1. Introduction The oil and gas reserves of shallow water are gradually being depleted, so it is necessary to move into ultra-deep waters for oil exploration. As a result, the design of marine risers has become an increasingly critical issue. The riser serves as the link between the platform and the wellhead on the sea base. Risers present many technical challenges, especially deep-water risers, which are used in harsh ocean-environment conditions. The risers in ultradeep water have complex dynamic characteristics. When parametric resonance occurs, risers encounter damage due to excessive

n Corresponding author at: State Key Lab of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China. Tel.: +86 21 34207009; fax: +86 216 2932 320. E-mail addresses: [email protected], [email protected] (H. Yang).

0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.05.002

stress or significant fatigue, leading to oil/gas spills that cause environmental pollution and significant economic losses (Cruz and Krausmann, 2008; Yang and Li, 2009). Recently, research has been focused on the parametric instability of offshore structures, such as the parametric rolling of ships, spar platforms, the tethers of tension-leg platforms (TLPs), and risers (Park and Jung, 2002; Tao et al., 2004; Ahmed et al., 2010; Chandrasekaran et al., 2006; Chatjigeorgiou and Mavrakos, 2002; Koo et al., 2004; Rho et al., 2005; Zhang et al., 2010). Researchers have focused on the Mathieu instability of long slender structures, such as tethers and risers. Patel and Park (1991) reported an investigation of the dynamics of tethers with reduced pre-tension to facilitate increases in payload compared to conventional TLP designs. However, when tether pre-tension is reduced, the dynamics of wave-induced time-varying axial forces becomes important. This time-varying axial force causes the tether to undergo

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H. Yang et al. / Ocean Engineering 70 (2013) 39–50

parametric oscillations described by harmonic signals. Chatjigeorgiou and Mavrakos (2002) investigated the non-linear dynamic response in the transverse direction of long slender structures subjected to parametric excitation at the top of the structure. Their analytical approach showed that the dynamic lateral response was governed by effects originating from the coupling of modes in the transverse direction. They also showed that the internal resonance originates from parametric excitation of a slender pipe conveying fluid for marine applications, focusing on a specific case corresponding to the frequency equal to twice the first lateral natural frequency of the structure (Chatjigeorgiou and Mavrakos, 2005). Zhang et al. (2002) investigated tension variations in a tendon of the TLP and showed that tension variations could lead to lateral motion. These studies assumed only harmonic imposed motions and referred only to Mathieu's equation. However, an investigation of parametric resonance in regular waves can reveal only the generic behaviour of the system, which can differ considerably from the behaviour in random seas. API RP 2RD suggests that TTRs with relatively stiff tensioning systems may experience tension fluctuations that are significant relative to the mean tension, leading to significant changes in the lateral stiffness (API RP 2RD, 1998). Therefore, the development of a reliable method for predicting parametric instability in nonharmonic motion is urgently needed. Parametric instability does occur in irregular waves (Hua and Wang, 2001). In recent years, several modern ships have had serious accidents due to parametric rolling (France et al., 2003; Hua et al., 2006). Because the stochastic Mathieu instability is practically non-ergodic, it is very difficult to experimentally model the behaviour (Maki et al., 2011). Several researchers have discussed parametric excitation with two frequencies (El-Bassiouny, 2005; To, 1991). Spyrou (2000) proposed a methodology to analyse the Mathieu instability for ship rolling. The method assumes that bi-chromatic waves need to be evaluated to understand the behaviour of a ship under the effect of a wave group containing two independent frequencies. Bulian et al. (2006) concluded that the presence or absence of ergodicity in ship rolling cannot be determined through an experimental approach. In the tested conditions, the parametric roll process should be considered ‘practically not ergodic’, which means that the run length needed to achieve a sufficiently small confidence interval for temporal averages is very long and may be impracticable for many facilities because of wave reflections. Therefore, numerical prediction tools may be helpful in determining practical limitations. Hua and Wang (2001) reported parametric rolling in roll-on/roll-off ships in irregular following waves. The parametric excitation resulting from wave loadings produced considerable variation in the metacentric height (GM). Zounes and Rand (1998) investigated an extension of Mathieu's equation in which the coefficients of the stiffness terms have two frequency components that are equivalent to the instability of bi-frequency waves in wave-driven parametric excitations. The stability diagrams of the bi-wave Mathieu equation presented in their paper are quite intricate. Because of the lack of communication between mathematicians and engineers, it has been difficult to use these diagrams in engineering fields. Chang (2008) recommended the investigation of parametric rolling in irregular waves to thoroughly understand the parametric rolling of ships. Radhakrishnan et al. (2007) observed the transverse instability of a tethered spherical buoy when the period of the wave generated was close to one-half of the natural period of the buoy, and suggested that the transverse instability occurs in the field when structures have a more complex form than that of model structures and the waves are irregular. Zhang et al. (2002) pointed out that it is more important to develop stability diagrams in irregular waves with JONSWAP, or PM, or other defined spectrums. Wang and Zou (2006) noted the continuing challenge of developing a stability diagram with

damping effects for irregular waves with a specified spectrum (Wang and Zou, 2006). The stability threshold of irregular waves noted by Rainey (1977) continues to be unresolved. The Mathieu stability of risers in regular waves may be induced in certain conditions depending on the magnitude of the tension fluctuation and the damping of lateral motion. Studying the instability of long slender structures in irregular waves may require the use of a Hill's equation-based stability analysis. However, because it is very difficult to predict parametric resonance in irregular wave conditions, stability analyses should be extended to ‘realistic' conditions. Therefore, the development of a reliable method to predict the parametric instability of a TTR in irregular waves is urgently needed. The aforementioned studies that were focused on tethers or risers discuss the parametric resonance or Mathieu instability under the assumption that the tension fluctuation or top motion is harmonic. However, the waves in realistic sea conditions are stochastic. Thus, when an offshore platform is subjected to waves, it will heave with the rhythm of the wave motion; as a result, the tension of the riser (or the top displacement of the riser) will fluctuate with the rhythm of the platform motion. The irregularity of ocean waves must be considered when studying the stochastic Mathieu problem and assessing the instability of the system. In the present study, the external excitation, which is a type of multifrequency function, is obtained from a simulation of the PiersonMoskowitz wave spectrum based on linear-wave theory and the general form of a Hill’s equation is derived in Section 2. In Section 3, Hill's equation is solved using the Bubnov–Galerkin procedure and the corresponding stability diagram is obtained. The difference between single-frequency and stochastic excitation is compared. The influence of the random-phase modulation on the shape of parametric resonance regions is evaluated in Section 4. Finally, in Section 5, engineering case studies are described to provide suggestions for TTR design in irregular waves.

2. Theory and method 2.1. Irregular wave theory Irregular waves are described by linear-wave theory, which is efficient for wave calculations. The parametric rolling of ships in irregular waves assumes that the parametric excitation is a linear stochastic process (Hua and Wang, 2001). According to linearwave theory, an irregular sea is assumed to be a wave system consisting of a number of regular wave components. The phase which lags between the regular wave components changes at random within ½π; π. A superposition of an unlimited number of cosine waves is used to describe the wave front of a fixed point. ∞

ηðtÞ ¼ ∑ an cos ðωn t þ εn Þ n¼1

ð1Þ

In Eq. (1), an is the amplitude of the wave components and ωn is the circular frequency of the wave components. The stochastic initial phase εn is in the range of ½π; π. The average energy in a frequency interval Δω is defined as follows: SðωÞ ¼

1 ωþΔω 1 2 a ∑ Δω ω 2 n

ð2Þ

Eq. (2) represents the energy within a unit frequency interval when Δω ¼ 1, namely the energy density. Therefore, SðωÞ denotes energy spectrum density. The two lines in Fig. 1 represent the P-M spectrum and the effective wave spectrum, respectively. The P-M spectrum is the theoretical spectrum used in the present study, the effective wave

H. Yang et al. / Ocean Engineering 70 (2013) 39–50

Power spectrum density S(f) (m2.s)

According to the stress–strain relation, the tension of the TTR can be expressed as follows: "
 # ∂u 1 ∂v 2 T ¼ EA þ ð8Þ ∂x 2 ∂x

Effective wave spectrum P−M spectrum

2.5

41

2

Substituting formula (8) into Eq. (5), the following equation can be obtained:  ∂v  ∂v ∂2 v ∂4 v ∂2 v ∂v   ¼0 ð9Þ M 2 þ 0:5C d ρw D  þ EI 4  T 2  mw ∂t ∂t ∂x ∂t ∂t ∂t

1.5

1

Because the tension varies in space owing to the wet weight, mw ¼ ∂T=∂x represents the wet weight per unit length. In the present study, the platform's geometric centre is treated as the reference point for wave motion. Tension fluctuation is assumed to follow the rhythm of wave motion, and thus, the riser tension can be defined as follows:

0.5

0 0

0.5

1

1.5

2

Frequency f (Hz) Fig. 1. Schematic of P-M spectrum and the corresponding effective wave spectrum.

∞

T ¼ T 0 þ ΔT ∑ an cos ðωn t þ φn Þ n¼0

spectrum is determined through linear-wave theory using the P-M spectrum and the amplitude an and circular frequency ωn in Eq. (1) is obtained in the meanwhile. The P-M spectrum adopted here is defined as follows: −4

SðωÞ ¼ Aω−5 e½−Bω



ð3Þ

ð10Þ

Where, T 0 is the mean tension of the TTR, ΔT is the amplitude of tension variation, ωn is the tension variation frequency of the TTR, and φn is the random initial phase. 2.3. Hill's equation derivation

In Eq. (3), A ¼ 123:9H 2s T −4 , B ¼ 495:8T −4 , T ¼ 0:711T p , H s is the significant wave height, and T p is the spectral peak period.

According to the boundary condition of the TTR, the lateral motion of the nth natural mode follows the following form:

2.2. Parametric problem of TTRs in irregular waves

To determine the stability of the special solution (11), a neighbouring solution, F n ðtÞ þ δF n ðtÞ is obtained by changing the initial conditions slightly to compare with the aforementioned solution. The stability of solution F n ðtÞ is judged according to whether or not the solution of Eq. (12) is completely bound. Substituting Eq. (11) into Eq. (9),   2 ∞ d δF n dδF n þ 2c þ a þ 2q ∑ a cos ðω t þ φ Þ δF n ¼ 0 ð12Þ n n n dz2 dz2 n¼0

vn ðx; tÞ ¼ F n ðtÞ sin ðnπx=lÞ; n ¼ 1; 2; 3; …

In the present investigation, the TTR system is simplified as a beam-column with two ends hinged; the stiffness and other material properties of the TTR system are unchanged along the length. The TTR is placed in a steady uniform flow and the unidirectional wave and current are considered. A nonlinear strain-displacement relation is used here to derive the static equilibrium equations of a TTR as a coupled beamcolumn:
 ∂u 1 ∂v 2 ε¼ þ ð4Þ ∂x 2 ∂x where, uðx; tÞ and vðx; tÞ are the axial and lateral displacements, of the riser, respectively. Compared to the cross-sectional rotation, the strain is assumed to be small, according to Kirchhoff's hypothesis. Incorporating inertial forces into Eq. (4), the following can be derived: (
"
 #)
 ∂2 v ∂2 ∂2 v ∂ ∂v ∂u 1 ∂v 2 þ M 2 þ 2 EI 2  ¼ f ðx; tÞ ð5Þ ∂x ∂x ∂x 2 ∂x ∂x ∂x ∂t Where, the riser mass M is the mass per unit length and composed of steel mass, added mass, and internal fluid mass, and is defined as follows: M ¼ πDhρs þ 0:25πD2 ρw þ ρi Ai

ð6Þ

Where, D is the diameter, h is the wall thickness, ρs is the density of steel, ρw is the density of sea water, ρi is the density of internal fluid, and Ai is the internal area. Considering the influence of flow when the riser system is vibrating in the current, f ðx; tÞ is the drag force, which is defined as follows:  ∂v  ∂v   ð7Þ f ðx; tÞ ¼ 0:5C d ρw D  ∂t ∂t where, C d is the hydrodynamic drag coefficient.

ð11Þ

Rewriting Eq. (12) into the conventional Hill equation using differential symbols, the following can be obtained:   ∞ x€ þ 2cx_ þ a þ 2q ∑ an cos ðωn t þ φn Þ x ¼ 0 ð13Þ n¼0

where, z ¼ ωt, ω is the basic frequency and a ¼ ðωn =ωÞ2 ; q ¼ ΔTðnπ=lÞ2 =2Mω2 ; c ¼ 2C d ρw DV max =3πMω where, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωn ¼ ðnπ=lÞ3 ½EIðnπ=lÞ3 þ T 0 ðnπ=lÞ þ mω =M is the lateral natural frequency of the TTR. Factor an is completely normalized, namely ∑∞ n ¼ 1 an ¼ 1. In the present study, the value of n is finite. During the derivation, a linearization process was carried out to simplify damping expression, where jdF n =dtj was substituted by V max , which is the largest velocity of riser transverse motion.

3. Comparison between stochastic and harmonic excitations 3.1. Parametric stability chart comparison based on harmonic and non-harmonic excitations The stability chart is a well-known tool for examining the properties of parametric instability, which consists of massive transition curves above which are unstable zones and below which are stable zones. Parametric instability analysis using the stability diagram

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H. Yang et al. / Ocean Engineering 70 (2013) 39–50

Stability chart for single−frequency (black)/multi−frequency (red)

Mathieu stability diagram corresponding to regular waves 50

100

damping = 0.0 90

45

80

40

70

35

Black −−−single−frequency Red −−−multi−frequency

multi−frequency

unstable

30

50

q

q

60

25

unstable 40

20 A

unstable

30

unstable

20

B

15 10

10 unstable

Design point

5 single frequency

0

0

10

20

30

40

50

60

a

0

10

20

30

40

50

60

a

Fig. 2. Stability diagram of single-frequency excitation.

Fig. 4. Parametric stability chart under single-frequency and multi-frequency excitations.

Hill stability diagram corresponding to irregular waves

both stability charts, the parametric plane is separated by transition curves, above which are unstable zones and below which are stable zones. Fig. 2 indicates that the areas of the unstable zones corresponding to single-frequency excitation are becoming larger and the correlated bell-shaped unstable zones are moving upwards, as their order increases. In the first four unstable zones in Fig. 3, the higher-order unstable zones are larger than lower-order ones. In zones above the fourth order, the unstable area does not increase as the order becomes higher. The first three unstable zones are far from the horizontal axis, whereas the other zones are close to it. For both Figs. 2 and 3, the order of unstable zones ranges from one to infinity along the positive x-axis. A comparison between Figs. 2 and 3 shows that a system predicted by single-frequency excitation tends to be more prone to fall into the first three unstable zones. From this point of view, single-frequency predictions are more conservative than multifrequency predictions, which can result in a waste of resources due to conservative designs. However, the stable zones that are predicted by single-frequency excitation can be higher-order unstable zones under multi-frequency excitation; this situation is dangerous if a system is predicted to be stable using singlefrequency excitation.

100

damping = 0.0 90 80

unstable

70 60

q

0

unstable

unstable

unstable

unstable

50 40 30 20 10 0

0

10

20

30

40

50

60

a Fig. 3. Stability diagram of multi-frequency excitation.

has several advantages: (a) the stability diagram allows for easy visualization of the change in stability as the parameters are varied; (b) multiple physical situations can be compared at the same time; and (c) the stability diagram can be very useful to support offshore structural design.

3.1.1. Comparison of stability charts from single-frequency and multi-frequency excitations In this section, the stability diagrams from single-frequency and multi-frequency excitations are developed, analyzed, and compared. Fig. 2 shows the stability chart under single-frequency excitation and Fig. 3 shows the stability chart under multi-frequency excitation. In general, The bottom of the bell-shaped unstable zone would touch the horizontal axis of the stability chart with no damping if the calculation accuracy is high (Poulin and Flierl, 2008). Due to numerical deviation arising from high-power index calculations, the stability chart does not intersect with axis. The calculation accuracy of the present study is sufficiently high since all zero damping transition curves touch the horizontal axis, which can be observed from Figs. 2 and 3. Simulation results are compared for two types of wave excitations: single-frequency excitation and excitation generated according to P-M sea spectra. There are significant differences between these two cases, although to some extent, they are quite similar. In

3.1.2. Dynamic response predictions from single-frequency and multi-frequency stability charts The dynamic responses predicted by single-frequency and multi-frequency stability charts are discussed in this section. It is well known that most of the vibration energy of a system is concentrated in lower-order vibration modes, so this part emphasizes the first seven unstable zones, and discussions are focused on the dynamic response and phase plane trajectory of a TTR system under single-frequency and multi-frequency excitations, respectively. For the convenience of comparison, the parametric stability diagrams under two kinds of excitation are overlapped, as shown in Fig. 4, and the system damping is 0.05 for both cases. The differences between the single-frequency and multi-frequency stability charts are more apparent in the superposition stability chart. For the first three zones, the unstable area for single-frequency zones is much larger than that for multi-frequency ones, whereas for higher-order zones, many stable single-frequency zones fall into unstable multi-frequency zones.

H. Yang et al. / Ocean Engineering 70 (2013) 39–50

43

Dynamic response for design point A (multi−frequency)

Table 1 Property of representative design points in superimposed parametric stability diagram.

8

6

A(a¼ 9.1872, q¼ 16.426) B(a¼ 35.922, q¼ 14.692)

Point position Third order zone Behavior( single-frequency) Unstable Behavior( multi-frequency) Stable

Sixth order zone Stable Unstable

x 1072 Dynamic response for design point A (single−frequency)

4

Amplitude (m)

Design point

1

2

0

−2

0.5 −4 0

Amplitude (m)

−0.5

−6

0

20

40

−1

60

80

100

time (s)

−1.5

Fig. 6. Dynamic response under multi-frequency excitation (Point A).

−2 −2.5 Dynamic response for design point B (single−frequency) −3

8

−3.5

6

−4

0

20

40

60

80

4

100

Amplitude (m)

time (s) Fig. 5. Dynamic response under single-frequency excitation (Point A).

2 0 −2 −4

Two representative design points corresponding to the zones mentioned above are chosen here for investigation. The properties of points A and B are listed in Table 1.

−6 −8 −10

0

20

40

60

80

100

time (s) Fig. 7. Dynamic response under single-frequency excitation (Point B).

x 105

Dynamic response for design point B (multi−frequency)

8 6 4

Amplitude (m)

3.1.2.1. Dynamic response comparison in time domain. Fig. 5 shows the dynamic response for design point A that is predicted by single-frequency excitation. The vibration amplitude increases to a large value within 100s, and therefore, the system is unstable. Fig. 6 shows the dynamic response for design point A that is predicted by multi-frequency excitation. The vibration amplitude decreases to a small value within 100s, and therefore, the system is stable. The stability properties indicated by Figs. 5 and 6 are congruent with the stability zone that point A belongs to, i.e. the stable zone for multi-frequency prediction and the unstable zone for single-frequency prediction, respectively, as shown in Fig. 4. Once an unstable vibration mode of a system occurs, extra damping is needed to suppress it, which means extra cost. It can be observed from Fig. 4 that the area of the unstable zones for single-frequency excitation is many times that of multi-frequency excitation. Thus, a single-frequency prediction of parametric stability is too conservative for the first three zones. Fig. 7 shows the dynamic response for design point B that is predicted by the single-frequency stability chart; the vibration amplitude decreases to a small value within 100s, and therefore, the system is stable. Fig. 8 illustrates the dynamic response for design point B that is predicted by the multi-frequency stability chart; the vibration amplitude becomes large within 100s compared to the initial amplitude, and therefore, the system is unstable. The stability properties shown in Figs. 7 and 8 agree with the stability zone where design point B is in: unstable zone for multi-frequency excitation and stable zone for single-frequency excitation, respectively. Therefore, the vibration mode of a system is predicted to be stable using single-frequency excitation, but it also belongs to the unstable zone predicted by multi-frequency

2 0 −2 −4 −6 −8

0

20

40

60

80

100

time (s) Fig. 8. Dynamic response under multi-frequency excitation (Point B).

excitation. In this situation, the parametric resonance would take place and the consequent increasing stress can prove dangerous for systems functioning in real sea conditions. Fig. 4 also shows that many single-frequency stable zones overlap with multi-frequency unstable zones, which means that a single-frequency prediction can

44

H. Yang et al. / Ocean Engineering 70 (2013) 39–50

involve significant danger, especially when the external excitation is irregular.

x 10

72

Phase plane trajectory for design point A (single−frequency)

5

initial point (1,0) initial point (2,0)

3 2 1 0 −1 −2 −3 −2

−1.5

−1

−0.5

0

0.5

1 x 1072

Generalized coordinates (m)

Fig. 9. Phase plane trajectory for design point A under single-frequency excitation. Phase plane trajectory for design point A (multi−frequency) 8

initial point (1,0) initial point (2,0)

6

Generalized velocities (m/s)

3.1.2.2. Phase plane trajectory of dynamic response. Fig. 9 shows the phase plane trajectory for design point A under single-frequency excitation. This trajectory is the reflection of the system's whole motion; the stability property is unstable, which is consistent with the unstable zone of single-frequency stability diagram that point A belongs to, because both the generalised coordinate and the velocity are approaching infinity. Fig. 10 shows the phase-plane trajectory for design point A under multi-frequency excitation; the reflected stability property is unstable, which is congruent with the stable zone of multi-frequency stability diagram where point A is in, because both the generalised coordinate and the velocity are approaching zero. Fig. 11 shows the phase-plane trajectory for design point B under single-frequency excitation, and the reflected stability property is stable, which is consistent with the stable zone of single-frequency stability diagram that A belongs to, because both the generalized coordinate and the velocity are approaching zero. Fig. 12 is the phase-plane trajectory for design point B under multi-frequency excitation, and the reflected stability property is stable, which is in agreement with the stable zone of multifrequency stability diagram where point A is in, because both the generalised coordinate and the velocity are increasing and approaching infinity. The initial points for both cases are the same: (1, 0) for the dashed line and (2, 0) for the solid line, respectively. Both figures show that the initial state has no influence on the motion property of the system, because the shapes of the reflected trajectories of both lines are similar.

Generalized velocities (m/s)

4

4 2 0 −2 −4 −6 −8

4. Property of stability under multi-frequency excitation −10 −3

−2

4.1. Comparison of damping impact on stability chart

0

1

2

3

Fig. 10. Phase plane trajectory for design point A under multi-frequency excitation.

Phase plane trajectory for design point B (single−frequency) 20

initial point (1,0) initial point (2,0)

15

Generalized velocities (m/s)

Fig. 13 shows a stability diagram under single-frequency excitation with damping; Fig. 14 shows a stability diagram under multifrequency excitation with damping. In both figures, damping varies from 0 to 0.3, in 0.1 intervals. A comparison of Figs. 13 and 14 shows some differences. First, the single-frequency and multi-frequency stability diagrams with damping initially look totally different, apart from the similarity that both have transition curves that separate the stable and unstable zones. Second, the overall influence of damping on the suppression of multi-frequency unstable zones is much more effective than on single-frequency unstable zones, namely, the unstable zones of multi-frequency stability chart are more vulnerable to damping effects than single-frequency ones. Finally, in the single-frequency stability chart, higher-order unstable zones are more vulnerable to damping effects than lower-order ones. Compared with single frequency stability chart, the damping has a different effect on the multi-frequency one. As can be observed from Fig. 14, the damping influence varies as the zone order changes. First, for the first and second order unstable zones, the damping impact is more effective on suppressing the second order unstable zone than the first order zone. Second, the power of damping to suppress unstable zones decreases from the second to fourth order unstable zones. In the fifth to seventh order unstable zones, the damping impact increases as the unstable zone number increases. The above comparison indicates that, in real sea conditions, a system's parametric stability property predicted by singlefrequency excitation may be totally different from that predicted by multi-frequency excitation, which is more reasonable prediction through the aforementioned comparisons and analyses.

−1

Generalized coordinates (m)

10 5 0 −5 −10 −15 −20 −4

−3

−2

−1

0

1

2

3

Generalized coordinates (m) Fig. 11. Phase plane trajectory for design point B under single-frequency excitation.

In engineering field, once a riser is designed the correlated system damping is determined. And there is a smallest critical damping value, namely the critical drag coefficient in most cases, to prevent the parametric instability of the system from being excited, namely to keep all design points staying in stable zones in

H. Yang et al. / Ocean Engineering 70 (2013) 39–50

2

100

initial point (1,0) initial point (2,0)

Red −−−Stochastic Phase 1 Green −−−Stochastic Phase 2 Black −−−Stochastic Phase 3

90 80

1 70

0.5

60

0

50

Stochastic Phase 3

q

Generalized velocities (m/s)

Hill stability chart comparison on different stochastic initial phase

x 106 Phase plane trajectory for design point B (multi−frequency)

1.5

45

Stochastic Phase 2

40

−0.5

30

−1 20

−1.5 −2 −4

Stochastic Phase 1

10 0

−3

−2

−1

0

1

2

Generalized coordinates (m)

3 x 105

Fig. 12. Phase plane trajectory for design point B under multi-frequency excitation.

0

10

20

30

40

50

60

a Fig. 15. Comparison of stability diagram corresponding to a stochastic initial phase.

Mathieu stability diagram with damping varying (single−frequency) 100 Red −−−damping = 0 Blue −−−damping = 0.1 Green −−−damping = 0.2 Black −−−damping = 0.3

90 80 70

unstable

q

60

unstable

50 40

damping=0.3

unstable damping=0.2

30

unstable

20

10 unstable damping=0

0

0

10

20

30

damping=0.1

40

50

60

a Fig. 13. Stability diagram under single-frequency excitation with varying damping.

Hill stability diagram with damping varying (multi−frequency) 100 Red −−−damping = 0 Blue −−−damping = 0.1 Green −−−damping = 0.2 Black −−−damping = 0.3

90 80

unstable

70

unstable

unstable

Fig. 16. Schematic diagram of a top-tensioned riser (TTR) in waves and current.

4.2. Influence of initial phase of linear wave on stability prediction

unstable

q

60 unstable 50 40

damping=0.3

30 20 damping=0.2

10 0

damping=0

0

10

20

30

damping=0.1

40

50

60

a Fig. 14. Stability diagram under multi-frequency excitation with varying damping.

Hill stability chart. If there is one or more than one design points being surrounded by unstable zones, extra damping needs to be introduced in order to obtain the smallest critical damping value, details are given in Section 5.

The impact of the stochastic initial phase of wave frequency components on the stability diagram is investigated. As in real sea conditions, the appearance of wave excitation depends on the variation of the random initial phase. Therefore, this section discusses the influence of the initial phase on the stability diagram. In the real situation, the initial phase is essential to estimate the wave, so a comparison among the three cases under different stochastic initial phases have been developed. The comparison is shown in Fig. 15, which shows the comparison among the three cases for wave estimations where the stochastic initial phase differs. Analysis of the stability diagram in Fig. 15, which results from multi-frequency excitation with a stochastic initial phase, shows that there is always one or two transition curves nearly overlap among the three cases, regardless of its location. All transition curves overlap when q o 20. Thus, when the stability of a system predicted by multi-frequency excitation with a stochastic initial phase and the value of q for the studied system is less than 20, the influence of phase varying can be neglected.

46

H. Yang et al. / Ocean Engineering 70 (2013) 39–50

5. Engineering case study 5.1. Stability assessment of a TTR system under multi-frequency excitation The schematic diagram of the mechanism of parametric excited vibration of a TTR is shown in Fig. 16. A mini-TLP in the Gulf of Mexico with eight risers in an 861 m water depth was selected to study the parametric instability issue. The corresponding configurations of the platform and its risers are displayed in Table 2. In real sea conditions, linear-wave theory is used to predict the stability properties of a system, particularly to obtain the stability diagram corresponding to a certain sea condition. Because most of a system's vibration energy is distributed in the first few vibration modes, the present study considers only the first nine vibration modes. The first nine vibration modes are described in Table 3. Parameters a and q of each vibration mode form a design point located in the parametric plane. The TTR system's stability is determined by judging whether or not each design point falls into Hill unstable zones. To simplify the analysis, the nine design points are plotted on the corresponding stability diagram in Fig. 17; the points are marked with triangles labelled from 1 to 9. None of these nine design points fall into unstable zones; thus, the system is stable in terms of the studied vibration modes. 5.2. Discussion of essential damping in extreme conditions to suppress the Hill instability The above study in Section 5.1 was based on normal operating conditions. However, parametric instability can occur when a TTR system encounters extreme operating conditions which are among certain conditions parametric resonance can be excited. When instability occurs, extra damping or other measures are needed to suppress it. This section evaluates the damping that is essential for suppressing the Hill instability under three extreme operating conditions, corresponding to three extreme tension fluctuation cases: case I, ΔT ¼ 0:6e4 kN; case II, ΔT ¼ 0:8e4 kN; and case III, ΔT ¼ 1:0e4 kN. Table 2 Configurations of Tension Leg Platform and its TTR. Items

Configuration

Design water depth (m) Draft (m) Displacement (tonnes) Number of risers Top pretension (tonnes) Riser diameter (O.D.) (m) Riser wall thickness (mm) Riser length (m) Total wet weight (tonnes)

861 34.5 22578 8 7085 0.6604 26.29 826.5 465.3

Figs. 18–20 show the positions of the corresponding design points in cases I–III, respectively. All three cases are unstable, which means that extra damping is essential for suppressing the Hill instability—that is, to shrink the unstable zones where some design points fall into and to prevent the related design points from being surrounded by transition curves. The lowest amount of extra damping required is displayed in Table 4. From Table 4, it can be observed that although the value of the tension fluctuation in case II is 1.33 times that in case I, the corresponding smallest damping needed in case II is 1.97 times that in case I. Furthermore, the tension fluctuation value in case III is only 1.67 times that in case I and the corresponding damping is 2.76 times that in case I. Thus, the corresponding smallest damping needed for suppressing the Hill instability does not increase with the same ratio as the related tension fluctuation does. Namely, in more severe operating conditions, it is much more difficult to control the resulting Hill instability. 5.3. Dynamic response analysis of TTR system under extreme operating conditions The response of a TTR system under normal operating conditions is significant, because most operating conditions are normal. While extreme operating conditions are also very important, they create situations—as is the case in many other study fields—where the object studied can be damaged. Therefore, TTR system under the extreme operating conditions is further investigated in this section, the operating conditions used are identical to the three extreme conditions described in Section 5.2. This section investigates the dynamic response of the TTR system, including the dynamic response in the time domain, the phase plane trajectory, and the power spectrum density. The operating condition studied is the same as that in Section 5.2 for cases I–III; there is only one design point falling into the unstable zone for case I and case II, whereas there are two such design points for case III. Data for all design points mentioned are displayed in Table 5. 5.3.1. Dynamic response in time domain Figs. 21–23 show the TTR system's final dynamic response in the time domain corresponding to cases I, II, and III, respectively. The amplitude for case I in Fig. 21 increases to 15 within 100 s, whereas the maximum amplitudes for case II (Fig. 22) and case III (Fig. 23) are 3  104 and 8  106, respectively. A comparison of Figs. 21–23 shows that the TTR system becomes more unstable from cases I to III. This conclusion is in good agreement with the

Mathieu stability assessment for damping c=0.08 (ΔT = 0.1e4 kN) 100 Design point

90 80 70

Table 3 Specifications of the first nine vibration modes of a TTR system.

q

60

Vibration mode n

Natural frequency wn

Damping c

a

q

1 2 3 4 5 6 7 8 9

0.359 0.716 1.075 1.436 1.802 2.172 2.548 2.930 3.320

0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08

0.597 2.372 5.344 9.545 15.018 21.822 30.025 39.711 50.975

0.033 0.132 0.297 0.529 0.826 1.189 1.619 2.115 2.676

50 40 30 20 10 2 3 0 1 0

4 10

6

5 20

7

8

30

40

9 50

60

a Fig. 17. Stability assessment of a TTR system under multi-frequency excitation.

H. Yang et al. / Ocean Engineering 70 (2013) 39–50

Hill stability diagram for Case I (ΔT = 0.6e4 kN)

47

Table 5 Data for design points that fall into unstable zones in the stability diagram.

100 Design point

90

Case I

Case II

Case III

a ¼50.9747, q¼ 16.0574 None

a¼ 50.9747, q ¼21.4099 None

a ¼15.0183, q¼ 8.2600 a ¼50.9747, q¼ 26.7623

80 70

q

60 50

Dynamic response for Case I 15

40 30

10

20 8

0

1 2 3 0

4

5

10

6 20

7

30

40

50

60

a Fig. 18. Hill stability assessment of TTR system under case I (ΔT ¼ 0.6e4 kN).

Amplitude (m)

10

9 5

0

−5

Hill stability diagram for Case II (ΔT = 0.8e4 kN)

−10

100 Design point

90

−15 80

0

20

40

60

80

100

time (s) 70

Fig. 21. Dynamic response prediction for case I.

q

60 50

x 10

4

Dynamic response for case II

3 40 30

2

9

10 0

1 2 3 0

4

6

5

10

20

7

30

8

40

50

60

a Fig.19. Hill stability assessment of TTR system under case II (ΔT¼ 0.8e4 kN).

Amplitude (m)

20

1

0

−1

Hill stability diagram for Case III (ΔT = 1.0e4 kN) 100

−2 Design point

90

−3

80

0

20

40

80

100

Fig. 22. Dynamic response prediction for case II.

60

q

60

time (s)

70

50 8

30

9

6

8 20

7 6

0

3 12 0

4

4

5

10

20

30

40

50

60

a Fig. 20. Hill stability assessment of TTR system under case III (ΔT ¼ 1.6e4 kN).

Amplitude (m)

10

Dynamic response for Case III

x 106

40

2 0 −2 −4

Table 4 Smallest damping needed to retain stability under three extreme cases.

−6

Case

Tension fluctuation

Smallest damping needed

Ratio

I II III

ΔT ¼ 0.6e4 kN ΔΤ ¼ 0.8e4 kN ΔΤ ¼ 1.0e4 kN

0.0862 0.1700 0.2382

1 1.97 2.76

−8

0

20

40

60

80

time (s) Fig. 23. Dynamic response prediction for case III.

100

48

H. Yang et al. / Ocean Engineering 70 (2013) 39–50

severity of the corresponding operating conditions: as the extreme operating condition becomes more severe, the dynamic response becomes more violent. 5.3.2. Phase plane trajectory of the entire motion process Figs. 24–26 show the phase plane trajectories of the TTR system corresponding to three cases, which reflect their entire motion responses, respectively. The state of motion corresponding to case I, which is reflected in the phase plane trajectory in Fig. 24, is completely consistent with the corresponding dynamic response in the time domain in Fig. 21; these correlations occur between Figs. 22 and 25 and between Figs. 23 and 26. As the instability becomes more violent from case I to case III, the motion diverges faster, so the phase plane trajectories in Figs. 24–26 become increasingly sparse as the operating conditions become more severe. In addition, the responses in Figs. 24–26 have good symmetry, which means that the appropriate motion state is periodic and a power spectrum density analysis is required to determine the corresponding response frequency. 5.3.3. Power-spectrum density analysis of dynamic response Figs. 27–29 show the power spectrum densities of the dynamic responses of TTR systems within 100s under three extreme operating conditions. In all three extreme operating conditions, 1.11 Hz is the most prominent frequency of the TTR dynamic response, which means that the vibration mode with a frequency

1.11 Hz is the main vibration mode of the TTR system. Meanwhile, the degree of instability can be observed from the order of magnitude of the power spectrum density, which is 105 in Fig. 27 for case I, 1011 in Fig. 28 for case II, and 1015 in Fig. 29 for case III; these values are fully consistent with the conclusions obtained from the analyses of the dynamic responses in the time domain and phase plane trajectory. However, differences among the three cases are apparent. Apart from the main frequency, 1.11 Hz, there are two additional super harmonic components in Fig. 27, two additional super harmonic components and one sub harmonic component in Fig. 28, and three additional super harmonic components and one sub harmonic component in Fig. 29, which can be seen from comparing the three power spectrum density figures. The amplitudes of the super-harmonic and sub-harmonic components become increasingly distinct as the operating conditions become more severe. For operating condition III, one super-harmonic component is visible.

5.4. Discussion of instability evaluation and suppression The instability of a system that is induced by parametric excitation, which results from wave action, can be evaluated using a stability chart and the corresponding dynamic response analysis

6

Phase plane trajectory for Case I

x 107

Phase plane trajectory for Case III initial point (1,0)

40

initial point (1,0)

4

Generalized velocities (m/s)

20 10 0 −10 −20

2

0

−2

−4

−30 −6 −1

−40 −6

−4

−2

0

2

4

−0.5

0

0.5

Generalized coordinates (m)

6

Generalized coordinates (m)

1 x 10

7

Fig. 26. Phase plane trajectory for case III.

Fig. 24. Phase plane trajectory for case I. x 105

Phase plane trajectory for case II

x 104

Power spectrum density for Case I

8

18

initial point (1,0)

Power spectrum density (m2.s)

6

Generalized velocities (m/s)

Generalized velocities (m/s)

30

4 2 0 −2 −4 −6 −8 −10 −1.5

1.11

16 14 12 10 8 6 4 2

−1

−0.5

0

0.5

Generalized coordinates (m) Fig. 25. Phase plane trajectory for case II.

1

1.5 x 10

4

0

0

0.5

1

1.5

2

Frequency (Hz) Fig. 27. Power spectrum density of dynamic response of a TTR under case I.

H. Yang et al. / Ocean Engineering 70 (2013) 39–50

x 1011

Power spectrum density (m2.s)

But the knowledge on how to distribute the above mentioned extra damping introducing measures based on the smallest drag coefficient highly depends on experimental tests. In addition, the aforementioned methods of structural parameter redesign can be utilized at the same time; for example, a composite material riser can be adopted, which can decrease the total mass significantly, and the corresponding ultimate strength of the TTR system can be improved greatly. However, top pre-tension can also improve the stability of a TTR system (Simos and Pesce, 1997; Chandrasekaran et al., 2006).

Power spectrum density for case II

2

1.11

1.5

1

Functional forms of parameters (a ¼ ðωn =ωÞ2 , q ¼ ΔTðnπ=lÞ2 = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Mω2 and ωn ¼ ðnπ=lÞ3 ½EIðnπ=lÞ3 þ T 0 ðnπ=lÞ þ mω =M ) can provide

0.5

0

49

0

0.5

1

1.5

2

Frequency (Hz) Fig. 28. Power spectrum density of dynamic response of a TTR under case II.

x 1015

Power spectrum density for Case III

a reasonable interpretation. As can be seen from the above functional forms, the natural vibration frequency ωn can increase as the top pre-tension T 0 increases, resulting in an increase in the corresponding value of a. Therefore, the parametric pair (a, q) will give a larger a with the corresponding q unchanged, causing the parametric pair (a, q) (which are design points 5 and 9 discussed in Section 5.3) to fall out of the unstable zones; Therefore, the ultimate effect is equal to the impact of damping.

2.5

Power spectrum density (m2.s)

1.11

6. Conclusions

2

1.5

1

0.5

0

0

0.5

1

1.5

2

Frequency (Hz) Fig. 29. Power spectrum density of dynamic response of a TTR under case III.

developed above. However, the ultimate purpose of this investigation of parametric responses is not only to study the corresponding mechanism but also to control the instability. It is true that a damping impact is an effective means of suppressing parametric excited instability, as has been discussed in detail in many studies (Zhang et al., 2002; Koo et al., 2004; Yang and Li, 2009). As shown in formula c ¼ 2C d ρw DV max =3πMω, the system damping is a function of many parameters, among which only dynamic drag coefficient Cd and outer diameter D are completely independent factors, other factors are either not independent or cannot be changed through riser design. While the function relation between outer diameter D and system damping is complex and nonlinear. As riser mass per unit length M ¼ πDhρs þ 0:25πD2 ρw þ ρi Ai appears as a part of the denominator of damping function, and its variation trend as a function of D is nonlinear and no apparent direct proportion or inverse proportion relationship can be found. Therefore, to obtain extra damping via changing the value of D depends on practical design situation. However, extra damping can be obtained through increasing the value of dynamic drag coefficient Cd, as system damping is in direct proportion to Cd. There are several physical options for designers to achieve the extra system damping, such as by adding dampers, attaching new damped materials, or installing helical strakes. Once the smallest drag coefficient is obtained, the correlated damping introducing measures can be taken during engineering design.

Hill's equation for a TTR functioning in an irregular wave has been derived for the first time using axial and lateral coupled motion equation. The Mathieu form equation cannot capture nonharmonic tension for a TTR operated in real sea conditions. Hill's equation is capable of representing non-harmonically varying tension in irregular wave conditions. Stability charts and the appropriate dynamic responses predicted by single-frequency excitation and multi-frequency excitation were compared. The present study explored an engineering case study using the stability chart as a guide. From the presented work, the following conclusions can be made: (1) Bubnov–Galerkin methodology is a useful tool for a stability analysis of Hill's equation. The analytical formulae can successfully characterize the behaviour of a Hill-type system. (2) In terms of a TTR operated in real sea conditions, the parametric stability predicted by single-frequency excitation and multifrequency excitation is significantly different. The Mathieu stability chart predicts that lower-order vibration modes are the primary modes to be excited, whereas higher-order vibration modes are easier to be induced according to the Hill stability chart. (3) When the ordinate value of bell-shaped transition curves in a stability chart is larger than a critical value, the impact of the stochastic initial phase on stability prediction is apparent. However, if the correlated value is less than the critical value, the influence of the initial phase disappears. (4) Stability chart can be used for predicting a TTR system's parametric stability. A TTR system's Hill instability can be induced under severe operating conditions. The more severe the operating condition, the more violent is the instability of a system. As the operating conditions of a TTR system become more severe, more vibration modes will be induced. (5) For a TTR system operated in real sea conditions, measures should be taken to reduce the amplitude of tension fluctuation, preventing parametric resonance of the system from being excited. Finally, this study is only preliminary and, to some extent, qualitative in nature; unfortunately, it is not yet supported by experimental results. In future research, the influence of the spectral peak period and the significant wave height on the parametric

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instability of a system will be explored, especially when the spectral peak frequency to the system's natural frequency satisfies multiple relationships. In addition, the corresponding reliability analysis of a system under parametric excitation will be investigated. Acknowledgments This work was financially supported by the National Natural Science Foundation of China (Grant No. 51009093). References Ahmed, T.M., Hudson, D.A., Temarel, P., 2010. An investigation into parametric roll resonance in regular waves using a partly non-linear numerical model. Ocean Eng. 37 (14–15), 1307–1320. API RP 2RD, 1998RD. Design of Risers for Floating Production Systems (FPSs) and Tension-Leg Platforms (TLPs), first ed. American Petroleum Institute, Washington DC. (2006 reaffirmed). Bulian, G., Francescutto, A., Lugni, C., 2006. Theoretical, numerical and experimental study on the problem of ergodicity and ‘practical ergodicity’ with an application to parametric roll in longitudinal long crested irregular sea. Ocean Eng. 33 (8–9), 1007–1043. Chandrasekaran, S., Chandak, N.R., Anupam, G., 2006. Stability analysis of TLP tethers. Ocean Eng. 33 (3-4), 471–482. Chang, B.C, 2008. On the parametric rolling of ships using a numerical simulation method. Ocean Eng. 35 (5–6), 447–457. Chatjigeorgiou, I.K., Mavrakos, S.A., 2002. Bounded and unbounded coupled transverse response of parametrically excited vertical marine risers and tensioned cable legs for marine applications. Appl. Ocean Res. 24 (6), 341–354. Chatjigeorgiou, I.K., Mavrakos, S.A., 2005. Nonlinear resonances of parametrically excited risers-numerical and analytic investigation for Ω ¼2ω1. Comput. Struct. 83 (8–9), 560–573. Cruz, A.M., Krausmann, E., 2008. Damage to offshore oil and gas facilities following hurricanes Katrina and Rita: an overview. J. Loss Prev. Process Ind. 21 (6), 620–626. El-Bassiouny, A.F., 2005. Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations. Mech. Res. Commun. 32 (3), 337–350. France, W.N., Levadou, M., Treakle, T.W., Paulling, J.R., Michel, R.K., Moore, C., 2003. An investigation of head-sea parametric rolling and its influence on container lashing systems. Mar. Technol. 40 (1), 1–19.

Hua, J.B., Wang, W.H., 2001. Roll motion of a roro-ship in irregular following waves. J. Mar. Sci. Technol. 9 (1), 38–44. Hua, J., Palmquist, M., Lindgren, G., 2006. An analysis of the parametric roll events measured onboard the PCTC AIDA. In: Proceedings of the Ninth International Conference on Stability of Ships and Ocean Vehicles, Rio de Janeiro, Brazil. Koo, B.J., Kim, M.H., Randall, R.E., 2004. Mathieu instability of a spar platform with mooring and risers. Ocean Eng. 31 (17–18), 2175–2208. Maki, A., Umeda, N., Shiotani, S., Kobayashi, E., 2011. Parametric rolling prediction in irregular seas using combination of deterministic ship dynamics and probabilistic wave theory. J. Mar. Sci. Technol. 16 (3), 294–310. Park, H.I., Jung, D.H., 2002. A finite element method for dynamic analysis of long slender marine structures under combined parametric and forcing excitations. Ocean Eng. 29 (11), 1313–1325. Patel, M.H., Park, H.I., 1991. Dynamics of tension leg platform tethers at low tension. Part I—Mathieu stability at large parameters. Mar. Struct. 4 (3), 257–273. Poulin, F.J., Flierl, G.R., 2008. The stochastic Mathieu's equation. Proc. Roy. Soc. Lond. Math. Phys. Sci. 464 (2095), 1885–1904. Radhakrishnan, S., Datla, R., Hires, R.I., 2007. Theoretical and experimental analysis of tethered buoy instability in gravity waves. Ocean Eng. 34 (2), 261–274. Rainey, R., 1977. The dynamics of tethered platforms. R. Inst. Naval Architects 120, 59–80. Rho, J.B., Choi, H.S., Shin, H.S., Park, I.K., 2005. A study on Mathieu-type instability of conventional spar platform in regular waves. Int. J. Offshore Polar Eng. 15 (2), 104–108. Simos, A.N., Pesce, C.P., 1997. Mathieu stability in the dynamics of TLP tether considering variable tension along the length. Offshore Eng. 29, 175–186. Spyrou, K.J., 2000. Designing against parametric instability in following seas. Ocean Eng. 27 (6), 625–653. Tao, L., Lim, K.Y., Thiagarajan, K., 2004. Heave response of classic spar with variable geometry. J. Offshore Mech. Arct. Eng. 126 (1), 90–95. To, C.W.S., 1991. Principal resonance of a nonlinear system with two-frequency parametric and self-excitations. Nonlinear Dyn., 419–444. Wang, T., Zou, J., 2006. Hydrodynamics in deepwater TLP tendon design. J. Hydrodyn. Ser. B 18 (3), 386–393. Yang, H.Z., Li, H.J., 2009. Instability assessment of deep-sea risers under parametric excitation. China Ocean Eng. 23 (4), 603–612. Zhang, L.B., Ayman, E., Paul, J., 2010. Mathieu instability of TTR tension fluctuation due to VIV. In: Proceedings of the Sixth International Offshore Pipeline Forum IOPF 2010–2002, Houston, Texas. Zhang, L.B., Zou, J., Huang, E.W., 2002. Mathieu instability evaluation for DDCV/ SPAR and TLP tendon design. In: Proceedings of the 11th Offshore Symposium, Society of Naval Architect and Marin Engineer (SNAME), Houston, Texas. Zounes, R.S., Rand, R., 1998. Transition curves for the Quasi-periodic Mathieu equation. SIAM J. Appl. Math. 58 (4), 1094–1115.