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Active heave compensation on drill-ships in irregular waves
Ocean Engng, Vol. 25, No. 7, pp. 541–561, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0029–8018/98 $19.00 + 0.00
Pergamon
PII: S0029–8018(97)00028–0
ACTIVE HEAVE COMPENSATION ON DRILL-SHIPS IN IRREGULAR WAVES Umesh A. Korde* 5901 Ardwick Drive, Memphis, TN 38119, U.S.A. (Received 1 December 1996; accepted in final form 12 February 1997) Abstract—This paper investigates a possible method for heave compensation on board deep-water drill-ships subjected to irregular-wave excitation. The proposed system exploits favorable interaction of coupled oscillators to achieve the desired results. This study examines an actively controlled compensator which performs well over a large wave-frequency bandwidth. Performance under certain operating conditions is investigated using a dynamic model. Simple mathematical arguments and frequency-domain computations in an irregular wave spectrum show the proposed heave compensation system to be effective within the bounds of linearity. 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
Drill-ships used in marine geological explorations are frequently required to operate in significantly deep waters. The drilling technology used in this context shares some features with oil-drilling technology employed in less deep waters. In drill-ships such as Joides Resolution for example (Cullen, 1992), the drill-bit is attached to a long drill-pipe which is driven from atop a derrick (‘top drive’). The derrick is located amidships, and the top drive is placed along the ship’s centerline. More recent designs of scientific-drilling ships also include a riser enclosing the drill-pipe and terminating in a blow-out preventer (JAMSTEC, 1995). It is well-known that wave-induced motions of a floating vessel impose significant loads on the drill-pipe/riser (Sarpkaya and Issacson, 1981). The loads arising from a vessel’s surge and sway can, to a great extent, be reduced by a combination of mooring constraints and thrusters associated with dynamic positioning. However, the effect of heave remains significant, and heave compensation systems become necessary in many cases. For instance, oil-drilling semi-submersibles often use a heave compensator mounted between the derrick and the top drive. Two types of heave compensators which are frequently used in offshore drilling are: (i) compressed-air-based compensator; and (ii) crown-mounted compensator. Details relating to these methods can be found in references such as JAMSTEC (1991). Ship forms frequently turn out to be more cost-effective for deep-water drilling than semi-submersibles or other platform types (see for instance Kenison and Hunt, 1996).
*Current address: Coastal Research Department, Japan Marine Science and Technology Center (JAMSTEC), 2– 15 Natsushima-cho, Yokosuka, 237, Japan. 541
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U. A. Korde
However, due to their greater water-plane areas, ship forms undergo appreciably greater motion in waves than do semi-submersibles; both in terms of magnitude and bandwidth (for example Morooka, 1996). Thus, especially for drill-ships operating in deep-water wave climates, it appears worthwhile to consider special heave compensation methods; preferably those that use active control for increased effectiveness over a wide frequency band. Advantages of such heave compensation would include: (i) better regulation of the tension on the drill-pipe/riser; and (ii) reduced fatigue loads on the drill-pipe/riser. This paper examines one such system. The present system is based on a physical phenomenon resulting from interaction between coupled mechanical oscillators. In addition, it exploits active control to minimize the vertical oscillations of the top drive over a significant wave-frequency bandwidth. The system could in principle be used on other types of drilling vessels, but is expected to be most effective on deep-water drillships where heave motions with large bandwidths may be a problem. The section following this Introduction outlines the principal features of the system. Section 3 sets up a dynamic model (that includes the compensator, drill-ship and an elastic drill-pipe) for a theoretical investigation of compensator performance. A control method is then outlined, and the overall dynamic response with and without control is computed. Linearity is assumed, and calculations are carried out in a frequency domain in a fully developed Pierson–Moskowitz spectrum. Results are discussed in Sections 4 and 5, and the principal conclusions are summarized in Section 6. 2. ACTIVELY CONTROLLED HEAVE COMPENSATION SYSTEM
As shown by Ogata (1978) and Meirovitch (1986), if an undamped spring–mass system is coupled with a second oscillating system being excited by a sinusoidal external force, then there exists an excitation frequency at which the second mass will remain stationary for any magnitude of excitation. Similar behavior can be exploited to stabilize one of three coupled oscillators, as is done in this work. In addition, it is also found that active control can be used to extend such behavior over a range of excitation frequencies. In this work, the oscillator to be maintained at rest is a spring-supported block Mc from which the drill-pipe/riser is driven. The drill-ship with mass Ms forms another oscillator, while the third oscillator is an undamped mass Mm. All masses are thought to move primarily in the vertical direction. Mass Mm is to be controlled using feedback in such a way that mass Mc remains stationary over a large frequency range. Such control would be useful in irregular waves where the exciting force FD on the ship Ms can be represented as a sum of sinusoids at different frequencies, amplitudes and phases. The system is configured as shown in Figs 1a and 1b. Block Mc supports the drill-pipe/riser. Mass Mm is spring-supported, and controlled from Mc by a linear actuator. The drill-pipe is rotated from Mc. Mc is supported on spring-loaded vertical guides, along which it is driven vertically from the derrick by a pair of linear actuators. The damping force acting on Mc does not have to be zero. However, the damping on Mm should be as small as possible. The required actuation is provided by linear motors or hydraulic rams. Feedback of the three oscillations is obtained as follows. An accelerometer mounted on Mm measures the acceleration of Mm. The vertical acceleration of Mc is measured by averaging outputs of four accelerometers fixed near four corners of the block. Vertical acceleration of the ship is sampled by averaging the measurements of four accelerometers fixed to the derrick near the actuators driving Mc and distributed
Heave compensation on drill-ships
543
Fig. 1. Schematic of the active heave compensation system.
so as to average out the effects of any small roll and pitch of the ship. Control signals for the two actuators may be evaluated using a dedicated Digital Signal Processor. Physical/mechanical system parameters relevant to this work are outlined in Section 4. 3. EQUATIONS OF MOTION
The modelled system (see Fig. 1) consists of the ship with rest mass Ms, support block Mc, undamped mass Mm, and the drill-pipe with a mass per unit length 0. The focus is on setting up a simple model to investigate the essential physical behavior of the system. For this reason, a number of simplifying assumptions are made regarding environmental loads and resulting motions. All oscillations are assumed to be small enough to justify use of linear differential equations. Consistent with the purpose of the compensator, the ship is thought to be oscillating primarily in heave, and other motions are considered small in comparison. For the same reason, current velocities are assumed to be small enough to allow horizontal oscillations of the drill pipe to be discounted. Two situations are studied: (I) when the drill-pipe is free at the bottom end and is not yet engaged in the borehole; and (II) when the pipe is engaged in the borehole (see Fig. 2) during drilling. Variations in tension due to vertical elastic oscillations of pipe are
Fig. 2. Two heave compensation situations studied in this work.
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U. A. Korde
included in the model for situation (II). For each situation, the dynamic response of the overall system to irregular wave excitation is first studied without control, and next using the control scheme outlined here. A fixed reference frame is placed at the location O shown in Fig. 2. The vertical motions of ship Ms and masses Mc and Mm are denoted xs(t), xc(t) and xm(t), respectively, where t denotes time. The elastic vertical motion of the pipe is denoted v(t,z), where z is measured along the positive Z axis. The pipe length is L and pipe tension at the top end is TL. 3.1. (I) With pipe disengaged from borehole The free-body diagrams for this situation are shown in Fig. 3. Here, ks is the hydrostatic spring assumed constant for small ship motions and given by ks = wgAWP, where w is
Fig. 3. Free-body diagrams for the coupled oscillators.
Heave compensation on drill-ships
545
the density of sea-water, g the acceleration due to gravity, and AWP the ship water-plane area. kc and km are the spring constants shown in Fig. 1b, and Cc is the damping rate along the vertical guides. FD is the diffraction heave force due to waves, and FR is the radiation force due to ship motion on the water surface. fm and fc are the control forces applied on masses Mm and Mc by their respective linear actuators. In this situation the pipe is free at the bottom end. The vertical wave-force for a pipe of the chosen diameter is found negligibly small. Consequently, TL is here just the weight of the pipe minus its buoyancy. Dynamic equilibrium in this situation implies Mmx¨m = ⫺ km(xm ⫺ xc) + fm Mcx¨c = ⫺ fm + fc ⫺ Cc(x·c ⫺ x·s) ⫺ kc(xc ⫺ xs) ⫺ TL + km(xm ⫺ xc) M x¨ = F ⫺ f ⫺ F ⫺ k x + C (x· ⫺ x· ) + k (x ⫺ x ) s s
D
c
R
s s
c
c
s
c
c
s
(1) (2) (3)
where
冕 ⬁
FR = M⬁x¨s + h()x·s(t ⫺ ) d
(4)
0
with h() denoting the causal impulse response function, and M⬁ the infinite-frequency added mass in heave for the ship. Substituting Equation (4) into Equation (3) and applying the Fourier transform to Equations (1), (2) and (3): ⫺ 2MmXˆm = ⫺ km(Xˆm ⫺ Xˆc) + ˆfm (5) ⫺ 2McXˆc = ⫺ ˆfm + ˆfc ⫺ iCc(Xˆc ⫺ Xˆs) ⫺ kc(Xˆc ⫺ Xˆs) ⫺ TˆL + km(Xˆm ⫺ Xˆc) (6) 2 2 ⫺ MsXˆs = FˆD ⫺ ˆfc + (M⬁ + M)Xˆs ⫺ iXˆs ⫺ ksXˆs + iCc(Xˆc ⫺ Xˆs) + kc(Xˆc ⫺ Xˆs)
(7)
where M and are the frequency-dependent added mass and radiation damping coefficients for the ship in heave, and are defined by
冕
冕
⬁
⬁
2 2 h() = F() cos d = ⫺ M() sin d
(8)
0
0
The symbol ˆ denotes Fourier transforms. 3.1.1. (I-A) Without control. Setting ˆf m = ˆfc = 0 for the case without control, Equations (5), (6) and (7) can be rewritten in matrix form ˆ = Fˆ, with AX A=
冤
⫺ 2Mm + km
⫺ km
⫺ km
⫺ Mc + kc + km + iCc
⫺ iCc ⫺ kc
0
⫺ iCc ⫺ kc
⫺ 2(Ms + M⬁ + M) + i( + Cc) + kc + ks
2
0
冥
(9a)
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U. A. Korde
冤冥 冤 冥
Xˆm ˆ = Xˆc X Xˆs
(9b)
0
Fˆ =
⫺ TˆL FˆD
(9c)
T0 where TˆL = ␦(). T0 is here the value (weight ⫺ buoyancy) for the pipe, and ␦ denotes 2 the Dirac delta function. The variations , M⬁ + M, and FˆD vs can be obtained from published results. Denoting the determinant of the matrix A as 兩A兩, we can express the complex amplitudes Xˆm, Xˆc, and Xˆs as
冑
1 Xˆm = ( ⫺ D12TˆL + D13FˆD) 兩A兩
(10a)
1 Xˆc = ( ⫺ D22TˆL + D23FˆD) 兩A兩
(10b)
1 Xˆs = ( ⫺ D32TˆL + D33FˆD) 兩A兩
(10c)
where the elements Dij of the adjoint of A are given by D12 = km[ ⫺ 2(Ms + M⬁ + M) + i( + Cc) + kc + ks]
(11a)
D13 = km(iCc + kc)
(11b)
D22 = ( ⫺ Mm + km)[ ⫺ (Ms + M⬁ + M) + i( + Cc) + kc + ks] 2
2
(11c) D23 = D32 = ( ⫺ 2Mm + km)(iCc + kc)
(11d)
D33 = ( ⫺ Mm + km)( ⫺ Mc + kc + km + iCc) ⫺ k
(11e)
兩A兩 = ( ⫺ 2Mm + km)(D11) + km( ⫺ D12)
(11f)
2
2
2 m
and
where D11 = ( ⫺ 2Mc + kc + km + iCc)[ ⫺ 2(Ms + M⬁ + M) + i( + Cc) + kc + ks] ⫺ (iCc + kc)2 Note from Equations (10b), (11c) and (11d) that Xˆc = 0 at a frequency at which ( ⫺ 2Mm + km) = 0, which is the uncoupled natural frequency of oscillator (Mm, km). This resembles the phenomenon analyzed by Meirovitch (1986) and Ogata (1978) for a twodegree-of-freedom system. For a passive system described by Equations (9a), (9b) and
Heave compensation on drill-ships
547
(9c), therefore, perfect heave compensation would be achieved if the excitation force km FˆD were a perfect sinusoid with frequency m = . Mm This suggests that if the predominant wave frequency in an incident wave spectrum were m, a passive system would greatly reduce the cyclic loads on the drill-pipe due to ship heave. In this paper, however, it is proposed that the coupled system be controlled by means of two control forces ˆf m and ˆf c such that the complex amplitude Xˆc is zero over a range of frequencies, rather than just at the uncoupled natural frequency m of the oscillator (Mm, km).
冪
3.1.2. (I-B) With control. We choose the control forces fm and fc in Equations (1), (2) and (3) such that f m = Mm(x¨m ⫺ x¨c) + km(xm ⫺ xc)
(12)
f c = kl(xc ⫺ xs)
(13)
This leads to the following equations in the time domain: Mmx¨c = 0
(14)
(Mc ⫺ Mm)x¨c + Mmx¨m + Ccx·c ⫺ Ccx·s + (kc ⫺ kl)xc + (kl ⫺ kc)xs = ⫺ TL
(15)
冕 ⬁
Msx¨s ⫺ Ccx·c + Ccx·s + (ks + kc ⫺ kl)xs + (kl ⫺ kc)xc + h()x·s(t ⫺ ) d = FD 0
(16) Fourier transformation leads to the following equations: ⫺ 2MmXˆc = 0
(17)
⫺ 2MmXˆm ⫺ 2(Mc ⫺ Mm)Xˆc + iCcXˆc ⫺ iCcXˆs + (kc ⫺ kl)Xˆc + (kl ⫺ kc)Xˆs = ⫺ TˆL
(18)
⫺ 2(Ms + M⬁ + M)Xˆs ⫺ iCcXˆc + i(Cc + )Xˆs + (ks + kc ⫺ kl)Xˆs + (kl ⫺ kc)Xˆc = FˆD
(19)
ˆ = Fˆ Rewriting in the form AX A=
冤
冥
0
⫺ 2Mm
0
⫺ 2Mm
⫺ 2(Mc ⫺ Mm) + kc ⫺ kl + iCc
⫺ iCc + kl ⫺ kc
0
⫺ iCc + kl ⫺ kc
⫺ (Ms + M⬁ + M) + i( + Cc) + ks + kc ⫺ kl
冤冥
Xˆm ˆ = Xˆc X Xˆs
2
(20a)
(20b)
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U. A. Korde
冤 冥 0
Fˆ =
⫺ TˆL FˆD
(20c)
In this case, Dij, the elements of adj(A), are D12 = 2Mm[ ⫺ 2(Ms + M⬁ + M) + i(Cc + ) + ks + kc ⫺ kl]
(21a)
D13 = ⫺ 2Mm[ ⫺ iCc + kl ⫺ kc]
(21b)
D22 = 0
(21c)
D23 = D32 = 0
(21d)
D33 = ⫺ M
(21e)
4
2 m
and 兩A兩 = ⫺ 4M2m[ ⫺ 2(Ms + M⬁ + M) + i(Cc + ) + ks + kc ⫺ kl]
(21f)
兩A兩 is nonzero for ⬎ 0. Equations (10a–c) and (21a–f) lead to the conclusion that, provided fm and fc are chosen according to Equations (12) and (13) TˆL [ ⫺ iCc + (kl ⫺ kc)]FˆD + 2 Xˆm = 2 Mm Mm[ ⫺ 2(Ms + M⬁ + M) + i(Cc + ) + ks + kc ⫺ kl] (22a) Xˆc = 0 Xˆs =
(22b)
FˆD ⫺ (Ms + M⬁ + M) + i(Cc + ) + ks + kc ⫺ kl 2
(22c)
Equations (22a–c) show that the effect of the chosen control is to lock the motion Xˆc of block Mc to zero for ⬎ 0. Force fm brings about the actual cancellation of Xˆc, while force fc restricts the motion of Mm in the presence of fm. Fig. 4 shows the essentials of a method to synthesize fm and fc from sensed accelerations of Mm, Mc, and Ms [reference may be made to Kuo (1987) or Raven (1987) for Laplace transforms and classical control theory]. Feedback-controlled linear actuators with small time-constants ␣ would be preferable, and the design of amplifiers 1, 2 and 3 will correspondingly require some attention. In addition, the response of amplifier 2 should be shaped so that its zero-frequency gain is not exactly equal to Mm/kA2 to avoid a steady drift resulting from 兩A兩→0 as →0. This measure will cause a slight drop in the low-frequency performance of the compensator. 3.2. (II) With pipe engaged in borehole When the drill-pipe is engaged in the borehole, it plays an important role in the dynamics of the overall system. The free-body diagrams for the three masses Ms, Mc, and Mm are essentially the same as in Fig. 3. However, the force TL is now determined by the pipe tension set by the ‘pipe-tensioner’ (mounted on block Mc here) and the tension variations caused by elastic oscillations of the pipe. It is therefore necessary to probe the dynamics of the pipe. The pipe diameter assumed in this work (d = 0.14 m) is small enough for the vertical wave-force to be negligible. Further, current velocities are assumed small enough
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549
Fig. 4. Block diagram for the control system outlined in this work.
to allow any current-induced forces to be neglected. In addition, the pipe buoyancy is assumed to balance the submerged portion of the pipe weight and changes in the pipe submerged volume during oscillations are neglected. Consequently, the only excitation of the pipe is provided by the vertical motion of the pipe-support block Mc. The equation of motion for a pipe element of length ⌬z can be written as (see Fig. 5)
Fig. 5. Free-body diagram for a pipe element as modeled here.
550
U. A. Korde
0⌬zvtt = ⫺ 0cd⌬zvt + T(z + ⌬z) ⫺ T(z)
(23)
with subscripts t and z denoting partial differentiation wherever they appear. Here 0 is the mass per unit length, and 0cd the damping due to viscous skin friction. Dividing throughout by ⌬z and letting ⌬z be very small, we have
0vtt = 0cdvt =
∂T ∂z
(24)
For small axial elastic oscillations T = Ea0vz + T0
(25)
where E is the modulus of elasticity and a0 the nominal cross-sectional area of the pipe. T0 is the desired steady tension applied to prevent pipe slackness. As a result, the equation vtt + cdvt ⫺
Ea0 v =0 0 zz
(26)
models pipe oscillations in this situation. It is subject to the following boundary conditions at z = 0 and z = L: v(t, 0) = 0
(27a)
v(t, L) = xc
(27b)
where L denotes the pipe length. The oscillations are thought to be steady state (i.e. continuing from t→ ⫺ ⬁). Next we effect a change of variables in Equation (26) with
(t,z) = v(t,z) ⫺
z x (t) L c
(28)
so that the boundary conditions become homogeneous. Thus, we solve
tt ⫹ cdt ⫺
z Ea0 z = ⫺ cd x·c ⫺ x¨c 0 zz L L
(29)
subject to
(t, 0) = 0
(30a)
(t, L) = 0
(30b)
The undamped, unforced equation corresponding to Equation (29) is
tt ⫺
Ea0 =0 0 zz
(31)
This can be solved using separation of variables (t,z) = ⌳(z)⌫(t) to obtain ⌳(z) = a1 sin
0
1
0
and
0
冪Ea z + b cos 冪Ea z 0
(32a)
Heave compensation on drill-ships
⌫(t) = A1 sin t + B1 cos t
551
(32b)
The boundary conditions lead to ⌳(0) = 0⇒b1 = 0, ⌳(L) = 0⇒a1 sin
0
冪Ea
L
=0
(33)
0
The second condition provides the eigenvalues n for the pipe oscillation, where
n =
n L
冪
Ea0
, n = 1,2,3,…
(34)
0
For example, for a 3832 m long steel pipe having a diameter of 14.5 cm, 1 = 4.1514 rad/s. The solution can then be expressed as
冘 (A sin t + B cos t) sin nL z ⬁
=
n
n
(35)
n=1
To find the forced elastic response to xc(t), we can expand the excitation terms on the right-hand side of Equation (29) in terms of eigenfunctions sin nz/L such that
冘 f sin nLz = ⫺ cLz x· ⫺ Lz x¨ ⬁
d
n
c
c
(36)
n=1
It follows from orthogonality of the eigenfunctions that 2 fn = L
冋
册冕 L
x¨c ⫺ cd · xc ⫺ L L
z sin
nz dz L
(37)
0
Integration leads to fn =
2 (c x· ⫹ x¨c) cos n n d c
(38)
This implies that fn = ⫺
2 (c x· ⫹ x¨c), n = 1,3,5,… n d c
(39a)
and fn =
2 (c x· ⫹ x¨c), n = 2,4,6,… n d c
(39b)
Next, letting the forced response of the pipe be
冘 C sin nL z ⬁
(t,z) =
n
n=1
we can substitute
(40)
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U. A. Korde
sin
inz nz i ⫺ inz = (e L ⫺ e L ) L 2
together with Equations (39a), (39b) and (40) in Equation (29), which is valid separately for each mode due to the orthogonality of the eigenfunctions. Some manipulation leads to Ea0n22 2 · Cn = ⫿ (c x· ⫹ x¨c) C¨n + cdCn + 2 0L n d c
(41)
The negative sign applies to n = 1,3,5,... and the positive sign to n = 2,4,6,.... The tension TL at the pipe-support can now be found from
冉 冊
TL = T0 + (Ea0vz)z = L = T0 + Ea0 z +
xc L
z=L
This leads to TL = T0 +
Ea0 L
冢 冘 ⫿ nC + x 冣 ⬁
n
(42)
c
n=1
Substituting Equation (42) in Equations (1), (2) and (3), we arrive at a model for the system with the drill-pipe engaged in the borehole. The corresponding equations are summarized below. Mmx¨m = ⫺ km(xm ⫺ xc) + fm Mcx¨c = ⫺ fm + fc ⫺ Cc(x·c ⫺ x·s) ⫺ kc(xc ⫺ xs)
冘
(43)
⬁
+ km(xm ⫺ xc) ⫺ T0 ⫺
Ea0 Ea0 x ⫿ nCn ⫺ L n=1 L c
(44)
冕 ⬁
Msx¨s = FD ⫺ fc ⫺ h()x·s (t ⫺ ) d ⫺ M⬁x¨s ⫺ ksxs + Cc(x·c ⫺ x·s) + kc(xc ⫺ xs) 0
(45) Ea0n22 2 · Cn = ⫿ (c x· ⫺ x¨c) C¨n + cdCn + 0L2 n d c
(46)
In this work, the first three modes of pipe oscillation only are considered significant. The result of this approximation is a system of six equations in six unknowns. 3.2.1. (II-A) Without control. Setting fm = fc = 0 allows study of the situation without control. The reduced system of equations can be written as Mmx¨m = ⫺ km(xm ⫺ xc)
(47)
Ea0 C1 Mcx¨c = ⫺ Cc(x·c ⫺ x·s) ⫺ kc(xc ⫺ xs) + km(xm ⫺ xc) ⫺ T0 + L
(48)
Heave compensation on drill-ships
⫺
553
2Ea0 3Ea0 Ea0 C2 + C3 ⫺ x L L L c
冕 ⬁
Msx¨s = FD ⫺ h()x·s(t ⫺ ) d ⫺ M⬁x¨s ⫺ ksxs + Cc(x·c ⫺ x·s) + kc(xc ⫺ xs) 0
(49) Ea02 2 · C = ⫺ (cdx·c ⫹ x¨c) C¨1 + cdC1 + 0L2 1
(50)
4Ea02 1 · C2 = (cdx·c ⫹ x¨c) C¨2 + cdC2 + 2 0L
(51)
9Ea02 2 · C3 = ⫺ (c x· ⫹ x¨c) C¨3 + cdC3 + 2 0L 3 d c
(52)
ˆ = Fˆ, and letting aij, 1 ⱕ Applying the Fourier transform, rewriting in matrix form AX i, j ⱕ 6, define the elements of A, we have a11 = ⫺ 2Mm + km
(53a)
a12 = a21 = ⫺ km
(53b)
a22 = ⫺ 2Mc + iCc + kc + km +
Ea0 L
a23 = a32 = ⫺ kc ⫺ iCc a24 = ⫺ a25 =
Ea0 L
(53f)
3Ea0 L
(53g)
a33 = ⫺ 2(Ms + M⬁ + M) + i(Cc + ) + ks ⫹ kc a42 =
2 (icd ⫺ 2)
a44 = ⫺ 2 + icd + a52 = ⫺
(53h) (53i)
Ea02 0L2
1 (icd ⫺ 2)
a55 = ⫺ 2 + icd +
(53d) (53e)
2Ea0 L
a26 = ⫺
(53c)
4Ea02 0L2
(53j) (53k) (53l)
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U. A. Korde
a62 =
2 (icd ⫺ 2) 3
a66 = ⫺ 2 + icd +
(53m)
9Ea02 0L2
(53n)
aij = 0, for all other i and j
(53o)
ˆ = [Xˆm,Xˆc,Xˆs,Cˆ1,Cˆ2,Cˆ3] can be found by The complex amplitudes forming the vector X solving the linear system of equations above. The vector Fˆ on the right side equals [0, ⫺ Tˆ0,FˆD,0,0,0]T, with the only nonzero components being Tˆ0 and FˆD. The constant T0 T0 ␦() in the frequency domain in terms of the Dirac delta function. leads to Tˆ0 = 冑2 T
3.2.2. (II-B) With control. Next we choose, as in situation (I-b) f m = Mm(x¨m ⫺ x¨c) + km(xm ⫺ xc) f c = kl(xc ⫺ xs) Substitution in Equations (42)–(46) followed by Fourier transformation leads to the equˆ = Fˆ, where the elements aij of A are now given by ation AX a12 = a21 = ⫺ 2Mm
(54a)
a22 = ⫺ 2(Mc ⫺ Mm) + iCc + (kc ⫺ kl) +
Ea0 L
a23 = a32 = (kl ⫺ kc) ⫺ iCc a24 = ⫺ a25 =
Ea0 L
(54e)
3Ea0 L
(54f)
a33 = ⫺ 2(Ms + M⬁ + M) + i( + Cc) + (ks ⫺ kl + kc) a42 =
2 (icd ⫺ 2)
a44 = ⫺ 2 + icd + a52 = ⫺
(54g) (54h)
Ea02 0L2
1 (cd ⫺ 2)
a55 = ⫺ 2 + icd +
(54c) (54d)
2Ea0 L
a26 = ⫺
(54b)
4Ea02 0L2
(54i) (54j) (54k)
Heave compensation on drill-ships
a62 =
555
2 (icd ⫺ 2) 3
a66 = ⫺ 2 + icd +
(54l)
9Ea02 0L2
(54m)
aij = 0, for all other i and j
(54n)
ˆ denotes the vector of six complex amplitudes of interest, and Fˆ denotes Once again, X the vector of exciting forces which is the same as in case (II-A) above. In particular, the complex amplitude Xˆc of block Mc can be inferred in terms of the determinant 兩A兩 (which is nonzero for ⬎ 0) and the elements Dij of the adjoint of A as ⫺ D22 ˆ D23 ˆ T0 + F Xˆc = 兩A兩 兩A兩 D
(55)
where
| |
0 0
0
0
0
0 a33 0
0
0
D22 = 0 0 a44 0
| |
| |
0
0 a55 0
0 0
0
0 a66
0
0
0
a21 a23 a24 a25 a26
0 = 0 and D23 = 0
0 0
0
0 a44 0
| |
0 =0
0
0
0 a55 0
0
0
0
0 a66
(56)
The chosen fm thus locks the motion xc of the pipe-support block Mc to zero for ⬎ 0 even while the pipe is engaged in the borehole. Once again, the role of fc here is to restrict xm, the magnitude of oscillations of the undamped mass Mm. 4. RESULTS
Calculations based on the models of Section 3 were carried out in the frequency domain. A standard Gauss-elimination-based equation solver was used. Results were obtained for the four situations examined above, namely: (I) drill-pipe disengaged from borehole, (a) without control and (b) with control; and (II) drill-pipe engaged in borehole, (a) without control and (b) with control. Control was applied according to Equations (12) and (13). For convenience, the drill-ship form was approximated by a rectangular cylinder, and corresponding curves for added mass M⬁ + M, radiation damping , and exciting force FD (per unit wave-amplitude) were obtained from Newman (1977) following necessary interpolation for the correct breadth-to-draft ratio. A strip-theory type approximation was used. The ship principal dimensions were L = 143.3 m, B = 21.3 m, and T = 7.5 m with a displacement Ms = 23464.0 tonne. The operating water depth was 3780 m. Parameters of the heave compensation system were: Mm = 1.0 tonne, Mc = 20.0 tonne, km = 250 kN/m, kc = 1000 kN/m, kl = kc, Cc = 0.1 kN s/m, cd = 0.01 m3/s. The pipe length L was 3832 m,
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U. A. Korde
and the pipe diameter was taken to be d = 0.14 m. A steel drill-pipe (specific gravity 7.8 and modulus of elasticity E = 2 × 108 kN/m2) was assumed. The tension T0 in (II) was taken to be 100 kN. The calculations were carried out in fully developed beam seas described by a Pierson– Moskowitz spectrum (e.g. see Newman, 1977) at a wind-speed of 50 knot. Fig. 6 shows the wave-amplitude variation for the assumed spectrum. Results based on this wave-amplitude variation are shown in Figs 7–11. The vertical oscillation amplitudes of ship Ms, block Mc, and mass Mm with the pipe disengaged from the borehole and without control are shown in Fig. 7. Fig. 8 shows the amplitudes of these masses with the pipe disengaged from the borehole but with control. For the case with the pipe engaged in the borehole, Fig. 9 illustrates the amplitudes of Ms, Mc, and Mm without control. For the same situation, Fig. 10 shows the response of the first three elastic modes of the pipe oscillations, C1,...,C3 3 nz . Finally, the oscillation amplitudes of Ms, Mc, and corresponding to ⬇ Cn sin L 0 Mm and the three pipe modes in the presence of control are illustrated in Fig. 11.
冘
5. DISCUSSION
It should be emphasized that the aim here is to control the motion of a block supported on board the drill-ship and not the motion of the drill-ship itself. The masses Mm and Mc oscillate in air as opposed to on the air–water free surface. Thus, the mass, spring constant, and damping rate for the oscillators (Mm,km) and (Mc,Cc,kc) are independent of frequency. Consequently, in principle, time-domain control to drive xc to zero requires knowledge of the present displacements and accelerations only. Thus, in theory it is not difficult to design
Fig. 6. Wave-amplitude variation used in computations.
Heave compensation on drill-ships
Fig. 7. System response with pipe free at the bottom, without control.
Fig. 8. System response with pipe free at the bottom, with control.
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Fig. 9. Vertical motion of Ms, Mc, and Mm with pipe engaged at bottom, without control.
Fig. 10. Elastic modes for vertical oscillation of pipe, without control.
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559
Fig. 11. System response with pipe engaged in borehole, with control.
the controller in Fig. 4. As mentioned in Section 3, however, careful consideration should be given to the design of amplifiers 1, 2, and 3. In particular, the response of amplifier 2 should be shaped to avoid zero-frequency instability with the control active. This would be at the expense of a slight drop in the low-frequency performance of the compensator. With the exception just mentioned, the amplifier gains are required precisely to produce the desired match with km and Mm. Further, the damping on Mm should be as close to zero as possible. Any unavoidable mechanical damping on Mm can in principle be balanced by an additional equal and opposite actuator force derived from velocity feedback, but this in turn would require that the residual damping be quantified exactly. In this context, it would seem helpful to estimate on-line any uncertainties or slow changes in the nominal values of Mm and km from acceleration and actuator force measurements, and to use these estimates for adaptive improvements to the controller performance. Equations (22a–c) and (57) imply that, when successfully implemented, such control would effectively reduce xc to zero both without and with the drill-pipe engaged in the borehole. This suggests that, when heave is the dominant vessel motion and when current velocities are small, the proposed heave compensation will be effective both during deployment/re-entry of the drill-pipe and during drilling. The results obtained in this work are discussed below. Fig. 6 shows the wave-amplitude variation associated with a Pierson–Moskowitz spectrum for a wind-speed of 50 knot. The peak wave amplitude of 2.25 m occurs near = 0.375 rad/s corresponding to a peak period of 16.75 s. Fig. 7 shows the vertical-amplitude response of the ship Ms and masses Mc and Mm to this wave-amplitude variation. The three amplitudes are seen to be almost identical over the whole frequency range. A small steady deflection brought about by the above-water weight of the pipe can be observed in the case of the mass Mc. The amplitudes
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peak at ⬇ 0.35 rad/s. As the graph indicates negligible relative motion among the masses Ms, Mc, and Mm, the dynamics in this case appear to be dominated by the greatest mass, namely the drill-ship. Smaller values of km and kc would allow greater relative motion and would prove useful in passive implementations of the proposed hardware. The figure further suggests a second peak near = 0.8 rad/s. This is an influence of the undamped ks natural frequency of the ship in heave, which is found using sn = to Ms + M⬁ + M be 0.788 rad/s. Fig. 8 shows the motion amplitudes in the presence of control forces fm and fc. While no significant change is observed in the ship heave amplitude, the amplitude of Mc (thick dark line) is seen to be zero over ⬎ 0 as expected. The ship heave amplitude peaks near 0.35 rad/s as before, while mass Mm is seen to have a maximum amplitude of 0.35 m near = 0.32 rad/s. This shift is most likely because the coupling effects among the three masses become appreciable under control. It can be inferred from this figure, further, that control achieves the theoretically expected results in this situation. The dynamics of the system are next studied with the drill-pipe engaged in the borehole. In this case the motion of Mc and Mm is restricted by the pipe to about half of what it would be with the pipe disengaged (see Fig. 9). The three modes of pipe motion are shown in Fig. 10. It is straightforward to infer the overall amplitude of the pipe at z = L using [v(L)]amp = [(L) + xc]amp, and this is found to be [v(L)]amp ⬇ 0.6 m. This results in an average axial strain of ⌬L/L = 0.6/3832 = 1.5657 × 10 ⫺ 4, which in turn causes an average axial stress ⬇ 320 kg/cm2, which is well within the elastic limit for steel. The first natural frequency for the pipe vertical elastic oscillations is 1 = 4.1514 rad/s, which is near the margin of the frequency range where sufficient wave energy would exist. It is interesting to note that the peak for Cˆ1 appears near 0.415 rad/s, while the peak for Xˆc is near 0.345 rad/s, and this again is likely an effect of coupling among Mm, Mc, and the pipe oscillation. The influence of the ship natural frequency of 0.788 rad/s can also be noticed in the response of the three pipe modes shown in the figure. Finally, Fig. 11 shows that with the pipe engaged in the borehole, and with control applied according to Equations (12) and (13), the amplitudes of mass Mc and the three pipe modes are ⬍ 1.0 × 10 ⫺ 8 m (or equal to zero for practical purposes) for ⬎ 0. This shows the proposed control to be yielding expected results also with the pipe engaged in the borehole. Once again, increased coupling effects are likely to have been the cause of a slight shift along the frequency axis in the maximum amplitudes of Mm and Ms.
冪
6. CONCLUSIONS
The purpose of this study was to investigate an actively controlled heave compensation system for deep-water drill-ships. The system is controlled such that motion of a block supporting the drill-pipe/riser is effectively locked to zero over the frequency band covered by irregular ocean waves. Because the body whose motion is being controlled is not in contact with the water surface, the required control method is straightforward in theory, and knowledge of current oscillations only is necessary. A notable aspect of the proposed system is that the control has minimal effect on the motion amplitude of the ship itself. This feature is useful because of the desirability of compliance in high waves. A simplified dynamic model which considers only vertical motion of the ship and the
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drill-pipe was used to test the compensator’s ability to achieve the objective for which it is designed. The effect of ship roll and pitch on the motion of Mc and horizontal oscillations of the pipe in response to current forces needs to be examined. Extensions of the proposed system to allow actuation and control of Mc in two horizontal directions are possible, and their effectiveness would be studied using a more elaborate model. It is conceivable that similar extensions could also be used to reduce vortex-induced horizontal elastic oscillations of the pipe in response to currents. While the present results indicate isolation of block Mc from ship heave under the proposed control over a wide frequency band, these are based on the assumption that ship heave is small enough to be treated by a linear model. Coupling effects among heave, pitch and roll, and performance when these motions are large, need to be studied before a definitive conclusion as to the effectiveness of this method in large waves can be made. It can be said, however, that with careful implementation, the proposed method will effectively isolate block Mc from ship heave in moderate waves, where the ship heave is not too large to preclude independent treatment under linear theory. REFERENCES Cullen, V. (1992/93) Ocean drilling program. Oceanus Winter, 31–33. JAMSTEC (1995) 1994 Annual Report. Japan Marine Science and Technology Center, pp. 18–20. JAMSTEC (1991) State of the Art of Deep-Ocean Floor Sampling Systems. Japan Marine Science and Technology Center, pp. 73–76 (in Japanese). Kenison, R. C. and Hunt, C. V. (1996) Why FPSOs for the Atlantic frontier. Proceedings of Offshore Technology Conference, OTC 8032, pp. 31–40. Kuo, B. C. (1987) Automatic Control Systems, 5th edn., Chapter 3. Prentice-Hall, Englewood Cliffs, NJ. Meirovitch, L. (1986) Elements of Vibration Analysis, 2nd edn., Chapters 3–6. McGraw-Hill, New York. Morooka, C. K. (1996) Design and implementation of an oil-drilling semisubmersible–riser–template system. Transcript of presentation at Institute of Industrial Science, University of Tokyo (in Japanese). Newman, J. N. (1977) Marine Hydrodynamics, pp. 295–300. MIT Press. Ogata, K. (1978) System Dynamics, pp. 456–459. Prentice-Hall, Englewood Cliffs, NJ. Raven, F. H. (1987) Automatic Control Engineering, 4th edn., Chapter 5. McGraw-Hill, New York. Sarpkaya, T. and Issacson, M. (1981) Mechanics of Wave Forces on Offshore Structures, pp. 354–366. Van Nostrand Reinhold, NY.
Pergamon
PII: S0029–8018(97)00028–0
ACTIVE HEAVE COMPENSATION ON DRILL-SHIPS IN IRREGULAR WAVES Umesh A. Korde* 5901 Ardwick Drive, Memphis, TN 38119, U.S.A. (Received 1 December 1996; accepted in final form 12 February 1997) Abstract—This paper investigates a possible method for heave compensation on board deep-water drill-ships subjected to irregular-wave excitation. The proposed system exploits favorable interaction of coupled oscillators to achieve the desired results. This study examines an actively controlled compensator which performs well over a large wave-frequency bandwidth. Performance under certain operating conditions is investigated using a dynamic model. Simple mathematical arguments and frequency-domain computations in an irregular wave spectrum show the proposed heave compensation system to be effective within the bounds of linearity. 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
Drill-ships used in marine geological explorations are frequently required to operate in significantly deep waters. The drilling technology used in this context shares some features with oil-drilling technology employed in less deep waters. In drill-ships such as Joides Resolution for example (Cullen, 1992), the drill-bit is attached to a long drill-pipe which is driven from atop a derrick (‘top drive’). The derrick is located amidships, and the top drive is placed along the ship’s centerline. More recent designs of scientific-drilling ships also include a riser enclosing the drill-pipe and terminating in a blow-out preventer (JAMSTEC, 1995). It is well-known that wave-induced motions of a floating vessel impose significant loads on the drill-pipe/riser (Sarpkaya and Issacson, 1981). The loads arising from a vessel’s surge and sway can, to a great extent, be reduced by a combination of mooring constraints and thrusters associated with dynamic positioning. However, the effect of heave remains significant, and heave compensation systems become necessary in many cases. For instance, oil-drilling semi-submersibles often use a heave compensator mounted between the derrick and the top drive. Two types of heave compensators which are frequently used in offshore drilling are: (i) compressed-air-based compensator; and (ii) crown-mounted compensator. Details relating to these methods can be found in references such as JAMSTEC (1991). Ship forms frequently turn out to be more cost-effective for deep-water drilling than semi-submersibles or other platform types (see for instance Kenison and Hunt, 1996).
*Current address: Coastal Research Department, Japan Marine Science and Technology Center (JAMSTEC), 2– 15 Natsushima-cho, Yokosuka, 237, Japan. 541
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However, due to their greater water-plane areas, ship forms undergo appreciably greater motion in waves than do semi-submersibles; both in terms of magnitude and bandwidth (for example Morooka, 1996). Thus, especially for drill-ships operating in deep-water wave climates, it appears worthwhile to consider special heave compensation methods; preferably those that use active control for increased effectiveness over a wide frequency band. Advantages of such heave compensation would include: (i) better regulation of the tension on the drill-pipe/riser; and (ii) reduced fatigue loads on the drill-pipe/riser. This paper examines one such system. The present system is based on a physical phenomenon resulting from interaction between coupled mechanical oscillators. In addition, it exploits active control to minimize the vertical oscillations of the top drive over a significant wave-frequency bandwidth. The system could in principle be used on other types of drilling vessels, but is expected to be most effective on deep-water drillships where heave motions with large bandwidths may be a problem. The section following this Introduction outlines the principal features of the system. Section 3 sets up a dynamic model (that includes the compensator, drill-ship and an elastic drill-pipe) for a theoretical investigation of compensator performance. A control method is then outlined, and the overall dynamic response with and without control is computed. Linearity is assumed, and calculations are carried out in a frequency domain in a fully developed Pierson–Moskowitz spectrum. Results are discussed in Sections 4 and 5, and the principal conclusions are summarized in Section 6. 2. ACTIVELY CONTROLLED HEAVE COMPENSATION SYSTEM
As shown by Ogata (1978) and Meirovitch (1986), if an undamped spring–mass system is coupled with a second oscillating system being excited by a sinusoidal external force, then there exists an excitation frequency at which the second mass will remain stationary for any magnitude of excitation. Similar behavior can be exploited to stabilize one of three coupled oscillators, as is done in this work. In addition, it is also found that active control can be used to extend such behavior over a range of excitation frequencies. In this work, the oscillator to be maintained at rest is a spring-supported block Mc from which the drill-pipe/riser is driven. The drill-ship with mass Ms forms another oscillator, while the third oscillator is an undamped mass Mm. All masses are thought to move primarily in the vertical direction. Mass Mm is to be controlled using feedback in such a way that mass Mc remains stationary over a large frequency range. Such control would be useful in irregular waves where the exciting force FD on the ship Ms can be represented as a sum of sinusoids at different frequencies, amplitudes and phases. The system is configured as shown in Figs 1a and 1b. Block Mc supports the drill-pipe/riser. Mass Mm is spring-supported, and controlled from Mc by a linear actuator. The drill-pipe is rotated from Mc. Mc is supported on spring-loaded vertical guides, along which it is driven vertically from the derrick by a pair of linear actuators. The damping force acting on Mc does not have to be zero. However, the damping on Mm should be as small as possible. The required actuation is provided by linear motors or hydraulic rams. Feedback of the three oscillations is obtained as follows. An accelerometer mounted on Mm measures the acceleration of Mm. The vertical acceleration of Mc is measured by averaging outputs of four accelerometers fixed near four corners of the block. Vertical acceleration of the ship is sampled by averaging the measurements of four accelerometers fixed to the derrick near the actuators driving Mc and distributed
Heave compensation on drill-ships
543
Fig. 1. Schematic of the active heave compensation system.
so as to average out the effects of any small roll and pitch of the ship. Control signals for the two actuators may be evaluated using a dedicated Digital Signal Processor. Physical/mechanical system parameters relevant to this work are outlined in Section 4. 3. EQUATIONS OF MOTION
The modelled system (see Fig. 1) consists of the ship with rest mass Ms, support block Mc, undamped mass Mm, and the drill-pipe with a mass per unit length 0. The focus is on setting up a simple model to investigate the essential physical behavior of the system. For this reason, a number of simplifying assumptions are made regarding environmental loads and resulting motions. All oscillations are assumed to be small enough to justify use of linear differential equations. Consistent with the purpose of the compensator, the ship is thought to be oscillating primarily in heave, and other motions are considered small in comparison. For the same reason, current velocities are assumed to be small enough to allow horizontal oscillations of the drill pipe to be discounted. Two situations are studied: (I) when the drill-pipe is free at the bottom end and is not yet engaged in the borehole; and (II) when the pipe is engaged in the borehole (see Fig. 2) during drilling. Variations in tension due to vertical elastic oscillations of pipe are
Fig. 2. Two heave compensation situations studied in this work.
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included in the model for situation (II). For each situation, the dynamic response of the overall system to irregular wave excitation is first studied without control, and next using the control scheme outlined here. A fixed reference frame is placed at the location O shown in Fig. 2. The vertical motions of ship Ms and masses Mc and Mm are denoted xs(t), xc(t) and xm(t), respectively, where t denotes time. The elastic vertical motion of the pipe is denoted v(t,z), where z is measured along the positive Z axis. The pipe length is L and pipe tension at the top end is TL. 3.1. (I) With pipe disengaged from borehole The free-body diagrams for this situation are shown in Fig. 3. Here, ks is the hydrostatic spring assumed constant for small ship motions and given by ks = wgAWP, where w is
Fig. 3. Free-body diagrams for the coupled oscillators.
Heave compensation on drill-ships
545
the density of sea-water, g the acceleration due to gravity, and AWP the ship water-plane area. kc and km are the spring constants shown in Fig. 1b, and Cc is the damping rate along the vertical guides. FD is the diffraction heave force due to waves, and FR is the radiation force due to ship motion on the water surface. fm and fc are the control forces applied on masses Mm and Mc by their respective linear actuators. In this situation the pipe is free at the bottom end. The vertical wave-force for a pipe of the chosen diameter is found negligibly small. Consequently, TL is here just the weight of the pipe minus its buoyancy. Dynamic equilibrium in this situation implies Mmx¨m = ⫺ km(xm ⫺ xc) + fm Mcx¨c = ⫺ fm + fc ⫺ Cc(x·c ⫺ x·s) ⫺ kc(xc ⫺ xs) ⫺ TL + km(xm ⫺ xc) M x¨ = F ⫺ f ⫺ F ⫺ k x + C (x· ⫺ x· ) + k (x ⫺ x ) s s
D
c
R
s s
c
c
s
c
c
s
(1) (2) (3)
where
冕 ⬁
FR = M⬁x¨s + h()x·s(t ⫺ ) d
(4)
0
with h() denoting the causal impulse response function, and M⬁ the infinite-frequency added mass in heave for the ship. Substituting Equation (4) into Equation (3) and applying the Fourier transform to Equations (1), (2) and (3): ⫺ 2MmXˆm = ⫺ km(Xˆm ⫺ Xˆc) + ˆfm (5) ⫺ 2McXˆc = ⫺ ˆfm + ˆfc ⫺ iCc(Xˆc ⫺ Xˆs) ⫺ kc(Xˆc ⫺ Xˆs) ⫺ TˆL + km(Xˆm ⫺ Xˆc) (6) 2 2 ⫺ MsXˆs = FˆD ⫺ ˆfc + (M⬁ + M)Xˆs ⫺ iXˆs ⫺ ksXˆs + iCc(Xˆc ⫺ Xˆs) + kc(Xˆc ⫺ Xˆs)
(7)
where M and are the frequency-dependent added mass and radiation damping coefficients for the ship in heave, and are defined by
冕
冕
⬁
⬁
2 2 h() = F() cos d = ⫺ M() sin d
(8)
0
0
The symbol ˆ denotes Fourier transforms. 3.1.1. (I-A) Without control. Setting ˆf m = ˆfc = 0 for the case without control, Equations (5), (6) and (7) can be rewritten in matrix form ˆ = Fˆ, with AX A=
冤
⫺ 2Mm + km
⫺ km
⫺ km
⫺ Mc + kc + km + iCc
⫺ iCc ⫺ kc
0
⫺ iCc ⫺ kc
⫺ 2(Ms + M⬁ + M) + i( + Cc) + kc + ks
2
0
冥
(9a)
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U. A. Korde
冤冥 冤 冥
Xˆm ˆ = Xˆc X Xˆs
(9b)
0
Fˆ =
⫺ TˆL FˆD
(9c)
T0 where TˆL = ␦(). T0 is here the value (weight ⫺ buoyancy) for the pipe, and ␦ denotes 2 the Dirac delta function. The variations , M⬁ + M, and FˆD vs can be obtained from published results. Denoting the determinant of the matrix A as 兩A兩, we can express the complex amplitudes Xˆm, Xˆc, and Xˆs as
冑
1 Xˆm = ( ⫺ D12TˆL + D13FˆD) 兩A兩
(10a)
1 Xˆc = ( ⫺ D22TˆL + D23FˆD) 兩A兩
(10b)
1 Xˆs = ( ⫺ D32TˆL + D33FˆD) 兩A兩
(10c)
where the elements Dij of the adjoint of A are given by D12 = km[ ⫺ 2(Ms + M⬁ + M) + i( + Cc) + kc + ks]
(11a)
D13 = km(iCc + kc)
(11b)
D22 = ( ⫺ Mm + km)[ ⫺ (Ms + M⬁ + M) + i( + Cc) + kc + ks] 2
2
(11c) D23 = D32 = ( ⫺ 2Mm + km)(iCc + kc)
(11d)
D33 = ( ⫺ Mm + km)( ⫺ Mc + kc + km + iCc) ⫺ k
(11e)
兩A兩 = ( ⫺ 2Mm + km)(D11) + km( ⫺ D12)
(11f)
2
2
2 m
and
where D11 = ( ⫺ 2Mc + kc + km + iCc)[ ⫺ 2(Ms + M⬁ + M) + i( + Cc) + kc + ks] ⫺ (iCc + kc)2 Note from Equations (10b), (11c) and (11d) that Xˆc = 0 at a frequency at which ( ⫺ 2Mm + km) = 0, which is the uncoupled natural frequency of oscillator (Mm, km). This resembles the phenomenon analyzed by Meirovitch (1986) and Ogata (1978) for a twodegree-of-freedom system. For a passive system described by Equations (9a), (9b) and
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547
(9c), therefore, perfect heave compensation would be achieved if the excitation force km FˆD were a perfect sinusoid with frequency m = . Mm This suggests that if the predominant wave frequency in an incident wave spectrum were m, a passive system would greatly reduce the cyclic loads on the drill-pipe due to ship heave. In this paper, however, it is proposed that the coupled system be controlled by means of two control forces ˆf m and ˆf c such that the complex amplitude Xˆc is zero over a range of frequencies, rather than just at the uncoupled natural frequency m of the oscillator (Mm, km).
冪
3.1.2. (I-B) With control. We choose the control forces fm and fc in Equations (1), (2) and (3) such that f m = Mm(x¨m ⫺ x¨c) + km(xm ⫺ xc)
(12)
f c = kl(xc ⫺ xs)
(13)
This leads to the following equations in the time domain: Mmx¨c = 0
(14)
(Mc ⫺ Mm)x¨c + Mmx¨m + Ccx·c ⫺ Ccx·s + (kc ⫺ kl)xc + (kl ⫺ kc)xs = ⫺ TL
(15)
冕 ⬁
Msx¨s ⫺ Ccx·c + Ccx·s + (ks + kc ⫺ kl)xs + (kl ⫺ kc)xc + h()x·s(t ⫺ ) d = FD 0
(16) Fourier transformation leads to the following equations: ⫺ 2MmXˆc = 0
(17)
⫺ 2MmXˆm ⫺ 2(Mc ⫺ Mm)Xˆc + iCcXˆc ⫺ iCcXˆs + (kc ⫺ kl)Xˆc + (kl ⫺ kc)Xˆs = ⫺ TˆL
(18)
⫺ 2(Ms + M⬁ + M)Xˆs ⫺ iCcXˆc + i(Cc + )Xˆs + (ks + kc ⫺ kl)Xˆs + (kl ⫺ kc)Xˆc = FˆD
(19)
ˆ = Fˆ Rewriting in the form AX A=
冤
冥
0
⫺ 2Mm
0
⫺ 2Mm
⫺ 2(Mc ⫺ Mm) + kc ⫺ kl + iCc
⫺ iCc + kl ⫺ kc
0
⫺ iCc + kl ⫺ kc
⫺ (Ms + M⬁ + M) + i( + Cc) + ks + kc ⫺ kl
冤冥
Xˆm ˆ = Xˆc X Xˆs
2
(20a)
(20b)
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U. A. Korde
冤 冥 0
Fˆ =
⫺ TˆL FˆD
(20c)
In this case, Dij, the elements of adj(A), are D12 = 2Mm[ ⫺ 2(Ms + M⬁ + M) + i(Cc + ) + ks + kc ⫺ kl]
(21a)
D13 = ⫺ 2Mm[ ⫺ iCc + kl ⫺ kc]
(21b)
D22 = 0
(21c)
D23 = D32 = 0
(21d)
D33 = ⫺ M
(21e)
4
2 m
and 兩A兩 = ⫺ 4M2m[ ⫺ 2(Ms + M⬁ + M) + i(Cc + ) + ks + kc ⫺ kl]
(21f)
兩A兩 is nonzero for ⬎ 0. Equations (10a–c) and (21a–f) lead to the conclusion that, provided fm and fc are chosen according to Equations (12) and (13) TˆL [ ⫺ iCc + (kl ⫺ kc)]FˆD + 2 Xˆm = 2 Mm Mm[ ⫺ 2(Ms + M⬁ + M) + i(Cc + ) + ks + kc ⫺ kl] (22a) Xˆc = 0 Xˆs =
(22b)
FˆD ⫺ (Ms + M⬁ + M) + i(Cc + ) + ks + kc ⫺ kl 2
(22c)
Equations (22a–c) show that the effect of the chosen control is to lock the motion Xˆc of block Mc to zero for ⬎ 0. Force fm brings about the actual cancellation of Xˆc, while force fc restricts the motion of Mm in the presence of fm. Fig. 4 shows the essentials of a method to synthesize fm and fc from sensed accelerations of Mm, Mc, and Ms [reference may be made to Kuo (1987) or Raven (1987) for Laplace transforms and classical control theory]. Feedback-controlled linear actuators with small time-constants ␣ would be preferable, and the design of amplifiers 1, 2 and 3 will correspondingly require some attention. In addition, the response of amplifier 2 should be shaped so that its zero-frequency gain is not exactly equal to Mm/kA2 to avoid a steady drift resulting from 兩A兩→0 as →0. This measure will cause a slight drop in the low-frequency performance of the compensator. 3.2. (II) With pipe engaged in borehole When the drill-pipe is engaged in the borehole, it plays an important role in the dynamics of the overall system. The free-body diagrams for the three masses Ms, Mc, and Mm are essentially the same as in Fig. 3. However, the force TL is now determined by the pipe tension set by the ‘pipe-tensioner’ (mounted on block Mc here) and the tension variations caused by elastic oscillations of the pipe. It is therefore necessary to probe the dynamics of the pipe. The pipe diameter assumed in this work (d = 0.14 m) is small enough for the vertical wave-force to be negligible. Further, current velocities are assumed small enough
Heave compensation on drill-ships
549
Fig. 4. Block diagram for the control system outlined in this work.
to allow any current-induced forces to be neglected. In addition, the pipe buoyancy is assumed to balance the submerged portion of the pipe weight and changes in the pipe submerged volume during oscillations are neglected. Consequently, the only excitation of the pipe is provided by the vertical motion of the pipe-support block Mc. The equation of motion for a pipe element of length ⌬z can be written as (see Fig. 5)
Fig. 5. Free-body diagram for a pipe element as modeled here.
550
U. A. Korde
0⌬zvtt = ⫺ 0cd⌬zvt + T(z + ⌬z) ⫺ T(z)
(23)
with subscripts t and z denoting partial differentiation wherever they appear. Here 0 is the mass per unit length, and 0cd the damping due to viscous skin friction. Dividing throughout by ⌬z and letting ⌬z be very small, we have
0vtt = 0cdvt =
∂T ∂z
(24)
For small axial elastic oscillations T = Ea0vz + T0
(25)
where E is the modulus of elasticity and a0 the nominal cross-sectional area of the pipe. T0 is the desired steady tension applied to prevent pipe slackness. As a result, the equation vtt + cdvt ⫺
Ea0 v =0 0 zz
(26)
models pipe oscillations in this situation. It is subject to the following boundary conditions at z = 0 and z = L: v(t, 0) = 0
(27a)
v(t, L) = xc
(27b)
where L denotes the pipe length. The oscillations are thought to be steady state (i.e. continuing from t→ ⫺ ⬁). Next we effect a change of variables in Equation (26) with
(t,z) = v(t,z) ⫺
z x (t) L c
(28)
so that the boundary conditions become homogeneous. Thus, we solve
tt ⫹ cdt ⫺
z Ea0 z = ⫺ cd x·c ⫺ x¨c 0 zz L L
(29)
subject to
(t, 0) = 0
(30a)
(t, L) = 0
(30b)
The undamped, unforced equation corresponding to Equation (29) is
tt ⫺
Ea0 =0 0 zz
(31)
This can be solved using separation of variables (t,z) = ⌳(z)⌫(t) to obtain ⌳(z) = a1 sin
0
1
0
and
0
冪Ea z + b cos 冪Ea z 0
(32a)
Heave compensation on drill-ships
⌫(t) = A1 sin t + B1 cos t
551
(32b)
The boundary conditions lead to ⌳(0) = 0⇒b1 = 0, ⌳(L) = 0⇒a1 sin
0
冪Ea
L
=0
(33)
0
The second condition provides the eigenvalues n for the pipe oscillation, where
n =
n L
冪
Ea0
, n = 1,2,3,…
(34)
0
For example, for a 3832 m long steel pipe having a diameter of 14.5 cm, 1 = 4.1514 rad/s. The solution can then be expressed as
冘 (A sin t + B cos t) sin nL z ⬁
=
n
n
(35)
n=1
To find the forced elastic response to xc(t), we can expand the excitation terms on the right-hand side of Equation (29) in terms of eigenfunctions sin nz/L such that
冘 f sin nLz = ⫺ cLz x· ⫺ Lz x¨ ⬁
d
n
c
c
(36)
n=1
It follows from orthogonality of the eigenfunctions that 2 fn = L
冋
册冕 L
x¨c ⫺ cd · xc ⫺ L L
z sin
nz dz L
(37)
0
Integration leads to fn =
2 (c x· ⫹ x¨c) cos n n d c
(38)
This implies that fn = ⫺
2 (c x· ⫹ x¨c), n = 1,3,5,… n d c
(39a)
and fn =
2 (c x· ⫹ x¨c), n = 2,4,6,… n d c
(39b)
Next, letting the forced response of the pipe be
冘 C sin nL z ⬁
(t,z) =
n
n=1
we can substitute
(40)
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U. A. Korde
sin
inz nz i ⫺ inz = (e L ⫺ e L ) L 2
together with Equations (39a), (39b) and (40) in Equation (29), which is valid separately for each mode due to the orthogonality of the eigenfunctions. Some manipulation leads to Ea0n22 2 · Cn = ⫿ (c x· ⫹ x¨c) C¨n + cdCn + 2 0L n d c
(41)
The negative sign applies to n = 1,3,5,... and the positive sign to n = 2,4,6,.... The tension TL at the pipe-support can now be found from
冉 冊
TL = T0 + (Ea0vz)z = L = T0 + Ea0 z +
xc L
z=L
This leads to TL = T0 +
Ea0 L
冢 冘 ⫿ nC + x 冣 ⬁
n
(42)
c
n=1
Substituting Equation (42) in Equations (1), (2) and (3), we arrive at a model for the system with the drill-pipe engaged in the borehole. The corresponding equations are summarized below. Mmx¨m = ⫺ km(xm ⫺ xc) + fm Mcx¨c = ⫺ fm + fc ⫺ Cc(x·c ⫺ x·s) ⫺ kc(xc ⫺ xs)
冘
(43)
⬁
+ km(xm ⫺ xc) ⫺ T0 ⫺
Ea0 Ea0 x ⫿ nCn ⫺ L n=1 L c
(44)
冕 ⬁
Msx¨s = FD ⫺ fc ⫺ h()x·s (t ⫺ ) d ⫺ M⬁x¨s ⫺ ksxs + Cc(x·c ⫺ x·s) + kc(xc ⫺ xs) 0
(45) Ea0n22 2 · Cn = ⫿ (c x· ⫺ x¨c) C¨n + cdCn + 0L2 n d c
(46)
In this work, the first three modes of pipe oscillation only are considered significant. The result of this approximation is a system of six equations in six unknowns. 3.2.1. (II-A) Without control. Setting fm = fc = 0 allows study of the situation without control. The reduced system of equations can be written as Mmx¨m = ⫺ km(xm ⫺ xc)
(47)
Ea0 C1 Mcx¨c = ⫺ Cc(x·c ⫺ x·s) ⫺ kc(xc ⫺ xs) + km(xm ⫺ xc) ⫺ T0 + L
(48)
Heave compensation on drill-ships
⫺
553
2Ea0 3Ea0 Ea0 C2 + C3 ⫺ x L L L c
冕 ⬁
Msx¨s = FD ⫺ h()x·s(t ⫺ ) d ⫺ M⬁x¨s ⫺ ksxs + Cc(x·c ⫺ x·s) + kc(xc ⫺ xs) 0
(49) Ea02 2 · C = ⫺ (cdx·c ⫹ x¨c) C¨1 + cdC1 + 0L2 1
(50)
4Ea02 1 · C2 = (cdx·c ⫹ x¨c) C¨2 + cdC2 + 2 0L
(51)
9Ea02 2 · C3 = ⫺ (c x· ⫹ x¨c) C¨3 + cdC3 + 2 0L 3 d c
(52)
ˆ = Fˆ, and letting aij, 1 ⱕ Applying the Fourier transform, rewriting in matrix form AX i, j ⱕ 6, define the elements of A, we have a11 = ⫺ 2Mm + km
(53a)
a12 = a21 = ⫺ km
(53b)
a22 = ⫺ 2Mc + iCc + kc + km +
Ea0 L
a23 = a32 = ⫺ kc ⫺ iCc a24 = ⫺ a25 =
Ea0 L
(53f)
3Ea0 L
(53g)
a33 = ⫺ 2(Ms + M⬁ + M) + i(Cc + ) + ks ⫹ kc a42 =
2 (icd ⫺ 2)
a44 = ⫺ 2 + icd + a52 = ⫺
(53h) (53i)
Ea02 0L2
1 (icd ⫺ 2)
a55 = ⫺ 2 + icd +
(53d) (53e)
2Ea0 L
a26 = ⫺
(53c)
4Ea02 0L2
(53j) (53k) (53l)
554
U. A. Korde
a62 =
2 (icd ⫺ 2) 3
a66 = ⫺ 2 + icd +
(53m)
9Ea02 0L2
(53n)
aij = 0, for all other i and j
(53o)
ˆ = [Xˆm,Xˆc,Xˆs,Cˆ1,Cˆ2,Cˆ3] can be found by The complex amplitudes forming the vector X solving the linear system of equations above. The vector Fˆ on the right side equals [0, ⫺ Tˆ0,FˆD,0,0,0]T, with the only nonzero components being Tˆ0 and FˆD. The constant T0 T0 ␦() in the frequency domain in terms of the Dirac delta function. leads to Tˆ0 = 冑2 T
3.2.2. (II-B) With control. Next we choose, as in situation (I-b) f m = Mm(x¨m ⫺ x¨c) + km(xm ⫺ xc) f c = kl(xc ⫺ xs) Substitution in Equations (42)–(46) followed by Fourier transformation leads to the equˆ = Fˆ, where the elements aij of A are now given by ation AX a12 = a21 = ⫺ 2Mm
(54a)
a22 = ⫺ 2(Mc ⫺ Mm) + iCc + (kc ⫺ kl) +
Ea0 L
a23 = a32 = (kl ⫺ kc) ⫺ iCc a24 = ⫺ a25 =
Ea0 L
(54e)
3Ea0 L
(54f)
a33 = ⫺ 2(Ms + M⬁ + M) + i( + Cc) + (ks ⫺ kl + kc) a42 =
2 (icd ⫺ 2)
a44 = ⫺ 2 + icd + a52 = ⫺
(54g) (54h)
Ea02 0L2
1 (cd ⫺ 2)
a55 = ⫺ 2 + icd +
(54c) (54d)
2Ea0 L
a26 = ⫺
(54b)
4Ea02 0L2
(54i) (54j) (54k)
Heave compensation on drill-ships
a62 =
555
2 (icd ⫺ 2) 3
a66 = ⫺ 2 + icd +
(54l)
9Ea02 0L2
(54m)
aij = 0, for all other i and j
(54n)
ˆ denotes the vector of six complex amplitudes of interest, and Fˆ denotes Once again, X the vector of exciting forces which is the same as in case (II-A) above. In particular, the complex amplitude Xˆc of block Mc can be inferred in terms of the determinant 兩A兩 (which is nonzero for ⬎ 0) and the elements Dij of the adjoint of A as ⫺ D22 ˆ D23 ˆ T0 + F Xˆc = 兩A兩 兩A兩 D
(55)
where
| |
0 0
0
0
0
0 a33 0
0
0
D22 = 0 0 a44 0
| |
| |
0
0 a55 0
0 0
0
0 a66
0
0
0
a21 a23 a24 a25 a26
0 = 0 and D23 = 0
0 0
0
0 a44 0
| |
0 =0
0
0
0 a55 0
0
0
0
0 a66
(56)
The chosen fm thus locks the motion xc of the pipe-support block Mc to zero for ⬎ 0 even while the pipe is engaged in the borehole. Once again, the role of fc here is to restrict xm, the magnitude of oscillations of the undamped mass Mm. 4. RESULTS
Calculations based on the models of Section 3 were carried out in the frequency domain. A standard Gauss-elimination-based equation solver was used. Results were obtained for the four situations examined above, namely: (I) drill-pipe disengaged from borehole, (a) without control and (b) with control; and (II) drill-pipe engaged in borehole, (a) without control and (b) with control. Control was applied according to Equations (12) and (13). For convenience, the drill-ship form was approximated by a rectangular cylinder, and corresponding curves for added mass M⬁ + M, radiation damping , and exciting force FD (per unit wave-amplitude) were obtained from Newman (1977) following necessary interpolation for the correct breadth-to-draft ratio. A strip-theory type approximation was used. The ship principal dimensions were L = 143.3 m, B = 21.3 m, and T = 7.5 m with a displacement Ms = 23464.0 tonne. The operating water depth was 3780 m. Parameters of the heave compensation system were: Mm = 1.0 tonne, Mc = 20.0 tonne, km = 250 kN/m, kc = 1000 kN/m, kl = kc, Cc = 0.1 kN s/m, cd = 0.01 m3/s. The pipe length L was 3832 m,
556
U. A. Korde
and the pipe diameter was taken to be d = 0.14 m. A steel drill-pipe (specific gravity 7.8 and modulus of elasticity E = 2 × 108 kN/m2) was assumed. The tension T0 in (II) was taken to be 100 kN. The calculations were carried out in fully developed beam seas described by a Pierson– Moskowitz spectrum (e.g. see Newman, 1977) at a wind-speed of 50 knot. Fig. 6 shows the wave-amplitude variation for the assumed spectrum. Results based on this wave-amplitude variation are shown in Figs 7–11. The vertical oscillation amplitudes of ship Ms, block Mc, and mass Mm with the pipe disengaged from the borehole and without control are shown in Fig. 7. Fig. 8 shows the amplitudes of these masses with the pipe disengaged from the borehole but with control. For the case with the pipe engaged in the borehole, Fig. 9 illustrates the amplitudes of Ms, Mc, and Mm without control. For the same situation, Fig. 10 shows the response of the first three elastic modes of the pipe oscillations, C1,...,C3 3 nz . Finally, the oscillation amplitudes of Ms, Mc, and corresponding to ⬇ Cn sin L 0 Mm and the three pipe modes in the presence of control are illustrated in Fig. 11.
冘
5. DISCUSSION
It should be emphasized that the aim here is to control the motion of a block supported on board the drill-ship and not the motion of the drill-ship itself. The masses Mm and Mc oscillate in air as opposed to on the air–water free surface. Thus, the mass, spring constant, and damping rate for the oscillators (Mm,km) and (Mc,Cc,kc) are independent of frequency. Consequently, in principle, time-domain control to drive xc to zero requires knowledge of the present displacements and accelerations only. Thus, in theory it is not difficult to design
Fig. 6. Wave-amplitude variation used in computations.
Heave compensation on drill-ships
Fig. 7. System response with pipe free at the bottom, without control.
Fig. 8. System response with pipe free at the bottom, with control.
557
558
U. A. Korde
Fig. 9. Vertical motion of Ms, Mc, and Mm with pipe engaged at bottom, without control.
Fig. 10. Elastic modes for vertical oscillation of pipe, without control.
Heave compensation on drill-ships
559
Fig. 11. System response with pipe engaged in borehole, with control.
the controller in Fig. 4. As mentioned in Section 3, however, careful consideration should be given to the design of amplifiers 1, 2, and 3. In particular, the response of amplifier 2 should be shaped to avoid zero-frequency instability with the control active. This would be at the expense of a slight drop in the low-frequency performance of the compensator. With the exception just mentioned, the amplifier gains are required precisely to produce the desired match with km and Mm. Further, the damping on Mm should be as close to zero as possible. Any unavoidable mechanical damping on Mm can in principle be balanced by an additional equal and opposite actuator force derived from velocity feedback, but this in turn would require that the residual damping be quantified exactly. In this context, it would seem helpful to estimate on-line any uncertainties or slow changes in the nominal values of Mm and km from acceleration and actuator force measurements, and to use these estimates for adaptive improvements to the controller performance. Equations (22a–c) and (57) imply that, when successfully implemented, such control would effectively reduce xc to zero both without and with the drill-pipe engaged in the borehole. This suggests that, when heave is the dominant vessel motion and when current velocities are small, the proposed heave compensation will be effective both during deployment/re-entry of the drill-pipe and during drilling. The results obtained in this work are discussed below. Fig. 6 shows the wave-amplitude variation associated with a Pierson–Moskowitz spectrum for a wind-speed of 50 knot. The peak wave amplitude of 2.25 m occurs near = 0.375 rad/s corresponding to a peak period of 16.75 s. Fig. 7 shows the vertical-amplitude response of the ship Ms and masses Mc and Mm to this wave-amplitude variation. The three amplitudes are seen to be almost identical over the whole frequency range. A small steady deflection brought about by the above-water weight of the pipe can be observed in the case of the mass Mc. The amplitudes
560
U. A. Korde
peak at ⬇ 0.35 rad/s. As the graph indicates negligible relative motion among the masses Ms, Mc, and Mm, the dynamics in this case appear to be dominated by the greatest mass, namely the drill-ship. Smaller values of km and kc would allow greater relative motion and would prove useful in passive implementations of the proposed hardware. The figure further suggests a second peak near = 0.8 rad/s. This is an influence of the undamped ks natural frequency of the ship in heave, which is found using sn = to Ms + M⬁ + M be 0.788 rad/s. Fig. 8 shows the motion amplitudes in the presence of control forces fm and fc. While no significant change is observed in the ship heave amplitude, the amplitude of Mc (thick dark line) is seen to be zero over ⬎ 0 as expected. The ship heave amplitude peaks near 0.35 rad/s as before, while mass Mm is seen to have a maximum amplitude of 0.35 m near = 0.32 rad/s. This shift is most likely because the coupling effects among the three masses become appreciable under control. It can be inferred from this figure, further, that control achieves the theoretically expected results in this situation. The dynamics of the system are next studied with the drill-pipe engaged in the borehole. In this case the motion of Mc and Mm is restricted by the pipe to about half of what it would be with the pipe disengaged (see Fig. 9). The three modes of pipe motion are shown in Fig. 10. It is straightforward to infer the overall amplitude of the pipe at z = L using [v(L)]amp = [(L) + xc]amp, and this is found to be [v(L)]amp ⬇ 0.6 m. This results in an average axial strain of ⌬L/L = 0.6/3832 = 1.5657 × 10 ⫺ 4, which in turn causes an average axial stress ⬇ 320 kg/cm2, which is well within the elastic limit for steel. The first natural frequency for the pipe vertical elastic oscillations is 1 = 4.1514 rad/s, which is near the margin of the frequency range where sufficient wave energy would exist. It is interesting to note that the peak for Cˆ1 appears near 0.415 rad/s, while the peak for Xˆc is near 0.345 rad/s, and this again is likely an effect of coupling among Mm, Mc, and the pipe oscillation. The influence of the ship natural frequency of 0.788 rad/s can also be noticed in the response of the three pipe modes shown in the figure. Finally, Fig. 11 shows that with the pipe engaged in the borehole, and with control applied according to Equations (12) and (13), the amplitudes of mass Mc and the three pipe modes are ⬍ 1.0 × 10 ⫺ 8 m (or equal to zero for practical purposes) for ⬎ 0. This shows the proposed control to be yielding expected results also with the pipe engaged in the borehole. Once again, increased coupling effects are likely to have been the cause of a slight shift along the frequency axis in the maximum amplitudes of Mm and Ms.
冪
6. CONCLUSIONS
The purpose of this study was to investigate an actively controlled heave compensation system for deep-water drill-ships. The system is controlled such that motion of a block supporting the drill-pipe/riser is effectively locked to zero over the frequency band covered by irregular ocean waves. Because the body whose motion is being controlled is not in contact with the water surface, the required control method is straightforward in theory, and knowledge of current oscillations only is necessary. A notable aspect of the proposed system is that the control has minimal effect on the motion amplitude of the ship itself. This feature is useful because of the desirability of compliance in high waves. A simplified dynamic model which considers only vertical motion of the ship and the
Heave compensation on drill-ships
561
drill-pipe was used to test the compensator’s ability to achieve the objective for which it is designed. The effect of ship roll and pitch on the motion of Mc and horizontal oscillations of the pipe in response to current forces needs to be examined. Extensions of the proposed system to allow actuation and control of Mc in two horizontal directions are possible, and their effectiveness would be studied using a more elaborate model. It is conceivable that similar extensions could also be used to reduce vortex-induced horizontal elastic oscillations of the pipe in response to currents. While the present results indicate isolation of block Mc from ship heave under the proposed control over a wide frequency band, these are based on the assumption that ship heave is small enough to be treated by a linear model. Coupling effects among heave, pitch and roll, and performance when these motions are large, need to be studied before a definitive conclusion as to the effectiveness of this method in large waves can be made. It can be said, however, that with careful implementation, the proposed method will effectively isolate block Mc from ship heave in moderate waves, where the ship heave is not too large to preclude independent treatment under linear theory. REFERENCES Cullen, V. (1992/93) Ocean drilling program. Oceanus Winter, 31–33. JAMSTEC (1995) 1994 Annual Report. Japan Marine Science and Technology Center, pp. 18–20. JAMSTEC (1991) State of the Art of Deep-Ocean Floor Sampling Systems. Japan Marine Science and Technology Center, pp. 73–76 (in Japanese). Kenison, R. C. and Hunt, C. V. (1996) Why FPSOs for the Atlantic frontier. Proceedings of Offshore Technology Conference, OTC 8032, pp. 31–40. Kuo, B. C. (1987) Automatic Control Systems, 5th edn., Chapter 3. Prentice-Hall, Englewood Cliffs, NJ. Meirovitch, L. (1986) Elements of Vibration Analysis, 2nd edn., Chapters 3–6. McGraw-Hill, New York. Morooka, C. K. (1996) Design and implementation of an oil-drilling semisubmersible–riser–template system. Transcript of presentation at Institute of Industrial Science, University of Tokyo (in Japanese). Newman, J. N. (1977) Marine Hydrodynamics, pp. 295–300. MIT Press. Ogata, K. (1978) System Dynamics, pp. 456–459. Prentice-Hall, Englewood Cliffs, NJ. Raven, F. H. (1987) Automatic Control Engineering, 4th edn., Chapter 5. McGraw-Hill, New York. Sarpkaya, T. and Issacson, M. (1981) Mechanics of Wave Forces on Offshore Structures, pp. 354–366. Van Nostrand Reinhold, NY.