Passive Heave Compensator Design and Numerical Simulation for Strand Jack During Lift Operation in Deep Water

Journal of Marine Science and Engineering

Article Passive Heave Compensator Design and Numerical Simulation for Strand Jack during Lift Operation in Deep Water

Yong Zhan *, Bailin Yi, Shaofei Wu and Jianan Xu

College of Mechanical & Electrical Engineering, Harbin Engineering University, Harbin 150001, China; [email protected] (B.Y.); [email protected] (S.W.); [email protected] (J.X.) * Correspondence: [email protected]; Tel.: +86-451-8256-9750

Abstract: In this paper, the passive heave compensator for the strand jack lifting system is studied. An analytical model is developed, which considers the nonlinear characteristic of the compensator stiffness thus as to predict its response under different parameters accurately. This analytical model helps to find the feasible gas volume of the compensator. The comparative analysis is carried out to analyze the influence of key design parameters on the dynamic response of the compensator. In order to evaluate the effectiveness of the compensator, a coupling model of the strand jack lifting system is derived. The compensator efficiency is evaluated in terms of the lifted structure displacement and the strand dynamic tension. The numerical simulations are performed to evaluate the effectiveness of the compensator. Numerical results show that the compensator is able to significantly decrease the tension variation in the strands and the motion of the lifted structure.

Keywords: passive heave compensator; strand jack; deepwater lift operation; nonlinear model

Citation: Zhan, Y.; Yi, B.; Wu, S.; Xu, J. Passive Heave Compensator Design 1. Introduction and Numerical Simulation for Strand Strand jacks are lifting devices, which are widely used in heavy liftings, such as Jack during Lift Operation in Deep offshore installation [1] and marine salvage [2]. A general strand jack consists of a hydraulic Water. J. Mar. Sci. Eng. 2021, 9, 714. cylinder and two anchors, as shown in Figure1. A number of strands are installed through https://doi.org/10.3390/jmse9070714 the strand jack and the load is attached to the end of the strands by a suitable block. The main operation cycle of the strand jack in the lifting process can be described as follows: Academic Editor: Anchor 1 grips the strands, then Anchor 2 releases the strands, and the jack extends its Constantine Michailides piston together with the load. Once the piston travels a predeﬁned distance, Anchor 2 grips the strands, then Anchor 1 releases the strands, and the jack reacts its piston without Received: 10 June 2021 load. The above operation cycle repeats to lift the load. The strand jack offers many Accepted: 26 June 2021 Published: 28 June 2021 advantages over ship cranes, such as small size, simple structure, large capacity, high efﬁciency, and good controllability. Therefore, it has broad application prospects in the

Publisher’s Note: MDPI stays neutral marine engineering field. The detailed information about the working principle of the with regard to jurisdictional claims in strand jack can be referred to in paper [3]. published maps and institutional affil- With the development of ocean resource exploitation, such as oil and gas, lifting iations. operations of structures have to be conducted in the deeper sea. Meanwhile, the lifted structures have become larger in size and heavier in weight in order to reduce the frequency of lifting operations and the period of underwater installation. Dealing with large size and heavy structures at a water depth above 300 m puts new challenges on the lifting equipment since the lifting operation of the strand jack is continuously influenced by the Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. wave-induced motion of the vessel. The heave motion of the lifted structure referred to This article is an open access article seabed results in excessive dynamic tension variations in the strand, which are directly distributed under the terms and exerted on the strand jack. This phenomenon is especially important in rough seas, where conditions of the Creative Commons the critical tension of the strand can be exceeded and leads to serious damage. Therefore, a Attribution (CC BY) license (https:// heave compensation system should be used to decrease the maximum tension in the strand creativecommons.org/licenses/by/ and the motion of the lifted structure, thus as to increase the safe operating sea-states of 4.0/). the strand jack.

J. Mar. Sci. Eng. 2021, 9, 714. https://doi.org/10.3390/jmse9070714 https://www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2021, 9, 714 2 of 18

heave compensation system should be used to decrease the maximum tension in the J. Mar. Sci. Eng. 2021, 9, 714 strand and the motion of the lifted structure, thus as to increase the safe operating2 ofsea- 16 states of the strand jack.

Figure 1.1. Structure of strand jack.jack.

Heave compensation is is an an effective effective techni techniqueque to todecouple decouple the the load load motion motion from from the thewave-induced wave-induced vessel vessel motion. motion. There There are three are three types types of heave of heave compensation compensation systems: systems: pas- passive,sive, active, active, and andsemi-active semi-active heave heave compensati compensationon systems systems [4]. The [4 passive]. The heave passive compen- heave compensationsation system uses system a fixed uses mass a ﬁxed of massgas as of th gase elastic as the medium elastic medium and store and or store release or releaseenergy energywhen the when dynamic the dynamic load is performed load is performed to keep toth keepe position the position of the load of the constant load constant with respect with respectto seabed to seabed[5]. The [5 passive]. The passive compensation compensation system system has the has advantages the advantages of consuming of consuming little littleexternal external power, power, simplicity, simplicity, and high and highreliability. reliability. Therefore, Therefore, it is itsuitable is suitable for large for large size sizeand andheavy heavy structure structure lifting lifting applications. applications. The Theactive active heave heave compensation compensation system system uses usesthe sen- the sensorsor to measure to measure the thevessel's vessel’s motion. motion. Then, Then, the actuator the actuator is controlled is controlled dynamically dynamically to com- to compensatepensate for the for vessel the vessel motion. motion. It has It high has highcompen compensationsation efficiency efﬁciency and is and accurate is accurate in posi- in positioning. However, there are some disadvantages to the active heave compensation tioning. However, there are some disadvantages to the active heave compensation system, system, such as high cost, low reliability, and high power consumption. The semi-active such as high cost, low reliability, and high power consumption. The semi-active heave heave compensation system is a hybrid system, which combines features of both passive compensation system is a hybrid system, which combines features of both passive and and active systems. More detailed information can be referred to in the literature [4]. active systems. More detailed information can be referred to in the literature [4]. The operational process is time-consuming, and the strand jack experiences heavy The operational process is time-consuming, and the strand jack experiences heavy load when lifting operation of large size, and heavy structure is carried out in deep water. load when lifting operation of large size, and heavy structure is carried out in deep water. Moreover, high reliability and low power consumption are the main considerations. Thus, Moreover, high reliability and low power consumption are the main considerations. Thus, the passive compensation system was considered in this paper. the passive compensation system was considered in this paper. The passive heave compensation system has a broad variety of applications, including offshore lifting [6], vertical array of hydrophone [7], and riser tensioner [8–10]. Therefore, passive heave compensation systems received much attention in recent years, and much effort has been made to design the compensators with good performance. In order to investigate the dynamic response of the passive compensator, several dynamic models J. Mar. Sci. Eng. 2021, 9, 714 3 of 18

The passive heave compensation system has a broad variety of applications, including offshore lifting [6], vertical array of hydrophone [7], and riser tensioner [8–10]. There- J. Mar. Sci. Eng. 2021, 9, 714 fore, passive heave compensation systems received much attention in recent years,3of and 16 much effort has been made to design the compensators with good performance. In order to investigate the dynamic response of the passive compensator, several dynamic models have been proposed in the literature. Nam [6] proposed a model based on linear stiffness havefor a beenpassive proposed compensator in the literature.without the Nam consider [6] proposedation of athe model damping based effect. on linear Both stiffness numeri- forcal asimulations passive compensator and experimental without the results consideration show that of the the dampingcompensator effect. is Botheffective numerical in de- simulationscreasing the and wire experimental tension variation results for show the that ship the crane compensator lifting operation. is effective Duc in [11] decreasing investi- thegated wire the tension crane variationvibration-reducing for the ship proble cranem liftingusing operation.a linear compensator Duc [11] investigated model and thepointed crane out vibration-reducing that the vertical motion problem of the using payload a linear and compensator the peak tension model of andthe wire pointed rope outwere that decreased the vertical by using motion the ofpassive the payload compensa andtor. the Based peak on tension a two-dimensional of the wire rope nonlinear were decreasedfinite element by usingmodel, the Liu passive [12] demonstrated compensator. that Based the passive on a two-dimensional compensator is effective nonlinear in finitereducing element the vibration model, Liudisplacement [12] demonstrated and keeps that the wireline the passive with compensator a nearly constant is effective tension in reducing the vibration displacement and keeps the wireline with a nearly constant independent of the heave motion of the ship. Further, it is more effective to keep wireline tension independent of the heave motion of the ship. Further, it is more effective to keep tension constant than motion compensation. When the passive compensator works un- wireline tension constant than motion compensation. When the passive compensator derwater, hydrostatic pressure variation has a significant negative influence on the stiff- works underwater, hydrostatic pressure variation has a significant negative influence on ness of the passive compensator. Therefore, depth compensation should be carried out in the stiffness of the passive compensator. Therefore, depth compensation should be carried such scenarios. Patens [13] and [14] proposed two depth compensated passive compensa- out in such scenarios. Patens [13] and [14] proposed two depth compensated passive tors, which are applicable underwater with constant stiffness. compensators, which are applicable underwater with constant stiffness. In this paper, the passive compensator for the strand jack lifting system is studied. In this paper, the passive compensator for the strand jack lifting system is studied. An analytical model is developed, which considers the nonlinear characteristic of the com- An analytical model is developed, which considers the nonlinear characteristic of the pensator stiffness. This mathematical model helps to find the feasible gas volume of the compensator stiffness. This mathematical model helps to find the feasible gas volume ofcompensator. the compensator. The comparative The comparative analysis analysis is carried is carried out to analyze out to analyze the influence the influence of key de- of keysign design parameters parameters on the ondynamic the dynamic response response of the compensator. of the compensator. In order Into orderverify tothe verify effec- thetiveness effectiveness of the compensator, of the compensator, a coupling a coupling model of model the strand of the jack strand lifting jack system lifting is system derived. is derived.Numerical Numerical results show results that show the thatcompensator the compensator is able to is significantly able to significantly decrease decrease the tension the tensionvariation variation in the strand in the and strand the and motion the motionof the lifted of the structure. lifted structure.

2.2. ConfigurationConfiguration of of the the Lifting Lifting System System and and Dynamic Dynamic Model Model 2.1.2.1. ConfigurationConfiguration ofof thethe LiftingLiftingSystem Systemand andWorking Working Principle Principle of of the the Compensator Compensator TheThe schematicschematic layout of of the the strand strand jack jack liftin liftingg system system can can be beseen seen in Figure in Figure 2. It2. con- It consistssists of five of five sub-systems: sub-systems: the vessel, the vessel, the pass theive passive compensator, compensator, the strand the strandjack, the jack, strands, the strands,and the andlifted the structure. lifted structure. The passive The passive compensator compensator hangs hangsbelow belowa hook. a hook.The rod The of rod the ofhydraulic the hydraulic cylinder cylinder of the ofcompensator the compensator is conne iscted connected via a hook via ato hook the strand to the jack. strand Several jack. Severalstrandsstrands suspend suspend the lifted the structur lifted structure,e, which is which submerged is submerged in water. in water.

FigureFigure 2.2.Schematic Schematic layoutlayout ofof thethe strandstrand jackjack liftinglifting system. system.

As shown in Figure3, the passive compensator can be broken down into three parts: a hydraulic cylinder, a piston accumulator, and additional gas bottles. The lower chamber of the hydraulic cylinder is interconnected to the oil chamber of the piston accumulator by a pipeline. The gas chamber of the piston accumulator is interconnected to the additional gas bottles, which are ﬁlled with the pressured nitrogen. The compressibility of the gas J. Mar. Sci. Eng. 2021, 9, 714 4 of 18

As shown in Figure 3, the passive compensator can be broken down into three parts: a hydraulic cylinder, a piston accumulator, and additional gas bottles. The lower chamber

J. Mar. Sci. Eng. 2021, 9, 714 of the hydraulic cylinder is interconnected to the oil chamber of the piston accumulator4 of 16 by a pipeline. The gas chamber of the piston accumulator is interconnected to the additional gas bottles, which are filled with the pressured nitrogen. The compressibility of the gas provides the desired elasticity for the compensator. In order to improve compensation provides the desired elasticity for the compensator. In order to improve compensation performance, soften stiffness is required. The piston accumulator is used for this purpose performance, soften stiffness is required. The piston accumulator is used for this purpose since it sincehas the it has advantage the advantage that larger that larger sizes sizes are areavailable, available, and and it itis is easy easy to to be be interconnected interconnected with thewith additional the additional gas bottles gas bottles [15]. [ 15].

Pipeline Additional Accumulator gas bottoles

Hydraulic cylinder

Figure 3.Figure Schematic 3. Schematic layout layout of the of compensator. the compensator. In static conditions, the gas volume needs to be adjusted accordingly thus as to set the Inpiston static ofconditions, the hydraulic the cylinder gas volume at the mid-stroke.needs to be As adjusted the vessel accordingly heaves upward, thusthe as forceto set the pistonacting of onthe the hydraulic rode of the cylinder cylinder at becomes the mid-stroke. larger due As to the inertiavessel ofheaves the lifted upward, structure. the force actingAs a result,on the therode rod of isthe stretched, cylinder and becomes thus the larger gas isdue compressed, to the inertia and of the the gas lifted pressure structure. Asincreases. a result, Asthe the rod vessel is stretched, heaves downward, and thus the the gas force is compressed, acting on the rodand becomesthe gas pressure smaller. increases.Then, As the the rod vessel is contracted. heaves downward, In the above th twoe force situations, acting the on piston the rod rod becomes of the hydraulic smaller. Then, thecylinder rod is moves contracted. in the opposite In the above direction two of situations, the vessel motion, the piston thus rod as to of compensate the hydraulic for the vessel motion. cylinder moves in the opposite direction of the vessel motion, thus as to compensate for the vessel2.2. motion. Dynamic Model of the Strand Jack Lifting System In order to accurately predict the lifting system dynamic behaviors of the lifting 2.2. Dynamicsystem, Model a coupling of the model Strand of Jack the strandLifting jack System together with the passive compensator and the In strand-suspendedorder to accurately structure predict is developed.the lifting system dynamic behaviors of the lifting system, a couplingIn modeling, model twoof the following strand issuesjack together are considered. with the The passive ﬁrst one compensator is that the suspended and the strand-suspendedstructure is lifted structure or lowered is developed. by the strands during the strand jack operation. In the case of deep-sea operations, the stiffness and mass of the strands have signiﬁcant impact on the In modeling, two following issues are considered. The first one is that the suspended dynamic behaviors of the lifting system during its response to the heave excitation, since structurethe is length lifted of or the lowered strands by is more the strands than 300 during m. Therefore, the strand the effect jack of operation. the elasticity In andthe case the of deep-seamass ofoperations, the strands the should stiffness be taken and into mass account of the in thestrands coupling have model. significant The second impact issue on the dynamicis that sincebehaviors the lifted of the structure lifting is system submerged during in water, its response the hydrodynamic to the heave force excitation, has to be since theconsidered length of due the to strands its large size.is more than 300 m. Therefore, the effect of the elasticity and the massThe of assumptions the strands belowshould are be made taken in theinto lifting account system in the coupling coupling model. model. The second issue1. isThe that hydraulic since the oil lifted is uncompressible. structure is submerged in water, the hydrodynamic force has to be2. consideredNeglecting due leakage to its large flows size. and seal friction forces of the hydraulic cylinder of The assumptionsthe compensator. below are made in the lifting system coupling model. 3. All the strands have the same dimensions and mechanical properties. 1. The hydraulic oil is uncompressible. 4. Neglecting damping effect in the strands. 2. Neglecting leakage flows and seal friction forces of the hydraulic cylinder of the compensator.Figure 4 shows the schematic diagram of the strand jack lifting system. mj is the mass 3. Allof the the strands strand jack. have As the previously same dime mentioned,nsions and a damped mechanical spring properties. model is used to simulate the dynamic behavior of the compensator with stiffness k and damping c . The strand is 4. Neglecting damping effect in the strands. h h modeled as lumped mass with elasticity. Moreover, the spring with stiffness ks refers to the Figurestiffness 4 shows of the the strands. schematicms represents diagram the of massthe strand of the strands,jack lifting which system. is a function mj is the of mass the of the strandstrand length,jack. As density, previously and number mentioned, of strands. a damped When thespring lifted model structure is used moves to insimulate water, the reaction forces acting on it is modeled as the added mass ma and the hydrodynamic

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the dynamic behavior of the compensator with stiffness kh and damping ch. The strand is

modeled as lumped mass with elasticity. Moreover, the spring with stiffness ks refers to the stiffness of the strands. ms represents the mass of the strands, which is a function of the strand length, density, and number of strands. When the lifted structure moves in water, the reaction forces acting on it is modeled as the added mass ma and the hydrody- J. Mar. Sci. Eng. 2021, 9, 714 5 of 16 namic damping cp, which is governed by the flow condition and the shape of the lifted structure. In this paper, the lifted structure is approximated as a cuboid for simplicity. According to DNV-RP-H103 [16], ma can be given as 0.15ml, where ml is the mass of the damping cp, which is governed by the ﬂow condition and the shape of the lifted structure. liftedIn this structure. paper, the lifted Thus, structure the total is approximated mass mp aslifted a cuboid by forthe simplicity. strand jack According is given to by mp = ms + ml + DNV-RP-H103 [16], ma can be given as 0.15ml, where ml is the mass of the lifted structure. ma. Thus, the total mass mp lifted by the strand jack is given by mp = ms + ml + ma.

FigureFigure 4. SchematicSchematic diagram diagram of the strandof the jack strand lifting jack system. lifting system. The vessel heave motion induced by the wave is assumed to be known and is modeled as aThe prescribed vessel displacement heave motion function inducedyb, which isby applied the wave to the baseis assumed of the compensator. to be known and is mod- y and yp are the absolute displacements of the base of strand jack and the lifted structure eledj as a prescribed displacement function yb, which is applied to the base of the compen- due to vessel motion, respectively. All displacement variables are deﬁned in the earth-ﬁxed sator.coordinate yj and frame. yp are the absolute displacements of the base of strand jack and the lifted structureAccording due to Newton’svessel motion, second law, respectively. the coupling All model displacement of the strand jackvariables lifting are defined in the system is given as: earth-fixed coordinate frame. .. . . According tom jNewton’syj + kh(yj − ysecondb) + ch(y jlaw,− yb) the + ks (couplingyj − yp) = 0model of the strand jack lifting sys- .. . (1) + ( − ) + = tem is given as: mpyp ks yp yj cpyp 0

.. . . 2.3. Mathematical Modeling of the+ Compensator− + − + − = In this section, am nonlinearj y j mathematicalk h ( y j y modelb ) c ofh the( y compensatorj y b ) k wass ( y developed,j y p ) 0

and a comparative analysis.. was carried out to analyze the influence. of key design parame- (1) ters on the dynamic responsem y of+ thek compensator.( y − y ) + c y = 0 The compensator wasp modeledp s by ap nonlinearj dampedp springp system. Figure5 shows a simplification of the compensator in order to study its stiffness and damping coefficients in a more straightforward way. Assuming that the hydraulic cylinder has a stroke of 2H, 2.3.piston Mathematical diameter of D Modelingp and piston of rod the diameter Compensator of Dr. The x-axis is along the axis of the cylinder with vertical upward displacement defined as positive, and the zero point of x-axis is atIn the this cylinder section, mid-stroke. a nonlinear Obviously, mathematical the value range of modelx is [−H of, H ].the compensator was developed, and a comparative analysis was carried out to analyze the influence of key design parameters on the dynamic response of the compensator. The compensator was modeled by a nonlinear damped spring system. Figure 5 shows a simplification of the compensator in order to study its stiffness and damping coefficients in a more straightforward way. Assuming that the hydraulic cylinder has a stroke of 2H, piston diameter of Dp and piston rod diameter of Dr. The x-axis is along the

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J. Mar. Sci. Eng. 2021, 9, 714 6 of 16 axis of the cylinder with vertical upward displacement defined as positive, and the zero point of x-axis is at the cylinder mid-stroke. Obviously, the value range of x is [−H, H].

Fp Moving platform

-H

Pi Pc k c x h h Pa

H Base platform

(a) (b)

Composition diagram of the compensator Simplified schematic diagram of the compensator

Figure 5. Schematic diagram of the compensator. Figure 5. Schematic diagram of the compensator. The cylinder was divided into two chambers by the piston, the gas chamber, and the The cylinder was divided into two chambers by the piston, the gas chamber, and the oil chamber. The gas chamber was open to the atmosphere. Thus, the pressure Pa in the gas oilchamber chamber. was The always gas chamber the constant was atmosphericopen to the atmosphere. pressure. In theThus, static the equilibrium pressure P condition,a in the gasthe chamber piston was always at the middle the constant of its atmospheric stroke. Pi0 and pressure.Vc0 are In the the initial static pressureequilibrium and con- initial dition,volume the ofpiston the gas was corresponding at the middle to of the its staticstroke. equilibrium Pi0 and Vc0 position, are the initial respectively. pressure and initial volume of the gas corresponding to the static equilibrium position, respectively. 2.3.1. Stiffness of the Compensator 2.3.1. StiffnessWhen the of the piston Compensator is at the static equilibrium position, unequal pressures on two sides ofWhen the piston the piston create is a forceat the thatstatic balances equilibrium the inertial position, force unequal of the liftedpressures structure. on two Let sidesFp be of the forcepiston acting create on a force the rod, that and balancesFp = (m thej + inertialmp)g. force of the lifted structure. Let Fp be Under equilibrium condition, Pi0 is given as below: the force acting on the rod, and Fp = (mj + mp)g. Under equilibrium condition, Pi 0 is given as below:0 Fp + Pa A P = d (2) i0 A d ， Fp + Pa A π 2 2 0 π 2 d where A = (D − D ),A = DP.i0 = (2) d 4 p r d 4 p A Under dynamic conditions, the piston of thed hydraulic cylinder performs a relative axial motion induced by wave, and the resulting gas volume increased or decreased. π 2 2 ， π 2 Therefore,Ad = the(D gasp − pressureDr ) A was= aD functionp of the piston position. 4 d 4 where The vessel motion was, in a short. period of 8–10 s. Thus, there was insufficient time forUnder the heat dynamic exchange conditions, between the gaspiston of theof the compensator hydraulic andcylinder the atmosphere. performs a Therefore,relative axialthe motion thermodynamics induced by of wave, the compression and the resulting and expansion gas volume of theincreased gas can or be decreased. modeled as Therefore,an adiabatic the gas process pressure [8]. was Defining a function the displacementof the piston position. of the piston relative to the static x equilibriumThe vessel position motion aswas. in The a short following period equation of 8–10 is s. valid: Thus, there was insufficient time for the heat exchange between the gas of then compensatorn and the atmosphere. Therefore, Pi(x)V (x) = Pi V (3) the thermodynamics of the compression andc expansion0 c0 of the gas can be modeled as an adiabatic process [8]. Defining the displacement of the piston relative to the static equilib- where, n is adiabatic coefficient (n = 1.4 for nitrogen); Pi(x) and Vc(x) are the instantaneous riumpressure position and as the x. The instantaneous following equation volume ofis thevalid: gas, respectively, when the piston position is at x. n n Thus, Pi(x) is given by: Pi (x)Vc (x)= Pi0Vc0 n (3) Vc0 Pi(x) = Pi0 (4) where, n is adiabatic coefficient (n = 1.4 for nitrogen);Vc( xP)i(x) and Vc(x) are the instantaneous

pressureThe and gas the volume instantaneous equals tovolumeVc(x) of= Vthec0 gas,+ x Arespectively,d. Substituting whenVc the(x) =pistonVc0 + positionx Ad into is atEquation x. (4) yields: n Thus, Pi(x) is given by: Vc0 Pi(x) = Pi0 (5) Vc0 + xAd

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The total force Fi provided by the compensator due to pressure differential across the piston can be written as follows:

F = (P (x) − P )A0 i i a d (6)

Substituting Pi(x) into Equation (6) gives Fi as

V n F = A0 P c0 − A0 P i d i0 d a (7) Vc0 + xAd

The stiffness can be derived by calculating the partial derivative of Fi with respect to the piston position x, as shown in Equation (8).

∂F k = − i (8) h ∂x Which can be expressed as:

Vn k = nP A0 A c0 (9) h i0 d d n+1 (Vc0 + xAd)

From Equation (9), it is seen that kh is a nonlinear function of Pi0, Vc0 and Ad for a given piston position x. In addition, the value of Ad is dependent on Dp and Dr. Therefore, the compensator stiffness kh can be optimized by choosing the value of the initial gas pressure, the initial gas volume, the rod diameter, and the piston diameter reasonably. The compensator stiffness for different values of Vc0 is shown in Figure6 when the piston was 500 mm in diameter, and the rod was 280 mm in diameter. Figure6 indicates that the compensator stiffness decreased with increasing Vc0 for a given value of x. Moreover, the larger the gas volume was, the smaller the stiffness changes with the piston displacement. This was due to the fact that the changes in gas volume resulting from piston motion were equal. Moreover, when the gas volume was larger, the resulting J. Mar. Sci. Eng. 2021, 9, 714 8 of 18 pressure change was smaller due to the relatively smaller change rate of gas volume, thus the stiffness change was smaller.

FigureFigure 6. 6.EffectEffect of ofGas Gas Volume Volume on on Stiffness. Stiffness.

AsAs observed observed in inFigure Figure 6,6 the, the compensa compensatortor stiffness stiffness increases increases with with decreasing decreasing x xforfor a a 3 constant V . In addition, it is highly nonlinear with respect to x. In the case of V = 5 m3 , constant Vc0. cIn0 addition, it is highly nonlinear with respect to x. In the case of Vc0c =0 5 m , × 5 × 5 × 5 − thethe stiffness stiffness values values were were 1.76 1.76 × 10105, 1.49, 1.49 × 10510 andand 1.27 1.27 × 105,10 respectively,, respectively, when when x = x−=2.5, 2.5,0 and0 and 2.5. 2.5.The The maximum maximum stiffness stiffness change change rate rate was was 38.6% 38.6% compared compared to tominimum minimum stiffness stiffness when the piston moves from the top dead to the bottom dead. Clearly, the compensator stiffness changed significantly during the motion of the cylinder piston. Therefore, the assumption was invalid for the studied problem in this paper that the compensator stiffness was constant due to small gas volume change. As a result, it was more accurate to use nonlinear stiffness to predict the compensator’s dynamic response. As previously mentioned, Ad is another key parameter affecting the stiffness in view

of the compensator design. Equation (9) suggests that as Ad decreases, the stiffness was

reduced. As a result, we can obtain different stiffness by changing the values of Dp and

Dr, since Ad is a function of Dp and Dr.

Figure 7 shows the compensator stiffness for different values of Dp when the piston

moves from bottom dead to top dead. As Dp increases, the compensator stiffness increased

rapidly. When the piston was at the mid-stroke, namely x = 0, an increase of Dp by 15%

will increase the value of kh by nearly 100% at maximum. Therefore, the piston diameter had more influence on the compensator stiffness than gas volume.

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when the piston moves from the top dead to the bottom dead. Clearly, the compensator stiffness changed signiﬁcantly during the motion of the cylinder piston. Therefore, the assumption was invalid for the studied problem in this paper that the compensator stiffness was constant due to small gas volume change. As a result, it was more accurate to use nonlinear stiffness to predict the compensator’s dynamic response. As previously mentioned, Ad is another key parameter affecting the stiffness in view of the compensator design. Equation (9) suggests that as Ad decreases, the stiffness was reduced. As a result, we can obtain different stiffness by changing the values of Dp and Dr, since Ad is a function of Dp and Dr. Figure7 shows the compensator stiffness for different values of Dp when the piston moves from bottom dead to top dead. As Dp increases, the compensator stiffness increased J. Mar. Sci. Eng. 2021, 9, 714 rapidly. When the piston was at the mid-stroke, namely x = 0, an increase of Dp 9by of 18 15%

will increase the value of kh by nearly 100% at maximum. Therefore, the piston diameter had more inﬂuence on the compensator stiffness than gas volume.

105 10 D =0.5m 9 p D =0.625m p D =0.75m 8 p

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x(m) 3 FigureFigure 7. Effect 7. Effect of piston of piston diameter diameter on onstiffness stiffness (Vc (0 V=c 40 m=3 4, mDr =, D0.28r = 0.28m). m).

FromFrom Figures Figures 6 and6 and 7,7 it, itcan can be be observed observed that that the the compensator compensator stiffness stiffness depends depends on on D V thethe values values of D ofp andp and Vc0. Thus,c0. Thus, they must they mustbe carefully be carefully chosen, chosen, or else orthe else compensator the compensator may notmay work. not work. 2.3.2. Optimization of the Gas Volume 2.3.2. Optimization of the Gas Volume The proposed compensator was modeled as a parallel spring-damper system in this The proposed compensator was modeled as a parallel spring-damper system in this paper. According to vibration theory, vibration isolation does not occur until the frequency paper. According to vibration theory, vibration isolation does not occur until the fre- ratio satisﬁes the following inequality. quency ratio satisfies the following inequality. ω ω ≥ µ (10) ωn≥ μ ω (10) n where ω√is the frequency of the vessel motion, ω is the natural frequency of the compensator, whereand ωµ =is the2. frequency of the vessel motion, ω is the natural frequency of the compensator, andThe µ natural = √2. frequency of the compensator can be calculated as follows: The natural frequency of the compensator scan be calculated as follows: kh ωn = (11) ω = khmv n (11) mv where mv = mj + mp. where mv = mj + mp.

From Equations (10) and (11), the compensator stiffness must satisfy Equation (12) thus as to mitigate the transmission of heave movement from the vessel to the lifted structure. π 2 4 mv k ≤ (12) h μ2T 2

where T is the period of the wave-induced vessel motion.

Replacing kh with its expression in Equation (3), yields:

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From Equations (10) and (11), the compensator stiffness must satisfy Equation (12) thus as to mitigate the transmission of heave movement from the vessel to the lifted structure.

4π2m k ≤ v (12) h µ2T2

where T is the period of the wave-induced vessel motion. Replacing kh with its expression in Equation (3), yields: r 1 1 β V ≥ + − (13) c0 2α 4α2 α

2 4π mv where α = 2 2 , β = (n + 1)xAd. nAdµ T (mv g+P2 Ad) Equation (9) illustrates that the compensator stiffness is a function of the piston position x. The value of the compensator stiffness is maximum when x = −H. Therefore, if the compensator stiffness of x = −H satisﬁes Equation (13), the compensator stiffness of other positions will satisfy Equation (13). Setting x = −H in Equation (13), yields: r 1 1 β− V ≥ + − H (14) c0 2α 4α2 α

where β−H= −(n + 1)HAd. Equation (14) gives lower bound of the Vc0. The compensator acts as a hydro-pneumatic spring. When the piston oscillates around its equilibrium position, the spring force exerted by the compensator changes dynamically. The range of the spring force variation was very important, since the spring force must balance the external load of the compensator. However, the cylinder stroke was limited. This means that the spring forces in the top dead and the bottom dead must be enough to balance the external load. Otherwise, the piston will hit the top end cover or the bottom end cover. It is very dangerous since the compensator is impulsively loaded when the piston hits the end cover. The resulting rapid increase in the strand tension is very dangerous to safety. In the normal operation of the compensator, the extreme load applied to the compensator is assumed to be Fp ± ξFp, where ξ is speciﬁed as a fraction of Fp. Therefore, the force exerted by the gas is required to be in the range of [Fp − ξFp,Fp + ξFp] in order to avoid hitting the end cover of the cylinder. Let [vd,vu] be the feasible value range of the initial gas volume vc0. if vc0 > vu, the increasing force induced by the compression of the gas from x = 0 to x = −H is less than ξFp. As a result, the piston will hit the bottom end cover of the cylinder. Similarly, if vc0 < vd, the decreasing force induced by the expansion of the gas from x = 0 to x = H is more than ξFp. As a result, the piston will hit the top end cover of the cylinder. Based on the above discussion, the following inequality must hold to avoid hitting the end cover of the cylinder.

0 P (−H)A ≥ mvg(1 + ξ) + Pa A i d d (15) P (H)A ≤ m g( − ξ) + P A0 i d v 1 a d

Solving Equation (15), yields:

ϕHAd Vc0 ≤ ϕ−1 (16) ϕHAd Vc0 ≤ 1−φ

1 1 ( + )+ n ( − )+ n where, ϕ = mv g 1 ξ Pa Ad , φ = mv g 1 ξ Pa Ad . Pi0 Ad Pi0 Ad J. Mar. Sci. Eng. 2021, 9, 714 11 of 18

J. Mar. Sci. Eng. 2021, 9, 714 10 of 16 In the view of the compensator design, the initial gas volume must be carefully chosen to satisfy Equations (13) and (16) thus that the desired stiffness is obtained, as well as avoidingIn the hitting view of thethe compensator end cover. design, Figure the initial 8 shows gas volume the mustfeasible be carefully region chosen for Vc0 and Dp. Values into the satisfy red Equations region are (13) feasible and (16) thus choices that the for desired the initial stiffness gas is obtained, volume as and well asthe piston diameter, avoiding hitting the end cover. Figure8 shows the feasible region for Vc0 and Dp. Values in whichthe red satisfies region are Equations feasible choices (7) for and the initial(10). gas volume and the piston diameter, which satisﬁes Equations (7) and (10).

FigureFigure 8. 8. Feasible region region for V forc0 and VcD0 pand. Dp.

Figure8 clearly indicates that if the smaller Dp is chosen, the desired stiffness will be achievedFigure using8 clearly less gas indicates volume. Otherwise, that if the more smaller gas volume Dp is is chosen, needed to the obtain desired the stiffness will be achievedidentical stiffnessusing less for the gas case volume. of the larger Otherwise,Dp. The size more of the gas compensator volume willis needed become to obtain the iden- smaller by decreasing the piston diameter Dp. However, decreasing the piston diameter Dp ticalresults stiffness in increasing for the the case pressure of the in the larger compensator, Dp. The which size raises of the the compensator requirement to the will become smaller strength of the cylinder. Therefore, there is a tradeoff process between the gas volume and by decreasing the piston diameter Dp. However, decreasing the piston diameter Dp results the piston diameter for an appropriate value of the stiffness. in increasingFigure9 shows the numerical pressure simulation in the results compensa of spring forcetor, versuswhich piston raises displacement the requirement to the strengthcurves for of four the different cylinder. values Therefore, of Vc0. The followingthere is parametersa tradeoff are process used in the between numerical the gas volume and thesimulations: piston diameterDp = 0.5 m, forD ran= 0.28appr mopriate and H = 2.5value m; m ofl = the 400 stiffness. t; ma = 0.15 ml; mj = 1 t; ms = 7.14 t; ξ = 0.075. The two horizontal pink dashed lines represent the upper bound andFigure the lower 9 bound shows of thenumerical external force simulation applied on theresults cylinder of pistonspring rod, force respectively. versus piston displace- 6 mentThe fourcurves curves for intersect four different at (0, 4.60 × values10 ), which of V isc0 the. The static following spring force parameters exerted by the are used in the nu- compensator. Since the initial gas volumes are different, there are significant differences in merical simulations: Dp = 0.5 m, Dr = 0.28 m and H = 2.5 m; ml = 400 t; ma = 0.15 m ; mj = 1 t; the range of the spring forces. Table1 lists the spring forces when the piston is at x = 2.5 and l 3 msx == 7.14−2.5. t; In ξ the = case0.075. of V Thec0 = 10two m ,horizontal the compensator pink stiffness dashed is too lines softer. represent As a result, the upper bound and piston will hit the end cover of the cylinder, for which Equations (13) and (15) do not hold. the lower bound of 3the external force applied on the cylinder piston rod, respectively. The In the case of Vc0 = 6 m , the piston will hit the top end cover of the cylinder since the spring 6 3 fourforce curvesFi at x = intersect 2.5 was less at than (0, the 4.60 force × required.10 ), which In the is case the of Vstaticc0 = 4.5 spring m , although force the exerted by the compensator.piston will Since not hit the the endinitial cover gas of thevolumes cylinder, are the compensationdifferent, there efficiency are significant was limited differences in the rangesince theof the initial spring gas volume forces. does Table not satisfy 1 lists Equation the spring (13). Vc 0forces= 5.8 is thewhen only the case piston that is at x = 2.5 and satisfies Equations (13) and (15). The piston will not hit the end cover of the cylinder. c0 3 6 6 x =Meanwhile, −2.5. In the maximum case of andV minimum= 10 m , springthe compensator force is Fp = [4.25 stiffness× 10 , 4.94 is× too10 ]. softer. As a result, the piston will hit the end cover of the cylinder, for which Equations (13) and (15) do not hold. 3 In the case of Vc0 = 6 m , the piston will hit the top end cover of the cylinder since the 3 spring force Fi at x = 2.5 was less than the force required. In the case of Vc0 = 4.5 m , although the piston will not hit the end cover of the cylinder, the compensation efficiency was limited since the initial gas volume does not satisfy Equation (13). Vc0 = 5.8 is the only case that satisfies Equations (13) and (15). The piston will not hit the end cover of the cylinder. 6 6 Meanwhile, the maximum and minimum spring force is Fp = [4.25 × 10 , 4.94 × 10 ].

J. Mar. Sci. Eng. 2021, 9, 714 12 of 18

Table 1. External loads and Spring forces at x = 2.5 and x = −2.5.

Vc0 = 4.5 Vc0 = 5.8 Vc0 = 6 Vc0 = 10 J. Mar.Spring Sci. Eng. force2021 at, 9 ,x 714 = −2.5 (N) 5.14 × 106 5.06 × 106 4.99 × 106 4.83 × 106 11 of 16 Spring force at x = 2.5 (N) 4.15 × 106 4.21 × 106 4.27 × 106 4.39 × 106

FigureFigure 9. Force-displacement 9. Force-displacement relationship relationship for for vari variousous values values of ofthe the initial initial gas gas volume. volume.

− 2.4.Table Dynamic 1. External Model loads of the and Strand Spring Jack forces with at the x = Passive 2.5 and Compensator x = 2.5.

2.4.1. Damping Model of the CompensatorVc0 = 4.5 Vc0 = 5.8 Vc0 = 6 Vc0 = 10 DampingSpring force forces at x = of−2.5 the (N) compensator5.14 × 10originate6 5.06 from× 10 two6 main4.99 sources:× 106 mechanical4.83 × 106 friction,Spring such force as friction at x = 2.5 due (N) to relative4.15 motion× 106 between4.21 × the106 piston4.27 and× 10the6 cylinder;4.39 × fluid106 resistance in the pipeline between the cylinder and the accumulator. Since the friction force2.4. between Dynamic the Model piston of the and Strand the cylinder Jack with was the relatively Passive Compensator small compared with the external load2.4.1. of the Damping compensator, Model it of was the Compensatornot taken into account in this paper. We will only consider the resistive effect along the pipeline. Damping forces of the compensator originate from two main sources: mechanical Following the Hagen–Poiseuille equation, the pressure drop along the pipeline can friction, such as friction due to relative motion between the piston and the cylinder; ﬂuid beresistance calculated in by: the pipeline between the cylinder and the accumulator. Since the friction force between the piston and the cylinder was relativelyμ small compared with the external load − = 32 Ltu of the compensator, it was not takenPi intoPc account2 in this paper. We will only consider(17) the resistive effect along the pipeline. Dt Following the Hagen–Poiseuille equation, the pressure drop along the pipeline can be where, P and P represents the instantaneous pressure of oil pressure in the hydraulic calculatedi by: c cylinder and the accumulator, respectively; Lt is32 theµL lengthu of the pipeline; µ is the dy- P − P = t namic viscosity of hydraulic oil; u is thei averagec flow2 rate of hydraulic oil through the(17) Dt pipeline; Dt is the diameter of the pipeline. where, Pi and Pc represents the instantaneous pressure of oil pressure in the hydraulic Thus, the damping force Fc in the pipeline is: cylinder and the accumulator, respectively; Lt is the length of the pipeline; µ is the dynamic viscosity of hydraulic oil; u is the average ﬂow rateμ of hydraulic oil through the pipeline; = − = 32 Ltu D is the diameter of the pipeline.Fc Ad (Pi Pc ) Ad (18) t D2 Thus, the damping force Fc in the pipeline is: t

The average flow rate of hydraulic oil through 32pipelineµL u is given by: = ( − ) = t Fc Ad Pi Pc 2 Ad (18) Dt The average ﬂow rate of hydraulic oil through pipeline is given by:

. . 4Ad(y − y ) = b j u 2 (19) πDt J. Mar. Sci. Eng. 2021, 9, 714 12 of 16

Substituting Equation (19) into Equation (18), the damping force Fc can be expressed as:

128µL . . = t 2( − ) Fc 4 Ad yb yj (20) πDt . . Differentiating Fc with respect to yb − yj, the damping coefficient ch can be expressed as:

dF 128µL = c = t 2 ch . . 4 Ad (21) d(yb − yj) πDt

2.4.2. Strand Model A simple model of strands was used in this paper, which only considers the strand’s equivalent stiffness neglecting the damping force and the hydrodynamic drag force of strands. This model is valid as the velocity of the lifted structure is small. The tension in one strand can be calculated as follows: δ T = Asσ = AsE (22) Ls

where E is the Young’s modulus of strand, As is the intersectional area of strand, Ls is the strand length, and δ is the strand extension. Then, the stiffness of one strand following from Hook’s law can be written as follows:

T AsE ks1 = = (23) δ Ls There are q strands used in the strand jack, and the geometric and mechanical parameters of all strands are assumed to be identical. Thus, they can be modeled as q springs in parallel with common stiffness. The resulting stiffness can be expressed as:

AsE ks = q (24) Ls It can be seen from Equation (24) that the strand properties, length, and quantity dominate the stiffness of the strands.

2.4.3. Hydrodynamic Damping Coefficient Excited by vessel motion, the lifted structure vibrates in water, which is a viscous fluid. Thus, the lifted structure experiences reacting force, viscous damping force due to the surrounding fluid. A convenient approximation that can be used to estimate the viscous damping force for the submerged lifted structure can be derived based on Morison’s equation. It can be expressed as:

1 F = − ρ|v |v C A (25) d 2 l l d l

where ρ is the density of seawater, vl is the velocity of the lifted structure in water, Al is the lifted structure’s nominal cross-sectional area in the vertical direction, Cd is the drag coefﬁcient. The hydrodynamic damping coefﬁcient cp is given by:

Fd 1 1 . cp = − = ρ|vl|Cd Al = ρ yp Cd Al (26) vl 2 2

3. Numerical Simulation To evaluate the compensator’s effectiveness and the inﬂuence of key parameters on the efﬁciency of the compensator, the numerical simulations were carried out. J. Mar. Sci. Eng. 2021, 9, 714 13 of 16

3.1. Numerical Setting In the simulations, the vessel motion was given by Equation (27).

2π y = A sin t (27) b T where, A = 2.1 m and T = 6 s denote the amplitude and frequency, respectively. The shape of the lifted structure was simplified as a cuboid with the dimensions of 3 5 × 1.2 × 0.5 m . The damping coefficient Cd equaled 0.95 according to DNV-RP-H103 [16]. The parameters of the strand and simulation used in the simulations are listed in Tables2 and3.

Table 2. Speciﬁcations of the strand.

Diameter (mm) Cross-Section Area (mm2) Young’s Modulus (GPa) Mass per Unit Length (kg/m) 15.2 139 195 1.19

Table 3. Simulation parameters.

List of Items Value

Mj, ton 1

Ml, ton 400 L, m 300 ρ, g/cm3 1.02

3.2. Numerical Results To illustrate the effectiveness of the proposed compensator, numerical simulations were performed for 2 cases, the compensator on and the compensator off. Speciﬁcations of the compensator used in the simulations are summarized in Table4.

Table 4. Speciﬁcations of the compensator.

List of Items Value 3 Vd0, m 5.8

dp, mm 500

dr, mm 280

dt, mm 50

lt, m 2

Figure 10 depicts the simulation results of the lifted structure motion and the dynamic tension in the strand. It is seen from Figure 10 that there was a signiﬁcant reduction in the displacement yj and yp and dynamic tension Tr when the compensator was used. This illustrates the effectiveness of the compensator. Moreover, it was noted that the displacement amplitude of yp was smaller than the displacement amplitude of yj in both cases. This indicates that the displacement of the lifted structure decreased due to the strand elasticity. J. Mar. Sci. Eng. 2021, 9, 714 15 of 18

Table 4. Specifications of the compensator.

List of Items Value Vd0, m3 5.8 dp, mm 500 dr, mm 280 dt, mm 50 lt, m 2

Figure 10 depicts the simulation results of the lifted structure motion and the dynamic tension in the strand. It is seen from Figure 10 that there was a significant reduction in the displacement yj and yp and dynamic tension Tr when the compensator was used. This illustrates the effectiveness of the compensator. Moreover, it was noted that the displacement amplitude of yp was smaller than the displacement amplitude of yj in both cases. This indicates that the displacement of the lifted structure decreased due to the J. Mar. Sci. Eng. 2021, 9, 714 14 of 16 strand elasticity.

(a) Numerical results of lifting structure motion.

(b) Numerical results of tension.

Figure 10. Numerical results. Figure 10. Numerical results. The compensator had a nonlinear damping feature due to the flow characteristic of the pipeline. The influences of damping characteristics on the compensator efficiency and dynamic tension were studied for the various diameters of the pipeline, as shown in Figure 10. In Figure 11, responses were plotted for 3 different diameters dt of pipeline. As the pipeline diameter increased, the corresponding amplitude values of yp and Tr decreased. This indicated that the smaller damping values were better since the damping value decreased and the compensation efficiency increased with increasing the diameter of the pipeline. The reason for increased compensation efficiency was that the smaller damping force produced a smaller time delay to the compensator. J. Mar. Sci. Eng. 2021, 9, 714 16 of 18

The compensator had a nonlinear damping feature due to the flow characteristic of the pipeline. The influences of damping characteristics on the compensator efficiency and dynamic tension were studied for the various diameters of the pipeline, as shown in Fig- ure 10. In Figure 11, responses were plotted for 3 different diameters dt of pipeline. As the

pipeline diameter increased, the corresponding amplitude values of yp and Tr decreased. This indicated that the smaller damping values were better since the damping value decreased and the compensation efficiency increased with increasing the diameter of the pipeline. The reason for increased compensation efficiency was that the smaller damping J. Mar. Sci. Eng. 2021, 9, 714 15 of 16 force produced a smaller time delay to the compensator.

(a)

Motion response of compensator with different pipe diameters

(b)

Tension response of compensator with different pipe diameters

Figure 11. Responses of the compensator with different dt. Figure 11. Responses of the compensator with different dt. 4. Conclusions This paper presented a nonlinear dynamic model of the passive compensator, incor- porating nonlinearity of the compensator stiffness. The comparative analysis is carried out to analyze the inﬂuence of key design parameters on the dynamic response of the compensator. The results indicate that it is more accurate to use nonlinear stiffness to predict the compensator’s dynamic response. Moreover, it is established that the piston diameter has more inﬂuence on the compensator stiffness than the gas volume via numer- J. Mar. Sci. Eng. 2021, 9, 714 16 of 16

ical simulations. The heave compensation performance of the strand jack lifting system employing a passive compensator is evaluated and compared to different conditions. It is concluded that the lifted structure displacement and the strand dynamic tension can be improved considerably using the passive compensator.

Author Contributions: Conceptualization, Y.Z., J.X., B.Y. and S.W.; methodology, Y.Z., J.X., B.Y. and S.W.; software, B.Y. and S.W.; validation, J.X., B.Y.; formal analysis, Y.Z., B.Y. and S.W.; investigation, Y.Z.; resources, Y.Z. and J.X.; data curation, S.W.; writing—original draft preparation, Y.Z.; writing— review and editing, Y.Z. and B.Y.; visualization, Y.Z. and S.W.; supervision, Y.Z. and J.X.; project administration, Y.Z.; funding acquisition, Y.Z. and J.X. All authors have read and agreed to the published version of the manuscript. Funding: The authors gratefully acknowledge the financial support from The Natural Science Foundation of Heilongjiang Province, China (no. E2018022), Information Technology of the People’s Republic of China-Floating Security Platform Project (the second stage), and the Opening Fund of Acoustics Science and Technology Laboratory (grant no. SSKF2020009). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data that support the findings of this study are available from the corresponding author upon reasonable request. Acknowledgments: The authors would like to thank the members of the Marine Electromechanical Systems Research Institute for their continued support and discussion. Conflicts of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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