Computational Model for Simulation of Lifeboat Free-Fall During Its Launching from Ship in Rough Seas

Journal of Marine Science and Engineering

Article Computational Model for Simulation of Lifeboat Free-Fall during Its Launching from Ship in Rough Seas

Shaoyang Qiu *, Hongxiang Ren * and Haijiang Li

Key Laboratory of Marine Dynamic Simulation and Control for Ministry of Communications, Dalian Maritime University, Dalian 116026, China; [email protected] * Correspondence: [email protected] (S.Q.); [email protected] (H.R.); Tel.: +86-1804-115-4892 (S.Q.); +86-1894-092-9166 (H.R.) Received: 17 June 2020; Accepted: 12 August 2020; Published: 20 August 2020

Abstract: In order to improve the accuracy of the freefall of lifeboat motion simulation in a ship life-saving simulation training system, a mathematical model using the strip theory and Kane’s method is established for the freefall of the lifeboat into the water from a ship. With the boat moving on a skid, the model of the ship’s maneuvering mathematical group (MMG) is used to model the motion of the ship in the waves. Based on the formula of elasticity and friction theory, the forces of the skid acting on the boat are calculated. When the boat enters the water, according to the analytical solution theory of slamming, the slamming force of water entry is solved. The simulation experiments are carried out by the established model. The results of the numerical simulation are compared with the calculation results of the hydrodynamics software Star CCM+ at water entry under initial condition A in the paper. The position and velocity of the center of gravity of the boat, the angle, and velocity and acceleration of pitch calculated by the two methods are in good agreement. There is a little difference between the values of translation acceleration calculated by the two methods, which is acceptable. This shows that our numerical algorithm has good accuracy. A qualitative analysis is performed to find the safe point of water entry under the condition of different wave heights and two situations of a ship encountering waves. Finally, the model is applied to the ship life-saving training system. The model can meet the system requirements and improve the accuracy of the simulation.

Keywords: lifeboat; freefall; ship motion; Kane’s method

1. Introduction

1.1. Motivation The lifeboat is the main life-saving equipment onboard ships. When a shipwreck accident occurs, the crew onboard can quickly escape from the ship by the lifeboat. During the freefall of the lifeboat, the hull of the boat usually slams into the water at high speed, with huge instantaneous impact pressure. If the crews make a mistake when launching the lifeboat, it can cause serious damage to the hull structure and threaten the personal safety of the crew [1]. According to the International Convention on Standards of Training, Certiﬁcation and Watchkeeping for Seafarers, 1978, as amended in 1995 (STCW 78/95), the crew must be trained and pass a lifeboat assessment before boarding [2]. Training is limited by factors such as time and costs. In recent years, virtual reality technology has developed rapidly, and has been used in training for marine life-saving [3,4]. In order to improve the immersion experience and the reality of the ship’s life-saving training system, a mathematical model is established for the motion of the lifeboat’s freefall during its launch from the ship.

J. Mar. Sci. Eng. 2020, 8, 631; doi:10.3390/jmse8090631 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 631 2 of 25

1.2. Related Work Re et al. [5–7] conducted a series of model experiments of a boat launched from a fixed platform. The main focus of the experimental evaluation was the performance of the boat in a range of different weather conditions. Hollyhead et al. [8] and Hwang et al. [9] performed experiments on launching boats from moving ships. They analyzed the motion parameters and the load of the boat. Due to the limitation of experimental conditions, they could only analyze parameters that affected the water entry motion of the boat and did their experiments under good environmental conditions. In addition to the model experiment, some scholars used a computational method to analyze the freefall of the boat, which was mainly divided into two aspects. First, computational fluid dynamics (CFD) technology was used to numerically simulate the water entry of the lifeboat, and the slamming load of the hull was analyzed by the simulated results [10–12]. The second method was to establish a mathematical model for the freefall of the lifeboat for predicting the lifeboat motion attitude, and analyzing the risk of injury to the crew [13–15]. The mathematical model of the water entry of the lifeboat’s freefall started from Karman’s momentum theorem, solving the problem of water impact [16]. Boef [17,18] applied Karman’s theory to the modeling of the lifeboat’s motion, and divided the force of the lifeboat entering the water into gravity, buoyancy, drag, and slamming force. Arai et al. [19,20] simplified Boef’s model. They divided the lifeboat’s motion into four stages, sliding phase, rotation phase, freefall phase, and water entry phase, and analyzed the local acceleration of the bow, midship, and stern. The calculation results of local acceleration were close to the model experiment data. Khondoker et al. [21–23] applied Arai’s model to analyze the parameters that affect the boat’s water entry. Karim et al. [24,25] applied Arai’s model and took the effect of regular waves into account when calculating speed of the boat entering the water. Raman-Nair and White [26] used multibody dynamics to analyze the entry of the boat from an offshore platform with simple movement. The rotation phase at skid exit is automatically modeled in this way. They regarded the skid as a slope, and the lifeboat as a cube on the slope and as a cylinder at water entry. When calculating the slamming force, they added the item of incident wave force in addition to the item of momentum theory, considering the effect of waves [27,28]. Dymarski and Dymarski [29] studied the model of a lifeboat released into the water from the stern of a ship. They also regarded the skid as a slope and the lifeboat as a cube on the slope, considering the reaction of the boat to the skid, but the motion of the ship was not considered. The detail of the algorithm was not given. In summary, the current mathematical models for the freefall of the lifeboat do not take motion of ships in waves into account, and the contact force between the boat and skid is not modeled according to their actual structure. The slamming force is calculated according to momentum theory. All algorithms are not compared with real boat experiments or fluid dynamics software.

1.3. Our Contributions This paper presents a computational model for the simulation of lifeboat freefall during its launching from a ship in rough seas. In order to consider the effect of the ship’s motion on lifeboat motion, the maneuvering mathematical group (MMG) model is used to simulate the motion of the ship in the waves. In order to improve the calculation accuracy, the contact force between boat and skid is calculated based on strip theory, according to the actual structure of the boat and skid. When the boat enters the water, the slamming force is solved based on the theory of energy. The added mass of the boat is calculated according to its size and depth of water entry. Finally, the equations of motion of a freefall lifeboat are formulated using Kane’s method. A series of simulation experiments are carried out by using the computational model established in this paper. The results of simulation experiments are compared with results of the fluid dynamics software Star CCM+. They show that our numerical algorithm has good accuracy. A qualitative analysis is performed to find a safe point of water entry under the condition of different wave heights and two situations of a ship encountering waves. J.J. Mar.Mar. Sci.Sci. Eng.Eng. 20202020,, 88,, 631x FOR PEER REVIEW 33 of 2525 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 3 of 25 Section 2 describes the mathematical model of the motion of boat on the skid. Section 3 Section 2 describes the mathematical model of the motion of boat on the skid. Section 3 describesSection the2 describes mathematical the mathematical model of boat model’s water of the entry. motion The of boat setup, on the results, skid. and Section analysi3 describess of the describes the mathematical model of boat’s water entry. The setup, results, and analysis of the thesimulation mathematical experiments model ofare boat’s presented water in entry. Section The 4 setup,. Section results, 5 describes and analysis an application of the simulation of the simulation experiments are presented in Section 4. Section 5 describes an application of the experimentsmathematical are model presented and Section in Section 6 gives4. Section conclusions.5 describes an application of the mathematical model mathematical model and Section 6 gives conclusions. and Section6 gives conclusions. 2. Motion of the Boat on the Skid 2. M Motionotion of the BoatBoat on the SkidSkid After the lifeboat is unhooked and released, the lifeboat slides down the skid away from the After the lifeboat is unhooked and released, the lifeboat slides down the skid away from the motherAfter ship the by lifeboat its own is gravity. unhooked When and the released, center of the gravity lifeboat of the slides boat down slides the out skid of the away lowest from point the mother ship by its own gravity. When the center of gravity of the boat slides out of the lowest point motherof the skid, ship the by itsboat own begins gravity. to rotate When owing the center to the of gravityvector of of the the gravity boat slides and out the of force the lowestof the skid point not of of the skid, the boat begins to rotate owing to the vector of the gravity and the force of the skid not theacting skid, on the the boat boat begins in a straight to rotate line. owing to the vector of the gravity and the force of the skid not acting onacting the boaton the in boat a straight in a straight line. line. 2.1. Coordinate System 2.1. Coordinate SystemSystem Figure 1 shows the central longitudinal section of the ship. There are three Cartesian coordinate Figure1 1 shows shows thethe centralcentral longitudinallongitudinal section section of of the the ship.ship. ThereThere areare threethree CartesianCartesian coordinatecoordinate systems: 표푥푦푧 is the inertial coordinate system, with unit vectors of the three axes 푁1, 푁2, 푁3 ; the systems:systems: 표푥푦푧oxyz is the the inertial inertial coordinate coordinate system, system, with with unit unit vectors vectors of ofthe the three three axes axes 푁1, 푁N21,,푁N32 ;, Nthe3; coordinate system of 표0푥0푦0푧0 is fixed on the ship; 표0 is located at the center of gravity of the ship, thecoordinate coordinate system system of 표0 of푥0푦o00푧x00 yis0z fixed0 is fixed on the on ship the; 표 ship;0 is locatedo0 is located at the center at the of center gravity of of gravity the ship, of the direction of 표0푥0 points to the bow, the direction of 표0푧0 points to the keel, and 푖0, 푗0, 푘0 are the thethe ship,direction the direction of 표0푥0 points of o0x0 topoints the bow, to the the bow, direction the direction of 표0푧0 of pointso0z0 points to the to keel, the keel,and and푖0, 푗0i,0푘,0j 0are, k0 arethe unit vectors of the axes. The coordinate system of 표1푥1푦1푧1 is fixed on the skid and 표1 is located at theunit unit vectors vectors of the of theaxes. axes. The Thecoordinate coordinate system system of 표1 of푥1o푦11x푧11y 1isz1 fixedis fixed on onthe theskid skid and and 표1 ois1 islocated located at the center of the upper end of the skid, with 표1푥1 pointing down along the skid and 표1푧1 pointing atthe the center center of ofthe the upper upper end end of ofthe the skid, skid, with with 표1o푥11x 1pointingpointing down down along along the the skid skid and and 표o11푧z11 pointingpointing up the perpendicular to the slope of the skid; 푖1, 푗1, 푘1 are the unit vectors of the three axes; 휙 is the up the perpendicular to to the the slope slope of of the the skid skid;; 푖i11,,푗j11,,푘k11 are the unit vectors of of the the three three axes; axes; 휙φ isis thethe angle between the plane of the skid and the horizontal plane. As shown in Figure 2, 퐺푏 is the center angle between the plane of the skid and the horizontal plane. plane. As As shown shown in in Figure Figure 2,2, 퐺G푏b isis thethe centercenter of gravity of the lifeboat, 표2푥2푦2푧2 is the coordinate system attached to the boat, 표2 is located at 퐺푏, of gravity of the the lifeboat, lifeboat, 표o2푥x2푦y2z푧2 isis thethe coordinatecoordinate system attached attached to to the the boat, boat, 표o22 is located at at G퐺푏b, and 푏1, 푏2, 푏3 are the unit vectors for the three axes. and b푏11,,b푏22,,b푏33are are the the unit unit vectors vectors for for the the three three axes. axes.

Z Z1 O1 1 X O1 X1  1 r  r O0  X0 O X 0  0  

Z0 Z0 X O X O

Z Z Figure 1. The three coordinate systems are the inertial coordinate system 표푥푦푧, the coordinate Figure 1. 1.The The three three coordinate coordinate systems systems are are the inertialthe inertial coordinate coordinate system systemoxyz, the표푥푦푧 coordinate, the coordinate system system 표0푥0푦0푧0 fixed on the ship, and the coordinate system 표1푥1푦1푧1 fixed on the skid. osystem0x0 y0z 0표ﬁxed0푥0푦0 on푧0 thefixed ship, on the and ship, the coordinate and the coordinate system o1 systemx1 y1z1 ﬁxed표1푥1푦 on1푧1 the fixed skid. on the skid.

FigureFigure 2.2. GeneralizedGeneralized coordinates q푞r ofof the the boat boat in in the the coordinate coordinate system system ﬁxedfixed onon thethe skidskid andand thethe 푞 푟 coordinateFigure 2. Generalized system attached coordinates to the boat 푟 of with the the boat unit in vectorsthe coordinateb (r = 1,system 2, 3). fixed on the skid and the coordinate system attached to the boat with the unit vectors r푏푟 (푟 = 1,2,3). coordinate system attached to the boat with the unit vectors 푏푟 (푟 = 1,2,3).

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2.2. Mathematical Model of Boat

The relative position of o0 and o1 remains unchanged in o0x0 y0z0. The vector o0o1 is represented by r and a is the angle of r with the horizontal line. Based on the theorem for composited velocity and acceleration, the velocity and acceleration of o1 are   v = v + r  o1 o0 ωo0  . × , (1)  ao = ao + ωo r + ωo (ωo r) 1 0 0 × 0 × 0 × where vo0 , ωo0 are the velocity and acceleration of the ship. When considering only the ship’s longitudinal motion, the ship’s pitch angle is θ ; u and w are components of the ship’s velocity in the direction of axis ox0 and oz0. The velocity and acceleration of o are 1  . .  vo = u θ r sina i0 + w + θ r cosa k0  1 − | | | |  . . .. . 2 . . .. . 2 . (2)  ao = u + wθ θ r sina + θ r cosa i + w uθ + θ r cosa + θ r sina k  1 − | | | | 0 − | | | | 0

The velocity and acceleration of Gb are

 . . . .  vG = vo + q + q2φ i1 + q q1φ k1  1 1 2 −  ...... 2 ...... 2 . (3)  a = ao + q + q φ + 2q φ q φ i + q q φ 2q φ q φ k  G 1 1 2 2 − 1 1 2 − 1 − 1 − 2 1 . . The angular velocity of the rigid body is irrelevant to the choice of the base point, so φ = θ. The angular velocity and acceleration of the boat are

 . .  =  ωb φ q3 N2  .. − .. . (4)  α = φ q N b − 3 2 The lifeboat is divided into n cross sections with equal thickness along the length of the boat. The coordinates of the center of each section Sk, (k = 1, ... ., n) are expressed as xsk , 0, 0 in the coordinate system of the boat, and lb is the length of boat. The thickness of each section is ts = lb/n ; the velocity at the center of each cross section is

. . . . . . . . vs = vo + (q + q φ + φxs sinq q xs sinq )i + q q φ φxs cosq + q xs cosq k . (5) k 1 1 2 k 3 − 3 k 3 1 2 − 1 − k 3 3 k 3 1 . Generalized coordinates are qr, generalized velocities are ur = qr(r = 1, 2, 3), partial velocities associated with points Gb, Sk and the angular velocity are written as follows [30]:

 ( )  i1 r = 1 r  v =  k1 (r = 2) , (6) G   0 (r = 3)

 ( )  i1 r = 1 r  v =  k1 (r = 2) , (7) Sk   x cosq k x sinq i (r = 3) Sk 3 1 − Sk 3 1  ( )  0 r = 1 b  ωr =  0 (r = 2) . (8)   N (r = 3) − 2 The force analysis of the boat at the skid is shown in Figure3. Forces acting on the boat are gravity G, supporting force Fn, and friction F f . J.J. Mar. Mar. Sci. Sci. Eng.Eng.2020 2020,,8 8,, 631x FOR PEER REVIEW 5 ofof 2525

Fn Fn

(a) (b)

FigureFigure 3.3.The The sideside viewview ((aa)) ofof forcesforces andand frontfront viewview ((bb)) of of supporting supporting forceforce actingacting onon thethe boatboat atat thethe skid.skid. TheThe lifeboatlifeboat isis locatedlocated onon thethe contactcontact rollersrollers onon bothboth sidessides of of thethe skid.skid. TheThe toptop pointpoint ofof thethe rollersrollers isis in in contact contact with with the the boat boat and and provides provides supporting supporting force force and and friction. friction.

TheThe generalizedgeneralized forceforce ofof gravitygravity and and inertia inertia are, are, respectively, respectively, 퐹푟 = 푣푟 ∙ 푚 𝑔푘, (9) 퐺 Fr =퐺 vr 푏m gk, (9) G G· b 푟 푟 푟 퐹∗ =r푣퐺 ∙ (r−푚푏푎퐺) + 휔 ∙r(−퐼2훼푏), (10) F = vG ( mbaG) + ω ( I2αb), (10) ∗ · − · − where 푚푏 is the mass of the boat, 𝑔 is the acceleration of gravity, and 퐼2 is the moment of inertia where mb is the mass of the boat, g is the acceleration of gravity, and I2 is the moment of inertia around around 표푦2. oy2. For the supporting force 퐹푛, this paper regards the contact roller as a spring with stiffness For the supporting force Fn, this paper regards the contact roller as a spring with stiffness coefficient 푘 and damping coefficient 푐 . The frictional contact force is modeled using a coefficient coefficient k and푟 damping coefficient c . The푟 frictional contact force is modeled using a coefficient of r r of 휇푟 . The coordinates of the center of each section 푆푘 (푘 = 1, . . . . , 푛) are (푞1 + 푥푠푘 푐표푠 푞3 , 0, 푞2 + µr . The coordinates of the center of each section Sk (k = 1, ... ., n) are q1 + xsk cosq3, 0, q2 + xsk sinq3 in 푥 푠푖푛 푞 ) in the boat coordinate system. We define them as follows: the푠푘 boat coordinate3 system. We define them as follows: 훼 = 푞 + 푥 푐표푠 푞 푠푘 1 푠푘 3 훽 α=s 푞= +q1푥+ x푠푖푛s cosq푞 3− 푧 { 푠푘 k 2 푠푘 k 3 푝. (11)  = +  βsk q2 xsk sinq3 zp . (11) 훾푠푘 = 1/2(|훽푠푘 | − 훽푠푘)  −  γsk = 1/2 βsk βsk The velocity of each section relative to its contacting− point is 푣 . The positions of each 푠푘/푠푘′ contactThe roller velocity are known; of each there section are relative푛 contact to its rollers contacting on each point side of is thevs / skid.s . The The positionscoordinates of of each the 푝 k k0 contacttop of the roller contact are known; roller on there each are sidenp ofcontact the axes rollers of 표 on1푥1 each and side 표1푧1 of at the the skid. skid The coordinate coordinates system of theare 푡 푡 top of the contact roller on each푠 side of the axes of 푠o1x1 and o1z1 at the skid coordinate system are 푥푝 , 푧푝(푖 = 1, . . . , 푛푝) as 푥푝 − 푐표푠 푞3 < 훼푠푘 < 푥푝 + 푐표푠 푞3 . The contacting force between 푆푘 and 𝑖 𝑖 ts 2 t𝑖s 2 xp , zp i = 1, ... , np as xp cosq < αs < xp + cosq . The contacting force between S and contact contacti rollers [31,32] is i − 2 3 k i 2 3 k rollers [31,32] is 퐹푠푘 = 2(푘푟 ∗ 훾푠 − 푐푟(푣푠 /푠 ∙ 푘1)sign(훾푠 ))푘1 − 2(휇푟푘푟훾푠 sign(v푠 /푠 ∙ 푖1))푖1 (12) 푘 푘 푘′ 푘 푘 푘 푘′ Fsk = 2 kr γsk cr vs /s k1 sign γsk k1 2 µrkrγsk sign vs /s i1 i1 (12) where ∗ − k k0 · − k k0 · where 푣 푘 = 푞̇ − 푞 휙̇ − 휙̇푥 푐표푠 푞 + 푞̇ 푥 푐표푠 푞 + 휙̇훼 푠푘/푠 ′ ∙ 1 2 . 1 . .푠푘 3 .3 푠푘 3 . 푠푘 {  푘 = + +  vs /s k1 q q1φ φXSk cosq3 q XSk cosq3 φαSk . (13)  k k0 2 ̇ . ̇ . 3 푣푠푘/푠 ′ ∙ 푖1 ·= 푞̇1 +. 푞−2휙 + 휙−푥푠푘 푠푖푛 푞3 − 푞̇3. 푥푠푘 푠푖푛 푞3 . (13)  푘 = +  vs /s i1 q1 q2φ φXSk sinq3 q3XSk sinq3 k k0 · − − The associated generalized force acting on the 푆푘 by the skid is The associated generalized force acting on the Sk by the skid is 푟 푟 퐹푠 = 푣푠 ∙ 퐹푠 (푟 = 1,2,3). (14) 푘 r 푘 r 푘 Fs = vs Fs (r = 1, 2, 3). (14) k k · k 푟 ∑ 푟 The combined generalized force of the skid that acted on the boat is 퐹푐 = 퐹푠푘 (푟 = 1,2,3). The P equation of the boat motion is Fr = Fr (r = ) The combined generalized force of the skid that acted on the boat is c sk 1, 2, 3 . The equation of the boat motion is 푟 푟 퐹 + 퐹∗ = 0(푟 = 1,2,3), (15) 푟 푟 푟 r r where 퐹 = 퐹푐 + 퐹퐺 . Finally, the fourthF -+orderF = Runge‒Kutta0 (r = 1, 2, 3) ,method is used to solve the differential (15) equation. ∗

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r = r + r where F Fc FG. Finally, the fourth-order Runge-Kutta method is used to solve the diﬀerential equation.

2.3. Mathematical Model of Ship Motion Based on the model of MMG [33], the coordinate system is as shown in Figure1. The ship dynamic equation is  .  ( + ) + = +  m mx u vr qw XH Xwave  . −  m + m v + ur pw = Y + Y  y H wave  . −  (m + mz) w + pv qu = ZH + Zwave  − , (16)  .  (Ixx + Jxx)p + Izz Iyy qr = KH + Kwave  . −  + + ( ) = +  Iyy Jyy q Ixx Izz pr MH Mwave  . −  (Izz + Jzz)r + Iyy Ixx pq = NH + Nwave − where the variables with subscript H are the forces and moments on the hull; the variables with subscript wave are wave forces. m is the mass of the ship; mx, my and mz are, respectively, added masses in the direction of the axes ox0, oy0and oz0 ; Ixx, Iyy and Izz are the rotational moment of inertia around axes ox0, oy0 and oz0; Jxx, Jyy and Jzz are the added moment of inertia around axes ox0, oy0, oz0 ; u, v, w are velocities in the direction of axes ox0, oy0and oz0; and p, q, r are the angular velocities around axes ox0, oy0, oz0. mz = m ; other added masses are calculated as follows:  !  c L L 2 dL  mx = 1 0.398 + 11.988c 1 + 3.73 d 2.89 b p 1 + 1.13 d + 0.175c p 1 + 0.541 d 1.107 p  m 100 b B B B b B B B2  − − , (17)  my Lp Lp Lp  = 0.882 0.54c 1 1.6 d 0.156(1 1.673c ) + 0.826 d (1 0.678 d ) 0.638 d (1 0.669 d ) m − b − B − − b B B B − B − B B − B where cb is the block coeﬃcient, d is the draft, B is the ship width, and Lp is the length between perpendiculars. Ixx, Iyy, Izz, Jxx, Jyy and Jzz are calculated as follows:   + = + B Lp  Ixx Jxx mB 0.3085 0.0227 d 0.0043 100  −  B 2  Iyy = Jyy = 0.83 0.25Lpcp m  2d , (18)  4.5 2 2.4  Izz = 1 + cb m + Lp + B /24   2 Lp  Jzz = 0.01m 33Lp 76.85c (1 0.784c ) + 3.43 (1 0.63c ) − b − b B − b where cp is the prismatic coeﬃcient. The kinematics equation is  .  x = µcosψcosθ + v(cosψsinθsinϕ sinψcosϕ) + w(cosϕsinθcosψ + sinϕsinψ)  . −  = + ( + ) + ( + )  y µsinψsinθ v sinψsinθsinϕ cosψcosϕ w cosϕsinθcosψ sinϕcosψ  .  z = µsinθ + vcosθsinϕ + wcosϕcosθ  . − (19)  ϕ = p + qsinϕ tan θ + rcosϕ tan θ  .   θ = qcosϕ rsinϕ  . −  qsinϕ rcosϕ ψ = cosθ + cosθ where x, y, z and ϕ, θ, ψ are displacements and Euler angles relative to the inertial coordinate system, respectively. J. Mar. Sci. Eng. 2020, 8, 631 7 of 25

The forces and moments on the hull are

 2 2 2  XH = Xuuu + Xvvv + Xvrvr + Xrrr   YH = Yvv + Yrr + Yr r r r + Yvrvr  | | |. |  .  ZH = Zww Zqq Zqq Zθθ  − − − − , (20)  KH = 2Kpp ∆ GM sinϕ YHzH  − − · . · −  .  MH = Mww Mww Mqq Mθθ  − − − −  NH = Nvv + Nrr + Nr r r r + Nvrvr + YHxc | | | |

. . where Xuu, Xvv, Xvr, Xrr, Yv, Yr, Yr r , Yvr, Zw, Zq, Zq, Zθ, Kp, Mw, Mw, Mq, Mθ, Nv, Nr, Nr r , Nvr, and Nvr | | | | are hydrodynamic derivatives, which are easily calculated according to the ship parameters [34]; ∆ is the ship displacement, GM is the metacentric height, zH is the coordinate of action point of YH in the direction of oz0, and xc is the distance between the center of gravity and the center. For the wave force, it is estimated, based on the Frude-Krenov assumption, that the hull is simpliﬁed to a box, and the six-degrees-of-freedom wave force and moment are as follows [34]:

 ( kBsin(χ) )  sin 2 kd L  Xwave = 2ρga e Bdsin(k cos(χ))sin(ωet)  kBsin(χ) − 2  2  kLcos(χ)  sin( )  Y = 2ρga 2 e kdLdsin(k B sin(χ))sin(ω t)  wave kBcos(χ) − 2 e  − 2  kBsin(χ) kLcos(χ)  sin( ) sin( )  Z = ρgak 2 e kdBLd 2 cos(ω t)  wave kBsin(χ) − kBcos(χ) e  2 2  kBsin(χ) kLcos(χ) (21)  sin( ) sin( )  K = ρgasin(χ) 2 e kdd2 2 sin(ω t)  wave kBsin(χ) − cos(χ) e  2   ( kBsin(χ) ) ( kLcos(χ) ) ( ) ( kLcos(χ) )  sin 2 kd 2sin 2 kLcos χ cos 2  Mwave = ρga ( ) e− d −2 2 sin(ωet)  ksin χ k cos (χ)  2  kBsin(χ) kLcos(χ) kLcos(χ)  sin( ) 2sin( ) kLcos(χ)cos( )  N = ga 2 e kdd 2 − 2 cos( t)  wave ρ ksin(χ) − k2cos2(χ) ωe 2 where a is the amplitude of the wave, k is the number of waves, ωe is the encounter frequency, χ is the encounter angle, ρ is the water density, and L is the ship waterline length. The motion parameters of the ship can be obtained by solving the diﬀerential Equations (16) and (19) using the fourth-order Runge-Kutta method.

3. Water Entry The geometric orientation relationship of the inertial and boat coordinate system is represented by β1, β2, β3. The transformation relationship between the two coordinate systems is as follows, where [NCb] is transformation matrix:      N1   b1    h i   N  = NCb  b   2   2 , (22)     N3 b3

 +   c2c3 s1s2c3 s3c1 c1s2c3 s3s1  hN bi  −  C =  c2s3 s1s2s3 + c1c3 c1s2s3 c3s1 , (23)  −   s s c c c  − 2 1 2 1 2 where si = sinβi, ci = cosβi, (i = 1, 2, 3). The generalized coordinates are   qb = β  i i  (i = 1, 2, 3). (24)  qb = OO→ b 3+i b · i J. Mar. Sci. Eng. 2020, 8, 631 8 of 25

The generalized velocities are   ub = ωb b  i i  · (i = 1, 2, 3), (25)  ub = vb b 3+i · i where ωb, vb are the velocity and angular velocity of the lifeboat, respectively. The relationships between generalized coordinates and generalized velocities are

 .  b = b + b + b  q1 u1 s2/c2 u2s1 u3c1   . b b b  q = u c1 u s1 , (26)  2 2 − 3  .  b = b + b  q3 u2s1 u3c1 /c2

 . b b b b b b  q4 = u u q + u q  4 − 2 6 3 5  . b  q = ub + ubqb ubqb . (27)  5 5 1 6 − 3 4  . b  q = ub ubqb + ubqb 6 6 − 1 5 2 4 The acceleration and angular acceleration of the boat are

. b αb = Nd ωb /dt = bd ωb /dt + ωb ωb = u b , (28) × i i . b . b . b ab = (u + ubub ubub)b + (u ubub + ubub)b + (u + ubub ubub)b . (29) 4 2 6 − 3 5 1 5 − 1 6 3 4 2 6 1 5 − 2 4 3 The partial angular velocities and velocities of the boat are ( b (r = 1, 2, 3) ωb = r , (30) r 0 (r = 4, 5, 6) ( b 0 (r = 1, 2, 3) vr = . (31) br 3 (r = 4, 5, 6) − The generalized inertial force is

b b b b F∗ = ω T∗ + v m a , (32) r r · r · − b  . b  [u I ubub(I I )]  − 1 1 − 2 3 2 − 3  . b  [u I ubub(I I )]  2 2 3 1 3 1  − − −  . b b b  [u I3 u u (I1 I2)] b  − 3 − 1 2 − Fr∗ =  . , (33)  ( b + b b b b)  mb u4 u2u6 u3u5  − −  . b b b b b  mb(u u u + u u )  − 5 − 1 6 3 4  . b  m (u + ubuB ubub) − b 6 1 5 − 2 4 where I1, I2, I3 is the moment of inertia around ox2, oy2, oz2. The generalized force caused by the gravity of the boat is

FG/b = m gN vb, (34) r b 3 · r J. Mar. Sci. Eng. 2020, 8, 631 9 of 25

  0    0   0 FW/b =  . (35) r   mb g(sin q2)  −  mb g(sinq1cosq2)  −  m g(sinq cosq ) − b 1 2 The generalized force caused by air drag is

FW/b = FW vb, (36) r b · r

W 1 b/W b/W F = ρwAwCw v v , (37) b −2 where vb/W is the vector difference between the lifeboat velocity and the wind velocity vb/W = vb vW , b/W − ρw is the air density, Aw is the projected area of the lifeboat in the plane perpendicular to v , and Cw is the coefficient of air drag. The generalized force caused by fluid drag is

Xn FD/b = FD/sk (r = 1, , 6), (38) r k=1 r ···

FD/sk = FD vsk , (39) r sk · r FD S vsk S S where sk is the ﬂuid drag acting on k and r is the partial velocity of k. The velocity of k is as follows: s s s sk = k + k + k v v1 b1 v2 b2 v3 b3, (40)

sk sk sk where v = u , v = u + xs u , v = u xs u . 1 4 2 5 k 3 3 6 − k 2 The partial velocities of Sk are   0 (r = 1)  s  xs b (r = 2) v k =  − k 3 . (41) r  x b (r = 3)  sk 2   br 3 (r = 4, 5, 6) −

The components of the ﬂuid drag acting on Sk are as follows [22]:

X3 D sk F = D bi, (42) sk i=1 i  sk 1 sm sm sm/R sm/R  D = ρaA c v v  1 2n 1 D1 1 1  −  sk 1 sk sk sk/R sk/R D = ρaA c v v , (43)  2 2 2 D2 2 2  −  sk 1 sk sk sk/R sk/R  D = ρaA c v v 3 − 2 3 D3 3 3 sk where ρa is the density of seawater and Ai is the area of sk perpendicular to bi below the water sm sk sk surface. The section sm has maximum area A = max(A )(k = 1 n), c is the ﬂuid drag coeﬃcient, 1 1 ··· Di sk/R and vi is the component of the velocity of the section sk relative to the wave surface on the axis of bi. The generalized force caused by buoyancy is

Xn FB/b = FB/sk (r = 1, , 6), (44) r k=1 r ···

FB/sk = FB vsk , (45) r sk · r J. Mar. Sci. Eng. 2020, 8, 631 10 of 25

B F = ρaVs gN , (46) sk − k 3 V S FB s where sk is the volume of k in water and sk is the buoyancy acting on k. The generalized force caused by the slamming force is

p/b Xn p/s F = F k (r = 1, , 6), (47) r k=1 r ···

p/s p F k = F vsk , (48) r sk · r X3 p sk F = p bi, (49) sk i=1 i p F s where sk is the slamming force that acted on section k. Previous scholars adopted the momentum theory to calculate the slamming force. When an object enters the water, part of its initial momentum will be transferred to the surrounding water. Assuming that the momentum conversion process is irreversible, the slamming force acting on the boat can be calculated by the rate of its momentum change: m0v0 = (m0 + ma)v, (50) ! p d d dma dv F = (m v) = (mav) = v + ma , (51) dt 0 −dt − dt dt where v0 is the velocity of the object before entering the water and v is the velocity of the object after entering the water; m0 is the mass of the object and ma is the added mass of the object. Another method is based on energy theory, which was derived by Wu [35]. It has more physical signiﬁcance and is widely used [36–38]. The relationship between slamming force and added mass is

p dv 1 dma F = ma v. (52) − dt − 2 dt

In order to consider the effect of the wave on the boat, the incident wave force Fin is added into the slamming force: in f F = ρaVsuba , (53) f where Vsub is the submerged volume and a is the fluid acceleration. p In this paper, the slamming force Fs is calculated as the sum of energy theory and incident wave k p force for the first time. psk are the components of F on the axis of b : i sk i

 s s /R dm h dv h  sk 1 f /sh 1 sh/R 1 sh  p = (ρaVs a (0.5 v + m ))  1 n h 1 − dt 1 dt 1  s s /R  dm k dv k sk f /sk 2 sk/R 2 sk , (54)  p = ρaVs a (0.5 v + m )  2 k 2 − dt 2 dt 2  sk sk/R  sk f /sk dm3 sk/R dv3 sk  p = ρaVs a (0.5 v + m ) 3 k 3 − dt 3 dt 3 f /s f /sk k where a is the wave surface acceleration at section sk at the coordinate system of the boat, ai f /sk sk are the components of a at the coordinate system of the boat, mi are the added mass of the cross sh sh section in the direction of bi, m1 is the added mass of the boat in the direction of b1, and v1 is the velocity component of the midsection sh between the bow and the center of gravity of the boat in the direction of b1. g g g s (x y z ) k have coordinates sk , sk , sk in an inertial coordinate system:

 g    x q + x  sk   4 sk   g  hN bi   ys  = C  q5 . (55)  gk     z   q  sk 6 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 25

푝 In this paper, the slamming force 퐹푠푘 is calculated as the sum of energy theory and incident 푠푘 푝 wave force for the first time. 푝𝑖 are the components of 퐹푠푘 on the axis of 푏𝑖: 푠 푠 /푅 푠 1 푓/푠 푑푚 ℎ 푠 /푅 푑푣 ℎ 푠 푝 푘 = (휌 푉 푎 ℎ − (0.5 1 푣 ℎ + 1 푚 ℎ)) 1 푛 푎 푠ℎ 1 푑푡 1 푑푡 1 푠 푠 /푅 푠 푓/푠 푑푚 푘 푠 /푅 푑푣 푘 푠 푝 푘 = 휌 푉 푎 푘 − (0.5 2 푣 푘 + 2 푚 푘 ) , (54) 2 푎 푠푘 2 푑푡 2 푑푡 2 푠푘 푠푘/푅 푠 푓/푠 푑푚 푠 /푅 푑푣 푠 푝 푘 = 휌 푉 푎 푘 − (0.5 3 푣 푘 + 3 푚 푘 ) { 3 푎 푠푘 3 푑푡 3 푑푡 3

푓/푠푘 where 푎 is the wave surface acceleration at section 푠푘 at the coordinate system of the boat, 푓/푠 푠 푘 푓/푠푘 푘 푎𝑖 are the components of 푎 at the coordinate system of the boat, 푚𝑖 are the added mass of 푠ℎ the cross section in the direction of 푏𝑖, 푚1 is the added mass of the boat in the direction of 푏1, and 푠ℎ 푣1 is the velocity component of the midsection 푠ℎ between the bow and the center of gravity of the boat in the direction of 푏1. 푔 푔 푔 푠푘 have coordinates (푥푠푘, 푦푠푘 , 푧푠푘) in an inertial coordinate system: 푔 푥푠푘 푞4 + 푥푠푘 푔 푁 푏 푞 (푦푠푘 ) = [ 퐶 ] ( 5 ). (55) 푔 푞6 푧푠푘

The depth of 푠푘 is 푔 ℎ푠푘 = 퐶31 ⋅ (푞4 + 푥푠푘) + 퐶32 ⋅ 푞5 + 퐶33 ⋅ (푞6 − ℎ0) − 휂(푥푠푘, 푡), (56) 푁 푏 where 퐶𝑖𝑖 are elements of the matrix of [ 퐶 ], the subscript is its location in the matrix, ℎ0 is the vertical distance from the center of gravity to the keel, ℎ0 is 1.15 m in Figure 4, and 휂(푥, 푡) is a J.known Mar. Sci. function Eng. 2020 ,describing8, 631 the wave surface. 11 of 25 푠푘 푠푘 푠푘 The added mass is 푚2 = 푚3 in this paper. 푚3 depends on the depth and the instantaneous half width of 푠푘 at the wave surface: The depth of sk is 휌휋퐶2(푥 , ℎ )/2 ℎ < 푑 푠푘 푠푘 푠푘 푠푘 1 푚3 = { 2 , g (57) hs = C q + x휌휋s 퐶+(C푥푠 , 푑q1)/+2C ℎ푠 (≥q 푑1h ) η x , t , (56) k 31 · 4 k 32푘 · 5 33 ·푘 6 − 0 − sk where 퐶(푥 , ℎ ) is half the width of 푠 at depth ℎ . When the depth of 푠 is greater than 푑 , where C are푠푘 elements푠푘 of the matrix of [NC푘b], the subscript푠푘 is its location in the matrix,푘 h is the vertical1 ii 푑 0 the added mass is calculated according to the half-width at depth 1 , as( shown) in Figure 4. We distance from the center of gravity to the푠푘 keel,푠h푘0 is 1.15 m in Figure4, and η x, t is a known function 푠 푑푚 푑푚 푑ℎ푠 푑ℎ푠 푠 /푅 take the derivative of 푚 푘 to time 3 = 3 푘 with 푘 = max (푣 푘 , 0). describing the wave surface.3 푑푡 푑ℎ 푑푡 푑푡 3 푠푘 A

d2 b3 3.68 m b1 1.15 m d1

A 7.4 m A-A

(a) (b)

Figure 4. Lifeboat cross section: ( a) longitudinal longitudinal section in center plane, (b) A-AA‒A crosscross section.section.

푚푠ℎ sk = 훼sk sk TheTo calculate added mass 1 is, definem2 m푠3푘 in this paper. m3 depends on the depth and the instantaneous half ℎ푔 ≥ 0 width of sk at the wave surface: 1 푠푘 훼 = { , (58) 푠푘 0 ℎ푔 < 0  2 푠푘 푔 푔 푔  ρπC xs , hs /2 where ℎ = 푧 − 휂(푥 , 푡). The lengthsk  of waterk entryk in thehs kdirection< d1 of 푏 is 푙 : 푠푘 푠푘 푠푘 m = ,1 1 (57) 3  2  ρπC x , d /2 hsk d1 sk 1 ≥ where C(xsk , hsk ) is half the width of sk at depth hsk . When the depth of sk is greater than d1, the added mass is calculated according to the half-width at depth d1 , as shown in Figure4. We take the derivative s s k k dh dh sk dm3 dm3 sk sk sk/R of m to time dt = dh dt with dt = max(v , 0). 3 sk 3 sh To calculate m1 , deﬁne αsk ( g 1 hs 0 α = gk ≥ , (58) sk h 0 sk < 0 g g g where h = z η(x , t). The length of water entry in the direction of b is l : sk sk − sk 1 1 Xn l1 = αs ts, (59) k=1 k ·   m 2(l /l )3 4(l /l )2 + 2.5(l /l ) l < l /2 sh =  ax 1 b 1 b 1 b 1 b m1  − , (60)  m /2 l1 lb/2 ax ≥ where max is the added mass of the boat in the direction of b1 when the boat is completely in the water.

2 kmπρalb(d1 + d2) m = , (61) ax 6

s s dm h dm h lb sh 1 1 dl1 where km is a coeﬃcient depending on . We take the derivative of m to time = (d1+d2) 1 dt dl1 dt dl1 = ( sh/R ) with dt max v1 , 0 . J. Mar. Sci. Eng. 2020, 8, 631 12 of 25

The acceleration of the section is X3 sk sk a = a bi, (62) i=1 i . . . . . sk = + sk sk = + + sk sk = + sk sk = ( 2 + 2) where a1 u4 z1 , a2 u5 xsk u3 z2 ,a3 u6 xsk u2 z3 ,z1 u2u6 u3u5 xsk u2 u3 , sk sk − − − z = u u + u u + xs u u ,z = u u + u u + xs u u . 2 − 1 6 3 4 k 1 3 3 1 5 2 4 k 1 3 So . s . b 1 v h = u + ubub ubub L u2 + u2 , (63) 1 4 2 6 − 3 5 − 2 f 2 3 where L f is the distance from the center of gravity of the boat to the bow. In summary, the equation of motion is

b G/b W/b D/b B/b p/b F∗ + F + F + F + F + F = 0 (r = 1, , 6). (64) r r r r r r ··· The motion parameters of the lifeboat can be solved by Equation (64) by the fourth-order Runge-Kutta method.

4. Results and Analysis

4.1. Experimental Setup and Implementation The basic information of the boat, ship, and skid is shown in Tables1–3. Air drag coe ﬃcient sk Cw = 0.5. Fluid drag coeﬃcient c = 1.2. Figure5 gives the structural dimension diagram of the ship Di and boat in the simulation experiment.

Table 1. Basic information of the boat.

Item (Unit) Value Length (m) 7.4 Width (m) 2.65 Draft (m) 0.74 Maximum height (m) 2.3 Maximum half width (m) 1.42 Distance between center of gravity and center (m) 0.02 I (kg m2) 34,825 2 · mass (kg) 6200 2 The projected areas of the boat at plane perpendicular to b1 (m ) 4.2 2 The projected areas of the boat at plane perpendicular to b2 (m ) 13.2 2 The projected areas of the boat at plane perpendicular to b3 (m ) 13.2

Table 2. Basic information of the ship.

Item (Unit) Value Length over all (m) 144.4 Waterline length (m) 133.55 Length between perpendiculars (m) 129 Width (m) 20.8 Draft (m) 4.4 Depth molded (m) 11.4 GM (m) 5.57 Distance between center of gravity and center (m) 2.3 Block coefficient 0.68 Water plane coefficient 0.83 Prismatic coefficient 0.693 Displacement (kg) 7,550,000 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 12 of 25

Table 2. Basic information of the ship.

Item (Unit) Value Length over all (m) 144.4 Waterline length (m) 133.55 Length between perpendiculars (m) 129 Width (m) 20.8 Draft (m) 4.4 Depth molded (m) 11.4 퐺푀 (m) 5.57 Distance between center of gravity and 2.3 center (m) Block coefficient 0.68 Water plane coefficient 0.83 Prismatic coefficient 0.693 Displacement (kg) 7,550,000 J. Mar. Sci. Eng. 2020, 8, 631 13 of 25

Table 3. Basic information of the skid. Table 3. Basic information of the skid. Item (Unit) Value Vertical distanceItem between (Unit) the low end of Value 3.5 Vertical distance betweenskid and the deck low (m) end of skid and deck (m) 3.5 InclinationInclination (°) (◦ )30 30 LengthLength (m) (m) 9 9 SlidingSliding distance distance of of the the center center of of gravity gravity (m (m)) 4.96 4.96

8 m 8

3.5 m 7 m 7

30° X0 69.9 m

(a) (b)

Figure 5. Structural dimension dimension diagram diagram of of the the ship ship and and boat: boat: (a ()a the) the size size of ofthe the longitudinal longitudinal section section in inthe the center center plane plane of the of the ship ship and and skid; skid; (b) (theb) the inclination inclination of the of theskid skid and and the theposition position of the of theboat boat at the at theskid. skid.

The simulation experiment of the whole algorithm was was realized realized by by MATLAB. MATLAB. The flow flow of the algorithm consists of two stages that are separated byby thethe timetime pointpoint ofof thethe boatboat outout ofof the skid. In the firstfirst stage, the motion of the skid is calculated according to parameters of ship motion solved by Equations (16) (16) and and (19). (19). The The generalized generalized contact contact force can force be cansolved be by solved Equation by (14), Equat theion generalized (14), the forcegeneralized caused force by gravity caused and by gravity inertia can and be inertia solved can by be Equations solved by (9) Equations and (10), (9) and and the (10), generalized and the velocitiesgeneralized and velocities generalized and coordinatesgeneralized ofcoordinates the boat can of bethe solved boat can by Equationbe solved (15).by Equation In the second (15). In stage, the thesecond generalized stage, the force generalized caused by force inertial, caused gravity, by inertial, air drag, gravity, fluid drag,air drag, buoyancy, fluid drag, and slammingbuoyancy, force and canslamming be solved force by Equationscan be solved (32), (34), by Equations (36), (38), (44) (32), and (34) (47);, (36), the (38),generalized (44) and velocities (47); the of the generalized boat can bevelocities solved of by the Equation boat can (64); be solved and the by generalized Equation (64 coordinates); and the generalized of the boat cancoordinates be solved of by the Equations boat can (26)be solved and (27). by For Equations the two (26) stages, and 20 (27). lifeboat For thesegments two stages, were used. 20 lifeboat For convenience segments were of analysis used. and For understanding,convenience of analysis the data and of the understanding, boat’s translation the data and of pitch the areboat’s shown translation in the coordinates and pitch are obtained shown byin the inertialcoordinates coordinate obtained system, by the rotating inertial 180coordinate◦ around system, the axis rotating of oy. 180° around the axis of 표푦. The results of a simulation experiment by the numerical algorithm in this paper were compared with thethe resultsresults of of Star Star CCM CCM++ at at the the initial initial condition condition A, asA, shown as shown in Table in Table4. The 4. version The version of Star of CCM Star+ usedCCM+ was used 13.02. was These 13.02 computations. These computations were performed were performed using an using overlapping an overlapping grid, as shown grid, as in Figureshown6 ina, in order to provide accurate wave representation at any location irrespective of the lifeboat position. The behavior of two fluids (air and water) in the same continuum is modeled by using the model of volume of fluid (Figure6b). Due to the existence of two fluids in di fferent phases, the Euler multiphase model is activated and the gravity model is used to take into account the gravity effect of the two fluids. The initial condition of water entry is obtained by our numerical algorithm: the boat enters the water at the crest of the wave, the pitch angle of the boat when entering the water is 50.7◦ with horizontal, vertical, and rotational velocity of 5.9 m/s, 11 m/s, and 0.46 rad/s. − J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 13 of 25

Figure 6a, in order to provide accurate wave representation at any location irrespective of the lifeboat position. The behavior of two fluids (air and water) in the same continuum is modeled by using the model of volume of fluid (Figure 6b). Due to the existence of two fluids in different phases, the Euler multiphase model is activated and the gravity model is used to take into account the gravity effect of the two fluids. The initial condition of water entry is obtained by our numerical algorithm: the boat enters the water at the crest of the wave, the pitch angle of the boat when J.entering Mar. Sci. Eng.the water2020, 8, 631is 50.7° with horizontal, vertical, and rotational velocity of 5.9 m/s, ‒11 m/s,14 ofand 25 0.46 rad/s.

Table 4. Details of initial condition A. Table 4. Details of initial condition A.

ItemItem (Unit) (Unit) ValueValue WaveWave height height (m) (m) 22 PeriodPeriod ofof wavewave (s) (s) 88 WindWind velocity velocity (m (m/s)/s) ( 0,(0 0,,0 0,0)) Ship’sShip’s initial initial position position (m) (m) ( 0,(0 0,,0 0,0)) Ship’sShip’s initialinitial velocity velocity (knot) (knot) ( 1,(1 0,,0 0,0)) Situation of ship encountering waves Situation of ship encountering waves 1 (0: head the wave; 1: follow the wave) 1 (0: head the wave; 1: follow the wave)

(a) (b)

Figure 6. Simulation experiment experiment diagram diagram of of Star Star CCM+: CCM+ (:(a)a overlapping) overlapping grid grid;; (b) ( bvolume) volume fraction fraction of ofwater. water.

4.2. Comparison and A Analysisnalysis of E Experimentalxperimental R Resultsesults The redred lineline isisthe the result result of of the the whole whole motion motion process process calculated calculated by by the the model model in this in this paper. paper The. The red bluered blue is the is resultthe result of the of waterthe water entry entry process process calculated calculated by Star by Star CCM CCM+. +. Figure 77 shows shows thethe trajectorytrajectory andand pitchpitch angleangle ofof thethe boat.boat. TheThe resultsresults showshow thatthat thethe trajectorytrajectory and pitch angle calculated by two methods are in good agreement. When the boat moves moves on on the the skid, skid, the pitchpitch angleangle begins begins to to change change with with no obvious no obvious logic. logic The. pitchThe pitch angle angle begins begins to increase to increase at the bottom at the ofbottom the skid. of the After skid. the After boat the leaves boat the leaves skid, the the ski pitchd, t anglehe pitch of theangle boat of gradually the boat gradually increases toincreases about 53 to◦. Theabout pitch 53°. angleThe pitch begin angle to decrease begin to afterdecreas increasese after increases in short time in short because time thebecause bow the of the bow boat of the hits boat the water.hits the The water center. The of center gravity of ofgravity the boat of the is sinking, boat is thesinking, bow ofthe the bow boat of isthe raised, boat is resulting raised, result in theing stern in hittingthe stern the hitting water. the The water buoyancy. The increases buoyancy with increases the increase with the of the increase depth of the the stern. depth Subsequently, of the stern. theSubsequently, pitch angle the of the pitch boat angle begins of tothe increase boat begins again, to and increase the lifeboat again, begins and the to ﬂoat.lifeboat After begins it comes to float. out ofAfterJ. Mar. the it water,Sci. comes Eng. the20 out20 bow, 8 of, x FORthe hits water,PEER the water REVIEW the bow again. hits the water again. 14 of 25

(a) (b)

FigureFigure 7.7. TrajectoryTrajectory ((aa)) andand pitchpitch ((bb)) ofof thethe lifeboatlifeboat freefallfreefall during during its its launch launch from from the the ship. ship.

Figure 8 shows the ratio of the acceleration of the center of gravity to 𝑔. When the boat is on the skid, the acceleration value of the center of gravity of the boat begins to vibrate. When the boat is at the low end of the skid, the vibration frequency of the acceleration value increases and the amplitude decreases. When the boat leaves the skid, the horizontal acceleration of the boat is zero and the vertical acceleration is 𝑔 before entering the water. The horizontal acceleration of the boat is negative in the initial stage of water entry because of the slamming force and the fluid drag force, and the acceleration becomes positive when the boat floats up because of its large buoyancy. The vertical acceleration of the boat is almost positive because of the upward force, and its negative value caused by gravity and fluid drag when the boat floats up. There is a little difference between the acceleration values calculated by the two methods, especially at the initial stage of water entry. The difference is acceptable. This phenomenon may be due to the fact that the deformation of the water caused by boat is not considered in this algorithm.

(a) (b)

Figure 8. Acceleration of center of gravity of the lifeboat in the horizontal (a) and vertical (b) direction.

Figure 9 shows the velocity of the center of gravity of the boat. The horizontal velocity of the boat gradually begins to increase at the skid, and remains the same in the air before entering the water; it begins to decrease due to the negative acceleration at the beginning of the water entry phase, and begins to increase when the boat floats up. The value of vertical acceleration is decreasing because the boat moves down the skid. The value of vertical acceleration decreases at the same rate because of g in the air before entering the water. Its value gradually increases until the sign is positive because of the slamming force and the buoyancy at the water entry phase; its sign becomes negative when the boat falls into the water again. The velocity values calculated by the two methods are in good agreement.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 14 of 25

(a) (b) J. Mar. Sci. Eng. 2020, 8, 631 15 of 25 Figure 7. Trajectory (a) and pitch (b) of the lifeboat freefall during its launch from the ship.

FigureFigure8 8shows shows the the ratio ratio of of the the acceleration acceleration of of the the center center of of gravity gravity to tog .𝑔 When. When the the boat boat is is on on the the skid,skid, the the acceleration acceleration value value of of the the center center of of gravity gravity of of the the boat boat begins begins to to vibrate. vibrate. When When the the boat boat is is at at thethe low low end end of the of skid, the skid, the vibration the vibration frequency frequency of the acceleration of the acceleration value increases value and increases the amplitude and the decreases.amplitude When decreases. the boat When leaves the the boat skid, leaves the horizontal the skid, accelerationthe horizontal of theacceleration boat is zero of andthe theboat vertical is zero accelerationand the vertical is g beforeacceleration entering is 𝑔 the before water. entering The horizontal the water. acceleration The horizontal of the acc boateleration is negative of the inboat the is initialnegative stage in of the water initial entry stage because of water of the entry slamming because force of the and slamming the fluid drag force force, and andthe fluid the acceleration drag force, becomesand the positiveacceleration when become the boats positive floats upwhen because the boat of its floats large up buoyancy. because of The its verticallarge buoyancy acceleration. The ofvertical the boat acceleration is almost positiveof the boat because is almost of the positive upward because force, of and the its upward negative force value, and caused its negative by gravity value andcaused fluid by drag gravity when and the fluid boat drag floats when up. There the boat is a float littles diup.fference There betweenis a little the difference acceleration between values the calculatedacceleration by thevalues two calculated methods, especiallyby the two at methods, the initial especially stage of water at the entry. initial The stage difference of water is acceptable. entry. The Thisdifference phenomenon is acceptable may be. This due phenomenon to the fact that may the bedeformation due to the fact of that the waterthe deformation caused by of boat the is water not consideredcaused by inboat this is algorithm.not considered in this algorithm.

(a) (b)

FigureFigure 8. 8.Acceleration Acceleration of center of center of gravity of gravity of the of lifeboat the lifeboat in the horizontal in the horizontal (a) and vertical (a) and (b ) vertical direction. (b) direction. Figure9 shows the velocity of the center of gravity of the boat. The horizontal velocity of the boat graduallyFigure 9 shows begins the to increasevelocity of at the center skid, and of gravity remains of thethe sameboat. inThe the horizontal air before velocity entering of the the water;boat gradually it begins to begins decrease to increase due to the at negativethe skid, acceleration and remains at the beginningsame in the of air the before water entryentering phase, the andwater; begins it begins to increase to decrease when the due boat to ﬂoats the negative up. The value acceleration of vertical at accelerationthe beginning is decreasing of the water because entry thephase, boat and moves begins down to theincrease skid. when The value the boat of vertical floats up. acceleration The value decreases of vertical at acceleration the same rate is becausedecreasing of gbecausein the air the before boat enteringmoves down the water. the skid Its value. The value gradually of vertical increases acceleratio until then signdecreases is positive at the because same rate of thebecause slamming of g force in the and air the before buoyancy entering at the the water water entry. Its phase; value its gradually sign becomes increase negatives until when the the sign boat is fallsJ.positive Mar. into Sci. theEng.because water2020, 8of, again.x theFOR slamming PEER The REVIEW velocity force values and the calculated buoyancy by at the the two water methods entry are phase; in good its sign agreement. becomes15 of 25 negative when the boat falls into the water again. The velocity values calculated by the two methods are in good agreement.

(a) (b)

FigureFigure 9.9. VelocityVelocity ofof centercenter ofof gravitygravity ofof thethe lifeboatlifeboat inin thethe horizontalhorizontal ((aa)) and and vertical vertical ( b(b)) direction. direction.

FigureFigure 1010 showsshows the velocity and acceleration of of pitch. pitch. When When the the boat boat is is on on the the skid, skid, the the value value of ofthe the velocity velocity of of the the pitch pitch begin beginss to to vibrate. vibrate. When When the the boat boat leaves leaves the the skid, skid, the velocity ofof thethe pitchpitch remains the same before entering the water. The velocity of the pitch decreases to a negative value when the boat enters the water. Its sign is positive when the boat’s bow begins to fall into the water again. The acceleration of the pitch has the same rule of change as the horizontal acceleration. The velocity and acceleration of pitch calculated by the two methods are in good agreement. In summary, the position and velocity of the center of gravity of the boat, the angle, and velocity and acceleration of pitch calculated by the two methods are in good agreement. There is a little difference between the values of translation acceleration calculated by the two methods. The difference is acceptable. This shows that the numerical algorithm in this paper has good accuracy.

(a) (b)

Figure 10. Velocity (a) and acceleration (b) of pitch of the lifeboat.

The boat experiences three water impacts (bow impact, stern impact, bow impact) in the previous analysis. In order to analyze the local acceleration of different positions in the boat, the local acceleration of the bow, midship, and stern is calculated by Equation (1). Figure 11 shows the acceleration of the bow (a,b), midship (c,d), and stern (e,f) in the boat coordinate system. The acceleration values in all figures are the ratio of actual values to the acceleration of gravity 𝑔. During the whole water entry process, the acceleration curves have three peaks because of three water impacts (Figure 11a,c) The first two peaks are close; there is a positive maximum acceleration

in the direction of 푏3 at the bow at the first impact as the first peak of the curve (Figure 11a). There is a positive maximum acceleration in the direction of 푏3 at the midship at the second impact as the second peak of the curve (Figure 11c). There are two contrary extreme values of the acceleration at

the stern in the direction of 푏3 (Figure 11e): a negative extremum due to the first impact, and a positive extreme value due to the second impact. The negative maximum acceleration in the

direction of 푏1 at the bow, midship, and stern is because of the first impact (Figure 11b,d,f). The maximum acceleration is about 3 𝑔 in the direction of 푏3, derived from the bow’s first impact.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 15 of 25

(a) (b)

Figure 9. Velocity of center of gravity of the lifeboat in the horizontal (a) and vertical (b) direction.

Figure 10 shows the velocity and acceleration of pitch. When the boat is on the skid, the value of the velocity of the pitch begins to vibrate. When the boat leaves the skid, the velocity of the pitch remains the same before entering the water. The velocity of the pitch decreases to a negative value

J.when Mar. Sci. the Eng. boat2020 enters, 8, 631 the water. Its sign is positive when the boat’s bow begins to fall into the16 water of 25 again. The acceleration of the pitch has the same rule of change as the horizontal acceleration. The velocity and acceleration of pitch calculated by the two methods are in good agreement. remainsIn thesummary same before, the entering position the and water. velocity The velocity of the center of the pitch of gravity decreases of the to a negativeboat, the value angle, when and thevelocity boat entersand acceleration the water. Itsof signpitch is calculated positive when by the the two boat’s methods bow beginsare in togood fall agreement. into the water There again. is a Thelittle acceleration difference ofbetween the pitch the has values the sameof translation rule of change acceleration as the horizontalcalculated acceleration.by the two methods. The velocity The anddifference acceleration is acceptable. of pitch calculatedThis shows by that the the two numerical methods algorithm are in good in agreement. this paper has good accuracy.

(a) (b)

FigureFigure 10. 10.Velocity Velocity ( a(a)) and and acceleration acceleration ( b(b)) of of pitch pitch of of the the lifeboat. lifeboat.

InThe summary, boat experiences the position three and velocity water impacts of the center (bow of impact, gravity stern of the impact, boat, the bow angle, impact) and velocity in the andprevious acceleration analysis. of In pitch order calculated to analyze by the local two methodsacceleration are of in gooddifferent agreement. positions Therein the is bo aat, little the dilocalfference acceleration between of the the values bow, of midship, translation and acceleration stern is calculated calculated by by Equation the two methods.(1). Figure The 11 dishowsfference the isacceleration acceptable. ofThis the shows bow that(a,b) the, midship numerical (c,d) algorithm, and stern in this (e,f) paper in the has boat good coordinate accuracy. system. The accelerationThe boat values experiences in all threefigures water are the impacts ratio (bowof actual impact, values stern to impact,the acceleration bow impact) of gravity in the previous𝑔. analysis.During In order the whole to analyze water the entry local process, acceleration the acceleration of different positions curves have in the three boat, peaks the local because acceleration of three ofwater the bow, impacts midship, (Figure and 11a,c) stern T ishe calculated first two peaks by Equation are close (1).; there Figure is a11 positive shows themaximum acceleration acceleration of the bowin the (a,b), direction midship of 푏 (c,d),3 at the and bow stern at (e,f)the first in the impact boat coordinateas the first peak system. of the The curve acceleration (Figure 11a values). There in all is figuresa positive are themaximum ratio of acceleration actual values in to the the direction acceleration of 푏 of3 gravityat the midshipg. at the second impact as the secondDuring peak the of wholethe curve water (Figure entry 11c). process, There the are acceleration two contrary curves extreme have threevalues peaks of the because acceleration of three at waterthe stern impacts in the (Figure direction 11a,c) of The 푏3 first(Figure two peaks 11e): aare negative close; there extremum is a positive due tomaximum the first acceleration impact, and in a thepositive direction extreme of b3 at value the bow due at to the the first second impact impact. as the firstThe peak negative of the ma curveximum (Figure acceleration 11a). There in is the a positivedirection maximum of 푏1 at accelerationthe bow, midship, in the direction and stern of bis3 atbecause the midship of the at first the impact second impact(Figure as 11b,d,f). the second The peakmaximum of the curveacceleration (Figure is 11 aboutc). There 3 𝑔 arein the two direction contrary of extreme 푏3, derived values from of the the acceleration bow’s first at impact. the stern in the direction of b3 (Figure 11e): a negative extremum due to the first impact, and a positive extreme value due to the second impact. The negative maximum acceleration in the direction of b1 at the bow, midship, and stern is because of the first impact (Figure 11b,d,f). The maximum acceleration is about 3 g in the direction of b3, derived from the bow’s first impact.

4.3. Qualitative Analysis According to the numerical calculation method used in this paper, the local acceleration extremum is calculated under diﬀerent initial conditions. The wave heights are 1, 2, 3, 4, and 5 m. By adjusting the initial phase of the wave, the points of water entry of the boat are shown in Figure 12. Figure 12a shows the ship heading the wave and Figure 12b shows the ship following the wave. When the ship is heading or following the wave, only the surge, heave, and pitch of the ship are considered. Other initial conditions (period of wave, wind velocity, ship’s initial position, ship’s initial velocity) are the same as in condition A. Figures 13–17 show the trajectory and pitch angle of the lifeboat when the ship is heading the wave. Figure 18 shows the maximum value of the acceleration in the directions of b1 and b3 at the bow (Figure 18a,b), midship (Figure 18c,d), and stern (Figure 18e,f) of the boat at eight diﬀerent points of water entry when the ship is heading waves. Figures 19–23 show the trajectory and pitch angle of the lifeboat when the ship is following waves. Figure 24 shows the maximum value of the acceleration in J. Mar. Sci. Eng. 2020, 8, 631 17 of 25

the directions of b1 and b3 at the bow (Figure 24a,b), midship (Figure 24c,d), and stern (Figure 24e,f) of the boat at eight diﬀerent points of water entry when the ship is following waves. As there are no experimental data for comparison, the numerical calculation results may not be very accurate, but can beJ. used Mar. Sci. in aEng. qualitative 2020, 8, x FOR analysis. PEER REVIEW 16 of 25

(a) (b)

(e) (f)

FigureFigure 11. 11.The The acceleration acceleration of of the the bow, bow, midship, midship, and and stern stern inin thethe coordinatecoordinate system of the lifeboat lifeboat in in J. Mar. Sci. Eng. 2020, 8, x FOR푏 PEER REVIEW 푏 17 of 25 thethe direction direction of ofb3 (a3, c,(ea),c and,e) and in thein the direction direction of bof1 (b1,d (,fb).,d,f).

4.3. Qualitative Analysis According to the numerical calculation method used in this paper, the local acceleration extremum is calculated under different initial conditions. The wave heights are 1, 2, 3, 4, and 5 m. By adjusting the initial phase of the wave, the points of water entry of the boat are shown in Figure 12. Figure 12a shows the ship heading the wave and Figure 12b shows the ship following the wave. When the ship is heading or following the wave, only the surge, heave, and pitch of the ship are

considered. Other initial conditions(a) (period of wave, wind velocity,(b) ship’s initial position, ship’s initial velocity) are the same as in condition A. Figure 12. Eight different diﬀerent points of water entry of the boat when the ship is heading the wave ( a) and following the wave (b).

Figures 13‒17 show the trajectory and pitch angle of the lifeboat when the ship is heading the wave. Figure 18 shows the maximum value of the acceleration in the directions of 푏1 and 푏3 at the bow (Figure 18a,b), midship (Figure 18c,d), and stern (Figure 18e,f) of the boat at eight different points of water entry when the ship is heading waves. Figures 19‒23 show the trajectory and pitch angle of the lifeboat when the ship is following waves. Figure 24 shows the maximum value of the acceleration in the directions of 푏1 and 푏3 at the bow (Figure 24a,b), midship (Figure 24c,d), and stern (Figure 24e,f) of the boat at eight different points of water entry when the ship is following waves. As there are no experimental data for comparison, the numerical calculation results may not be very accurate, but can be used in a qualitative analysis. The ship heads the wave. As shown in Figures 15‒17a, when the wave height exceeds 3 m, the boat will move to the side of the ship after water entry at P7 and P8. As shown in Figures 16a and 17a, when the wave height exceeds 4 m, the same phenomenon appears at point P6. As shown in Figures 13‒17b, at points P1 and P8, the pitch angle of the boat has a large range of change. As shown in Figure 18, in the direction of 푏1, the maximum values of acceleration at the bow, midship, and stern of the boat appear at point P4; the minimum value generally appears at P1 and P8. The maximum values at the bow and midship are almost the same, slightly lower than the stern. In the direction of 푏3, the maximum absolute values of acceleration at the bow and midship of the boat appear at points P3 and P2; the minimum value generally appears at points P7 and P8. There is no obvious rule about the values of acceleration at the stern; the acceleration is the largest at the bow and the smallest at the midship.

(a) (b)

Figure 13. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 1 m.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 17 of 25

(a) (b)

Figure 12. Eight different points of water entry of the boat when the ship is heading the wave (a) and following the wave (b).

J.obvious Mar. Sci. Eng.rule2020 about, 8, 631 the values of acceleration at the stern; the acceleration is the largest at the18 ofbow 25 and the smallest at the midship.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 25 J. Mar. Sci. Eng. 2020, 8, x FOR PEER(a) REVIEW (b) 18 of 25 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 25 FigureFigure 13.13. TrajectoryTrajectory ((aa)) andand pitchpitch ((bb)) ofof thethe lifeboatlifeboat freefallfreefall when when the the wave wave height height is is 1 1 m. m.

(a) (b) (a) (b) Figure 14. Trajectory(a) (a) and pitch (b) of the lifeboat freefall when the(b wave) height is 2 m. Figure 14. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 2 m. FigureFigure 14.14. TrajectoryTrajectory ((aa)) andand pitchpitch ((bb)) ofof thethelifeboat lifeboat freefallfreefall when when the the wave wave height height is is 2 2 m. m.

(a) (b) (a) (b) Figure 15. Trajectory(a) (a) and pitch (b) of the lifeboat freefall when the(b wave) height is 3 m. FigureFigure 15. 15.Trajectory Trajectory ( a(a)) and and pitch pitch ( b(b)) of of thethe lifeboatlifeboat freefallfreefall whenwhen thethe wavewave height is 3 m. Figure 15. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 3 m.

(a) (b) (a) (b)

FigureFigure 16. 16.Trajectory Trajectory ( a(a)) and and pitch pitch ( b(b)) of of the the lifeboat lifeboat freefallfreefall whenwhen thethe wavewave heightheight isis 4 m. Figure 16. Trajectory(a) (a) and pitch (b) of the lifeboat freefall when the(b wave) height is 4 m. Figure 16. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 4 m.

(a) (b) (a) (b)

(a) (b)

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 25

(a) (b)

Figure 14. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 2 m.

(a) (b)

Figure 15. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 3 m.

(a) (b)

J. Mar. Sci. Eng.Figure2020, 16.8, 631 Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 4 m. 19 of 25

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 19 of 25 (a) (b) Figure 17. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 5 m. Figure 17. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 5 m.

(a) (b)

(e) (f)

FigureFigure 18. 18.The The maximummaximum accelerationacceleration ofof thethe bow,bow, midship, midship, and and sternstern inin thethe coordinatecoordinate systemsystem ofof thethe

lifeboatlifeboat in in the the direction direction of ofb 3푏(3a ,(ca,e,c),,e and), and in in the the direction direction of ofb1 (푏b1,d (,fb),,d at,f), di atﬀ erentdifferent wave wave heights. heights.

The ship follows the wave. As shown in Figures 21‒23a, when the height of the wave exceeds 3 m, the boat will move to the side of the ship after water entry at points P1 and P8. As shown in Figures 22‒23a, when the height of the wave exceeds 4 m, the same phenomenon appears at point P7. As shown in Figures 19‒23b, the pitch angle of the boat has a large range of change at points P5

and P6. As shown in Figure 24, in the direction of 푏1, the maximum values of acceleration at the bow and midship appear at points P6 and P7, and the minimum values generally appear at points P2 and P3. The maximum values of acceleration at the stern appear at points P4 and P5, and the minimum values generally appear at points P1 and P2. The maximum values at the bow and midship of the

boat are almost the same, and slightly lower than for the stern. In the direction of 푏3, the maximum values of acceleration at the bow and midship of the boat appear at points P4 and P5; the minimum value generally appears at points P1 and P8. The maximum absolute values of the two extremal values at the stern appear at P4 and P5. The minimum values generally appear at P1 and P8; the acceleration is the largest at the bow and the smallest at the midship.

J. Mar.J. Mar. Sci. Sci. Eng. Eng.2020 2020, 8, ,8 631, x FOR PEER REVIEW 2020 of of 25 25 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 20 of 25

(a) (b) (a) (b) Figure 19. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 1 m. FigureFigure 19.19. TrajectoryTrajectory ((aa)) andand pitchpitch ((bb)) ofof thethe lifeboatlifeboat freefallfreefall when when the the wave wave height height is is 1 1 m. m.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER(a) REVIEW (b) 21 of 25 J. Mar. Sci. Eng. 2020, 8, x FOR PEER(a) REVIEW (b) 21 of 25 FigureFigure 20. 20.Trajectory Trajectory ( a(a)) and and pitch pitch ( b(b)) of of the the lifeboatlifeboat freefallfreefall whenwhen thethe wavewave height is 2 m. Figure 20. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 2 m.

(a) (b) (a) (b) Figure 21. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 3 m. FigureFigure 21. 21.Trajectory Trajectory ( a(a)) and and pitch pitch ( b(b)) of of the the lifeboatlifeboat freefallfreefall whenwhen thethe wavewave heightheight is 3 m.

(a) (b) (a) (b) FigureFigure 22. 22.Trajectory Trajectory ( a(a)) and and pitch pitch ( b(b)) of of the the lifeboatlifeboat freefallfreefall whenwhen thethe wavewave heightheight is 4 m. Figure 22. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 4 m.

(a) (b) (a) (b) Figure 23. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 5 m. Figure 23. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 5 m.

(a) (b) (a) (b)

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 21 of 25

(a) (b)

Figure 21. Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 3 m.

(a) (b)

J. Mar. Sci. Eng.Figure2020, 22.8, 631 Trajectory (a) and pitch (b) of the lifeboat freefall when the wave height is 4 m. 21 of 25

J. Mar. Sci. Eng. 2020, 8, x FOR( PEERa) REVIEW (b) 22 of 25 FigureFigure 23.23. TrajectoryTrajectory ((aa)) andand pitchpitch ((bb)) ofof thethe lifeboatlifeboat freefallfreefall when when the the wave wave height height is is 5 5 m. m.

(a) (b) (a) (b)

(e) (f)

FigureFigure 24. The 24. The maximum maximum acceleration acceleration of of the the bow, bow, midship, midship, and and stern stern in in the the coordinate coordinate systemsystem of the lifeboat in the direction of 𝑏 (a,c,e), and in the direction of 𝑏 (b,d,f), at different wave heights. lifeboat in the direction of b3 (a,c,e), and in the direction of b1 (b,d,f), at diﬀerent wave heights.

Therefore, when the ship is heading the wave, P7 and P8 can be selected to enter the water at a low wave. In order to prevent the boat from moving to the side, one can choose point P1 at a high wave, which will withstand a large change in pitch. When the ship is following the wave, P1 and P8 can be selected at a low wave. In order to prevent the boat moving to the side, one can choose P2 and P3 at a high wave. In summary, it is safer to select points at the crest and behind the crest when entering the water.

J. Mar. Sci. Eng. 2020, 8, 631 22 of 25

The ship heads the wave. As shown in Figures 15–17a, when the wave height exceeds 3 m, the boat will move to the side of the ship after water entry at P7 and P8. As shown in Figures 16a and 17a, when the wave height exceeds 4 m, the same phenomenon appears at point P6. As shown in Figures 13–17b, at points P1 and P8, the pitch angle of the boat has a large range of change. As shown in Figure 18, in the direction of b1, the maximum values of acceleration at the bow, midship, and stern of the boat appear at point P4; the minimum value generally appears at P1 and P8. The maximum values at the bow and midship are almost the same, slightly lower than the stern. In the direction of b3, the maximum absolute values of acceleration at the bow and midship of the boat appear at points P3 and P2; the minimum value generally appears at points P7 and P8. There is no obvious rule about the values of acceleration at the stern; the acceleration is the largest at the bow and the smallest at the midship. The ship follows the wave. As shown in Figures 21–23a, when the height of the wave exceeds 3 m, the boat will move to the side of the ship after water entry at points P1 and P8. As shown in Figures 22 and 23a, when the height of the wave exceeds 4 m, the same phenomenon appears at point P7. As shown in Figures 19–23b, the pitch angle of the boat has a large range of change at points P5 and P6. As shown in Figure 24, in the direction of b1, the maximum values of acceleration at the bow and midship appear at points P6 and P7, and the minimum values generally appear at points P2 and P3. The maximum values of acceleration at the stern appear at points P4 and P5, and the minimum values generally appear at points P1 and P2. The maximum values at the bow and midship of the boat are almost the same, and slightly lower than for the stern. In the direction of b3, the maximum values of acceleration at the bow and midship of the boat appear at points P4 and P5; the minimum value generally appears at points P1 and P8. The maximum absolute values of the two extremal values at the stern appear at P4 and P5. The minimum values generally appear at P1 and P8; the acceleration is the largest at the bow and the smallest at the midship. Therefore, when the ship is heading the wave, P7 and P8 can be selected to enter the water at a low wave. In order to prevent the boat from moving to the side, one can choose point P1 at a high wave, which will withstand a large change in pitch. When the ship is following the wave, P1 and P8 can be selected at a low wave. In order to prevent the boat moving to the side, one can choose P2 and P3 at a high wave. In summary, it is safer to select points at the crest and behind the crest when entering the water.

5. Application This paper applies an established mathematical model to a ship’s lifesaving training system, improving the immersion and reality of the system. The system takes the Panama bulk carrier as a physicalJ. prototype,Mar. Sci. Eng. 2020 and, 8, x FOR uses PEER 3D REVIEW Studio Max (3ds Max) to build 3D models of the 23 ship, of 25 lifeboat, and skid. The system constructs a virtual scene of a ship life-saving drill, and releases the lifeboat through askid. three-dimensional The system constructs virtual a virtual operation, scene asof showna ship life-saving in Figure drill, 25. and releases the lifeboat through a three-dimensional virtual operation, as shown in Figure 25.

(a) (b)

Figure 25. FigureScene 25. of Scene virtual of humanvirtual human operating operating lifeboat lifeboat (a )(a and) and lifeboat lifeboat moving on on the the skid skid (b) (inb )ain a ship’s ship’s lifesaving training system. lifesaving training system. 6. Conclusions This paper presents a computational model for the simulation of lifeboat freefall during its launching from a ship in rough seas. The model is applied to the ship’s lifesaving training system. We can draw the following conclusions:

(1) The mathematical model in this paper can simulate the entire process of the water entry of the ship’s lifeboat and can acquire the parameters of the boat’s trajectory, pitch angle, velocity, local acceleration, etc. (2) The results of a numerical simulation experiment are compared with the calculation results of the hydrodynamics software Star CCM+ at water entry under initial condition A. It shows that our numerical algorithm has good accuracy. The model can be applied to other ships by adjusting the parameters. (3) Under different wave heights and two situations of the ship encountering waves, a qualitative analysis is performed to determine the safe point of water entry. It is safer to select points at the crest and behind the crest when entering the water. (4) Since the motion of the boat on the skid is only three-degrees-of-freedom, the effects of the ship’s roll, sway, and yaw are not considered; only the simulation experiments of the ship heading and following the wave are analyzed. Future research will consider the effect of the ship’s roll, sway, and yaw.

Funding: The authors are thankful for the financial support provided by National Natural Science Foundation of China (Grant No. 51679024), the Fundamental Research Funds for the Central Universities (Grant No. 3132020372), and Natural Science Foundation Guidance Project of Liaoning Province (Grant No. 2019-ZD-0152).

Conflicts of Interest: The authors declare no conflict of interest.

References

J. Mar. Sci. Eng. 2020, 8, 631 23 of 25

6. Conclusions This paper presents a computational model for the simulation of lifeboat freefall during its launching from a ship in rough seas. The model is applied to the ship’s lifesaving training system. We can draw the following conclusions: (1) The mathematical model in this paper can simulate the entire process of the water entry of the ship’s lifeboat and can acquire the parameters of the boat’s trajectory, pitch angle, velocity, local acceleration, etc. (2) The results of a numerical simulation experiment are compared with the calculation results of the hydrodynamics software Star CCM+ at water entry under initial condition A. It shows that our numerical algorithm has good accuracy. The model can be applied to other ships by adjusting the parameters. (3) Under different wave heights and two situations of the ship encountering waves, a qualitative analysis is performed to determine the safe point of water entry. It is safer to select points at the crest and behind the crest when entering the water. (4) Since the motion of the boat on the skid is only three-degrees-of-freedom, the effects of the ship’s roll, sway, and yaw are not considered; only the simulation experiments of the ship heading and following the wave are analyzed. Future research will consider the effect of the ship’s roll, sway, and yaw.

Author Contributions: S.Q. led the writing of the manuscript, contributed to data analysis and research design, and serves as the corresponding author. H.R. supervised this study, contributed to the funding acquisition, and serves as the corresponding author. H.L. contributed to the investigation and corrected the format of the paper. All authors have read and agreed to the published version of the manuscript. Funding: The authors are thankful for the financial support provided by National Natural Science Foundation of China (Grant No. 51679024), the Fundamental Research Funds for the Central Universities (Grant No. 3132020372), and Natural Science Foundation Guidance Project of Liaoning Province (Grant No. 2019-ZD-0152). Conflicts of Interest: The authors declare no conflict of interest.

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