applied sciences
Article Parameter Optimization for Ship Antiroll Gyros
Yuanyuan Zhu, Shijie Su * , Yuchen Qian, Yun Chen and Wenxian Tang
School of Mechanical Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China; [email protected] (Y.Z.); [email protected] (Y.Q.); [email protected] (Y.C.); [email protected] (W.T.) * Correspondence: [email protected]; Tel.: +86-135-1169-6751
Received: 26 November 2019; Accepted: 14 January 2020; Published: 16 January 2020
Abstract: Ship antiroll gyros are a type of equipment used to reduce ships’ roll angle, and their parameters are related to the parameters of a ship and wave, which affect gyro performance. As an alternative framework, we designed a calculation method for roll reduction rate and considered random waves to establish a gyro parameter optimization model, and we then solved it through the bacteria foraging optimization algorithm (BFOA) and pattern search optimization algorithm (PSOA) to obtain optimal parameter values. Results revealed that the two methods could effectively reduce the overall mass and floor space of the antiroll gyro and improved its antirolling effect. In addition, the convergence speed and antirolling effect of the BFOA were better than that of the PSOA.
Keywords: parameter optimization; ship antiroll gyro; roll reduction rate
1. Introduction The stability of a ship greatly influences its crew and built-in equipment [1]. Therefore, reducing the roll motion of a ship is crucial. Compared with other ship antiroll products, antiroll gyros have advantages, such as easy installation, low energy consumption, and antiroll capability, at any speed of the ship [2,3]. In 1904, Schlick first proposed placing a large gyro on a ship to provide a roll-damping moment [4]. In 1917, Orden modified the structure of the ship antiroll gyro to make the structure simpler [5]. In 1925, Thompson introduced a new antiroll gyro that reduced the gyro’s energy consumption [6]. Perez and Steinmann proposed using several small antiroll gyros to distribute the overall capsizing moment of the ship, monitor it, and adjust the number of gyros according to the variable navigational conditions [7,8]. With scientific progress, research is no longer limited to structural optimization but improving the performance of gyros around the control method [9–11]. The rotating gyro rotor produces a damping moment opposite to the swaying direction of the ship during precession. Therefore, various gyro parameters produce various damping moments and antirolling effects. However, research has been limited to the optimization of the gyro’s structure and control method and the lack of parameter optimization. Similar to antiroll gyros, tuned mass dampers (TMDs) are often used to reduce vibration in high-rise buildings and bridges. Researchers show that parameter optimization is an effective means to improve damping performance [12–14]. Xin took the minimum standard deviation of the fore-aft displacement at the top of a tower as the control objective and optimized the mass, damping, and stiffness coefficients of the TMD. The results indicated that the fore-aft displacement was reduced by 54.5% through the parameter optimization method [14]. Scholars have proposed some classical optimization algorithms and intelligent optimization algorithms to solve different optimization problems [15–18]. Pattern search optimization algorithm (PSOA) is one of the classical algorithms. It is a method for solving optimization problems that do not require any information about the gradient of the objective function [15]; however, this method tends
Appl. Sci. 2020, 10, 661; doi:10.3390/app10020661 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 661 2 of 15 to fall into a locally optimal solution. In contrast, the bacteria foraging optimization algorithm (BFOA) is a new swarm intelligence optimization algorithm. It has the advantages of simple realization, group parallel search, fast convergence speed, and easy to jump out of the local optimal solution [16], so it has been widely used in many engineering fields. Referring to the parameter optimization of TMDs and considering the interaction between the wave, ship, and gyro, this paper established the joint dynamical equation of ships and antiroll gyros under random waves. Most control objectives in TMDs are intuitive, such as displacement. However, our objective, the roll reduction rate, needs to be calculated through the roll angles’ mean value over a period of time. Therefore, considering the nonreal time of the roll reduction rate, we proposed resolving the roll reduction rate through continuous iteration and designed its calculation method. In addition, considering the lack of research on gyro’s parameter optimization, we established a gyro’s parameter optimization model and then solved it through the BFOA and PSOA to obtain optimal parameter values.
2. Dynamical Model of Ship’s Antiroll Gyro
2.1. Mathematical Model of Random Waves The interference moment of random waves was mainly related to the wave slope angle α(t), and the essence of α(t) was to convert the spectrum of waves into that of α(t). Where α(t) is the maximum inclination of the wave surface on a vertical section orthogonal to the crest, and the wave slope angle α(t) [19] was defined as follows: s N Z ω X ei α(t) = 2 S (ωe)dωe cos(ωe + ε ) sin(χ), i = 1, 2, (1) σ i i ωei 1 ··· i=1 − where ωe is the encounter frequency, εi is the random phase angle uniformly distributed between 0 and 2π, N is the number of selected harmonics, and χ is the course angle. Sσ(ωe) is the spectrum function of wave slope angle, which could be obtained with the density of the wave energy spectrum Sζ(ω):
ω4 ( ) g2 Sζ ω S (ω ) = (2) σ e 2ω 1 + g u cos χ where g is the gravitational acceleration, u is the ship sailing speed, and ω is the harmonic angular frequency. The two-parameter spectrum proposed by the International Towing Tank Conference (ITTC) was used as the density of the wave energy spectrum Sζ(ω) [20]:
2 691 173h − 1/3 ω4T4 S (ω) = e 1 (3) ζ 5 4 ω T1 where h1/3 is the significant wave height and T1 is the mean period of the wave.
2.2. Ship Rolling Mathematical Model under Random Wave Excitation .. The stress of the ship is shown in Figure1. α(t) is the wave slope angle, Iφφ + Jφφ φ is the mass . inertia moment, R φ is the roll-damping moment, K(φ) is the roll-restoring moment, and M(χ, ω, t) is the wave excitation moment. Appl. Sci. 2020, 10, 661 18 of 19
2.2. Ship Rolling Mathematical Model Under Random Wave Excitation The stress of the ship is shown in Figure 1. α ()t is the wave slope angle, ()IJφφ+ φφ φ is the φ φ Appl.mass Sci. inertia2020, 10 moment,, 661 R() is the roll-damping moment, K () is the roll-restoring moment,3 ofand 15 M ()χω,,t is the wave excitation moment.
Figure 1. The stress of the ship.
Based onon the the Mathieu Mathieu equation equation [21 [21],], considering considering the nonlinearthe nonlinear damp damp and ship and nonlinear ship nonlinear restore moment,restore moment, the ship the rolling ship motionrolling equationmotion equation was established was established as follows as follows [22]: [22]: .. . I + J φ + R φ + K(φ) = M(χ, ω, t) (4) φφ +++=φφ φφ φ χω ()IφφJRKMt φφ () () (,, ) (4) where Iφφ is the ship’s moment of inertia, Jφφ is the moment of inertia of additional mass, φ is the roll .. φ angle,where φIisφφ the is the roll ship’s angular moment acceleration, of inertia, and χJisφφ the is the course moment angle. of inertia of additional mass, is . the rollThe angle, roll-damping φ is the moment roll angularR φ acceleration,was calculated and as theχ is linear the course damping angle. plus cubic damping [22]: The roll-damping moment R()φ was . calculated. as. 3the linear damping plus cubic damping R φ = c0 φ + c30 φ (5) [22]: 1 . where φ is the roll angular velocity, and c0 and c0 are the damping moment coefficients. 1 ()φφφ=+3 ''3 To simplify the calculation, the roll-restoringRcc13 moment was approximated to a fifth-degree(5) polynomial [22]: 3 5 φ K(φ) = K' 1φ + K3' φ + K5φ (6) where is the roll angular velocity, and c1 and c3 are the damping moment coefficients. whereToK 1simplify, K3, and theK5 arecalculation, the restoring the momentroll-restoring coeffi cients.moment was approximated to a fifth-degree polynomialThe wave [22]: excitation moment was expressed as a function of wave slope angle [22]:
M()φφφφ(χ=+, ωe, t) = Dh35 +α(t) (7) KKKK13 5 (6) where D is the ship’s displacement, h is the transverse metacentric height, and α(t) is the wave slopewhere angle.K1 , K3 , and K5 are the restoring moment coefficients. TheThe wave ship excitation rolling mathematicalmoment was expressed model couldas a function be transformed of wave slope into: angle [22]:
.. . . 3 3 5 φ = c1φ c3φ kMtDh1φ()χω,,k3φ = k5φ α ()+ tDhα(t)/ Iφφ+Jφφ (8) − − − − e − (7) where ci = c0/ Iφφ+Jφφ , i = 1, 3 and kj = Kj/ Iφφ+Jφφ , j = 1, 3, 5. where D isi the ship’s displacement, h is the transverse metacentric height, and α ()t is the 2.3.wave Joint slope Dynamical angle. Equation of Ship and Antiroll Gyro The ship rolling mathematical model could be transformed into: As displayed in Figure2, the antiroll gyro was mounted on the ship deck. The antiroll gyro φφφφφφ−− 335 −− − αO() consisted of a precession axis,=c frame,13 c rotor, k 1 and k rotor 3 spindle.k 5 +Dhtξηζ ()is I+Jφφ the φφ absolute coordinate system,(8) and φ is the roll angle of the hull around Oζ. Oxyz is the relative coordinate system, and Oz is the rotating axis of the rotor. Oy is the rotating axis of the outer frame, and β is the precession angle of the gyro around Oy. Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 19
' cc==()I+J,φφ φφ i 1,3 kK==()I+J,φφ φφ j 1,3,5 where ii and jj .
2.3. Joint Dynamical Equation of Ship and Antiroll Gyro As displayed in Figure 2, the antiroll gyro was mounted on the ship deck. The antiroll gyro consisted of a precession axis, frame, rotor, and rotor spindle. Oξηζ is the absolute coordinate system, and φ is the roll angle of the hull around Oζ . Oxyz is the relative coordinate system, and Appl.Oz is Sci. the2020 ,rotating10, 661 axis of the rotor. Oy is the rotating axis of the outer frame, and β is4 ofthe 15 precession angle of the gyro around Oy .
Figure 2. StructureStructure and and principle of the antiroll gyro.
The mass of the gyro’s outer frame was neglected, and the influence influence of the ship’s movement on other degreesdegrees ofof freedomfreedom on on the the gyro gyro was was not not considered. considered. The The rotor rotor was was an axisymmetric an axisymmetric rigid bodyrigid whose moment of inertia around Oz was Iz, the rotor speed was constant at ω0, and theω momentum body whose moment of inertia around Oz was I z , the rotor speed was constant at 0 , and the moment constant of Ox and Oy was J. According to the Euler equation of the motion of rigid bodies [23], momentum moment constant of Ox and Oy was J . According to the Euler equation of the the motion equation of the antiroll gyro relative to Oxyz was as follows: motion of rigid bodies [23], the motion equation of the antiroll gyro relative to Oxyz was as .. . follows: Mx = Jφ cos β + h0β .. . . 2 (9) My = Jβ + Jφ sin β cos β h0φ cos β MJ=+φβ. cos h β − Mzx = Iω = 0 0 0 MJJ=+βφ 2 sin β cos β − h φβ cos y 0 . (9) where Mx, My, and Mz are the componentsMI==ω of0 the resultant external torque on Oxyz, β is the precession .. z 0 angular velocity, β is the precession angular acceleration, and h0= Izω0 is the momentum moment constant of the gyro. If Oy was considered as the input axis, then input torque M caused the gyro to y β where M x , M , and Mz are the components of the resultant external torque on Oxyz , is the precess, and theny a torque Mx output on Ox occurred. The output torque Mx from the above motion precessionequation was angular projected velocity, into O ξηζβ . is Given the theprecession stability ofan thegular high-speed acceleration, spinning and gyro,h=I theω angular is the . 00z momentumvelocity φ was moment considerably constant smaller of the than gyro.ω 0If, andOy itswas second considered derivative as the was input ignored. axis, Equationthen input (9) torque could thenM becaused simplified the gyro as follows: to precess, and then a torque M output on Ox occurred. The output y . x M = h torque M from the above motion equation ξ was0β cosprojectedβ into Oξηζ . Given the stability of the x . Mζ = h0β sin β (10) .. φ . ω high-speed spinning gyro, the angular velocity was considerably smaller than 0 , and its My = Jβ h0φ cos β second derivative was ignored. Equation (9) could− then be simplified as follows: The damping device was added in the precession direction of the antiroll gyro, and the precession of the gyro was restricted appropriately depending= ββ on the characteristics of damping to improve the Mhξ 0 cos . antirolling effect. The total damping torque My could be expressed as My = Cβ, and C was defined as M = hsinββ the gyro’s damping coefficient in units of Nsζ /m, which0 was the ratio of the damper’s damping force(10) =−βφβ installed in the gyro to the movement speedMhy0 of theJcos damper’s piston rod. Furthermore, the motion equation of the ship and the mathematical model of the antiroll gyro couldThe be simultaneouslydamping device established was added to obtainin the the precession ship antiroll direction gyro’s motionof the equation:antiroll gyro, and the precession of the gyro was restricted appropriately depending on the characteristics of damping to .. . . 3 . 3 5 φ = c1φ c3φ k1φ k3φ k5φ h0β cos(β)/ Iφφ+Jφφ + Dhα(t)/ Iφφ+Jφφ .. − . − − . − − − (11) β= h φcos(β)/J Cβ/J 0 − 3. Study of the Antirolling Characteristics of the Gyro Following Equation (11), this Section describes the antirolling characteristics of a public service ship and its supporting antiroll gyro and the influence of various parameters of the antiroll gyro on the roll angle of the ship. The ship is presented in Figure3, and the parameters of it are presented in Table1. The structure and parameters of the supporting antiroll gyro are presented in Table2 and Appl. Sci. 2020, 10, 661 20 of 19
= β improve the antirolling effect. The total damping torque M y could be expressed as M y C , and C was defined as the gyro’s damping coefficient in units of Ns/m, which was the ratio of the damper’s damping force installed in the gyro to the movement speed of the damper’s piston rod. Furthermore, the motion equation of the ship and the mathematical model of the antiroll gyro could be simultaneously established to obtain the ship antiroll gyro’s motion equation:
φφφφφφββ−− 335 −− − − ()() α()() =c13 c k 1 k 3 k 5 h 0 cos I+J+Dhtφφ φφ I+J φφ φφ (11) βφ () β− β =h0 cos JC J
3. Study of the Antirolling Characteristics of the Gyro
Appl. Sci.Following 2020, 10, 661Equation (11), this Section describes the antirolling characteristics of a public service20 of 19 shipAppl. Sci.and2020 its, supporting10, 661 antiroll gyro and the influence of various parameters of the antiroll gyro5 of on 15 = β theimprove roll angle the antirolling of the ship. effect. The ship The istotal presented damping in torqueFigure 3,M andy could the parameters be expressed of it as are M presentedy C , and in TableFigureC was 1.4. Thedefined The structure gyro’s as the rotor an gyro’sd hadparameters andamping axisymmetric of thecoefficient supporting structure, in units antiro and ofll the Ns/m,gyro intermediate are which presented was rotating the in Tableratio shaft of2 and andthe Figurerotatingdamper’s 4. outer Thedamping gyro’s ring were force rotor connected installed had an inaxisymmetric by the rib gyro welding. to thsteructure, movement and speedthe intermed of the damper’siate rotating piston shaft rod. and rotatingFurthermore, outer ring thewere motion connected equation by rib of welding. the ship and the mathematical model of the antiroll gyro could be simultaneously established to obtain the ship antiroll gyro’s motion equation:
φφφφφφββ−− 335 −− − − ()() α()() =c13 c k 1 k 3 k 5 h 0 cos I+J+Dhtφφ φφ I+J φφ φφ (11) βφ () β− β =h0 cos JC J
3. Study of the Antirolling Characteristics of the Gyro Following Equation (11), this Section describes the antirolling characteristics of a public service ship and its supporting antiroll gyro and the influence of various parameters of the antiroll gyro on Figure 3. Marine surveillance ship and antiroll gyro. the roll angle of the ship. TheFigure ship 3. isMarine presented surveillance in Figure ship 3,and and antiroll the parameters gyro. of it are presented in Table 1. The structure and parametersTable of the 1. Ship’ssupporting parameters. antiroll gyro are presented in Table 2 and
Figure 4. The gyro’sParameter rotor had an axisymmetric Value structure, Parameterand the intermediate rotating Value shaft and rotating outer Totalring length wereL (m)connected by 44.8 rib welding. Ship’s roll-damping coefficient c1 (Ns/m) 843 Molded breadth b (m) 8.9 Ship’s roll-damping coefficient c3 (Ns/m) 6589 2 Molded depth H (m) 4 Roll-restoring moment coefficient k1 (Ns /m) 10,791 2 Displacement d (t) 594 Roll-restoring moment coefficient k3 (Ns /m) –9284 2 Initial metacentric height h (m) 1.09 Roll-restoring moment coefficient k5 (Ns /m) 894 2 6 Total moment of inertia Js (Nms ) 6 10 ×
Table 2. Parameters of the antiroll gyro.
Parameter Value Parameter Value
Rotor inner diameter D1 (m) 1.07 Gyro’s damping coefficient C (Ns/m) 25,300 Rotor outer diameter D2 (m) 1.24 Motor speed ωm (r/min) 1255 Figure 4. Structure of supporting antiroll gyro rotor. 3 Rotor shaft diameter D3 (m) 0.18 Rotor material density ρ (kg/m ) 7850 Rotor thickness H1 (m) 0.70 Rotational momentum moment h0 (Nms) 104,100 Shaft length H (m) 1.05 Precession momentum moment J (Nms) 25,100 2 Ribbed plate thickness HFigure3 (m) 3. Marine 0.05 surveillance ship and antiroll gyro.
Figure 4. Structure of supporting antiroll gyro rotor. Figure 4. Structure of supporting antiroll gyro rotor. A comparison and analysis of the motion equation of the rolling ship and antiroll gyro indicated that the antirolling effect of the ship was related to the rotational momentum moment h0, the precession momentum moment J, and the gyro’s damping coefficient C.
3.1. Influence of Gyro’s Damping Coefficient on Roll Reduction Rate By substituting values of gyro’s damping coefficients into Equation (11) to solve the differential equation, the variation in the ship roll reduction rate under corresponding gyro’s damping coefficients could be obtained. Where the roll reduction rate means the rate at which the roll angle decreases when the gyro is working compared to that when the gyro is not working, and it would be defined qualitatively in Equation (18). As displayed in Figure5, when the gyro’s damping coe fficient increased, Appl. Sci. 2020, 10, 661 6 of 15 the roll reduction rate gradually increased. When the gyro’s damping coefficient C = 16, 000 Ns/m, the roll reduction rate reached the maximum value and then began to decline. Considering that the mean value period has little influence on the wave slope angle, we only studied the influence of gyro’s damping coefficients under different significant wave heights on roll reduction rate. Usually, when the wave height is higher than 5 m, the surveillance ship will not cruise. So the roll reduction rate under different gyro’s damping coefficients was solved in five cases with wave heights of 1–5 m, and the significant wave height of 5 m was taken as the subsequent optimization condition. The five coordinates in Figure6 were, respectively, the maximum values of the roll reduction rate when the wave height was 1–5 m. Figure6 illustrates that as the significant wave height changed, theAppl. optimal Sci. 2020, interval10, 661 of gyro’s damping coefficient changed very little. Therefore, the gyro’s damping22 of 19 coeAppl.ffi Sci.cient 2020 was, 10, 661 taken as a constraint condition in the subsequent simulation. 22 of 19
Figure 5. Roll reduction rate under different gyro’s damping coefficients. Figure 5. Roll reduction rate under different gyro’s damping coefficients. Figure 5. Roll reduction rate under different gyro’s damping coefficients.
Figure 6. Roll reduction rate under different gyro’s damping coefficients and wave heights.
3.2. Influence of Rotational and Precession Momentum Moment on Roll Reduction Rate Figure 6. Roll reduction rate under different gyro’s damping coefficients and wave heights. By substitutingFigure 6. Roll di reductionfferent values rate under of h different0 and J intogyro’s Equation damping (11) coefficients for the and solution, wave heights. the variation in 3.2. Influence of Rotational and Precession Momentum Moment on Roll Reduction Rate roll reduction rate with the corresponding h0 and J is illustrated in Figure7, which shows that the roll reduction rate increased with increasing h0 and decreased with increasing J. 3.2. InfluenceBy substituting of Rotational different and Precession values of Momentumh0 and J Moment into Equation on Roll Reduction(11) for the Rate solution, the variation To identify the relationship among roll reduction rate, rotor diameter, and thickness more directly, in rollBy reduction substituting rate different with the values corresponding of h and hJ0 and into EquationJ is illustrated (11) for inthe Figure solution, 7, which the variation shows we determined the roll reduction rate with0 various rotor diameter and thickness in Figure8, which that the roll reduction rate increased with increasingh h0 J and decreased with increasing J . indicatedin roll reduction that an increaserate with in rotorthe corresponding diameter resulted 0 inand an increase is illustrated in gradual in roll Figure reduction 7, which rate. shows When To identify the relationship among roll reduction rate, rotor diameter, and thickness more thethat rotor the roll diameter reduction was rate 0.5 m, increased as the rotor with thickness increasing increased, h0 and the decreased roll reduction with increasing rate increased J . first and thendirectly,To decreased identifywe determined before the relationship reaching the roll a peakreduction among of 70% rollrate when redu with thection various rotor rate, thickness rotor rotor diameter wasdiameter, 0.75 and m. and Withthickness thickness the increase in Figure more of rotordirectly,8, which diameter, we indicated determined the influence that the an of rollincrease rotor reduction thickness in rotor rate on wi thedithameter roll various reduction resulted rotor rate diameter in decreasedan increase and gradually.thickness in gradual However,in Figure roll thereduction8, which increase indicatedrate. of rotor When diameterthat the an rotor andincrease thicknessdiameter in rotor ledwas to di0.5 anameter increasem, as resulted the in gyro’srotor in powerthicknessan increase consumption increased, in gradual andthe floorroll space.reduction Therefore, raterate. increasedWhen it was the necessary first rotor and todiameter then establish decreased was a parameter 0.5 before m, as optimization reachingthe rotor a thickness model,peak of set 70% increased, up when constraints, the the rotor androll determinereductionthickness wasrate the 0.75 optimalincreased m. With solution first the and increase of gyro then parameters ofdecreased rotor diam inbeforeeter, consideration thereaching influence ofa variouspeak of rotor offactors. 70%thickness when on the the rotor roll thicknessreduction wasrate 0.75decreased m. With gradually. the increase However, of rotor the diam increaseeter, the of rotorinfluence diameter of rotor and thickness thickness on led the to roll an reductionincrease in rate gyro’s decreased power gradually. consumption However, and floor the inspace.crease Therefore, of rotor diameter it was necessary and thickness to establish led to an a increaseparameter in optimizationgyro’s power model, consumption set up andconstraint floor space.s, and Therefore,determine it the was optimal necessary solution to establish of gyro a parameterparameters optimizationin consideration model, of various set up factors. constraint s, and determine the optimal solution of gyro parameters in consideration of various factors. Appl. Sci. 2020, 10, x FOR PEER REVIEW 23 of 19 Appl. Sci. 2020, 10, 661 7 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 23 of 19
Figure 7. RollRoll reduction reduction rate rate with with various various rotational and precession momentum moments. Figure 7. Roll reduction rate with various rotational and precession momentum moments.
Figure 8. Roll reduction rate with various rotor diameter and thickness. Figure 8. Roll reduction rate with various rotor diameter and thickness. 4. Mathematical Model for Parameter Optimization of Antiroll Gyro Figure 8. Roll reduction rate with various rotor diameter and thickness. 4. MathematicalAccording to Model the study for Parameter in Section3 Optimization, there was a complicated of Antiroll nonlinearGyro relationship between the roll4. Mathematical reductionAccording rate to Model andthe thestudy for rotor’s Parameter in Section size. In Optimization addition,3, there was since aof thecomplicated Antiroll gyro is Gyro restricted nonlinear by therelationship ship, the parameter between theoptimization rollAccording reduction is needed to ratethe toandstudy obtain the in rotor’s theSection optimal size. 3, there parameters.In addi wastion, a complicatedIn since this Section,the gyro nonlinear the is shiprestricted rollrelationship reduction by the ship,between rate andthe parameterthethe rotorroll reduction mass optimization were rate modeled and is neededthe as rotor’s objective to obtainsize. functions. In the addi option,timal Then, since parameters. the shipthe gyro space, Inis powerthisrestricted Section, drive, by the andthe shipship, material rollthe reductionstrengthparameter were rate optimization consideredand the rotor is as needed constraintsmass were to obtain modeled to form the the asop optimizationobjectivetimal parameters. functions. model. InThen, this theSection, ship space,the ship power roll drive, and material strength were considered as constraints to form the optimization model. 4.1.reduction Establishing rate and the Mathematicalthe rotor mass Model were of modeled the Ship Rollas objective Reduction functions. Rate Then, the ship space, power drive, and material strength were considered as constraints to form the optimization model. 4.1. EstablishingThe antirolling the Mathematical capability ofModel gyro of could the Ship be Roll directly Reduction evaluated Rate by the ship roll reduction rate. The solution of the ship roll reduction rate required continuous iteration, and its iterative process is 4.1. EstablishingThe antirolling the Mathematical capability of Model gyro ofcould the Ship be Rolldirectly Reduction evaluated Rate by the ship roll reduction rate. presented in Figure9. The solutionThe antirolling of the ship capability roll reduction of gyro rate could required be directly continuous evaluated iteration, by the and ship its rolliterative reduction process rate. is presentedThe1. solutionWhen in the Figureof gyrothe ship9. was roll in a reduction nonworking rate state required continuous iteration, and its iterative process is presented in Figure 9. According to Equation (8), the ship’s mathematical model could be established as follows: .. . . φ = c φ c φ 3 k φ k φ 3 k φ 5 + Dhα(t)/ I +J (12) b − 1 b − 3 b − 1 b − 3 b − 5 b φφ φφ where φb is the ship’s roll angle when the gyro is not in operation. . Let x = φb. Equation (12) could be transformed into an initial value problem of first-order differential equations: x(0) = φb1 = x0, φb(0) = φb0 . ( ) = ( ) (13) φb t x t . x(t) = f1(t, φb(t), x(t)) where f (t, φ (t), x(t)) = c x(t) c x(t)3 k φ (t) k φ (t)3 k φ (t)5 + Dhα(t)/ I +J . 1 b − 1 − 3 − 1 b − 3 b − 5 b φφ φφ Appl. Sci. 2020, 10, 661 8 of 15
By solving the differential equation for time t, the following equation could be obtained: . φb(0) = φb0, φ (0) = x0 b φ = φ + lx bk b(k 1) k (14) − xk = xk 1 + l f1(tk, φbk, xk), k = 1, 2 − ··· tk = tk 1 + l − where l is the iterative step length of time t, and k is the number of iterations. Then, the expression of theAppl. roll Sci. angle2020, 10φ, b661changing with time t could be obtained from Equation (14), φb = φb(k 1) + lxk24. of 19 −
Discretize the joint dynamical equation of ship and anti-roll gyro
The gyro is in non-working state The gyro is in working state
Initialize Initialize the ship roll angle φ : φφ()0 = the ship roll angle φ : φ ()0 = φ a aa0 b bb0 φ φ ()= the ship roll angular velocity a : a 0 x0 φ φ ()= the gyro processional angle β : ββ()0 = the ship roll angular velocity b : b 0 x0 0 β β ()= the gyro processional angular velocity : 0 y0
Iterative The iterative step length of time t is l , the iterative number is k =+ ttkk−1 l
Circulation Solve Circulation
φφ=+ akak()−1 lx k φφ=+lx bkbk()−1 k ββ=+ kk−1 ly k =+ = xxkk−−11 lxk k,1,2 =+ = xxkk−1 lxk k,1,2 =+ = yykk−1 lyk k,1,2
Roll reduction rate SS()φφ− () TT =×ab100% S ()φ a Figure 9. Solution process of ship roll reduction rate. Figure 9. Solution process of ship roll reduction rate. 2. When the gyro was in a working state 1. When the gyro was in a nonworking state AccordingAccording toto EquationEquation (11), (8), the ship’s mathematical mathematical model model of the shipcould antiroll be established gyro could as befollows: established as follows: .. . . φφφφφφα=c−− c 335 −− k k. − k +Dht ()() I+J 3 b1b3b1b3b5b3 5 φφ φφ (12) φa = c1φa c3φa k1φa k3φa k5φa h0β cos(β)/ Iφφ + Jφφ + Dhα(t)/ Iφφ + Jφφ .. −. − − . − − − (15) β = h φ cos(β)/J Cβ/J φ 0 a − where b is the ship’s roll angle when the gyro is not in operation. where φa is= theφ ship’s roll angle when the gyro is in operation. Let x .b . Equation .(12) could be transformed into an initial value problem of first-order = = differentialLet x equations:φa and y β. Equation (15) could be transformed into an initial value problem of first-order differential equations: xx()0,0==φφφ () = x(0) = φa1 = x0, φabbb(100) = φa0 0 ( ) = =φ ()txt=( ) ()= y 0 β1 by0, β 0 β0 (13) . . (16) φ (t) = x(t),()x(t=) =()f φ(t, ()φ (t) (), x(t), β(t), y(t)) a xt f1 t1,,b ta xt . . β(t) = y(t), y(t) = f2(t, φa(t), x(t), β(t), y(t)) 335 ft(),,=ccφφφφα() txt () −− xt() xt () − k () t − k () t − k () t +DhtI+J ()()φφ φφ where 1b 1 3 1b3b 5b . By solving the differential equation for time t , the following equation could be obtained:
φφφ()== () bbb0,000x φφ=+ bkbk()−1 lx k (14) =+()φ = xxkk−11 lft kbkk,, xk , 1,2 =+ ttkk−1 l Appl. Sci. 2020, 10, 661 9 of 15
where,
3 3 5 f1(t, φa(t), x(t), β(t), y(t)) = c1x(t) c3x(t) k1φa(t) k3φa(t) k5φa(t) h0 y(t) cos(β(t))/ I + J − − − − − − φφ φφ +Dhα(t)/ Iφφ + Jφφ , f (t, x(t), β(t), y(t)) = h x(t)cos(β(t))/J Cy(t)/J. 2 0 − By solving the differential equation for time t, the following equation could be obtained: . φa(0) = φa0, φ (0) = x0 . a β(0) = β0, β(0) = y0 φ = φ + lx , β = β + ly ak a(k 1) k k k 1 k (17) − − xk = xk 1 + l f1(tk, φak, xk, βk, yk), k = 1, 2 − ··· y = y + l f (t , x , β , y ), k = 1, 2 k k 1 2 k k k k − ··· tk = tk 1 + l − where l is the iterative step to determine the length of time t, and k is the number of iterations. Then, the expression of roll angle φa changing with time t could be obtained from Equation (17), φa = φa(k 1) + lxk. − Expression of Ship Roll Reduction Rate The ship roll reduction rate TT was defined as follows:
S(φa) S(φb) TT = − 100% (18) S(φa) × s s k 2 k 2 ( ) 1 P ( ) 1 P where S φb = k φbi φb , S φa = k φai φa , φb is the average of φb from φb1 to φbk, i=1 − i=1 − and φa is the average of φa from φa1 to φak.
4.2. Objective Functions Different ships have different working environments and different requirements for their antiroll gyros, which can be roughly summarized by two points: (1) the efficiency principle, which refers to the antirolling effect that antiroll gyro can achieve and is typically expressed as the roll reduction rate; and (2) the lightweight principle, which refers to the gyro’s overall mass that is as small as possible because the mass of the antiroll gyro is mainly concentrated on the rotor and can also be expressed as the mass of the rotor. 1. Highest roll reduction rate The roll reduction rate was expressed in Equation (18), and the objective function was expressed as follows: minZ = 1 TT (19) 1 − 2. Minimum rotor mass The structure and size of the rotor are displayed in Figure4. Therefore, the mass of the rotor could be expressed as follows:
Ms = ρV = ρ[H (D D ) + 2H D + H D ] (20) 1 2 − 1 3 1 2 3 where V is the volume of the rotor in Figure4. The objective function was expressed as follows:
minZ2 = Ms (21) Appl. Sci. 2020, 10, 661 10 of 15
4.3. Constraint Conditions The dimensional constraint of the gyro rotor was obtained based on the maximum allowable mounting size of the gyro, given the assumption that the gyro’s maximum allowable mounting size is h hw h , where h , hw, and h are the length, width, and height of the mounting size. As shown in l × × h l h Figure4, the rotor’s inner diameter D1, the rotor’s outer diameter D2, the rotor’s shaft diameter D3, the rotor thickness H1, and the ribbed plate thickness H3 met the following constraints: ( D < D < min(h , h , h ), a < H < min(h , h , h ) 2 1 l w h 4 1 l w h (22) a1 < D2, a2 < D3 < a3, a5 < H3 < a6 where a , i 1, 2, , 6 are all positive, and their specific values were selected according to the actual i ∈ { ··· } size of the gyro’s rotor. The dimensional constraint of motor speed ωm was obtained based on the available power of the 9550P ship’s electrical system. As ωm = T , where P is the rated power and T is the rated torque, given the assumption that the power cap is Ps, the ω met the following constraints:
9550Ps a < ωm (23) 7 ≤ T where a7 is positive, and its specific value was selected empirically between 1050 and 1150. The constraint of the gyro’s damping coefficient was expressed as a C a , in which a and a 8 ≤ ≤ 9 8 9 were obtained from Figure6. The constraint of the mass of the gyro rotor was Ms a , where a was ≤ 10 10 selected according to gyro’s size. The constraint of the roll reduction rate was TT a , where the ≥ 11 specific value of a11 was given by the ship’s designer. Then, the constraint conditions can be summarized as follows: D < D < min(h , h , h ), a < D 2 1 l w h 1 2 a4 < H1 < min(hl, hw, hh), a2 < D3 < a3 9550Ps (24) a5 < H3 < a6, a7 < ωm ≤ T a C a , Ms a , a TT 8 ≤ ≤ 9 ≤ 10 11 ≤ 5. Comparative Analysis of Parameter Optimization Results
5.1. Principles of PSOA and BFOA PSOA is a direct search method, which only uses the function value instead of the derivative, so it is very effective in solving the optimization problem of functions that are not differentiable or difficult to differentiate. As shown in Figure 10, PSOA searches a set of points around the current point, looking for one where the value of the objective function is lower than the value at the current point. The equation of roll reduction rate is nonlinear, which is difficult to solve using the gradient-based optimization algorithm. So, PSOA is a feasible method to achieve the optimal object of this paper. However, PSOA tends to fall into the local optimal solution, and the selection of initial value has a great impact on the result. To overcome these shortcomings, we further applied a parallel search method, BFOA. BFOA is a new swarm intelligence optimization algorithm, which can be summarized as searching for food, moving the location, and digesting food. As shown in Figure 11, BFOA achieves optimization through three behaviors: Chemotaxis, replication, and dispersal. The chemotaxis can ensure the local search ability of bacteria, the replication can accelerate the search speed of bacteria, and the dispersal can enhance the global optimization ability of the algorithm. Appl. Sci. 2020, 10, 661 28 of 19
Then, the constraint conditions can be summarized as follows:
<<() < DD21min hhhaD l ,wh , , 12 aH< 5. Comparative Analysis of Parameter Optimization Results 5.1. Principles of PSOA and BFOA PSOA is a direct search method, which only uses the function value instead of the derivative, so it is very effective in solving the optimization problem of functions that are not differentiable or difficult to differentiate. As shown in Figure 10, PSOA searches a set of points around the current point, looking for one where the value of the objective function is lower than the value at the current point. The equation of roll reduction rate is nonlinear, which is difficult to solve using the gradient-based optimization algorithm. So, PSOA is a feasible method to achieve the optimal object of this paper. However, PSOA tends to fall into the local optimal solution, and the selection of initial value has a great impact on the result. To overcome these shortcomings, we further applied a parallel search method, BFOA. BFOA is a new swarm intelligence optimization algorithm, which can be summarized as searching for food, moving the location, and digesting food. As shown in Figure 11, BFOA achieves optimization through three behaviors: chemotaxis, replication, and dispersal. The chemotaxis can ensureAppl. Sci. the2020 local, 10, 661 search ability of bacteria, the replication can accelerate the search speed of bacteria,11 of 15 and the dispersal can enhance the global optimization ability of the algorithm. Figure 10. Schematic diagram of the pattern search optimization algorithm. Appl. Sci. 2020, 10, xFigure FOR PEER 10. SchematicREVIEW diagram of the pattern search optimization algorithm. 29 of 19 Figure 11. Schematic diagram of the bacteria foraging optimization algorithm. Figure 11. Schematic diagram of the bacteria foraging optimization algorithm. 5.2. Results and Analysis of the Highest Roll Reduction Rate 5.2. Results and Analysis of the Highest Roll Reduction Rate By taking the ship and its supporting antiroll gyro provided in Section3 as an example, this Section conductedBy taking a simulation the ship analysisand its supporting on the parameter antiroll optimization. gyro provided Given in Section the ship’s 3 as size, an theexample, parameter this optimizationSection conducted model a with simulation the highest analysis roll reductionon the pa raterameter as the optimization. objective could Given be writtenthe ship’s as follows:size, the parameter optimization model with the highest roll reduction rate as the objective could be written as follows: =− minZTT1 1 ≤≤ ≤≤ 0.8DD12 1.2, 1 1.4 ≤≤ ≤≤ 0.15DH31 0.22, 0.5 0.9 ≤≤ ≤≤ω (25) st. . 0.03 H3 0.06,1100m 1300 10000≤≤CMs 40000, ≤ 2000 ≤≥ 0.75TT , D21 - D 0.06 The relationship between the roll reduction rate and the iterations of the PSOA and BFOA is illustrated in Figure 12, and the solution time and results of the two algorithms are shown in Figure 13. Figures 11 and 12 indicate that the convergence speed of the BFOA was faster, and its roll reduction rate was higher than that of the PSOA. With the same mass of 2000 kg, the roll reduction rate obtained through the BFOA reached 79.6%, higher than that of the PSOA (78.5%). Appl. Sci. 2020, 10, 661 12 of 15 minZ1 = 1 TT − 0.8 D 1.2, 1 D 1.4 1 2 ≤ ≤ ≤ ≤ 0.15 D3 0.22, 0.5 H1 0.9 ≤ ≤ ≤ ≤ (25) s.t. 0.03 H 0.06, 1100 ω 1300 3 m ≤ ≤ ≤ ≤ 10, 000 C 40, 000, Ms 2000 ≤ ≤ ≤ 0.75 TT, D D 0.06 ≤ 2 − 1 ≥ The relationship between the roll reduction rate and the iterations of the PSOA and BFOA is illustrated in Figure 12, and the solution time and results of the two algorithms are shown in Figure 13. FiguresAppl. Sci. 202011 and, 10, 66112 indicate that the convergence speed of the BFOA was faster, and its roll reduction30 of 19 rate was higher than that of the PSOA. With the same mass of 2000 kg, the roll reduction rate obtained throughAppl. Sci. 2020 the, 10 BFOA, 661 reached 79.6%, higher than that of the PSOA (78.5%). 30 of 19 Figure 12. The iterative curve of roll reduction rate. Figure 12. The iterative curve of roll reduction rate. Figure 12. The iterative curve of roll reduction rate. Figure 13. Comparison of the results between the two algorithms. The comparisonFigure between 13. Comparison the parameters of the of results the gyro between before the and two after algorithms. the BFOA’s optimization of the roll reduction rate is presented in Table3. The gyro’s damping coe fficient C increased by 13.2%, Figure 13. Comparison of the results between the two algorithms. whichThe was comparison within the optimalbetween interval the parameters in Figure of6. Thethe gyro inner before diameter andD after1 became the BFOA’s larger, andoptimization the outer diameterof the rollD reductionbecame slightlyrate is presented smaller, which in Table caused 3. The the rotationalgyro’s damping momentum coefficient momentC increased of the gyro by The comparison2 between the parameters of the gyro before and after the BFOA’s optimization to13.2%, decrease. which The was rotor within thickness the optimalH increased, interval whichin Figure caused 6. The the inner precession diameter momentum D1 became moment larger, of of the roll reduction rate is presented1 in Table 3. The gyro’s damping coefficient C increased by theand gyrothe toouter increase. diameter It is consistentD became with slightly Figure 6smaller, that the increasewhich caused of rotational the rotational momentum momentum moment 13.2%, which was within the2 optimal interval in Figure 6. The inner diameter D became larger, and the decrease of precession momentum moment contributed to the increase of roll1 reduction rate. moment of the gyro to decrease. The rotor thickness H1 increased, which caused the precession and the outer diameter D2 became slightly smaller, which caused the rotational momentum Althoughmomentum the moment mass only of the decreased gyro to by increase. 20.6%, theIt is optimization consistent with of these Figure parameters 6 that the made increase the roll of moment of the gyro to decrease. The rotor thickness H increased, which caused the precession reductionrotational ratemomentum increase moment to 79.6%. and the decrease of precession1 momentum moment contributed to themomentum increase ofmoment roll reduction of the gyrorate. Althoughto increase. the It mass is consistent only decreased with Figureby 20.6%, 6 that the optimizationthe increase of theserotational parameters momentum made moment the roll reandduction the decrease rate increase of precession to 79.6%. momentum moment contributed to the increase of roll reduction rate. Although the mass only decreased by 20.6%, the optimization of these parameters made the roll reduction rate increase to 79.6%. Appl. Sci. 2020, 10, x FOR PEER REVIEW 31 of 19 Table 3. Gyro parameters before and after optimization for the highest roll reduction rate. Before After Changing Parameter Variation Optimization Optimization Rate Inner diameter D1 (m) 1.070 1.104 +0.034 +3.2% Appl. Sci. Outer2020, 10 diameter, 661 D2 (m) 1.240 1.233 −0.007 −0.6%13 of 15 Shaft diameter D3 (m) 0.180 0.176 −0.004 −2.2% Rotor thickness H (m) 0.700 0.718 +0.018 +2.6% Table 3. Gyro parameters1 before and after optimization for the highest roll reduction rate. Ribbed plate thickness H3 (m) 0.050 0.045 −0.005 −10.0% Parameter Before Optimization After Optimization Variation Changing Rate Gyro’s damping coefficient C (Ns/m) 25,300 28,648 +3348 +13.2% Inner diameter D1 (m) 1.070 1.104 +0.034 +3.2% MotorOuter diameterspeed Dω(m)(r/min) 1.2401255 1.233 1300 0.007 +45 0.6%+3.6% 2 − − Shaft diameter D (m) 0.180 0.176 0.004 2.2% Mass Ms3 (kg) 2520 2000 − −520 − −20.6% Rotor thickness H1 (m) 0.700 0.718 +0.018 +2.6% Ribbed plate thickness H3 (m)TT 0.050 0.045 0.005 10.0% Roll reduction rate (%) 73.7 79.6 − +5.9 − +8.0% Gyro’s damping coefficient C (Ns/m) 25,300 28,648 +3348 +13.2% Motor speed ω (r/min) 1255 1300 +45 +3.6% Mass Ms (kg) 2520 2000 520 20.6% 5.3. Results and Analysis of the Minimum Rotor Mass − − Roll reduction rate TT (%) 73.7 79.6 +5.9 +8.0% The parameter optimization model with the minimum rotor mass as the objective could be 5.3.rewritten Results as and follows: Analysis of the Minimum Rotor Mass The parameter optimization model with the minimum rotor mass as the objective could be min ZMs= rewritten as follows: 1 ≤≤ ≤≤ min Z1=0.8MsDD12 1.2, 1 1.4 ≤≤ ≤≤ 0.150.8 DH31D1 0.22,1.2, 0.5 1 D2 0.91.4 ≤ ≤ ≤ ≤ 0.80.15≤≤HHD 1.2，0.22, 0.03 0.5 ≤≤H 0.060.9 233 1 (26) st.. ≤ ≤ ≤ ≤ 0.8 ≤≤Hω2 1.2,， 0.03 ≤≤H3 0.06 (26) 1100m 1300 8000C 60000 s.t. ≤ ≤ ≤ ≤ 1100 ωm 1300, 8000 C 60, 000 Ms≤≤≤2000,≤ 0.75 TT ≤ ≤ Ms≥≥2000, 0.75 TT DD21-0.06,≤ HH 2-0.3 1≤ D D 0.06, H H 0.3 2 − 1 ≥ 2 − 1 ≥ The relationship between between the the rotor rotor mass mass and and iterations iterations of of the the PSOA PSOA and and BFOA BFOA is isillustrated illustrated in inFigure Figure 14, 14 and, and the the solution solution time time and and results results of of the the two two optimization optimization algorithms algorithms are are presented in Figure 1515.. FiguresFigures 14 14 and and 15 15 indicate indicate that that the the convergence convergence speed speed of theof BFOAthe BFOA was was faster, faster, and itsand rotor its massrotor wasmass smaller was smaller than that than of that PSOA. of PSOA. With the With same th rolle same reduction roll reduction rate of 75%, rate theof 75%, rotor the mass rotor obtained mass withobtained the BFOAwith the was BFOA 1560 was kg, smaller 1560 kg, than smaller that withthan thethat PSOA with the (1680 PSOA kg). (1680 kg). Figure 14. The iterative curve of rotor mass. Figure 14. The iterative curve of rotor mass. The comparison between the parameters of the gyro before and after the BFOA optimization of rotor mass is presented in Table4. As the rotor mass was directly related to the rotor size, the inner diameter D1, outer diameter D2, and rotor thickness H1 all had obvious changes, especially the rotor thickness, which decreased by 28.6%, higher than that of Section 5.2 (2.6%). In addition, the gyro’s damping coefficient C increased by 8.4%, smaller than that of Section 5.2 (13.2%), but still conformed to the optimal interval in Figure6. Although the roll reduction rate only increased to 75.1% in this optimization, the objective function, rotor mass decreased by 960 kg. Appl. Sci. 2020, 10, 661 14 of 15 Appl. Sci. 2020, 10, 661 32 of 19 Figure 15. Comparison of the results between the two algorithms. Table 4. GyroFigure parameters 15. Comparison before andof the after results optimization between the for thetwo minimumalgorithms. rotor mass. Parameter Before Optimization After Optimization Variation Changing Rate The comparisonInner diameter D1 (m)between the parameters 1.070 of the gyro 1.200before and after+ the0.130 BFOA optimization+12.1% of Outer diameter D2 (m) 1.240 1.263 +0.023 +1.9% rotor massShaft is diameterpresentedD (m) in Table 4. As 0.180 the rotor mass was 0.150 directly related to0.030 the rotor size,16.7% the inner 3 − − Rotor thickness H1 (m) 0.700 0.500 0.200 28.6% diameter D1 , outer diameter D2 , and rotor thickness H1 all had obvious− changes, especially− the Ribbed plate thickness H3 (m) 0.050 0.058 +0.008 +16.0% rotorGyro’s thickness, damping coe whichfficient C decreased(Ns/m) by 25,30028.6%, higher than 27,419that of Section 5.2+2119 (2.6%). In addition,+8.4% the Motor speed ω (r/min) 1255 1300 +45 +3.6% Mass Ms (kg) 2520 1560 960 38.1% gyro’s damping coefficient C increased by 8.4%, smaller than that of Section− 5.2 (13.2%),− but still Roll reduction rate TT (%) 73.7 75.1 +1.4 +1.9% conformed to the optimal interval in Figure 6. Although the roll reduction rate only increased to 75.1% in this optimization, the objective function, rotor mass decreased by 960 kg. 6. Conclusions Table 4. Gyro parameters before and after optimization for the minimum rotor mass. Little work has been conducted for optimizing gyro parameters. In the present study, we designed Before After Changing a parameterParameter optimization method for antiroll gyros of ships. The specific contributionsVariation of the present work are as follows: Optimization Optimization Rate WeInner established diameter aD joint1 (m) dynamical equation1.070 of ships and antiroll1.200 gyros and+ analyzed0.130 the+ influence12.1% of gyro’sOuter damping diameter coe Dffi2 (m)cient C, rotational momentum1.240 moment h1.2630, and precession+0.023 momentum +1.9% moment J on rollShaft reduction diameter rate.D3 (m) Following this,0.180 we proposed a calculation0.150 method−0.030 for the roll− reduction16.7% rate. Then,Rotor thickness taking the H minimum1 (m) rotor mass0.700 and highest roll reduction0.500 rate as−0.200 the objective −28.6% function, and the ship space, power drive, and material strength as the constraints, we established a gyro Ribbed plate thickness H3 (m) 0.050 0.058 +0.008 +16.0% parameterGyro’s damping optimization coefficient model. C Finally, we used the PSOA and BFOA to solve the two-parameter 25,300 27,419 +2119 +8.4% optimization(Ns/m) models. OurMotor simulation speed ω (r/min) results revealed that1255 antirolling characteristics,1300 such as roll+45 reduction+ rate3.6% and rotor mass,Mass improved Ms (kg) e ffectively through gyro2520 parameter optimization.1560 In addition,−960 the convergence−38.1% speedRoll of reduction the BFOA rate was TT faster(%) than that of the73.7 PSOA, and the antirolling 75.1 characteristics+1.4 obtained+1.9% by the BFOA were better than those obtained by the PSOA, and this was due to the excellent global search 6.ability Conclusions of the BFOA. AuthorLittle Contributions: work has beenConceptualization, conducted for Y.Z. optimizing and S.S.; methodology, gyro parameters. Y.Z.; software, In the Y.Z.;present validation, study, Y.Z.,we designedS.S. and Y.Q.; a parameter formal analysis, optimization Y.C.; investigation, method for W.T.; antiroll data gyros curation, of ships. Y.Z. and The S.S.; specific writing—original contributions draft of preparation,the present Y.Z.work and are Y.Q.; as writing—reviewfollows: and editing, Y.C. and S.S.; visualization, Y.Z.; supervision, S.S.; project administration, S.S.; funding acquisition, S.S. and Y.Z. All authors have read and agreed to the published version of theWe manuscript. established a joint dynamical equation of ships and antiroll gyros and analyzed the influence of gyro’s damping coefficient C , rotational momentum moment h , and precession Funding: This research was funded by the Postgraduate Research and Practice Innovation0 Program of Jiangsu momentumProvince (Grant moment No. KYCX19_1662) J on roll andreduction the Industry-University rate. Following Joint this, Research we proposed Project of a Jiangsu calculation Province method (Grant forNo. the BY2019038). roll reduction rate. Then, taking the minimum rotor mass and highest roll reduction rate as Acknowledgments:the objective function,The and authors the ship would space, power like to driv thanke, and the material editors strength and the as reviewersthe constraints, for their we professionalestablished suggestions.a gyro parameter optimization model. Finally, we used the PSOA and BFOA to solve theConflicts two-parameter of Interest: optimizationThe authors declare models. no conflict of interest. Our simulation results revealed that antirolling characteristics, such as roll reduction rate and rotor mass, improved effectively through gyro parameter optimization. In addition, the convergence speed of the BFOA was faster than that of the PSOA, and the antirolling characteristics Appl. Sci. 2020, 10, 661 15 of 15 References 1. Liang, L.; Zhao, P.; Zhang, S.; Ji, M.; Yuan, J. Simulation analysis of fin stabilizer on ship roll control during turning motion. Ocean Eng. 2018, 164, 733–748. 2. 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