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 0 N Z ! 1 BX ei C α(t) = B 2 S (!e)d!e cos(!e + " )C sin(χ), i = 1, 2, (1) @B σ i i AC !ei 1 ··· i=1 − where !e is the encounter frequency, "i 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(!e) is the spectrum function of wave slope angle, which could be obtained with the density of the wave energy spectrum Sζ(!): !4 ( ) g2 Sζ ! S (! ) = (2) σ e 2! 1 + g u cos χ where g is the gravitational acceleration, u is the ship sailing speed, and ! 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(!) [20]: 2 691 173h − 1/3 !4T4 S (!) = e 1 (3) ζ 5 4 ! 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, !, 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, !, 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 theis the roll ship’s angular moment acceleration, of inertia, and χJisφφ theis 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=+, !e, 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.