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 [9,10,11].
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 [12,13,14]. 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 [15,16,17,18]. 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 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 angleα(t), and the essence ofα(t)was to convert the spectrum of waves into that ofα(t). Whereα(t)is the maximum inclination of the wave surface on a vertical section orthogonal to the crest, and the wave slope angleα(t) [19] was defined as follows:
α(t)=(∑i=1N2∫ωei−1ωeiSσ(ωe)dωecos(ωe i+εi))sin(χ),i=1,2,⋯
whereωeis the encounter frequency,εiis the random phase angle uniformly distributed between0and2π,Nis the number of selected harmonics, andχis the course angle.Sσ(ωe)is the spectrum function of wave slope angle, which could be obtained with the density of the wave energy spectrumSζ(ω):
Sσ(ωe)=ω4g2Sζ(ω)1+2ωgucosχ
wheregis the gravitational acceleration,uis the ship sailing speed, andωis 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 spectrumSζ(ω) [20]:
Sζ(ω)=173h1/32ω5 T1 4e−691ω4 T14
whereh1/3is the significant wave height andT1is the mean period of the wave.
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,(Iϕϕ+Jϕϕ)ϕ¨is the mass inertia moment,R(ϕ˙)is the roll-damping moment,K(ϕ)is the roll-restoring moment, andM(χ,ω,t)is the wave excitation moment.
Based on the Mathieu equation [21], considering the nonlinear damp and ship nonlinear restore moment, the ship rolling motion equation was established as follows [22]:
(Iϕϕ+Jϕϕ)ϕ¨+R(ϕ˙)+K(ϕ)=M(χ,ω,t)
whereIϕϕis the ship’s moment of inertia,Jϕϕis the moment of inertia of additional mass,ϕis the roll angle,ϕ¨is the roll angular acceleration, andχis the course angle.
The roll-damping momentR(ϕ˙) was calculated as the linear damping plus cubic damping [22]:
R(ϕ˙)=c1′ϕ˙+c3′ ϕ˙3
whereϕ˙is the roll angular velocity, andc1′andc3′are the damping moment coefficients.
To simplify the calculation, the roll-restoring moment was approximated to a fifth-degree polynomial [22]:
K(ϕ)=K1ϕ+K3 ϕ3+K5 ϕ5
whereK1,K3, andK5are the restoring moment coefficients.
The wave excitation moment was expressed as a function of wave slope angle [22]:
M(χ,ωe,t)=Dhα(t)
whereDis the ship’s displacement,his the transverse metacentric height, andα(t)is the wave slope angle.
The ship rolling mathematical model could be transformed into:
ϕ¨=−c1ϕ˙−c3 ϕ˙3−k1ϕ−k3 ϕ3−k5 ϕ5+Dhα(t)/(Iϕϕ +Jϕϕ)
whereci=ci′/(Iϕϕ +Jϕϕ),i=1,3andkj=Kj/(Iϕϕ +Jϕϕ),j=1,3,5.
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.Oξηζis the absolute coordinate system, andϕis the roll angle of the hull aroundOζ.Oxyzis the relative coordinate system, andOzis the rotating axis of the rotor.Oyis the rotating axis of the outer frame, andβis the precession angle of the gyro aroundOy.
The mass of the gyro’s outer frame was neglected, and the influence of the ship’s movement on other degrees of freedom on the gyro was not considered. The rotor was an axisymmetric rigid body whose moment of inertia aroundOzwasIz, the rotor speed was constant atω0, and the momentum moment constant ofOxandOywasJ . According to the Euler equation of the motion of rigid bodies [23], the motion equation of the antiroll gyro relative toOxyzwas as follows:
{Mx=Jϕ¨cosβ+h0β˙My=Jβ¨+Jϕ˙2sinβcosβ−h0ϕ˙cosβMz=Iω˙0=0
whereMx,My, andMzare the components of the resultant external torque onOxyz,β˙is the precession angular velocity,β¨is the precession angular acceleration, andh0 =Iz ω0is the momentum moment constant of the gyro. IfOywas considered as the input axis, then input torqueMycaused the gyro to precess, and then a torqueMxoutput onOxoccurred. The output torqueMxfrom the above motion equation was projected intoOξηζ. Given the stability of the high-speed spinning gyro, the angular velocityϕ˙was considerably smaller thanω0, and its second derivative was ignored. Equation (9) could then be simplified as follows:
{Mξ=h0β˙cosβMζ=h0β˙sinβMy=Jβ¨−h0ϕ˙cosβ
The damping device was added in the precession direction of the antiroll gyro, and the precession of the gyro was restricted appropriately depending on the characteristics of damping to improve the antirolling effect. The total damping torqueMycould be expressed asMy=Cβ˙, andCwas 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:
{ϕ¨=−c1ϕ˙−c3 ϕ˙3−k1ϕ−k3 ϕ3−k5 ϕ5−h0β˙cos(β)/(Iϕϕ +Jϕϕ)+Dhα(t)/(Iϕϕ +Jϕϕ)β¨=h0ϕ˙cos(β)/J−Cβ˙/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 the roll angle of the ship. The ship is presented in Figure 3, and the parameters of it are presented in Table 1. The structure and parameters of the supporting antiroll gyro are presented in Table 2 and Figure 4. The gyro’s rotor had an axisymmetric structure, and the intermediate rotating shaft and rotating outer ring were connected by rib welding.
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 momenth0, the precession momentum momentJ, and the gyro’s damping coefficientC.
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 Figure 5, when the gyro’s damping coefficient increased, the roll reduction rate gradually increased. When the gyro’s damping coefficientC=16,000Ns/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 of1–5 m, and the significant wave height of 5 m was taken as the subsequent optimization condition. The five coordinates in Figure 6 were, respectively, the maximum values of the roll reduction rate when the wave height was1–5 m. Figure 6 illustrates that as the significant wave height changed, the optimal interval of gyro’s damping coefficient changed very little. Therefore, the gyro’s damping coefficient was taken as a constraint condition in the subsequent simulation.
3.2. Influence of Rotational and Precession Momentum Moment on Roll Reduction Rate
By substituting different values ofh0andJinto Equation (11) for the solution, the variation in roll reduction rate with the correspondingh0andJ is illustrated in Figure 7, which shows that the roll reduction rate increased with increasingh0and decreased with increasingJ.
To identify the relationship among roll reduction rate, rotor diameter, and thickness more directly, we determined the roll reduction rate with various rotor diameter and thickness in Figure 8, which indicated that an increase in rotor diameter resulted in an increase in gradual roll reduction rate. When the rotor diameter was 0.5 m, as the rotor thickness increased, the roll reduction rate increased first and then decreased before reaching a peak of 70% when the rotor thickness was 0.75 m. With the increase of rotor diameter, the influence of rotor thickness on the roll reduction rate decreased gradually. However, the increase of rotor diameter and thickness led to an increase in gyro’s power consumption and floor space. Therefore, it was necessary to establish a parameter optimization model, set up constraints, and determine the optimal solution of gyro parameters in consideration of various factors.
4. Mathematical Model for Parameter Optimization of Antiroll Gyro
According to the study in Section 3, there was a complicated nonlinear relationship between the roll reduction rate and the rotor’s size. In addition, since the gyro is restricted by the ship, the parameter optimization is needed to obtain the optimal parameters. In this Section, the ship roll reduction rate and the rotor mass were modeled as objective functions. Then, the ship space, power drive, and material strength were considered as constraints to form the optimization model.
4.1. Establishing the Mathematical Model of the Ship Roll Reduction Rate
The antirolling capability of gyro could be directly evaluated by the ship roll reduction rate. The solution of the ship roll reduction rate required continuous iteration, and its iterative process is presented in Figure 9.
1. When the gyro was in a nonworking state
According to Equation (8), the ship’s mathematical model could be established as follows:
ϕ¨b=−c1 ϕ˙b−c3 ϕ˙b 3−k1 ϕb−k3 ϕb 3−k5 ϕb 5+Dhα(t)/(Iϕϕ +Jϕϕ)
whereϕbis the ship’s roll angle when the gyro is not in operation.
Letx=ϕ˙b. Equation (12) could be transformed into an initial value problem of first-order differential equations:
{x(0)=ϕb1=x0,ϕb(0)=ϕb0ϕ˙b(t)=x(t)x˙(t)=f1(t,ϕb(t),x(t))
wheref1(t,ϕb(t),x(t))=−c1x(t)−c3x(t)3−k1 ϕb(t)−k3 ϕb (t)3−k5 ϕb (t)5+Dhα(t)/(Iϕϕ +Jϕϕ).
By solving the differential equation for timet, the following equation could be obtained:
{ϕb(0)=ϕb0,ϕ˙b(0)=x0ϕbk=ϕb(k−1)+lxkxk=xk−1+lf1(tk,ϕbk,xk),k=1,2⋯tk=tk−1+l
wherelis the iterative step length of timet, andkis the number of iterations. Then, the expression of the roll angleϕbchanging with timetcould be obtained from Equation (14),ϕb=ϕb(k−1)+lxk.
2. When the gyro was in a working state
According to Equation (11), the mathematical model of the ship antiroll gyro could be established as follows:
{ϕ¨a=−c1 ϕ˙a−c3 ϕ˙a 3−k1 ϕa−k3 ϕa 3−k5 ϕa 5−h0β˙cos(β)/(Iϕϕ+Jϕϕ)+Dhα(t)/(Iϕϕ+Jϕϕ)β¨=h0 ϕ˙acos(β)/J−Cβ˙/J
whereϕais the ship’s roll angle when the gyro is in operation.
Letx=ϕ˙aandy=β˙. Equation (15) could be transformed into an initial value problem of first-order differential equations:
{x(0)=ϕa1=x0,ϕa(0)=ϕa0y(0)=β1=y0,β(0)=β0ϕ˙a(t)=x(t),x˙(t)=f1(t,ϕa(t),x(t),β(t),y(t))β˙(t)=y(t),y˙(t)=f2(t,ϕa(t),x(t),β(t),y(t))
where,
f1(t,ϕa(t),x(t),β(t),y(t))=−c1x(t)−c3x(t)3−k1 ϕa(t)−k3 ϕa (t)3−k5 ϕa (t)5−h0y(t)cos(β(t))/(Iϕϕ+Jϕϕ)+Dhα(t)/(Iϕϕ+Jϕϕ),f2(t,x(t),β(t),y(t))=h0x(t)cos(β(t))/J−Cy(t)/J.
By solving the differential equation for timet, the following equation could be obtained:
{ϕa(0)=ϕa0,ϕ˙a(0)=x0β(0)=β0,β˙(0)=y0ϕak=ϕa(k−1)+lxk,βk=βk−1+lykxk=xk−1+lf1(tk,ϕak,xk,βk,yk),k=1,2⋯yk=yk−1+lf2(tk,xk,βk,yk),k=1,2⋯tk=tk−1+l
wherelis the iterative step to determine the length of timet, andkis the number of iterations. Then, the expression of roll angleϕachanging with timetcould be obtained from Equation (17),ϕa=ϕa(k−1)+lxk.
Expression of Ship Roll Reduction Rate
The ship roll reduction rateTTwas defined as follows:
TT=S(ϕa)−S(ϕb)S(ϕa)×100%
whereS(ϕb)=1k∑i=1k(ϕbi−ϕ¯b)2,S(ϕa)=1k∑i=1k(ϕai−ϕ¯a)2,ϕ¯bis the average ofϕbfromϕb1toϕbk, andϕ¯ais the average ofϕafromϕa1toϕak.
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:
minZ1=1−TT
2. Minimum rotor mass
The structure and size of the rotor are displayed in Figure 4. Therefore, the mass of the rotor could be expressed as follows:
Ms=ρV=ρ[H1(D2−D1)+2H3 D1+H2 D3]
whereV is the volume of the rotor in Figure 4.
The objective function was expressed as follows:
minZ2=Ms
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 ishl×hw×hh, wherehl,hw, andhh are the length, width, and height of the mounting size. As shown in Figure 4, the rotor’s inner diameterD1, the rotor’s outer diameterD2, the rotor’s shaft diameterD3, the rotor thicknessH1, and the ribbed plate thicknessH3met the following constraints:
{D2<D1<min(hl,hw,hh),a4<H1<min(hl,hw,hh)a1<D2,a2<D3<a3,a5<H3<a6
whereai, i∈{1,2,⋯,6}are all positive, and their specific values were selected according to the actual size of the gyro’s rotor.
The dimensional constraint of motor speedωmwas obtained based on the available power of the ship’s electrical system. Asωm=9550PT, wherePis the rated power andTis the rated torque, given the assumption that the power cap isPs, theωmet the following constraints:
a7<ωm≤9550PsT
wherea7is positive, and its specific value was selected empirically between 1050 and 1150.
The constraint of the gyro’s damping coefficient was expressed asa8≤C≤a9, in whicha8anda9 were obtained from Figure 6. The constraint of the mass of the gyro rotor wasMs≤a10, wherea10was selected according to gyro’s size. The constraint of the roll reduction rate wasTT≥a11, where the specific value ofa11was given by the ship’s designer.
Then, the constraint conditions can be summarized as follows:
{D2<D1<min(hl,hw,hh),a1<D2a4<H1<min(hl,hw,hh),a2<D3<a3a5<H3<a6,a7<ωm≤9550PsTa8≤C≤a9,Ms≤a10,a11≤TT
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.
5.2. Results and Analysis of the Highest Roll Reduction Rate
By taking the ship and its supporting antiroll gyro provided in Section 3 as an example, this Section conducted a simulation analysis on the parameter optimization. Given the ship’s size, the parameter optimization model with the highest roll reduction rate as the objective could be written as follows:
{minZ1=1−TTs.t.{0.8≤D1≤1.2, 1≤D2≤1.40.15≤D3≤0.22, 0.5≤H1≤0.90.03≤H3≤0.06,1100≤ωm≤130010,000≤C≤40,000,Ms≤20000.75≤TT,D2−D1≥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. Figure 11 and Figure 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%).
The comparison between the parameters of the gyro before and after the BFOA’s optimization of the roll reduction rate is presented in Table 3. The gyro’s damping coefficientC increased by 13.2%, which was within the optimal interval in Figure 6. The inner diameterD1became larger, and the outer diameterD2became slightly smaller, which caused the rotational momentum moment of the gyro to decrease. The rotor thicknessH1 increased, which caused the precession momentum moment of the gyro to increase. It is consistent with Figure 6 that the increase of rotational momentum moment and the decrease of precession 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%.
5.3. Results and Analysis of the Minimum Rotor Mass
The parameter optimization model with the minimum rotor mass as the objective could be rewritten as follows:
{minZ1=Mss.t.{0.8≤D1≤1.2, 1≤D2≤1.40.15≤D3≤0.22, 0.5≤H1≤0.90.8≤H2≤1.2, 0.03≤H3≤0.061100≤ωm≤1300, 8000≤C≤60,000Ms≤2000, 0.75≤TTD2−D1≥0.06, H2−H1≥0.3
The relationship between the rotor mass and iterations of the PSOA and BFOA is illustrated in Figure 14, and the solution time and results of the two optimization algorithms are presented in Figure 15. Figure 14 and Figure 15 indicate that the convergence speed of the BFOA was faster, and its rotor mass was smaller than that of PSOA. With the same roll reduction rate of 75%, the rotor mass obtained with the BFOA was 1560 kg, smaller than that with the PSOA (1680 kg).
The comparison between the parameters of the gyro before and after the BFOA optimization of rotor mass is presented in Table 4. As the rotor mass was directly related to the rotor size, the inner diameterD1, outer diameterD2, and rotor thicknessH1 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 coefficientC increased by 8.4%, smaller than that of Section 5.2 (13.2%), but still 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 Little work has been conducted for optimizing gyro parameters. In the present study, we designed a parameter optimization method for antiroll gyros of ships. The specific contributions of the present work are as follows:
We established a joint dynamical equation of ships and antiroll gyros and analyzed the influence of gyro’s damping coefficientC, rotational momentum momenth0, and precession momentum momentJon roll reduction rate. Following this, we proposed a calculation method for the roll reduction rate. Then, taking the minimum rotor mass and highest roll reduction rate as the objective function, and the ship space, power drive, and material strength as the constraints, we established a gyro parameter optimization model. Finally, we used the PSOA and BFOA to solve the two-parameter optimization models.
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 obtained by the BFOA were better than those obtained by the PSOA, and this was due to the excellent global search ability of the BFOA.
Enlarge this image.Figure 6. Roll reduction rate under different gyro's damping coefficients and wave heights.
Enlarge this image.Figure 7. Roll reduction rate with various rotational and precession momentum moments.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Total lengthL(m) | 44.8 | Ship’s roll-damping coefficientc1(Ns/m) | 843 |
| Molded breadthb(m) | 8.9 | Ship’s roll-damping coefficientc3(Ns/m) | 6589 |
| Molded depthH(m) | 4 | Roll-restoring moment coefficientk1 (Ns2/m) | 10,791 |
| Displacementd(t) | 594 | Roll-restoring moment coefficientk3 (Ns2/m) | –9284 |
| Initial metacentric heighth(m) | 1.09 | Roll-restoring moment coefficientk5 (Ns2/m) | 894 |
| Total moment of inertiaJs (Nms2) | 6 × 106 |
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Rotor inner diameterD1(m) | 1.07 | Gyro’s damping coefficientC(Ns/m) | 25,300 |
| Rotor outer diameterD2(m) | 1.24 | Motor speedωm(r/min) | 1255 |
| Rotor shaft diameterD3(m) | 0.18 | Rotor material densityρ (kg/m3) | 7850 |
| Rotor thicknessH1(m) | 0.70 | Rotational momentum momenth0(Nms) | 104,100 |
| Shaft lengthH2(m) | 1.05 | Precession momentum momentJ(Nms) | 25,100 |
| Ribbed plate thicknessH3(m) | 0.05 |
| Parameter | Before Optimization | After Optimization | Variation | Changing Rate |
|---|---|---|---|---|
| Inner diameterD1(m) | 1.070 | 1.104 | +0.034 | +3.2% |
| Outer diameterD2(m) | 1.240 | 1.233 | −0.007 | −0.6% |
| Shaft diameterD3(m) | 0.180 | 0.176 | −0.004 | −2.2% |
| Rotor thicknessH1(m) | 0.700 | 0.718 | +0.018 | +2.6% |
| Ribbed plate thicknessH3(m) | 0.050 | 0.045 | −0.005 | −10.0% |
| Gyro’s damping coefficientC(Ns/m) | 25,300 | 28,648 | +3348 | +13.2% |
| Motor speedω(r/min) | 1255 | 1300 | +45 | +3.6% |
| MassMs(kg) | 2520 | 2000 | −520 | −20.6% |
| Roll reduction rateTT(%) | 73.7 | 79.6 | +5.9 | +8.0% |
| Parameter | Before Optimization | After Optimization | Variation | Changing Rate |
|---|---|---|---|---|
| Inner diameterD1(m) | 1.070 | 1.200 | +0.130 | +12.1% |
| Outer diameterD2(m) | 1.240 | 1.263 | +0.023 | +1.9% |
| Shaft diameterD3(m) | 0.180 | 0.150 | −0.030 | −16.7% |
| Rotor thicknessH1(m) | 0.700 | 0.500 | −0.200 | −28.6% |
| Ribbed plate thicknessH3(m) | 0.050 | 0.058 | +0.008 | +16.0% |
| Gyro’s damping coefficientC(Ns/m) | 25,300 | 27,419 | +2119 | +8.4% |
| Motor speedω(r/min) | 1255 | 1300 | +45 | +3.6% |
| MassMs(kg) | 2520 | 1560 | −960 | −38.1% |
| Roll reduction rateTT(%) | 73.7 | 75.1 | +1.4 | +1.9% |
Author Contributions
Conceptualization, Y.Z. and S.S.; methodology, Y.Z.; software, Y.Z.; validation, Y.Z., S.S. and Y.Q.; formal analysis, Y.C.; investigation, W.T.; data curation, Y.Z. and S.S.; writing-original draft preparation, Y.Z. and Y.Q.; writing-review 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 the manuscript.
Funding
This research was funded by the Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant No. KYCX19_1662) and the Industry-University Joint Research Project of Jiangsu Province (Grant No. BY2019038).
Acknowledgments
The authors would like to thank the editors and the reviewers for their professional suggestions.
Conflicts of Interest
The authors declare no conflict of interest.
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Yuanyuan Zhu
,
Shijie Su
*,
Yuchen Qian
,
Yun Chen
and
Wenxian Tang
School of Mechanical Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China
*Author to whom correspondence should be addressed.
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