Ship shafting alignment technology and hull deformation based on improved genetic algorithm and shipbed calibration

3.1. The ship shafting alignment optimization model based on ship bed calibration

Shipyard is the working platform for ship repair in the shipyard, which is the site facility for ship repair after manual processing. The ship shafting is generally calibrated based on the shipyard, which is an important part of the power plant. The axial pushing force generated by the rotation of the propeller is transmitted to the thrust bearing through the shafting, and then the thrust bearing directly acts on the hull, forcing the ship to move forward or backward [16, 17]. The arrangement of ship shafting is shown in Fig. 1.

Fig. 1Shift shifting layout

From No. 1 to No. 12 in Fig. 1 are respectively propeller, bearing, tail shaft, coupling, intermediate shaft, thrust bearing, reducer, flywheel, main engine, elastic coupling, thrust shaft and intermediate bearing. The shafting alignment means that the static load distribution of each bearing is reasonable and meets requirements of manufacturer by adjusting the height of each bearing [18, 19]. The quality of ship shafting alignments is important in normal operation of shafting and main engine. The shafting alignment calculation by the finite element method is to carry out optimal modeling of ship shafting [20]. The method principles are to disperse continuous solution area into a group of element combinations connected by finite nodes, and use the approximate function slices to represent unknown functions to be solved in all solution areas. The finite element method also requires assumptions of continuity, homogeneity, isotropy, small deformation, and load linearly related to deformation. The basic flow is shown in Fig. 2.

Fig. 2Basic flow of final element method

The basic flow in Fig. 2 includes element division, element stiffness matrix, load treatment, system equation, boundary condition treatment, and final solution equation. The load treatment is to discretize the whole system into element $n$ and generate respective element procedure during finite element modeling of the shaft system, so as to conduct system integration. The system equation is shown in Eq. (1):

$\left[K_{n\times n}\right]$ is the system stiffness matrix. $\left[a_{n\times 1}\right]$ and $\left[P_{n\times 1}\right]$ respectively represent the node displacement vector and the system load vector. The purpose of shafting alignment optimization is to obtain a reasonable bearing height displacement value. For the shafting assembled based on the slipway, hull deformation and temperature rise of main engine shall be taken into consideration. Therefore, the shafting optimization model based on the feasible solution of the optimal design space has the characteristics of unknown, multi-dimensional, multi-peak, etc. Due to the cantilever action of propeller at the end of ship shaft system, the rear bearing capacity of tail tube is the largest. And because of the uneven pressure distribution, the bearing of rear shaft of tail tube is easy to form unilateral load and accelerate the bearing wear at this place. Therefore, the rear bearing capacity of tail tube is taken as the optimization objective function in the study, as shown in Eq. (2):

where, $R_{10}$ is counter-force of the rear bearing of the tail tube calculated by shafting alignment. $K_{1i}$ and $X_i$ respectively represent the counter-force influence coefficient and bearing displacement. The load uniformity shall be measured considering the ship launching deformation, thermal expansion and other working conditions of main engine, bearing load uniformity and bending moment. Eq. (2) is the objective optimization function of the berth shaft system, shaft system cold state under pressure, hot state under pressure, full load cold state, and full load hot state, while the objective function of load uniformity is shown in Eq. (3):

where, $Q$ is the load degree, and $R_i$ represents the supporting reaction of bearing $i$. As there are up to six objective functions, it is difficult to optimize at the same time. Therefore, a three-objective function optimization model is established by converting the slipway shafting, full-load cold state and ballast cold state into constraint conditions, as shown in Eqs. (4)-(7):

where, $f_1\left(x\right)$, $f_2\left(x\right)$, and $f_3\left(x\right)$ represent objective optimization functions of ballast thermal state, full-load thermal state and load uniformity. To ensure that the bearing load cannot be negative at this time, the minimum load requirements shall be met, as shown in Eq. (5):

where, $R_{\mathrm{m}\mathrm{i}\mathrm{n}}^i$ and $R_{\mathrm{m}\mathrm{a}\mathrm{x}}^i$ respectively represent the minimum allowable load and maximum allowable load of the bearing $i$. Eq. (6) is the determination of $R_{\mathrm{m}\mathrm{a}\mathrm{x}}^i$:

where, ${\left[p\right]}_i$, $l_i$, and $d_i$ respectively represent the allowable pressure of the bearing $i$, the length of the bearing lining and outer diameter of middle journal. The minimum allowable load $R_{\mathrm{m}\mathrm{i}\mathrm{n}}^i$ is shown in Eq. (7):

where, $W_l$ and $W_r$ respectively represent the dead weight of the left and right span shafts of the bearing $i$. $\sum P$ represent the sum of the external load. Due to the limitation of the given shafting environment, the optimal model of ship shafting needs to be improved. Therefore, the variation of load force caused by the propeller convolution effect on the rear bearing of the stern shaft is also considered. The propeller convolution effect refers to the deflection of the propeller with the rotation of the shaft, the direction of its momentum moment vector changes constantly, and inertia force and moment on shaft are generated, as shown in Fig. 3.

Fig. 3Propeller friction effect

In Fig. 3, $w_s$ and $\mathrm{\Omega }$ are the rotational angular velocity and the rotational angular velocity, $Fr$ and $r$ respectively represent the inertia force and central deflection of the propeller, $M$ represent the inertia moment, and $\theta$ represent the rotational angle of the rotating shaft. The solution of inertia force $Fr$ and inertia moment $M$ is shown in Eq. (8):

where, $m_p$ is the propeller mass, $J_p$ and $J_d$ represent the inertia polar moment and inertia radial moment of propeller disc, and $w$ is the absolute angular velocity.

3.2. The design of NSGA-Ⅱ algorithm based on NSGA algorithm

Ship shafting system is a complex stressed system. To realize the shafting alignment on the berth, it must meet certain requirements under the berth and floating state. And the bearing force of the stern shaft shall be reduced as far as possible and the load shall be evenly distributed. Currently, the optimization of shafting alignment can be categorized as a multi-objective function optimization problem. It is important to note that the multi-objective optimization problem differs from a single-objective optimization problem. When there are multiple objective functions, it is important to find a solution to optimize all the objective functions at the same time due to the conflict between the objectives. Therefore, there is usually a Pareto optimal solution. To address the multi-objective function optimization, Pareto evaluation ranking method based on the relationship between the advantages and disadvantages of the solution can be used. The representative Pareto ranking method is NSGA algorithm. NSGA takes the non-dominated layer as its initial fitness value, and then uses the selection, cross, mutation and other operations of traditional genetic algorithm to generate sub-generation individuals. Finally, for the individuals on the same non-dominated layer, shared niche technology is used to re-designate virtual fitness value, so that Pareto optimal solution is evenly distributed. The Euclidean distance $d_{ij}$ between individuals in the same non-governing layer shall be obtained first for the calculation of fitness value, as shown in Eq. (9):

where, $L$ is the number of target variables. $x_l^u$ and $x_l^d$ represent the upper and lower boundaries of non-governing layer $l$. $x_l^i$ and $x_l^j$ are individuals $i$ and $j$ of the non-governing layer $l$. Then the sharing function is used to calculate the relationship between individual $x_i$ and other individuals in the biotope population, as shown in Eq. (10):

where, ${\theta }_{share}$ is the shared radius, $\alpha$ is the constant, and the small population number of the individual $x_i$ is shown in Eq. (11):

where, $c_i$ is the Niche Count of $x_i$, and $n_m$ is the individual $n_m$ on the non-governing layer $m$. Then the shared fitness value $f_m^i$ of $x_i$ is calculated in Eq. (12):

where, $f_m$ represents the individual fitness value of layer $m$. Since the determination of ${\theta }_{share}$ needs to be determined manually, it often has a great contingency in the actual application, reducing the running efficiency and speed of the algorithm. An elite strategy is introduced to expand the NSGA algorithm to obtain the NSGA-Ⅱ. Fig. 4 is the composition of NSGA-II.

Fig. 4Composition of NSGA-Ⅱ

NSGA-II in Fig. 4 mainly consists of fast non-dominant sorting, crowding degree and crowding degree comparison, crowding degree concept introduction, and elite strategy introduction. Fast non-dominant sorting refers to reducing the time complexity from $O\left(m{N}^3\right)$ to $O\left(m{N}^2\right)$ and determining the population value $front$ by investigating the number of dominated individual $p$ and the dominated individuals set. Congestion degree refers to the average side length of the largest rectangle covering individual $i$ but excluding other individuals in the same non-dominated layer, and its calculation is shown in Eq. (13):

where, $nd$ represents the degree of crowding. $f_j^{n+1}$ and $f_j^{n-1}$ respectively represent $j$ objective function value of $i+1$ and $i-1$. $f_j^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $f_j^{\mathrm{m}\mathrm{i}\mathrm{n}}$ respectively represent the maximum and minimum values of $f_j$. Crowding degree comparison operator shall be introduced to redefine the non-dominated relation of population after fast non-dominated sorting and crowding degree calculation. In the study, individuals $i$ and $j$ are selected for comparison. If any following conditions are met, individual $i$ is superior to individual $j$, as shown in Eq. (14):

where, $i_{rank}$ and $j_{rank}$ respectively represent the non-governing layer between individuals $i$ and $j$. $i_{dis\mathrm{t}\mathrm{a}\mathrm{n}ce}$ and $j_{dis\mathrm{t}\mathrm{a}\mathrm{n}ce}$ respectively represent crowding distance between $i$ and $j$. Elitist strategy is to select the population $R_i$ composed of parent $C_i$ and descendant $D_i$ as the new parent $C_{i+1}$. It eliminates the infeasible scheme, and sorts the population of the whole layer into $C_{i+1}$ according to the non-dominant order value. When the number of the whole layer reaches $N$, individuals are added into $C_{i+1}$ according to the crowding degree $nd$ to ensure that the population number is $N$. Weighted evaluation method shall be used to evaluate Pareto optimal solution. First, normalization treatment shall be carried out for attribute value of objective function, as shown in Eq. (15):

where, $f_m\left(n\right)$ represents the dimensionless utility value. $a_m^{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $a_{mn}$ respectively represent the minimum value of objective function $m$ and all data. To reflect the order and mode selection of work execution, NSGA-II uses two-layer encoding to represent individuals, and its algorithm flow is shown in Fig. 5.

Fig. 5NSGA-Ⅱ algorithm flow diagram

The algorithm flow in Fig. 5 is as follows: first, the solution approximates the Pareto optimal solution through fast non-dominant sorting. Then the population individual crowding degree $nd$ is calculated and the appropriate individual is selected to generate a new parent population. Next, it determines whether the maximum evolution algebra is reached through selection, cross, mutation and other operations, so as to obtain the final Pareto optimal solution set.

3.3. Ship shafting optimization model results analysis based on NSGA-Ⅱ algorithm

To verify the feasibility of NSGA-Ⅱ by ship shafting optimization model, the study first compares NSGA-Ⅱ performance, and then applies ship optimization model to the propulsion shafting of a ship for practical application analysis. This chapter focuses on NSGA-Ⅱ performance and ship propulsion shaft optimization.

3.4. NSGA-Ⅱ performance analysis

To verify the effectiveness of NSGA-Ⅱ, MATLAB 2017a software is used for programming, and the Heuristic Particle Swarm Optimization (HPSO) and Ant Colony (AC) algorithms are used for comparative experiments. The experiment parameters of NSGA-Ⅱ algorithm are set as population $N=$ 50, and the cross probability and variation probability are 0.8 and 0.2, respectively. The population size is 100 with a maximum iteration of 200. The test hardware environment: the host memory is 8 G, the processor is 8-core Intel (R) Core (TM) i7-7700 CPU @3.6 GHz, and the operating system is Windows 10. The performance indicators are adaptive value, Inverse Generation Distance (IGD) and Hyper Volume (HV). Smaller IGD means better algorithm convergence and distribution [19, 20].

Fig. 6Fitting results of four algorithms

Fig. 6 shows the adaption values results. As the iteration number increases, the adaption values of the four algorithms are increased. After reaching a certain number of iterations, the curve of each adaption value tends to be flat. NSGA-Ⅱ algorithm has an adaptive value of 74.57 when the iteration number is 25 epoch. When the iteration number of NSGA algorithm is 34 epoch, the adaptive value is 58.77. HPSO algorithm has an adaptive value of 56.74 when the iteration number is 43 epoch. When the iteration number of AC algorithm is 62 epoch, the adaptive value is 52.33.

Fig. 7 shows the HV and IGD results. Fig. 7(a) shows the HV results. With the increase of the evaluation times, four HV values increase. When the evaluation times are 5000, the HV value of NSGA-II is 0.41, the HV value of NSGA is 0.38, the HV value of HPSO is 0.36, and the HV value of AC is 0.34. Fig. 7(b) shows the IGD results. As the evaluation times increase, the IGD values of four algorithms decline. When the evaluation times are 1000, the IGD values of NSGA-II are 0.92, and the IGD values of NSGA, HPSO, and AC algorithms are 1.25, 1.74, and 1.95, respectively. When the evaluation times are 5000, the IGD values of NSGA-II, NSGA, HPSO, and AC algorithms are 0.03, 0.06, 0.08, and 0.89, respectively. To further verify the performance of NSGA-II, Camel, Schaffer, and Goldstein-Price functions are used to test the four algorithms in Table 1.

Fig. 7HV and IGD results of four algorithms

a) HV results of four algorithms

b) IGD results of four algorithms

In Table 1, the convergence time of NSGA, HPSO and AC algorithms for Camel function is 0.0392 ms, 0.0422 ms, and 0.0511 ms, respectively, which is lower than 0.0341 ms of NSGA-II. The convergence value of NSGA-Ⅱ for Schaffer function is 1.0000, and the convergence values of NSGA, HPSO, and AC algorithms are 0.9998, 0.9990, and 0.9984, respectively. For Goldstein Price function, the convergence time and experimental error of NSGA-Ⅱ algorithm are significantly lower than those of other three algorithms. Based on the above results, NSGA-Ⅱ algorithm has excellent performance and can significantly improve the effect of ship shafting alignment optimization.

3.5. Ship shafting optimization model application analysis based on NSGA-Ⅱ algorithm

To verify practicability of NSGA-Ⅱ-based ship shafting optimization model, the propulsion shafting of a ship is selected for application analysis. The shafting is connected with the main engine by a tail shaft and an intermediate shaft. The tail shaft is supported by two bearings, and intermediate shaft is provided with one bearing. Table 2 shows the basic parameters of the shafting.

In Table 2, there are 7 bearings in the propulsion shaft of a ship. The maximum load of bearing 1 is 245082.4 N, close to the maximum allowable pressure of 267000 N, and its value is several times that of other six bearings. The negative load of No. 2 bearing is –25351.4 N, and the vertical position is 0.0051 mm. Meanwhile, the load difference between adjacent bearings is very uneven, which affects the installation quality of shafting of a ship. When the relative rigidity of the shafting increases, the hull deformation of a ship will be more sensitive. To explore the influence of hull deformation on shafting, the bearing force of each bearing under ballast and full load is analyzed.

Fig. 8Impact of ball and full load deformation on axial system action force

a) Influence of ballast deformation on axle system reaction force

b) Influence of full load deformation on axle system reaction force

Fig. 8 shows the deformation influence on shaft counter-force of each bearing under ballast and full load. Fig. 8(a) shows the ballast deformation influence on the shaft counter-force. When the shaft counter-force of other bearings is positive, the shaft counter-force of 4# bearing under cold and hot alignment is –47514 N and –18973 N, respectively. Fig. 8(b) shows the influence of full-load deformation on the shaft system reaction. The shaft system reaction of No. 3 bearing under cold alignment is 14523 N, and the shaft system reaction under hot alignment is –8741 N. From Fig. 8, the deformation of the hull makes the bearing load meeting the original alignment requirements negative, which is not conducive to the safe operation of the ship's shafting. To solve the above problems, the deformation data of the ship hull are integrated into the optimal scheme, and the position of the intermediate bearing of the shafting is adjusted, so that the optimal result can be obtained for a ship’s shafting.

Fig. 9Impact response display change and optimization results after position adjustment

a) Impact response displacement change

b) Comparison of optimal load between ballast and full load

Fig. 9 is the optimization result of the impact response displacement change and the adjusted position of the intermediate bearing, and Fig. 9(a) is the impact response displacement change curve of the intermediate bearing. When the axial position $x=$9.21 m, the impact displacement is 0.1315 mm, where the impact peak load is the minimum. Fig. 9(b) shows the optimization results when the intermediate bearing is adjusted to (9.21 m, 0.1315 mm). Under the ballast, the load of 4# bearing before adjustment is 74010 N, and its load is 49991 N after adjustment. Under the full load, the load of 4 # bearing before and after adjustment is 124023 N and 78453 N, respectively. According to Fig. 8, the load non-uniformity is greatly improved after adjusting intermediate bearing position. When the ship runs normally in the water, the gyroscopic effect of the propeller always exists. Based on the above, the thermal state of ballast and full load is used to analyze the shafting alignment under the gyroscopic effect.

Fig. 10Impact of gyro effect on axle alignment in full load hot state and ballast hot state

a) Full-load thermal gyro effect

b) Ballast thermal gyro effect

Fig. 10 shows the influence of gyro effect on shafting alignment under full load thermal state and ballast thermal state. Fig. 10(a) shows the influence of gyro effect on shafting alignment under full load thermal state. When the gyro effect is not included in bearing 4, the shaft reaction force is 141971 N, and when the gyro effect is included in bearing 4, the shaft reaction force is 137487 N. Fig. 10(b) shows the influence of gyroscopic effect on shafting alignment under ballast thermal state. The shaft reaction of bearing 4 without and with gyro effect is 140784 N and 136458 N, respectively. To sum up, the proposed shafting alignment optimization model has theoretical feasibility and can meet the safe operation requirements of the ship.