Multi-objective Optimization Design of a Heavy Duty Folding Mechanism and Self-discharging Equipment Development

 Abstract: In this paper, we investigated the technical problem of the recovery of overlength and heavy load conveying booms of self-unloading ships. A method of folding the conveying boom with a hydraulic-four-bar mechanism is presented, and by using a mathematical model for the optimization of folding speed stationary with ADAMS software, the optimization data and results were obtained. The multi-objective optimization index is introduced, and the multi-objective optimization problem is discussed. The results of the multi-objective optimization showed that parameters such as angular velocity and the change of angular acceleration of the conveyor boom were optimized. The paper has manufactured the connecting rod mechanism, and developed the self-discharging folding conveyance equipment. Through practical application, we determined that the developed folding conveying equipment had the advantages of smooth movement and high folding efficiency.


Introduction
The traditional self-unloading sand-ship often uses a fixed overhanging boom to unload sand.The key technical problems with this, such as the large size, unreturnable belt frame and poor safety, need to be solved urgently.In order to solve these problems, researchers have obtained a series of results, which have been summed into four solutions: telescopic, rotary, folding and flip.Among them, the folding boom recovery scheme was first proposed in [1], which uses the hydraulic-four-bar folding mechanism to realize the folding recovery of a 43.8 meter two-section boom (also the object of this paper).The comprehensive theory of linkage mechanisms and multi-objective optimization design have been classical research focuses in the field of mechanical design.Optimization design has been widely studied in optimization theory, mechanical properties, and material manufacturing [2][3][4][5][6][7].Alaa Hassan et al. [8] used the non-dominated sorting genetic algorithm version II (NSGA-II) to optimize the robot gripper design.Qiu et al. [9] provided a simple design program to optimize the design of truss beams.Yalcin et al. [10] introduced an improved optimization algorithm, which can quickly obtain the scheme of Mechanically Stabilized Earth Walls (MSEW).Rao et al. [11] proposed Rao algorithms and discussed the performance of Rao algorithms in the optimization of mechanical system parts design.Bai et al. [12] proposed a design optimization method for satellite antenna that takes into account the gap node of a biaxial drive mechanism, and used the generalized reduced gradient (GRG) algorithm to significantly reduce the peak acceleration of satellite antenna and the contact force of the gap node.Cicero et al. [13] adopted the method of topology optimization to optimize the design of the compliant mechanism, and solved the problem of hinges (single-node connections) in the design of the flexible mechanism.Li-Quan Wang et.al.[14] presented an optimal design method to optimize a novel subsea pipeline mechanical connector.Ya-Li MA et al. [15]optimized the bed structure of gantry-type machining center presented by using a light-weight design method.The solution-region method can be used to optimize the linkage mechanism [16][17], and there are still many cases where kinematics requirements need to be considered [18][19][20].Gabardi et al. [21] conducted kinematics analysis of the 4-UPU fully parallel manipulator to maximize the performance parameters in the design workspace.Sajid Nisar et al. [22] proposed a new remote center of motion (RCM) mechanism design for minimally invasive surgery (MIS) robotic manipulators, and optimized and reduced the size of the mechanism.Matteo Russo et al. [23] optimized the parallel mechanism with 3-UPR architecture for a robotic leg application by using four different objective functions.Robert et al. [24] proposed an optimization method that enables kinematic and dynamic optimization, combined with velocity profiling of the motor/drive system.Zhang et al. [25] used genetic algorithm(GA) to optimize the kinematics property of the double blades.Liu et al. [26] used a novel optimization method to optimize the locomotion and manipulation of an 18 DOFs tetrahedron-based mechanism.
To solve the problem of self-unloading sand-ship super-long overloading transportation boom recovery, this paper proposes a hydraulic, folding transportation boom of four bar linkage.The mathematical model for folding speed stability, the hydraulic oil cylinder force optimization problem, and the multi-objective optimization problem are discussed, and made the linkage, developed a self-unloading sand-ship conveying equipment.Consider that the folding arm can flip 180°; the four-bar mechanism satisfies the bar length condition, and rotating pair B is the rotating pair.Set as the shortest bar.According to the shortest bar condition and the bar length condition, the constraint conditions are as follows:  Through kinematic analysis and calculation, we deduced that the reversal angular velocity and angular acceleration of the folding arm were as follows: Among them:  (16) When working, the turning speed of the folding arm was stable.By optimizing the maximum value of angular acceleration, the stability of the velocity was optimized indirectly.The objective function of the angular velocity stability of the folding arm was as follows: In addition to the stability of the turning angular velocity, the force condition of the hydraulic cylinder was considered.The maximum thrust force of the hydraulic cylinder determined the diameter, which therefore affected the manufacture of the mechanism.Currently, when the complete folding state was about to be expanded, the hydraulic cylinder received a maximum thrust.
The force analysis diagram of the connecting rod mechanism is shown in Fig. 4. The thrust of the hydraulic cylinder is FE, the folding arm weight is G, and the total length of the folding arm is L1.Through force analysis and calculation, the thrust (or tension) of the hydraulic cylinder could be obtained as follows: ) sin( sin 2 sin ) ( In the process of flipping, the gravity direction was unchanged, so Eq. ( 18) needed to be multiplied by the coefficients of cos(θ2+θ2') and θ2' to make the value of θ2+θ2' equal to 0. Then Eq. ( 19) could be used to represent the force formula of the hydraulic cylinder in the whole folding process.When its value was regular, it was thrust, and when it was negative, it was tension.
In practical engineering, the smaller the maximum pressure required for the thrust and tension of the hydraulic cylinder, the more economical the hydraulic cylinder.Suppose the pressure required for the thrust was P1, the pressure required for tension was P2, and the maximum pressure required for hydraulic cylinder expansion and folding was Pmax.The difference between the pressure required for the maximum thrust and the pressure required for the maximum tension was P, the diameter of the hydraulic cylinder was D, and the diameter of the rod was d.According to Eq. 19, the following could be obtained:

Optimal design for speeds stationarity
According to the optimization model for the angular velocity stability of a four-bar mechanism, we could get the optimization result of the velocity stability.However, in the actual optimization design process, too many design variables and objective functions made it difficult to express the optimization, and to achieve the optimization result.Therefore, we used ADAMS simulation software to optimize the speed stationarity of four-bar linkage.The initial calculation size of the four-link mechanism is shown in Table 1, and its parametric model is shown in Fig. 5.   2. •6• In Table 2 we see the impact of each design variable on the folding speed when the initial value was obtained, and we see that the sensitivity of yA, yB, yE to the folding speed was highest.Therefore, yA, yB, yE were selected as the key design variables, and Eq. 17 was used as the optimization objective function for optimization analysis.The optimization analysis results are shown in Table 3: Table 3 shows that after optimization, the maximum angular acceleration of the folding arm was reduced by 64.4% in the folding process, and the stability of the folding speed of the folding arm was optimized.In order to see the stability optimization of the folding speed of the folding arm on the whole, the comparison diagram of folding angular velocity and angular acceleration of the folding arm before and after the optimization was introduced, as shown in Figs. 6 and 7.As can be seen from Figs. 6 and 7, both the folding angular velocity and the maximum folding angular acceleration of the folding arm decreased significantly after optimization, and the stability of the folding arm was optimized.
Fi (X) were the objective functions, Fi (X0) were the initial values of each objective function, ni was the weight factor, and n1+n2+...+ni=1 .
Using the optimization index K and simulation data, the optimization performance was calibrated more comprehensively.The expression for the speed stationarity optimization index K was as follows: ωf was the angular velocity at the end of the folding arm expansion under different design variables, ωmax was the maximum angular velocity in the expansion process, ωmin was the minimum angular velocity in the expansion process, and ω0f ,ω0max , and ω0min were the corresponding initial values before optimization.The initial value and the weight factor are seen in Table 4.
By looking at the study on the diagonal acceleration sensitivity of the design variables in Table 2, it can be seen that the diagonal velocities of points yE ,yA, yB had a great influence; the influence of the three variables on the optimization index K1 was mainly studied.The influence of yE to yA and yB on the optimization index K1 is shown in Figs. 8 and 9.
Table 4 The value of the weight factor and the initial value of the related angular velocity According to Figs. 8 and 9, we could not only obtain the optimal optimization solution under the relevant variables, but also found the relatively ideal optimization size data according to the corresponding optimization indexes.When point A was on the rack, the smaller the yA value was, the more compact the overall structure was.The contour map of the influence of variables yE and yB on K1 is shown in Fig. 10.In Fig. 10, we see that when yE=544.8 and yB=852.3,K1 reached a maximum value of 1.019, and the optimal optimization results are shown in Table 5.    6.
By Eq. ( 19), the maximum thrust or tension required for the hydraulic cylinder is in the starting or stopping position.Using Adams simulation data and Eq. ( 27 7.In practice, the folding arm was not horizontal, and the speed of the oil cylinder was low; the optimized pressure value was greater than the actual pressure value.

Prototype manufacturing and testing
Prototype manufacturing is divided into the 3D printing model and physical prototype.Through actual motion analysis of the 3D printed model, the parameters meet the expectations of the optimal design.Through cooperation with Hunan Xinghuo Machinery Manufacturing Co., the self-discharging transport equipment was manufactured, as shown in Fig. 12.By using the special test platform, we see that the folding mechanism folding time was less than 12 minutes, and the maximum pressure of the hydraulic cylinder was less than 18 MPa, in line with the standards of transport equipment of self-discharging ships.The actual working test of the prototype showed that the folding time was the same as the pressure of the hydraulic cylinder, the conveying volume of the conveyor was the same as that of the traditional fixed conveyor, the conveying bandwidth was 1,400 mm, and the conveying capacity was more than 4,000 tons/hour, thus achieving all the expected functions.The angular velocity curves of folding arms before and after optimization The angular acceleration curves of folding arms before and after optimization

2 Mathematical modeling of the optimization of the four-bar mechanism 2 . 1 Fig. 1 .
Fig. 1.Schematic diagram of four-bar mechanism motion.

2 . 2
The objective function Kinematic analysis is the basis of studying the dynamic characteristics of the mechanism.Figs. 2 and 3 show the motion analysis diagram of the four-bar mechanism.(a)Speed analysis diagram of hydraulic rod.(b) Speed analysis diagram of four-bar mechanism.

Fig. 6 .Fig. 7 .
Fig. 6.The angular velocity curves of folding arms before and after optimization

4. 2
Multi-objective optimization design of a four-bar mechanismThe force on the hydraulic cylinder and the folding speed multi-objective optimization index K2 were expressed as follows: relevant data and weight factors of the hydraulic cylinder are shown in Table ), the value of optimization index K2 can be obtained.The variables used were yE and xF, and yA and yB took the optimized values in Section 4.1 (yA = 425, yB = 852.3).When yE = 519.8and xF = 4255, the maximum value of K2 was 12.340, and the optimization results are shown in Table

Fig. 11 .
Fig. 11.3D printed model of each part of hydraulic connecting rod mechanism

Figure 2 Speed
Figure 2

Figure 4 Force
Figure 4

Table 1
Initial dimensions of four-bar linkage

Table 2
Design variables and sensitivity analysis results of four-bar linkage.

Table 3
Optimization analysis results

Table 5
Optimization results of velocity stationarity

Table 6
Hydraulic cylinder related data and weight factor value