Modeling and Optimization of Pulling Point Position of Luffing Jib on Portal Crane

Portal crane is the most commonly used equipment for cargo handling of large mixed loading ships with its advantages of flexible and convenient operation, wide adaptability, and high loading and unloading efficiency. +e reasonable modeling and optimization of the pulling point position of luffing jib of portal crane can reduce the rack force of portal crane and the power consumption output of the rack and pinion during the luffing process. Based on penalty function optimization, the interior point method is used to optimize the pulling point position of luffing jib. Compared with the initial design, the race force of the luffing jib is reduced to a certain extent. In addition, the consistency between the finite element analysis results and the optimization results can be verified, and the effectiveness of the optimization design is also proved through the finite element analysis of portal crane.


Introduction
Portal crane promotes the development of international logistics industry, and the demand of port handling in international logistics industry is also increasing. e increasing demand of portal crane in port handling also puts forward higher requirements for the operation performance and precision manufacturing of portal crane. At present, as the luffing system of crane, most of schemes of portal crane jib system adopt the horizontal luffing system, which makes the lifted weight move approximately horizontally in the process of luffing. e live counterweight behind the herringbone frame is used to balance the lifting weight so that crane can carry out luffing operation with lower driving force. e reasonable optimization design of rack pulling point position can reduce the driving force required by crane luffing operation and the power consumption output of rack and pinion work. e mixed penalty function is adopted to optimize the upper structure of jib cranes, which realizes the reduction in maximum luffing resistance moment and moment of force during crane operation [1]. e vibration testing technology is added to the fault diagnosis technology of portal crane, and the vibration testing and modal analysis of portal crane are carried out by using environmental vibration modal analysis in order to improve the safety and reliability of portal crane [2]. e transient dynamic simulation on the arm frame structure of crane is carried out, and the dynamic response of the arm frame is obtained at the same time [3]. e optimization design of the luffing mechanism is carried out, and the design scheme is formulated according to its characteristics, which achieves considerable optimization effect [4]. e optimization design of the four-bar luffing mechanism of portal crane is carried out by the method of combining genetic algorithm and finite element analysis to improve the design quality [5]. e electromechanical coupling dynamics of portal crane hoisting system is studied, and the electromechanical coupling model of portal crane hoisting system is established, which provides theoretical reference and basis for the matching and optimization of its electromechanical system parameters [6]. e spreader rotating parts of portal crane are studied, and the concept of skew angle control is proposed, which is successfully applied to port crane [7]. e deformation and stress distribution of portal crane trunk girder are analyzed by the finite element analysis method, and some reasonable suggestions and methods for structural optimization are given [8]. e BP neural network algorithm is used to calculate the lifting weight of portal crane and the instantaneous output power of lifting motor, and its effectiveness is proved by experiments [9]. e seismic performance of port quayside crane can be analyzed by a nonlinear static analysis method [10]. e braking process of portal crane under the action of pulsating wind is simulated, and the windproof effect of its windproof brake under complex wind load is analyzed [11]. Topology and shape optimization can be used to design the structure of mobile port crane [12]. A new three-dimensional position control scheme for mobile port crane is proposed, which can be used for the sway control in the process of cargo lifting [13]. A portal crane model is established through Solid-Works, and Adams software is used to obtain the dynamic characteristics of hinge point force in the process of crane luffing process [14]. A simulation experiment platform of port portal crane is established based on virtual technology [15]. e research process of this paper is also inspired by the relevant previous results [16][17][18][19][20][21][22].
Here, the penalty function method based on the interior point method is used to optimize the position of the rack pulling point of the portal crane so that the driving force required by the rack when the portal crane is hoisted and luffing is reduced to a lower value, and at the same time, the luffing mechanism is more energy-saving and environmentfriendly. In addition, the finite element model of portal crane is built and analyzed. By comparing the finite element analysis results with the optimization results, the feasibility and accuracy of the optimization of the rack pulling point of portal crane can be verified.

Principle of Penalty Function
Optimization Algorithm

Penalty Function Method.
e basic idea of penalty function method is to construct a new objective function by combining the objective function and all constraint functions in the constrained optimization problem to solve the constrained optimization problem.
For constrained optimization problem, (1) Penalty function method constructs the following function: where P(x) is some form of universal function defined by all constraint functions g u (x) and h v (x); r is the penalty factor that varies according to a given rule; and rP(x) is the penalty term, whose function is to promote the optimal solution to the feasible solution. Penalty function method transforms the original constrained optimization problem into unconstrained optimization problem, that is, When the iteration point violates the constraint conditions, the value of the rP(x) term becomes larger to urge the iteration point to move only in the direction of satisfying the constraint conditions.

Interior Point Penalty Function Method.
Interior point method is to define the new objective function in the feasible region so that its initial point and iteration point meet the requirement of existing in the feasible region.
Its constraint function P(x) is constructed as e penalty factor r should satisfy c and lim k⟶∞ r (k) � 0.
erefore, the penalty function form of interior point method is Figure 1, the balance weight is calculated according to the balance principle of moving center of gravity, and it can be concluded as

Live Counterweight Mechanical Model. As shown in
where G d is the balance weight, G b is the weight of arm frame, G l is the weight of large tie rods, G x is the weight of the trunk girder, h d is the lifting drop of the balance weight, h b is the lifting drop of the center of gravity of arm frame, and h x is the lifting drop of the trunk girder.

Calculation of Rack Force F by Mechanical Analysis.
Rack force of portal crane mainly consists of the following aspects: (1) As shown in Figure 2, rack resistance caused by the weight of the jib system is not fully balanced: where G g is the weight of tie rods of the balance system, G li is the weight of the balance beam, r f is the moment arm of force of rack against the arm frame point O, and N x is the force of G x + 1/2G l (i.e., the gravity of large tie rods and trunk girder) to the hinge point. (2) e rack resistance caused by wind load on the jib system is where p x is the wind load of trunk girder, p b is the wind load of the arm frame, h x is the moment arm of force of trunk girder against the arm frame point O, and h b is the moment arm of force of wind load against the arm frame point O. (3) As shown in Figure 3, rack resistance is caused by deflection angle α of the lifting rope to the plumb line due to wind load acting on the jib system, centrifugal force and luffing of the jib system during rotation, inertial force caused by the article during rotation, and braking during operation: where H is the resultant force of the horizontal force and the resulting large pull rods force S T which acts on the hinge point like trunk girder and arm frame, h a is the moment arm of force of resultant force H to the lower hinge point of the arm frame, and P Q is the weight load. Finally, it is concluded that the rack force F is 3.3. Maximum Force on Rack F max . Due to the luffing effect, the arm frame elevation angle θ of portal crane can be changed within a certain range. By dividing the elevation range of arm frame into N equal parts, a total of (N + 1) arm frame elevation angles {θ 1 , θ 2 , θ 3 , . . . , θ N+1 } can be obtained. At the same time, the value of F under a total of (N + 1) different arm frame elevation angles can be obtained, and then, the rack maximum force is Figure 4, this paper studies a port portal crane model, with an operating capacity of 9.5-33 m, a lifting capacity 25 t in the whole process, a height 50.4 m in nonworking dimensions, a portal gauge 10.5 m, and a dead weight 417.3 t.

Design Variable.
e pulling point position of the rack is mainly optimized, so the optimal design variable is x � (x 1 , x 2 ), where x 1 is the vertical distance from the hinge point of the rack to the arm frame and x 2 is the vertical distance from the hinge point of the rack to the rotation point of the arm frame. e unit of variables is mm.
According to the existing design, the reasonable boundary of design variables is given:

Constraint Condition
(1) Angle between arm frame and horizontal line is 43 < θ < 78, and the unit of angle is degree. (2) When the angle of the arm frame is at the maximum, the rack pulling point should not interfere with the buffer installed, so it needs to satisfy x 3 > 1500, and the unit of x 3 is mm.

e Objective Function.
e objective is to minimize the driving force required by the rack when the portal crane is lifting and luffing, in order to make the luffing mechanism more energy-saving and environment-friendly, so the objective function is Mathematical Problems in Engineering

Result Analysis
e optimized results and the initial design results are shown in Table 1. Compared with the initial design scheme, the optimized maximum rack force f(x) is reduced from 412 kN to 397 kN, which is about 3.64% lower and has a certain optimization effect. Figure 5 is the curve of rack pulling force of portal crane under different arm frame angles, and its horizontal coordinate is the elevation angle position after the arm frame angle is divided into 10 equal parts.
From Figure 5, it can be clearly seen that the rack pulling force tends to decrease first and then increase both the initial design and the optimized treatment, with the maximum value appearing at the position of maximum elevation angle (position 1 in the figure) and the minimum value appearing at around position 10. e optimized rack force is lower than the value before optimization under the elevation angle of each arm frame. For portal crane, the reduction in the overall rack pulling force in the crane luffing process makes the crane more energy-saving and environment-friendly. e establishment of finite element model for simulation analysis can obtain a more real stress situation and verify the accuracy of the optimized design effect. Figures 6 and 7 show the maximum rack force and minimum rack force obtained by calculating in the process of crane luffing under different settings and sampling points x 1 and x 2 , respectively.
From Figure 6, the trend of the maximum rack force with different x 1 and x 2 can be clearly known, where x 1 can cause a larger change in the maximum rack force. At the same time, it can be verified that the optimized maximum rack force is also at the lowest point of the whole curved surface, which is consistent with the results of finite element analysis. Figure 7 shows that the minimum rack force of crane is greatly affected by the change in x 2 . Since the objective function of the optimized design is the maximum rack force, the optimized minimum rack force cannot reach the minimum value in the whole range of design variables. However, compared with the initial design, the optimized minimum rack force is still lowered to a certain extent.      Mathematical Problems in Engineering

Conclusion
By optimizing the luffing jib pulling point position of portal crane, the maximum rack force of portal crane can be reduced to the lowest value under the condition of full-load luffing of goods. At the same time, the lower minimum rack force under full load is also obtained, and a good optimization effect is obtained, thus saving the output of rack operation power consumption, simultaneously, making portal crane safer and more energy-saving and environment-friendly. e optimization results of the luffing jib pulling point position of portal crane are verified jointly with the finite element analysis results. rough finite element modeling and mechanical analysis, the effectiveness of the optimization results can be proved.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.  Mathematical Problems in Engineering