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

Authors: Qianqian Jiao; Yixiao Qin; Yuehuang Han; Jinpeng Gu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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. The 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.


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## Body

## 1. Introduction

Portal crane promotes the development of international logistics industry, and the demand of port handling in international logistics industry is also increasing. The 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. The 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. The 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.The 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]. The 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]. The 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]. The 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]. The 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]. The 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]. The 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].The 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]. The 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]. The seismic performance of port quayside crane can be analyzed by a nonlinear static analysis method [10]. The 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 SolidWorks, 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]. The research process of this paper is also inspired by the relevant previous results [16–22].Numerical calculations are widely used in scientific research and engineering applications [23–25], which can realize optimal design and high-precision analysis of structures. A new and effective numerical method developed in recent years is the meshless method, including diffuse element method [26], smoothed particle method [27], reproducing kernel particle method [28–31], element-free Galerkin method [32–41], finite point method [42], natural element method [43], radial basis function method [44], mesh-free kp-Ritz method [45], complex variable meshless method [46–48], and all kinds of meshless boundary integral equation method [49–51]. In addition, numerical methods also include weighted residual method, finite difference method, finite element method, and boundary element method. The finite element method adopted in this paper is one of the most widely used methods with the advantages of simple, intuitive, and high calculation efficiency.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 environment-friendly. 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.


## 2. Principle of Penalty Function Optimization Algorithm

### 2.1. Penalty Function Method

The 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)minfx,x∈Rns.t.gux≤0,u=1,2,…,q,hvx≤0,v=1,2,…,p<n.Penalty function method constructs the following function:(2)φx,r=fx+rPx,where Px is some form of universal function defined by all constraint functions gux and hvx; r is the penalty factor that varies according to a given rule; and rPx 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,(3)minφx,r=fx+rPx.When the iteration point violates the constraint conditions, the value of therPx term becomes larger to urge the iteration point to move only in the direction of satisfying the constraint conditions.


### 2.2. 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 functionPx is constructed as(4)Px=−∑u=1q1gux.The penalty factorr should satisfy c and(5)limk⟶∞rk=0.Therefore, the penalty function form of interior point method is(6)φx,rk=fx−rk∑u=1q1gux.


## 2.1. Penalty Function Method

The 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)minfx,x∈Rns.t.gux≤0,u=1,2,…,q,hvx≤0,v=1,2,…,p<n.Penalty function method constructs the following function:(2)φx,r=fx+rPx,where Px is some form of universal function defined by all constraint functions gux and hvx; r is the penalty factor that varies according to a given rule; and rPx 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,(3)minφx,r=fx+rPx.When the iteration point violates the constraint conditions, the value of therPx term becomes larger to urge the iteration point to move only in the direction of satisfying the constraint conditions.


## 2.2. 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 functionPx is constructed as(4)Px=−∑u=1q1gux.The penalty factorr should satisfy c and(5)limk⟶∞rk=0.Therefore, the penalty function form of interior point method is(6)φx,rk=fx−rk∑u=1q1gux.


## 3. Mechanical Analysis Model of Rack

### 3.1. Live Counterweight Mechanical Model

As shown in Figure1, the balance weight is calculated according to the balance principle of moving center of gravity, and it can be concluded as(7)Gd≈Gbhb+Gx+1/2Glhxhd,where Gd is the balance weight, Gb is the weight of arm frame, Gl is the weight of large tie rods, Gx is the weight of the trunk girder, hd is the lifting drop of the balance weight, hb is the lifting drop of the center of gravity of arm frame, and hx is the lifting drop of the trunk girder.Figure 1
Schematic diagram of live counterweight-lever balance.


### 3.2. Calculation of Rack ForceF by Mechanical Analysis

Rack force of portal crane mainly consists of the following aspects:(1)
As shown in Figure2, rack resistance caused by the weight of the jib system is not fully balanced:(8)Ff1=Gdrdr1/r2+Glirlir1/r2−1/2Ggrgr1/r2+1/2Gglg+Gbrb+Nxrxrf,whereGg is the weight of tie rods of the balance system, Gli is the weight of the balance beam, rf is the moment arm of force of rack against the arm frame point O, and Nx is the force of Gx+1/2Gl (i.e., the gravity of large tie rods and trunk girder) to the hinge point.(2)
The rack resistance caused by wind load on the jib system is(9)Ff2=pxhx+pbhbrf,wherepx is the wind load of trunk girder, pb is the wind load of the arm frame, hx is the moment arm of force of trunk girder against the arm frame point O, and hb is the moment arm of force of wind load against the arm frame point O.(3)
As shown in Figure3, 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:(10)Ff3=Hharf,where H is the resultant force of the horizontal force and the resulting large pull rods force ST which acts on the hinge point like trunk girder and arm frame, ha is the moment arm of force of resultant force H to the lower hinge point of the arm frame, and PQ is the weight load.Figure 2
External load of jib system.Figure 3
External load of jib system.Finally, it is concluded that the rack forceF is(11)F=−Ff1+Ff2+Ff3.


### 3.3. Maximum Force on RackFmax

Due to the luffing effect, the arm frame elevation angleθ 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 {θ1,θ2,θ3,…,θN+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(12)Fmax=maxFθ1,Fθ2,Fθ3,…,FθN+1.


## 3.1. Live Counterweight Mechanical Model

As shown in Figure1, the balance weight is calculated according to the balance principle of moving center of gravity, and it can be concluded as(7)Gd≈Gbhb+Gx+1/2Glhxhd,where Gd is the balance weight, Gb is the weight of arm frame, Gl is the weight of large tie rods, Gx is the weight of the trunk girder, hd is the lifting drop of the balance weight, hb is the lifting drop of the center of gravity of arm frame, and hx is the lifting drop of the trunk girder.Figure 1
Schematic diagram of live counterweight-lever balance.


## 3.2. Calculation of Rack ForceF by Mechanical Analysis

Rack force of portal crane mainly consists of the following aspects:(1)
As shown in Figure2, rack resistance caused by the weight of the jib system is not fully balanced:(8)Ff1=Gdrdr1/r2+Glirlir1/r2−1/2Ggrgr1/r2+1/2Gglg+Gbrb+Nxrxrf,whereGg is the weight of tie rods of the balance system, Gli is the weight of the balance beam, rf is the moment arm of force of rack against the arm frame point O, and Nx is the force of Gx+1/2Gl (i.e., the gravity of large tie rods and trunk girder) to the hinge point.(2)
The rack resistance caused by wind load on the jib system is(9)Ff2=pxhx+pbhbrf,wherepx is the wind load of trunk girder, pb is the wind load of the arm frame, hx is the moment arm of force of trunk girder against the arm frame point O, and hb is the moment arm of force of wind load against the arm frame point O.(3)
As shown in Figure3, 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:(10)Ff3=Hharf,where H is the resultant force of the horizontal force and the resulting large pull rods force ST which acts on the hinge point like trunk girder and arm frame, ha is the moment arm of force of resultant force H to the lower hinge point of the arm frame, and PQ is the weight load.Figure 2
External load of jib system.Figure 3
External load of jib system.Finally, it is concluded that the rack forceF is(11)F=−Ff1+Ff2+Ff3.


## 3.3. Maximum Force on RackFmax

Due to the luffing effect, the arm frame elevation angleθ 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 {θ1,θ2,θ3,…,θN+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(12)Fmax=maxFθ1,Fθ2,Fθ3,…,FθN+1.


## 4. Optimization of Pulling Point Position of Luffing Jib

### 4.1. Original Design Parameters

As shown in Figure4, 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.Figure 4
Structural diagram of portal crane.


### 4.2. Optimization of Mathematical Model

#### 4.2.1. Design Variable

The pulling point position of the rack is mainly optimized, so the optimal design variable isx=x1,x2, where x1 is the vertical distance from the hinge point of the rack to the arm frame and x2 is the vertical distance from the hinge point of the rack to the rotation point of the arm frame. The unit of variables is mm.According to the existing design, the reasonable boundary of design variables is given:(13)1000<x1<2500,5500<x2<7400.


#### 4.2.2. Constraint Condition

(1)
Angle between arm frame and horizontal line is43<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 satisfyx3>1500, and the unit of x3 is mm.


#### 4.2.3. The Objective Function

The 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(14)fx=Fmax.


## 4.1. Original Design Parameters

As shown in Figure4, 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.Figure 4
Structural diagram of portal crane.


## 4.2. Optimization of Mathematical Model

### 4.2.1. Design Variable

The pulling point position of the rack is mainly optimized, so the optimal design variable isx=x1,x2, where x1 is the vertical distance from the hinge point of the rack to the arm frame and x2 is the vertical distance from the hinge point of the rack to the rotation point of the arm frame. The unit of variables is mm.According to the existing design, the reasonable boundary of design variables is given:(13)1000<x1<2500,5500<x2<7400.


### 4.2.2. Constraint Condition

(1)
Angle between arm frame and horizontal line is43<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 satisfyx3>1500, and the unit of x3 is mm.


### 4.2.3. The Objective Function

The 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(14)fx=Fmax.


## 4.2.1. Design Variable

The pulling point position of the rack is mainly optimized, so the optimal design variable isx=x1,x2, where x1 is the vertical distance from the hinge point of the rack to the arm frame and x2 is the vertical distance from the hinge point of the rack to the rotation point of the arm frame. The unit of variables is mm.According to the existing design, the reasonable boundary of design variables is given:(13)1000<x1<2500,5500<x2<7400.


## 4.2.2. Constraint Condition

(1)
Angle between arm frame and horizontal line is43<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 satisfyx3>1500, and the unit of x3 is mm.


## 4.2.3. The Objective Function

The 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(14)fx=Fmax.


## 5. Result Analysis

The optimized results and the initial design results are shown in Table1. Compared with the initial design scheme, the optimized maximum rack force fx is reduced from 412 kN to 397 kN, which is about 3.64% lower and has a certain optimization effect.Table 1
Comparison of optimization results with initial design.
Design schemeDesign variablesObjective functionOptimum proportionx1mmx2mmfx,kNOriginal21786364412—Optimized250064223973.64%Figure5 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.Figure 5
Rack force at each position.From Figure5, 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.The 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.The establishment of finite element model for simulation analysis can obtain a more real stress situation and verify the accuracy of the optimized design effect. Figures6 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 x1 and x2, respectively.Figure 6
Maximum rack forcefx.Figure 7
Minimum rack forcefx.From Figure6, the trend of the maximum rack force with different x1and x2 can be clearly known, where x1 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.Figure7 shows that the minimum rack force of crane is greatly affected by the change in x2. 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.


## 6. 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. The optimization results of the luffing jib pulling point position of portal crane are verified jointly with the finite element analysis results. Through finite element modeling and mechanical analysis, the effectiveness of the optimization results can be proved.



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*Source: 4627257-2021-04-08.xml*