Foreword
The roadway stacker crane is the core equipment of the automated three-dimensional warehouse. With the progress of science and technology and the improvement of production efficiency, the overall performance of the three-dimensional warehouse has gradually improved, including the increase of the rated load, the increase of the running speed and the increase of the size of the transported goods. When the stacker starts, accelerates, decelerates and stops, due to the gravity of the roadway stacker itself and the goods, it will generate a huge inertia force in the mechanical structure, and cause stress concentration, so that the key structural parts such as the cargo platform, column or beam will be bent and torsionally deformed, which will cause its fatigue phenomenon in the long run, and ultimately affect the stable operation of the three-dimensional warehouse. Therefore, in order to ensure the rigidity and motion stability of the stacker, it is very important to carry out dynamic and static analysis and structural optimization of the stacker. The author takes a certain type of double-column stacker as the research object, conducts static analysis and modal analysis, and then optimizes the gantry structure of the stacker crane through multi-objective optimization, so that the overall performance of the stacker crane can be optimized. 1. OverviewThe structure of the double-column roadway stacker crane is shown in Figure 1. 
Fig.1 Structural diagram of double-column roadway stackerupper beam; 2-left column; 3-lifting mechanism; 4-a horizontal walking mechanism; 5 lower beams; 6-operating table; 7-fork telescopic mechanism; 8-one cargo platform; 9-right column Figure 1 main parameters are as follows: rated load 3000kg, cargo platform mass 770kg, lifting mechanism mass 250kg. Under standard working conditions, the horizontal running speed is 2m s, and the horizontal running acceleration is 03m s2, lifting speed 03m s, lifting acceleration 03m/s2。2. Finite element analysis pretreatment2.1 Model simplification and importDue to the complex structure of the double-column stacker crane and the large variety and number of parts, the necessary simplification of the model was required before the finite element analysis could be carried out. The author focuses on the stress deformation of key core components such as gantry structure and cargo platform, and simplifies some parts that do not affect the main research, such as electrical equipment, control cabinets, belts, bearings, etc. At the same time, the model is simplified by removing extraneous chamfers and threaded holes from the model. The author uses SOLIDWORKS to complete the above simplification work to reduce the number of meshes and the amount of computer calculations in the later stage. When simplification is complete, import the model into Workbench. 2.2 Material Properties and Element Properties DefinitionsDue to the good welding performance of Q235A structural steel, this material is used in the structural components of this model stacker. The author defines the material properties as: elastic modulus 210gpa, density 7800kg m3, yield strength 235mpa, Poisson's ratio of 026。In addition, the metal frame of the stacker crane is a regular square tube or steel plate, so when meshing, the adaptive meshing with perfect performance is used, that is, the system decides to use tetrahedra or hexahedral for meshing according to the specific conditions of the model. The material thickness of the stacker gantry mechanism (upper beam, lower beam, left column, right column) is basically 10mm, and the material thickness of the cargo platform is basically 8mm, and the thickness of the two is different. The author uses 50mm, 45mm, 40mm, 35mm and 30mm decreasing grid sizes to divide the gantry structure, and uses 30mm, 25mm, 20mm, 15mm and 10mm decreasing grid sizes to divide the cargo platform and forks. After several rounds of calculation experiments, it can be found that when the grid size of the gantry structure is 40mm, and the grid size of the cargo platform and fork is 20mm, the stress and deformation of the stacker crane basically converge to a certain interval. Therefore, according to the above analysis results, the author sets the maximum size of the grid of the gantry structure (upper beam, lower beam, left column, right column) to be 40mm, and the maximum size of the grid of the cargo platform and fork is 20mm. The overall meshing results are shown in Figure 2. 
Figure 2 Meshing2.3 Constraints and load applicationThe author first stipulates that the direction of the stacker crane moving along the roadway is the X direction, the lifting direction of the cargo platform is the Y direction, the telescopic direction of the fork is the Z direction, and the coordinate system is shown in Figure 2. Due to the different forces and deformations of the stacker crane under different working conditions, the situation with large force deformation should be selected for analysis. When the load carrier is at its highest position and the forks are fully extended, the stacker crane is in a more dangerous state. At this time, the three degrees of freedom of the lower beam x, y, and z should be constrained. In addition, the upper beam retains the freedom of movement of the y-axis and the rotation of the z-axis, and should also constrain the degrees of freedom in other directions. By applying the stacker crane to the negative direction along the y-axis, the size is 9The gravitational acceleration of 8m s2 realizes the gravity loading of the stacker; An equivalent amount of force is applied to the pallet in the negative direction of 3000kg along the y-axis; The 6 fixed pulleys of the upper cross beam bear the weight of the cargo platform and the cargo, therefore, the force of each fixed pulley is distributed according to the actual force ratio; The two large pulleys on the load carrier pull the lifting work of the load carrier, so that they divide the weight of the load bed and the load equally. 3. Finite element analysis of stacker3.1 Static stress analysis3.1.1. Deformation analysis: The author uses the static structural module in the workbench software to obtain the deformation contour diagram of the whole stacker, as shown in Figure 3. 
Fig.3. The overall deformation contour of the stackerUnder full load conditions, the maximum deformation of the whole structure of the stacker crane is 1612mm。The deformation contour of the cargo carrier is shown in Figure 4. 
Fig.4. Deformation diagram of the cargo platformIt can be seen from 4 figures: the maximum deformation position of the stacker crane is also the maximum deformation position of the cargo platform, which appears in the front cross beam of the cargo platform; At this moment, the fork telescopic structure is fully extended, and the goods are located at the top of the goods, and the cargo platform is to maintain the balance in the horizontal direction, bear the bending moment effect brought by the goods, and the front beam position of the cargo platform in contact with the front end of the fork is deformed the most, but its deformation is small, therefore, it has no effect on the normal use of the stacker. In addition, the deformation of the load carrier is symmetrical. The deformation contour diagram of the gantry structure is shown in Figure 5. 
Fig.5. Deformation contour of gantry structureThe maximum deformation of the gantry structure occurs in the middle of the left and right columns, and the size is 05432mm, the reason is that the bending moment generated by the gravity of the cargo has a tensile effect on the column. In addition, the deformation of the left and right columns is still symmetrical, which also indirectly indicates that the entire stacker crane structure is symmetrical, and there will be no risk of dumping caused by the offset of the center of gravity. "JBT7016-2017 Roadway Stacking Crane" makes performance requirements for the static stiffness value of the stacker column, that is, "when the lifting height is not more than 10m, its static stiffness value should not be greater than H 2000 (H is the full height of the stacker); When the lifting height is greater than 10m, its static stiffness value should not be greater than h 1500", and the definition of static stiffness is the ability of the structure to resist deformation under specific static disturbances, which is generally measured by the deformation of the structure under static load, so the maximum deformation of the stacker crane is selected as the index to test the static stiffness of the stacker. According to this national standard design principle, when the height of the stacker crane is not more than 10m, the static stiffness value should not be greater than h 2000 = 37mm (the total height of this model stacker crane is 7.)4m)。According to the above analysis, the static stiffness value of this model stacker crane in the case of full load is 05432mm, which does not exceed the allowable static stiffness value of the stacker crane 37mm, to meet the requirements of stiffness. 3.1.2 Stress analysis The equivalent stress contour diagram of the double-column roadway stacker structure is shown in Figure 6. 
Fig.6. The overall stress contour of the stackerUnder full load conditions, the maximum value of the equivalent stress force of the stacker crane is 16853mpa。The equivalent stress of the loading dock is shown in Figure 7. 
Fig.7. Stress contour of the loading platformIn Fig. 7: the maximum stress is located at the junction of the lower rectangular tube and the vertical frame of the cargo carrying platform, because the bending moment generated by the protruding goods to the cargo carrying platform makes the rectangular pipe be squeezed, and this connection is originally designed to be connected at right angles, therefore, there is a stress singularity in the finite element calculation, and the calculation result cannot be stabilized to a certain interval with the improvement of the fineness of the grid, and the fillet needs to be added to achieve precise calculation; The maximum stress is less than the allowable stress of Q235 s=235MPa, so the strength of the cargo platform meets the requirements. The equivalent stress of the gantry structure is shown in Figure 8. 
Fig.8. Stress contour of gantry structureSince the lower beam bears the weight of the entire stacker crane (including the cargo platform and the cargo), the maximum stress of the gantry structure occurs at the junction of the column and the lower beam, and the size is 17811mpa。The national standard "GBT3811-2008 Crane Design Code" has made relevant provisions for the strength verification of the gantry structure of the stacker. The stress safety factor s of the frame structure is 14. Allowable stress [ ]= s [s]=167MPa, the maximum stress of the gantry structure is far less than the allowable stress of the material, and the strength meets the requirements. 3.2 Modal analysisThe operational stability of the stacker crane is determined by its dynamic characteristics, so it needs to be analyzed in the design and verification phase of the stacker. The author used ANSYS finite element software to analyze the modality of the stacker crane under no-load operation, which can not only obtain the natural frequency and corresponding mode shape of the stacker crane at different stages, but also provide data support for the subsequent structural optimization. Since the influence of the higher-order frequency on the structure of the stacker crane is small, the research significance is not great, so under the selected working conditions, the author only calculates the first 6 natural frequencies and the corresponding mode shapes of the stacker, as shown in Figure 9. 
Fig.9. Mode shape diagram of the first 6th order of the stackerAnalysis of Figure 9 shows that the first-order frequency is 14At 142Hz, the column is bent and deformed, which is manifested as a swing along the Z direction; The second-order frequency is 2469Hz, the main deformation of the stacker crane occurs on the column, which is manifested as a swing along the X direction, and at the same time, because the cargo platform is in contact with the column, it will be accompanied by deformation; The third-order frequency is 24828Hz, the column produces bending deformation along the X direction; The fourth-order frequency is 33At 612Hz, the right column is greatly deformed, which is manifested as torsion along the Y direction; The fifth-order frequency is 39At 352Hz, the maximum deformation occurs in the middle of the left and right columns, which is 10044mm;The sixth-order frequency is 43At 367Hz, the deformation occurs mainly in the loading dock and is manifested as bending in the XOY plane. From the above analysis, it can be seen that the vibration deformation of the first six mode shapes is small, and the natural frequency is high. Modal analysis is an effective means to avoid the resonance phenomenon of the machine in the structural design verification stage. The upper and lower roadways of the automated warehouse stacker crane are not a complete section of steel rails, but are welded together by many sections of steel rails. Due to the uneven weld of the guide rail, the external vibration of the stacker crane is mainly in the contact between the wheel and the weld when the stacker crane moves in the horizontal direction. According to the "JBT9018-2011 Automated Three-dimensional Warehouse Design Code", the horizontal speed of the stacker crane is stipulated: the speed is between 24m min and 250m min, and the author's project stipulates that the horizontal running speed of the stacker crane is 2m s, which is located in this interval. Since the distance between the bottom wheels of the stacker crane is 3420mm, the excitation frequency is 058Hz, which is much smaller than the first-order natural frequency of 14142Hz, so the stacker crane can effectively avoid the occurrence of resonance. Fourth, structural optimizationFrom the above dynamic and static analysis results, it can be obtained: the strength and stiffness of the stacker crane meet the design requirements under standard working conditions. As the core component, the gantry structure bears the largest load, accounting for more than 65% of the mass. Therefore, under the premise of ensuring the rigidity of the gantry structure, the author will take the lightweight of the gantry structure as the main optimization goal to improve the gantry structure of the stacker. 4.1 Sensitivity analysis of dimensional parameters4.1.1. The gantry structure of the model parametric double-column stacker crane is composed of four parts: the left column, the right column, the upper beam and the lower beam. The author will adjust the size of each component to achieve the overall optimal performance of the stacker. The optimization of gantry structural parameters requires a parameterized structural finite element model. In the ANSYS optimization analysis process, the model establishment and result extraction are all achieved through parameters, which can effectively improve the computational efficiency without manually modifying the model when performing a large number of dimensional iterative calculations. By importing the model into Solid Works, the author uses the software geometric parameter setting function to set the dimensions that have a great impact on the objective function as variables. Then, the parametric model was imported into Workbench for static analysis, and the maximum stress, maximum deformation, and mass were set as optimization goals. The design variables of the gantry structure of the stacker crane are shown in Figure 10. 
Fig.10. Dimensions of the parametric modelP1-upper beam top width; P2-upper beam bottom plate thickness; P3 - upper beam rib spacing; P4 a thickness of the roof plate of the upper beam; P5-upper beam outer plate thickness; P6-upper beam rib thickness; P7 - column thickness; P8-column cross-sectional length; P9-column cross-sectional width; P10 below the distance between the bottom plate and the top plate in the middle of the beam; P11 below the width of the roof in the middle of the beam; P12 below the thickness of the roof in the middle of the beam; p13 below the thickness of the beam ribs; pi4 below the thickness of the bottom plate of the beam; P15 below the spacing of the ribs of the beam; P16 below the thickness of the bottom plate of the beam 41.2 Sensitivity analysisThe author sets the maximum stress, structural quality and maximum deformation of the gantry structure of the stacker crane as the response target, uses the Re-Sponse Surface module in the Workbench software to analyze the sensitivity of each variable to the response target, and obtains the influence degree of 16 variables (P1 P16) on the response target through multiple sampling and fitting analysis, that is, the sensitivity, and plots it as a sensitivity curve, as shown in Figure 11.
Fig.11 Sensitivity of design variablesFig. 11(ac) shows the sensitivity of each variable to the maximum stress, maximum mass, and maximum deformation in response to the target, respectively. As can be seen from Figure (a), P14 has the greatest influence on the maximum stress of the stacker; In Fig. (b), P7 and P8 have a significant effect on the structural quality of the gantry structure of the stacker. In Figure (c), P8 has the greatest impact on the maximum deformation of the stacker, followed by P7. Therefore, the three variables p7, p8 and p14 in the design parameters have a great influence on the values of the three response objective functions, and the author chooses them as the optimization variables. 4.2. Optimization model establishmentBased on the structural model of the double-column stacker, the author establishes a multi-objective optimization model with the column thickness P7, the column width P8 and the lower beam bottom plate thickness P14 as the design parameters, and the stacker quality, structural strength and stiffness as the optimization goals
Where: m a stacker crane mass; The maximum deformation of the structure of a stacker crane at smax; max a stacker crane maximum stress value; [ The allowable stress value in the design code of a stacker crane is 167MPa;] [s] The allowable deformation value of the neutral column in the design code of a stack crane is 37 mm (see 3.)11);pi - design parameters, i=1,2,16。4.3. Optimization result analysisIn order to improve the optimization efficiency of the double-column stacker, the author uses the best filling space (OSF) to design the experiment, so that the sample points are evenly distributed in the design interval and the design parameters are calculated with fewer experimental points. Secondly, the Kriging** model was used to simulate the input-output function model to achieve an accurate description of the nonlinear function. Then, the MOGA multi-objective optimization method is used to reduce the optimization calculation time and ensure the convergence stability of the optimization. The MOGA parameters are set as follows: the initial sample size is 3000, the maximum allowable pareto percentage is 70%, and the convergence stability rate is 2%. Through the multi-objective optimization of Workbench, three Pareto optimal design schemes were obtained, as shown in Table 1. Table 1 Three optimal pareto solutions
Analysis of the three optimization schemes in Table 1 shows that scheme 1 has the best optimization result for the maximum deformation value of the gantry structure, which is 2 lower than the initial value53%, scheme 3 and scheme 2 increase the deformation; Scheme 2 has the best optimization effect on the maximum stress of the gantry structure, which is 7. lower than the initial value40%, followed by the optimization effect of scheme 1 and scheme 3; Scheme 3 has the most obvious effect on the quality optimization of the gantry structure, which is 603%, scheme 2 second, scheme 1 gantry structure quality increased instead. According to the analysis of the optimization objectives and above, it can be seen that the optimization effect of the third group of schemes is the best. By comparing the optimal solution with the initial solution after rounding the third group of schemes, the comparison table before and after optimization can be obtained, as shown in Table 2. Table 2 Comparison table before and after optimization
Although the maximum stress of the gantry structure of the stacker crane increases by 482%, deformation increased by 902%, but its strength and stiffness are still within the allowable range, and the mass is reduced by 983%, so as to realize the lightweight of the gantry structure of the stacker. 5. Concluding remarksIn view of the problems existing in the double-column stacker, the author takes a certain model of double-column stacker as an example to optimize the gantry structure of the stacker crane to optimize the overall performance of the stacker, that is, the static method and modal analysis method of the double-column stacker are analyzed by using workbench software, and the force, deformation and vibration of the structure are studied. The multi-objective optimization method was used to optimize the gantry structure.
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Author: Ma Chaopeng, State Key Laboratory of Advanced Design and Manufacturing of Automobile Body, Hunan University, Dajie Intelligent Technology (Guangdong)**Xie Hui**: Mechanical and Electrical Engineering
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