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As the main load-bearing structure of heavy machine tools, cranes, and other high-end equipment, the large-scale box structures usually bear moving loads, and the results of direct topology optimization usually have some problems: the load transfer skeleton is difficult to identify and all working conditions are difficult to consider comprehensively. In this paper, a layout design method of stiffened plates for the large-scale box structures under moving loads based on multiworking-condition topology optimization is proposed. Based on the equivalent principle of force, the box structures are simplified into the main bending functional section, main torsional functional section, and auxiliary functional section by the magnitude of loads and moments, which can reduce the structural dimension and complexity in topology optimization. Then, the moving loads are simplified to some multiple position loads, and the comprehensive evaluation function is constructed by the compromise programming method. The mathematical model of multiworking-condition topology optimization is established to optimize the functional sections. Taking a crossbeam of superheavy turning and milling machining center as an example, optimization results show that the stiffness and strength of the crossbeam are increased by 17.39% and 19.9%, respectively, while the weight is reduced by 12.57%. It shows that the method proposed in this paper has better practicability and effectiveness for large-scale box structures.

The large-scale box structures are widely used in heavy equipment such as machine tools, cranes, and so on. They usually bear the moving loads with the continuous change of position and its stiffness, strength, and weight directly affect the mechanical performance of the equipment. For example, the crossbeam of a machine moves on the gantry and bears the loads from the cutting components. The stiffness directly affects the machining accuracy [

The box structures are composed of external skin and internal stiffened plates, and the essence of lightweight design is to optimize the layout and size parameters of the internal stiffened plates [

Topology optimization is a method that is used to seek an optimal configuration for the hollow location and number in some certain and continuous regions. It has been widely used in structural lightweight design [

Based on the above problems, this paper proposes a layout design method for the large-scale box structures under moving loads based on multiworking-condition topology optimization, which transforms the 3D topology optimization with moving loads of the large-scale box structures into the 2D topology optimization with multiple position loads. Section 2 describes the existing problems of topology optimization for large-scale box structures. The design method of structural decomposition process based on the functional sections and multiworking-condition topology optimization are introduced, respectively, in Section

SIMP topology optimization [

The optimization problem could be solved using several different approaches, such as Optimality Criteria (OC) methods, Sequential Linear Programming (SLP) methods, or the Method of Moving Asymptotes (MMA) and others [_{e} is found from the optimality condition as follows:

A large number of stiffened plates are usually arranged in the large-scale box structures to improve the stiffness and strength, so the essence of lightweight is to optimize the layout of stiffened plates. Taking a typical box structure with 1000 × 200 × 200 mm shown in Figure

Diagram of a large-scale box structure.

Optimization results. (a) Case 1, (b) case 2.

Due to the large aspect ratio of the structure shown in Figure

Therefore, we can get a conclusion: the direct topology optimization of the large-scale box structures has some problems: the load transfer skeletons are difficult to identify and all working conditions are difficult to consider comprehensively, which cannot provide guidance for the rational layout design of the internal stiffened plates. So, it is necessary to develop new design ideas to optimize large-scale box structures.

According to the above problems, we assume that the material redundancy problem may be solved if we convert the 3D topology optimization problem into a 2D problem. The moving loads can be simplified to multiple position loads, which will restore the actual working condition of the box structures.

The decomposition method of the functional sections refers to a method of decomposing the complex 3D structure into 2D sections by analyzing loads of complex working conditions. For the structures with complex working conditions, the loads can be simplified according to the equivalent principle of force in theoretical mechanics. As shown in Figure _{x}, _{y}, _{z}) and moments (_{x}, _{y}, _{z}) are decomposed into three coordinate axes.

Diagrammatic sketch of load equivalent.

For the convenience of calculation, the _{x} > _{y} > _{z}, the XY plane is defined as the main bending functional section and the _{x} > _{y} > _{z}, the

After determining the functional sections, the bearing types and boundary conditions of each functional section are analyzed and simplified. The Finite Element Model (FEM) is established, and topology optimization can be carried out by equation (

In this paper, the moving loads on the large-scale box structures are equivalent to several working conditions according to the position of loads. It is necessary to unify the optimization objectives of each working condition to consider the load transfer skeletons at any position in the process of topology optimization, which is a problem of multiobjective topology optimization.

A comprehensive evaluation function is needed to transform the multiobjective problem into a single objective problem. The linear weighting method is usually used to make it in the traditional multiobjective topology optimization. But it is to calculate weight average value for all functions and cannot reflect the prominent influence from some certain functions, which does not guarantee that all functions obtain the relative optimal solutions. The compromise programming method [

So, the objective function of the static multiworking-condition stiffness is established by the compromise programming method, as shown in equation (

In this paper, a large-scale crossbeam of superheavy turning and milling machining center is taken as an example to verify the proposed method. The turning and milling machining center is composed of crossbeam, sliding parts, workbench, slide carriage, machine tool bed, portal frame, and other components, as shown in Figure

Superheavy turning and milling machining center.

The crossbeam is installed on the portal frame, which is the supporting part of the sliding parts and also the main part bearing the cutting forces. Its static and dynamic performance directly affects the accuracy of the machine tool. The crossbeam is 10.4 m long, 7.5 m span, 1.28 m wide, 1.8 m high, and 40058 kg weight, and it is welded by a Q235 steel plate. It belongs to a typical box structure because of its large volume and mass, large inertia load in the process of moving up and down. To improve the overall mechanical performance of the crossbeam, the decomposition method of functional sections and multiworking-condition topology optimization are used to optimize the layout of the internal stiffened plates.

The crossbeam and sliding parts not only contain many parts but also have complex features. The efficiency and accuracy of the Finite Element Analysis (FEA) for all parts are very low, which is easy to lead to errors in simulation analysis. In order to improve the calculation efficiency, the features such as small holes, small corners, small gaps, and welds on the crossbeam are ignored, and a simplified model is established, as shown in Figure

Simplified model of the crossbeam.

The boundary condition of the crossbeam can be equivalent to a simply supported beam. The sliding parts slide on one side of the crossbeam, and the crossbeam bears offset moving loads. Therefore, the loads that cause deformation of the crossbeam include self-gravity, gravity of the sliding parts, distortion caused by offset loads, and cutting forces. The cutting forces of the turning and milling machining center under a typical working condition are shown in Table

Cutting forces under a typical working condition.

Main cutting force _{c} | Feeding force _{f} | Radial force _{r} | |
---|---|---|---|

Value (N) | 74760 | 41118 | 29904 |

Simplified boundary conditions of the crossbeam.

The model is imported into the FEA software ANSYS. The tetrahedral element is used to divide the parts with complex features, while the hexahedral element (hexdominant method) is used to divide other parts. The loads and constraints shown in Figure

Finite element model (

As the sliding parts move to different positions on the crossbeam, the static response is different. 11 key positions are selected along the

The response curve of the static maximum displacement.

The results of FEA (

In this section, the decomposition method of functional sections is used to reduce the dimension of the crossbeam in topology optimization. Firstly, the coordinate system (shown in Figure

Equivalent loads at the center of mass of the crossbeam.

According to the equivalent principle of force, loads of the crossbeam under the typical cutting condition mainly include self-gravity _{1}, gravity _{2} of the sliding parts, cutting forces (main cutting force _{c}, radial force _{r}, feeding force _{f}), and moments _{x}\_{y}\_{z} from above loads on the center of mass. It can be seen from Section _{c} > _{f} > _{r} (Table

Based on the principle of determining functional sections proposed in Section

The main bending functional section is shown in Figure

Main bending functional section. (a) YZ plane. (b) Equivalent working condition.

The main torsional functional section is shown in Figure

Main torsional functional section. (a) XZ plane, (b) equivalent working condition.

In this section, topology optimization of the main bending functional section under nine working conditions (

In addition,

The compliance of the

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

2.66 | 4.07 | 6.14 | 8.69 | 9.82 | 8.69 | 6.14 | 4.08 | 2.67 | |

10.39 | 14.23 | 18.26 | 21.38 | 22.55 | 21.38 | 18.26 | 14.23 | 10.39 |

Taking the relative density as the design variable and a material removal rate of 30% as the constraint condition, aiming at the minimum comprehensive evaluation value of the static stiffness under multiple working conditions, the mathematical model of multiworking-condition topology optimization is established as shown in the following equation:

The optimization result is shown in Figure

The optimization result.

The layout of internal stiffened plates.

In order to show the advantages of multiworking-condition topology optimization, the topology optimization of the fifth working condition (_{gk5} in Figure

Optimization result of the fifth working condition.

The results of topology optimization for a main torsional functional section are shown in Figure

The optimization result of the XZ section.

New cross section of the crossbeam.

Based on the above results, the new model of the crossbeam is established as shown in Figure

The new model of the crossbeam.

In order to verify the effect of multiworking-condition topology optimization of the crossbeam, taking the new model as the research object and applying the same boundary conditions (the same as Figure

The displacement of 11 working conditions.

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

Location | 0 | 0.04083 | 0.03227 | 2700 | 3600 | 3700 | 4500 | 5400 | 6300 | 7200 | 7400 |

Initial model (mm) | 900 | 0.04998 | 0.04182 | 0.11514 | 0.12850 | 0.12898 | 0.12279 | 0.10020 | 0.06001 | 0.04068 | 0.03808 |

New model (mm) | 1800 | 0.08456 | 0.07074 | 0.09542 | 0.10642 | 0.10655 | 0.09726 | 0.07365 | 0.04323 | 0.03319 | 0.03259 |

The comparison of the displacement response.

The results show that the overall displacements of the crossbeam become smaller than the initial model, which indicates that the structural stiffness is strengthened. Figure

The results of FEA for the new model. (a) Displacement distribution. (b) Stress distribution.

Compare the static mechanical performances when the sliding parts are in the middle of crossbeam (

Comparison of mechanical performances.

Max displacement (mm) | Max stress (MPa) | Mass (kg) | |
---|---|---|---|

Initial model | 0.12898 | 12.777 | 40058 |

New model | 0.10655 | 10.234 | 35023 |

Variation | −17.39% | −19.9% | −12.57% |

In this paper, a layout design method for the large-scale box structures under moving loads based on multiworking-condition topology optimization is proposed to solve the problem of difficult identification and complex moving loads. According to the magnitude of loads and moments, the complex 3D structure is transformed into 2D functional sections including the main bending functional section, the main torsional functional section, and the auxiliary functional section, which makes the topology optimization simplify from 3D to 2D. The complex moving loads are equivalent to several working conditions, and the comprehensive evaluation function is constructed by using the compromise programming method, which solves the problem of load sickness under the topology optimization of extreme single-working-condition and avoids that the result is only the local optimal solution rather than the global optimal solution. Taking a crossbeam of superheavy turning and milling machining center as an example, the optimization results show that the stiffness and strength of the crossbeam are increased by 17.39% and 19.9%, respectively, while the weight is reduced by 12.57%. It shows that the method proposed in this paper has better practicability and effectiveness for large-scale box structures.

The data can be obtained from the corresponding author.

The authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work, and there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in this manuscript.

This work was supported by the Chinese National Key Research and Development Program (2016YFC0802900) and the Key Natural Science Projects of Hebei Provincial Department of Education (ZD2020156).