Lightweight design of double-head machine tool beam based on the adaptive multi-objective method

2.1. Software and devices

Experimental environment configuration for this article: Using SolidWorks 2020 software and ANSYS Workbench 2020 R2 software, running on a Windows 11 64 bit operating system, with a 13th Gen Intel (R) Core (TM) i9-13900 2GHz processor, the GPU is NVIDIA GeForce PTX3060, and a computer running memory of 32GB.

2.2. Establishment of 3D models

One head of the original double-head machine tool is a saw blade sucker assembly, and the other head is a machining center. It is properly simplified in SolidWorks software to delete unimportant parts, round holes, threads and chamfers [11]. In order to reduce the calculation time, two heads and the traveling frame for assembling the head are deleted. The simplified model is shown in Fig. 1.

Fig. 1Simplified model of the entire Double-head machine tool

2.3. Establishment of finite element model

The simplified 3D model of the two-head machine tool is connected to the ANSYS Workbench software. First, set the material of each part as shown in Table 1, and set the contact relationships between the parts. For the guide rail and slider, use No Separation (the contact surface between the parts can move but not separate), and for other parts, use Bonded (the contact surface between the parts is fixed). For the part of the machine bed that contacts the ground, use a fixed geometric shape to connect it to the ground [12].

Next, the grid division mode and size of each part are set. The simplified guide rail slider has a regular shape, and 10 mm hexahedral mesh is used for the slider and guide rail, and 10 mm tetrahedral mesh is used for the two marching frames. The shape of the base and the beam of the machine tool is complex, and the tetrahedral mesh of 50 mm is used for the base and 30 mm for the beam, and the rest is automatically meshed, and the mesh of the load attachment is encrypted. Finally, 1041697 nodes and 544,422 cells were generated. The standard earth gravity is set for the entire machine tool, and the 16 positioning holes and bolt holes on the base of the machine tool are fixed support constraints. The head weight of 1552.3 N is applied to the slider on the moving frame of the machining center, and the head weight of 2559.2 N is applied to the slider on the moving frame of the saw blade.

2.4. Static simulation and modal simulation

The close distance between the two spindle heads is to simulate the working condition when the load of the two spindle head is in the middle of the beam in the limit working condition. The vertical deformation cloud diagram and stress cloud diagram of the entire machine are shown in Fig. 2. According to the simulation results, due to the large span of the beam and the load of the two machine heads, the maximum stress is at the beam, and the maximum deformation is located in the middle of the beam. The maximum vertical deformation of the entire machine is 1.7956 mm, and the maximum vertical deformation of the beam is 1.7864 mm.

The entire machine undergoes fully constrained modal analysis, as illustrated in Fig. 3. The first natural frequency reveals that the crossbeam rotates around the $Y$-axis, the second indicates that the crossbeam experiences a vertical downward force on the $Y$-axis, and the third demonstrates that the crossbeam swings around the $X$-axis. Consequently, it is evident that the greatest deformation in the modal analysis occurs at the crossbeam. By combining the simulation results from both the static analysis and modal analysis of the entire machine, it is clear that the crossbeam is a crucial component, offering potential for design optimization.

Fig. 2Static characteristic diagram of two-head machine tool

a) Vertical deformation cloud diagram

b) Equivalent stress cloud diagram

Fig. 3Dynamic characteristic diagram of two-head machine tool

a) First-order frequency diagram

b) Second-order frequency diagram

c) Third-order frequency diagram

3.1. SIMP variable density method

The variable density method is to assume that the density of the material is variable, and then the element density is used as the design variable, so that the topology optimization problem is transformed into the material optimization problem, and some algorithms are used to solve the problem to achieve the optimal distribution of materials [13]. In the field of topology optimization based on variable density method, SIMP interpolation model and RAMP interpolation model are widely used. Among them, SIMP interpolation model has a slightly better penalty effect on intermediate density than RAMP interpolation model [14]. The SIMP interpolation method introduces continuous variable $x$, coefficient $\alpha$ and intermediate density unit into the discrete model in 0-1, so as to transform the discrete optimization problem into a continuous optimization problem [15]:

where, $E_i$ is the elastic modulus of element $i$, $x_i$ is the pseudo-density of the structural element, $\alpha$ is the correction factor, $E_0$ is the elastic modulus of the structure, and $E_{min}$ is the elastic modulus of the deleted material element. Therefore, the mathematical model of variable density method based on SIMP can be expressed as [16]:

where, $k_0$ is the stiffness matrix of the element with material density 1, $u_i$ is the displacement column vector of the material element, $x_i$ is the relative density of the material element, $v_i$ is the relative volume of the material element, and $x_{min}$ is the minimum material element density, which is usually taken as 0.001.

3.2. Topology optimization design of the beam

The original cross section of the beam is shown in Fig. 4. As the previous analysis shows, the deformation of the beam is the largest in the static characteristic analysis of the entire machine, so the beam is simulated separately, the fixed constraint is set at both ends of the beam, the simulated beam is fixed on the slider of the base, and the tetrahedral mesh of 20 mm is used to divide the beam, and the weight of the machining center marching frame head and the saw blade marching frame head are applied to the beam. Import the beam into the topology optimization module of Workbench for material redistribution, select density-based topology optimization as the optimization type, and retain 70 % of the original weight. The topology optimization results are shown in Fig. 5.

Fig. 4Cross section of the original beam

Fig. 5Optimization of the beam topology

The beam shown in the simulation result of topology optimization is difficult to be used directly, and this is in the limit working condition. The topology optimization result will change in the ordinary working condition, so the result is only used for reference. Because the beam is mainly subjected to vertical load, the beam is redesigned. Under the vertical load generated by the two spindle heads, the I-beam experiences compressive force along the upper flange and tensile force along the lower flange, while the central web plate remains largely unaffected. Leveraging their superior vertical compressive performance, six different beam structures were developed based on the I-beam concept, as depicted in Fig. 6.

Fig. 6Different sections of the beams

a) “Multiple X stiffener + single cylinder” beams

b) “X stiffener” beams

c) “X stiffener + single cylinder” beams

d) “Cross stiffener” beams

e) “Cross stiffener + single cylinder” beams

f) “Cross stiffener + double cylinder” beams

3.3. Static and mechanical analysis of six types of I-beam

Six different sections of I-beam were connected to the Workbench, and the static analysis module was used to conduct static simulation analysis with the same Settings as the original beam. The weight and maximum deformation of the seven types of beam were shown in Table 2. The static simulation results show that the weight and maximum deformation of “Cross stiffener”, “X stiffener” and “Cross stiffener + single cylinder” beams are required for this optimization design.

The above three kinds of beams were simulated for the static characteristics of the entire machine respectively, and the simulation results of the entire machine were not satisfactory. According to the comparison between the three simulation results and the original version, the beam surface that should be equipped with the guide rail on the front side was found to be complete, so as to realize the uniform force of the guide rail in front of the beam. Therefore, the section diagram of the new version of the I-beam was designed by combining the first six kinds of beams, as shown in Fig. 7. The static characteristics of the original beam and the redesigned beam after topology optimization are analyzed, and the comparison results are shown in Table 3. The weight and deformation of the new version of the beam are reduced to a certain extent.

4.1. Design variables

Keep the position of the original beam installation motor, slide rail, rack and bolt hole matched with the base. Select 13 parameters that have a great impact on the performance of the beam in Fig. 7. The specific selected position, size, upper and lower limits are shown in Table 4.

Fig. 7New version of the beam section and design variables

4.2. Sensitivity analysis

Sensitivity analysis reflects the sensitivity of design variables or parameter changes to the objective function. In mathematics, sensitivity analysis is interpreted as: if the objective function is derivable, the first-order sensitivity of the function is expressed as in a continuous system [17]:

The above formula is called the first-order differential sensitivity formula, where $F\left(x\right)$ is the performance function in the finite element simulation such as static deformation, weight, equivalent stress, etc., and $x_i$ is the design variable parameter such as each key size.

The solving efficiency of multi-objective optimization mathematical model is related to the number of parameters. It is usually necessary to evaluate the influence degree of different parameters on the optimization objective and eliminate the parameters with small impact, so as to improve the computational efficiency of the optimization process [18]. In this paper, the optimal space-filling design was used to generate 274 groups of experimental groups, and the influence degree of 13 design variables on the three objective functions was obtained after calculation. Fig. 8 is the bar chart of sensitivity analysis.

Fig. 8Bar chart of sensitivity analysis

It can be seen from the Fig. 8 that the parameters P1, P3, P9, P11, P12 and P13 have a great influence on the weight, and P11 is negatively correlated, while the rest are positively correlated. The parameters P1, P3, P4, P11, P12, P13 have a great influence on the deformation, and P13 is positive correlation, and the rest are negative correlation. The parameters P1, P3, P4, P11 and P12 have a great influence on the equivalent effect, and P1, P3 and P4 are positively correlated, while the rest are negatively correlated. P2, P5, P8 and P10 will be eliminated in the subsequent design process, and the remaining 9 design variables will be retained to participate in the subsequent multi-objective optimization of the beam, so as to improve the calculation efficiency.

4.3. Adaptive multi-objective optimization

Multi-objective optimization problem refers to the optimization process of the optimization objective is not single, and in general, there are contradictions between each objective function, so the solution is to find a “satisfactory solution”, rather than the “optimal solution”, the solution of multi-objective optimization problem is more than one, but a set of solutions, this set of solutions is called Pareto solution set [19].

Taking the minimum weight, minimum deformation and minimum equivalent stress of the beam as the optimization objective, taking the weight, maximum deformation and maximum equivalent stress of the beam as the constraint conditions, and taking the 9 dimensional parameters of the beam as the design variables, the multi-objective optimization mathematical model is established, which can be expressed as follows:

where $x_1,x_2,···,x_n$ is the combination of design variables, $F_1$, $F_2$ and $F_3$ represent the three objective functions of minimum beam weight, minimum beam deformation and minimum beam equivalent stress respectively, $f_1$, $f_2$ and $f_3$ represent the three constraints of beam weight, beam deformation and beam equivalent stress respectively, $S_1$, $S_2$ and $S_3$ are the constraint values, and $a_1,a_2,···,a_n$ represents the lower limit of the design variable value. $b_1,b_2,···,b_n$ represents the upper limit of the value of the design variables, and the range of the upper and lower limits is shown in Table 2. The value of $n$ is 9, which represents nine design variables.

Compared with the traditional GA, NSGA-II introduces a fast non-dominated sorting method, an elitist maintenance strategy, and an efficient crowding distance estimation process. These steps significantly improve the convergence rate of the iterative process, reduce the computational complexity of the algorithm, and effectively ensure the diversity of the population [20]. The adaptive multi-objective approach is a variant of the popular NSGA-II, Non-dominated Sorting Genetic Algorithm II, based on the controlled elite concept. It supports multiple objectives and constraints to find the global optimal solution in an adaptive loop iteration. In this paper, 78 samples are initially generated, 78 samples are generated in each iteration, and three candidate samples are found in a maximum of seven iterations.

4.4. Optimization calculation results and analysis

The 3D diagram illustrating the trade-off among geometric mass, equivalent stress, and maximum deformation of the crossbeam is presented in Fig. 9. It displays the Pareto front across all four stages. The optimization points shift from blue to green, then from green to yellow, and ultimately to red, indicating the convergence of the optimal solutions for the three objective functions.

Fig. 9Trade-off diagram for multi-objective optimization of beams

After the adaptive multi-objective method is used to calculate, three groups of better solutions are obtained as shown in Fig. 10. According to the comparison of the three groups, the weight is in the state of optimal solution under the premise of satisfying the appropriate deformation amount and equivalent stress. Therefore, the third group is selected as the “satisfactory solution”, that is, the Pareto solution.

Fig. 10Optimization design candidate solution set diagram

Because the data of the optimized Pareto solution set have decimal points, the optimized data are corrected, the corrected values are shown in Table 5, and changed into the parameters of the beam. The static simulation of the entire machine is carried out according to the same boundary conditions and grid division, and the results are shown in Fig. 11. The stiffness and strength of the beam are improved. Table 6 shows the comparison of performance parameters before and after optimization and the original beam.

By comparing the static characteristics of the beam before and after optimization, it can be seen that the weight of the beam is reduced by 8.88 %, the vertical deformation of the beam in the entire machine simulation is reduced by 11.29 %, and the equivalent stress of the entire machine is reduced by 28.33 % under the condition that the assembly parts are unchanged.

Fig. 11Static characteristic diagram of optimized two-head machine tool

a) Vertical deformation cloud diagram

b) Equivalent stress cloud diagram