1 design method

1.1 Experimental design method and establishment of response surface approximation model

The experimental design method is a sampling strategy used to study the influence of design parameters on the design of the model. The experimental design is an important part in the process of constructing the approximate model, which determines the number of sample points needed to construct the approximate model and the spatial distribution of these points.

The experimental design was carried out by using the central composite design method (CCDs), which was first proposed by Box and Wilson in 1951. Because of its sequential nature and high prediction efficiency, this design method has been widely used in practical design. The central composite design method can fully consider the interaction between experimental design variables while providing fewer experiments.

The approximation model establishes a functional relationship between the design variables and the response through mathematical statistics and experimental design methods to approximate complex practical problems. Commonly used approximate models include polynomial response surfaces, radial basis function models, Kriging models, and so on. The response surface method is a set of statistical processing methods for dealing with multivariate problem modeling and analysis. It can fit the whole data set as a whole when the uncertain data set tends to be a curve, which can be well solved. Multiple design variables simultaneously affect the optimization of the target. Therefore, we choose to establish a response surface approximation model for optimization.

1.2 Optimization process

The optimization flow chart is shown in Figure 1. Firstly, the design variables and their variation spaces are determined according to the actual problems. The central composite design method is used for the experimental design, and the sample points are collected according to the experimental design results. Then, the finite element analysis is performed on each set of sample sets. The response of the structure. A second-order response surface is established by fitting the functional relationship between the design variable and the system response, and the optimization surface is solved on the response surface. According to the mathematical model of the multi-objective optimization problem established by the free random method, the optimal solution is found on the entire second-order response surface, and the optimal solution found by the free-random method is used as the new sample space. Solve for further optimization and verify the true value of the optimal solution. If the accuracy requirement is met, the optimization solution process ends, and if not satisfied, the modified response surface is further iteratively solved. The accuracy of the response surface directly affects the pros and cons of the result. The error value of the approximate model that is required to fit is within 1%.

1.3 Multi-objective optimization problem processing method

What is often encountered in optimization design is the multi-objective optimization problem. For the multi-objective optimization processing methods, there are mainly evaluation function method, hierarchical solution method and target planning method. The evaluation function method solves by transforming a multi-objective problem into a single-objective problem by constructing an appropriate evaluation function.

The compromise method is a processing method of the evaluation function. Its basic idea is that the designer selects one of the multiple targets as the objective function for optimization, and treats the other objective functions as the feasible constraints, setting an undesired or less than Value, convert multi-objective problems into single-objective problems for optimization.

The paper uses the compromise method to optimize the design of the bed structure. The optimization model is:

Max k(1)st

F-[f][f]≤εX li≤X i≤X ui(i = 1,2,...,n)

Where: ε ??? allows the amount of change in the fundamental frequency of the bed during the optimization process; k ??? machine tool stiffness; [f] ??? original bed structure of the fundamental frequency; f ??? optimized bed structure The fundamental frequency; X i??? design variables; X ui, X li ??? the upper and lower limits of the design variables.

Equation (1) shows that the optimal value of the design variable X i(i = 1, 2, ..., n) is sought, so that the structural stiffness is maximum under the constraint that the fundamental frequency is close to the original structure.

1.4 optimization algorithm

Genetic algorithm is a highly parallel, random, adaptive search algorithm developed from the natural selection and evolution of the biological world. It uses a group search technique to solve the problem of a group of problems by generating a new generation of populations by applying a series of genetic operations such as selection, crossover and mutation to the current population, and gradually evolving the population to a state containing an approximate optimal solution. . Using genetic algorithm to solve the optimization model can solve the combinatorial optimization problem of arbitrary dimensional function, and can find the global optimal solution in the response surface. At the same time, the genetic algorithm can solve the problem of discrete variables well. The genetic algorithm solution flow chart is shown in 2.

2 examples

2.1 Research objects

A type of cylindrical grinding machine bed is cast from gray cast iron. The bed is 2 900 mm long, 1 170 mm wide, 707 mm high and 15 mm thick. The front bed has a V-flat rail with a total length of 2 800 mm. The front bed has 8 diaphragms and 1 level. Horizontal ribs. The back bed has two horizontal and vertical partitions. The thickness of the diaphragm inside the bed is 15 mm and the bottom of the bed is supported by 11 horns. A plurality of sand holes are arranged on the original bed. The finite element analysis results show that the stiffness of the original bed structure is 1.64×10 5N/mm, and the fundamental frequency is 212.8 Hz. In order to optimize the design, all the sand holes inside the bed are all Fill in and design the initial model for finite element analysis. The bed stiffness is 1.68×10 5N/mm and the fundamental frequency is 214 Hz.

2.2 Design Variables

The spacing of the internal diaphragms of the bed (x 1, x2, x 3, x 4), the thickness of the diaphragm x 5 and the spacing of the horns (x 6, x7, x8) are design variables, and they are studied on the bed. The influence of structural performance, design variables are shown in 4.

2.3 optimization solution

The optimization process is performed using the optimization process shown, and the design space of each variable is as shown. The central composite design method is used to generate the sample points of the design variables required for the 81 sets of experimental design, and the relationship between each set of design variables and the fundamental frequency and stiffness response is analyzed by the finite element method. By means of the least squares method, according to the experimental design results, the second-order response surface between the fundamental frequency and the stiffness and the design variable is constructed respectively. The maximum relative residual distribution of the two response surfaces is 0.61% and 0.74%, so the approximate model satisfies the accuracy. The requirements can be optimized instead of the actual model. The multi-objective optimization problem is processed according to formula (1) by using the compromise method, and the genetic algorithm is used to solve the optimization problem.

Design space design variable /mm value range/mm initial value/mm x 1 200～400 250 x 2 400～600 575 x 3 700～1 100 950 x 4 200～400 250 x 5 12～15 15 x 6 700～1 200 873 x 7 700～900 870 x 8 200～900 500

2.4 Optimization results

The optimized result is as shown in 2, the stiffness is increased from the original 1.68×105 N/mm to 1.73×105N/mm, which is increased by 3.04%. The fundamental frequency is reduced from the original 214 Hz to 212.8 Hz, which is reduced by 0.56. %.

2Optimization of dynamic and static performance of bed structure before and after optimization Design variables/mm Optimized stiffness before optimization/(N/mm) fundamental frequency/Hz Optimization before optimization After optimization optimization x 1 250 345 x 2 575 441 x 15 x 500 213

3 topology optimization

The bed structure needs to open the sand hole in a reasonable position, on the one hand, it is convenient for sanding in the casting process, and on the other hand, the bed body quality can be alleviated. The position of the sand hole must be such that after the opening, the rigidity and fundamental frequency of the structure meet the design requirements of the bed. Structural topology optimization can result in a reasonable material distribution in a given spatial structure. In this paper, the density structure is used to optimize the bed structure to obtain a reasonable sand hole location. Artificially introduced a hypothetical variable density material, the pseudo-density of each unit is a design variable, transforming the structural topology optimization problem into the material optimal distribution design problem, and applying the optimization criterion method to solve the optimal distribution of materials. The mathematical model of topology optimization is: max K(2)st 0<βi≤1 M≤M limit where: K??? stiffness; βi??? pseudo-density of element i; M??? ;M lmit??? The optimized structure quality set.

After the optimization, the bed quality is reduced by 10%, and the bed structure is topologically optimized by the density method. The optimized topology is shown in the figure. The gray in the figure represents the place where the bed material can be deleted, but the gray material only means that it can be theoretically deleted. Whether or not to delete the material needs to be combined with the actual needs of the bed structure. For example, if the material of the bed tank can be deleted, but it cannot be removed through the opening, the method of reducing the wall thickness can be selected to achieve the purpose of reducing the quality. Some structures show that the material needs to be retained through the topology diagram, but the actual bed structure needs to open some holes, and only some materials can be properly removed to meet the actual requirements. In combination with the actual structure of the bed structure and the actual needs of the bed structure, appropriate horizontal sand holes are opened in the horizontal partition plate and the bottom surface of the bed inside the bed body, and the wall thickness of the bottom surface of the bed body is reduced, and the bed structure is deleted after the material is removed. The structure is as shown in 6.

The finite element analysis of the optimized structure is carried out, and the structural performance of the optimized bed is as shown. The rigidity of the bed structure is 1.74×10 5N/mm, which is 1×10 4N/mm larger than the stiffness of the original bed structure, which is increased by 5.8%. Although the fundamental frequency is 204.5 Hz, the fundamental frequency of the original bed structure is higher than that of the original bed structure. The drop was 7.3 Hz, but the overall bed quality was reduced from the original 1 496.6 kg to 1 424. 3 kg reduced by 4.83%.

3 Topology optimization before and after bed structure performance comparison bed structure stiffness / (N / mm) fundamental frequency / Hz mass / kg before optimization 1. 64 × 10 5 212.8 1 496.6 after optimization 1. 74 × 10 5 205.5 1 424.3 change 5.8% (+) 3.43% (-) 4.83% (+) Note: + means rising; - means falling

4 Conclusion

The experimental design and approximate model method were used to optimize the spacing of the inner diaphragm of the bed, the thickness of the diaphragm and the position of the horn. After determining the spacing of the diaphragm and the position of the horn and the thickness of the diaphragm, the topology of the bed structure was optimized by the density method to determine the reasonable location of the sand hole. Research shows that such an optimization strategy not only can effectively improve the mechanical performance of the bed, but also has a high design efficiency. The proposed method was verified by taking a certain type of grinder bed as an example. After optimization, the structural rigidity of the bed increased by 5.8%, and the bed quality was reduced by 4.83%.

(Finish)

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