外文文献2010_Optimal design approach for eco-efficient machine tool bed_翻译.pdf

Optimal design approach for eco-effi cient machine tool bed Xiaohong DingYelin ChenWei Liu Received: 15 October 2010/Accepted: 21 December 2010/Published online: 1 January 2011 ? Springer Science+Business Media, B.V. 2010 AbstractAn optimal design approach of machine tool bed with the aim of obtaining an eco-effi cient machine structure is studied. The suggested method includes three phases. The fi rst is the layout design optimization of stiffener plates inside bed. In order to improve the design effi ciency, a simplifi ed design model called fi ber model is suggested, and the layout of the stiffener plates inside bed is optimized by changing a 3-dimensional topology design optimiza- tion problem to a 2-dimensional problem. The second is the detailed sizing optimization of stiffener plates and supporting blocks under the structure based on the initial optimal model resulted from phase one. Finally, a topology design optimization process is imple- mented to obtain a reasonable distribution of manu- facturing holes in bed structure. By considering the manufacturing requirement, an optimal bed structure is obtained. The validity of the suggested method is confi rmed by a typical cylindrical grinding machine tool bed, and the result shows that the suggested method is effective, and the optimal structure has much better mechanical and economical performance by comparing with the original structures. KeywordsStructural optimization ? Stiffener plate layout ? Machine tool bed ? Structural eco-effi ciency 1 Introduction The precision of machine tool has been highly improved to meet the development of every walk of industry, such as aerospace, automobile, ship, and so on. Precision is the guideline of the machine design and assembly, the machine elements manufacturing, the testing procedures and the use of auxiliary systems. The factors affecting the precision involve the structure deformations under the action of cutting and inertial forces, assembly errors of the machine mobile components on tool position, the dynamic behaviour of the system under the excitation of cutting forces, friction and backlash effects in the guideways and drive trains, non-deformability with respect to heating from thermal sources, etc. (Lo pez de Lacalle and Lamikiz 2009). Among these factors, stiffness error caused by the structural deformation of machine tool is one of the most important factors. The structure of machine tools holds all machine components and at the same time withstands the forces coming from the process, thus it should maintain enough stiffness to keep the required precision. One important structural component is the bed, where all others components rest. Since lighter machine usually means less stiff machines, the X. Ding ( Ma et al. 2006; Ryu et al. 2002; Jarmai and Farkas 2001). From the view of traditional theory of material strength, the vertical stiffener plates distributed on the bending plane can improve the bending strength and stiffness, the transverse stiffener plates distributed perpendicularly to the torsion vector can improve the torsion strength and stiffness, and the inclined stiffener plates can improve both torsion and bending strength and stiffness simultaneously. The current layout design optimization approaches of the stiffener plates inside box structures are usually based on the aforementioned basic theories combined with the designers experiences (Liu 2008; Ye 1986; Yang et al. 2006), or are part layout optimization on the condition that several layout forms of stiffener plates have been determined in advance (Zhang et al. 2008; Xu et al. 2001). On the other hand, by the develop- ment of the structural design optimization methods, the stiffener layout design optimization of plate and shell structures have been studied intensively (Ding and Yamazaki 2005; Lam and Santhikumar 2003; Luo and Gea 1998). However, the study of the layout design optimization of the stiffener plates inside a box structure is not enough. An alternative approach of the problem is to solve a 3-dimensional topology design optimization problem, that is, a box structure is fi lled with material inside fi rst, by utilizing the structural topology design optimization methods, such as density method, the optimum layout of inside stiffener plates can be obtained (Chang et al. 2008). But because most of the bed structures of machine tool have big volume, the design effi ciency of 3- dimensional topology optimization is very low. Moreover, because of the complexity of the stiffener layout form, the optimum design results need to be post-processed in order to distinguish the real layout form of the stiffener plates. This paper suggests a novel and effective optimal design approach of machine tool bed structure, which involves three phases. The fi rst phase is the stiffener plates layout design optimization of bed structures by a reasonable simple model called fi ber model. The suggested method changes a 3-dimensional topology design optimization problem to a 2-dimensional problem, so the design effi ciency is improved greatly. The second is the detail sizing optimization of stiffener plates and supporting blocks under the bed based on the initial optimal model resulted from phase one. Finally, a topology design optimization process is implemented to obtain a reasonable distribution of manufacturing holes in bed structure. By considering the manufacturing requirement, a new bed model is obtained. A typical cylindrical grinding machine tool bed is selected as a design example, and the result confi rms the validity and effectiveness of the suggested method. 2 Design method A traditional cylindrical grinding machine tool is shown in Fig. 1a, in which the bed of machine tool provides a basis for, and a connection between, the spindles and slides. The basic process of cylindrical grinding consists in a grinding wheel rotating in parallel to the workpiece, which is supported between the spindle box and the tailstock. The longitudinal feed is given by moving the working table longitu- dinally where the workpiece rests. Thus, the loads applied on the bed structure include the self-weight of the bed, and a moving load of working table caused by the cutting force and the inertial loads of all components on it. These loads cause a combined deformation with torsion, bending and compression of the bed structure. The distortion and vibration of the bed under high loads due to either the cutting forces or the inertial loads must be kept to a minimum for a required processing precision. The bed structure is shown in Fig. 1b, which consists of two parts, one is the front bed and the other is the rear bed. Both of the front and rear beds are generally composed of outer panels with inside stiffener plates. The oil box is in the central part of both the front and rear bed. 352X. Ding et al. 123 The factors affect the stiffness and weight of the structure include layout of the inner stiffener plates, position of the supporting block under the structure, geometric sizes of the structure, and positions and sizes of manufacturing holes. By considering these factors, an optimal design approach involving three phases is suggested. The fi rst phase is aim at layout design optimization of the inner stiffener plates. After determining the layout of inner stiffener plates, the detailed sizing optimization of inner stiffener plates and supporting blocks under the bed is implemented to make it possible to improve the mechanical performance of the bed, which is referred to as the second phase. In order to determine a reasonable positions of manufacturing holes and further decrease the weight of the bed, a topology optimization of the whole structure is done, which is the third phase. 2.1 Layout design optimization of stiffener plate inside bed Because the bed structure is a box structure with inner stiffener plates supported on the underside panel and loaded on the upper panel, let us consider a layout design optimization problem of a box struc- ture. A fi ber model is suggested in this paper, which means a stiffener plate inside a box structure is formed from numerous parallel fi bers. Figure 2 is the representation of a fi ber model of a box structure with two diagonal stiffener plates, in which Fig. 2a is the real structure and Fig. 2b is its fi ber model. In order to determine an optimal layout of the stiffener plates inside a box structure, an initial model of the structure should be fi lled with material. Thus, the initial fi ber model is a hollow box fi lled with infi nitude parallel fi bers inside it. The FEA fi ber model of an initial box structure is composed of shell elements and beam elements, as shown in Fig. 3. The outer panels are discretized into shell elements, in which the loaded panel and the supported panel should have the same grids. Beam elements inside are formed by two corresponding nodes on the loaded panel and the supported panel, which represents the parallel fi bers in the fi ber model. Figure 4 shows the detail design fl ow. Firstly, the design object is simplifi ed to a hollow box, which is discretizedintosomeshellelements.Itisnotedthatthe loaded panel and the supported panel should have the 1-bed; 2-spindle box 3-working table 4-grinding wheel 5-grinding wheel box 6-tail stock 7-operation handle (a)(b) Oil box Rear bed Front bed Fig. 1 Structure of a cylindrical grinding machine tool and its bed structure. a Structure of a cylindrical grinding machine tool, b typical structure of a machine tool bed Fig. 2 Fiber model representation. a Box structure with diagonal stiffener plate, b Corresponding fi ber model Fig. 3 FEA fi ber model of an initial box structure Optimal design approach for eco-effi cient machine tool bed353 123 same grids. Secondly, beam elements are formed by connecting the corresponding nodes between the loaded panel and the supported panel, and the fi ber model is constructed. Thirdly, the boundary and load conditions are applied, and the static fi nite element analysis is carried out. Fourthly, the loaded panel is set to be the design area, and the density method is adopted. As a result, the optimum material distribution ofloadedpanelisobtained,whichdecidestheexistence of inside beams, that is, a beam is remained if its end connectedtotheloadedpanelisinsidethehighmaterial density area, and it is removed if its end connected to the loadedpanelisinside thelowmaterialdensityarea. The stiffener plates are formed by connecting all remained beams. Consequently, the 3-dimensional topology design optimization problem is changed to a 2-dimensional design optimization problem. A typical cylindrical grinding machine tool bed is selected as an example, as shown in Fig. 1. Because the deformation of the guiding rail on the front bed is the most important factor to affect the precision of the grinding machine tool, the front bed structure is the only study object in this paper. The initial FEA model of the front bed structure is constructed, in which the central part where the oil box rests is not set to be a design space, as shown in Fig. 5a. Figure 5b is its corresponding fi ber model applied with a moving torsion load. It is noted that only a torsion load is considered here to design the layout of the inner stiffener plates, because most of the compression and bending loads are resisted by the outer panels of the box structure. The torque is produced by a couple of opposite moments spaced the length of the workpiece apart, and is moved longitudinally. The loaded panel excluding the area of the oil box is set to be the design area, and topology design optimization is carried out to determine the optimal material distribution. The density method is adopted. The shell element densities are set to be the design variables, which are changed from 0 to 1 continu- ously. The optimal design model of the density method is formulated by Eq. 1 discretization Loads and constrains Topology optimization of loaded panel Stiffener plate layout Fiber model Fig. 4 Design fl owchart Fig. 5 FEA model and its corresponding fi ber model of the front bed structure. a FEA model, b fi ber model and moving loading 354X. Ding et al. 123 min w compq1;q2;.;qn subject to : Vmax?V? 0?qi?1i 1;2;.n 1 where w_comp is the weighted compliance of the structurebyconsideringthemovingloadcondition.The weightedcoeffi cientofeachloadingpositionisassumed to be a same value. Vmaxis the maximum volume of the structure, and V is the allowable volume specifi ed in advance. qiis the density of element i, and n is the number of the elements in the design space. The optimal material distribution of the upper panel of the bed structure is shown in Fig. 6a and b. By connecting the fi bers of which ends in the loaded panel are located on the high density area, a layout of the stiffener plates can be obtained, as shown in Fig. 6c. It is found that three transverse stiffener plates are rest in the left side and one transverse stiffener plate rests in the right side. 2.2 Sizing optimization of stiffener plates inside bed and supporting blocks under the structure The result of the fi rst phase is an initial optimal model with a rough layout of the inner stiffener plates, detailed sizing optimization of a real structure with most of geometric features should be carried out to determine the spacing distance of the stiffener plates and the supporting blocks, as well as the thickness of the stiffener plates. The design variables are illus- trated in Fig. 7, and their size ranges and initial values are listed in Table 1. Design variables x1x4 are the spacing distance of the inner stiffener plates and x5is the thickness of the stiffener plates. Design variables x6x8are the spacing distance of the supporting blocks. Total 11 supporting blocks are selected according to the practical requirement. There are two key performance indexes of a bed structure, in which one is the natural frequency, another isthestaticstiffness.Whenthebedstructure is loaded by the combining loads including the cutting force, weights of all components on it and the self- weight, the transverse deformation of the guiding rail is the most important factor to affect the processing precision. The transverse stiffness Kpof the bed structure is evaluated by the ratio of the transverse cutting force Fpand the mean transverse deformation of the guiding rail, as formulated by Eq. 2 Kp Fp P n i1 uip=n 2 where up i is the transverse displacement of node i, n is the number of nodes on the guiding rail. (a) (b) (c) Fig. 6 Optimal material distribution of the loaded panel and layout of inner stiffener plates. a Optimal material distribution of the loaded panel (3-dimension), b optimal material distribution of the upper panel (top view), c layout of inner stiffener plates (a)(b)(c) x3 x2 x4 x8 x6x7 x5 x1 Fig. 7 Design variables of sizing optimization. a Spacing distance of stiffener plates, b spacing distance of supporting blocks, c thickness of stiffener plate Optimal design approach for eco-effi cient machine tool bed355 123 It is said that the low order frequency and the static stiffness of a structures correlate. Thus, the natural frequency is set to be the design objective, and the static transverse stiffness will be calculated after the optimization to verify the result. The optimal model of the sizing optimization is formulated by Eq. 3 Maximizef s:t:xli?xi?xu i i 1;2;.;8 3 where f is the natural frequency of the structure, and xi l and xi u are the low and upper limited values of the design variables, respectively. In order to improve the design effi ciency, an approximate optimization approach is adopted. The optimization process is shown in Fig. 8, which includes three modules. First, a CAD model of the bed structure is constructed, and the design vari- ables (x1x8) and the response variables (f) are parametrized. Secondly, a meta model is con- structed by DOE (Design of Experiment) method, in which total 81 groups of the design points are generated by the central composite design method in the specifi ed design space. On the basis of the obtained data of the design variables and the responses, 2-order response surface is obtained, as formulated by Eq. 4. f b0 X 8 i1 bixi X 8 i1 biix2 i X 8 ij bijxixj i; j 1;2;3.84 where coeffi cients bi, bii, and bij(i, j = 1,2,3,8) are determined by the least-squares analysis. The adjusted coeffi cient of determination of the fi nally obtained response surface of nature frequency f is 0.9953, which means that the obtained response surface has enough precision. The optimization module is based on the obtained response surface, and the multi-objective genetic algorithm is adopted as the optimization method. The design results are listed in Table 2. It is found that the natural frequency is improved by 1.08%, and the transverse stiffness is improved by 2.50%. Table 1 Design variables Design variablesx1x2x3x4x5x6x7x8 Size range (mm)200400400600700110020040012157001200700900200900 Initial value (mm)25057595025015873870500 Analysis module Optimization module DOE/RSM module Post Process Multi-objective Genetic algorithm DOE Response surface model Modeling Parametrizing FEA Fig. 8 Sizing design optimization process 356X. Ding et al. 123 2.3 Topology design optimization of manufacturing holes The bed structure is made by casting, so some manufacturing holes need to be excavated to put out sands in the manufacturing process. A topology optimization is implemented on the basis of the optimal result of the above two phases. The density method is adopted. The design objective is the maximum static stiffness of the structure. Figure 9 show the design result when the volume of the structure decreases 10%, in which the red parts can be removed, and the gray parts should be remained. Figure 9a is the material distribution of the bottom of the structure and Fig. 9