Guangxi University, Nanning , China *Corresponding author

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1 2017 2nd International Conference on Applied Mechanics and Mechatronics Engineering (AMME 2017) ISBN: Topological Optimization of Gantry Milling Machine Based on Finite Element Method Bo-tao DU 1,2, Yi-jun CHEN 1, Peng YU 1, Wei HUANG 3 and Li-xin HUANG 1,4,* 1 College of Civil Engineering and Architecture, Guangxi University, Nanning , China 2 Automotive Engineering Institute, Guangzhou Automobile Group Co., Ltd, Guangzhou , China 3 College of Mechanical Engineering, Guangxi University , China 4 The Key Laboratory of Disaster Prevention and Structural Safety of the Education Ministry, Guangxi University, Nanning , China *Corresponding author Keywords: Topological optimization, Gantry milling machine, Finite element method, ANSYS. Abstract. Based on the finite element method (FEM), gantry milling machine was topologically optimized. Complex gantry milling machine design was firstly simplified in order to make it suitable for the FEM software ANSYS modeling. Then, the primary principal stress and the modal were analyzed to set the foundation for the topological optimization of gantry milling machine. The topological optimization problem was formulated as the problem of minimizing the objective function that was defined as the function based on the flexibility and the natural frequency, respectively. The optimized gantry milling machine configuration was obtained with the stiffness increased by 37.83%, the volume decreased by 15.28%, and the first order natural frequency increased by 70.88%. The final configuration scheme has better mechanical properties and improvements in the material economy of gantry milling machine. Introduction With the development of advanced manufacturing technology, higher requirements have been put forward for the static and dynamic performance of milling machine, and in recent years there are a lot of researches based on this aspect [1-3]. Columns and beams are two important parts of the gantry milling machine, and their design has a crucial effect on the static and dynamic performance of the structure. In general, the three indicators of machine performance are: stiffness, weight, dynamic performance, the data corresponding to the three indicators is the cutter head displacement with load, the volume of the unit, the value of first order natural frequency. In this study, a certain type of gantry milling machine is investigated. Based on the finite element method (FEM), the column and beam is topologically optimized with flexibility and natural frequency as the optimization target. It is hoped that the results of these studies will be helpful for the prediction of static and dynamic performance of milling machines and the improvement of product design. Topological Optimization Model of Gantry Milling Machine The topological optimization is defined as finding the optimal material distribution scheme in a given design space, so that the structure is optimal under the condition that the state variables are satisfied. The topological optimization problem is characterized by the need to determine the material distribution problem on structure, such as where to open the hole, what form of hole to shape, how many holes to open, how to connect the different sections, and so on [4]. The topological optimization of continuum is defined in two or three dimensional region. The optimal material distribution scheme is obtained by selecting different design parameters and objective functions under the constraint equations. The topological optimization can be formulated as mathematical optimization problem. 74

n Minimize F X, X R (1) Subject to 0 1, 2,, gi X i m (2) 0 1, 2,, ; hj X j pp n (3) T where F X is the objective function, X x1, x2,, x n is the design variables, n is the number of the design

In our study, the objective function for the gantry milling machine optimization design can be defined as the flexibility or the first order natural frequency.

2 n Minimize F X, X R (1) Subject to 0 1, 2,, gi X i m (2) 0 1, 2,, ; hj X j pp n (3) T where F X is the objective function, X x1, x2,, x n is the design variables, n is the number of the design variables, gi X and hj X are the constraint functions. In our study, the objective function for the gantry milling machine optimization design can be defined as the flexibility or the first order natural frequency. The constraint function is defined as the volume function with the minimum requirement to decrease volume by 50%. Finite Element Model for Gantry Milling Machine In this study, unless otherwise specified, the following values and conditions are used in the process of the gantry milling machine topological optimization. According to the simplified drawing of the gantry milling machine, a finite element model can be established by ANSYS standard finite element program, as shown in Figure 1. The reinforcement rib of the column inner wall is simulated by beam element BEAM189; L/H>10 plate is simulated by shell element SHELL281; connecting part of the saddle and the beam is simulated by rigid beam element MPC184; saddle and ram are simulated by particle element MASS21; cutter head is simulated by MASS21, and load will be at this part; the rest of the solid element is simulated by SOLID186. Figure 1. Finite element model of gantry milling machine. Figure 2. Generate the constraint equations. As shown in Figure 2, between the column and the beam, between the column and the bed, the constraint equations in ANSYS are used to link those marked parts so that the freedom of nodes is synchronized. The bottom of the column and the bed are fastened on the foundation by bolts, so the simulation is made by full restraint. The material of this gantry milling machine is cast iron HT250 with elastic modulus E= MPa, Poisson's ratio μ=0.156, and density ρ= kg/ (mm) 3. According to the cutting test specification of Guilin Machine Tool Co., Ltd. and the machine tool parameters used in this paper, the cutter head will produce the maximum cutting force of N. Refer to Machine Tool Design Manual, the ratios of the main cutting force to other cutting forces can be obtained as follows FX F 0.3, FY F 0.55, FZ F 1.0 where FX is the horizontal component of force along the feed direction, FY is the force in the spindle direction of the milling machine, FZ is a component of the vertical. F takes the value of 13000N and according to the above ratios, it can be given by FX=3900N, FY=7150N and FZ= 13000N [5]. 75

Mechanical Characteristics of Gantry Milling Machine The gantry milling machine is made of cast iron, and the stress is only concerned with its primary principal stress.

3 Mechanical Characteristics of Gantry Milling Machine The gantry milling machine is made of cast iron, and the stress is only concerned with its primary principal stress. The results of FEM are shown in Figure 3 and the maximum principal stress is 4.666MPa, which is far less than the yield strength of HT250. Therefore, this is a small linear elastic deformation, and the majority of the regional stress is very small. It means that the gantry milling machine has great potential for optimization. Figure 3. Contour diagram of the first principal stress. The results of machine modal analysis are also shown in the Table 1. Table 1. The first 4 modes of the structure. Order Natural Frequency [Hz] Vibration Mode Beam swings back and forth Beam swings left and right Beam twists around the Y axis Middle of the beam shake up and down Topological Optimization Design of Gantry Milling Machine Topological optimization design of gantry milling machine is divided into two design steps: minimization of flexibility and maximization of fundamental frequency. Thus, general material distribution of the structure can be obtained, and material having density greater than 0.5 is reserved. Initial Topological Optimization with Minimum Flexibility as the Objective Function This problem is a linear elastic static analysis with multistep loading, and the corresponding load and boundary conditions need to be applied. The prototype of the gantry milling machine as shown in Figure 4 is used as the initial design space. Lead rail, saddle, and ram are functional parts, which cannot be optimized, while the rest are optimized areas. The flexibility function is selected as an optimized objective function with the volume function defined as a constraint function and reducing the volume by 50%. Optimization results shown in Figure 5, it s obvious that the upper right corner of the beam can be considered to cut off. Because the elements in left side of the column cannot be omitted, the topological optimization design space is considered to expand and therefore the volume of column increases. 76

Figure 4. The initial design space of gantry milling machine. Figure 5. Pseudo-densities of elements.

The optimized objective function and the constraint function are respectively redefined by the flexibility function and the volume function with reducing the volume by 50%.

volume. It s necessary to pay attention to avoiding larger column in order to keep cutter head process properly. Figure 6. New design space. Figure 7. Pseudo-densities of elements.

structure as the objective function, and only the boundary conditions need to be applied without the corresponding load.

4 Figure 4. The initial design space of gantry milling machine. Figure 5. Pseudo-densities of elements. Further Topological Optimization with Minimum Flexibility as the Objective Function The design space is redefined as shown in Figure 6. The optimized objective function and the constraint function are respectively redefined by the flexibility function and the volume function with reducing the volume by 50%. The optimization results as shown in Figure 7, it s obvious that the upper right corner of the beam can be cut off and the right side of column can be weakened, the left side need to increase the volume. It s necessary to pay attention to avoiding larger column in order to keep cutter head process properly. Figure 6. New design space. Figure 7. Pseudo-densities of elements. Topological Optimization with Maximum First Order Natural Frequency as the Objective Function Topological optimization analysis based on modal frequencies is to take the natural frequency of structure as the objective function, and only the boundary conditions need to be applied without the corresponding load. The function based on the first order natural frequency is defined as an optimized objective function. The volume function is defined as a constraint function with reducing the volume by 50%. Optimization results shown in Figure 8, it s obvious that the beam without the upper right corner and the column with large bottom can increase the first order natural frequency. From the above, it can be seen that cutting upper corner of beam and increasing the volume of the column bottom help the gantry milling machine decrease its flexibility and increase its first order natural frequency. According to the above characteristics with economical consideration, the optimized gantry milling machine configuration shown in Figure 9 can be obtained with the stiffness increased by 37.83%, the volume decreased by 15.28%, and the first order natural frequency increased by 70.88%. 77

The configuration characteristics which are beneficial to improve gantry milling machine performance are obtained, and according to those characteristics, the configuration scheme of gantry milling

5 Figure 8. Pseudo-densities of elements. Figure 9. Final scheme. Summary In this paper, the gantry milling machine is topologically optimized with the flexibility and natural frequency as the objective function based on the finite element method. The configuration characteristics which are beneficial to improve gantry milling machine performance are obtained, and according to those characteristics, the configuration scheme of gantry milling machine with better mechanical properties and economy is summarized. Acknowledgement The research work was financially supported by the National Natural Science Foundation of China ( ). Those supports are gratefully acknowledged References [1] Zhou Houming, Wang Chengyong, Zhao Zhenyu, Dynamic characteristics of conjunction of lengthened shrink-fit holder and cutting tool in high-speed milling, Journal of Materials Processing Technology, 2008, 207: [2] R. Jalili Saffar, M.R. Razfar, O. Zarei, Simulation of three-dimension cutting force and tool deflection in the end milling operation based on finite element method, Simulation Modeling Practice and Theory, 2008, 16: [3] Yüksel, E., Budak, E., & Ertürk, A.S., The effect of linear guide representation for topology optimization of a five-axis milling machine, Procedia Cirp, 2017, 58: [4] Martin P. Bendsoe, Ole Sigmund, Topology Optimization. Springer, 2004, 1-4. [5] I. Mane, V. Gagnol, B.C. Bouzgarrou, Stability-based spindle speed control during flexible workpiece high-speed milling, International Journal of Machine Tools & Manufacture, 2008, 48:

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