立式加工中心主軸組件的結(jié)構(gòu)設(shè)計(jì),立式加工中心主軸組件的結(jié)構(gòu)設(shè)計(jì),立式,加工,中心,主軸,組件,結(jié)構(gòu)設(shè)計(jì) F.-C. Chen Department of Mechanical Engineering Ctiang Gung University Tao-Yuan 333, Taiwan, R.O.C. H.-S. Yan Department of Meclianical Engineering National Ctieng Kung University Tainan 701, Taiwan, R.O.C. A Methodology for the Configuration Synthesis of Machining Centers with Automatic Tool Changer The purpose of this paper is to present a design methodology for the configuration synthesis of machining centers with automatic tool changer to meet the required topology and motion characteristics. According to the concept of coordinate system, graph theory, generalization, specialization, and motion synthesis, this design methodology is proposed and computerized, and the machining centers with automatic tool changer up to eight links are synthesized. As the result, for the machining centers with drum type tool magazine, the numbers of configurations of machining centers with 6, 7, and 8 links are 2, 13, and 20, respectively. Similarly, for the machining centers with linear type tool magazine, the numbers of configurations of machining centers with 5, 6, 7, and 8 links are , 5, 20, and 60, respectively. Furthermore, this work provides a systematic approach for synthesizing spatial open-type mechanisms with topology and motion requirements. Introduction Machining center kinematics may be considered as an open-type mechanism, and they have special functions with specific topology characteristics. The problems associated with the creative design of planar mechanisms have been the subject of a number of studies (Johnson, 1965; Freudenstein and Maki, 1979, 1983; Erdman, et al., 1980; Yan and Hsu, 1983; Yan and Chen, 1985; Yan, 1992) over the pa,st years. However, design methodologies for the structural synthe sis of open-type mechanisms with spatial motions are not available. In the past years, just a few articles focused on the configuration design of machining centers. Sugimura et al. (1981) used analyt ical approach to investigate the machine tool design. Ito and Shinno (1982, 1983, and 1987) generated the structural configu ration of machine tools by using directed graphs. Reshetov and Portman (1988) proposed the configuration code for synthesizing the machine tool configurations with the same shaping function. The concept of configuration code was widely used on the con figuration synthesis of 5-axis machine tools (Ishizawa, et al, 1991; Sakamoto and Inasaki, 1992). However, automatic tool changers were not considered. The system that automatically performs tool changes between the spindle and the tool magazine of a machining center is called automatic tool changer (ATC). ATC plays an important role in reducing the machine idle time and therefore increases productiv ity in machining process. The propose of this paper is to present a design methodology for the systematic generation of all possible configurations of machin ing centers with automatic tool changer, that are open-type spatial mechanisms subject to topology and motion constraints. Terminology and Notation For the terminology and notation presented in this paper, we follow the graph theoretic terminology and notation in references (Harry, 1969; Deo, 1974). Graph A graph consists of a set of objects V = V, V2, V3, . . .) called vertices, and another set = e, e2, e, . ., whose elements are called edges, such that each edge e* unordered pair (V, V) of vertices. is identified with an Incidence When vertex y, is an end vertex of edge Cj, V, and Cj are said to be incident with each other. In the graph shown in Fig. 1(a), vertex V| is incident with edge e 1 and vertex V2 is incident with edges e, 62, 5, and e. Adjacency Two nonparallel edges are said to be adjacent if they are incident with a common vertex. Similarly, two vertices are said to be adjacent if they are the end vertices of the same edge. In Fig. l(fl), vertices V, and V2 are adjacent, and so are edges e, and 2- Degree of vertex The number of edges incident with vertex V, is called the degree, d(V,), of vertex V;. In Fig. 1(a), the degree of vertex V, is four. Pendant Vertex A vertex of degree one is called a pendent vertex. In Fig. vertex V, is a pendent vertex. I(), Contributed by the Mechanisms Committee for publication in the JOURNAL OF MRCHANICAL DESIGN. Manuscript received Oct. 1996; revised Jun. 1999. Associate Technical Editor: K. Kazerounian. Tree Graph A tree graph is a connected graph with no loop. Figure 1 (b) shows a tree graph with five vertices and four edges. Branch Vertex A cut-point of a graph is where the removal increases the number of components. A branch vertex V,- of a tree graph is a cut point whose removal results maximal subtrees containing Vi as an end- point. In Fig. h), vertex V, is a branch vertex. Path A walk is defined as a finite alternating sequence of vertices and edges. An open walk in which no vertex appears more than once is called a path. In Fig. 1(a), V,eiV2e5V4e4V5 is a path. Edge Sequence Edge sequence, jfiy, in a tree graph is defined as the edges connecting vertices ( and 7. In Fig. lb), ,5 denotes the edge sequence e,e2e4. Journal of Mechanical Design Copyright 1999 by ASI/IE SEPTEMBER 1999, Vol. 121 / 359 Downloaded 26 Nov 2008 to 222.190.117.209. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfm 64 02 (a) 01000 10111 01010 01101 .01010. V, Vi V; (b) 11233 12 122 213 11 32112 ,3 2 1 2 IJ 63 (c) (d) Fig. 1 Terminology and notation of graphs Length of path The number of edges in a path is called the length of path. In Fig. 1(a), the length of the path VeyeVeW is three. Rooted Tree A tree is connected graph with no loop, A tree in which one vertex (called the root) is distinguished from all the others is called a rooted tree. Figure 1 b) shows a rooted tree where vertex V2 is the root. Adjacent matrix Adjacent matrix of a graph with n vertices is an n X n symmetric matrix X = x, such that: 1. x,j = 1, if there is an edge between ith andjth vertices; and 2. X;j = 0, if there is no edge between them. Figure 1(c) shows the adjacent matrix of the graph shown in Fig. 1(a). Distance Matrix Distance matrix of a tree graph with n vertices is defined as an X n symmetric matrix D = rfy such that: 1. If i = j, dij represents the number of edges incident with vertex i; and 2. If ; t j, dij represents the length of path between vertices i and;. Figure 1(d) shows the distance matrix of the graph shown in Fig. Kb). Design Process Figure 2 shows the proposed design process for the configura tion design of open-type mechanism. The idea of the design process starts by investigating existing mechanisms and concludes their topology and motion characteristics. The second step of the design process is to describe the topological structures of existing mechanisms based on the concept of tree-graph representation, link adjacent matrix, and mechanism allocation matrix. The third step of the design process is to transform these mechanisms into their corresponding generalized tree graphs according to the pro cess of generalization. The fourth step of the design process is to obtain all possible tree graphs with the given numbers of edges and vertices based on graph enumeration. The fifth step of the design process is to assign certain type of links and joints into available tree graphs subject to required topology constraints through the process of specialization. The last step of the design process is to obtain the atlas of mechanisms subject to specified motion con straints. 360 / Vol. 121, SEPTEMBER 1999 In the following, a machining center with automatic tool changer is taken as an example to illustrate this design methodol ogy in detail. Existing Mechanisms The first step of the design process is to study existing mecha nisms and conclude their topology and motion characteristics, A machining center is a machine tool consisting of four basic components: a spindle, a tool magazine, a tool change mechanism, and a machine tool structure including motion of power axes. The machine tool structure largely determines the accuracy of ma chined surface, stiffness, and dynamic quality. The spindle rotates the tool to machine the workpiece to the desired surface. The tool magazine stores the tools and moves them to suitable positions for use in machining operations. The tool change mechanism executes tool changes between the tool magazine and the spindle. The simplest ATC is a design without a tool change mechanism, and the relative motions between the tool magazine and the spindle achieve tool change motions. Figures 3(a) and (b) show two 3-axis horizontal machining centers with drum type and linear type tool magazines, respectively. To represent and analyze the topological structures and motion characteristics of machining centers, a coordinate system is defined to describe the allocation of each motion axis of the machining centers based on International Organization for Standardization (ISO, 1974) nomenclature. This standard coordinate system is right-handed rectangular Cartesian one, related to a workpiece mounted in a machine and aligned with the principal linear sideways of that machine. The positive direction of movement of a component of a machine is that which causes an increasing positive dimension of the work- piece. The schematic drawings of horizontal machining centers appended to ISO standard are shown in Fig. 3. By analyzing available existing 3-axis horizontal machining centers without tool change mechanism, we conclude their topol ogy and motion characteristics (Yan and Chen, 1995) as follows. Existing Mechanisms Coordinate Systems Topology Qiaracteristics Motion Characteristics Topological Structure Representations Generalization Generalized Tree Graphs Graph Enumeration Alias of Tree Graphs Specialization Topology Constraints Specialized Tree Graphs Motion Synthesis Motion Constraints Atlas of Mechanisms Fig. 2 Design process Transactions of the ASME Downloaded 26 Nov 2008 to 222.190.117.209. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfm Tool maga (a) Spindle head Tool magaz ng table (b) Fig. 3 Three-axis liorizontal macliining centers Topologv Characteristics 1. 2. 3. 4. 6. They are spatial open-type Mechanisms with multiple degrees of freedom. They have one fixed link (frame). They have a spindle which is a singular link. The J have a working table where the number of joints be tween the spindle and the working table is four. They have a tool magazine which is a singular link branching from the link located from the frame to the spindle head. The singular links could be the spindle, the tool magazine, or the working table. And, the maximum number of singular links must be three. The joint incident with the spindle must be a revolute pair. The joints between the spindle head and the working table must be prismatic pairs. The joints between the tool magazine and the branching link are revolute, prismatic, or cylindrical pairs. And, if there is a revolute pair or a cylindrical pair, it must be incident with the tool magazine. Fig. 4 Tooi change motions Motion Characteristics 1. The spindle head has three relative motions in Y, Z, and X directions continuously with respect to the working table. 2. The ATC uses the relative motions between the tool magazine and the spindle to exchange tools. Figure 4 shows the tool change motions of the two machining centers shown in Fig. 3, where S and M represent the spindle and the tool magazine respectively, P and R represent the prismatic and revolute pairs respectively, X, Y, and Z represent the motion directions of kinematic pairs, and the circle with a number in it represent the motion sequence. 3. In order to achieve tool change motions, the relative degrees of freedom between the tool magazine and the spindle head must be at least three. Topological Structure Representations The second step of the design process is to describe existing mechanisms based on the concept of tree-graph representation, link adjacent matrix, and mechanism allocation matrix. The topological structure of a mechanism is characterized by its numbers and types of links and joints, and the incidence between them. Here, we use tree-graph representation, link adjacent matrix, and mechanism allocation matrix for the representations of the topological structures of machining centers. Tree-graph representation Based on the defined coordinate system, the mechanism can be described by representing its links and joints with vertices and edges, respectively, in which two vertices (edges) are adjacent whenever the corresponding links (joints) of the mechanism are adjacent. Table 1 Graph representations of links Link Symbol Frame Spindle Working table Tool magazine Connecting rod OFr S T M L, L, Journal of Mechanical Design SEPTEMBER 1999, Vol. 121 / 361 Downloaded 26 Nov 2008 to 222.190.117.209. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfm Table 2 Joint Revolute joint Prismatic joint Cylindrical joint Graph representations of joints Degree of freedom 1 1 2 Symbol R, R, R,. P, P P,. C, Cy C, . According to this representation, the links of a machining center are represented by vertices with the link name as shown in Table 1, and the joints of a machining center are represented by edges with the name of the type of joints as shown in Table 2. The joint name has a subscript that denotes the allocation of motion axis of that joint. If the motion axis of a revolute joint is parallel to X axis, this joint is denoted as R; if the motion axis of a revolute joint is parallel to the Y axis, this joint is denoted as Ry, and so on. Figures 5(a) and b) show the corresponding tree-graph representations of the two machining centers shown in Fig. 3. Link Adjacent Matrix Link adjacent matrix (LAM) is defined to represent the topo logical structure of a mechanism with N links and J simple joints. It is a symmetrical matrix of N hy N order. If represents the elements of LAM, then; 1. fly = L, if i = j; 2. Oij = 0, if i = j and link i is not adjacent to link 7; 3. a = Jt, if i j and link ; is adjacent to link7. where subscripts i and j are the row and the column subscripts, respectively; L represents the name of links; Jt denotes the type of joint incident with links ; and j; and DOF denotes the relative degree of freedom between links / and j. For the two machining centers shown in Fig. 3, their corresponding LAMs are shown in Figs. 5(c) and (d). M Cz I -7 Px PzPy M R fl Pv (a) (b) Mechanism Allocation Matrix In order to represent the structure of machining centers clearly, we present the concept of mechanism allocation matrix (MAM), by modifying the definition of LAM, as follows: 1. b,j = L, if; = j 2. by = 0, if i i= j and link ( is not adjacent to link; 3. by = Jt, if i j and link ; is adjacent to link7. where Ax represents the joint axis allocation in the coordinate system. For the two machining centers shown in Fig. 3, their corresponding MAMs are shown in Figs. 5(e) and (/). Generalized Tree Graphs The third step of the design process is generalization. The purpose of generalization is to transform the existing mechanisms, which involve various types of links and joints, into their corre sponding generalized tree graphs. The process of generalization is based on a set of generalizing rules. These generalizing rules are derived according to defined generalizing principles. The general izing principles and rules are described in detail in references (Yan and Chen, 1985; Yan and Hwang, 1988). For the tree graph of the machining center shown in Fig. 5(a), the generalization is carried out by the following steps: 1. The fixed link is released and generalized into a ternary link. 2. The links, such as the spindle, the tool magazine, or the working table, are all generalized into singular links. 3. The other links are generalized into binary links. 4. The prismatic pairs are generalized into revolute pairs. where singular, binary, and ternary links are the links incident with 1, 2, and 3 joints, respectively. Figure 6(a) shows its corresponding generalized tree graph. Similarly, the generalized tree graph of the machining center shown in Fig. 5(b) is shown in Fig. 6b). Atlas of Tree Graphs The fourth step of the design process is to generate all possible tree graphs with the given numbers of edges and vertices. The explicit numbers of tree graphs through A vertices can be obtained from the counting series, i.e., the generating function (Deo, 1974): Tix) = X + x + x + 2x + 3x + 6x+ Ix + 23 + 47: -I- lOex + 235A: + 551X + . . . (1) The atlas for the tree graphs counted in the first 10 terms of Eq. (1) may be found in reference (Harry, 1969). S R IL, 0 1 0 0 0 0 0 0 .0 0 0 0 0 0 P 0 0 0 Fr P 0 P 1 L, P 0 0 TO 1 0 OL3 0 0 0 2 0 0 0 0 0 c M. (c) S R L, I 0 0 0 0 0 0 0 0 0 P 0 T P 1 M (d) Specialized Tree Graphs The fifth step of the design process is specialization. The process of specialization is to assign specific types of links and joints into available atlas of tree graphs to generate all possible specialized tree graphs subject to topology constraints. Design requirements and constraints of links and joints of machining centers in their corresponding tree graphs are listed. Then, the assignment rules of links and joints are concluded. Finally, all possible topology struc tures are obtained after specialization. S R zL| 0 y 0 0 0 0 0 0 0 0 0 p Fr z 0 x 0 0 0 0 0 0 0 POP L, P 0 X T 0 0 OL, 0 0 z 0 0 0 0 0 c M S R zL, 0 y 0 0 0 0 0 0 0 P L, z 0 0 0 0 P L, X 0 0 0 0 0 0 0 P 0 T P xM (e) (f) Fig. 5 Topological structure representations 362 / Vol. 121, SEPTEMBER 1999 * (a) (b) Fig. 6 Generalized tree graphs Transactions of thie ASME Downloaded 26 Nov 2008 to 222.190.117.209. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfm Adjacent matrix Distant matrix Findi, where 2*1/ . i-l,2,.orn. Findi, where di=4 , i=l,2,.orn, Set x = T k=0 Find i, where dsi=4-k see http:/www.asme.org/terms/Terms_Use.cfm Link adjacent matrix I . = (z,y,.x) I Fmdb, whered +d, = 4 otherwise continue. Step 2: If there is no degree of freedom between the spindle head and the branch vertex which can be assigned as the axis of tool change motion, assign the degree of freedom between the branch vertex and the tool magazine as the tool change motion, go to step 5; otherwise continue. Step 3: If there is a degree of freedom between the spindle head and the branch vertex which can he assigned as the axis of tool change motion, assign the degree of freedom between the branch vertex and the tool magazine or between the branch vertex and the spindle head as the tool change motion, and go to step 5. Step 4: If there is a degree of freedom between the branch vertex and the spindle head that can be assigned as the tool change motion, go to step 5; otherwise delete this spe cialized tree graph. Table 4 No. of configurations of macliining centers No. of links 5 6 7 Drum type magazine 0 2 13 Linear type magazine 1 5 20 8 20 60 364 / Vol. 121, SEPTEMBER 1999 Transactions of the ASME Downloaded 26 Nov 2008 to 222.190.117.209. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfm TR. S h S P S 1 R py mv p R p, Rz- (a) (b) (c) R. S T R, R Pv Rz (d) M R, R Py Rz s R, R P, R; (f) r* M Cz Px S T Px Pz Py Rz (g) pp *p *R f, ni fz fy z s (h) oap R n * n * FrR Py R (i) M n p RFTR Py Rz (k) M RZR Py f Cz f S Fr PxPzTry R (m) R( s T Px PzPy R (n) Px Pz Py f;Rz (o) Fig. 9 Tree-graph representations of machining centers Step 5: Continue to complete the tool change motion synthesis, go to step 1; otherwise stop. If there is a redundant degree of freedom between the branch vertex and the tool magazine that is not assigned as the tool change motions, delete this specialized tree graph. According to tool change motion sequences, we can assign acceptable speciaUzed tree graphs to their corresponding mecha nisms step by step. An