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As shown in Figure 2(b), the minimum criti-cal values of the axial cutting depth of stable zoneunder every order modal (blue straight line) changewith the values of dynamic stiffness kdr, where kdr=1/|Im(v)r|=2zrkr. The minimum critical values of theaxial cutting depth of stable zone under second-ordermodal and multi-modal are consistent basically. Thedynamic stiffness under the second-order modal is thelowest (see Table 1). Therefore, the milling stability ofmachine tools at some position is decided by someorder modal parameter which has the lowest dynamicstiffness.Establishment of response surface modelThe method of response surface can rapidly obtain thefunctional relationship between independent variableand dependent variable by few test sample data.Generally, the response surface model is the quadraticpolynomial modelf (X)=k0 +X ni=1kixi +X ni=1kiix2i+X ni=2X i 1j=1kijxixj ð8Þwhere X=(x1, x2,., xn) and xi (i=1, 2,., n) are thedesign variables. k0, ki,kii, and kij are weight coeffi-cients. L=(n + 1)(n + 2)/2 is the number of weightcoefficient.X is the design variable matrix composed by designsamples, K is the weight coefficient matrix, and F is thematrix of the minimum critical values of the axial cut-ting depth obtained by equation (7). e is the errormatrix between the theoretical calculated value F andapproximate value F# calculated by response surfacemethodF=KX+e=F0+ee=F KX ð9ÞThe desired weight coefficient matrix can beobtained by least square methodmin S(K)= e kk2=(F KX)T(F KX) ð10ÞdS(K)dki=0 ð11ÞrS(K)=r( e kk)2=r (KX F)T(KX F) =2(KX F)TT=0ð12ÞK=(XTX) 1XTF ð13ÞApplied caseFinite element model of machine toolsFigure 3 is the three-dimensional computer-aideddesign (CAD)/computer-aided engineering (CAE)model of the machine tool with box-in-box structure.The column and bed are the fixed parts, and they areconnected by bolts. The moving part 1 is composed ofFigure 2. Stability lobes’ diagrams: (a) zone pisions of stabilitylobes’ diagram and (b) stability lobes’ diagram under Three ordermodal parameters.expressed according to the mechanical vibration as the spindle and the spindle box. It can slide on the slid-ing carriage in the Y-direction.
The moving part 2 iscomposed of the moving part 1 and the sliding carriage.It can slide on the column in the X-direction. The mov-ing part 3 is composed of the worktable and the slidingtable. It can slide on the bed in the Z-direction. Thetransmissions have two types, screw–nut and guide–sli-der. The key point for accurate modeling of a machinetool structure is the simulation of the interfaces in theabove two types of transmissions. The finite elementmodels of the transmissions are shown in Figure 4.Figure 4(a) is the screw–nut transmission. Line 1 isthe beam element which represents the screw and Nod1 represents the nut. The rolling interface between thescrew and the nut is composed of Nod 1 and Nod 2(Nod 2 is on Line 1). Nod 1 and Nod 2 are connectedby bushing element which can simulate the spring-damping unit in SAMCEF software. Nod 1 and thenut fitting face are connected by mean element whichcan simulate mean force transmittance. The bearingelement is composed of Nod 3 and Nod 4; Nod 3 isalso on Line 1. Nod 4 and the bearing block fitting face1 are connected by mean element. The same connectionway is applicable for Nod 5, Nod 6, and the bearingblock fitting face 2.Figure 4(b) is the guide–slider transmission. Line 1 isthe beam element which represents the guide; the guideis fixed on the guide fitting face. Nod 1 represents theslider. The slider and the slider fitting face are con-nected by mean element. The sliding interface betweenthe guide and slider is composed by Nod 1 and Nod 2(Nod 2 is on Line 1). Nod 1 and Nod 2 are also con-nected by bushing element.The material properties of the components areshown in Table 2 and the stiffness values of interfacecan be obtained by consulting INA (German INA bear-ing company) series product manuals according to thework condition (see Table 3). The stiffness of guide–slider interface in feed direction is far smaller than theother two supporting directions and can be identified as0. The radial stiffness of screw–nut interface is notoffered in INA series product manuals; actually, theaxial stiffness of screw–nut interface is more concerned,because the radial stiffness is large enough to support follows the load; in the finite element method (FEM), thedegree of freedom in radial direction is constrained, sothe radial stiffness of screw–nut interface is marked as‘‘2.’’ The values of axial stiffness in Table 3 are the val-ues under the preload according to the data offered inproduct manuals. The axial stiffness and radial stiffnessof the bearing fixed in the bearing block are, respec-tively, 1.81 3 106and 3.62 3 106N/m.Division of spatial specific positionsThe structure of machine tool on the XOY plane is bilat-eral symmetrical, so when the moving part 2 is on the leftplane of the column, the dynamic characteristics ofmachine tools are the same as that on the right plane ofthe column. This study chooses the right plane as theinvestigated object. Separately set three specific positions(the original point, the middle point, and the end point)on the strokes of the moving part 1, the moving part 2,and the moving part 3. There are totally 27 specific com-bination positions in the right workspace (see Figure 5).The position variables’ groups of moving parts aredefined as X={0,0.5,1}, Y={0,0.5,1}, andZ={0,0.5,1}. X represents the ratio of the distancefrom the moving part 2’s position to the original pointand the stroke length. Y represents the ratio of the dis-tance from the moving part 1’s position to the originalpoint and the stroke length. Z represents the ratio ofthe distance from the moving part 3’s position to theoriginal point and the stroke length. Therefore, thespatial position variables can be expressed as{X,Y,Z}={(0,0,0), (0,0.5,0), (0,1,0), (0.5,0,0),(0.5,0.5,0), (0.5,1,0), (1,0,0), (1,0.5,0), (1,1,0), (0,0,0.5),(0,0.5,0.5), (0,1,0.5), (0.5,0,0.5), (0.5,0.5,0.5), (0.5,1,0.5),(1,0,0.5), (1,0.5,0.5), (1,1,0.5), (0,0,1), (0,0.5,1), (0,1,1),(0.5,0,1), (0.5,0.5,1), (0.5,1,1), (1,0,1), (1,0.5,1), (1,1,1)}.Dynamic characteristics of machine tools at differentspecific positionsOne of the key points for accurate simulation ofmachine tool structural dynamic characteristics isdamping values of interface. This article ascertains auniform damping ratio z at some specific position usingthe method in the literature;4z=0.024 is obtained bymultiple simulation. Figure 6(a) shows the test andmeasurement instruments of dynamic characteristics ofmachine tools. Figure 6(b)–(e) shows the analytic andmeasured results of dynamic characteristics of themachine tool at position b4 (0,1,0.5). The results ofsimulation and experiment are shown in Table 4.Compared with the results of dynamic characteristicsexperiment, the error of each order natural frequencyand modal stiffness of simulation is controlled in therange of 5%(see Table 4), and that of each order modaldamping ratio of simulation is controlled about therange of 10% (see Table 4). The machine tool finite ele-ment model can be used in subsequent research.Moreover, dynamic characteristics of machine tools atnine different specific positions have been obtained; theresults show that the positions influence on the dynamiccharacteristics of machine tool at both X- and Y-direc-tions (Figure 7).Establishment of response surface modelThe spatial position of machine tools can be shown bythree independent variables x, y, and z; each indepen-dent variable has three specific values 0, 0.5, and 1.Therefore, three factors and three levels of orthogonalexperiment need to be designed. In order to ensure theaccuracy of experiment and reduce the cost of com-puter operation, the Box–Behnken11response surfacedesign method is chosen in this study. Figure 8 is theTable