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The dynamic characteristic of machine tools’ structures isan important evaluation criterion. The present machinetools’ industry requires rapid analysis of structuraldynamics characteristics in virtual environment. Cuttingstability can effectively reflect the dynamic characteristicsof machine tools’ structure. Therefore, it is necessary toset the cutting stability as an evaluation criterion of thedynamic characteristics of machine tools’ structure tomodify and optimize the machine tools’ structure. On theother hand, the method of response surface can rapidlyobtain the functional relationship between independentvariable and dependent variable by few test sample dataand has been applied in many fields as a rapid modelingmethod. In order to evaluate the milling stability ofmachine tools comprehensively and rapidly, this articlepresents a method of rapid evaluation of machine toolswith position-dependent milling stability based onresponse surface model. First, several specific positionscombined are selected with spindle box, slide carriage,and worktable to calculate the minimum critical valuesof the cutting depth. Second, the response surface modelis established to obtain the minimum critical values ofthe cutting depth at arbitrary position combination.Finally, the chart of the minimum critical axial cuttingdepth values of machine tools in the whole workspacehas been drawn in the form of four-dimension, so thatthe dynamic characteristics of machine tools can be rap-idly analyzed according to the chart, and then machinetools structure can be effectively optimized. The flowchart of rapid evaluation of machine tools with position-dependent milling stability based on response surfacemodel is shown in Figure 1,
whereMrepresents the num-ber of specific positions. This method not only avoidscomplicated theoretical calculation but also is easy to becomprehended and operated.Modeling of milling stabilityThe present milling stability of machine tools can bepredicted mainly based on the analytical model devel-oped by Altintas and colleagues8,9In their approach,the time-varying force coefficient of the dynamic millingprocess model was approximated by Fourier-seriescomponents. Following this, the stability relationshipbetween the chatter-free axial cutting depths aplim andthe spindle speed n was obtained.The speed-dependent transfer function H(jv) repre-senting the ratio of the Fourier transform of the displa-cement X(jv) at the tool tip over the dynamic cuttingforce F(jv) can be derived by Gagnol et al.10as followsH(jv)= X(jv)F(jv)=Re(v)+jIm(v) ð1Þap lim = 1NKtKrRe(v)ð2Þf=p 2 tan 1 Im(v)Re(v)ð3Þn= 60vN(2kp+f), k=lobes(0, 1, 2, ... ) ð4ÞIn the above equations, H(jv)=Re(v)+ jIm(v); Reand Im are, respectively, the real and imaginary parts ofthe transfer function of the spindle tool tip. Kt is thecutting force coefficients in the tangential direction tothe cutter, and Kr is the ratio of the normal and tangen-tial cutting force coefficient. N is the number of cutterteeth and k is the lobe number.In a similar way, the chatter-free axial cutting depthsaplim in multiple mode system can be expressedaplim = 1NKtKrRe(v)rð5Þr is the number of the modal order; the real part of the Re(v)r = 1 vr2kr (1 vr2)2+(2zrvr)2hi ð6Þvr=v/vnr is the frequency ratio at the rth ordermodal, v is the incentive angular frequency of themilling system, and vnr is the natural angular frequencyof machine tools at the rth order modal. v2nr=kr=mr;zr=cr/(2mrvnr); mr,kr,cr, and zr are, respectively,modal mass, modal stiffness, modal damping, andmodal damping ratio, where kr=1/2zr|Im(v)r|. (2zrvr)2can be ignored because the order of magnitude of zr isusually 1022;when0\vr\1or vr .1, the order ofmagnitude of (2zrvr)2is 1024–1026, which is far smallerthan that of (12(vr)2)2. In order to simplify the calcu-lation, equation (5) can be transformed intoap lim = krNKtKr1 vvnr 2"#ð7ÞTo predict the machining stability, a two-tooth car-bide cutter was employed to machine the stock materialof Al7075. The radius of cutter is 12mm, the radialcutting depth is 6mm, and the milling method is down-milling. The cutting resistance coefficients were cali-brated as Kt=796N/mm2and Kr=0.21 by millingexperiment. Supposing, machine tool has three ordermodal parameters (see Table 1) at some position. Thestability diagrams of machine tools are shown inFigure 2(a). The zone under the blue straight line is theabsolute stable zone. The zone between the bluestraight line and the curve is the conditional stablezone. The zone on the curve (the critical values of theaxial cutting depth) is the chatter zone. The operatorsof machine tools often choose the low spindle speedwhen milling the steel or hard materials. In the mean-time, the values of the axial cutting depth in the abso-lute stable zone are chosen because the conditionalstable zone is narrow under low-speed milling. The bluetransfer function at the rth order modal (Re(v)r) can be straight line corresponding to the values represent theminimum critical values of the axial cutting depthaplim(min).