[机械模具数控自动化专业毕业设计外文文献及翻译]【期刊】手摇砂轮机自动磨削系统的建模与仿真-外文文献

ORIGINAL ARTICLEC. H. Liu A. Chen Y.-T. Wang C.-C. A. ChenModelling and simulation of an automatic grinding systemusing a hand grinderReceived: 14 November 2002/ Accepted: 11 March 2003/Published online: 10 March 2004C211 Springer-Verlag London Limited 2004Abstract A grinding process model for an automaticgrinding system with grinding force control is developedinthispaper.Thisgrindingsystemutilisesanelectrichandgrinder, driven by a CNC machine centre and a forcesensor for force measurement. This model includes com-pliance of the grinding system and is initially representedby a series of springs. The stiness of each component isestimatedinthisstudyanditisfoundthatthemodelmaybe simplified into a single spring-mass system. A corre-sponding PID controller is designed for the purpose ofgrinding force control, which calculates the appropriateCNC spindle displacement according to the force mea-sured by the force sensor. Computer simulation resultsshow that the system settling time is less than 0.25 s.Keywords Applications grinding Grinding systemsimulation Automatic grinding system1 IntroductionIt is known that force control may improve grindingresults of mould and dies 1, 2, 3, 4, 5, 6, and hence forcecontrol has become an important procedure for grindingand surface finishing processes. Recently, electric handgrinders have become popular surface finishing tools ofmoulds and dies. They have already been included inautomatic surface finishing systems 7, 8, in which handgrinders are driven by CNC machine centres. However,corresponding force control techniques for these systemshave not been developed. In this study, therefore, agrinding process model for an automatic grinding sys-tem using hand grinders is proposed, and a corre-sponding PID controller has been designed. The systemis similar to that of Chen and Due 7, and Hsu 8, buta force sensor is placed under the work-piece for forcemeasurement. The system is shown in Fig. 1. The modelproposed in this paper has been implemented in thegrinding system 9 and grinding force control experi-ments have been performed.Several grinding force models have been proposed. Agood summary of these models, up to the year 1992, isgiven by To nsho et al. 10. In these models basicallynormal grinding forces are related to cutting speed,various forms of working engagement and wheel diam-eters. Recently Ludwick et al. 11, and Jenkins andKurfess 12 suggested the modelQ KpFNC0FTHV 1where Q is material removal rate, FNis normal force,FTHis the threshold value of FN, V is the relative speed,and Kpis a proportion constant. This model is a com-bination of the model proposed by Hahn and Lindsay13, in which normal grinding force is proportional tomaterial removal rate, and the Preston equation (see14), which states that normal force is inversely pro-portional to he relative speed between wheel and work-piece. This model has been adopted by Jenkins andKurfess 15, 5, and Hekman and Liang 16 forgrinding force estimation. A similar model was sug-gested by Kurfess, et al. 17, Whitney et al. 18 andKurfess and Whitney 19. In a series of studies for weldbead grinding systems, they utilised the expressionQ K1P C0K22where the power P is the product of the grinding forceand the relative speed.C. H. Liu (&) Y.-T. WangDepartment of Mechanical Engineering, Tamkang University,Tamsui, Taipei Shien, Taiwan ROC 251E-mail: .twA. ChenDesktop Display Business Unit, Product DevelopmentDepartment, AU Optronics Corporation,23 Li-Hsin Rd. Science-Based Industrial Park,Hsinchu 300, Taiwan ROCC.-C. A. ChenDepartment of Mechanical Engineering,National Taiwan University of Science and Technology,#43, Sec.4, Keelung Rd., Taipei, Taiwan ROC 106Int J Adv Manuf Technol (2004) 23: 874881DOI 10.1007/s00170-003-1736-5All the models discussed above deal with grindingwheels. So far there is no process model for electric handgrinders using spherical tools. For such grinding pro-cesses, previous results 20 show that rotation speed haslittle contribution to grinding forces; hence velocity V inEq. 1 is not included in the present model. Also, in allthe previous models machine compliance was not con-sidered. In this study, compliance of the grinding systemshown in Fig. 1 is included in the model.2 Grinding process modelThe grinding system shown in Fig. 1 is modelled afterthe system shown in Fig. 2. In this figure, M and k1represent mass and stiness of the electric hand grinder(including link and holder in Fig. 1) respectively, k2isstiness of the material removal process, which is de-fined as the ratio of normal grinding force to grindingdepth (assumed to be a constant). The symbols m and k3denote mass and stiness of the work-piece, and k4isstiness of the force sensor. Displacements x1, x2, x3andx4are measured from static equilibrium positions; hencegravity forces may be neglected. In Fig. 2, P is the forceimposed by the CNC machine centre. Also, in thepresent model, the ratio of the normal grinding force tothe grinding depth is assumed to take a constant valuek2, in reality, however, the relation between the grindingforce and the grinding depth is expected to be nonlinear.The symbol Fdis used to represent the nonlinear term(i.e. real grinding force = k2x2+Fd).Springs k1, k2, and k3are in series and they areequivalent to a spring with the equivalent spring con-stant K. Estimations of k1, k2, k3and K are shown in theappendix and it is found that the equivalent springconstant K is dominated by k2. Generally k2is much lessthan the stiness of the force sensor k4, hence one mayexpect that k2C29K. For example, the ratio k4/K for thecurrent system is larger than 105. With this expectation(i.e. k4C29K) in mind, it may be assumed that k4fiandx4fi 0. Then the model shown in Fig. 2 may be sim-plified to the one shown in Fig. 3.Eqation of motion of the system isPtFdtC0FtMx13whereFtKx14is the spring force. Dierentiating both sides of Eq. 4,one may obtainFtKx15Substituting Eq. 5 into Eq. 3, one getsPtFdtC0FtMKFt 6Fig. 1 Grinding systemFig. 2 Modelling of the grinding systemFig. 3 Simplified model875Taking the Laplace transform for both sides of Eq. 6,also assuming zero initial conditions, one may obtainPsFdsC0FsMs2KFs 7From this equation, the block diagram of the processmodel may be drawn, as shown in Fig. 4.In this diagram, the input P(s) is the force applied byCNC machine centre, the disturbance Fd(s) is the non-linear grinding force defined above, the output F(s) is theforce measured bytheforce sensor.Thetransferfunctionof the block diagram is defined by GP(S)=F(S)/P(S).Using Eq. 7, also assuming Fd(s)=0 one may obtainGPSFSPS1MKs2183 Controller designIn this study, a PID controller is utilised and is rep-resented as GcSKpKiSKdS. The block diagramof the grinding system is shown in Fig. 5. In this fig-ure, F*(S) is the input force command, F(s) is the valuemeasured by the force sensor, and e(S)=F*(s)C0F(S)iserror. In order to investigate if the system may becontrolled under the steady state condition, the dis-turbance Fd(s) is set to zero. From Fig. 5 and Eq. 8one may show that the system transfer function G(s)inFig. 5 is given byGsFsFC3sKds2KpsKiMKs3Kds2 1KpC0C1sKi9In steady state, s fi 0, from Eq. 9 it is assumed thatlims!0FsFC3sKiKi 1 10This means F(s)=F*(s), hence the system may becontrolled in the steady state.Since F(t)=Kx1(t), dierentiating both sides of thisexpression, one finds_FtK_x1 KV t 11where V(t) is velocity of the spindle. Taking the Laplacetransform of this equation, one findsSF SKSx1SKV S 12Hence the term KdSF(S) in Fig. 5 may be replaced bythe expression KdKV. Also, SF*(S)=0 for the conditionof constant grinding force, the block diagram shown inFig. 5 may be replaced by the diagram shown in Fig. 6.The transfer function of this diagram may be shownto beFig. 5 Grinding system blockdiagramFig. 6 Modified grindingsystem block diagramFig. 4 Block diagram of the process876GsFsFC3sKpKsKiKMs3KdKs2 KpK KC0C1sKiK13The numerator polynomial in the current transferfunction is one degree less than the transfer functiondefined by Eq. 9, which means the current transferfunction has less zero point and also less influence upona pole. Also, in the block diagram of Fig. 5 the error e(s)is dierentiated once, implying that the induced noisewill be amplified, and this does not happen in the currentblock diagram. Therefore in the following discussion,the block diagram shown in Fig. 6 is used.In order to determine controller gains, the firstobvious point is that the denominator of Eq. 13 is apolynomial of the third order, and can be written in theformgSC17MS3KdKS2 KpK KC0C1S KiK S aS a jbS a C0jb 14Since it is relatively dicult to analyse a third ordersystem, the purpose now is to approximate this systemwith a second order one, by assuming the two complexroots are dominant roots. Requiring that percent over-shoot of the second order system does not exceed 3%,the following relation may be obtained 210:03 eC0pfC14 1C0f2p15which implies pf.1C0 f2p 3:057, or f=0.75. As thefirst attempt to determine system gains, settling time TSis arbitrarily set to 0.5 s. This means 21TS4fxn 0:5s 16thus fxn=8, and xn=10.67 rad/s.The angle h (see Fig. 7) take the value cosC01f or 41.4C176.Therefore the complex poles of the characteristic equa-tion (i.e. the equation g(S)=0, see Eq. 14) areS=C08j7.05. The idea is to keep the third pole awayfrom these two complex poles. The value S=C015 isarbitrarily chosen, hence Eq. 14 can be written in theformgSS331S2353:7S 1705:5 0 17Comparing coecients of the last equations to thedenominator of Eq. 13, the following relations are ob-tainedKdKM 31 18aKpK KM 353:7 18bandKiKM 1705:5 18cThe value of K is estimated in the appendix to beK=8.74103(N/m), and mass of the hand grinder (withholder and link) is M=5 kg, substituting these valuesinto Eqs. 18a, 18b, 18c, Kd=0.0178, Kd=C00.797, andKi=0.98 may be obtained. Since system gains cannot benegative, a second trial, with new values of settling timeTsand natural frequency xn, is necessary.For the second trial, the settling time is assumed to be0.25 s, i.e. TS 4= fxn0:25 swhich means xn 21:3,and fxn 16. Following the same steps as in the firsttrial, the two complex poles may be determined to beS=C016j14.1. Now assuming that the third pole is atthe point S=C045, then the characteristic equation isS377S21895S 20466 0 19and after comparing corresponding coecients, onefindsKdKM 77 20aKpK KM 1895 20bandKiKM 20466 20cwhich means Kd=0.0443, KP=0.0891, and Ki=11.762.The zero of the system is the root of the equation inthe numerator of Eq. 13, i.e. the equationKpKS KiK 0 21Substituting the above values into this equation, onemay find that the zero is the point S=C0132.0. Hence thetransfer function defined by Eq. 13 may be written in theformGSFSFC3SKpK=MSKi=KPMS377S21895S20466155:034 S132S45S16j14:1S16C0j14:122Note that in steady state, G(S) approaches 1 asS fi 0.Fig. 7 Root locus of the characteristic equation8774 Simulation results and discussionsThe system response may be obtained using the com-mercial software Matlab. In the simulation procedure,the spring constant K=8.74103(N/m), mass of thehand grinder (with holder and link) M=5 kg, systemgains are Kd=0.004, Kp=0.089 and Ki=11.762. Thesampling time is chosen to be 0.001 s.The loop transfer function of block diagram shown inFig. 6 isLSKpKS KiKMs3KdKS2KS23With the system gains just obtained, the root-locusdiagram corresponding to loop transfer function 23 maybe drawn and is shown in Fig. 8. The three poles ofEq. 22 are shown in this diagram. It can seen that boththe zero and the third pole are much farther apart fromthe imaginary axis than the dominant complex roots,making them have negligible influence on the system,hence the system may be approximated by a secondorder one.With the gain values just obtained, the Bode diagramfor the transfer function G(s) defined by Eq. 13 is drawn(the Bode diagram of the process model without con-trollers is given in Fig. 9) and is shown in Fig. 10.Comparing this diagram to the Bode diagram of theprocess model without controllers, (i.e. Fig. 9) one mayfind that the resonance has been greatly reduced. Also,while the phase margin shown in Fig. 9 is approximately180C176, which corresponds to an unstable state, the phaseangle shown in Fig. 10 is approximately 45C176.Due to the fact that the grinding force control systemtakes step input, step responses for the grinding processwith and without controller are compared. Figure 11shows step responses without controller, and Fig. 12shows step response with the controller just designed hasFig. 8 Root-locus diagram of open loop systemFig. 9 Bode diagram of the grinding processFig. 10 Bode diagram of the grinding process with controllerFig. 11 Step response of the grinding process878been imposed. In the system without controller, grindingforce oscillates with no steady state and in the systemwith force control; the settling time is less than 0.25 swith a percent overshoot of less than 3%. These systemgains (i.e. Kd=0.044, KP=0.089, and Ki=11.762), havebeen utilised in the automatic grinding system with forcecontrol; results show that surface roughness may be re-duced 9.5 ConclusionsIn this study, a grinding process model for the forcecontrol system utilises a CNC machine centre, an electrichand grinder and a force sensor has been proposed. Thisforce control system is represented by a spring-masssystem and according to estimated stiness of eachcomponent, the spring-mass system may be furthersimplified. A PID controller base on the simplified sys-tem has been designed. Controller gains are estimated byusing the commercial software Matlab. Estimation re-sults show that a settling time of less than 0.25 s and apercent overshoot of less than 3% may be obtained.Acknowledgements The authors gratefully acknowledge that thisstudy was supported by the National Science Council of ROCunder grant no. NSC 89-2218-E032030.Appendix: Estimations of stiffness constantsStiness of hand grinder (with link and holder): k1In Fig. 13, segment AB represents the link (see Fig. 1)which has a length a and makes an angle h with theZ-axis. The electric hand grinder is presented by segmentBC which is normal to segment AB and the length ofwhich is b. As the normal grinding force P is applied,the corresponding displacement of member ABC (i.e. thecombined parts of link, holder, and hand grinder) at thepoint C is DZ, which is to be estimated in two steps.First, segment AB is tightly fixed to the spindle and maybe represented by a cantilever beam, as shown in Fig. 14.The angle at point B due to the force Psinh and themoment Pbcosh ishbPbacoshEaIaC0Pa2sinh2EaIa24where Eaand Iaare modulus of elasticity and areamoment of inertia of the segment AB, respectively.Secondly, segment BC is also modelled by a cantileverbeam but it has an initial inclination angle hb, as shownin Fig. 15. The deflection at C of this beam due to theload is given byCC0Pb3cosh3EbIb25Fig. 13 Simplified model of hand grinder (with holder) forcalculating k1Fig. 14 Link AB is modelled by a cantilever beamFig. 12 Step response of the grinding process with controller879where Eband Ibdenote modulus of elasticity and areamoment of inertia of the segment BC respectively. Thetotal displacement at C in the direction of Z may beapproximated by the relationdc CC0bhbPb3cosh3EbIbPab 2bcosh C0asinh2EaIa26and thus the displacement at C in the Z direction isDZ dccosh Pb3cos2h3EbIbPabcosh 2bcosh C0asinh2EaIa27Link AB is made of stainless steel with the modulusof elasticity Ea=210 GPa. The hand grinder is a com-bination of stainless steel and aluminum alloy. The va-lue Eb=70 GPa is used and later one will see that anyvalue between 70 Gpa (aluminum) and 210 GPa(steel) may also be used and has very little eect on thefinal result. Diameters of segment AB and BC areda=0.036 m and db=0.026 m respectively. Using therelation I pd4C1464, one may determine Ia=8.2 10C08mandIb=2.2 10C08m respectively. Substi-tuting these values, together with the lengths a=0.13 m,b=0.195 m and the angle h=30C176 into Eq. 27 one mayobtain Z=1.38 10C06P. Then the stiness of the firstspring is k1=P/DZ=7.25 105N/m.Stiness of material removal process: k2Grinding force for a certain grinding depth may beobtained by using the force sensor shown in Fig. 1(Kistler_5295A). As feed rate is 20 mm/min, rotationspeed is 20,000 rpm, and tool diameter is 9.5 mm, nor-mal grinding forces are measured with various grindingdepths. Average grinding forces versus grinding depthsare plotted in Fig. 16. The relation is roughly linear andthe slope is taken to be the stiness k2, hence approxi-mately k2=8.84 103N/m.Stiness of work-piece: k3Stiness of the work-piece may be estimated by using thefinite element method. For example, specimens used byChen 9 have the size 40mm 40mm 15 mm, and aremade of SKD61 steel. Using the commercial softwareI-DEAS version 7.0, the displacement at the centre as aforce 100 N is applied is calculated to be 1.510-7m.Hence the stiness k3=6.67 108N/m.Equivalent stiness K for the seriesof springs k1, k2, and k3The equivalent stiness K is given by1K1k11k21k328Substituting values of k1,k2,andk3into Eq. 28, it isfound that K=8.74 103N/m. One may notice thatboth k1, and k3are much larger than k2, hence the right-hand side of Eq. 28 and also the equivalent stiness K,are dominated by k2.Stiness of the force sensor: k4The operation manual of the force sensor (Kis-tler_5295A) gives the value k4=2.6 109N/m.References1. Hahn RS (1964) Controlled-force grindinga new techniquefor precision internal grinding. J Eng Ind 86:2872932. S