新型絲杠磨床砂輪修整器設(shè)計

新型絲杠磨床砂輪修整器設(shè)計,新型絲杠磨床砂輪修整器設(shè)計,新型,磨床,砂輪,修整,設(shè)計 ORIGINAL ARTICLE C. H. Liu A. Chen Y.-T. Wang C.-C. A. Chen Modelling and simulation of an automatic grinding system using a hand grinder Received: 14 November 2002/ Accepted: 11 March 2003/Published online: 10 March 2004 C211 Springer-Verlag London Limited 2004 Abstract A grinding process model for an automatic grinding system with grinding force control is developed inthispaper.Thisgrindingsystemutilisesanelectrichand grinder, driven by a CNC machine centre and a force sensor for force measurement. This model includes com- pliance of the grinding system and is initially represented by a series of springs. The stiness of each component is estimatedinthisstudyanditisfoundthatthemodelmay be simplified into a single spring-mass system. A corre- sponding PID controller is designed for the purpose of grinding force control, which calculates the appropriate CNC spindle displacement according to the force mea- sured by the force sensor. Computer simulation results show that the system settling time is less than 0.25 s. Keywords Applications grinding Grinding system simulation Automatic grinding system 1 Introduction It is known that force control may improve grinding results of mould and dies 1, 2, 3, 4, 5, 6, and hence force control has become an important procedure for grinding and surface finishing processes. Recently, electric hand grinders have become popular surface finishing tools of moulds and dies. They have already been included in automatic surface finishing systems 7, 8, in which hand grinders are driven by CNC machine centres. However, corresponding force control techniques for these systems have not been developed. In this study, therefore, a grinding process model for an automatic grinding sys- tem using hand grinders is proposed, and a corre- sponding PID controller has been designed. The system is similar to that of Chen and Due 7, and Hsu 8, but a force sensor is placed under the work-piece for force measurement. The system is shown in Fig. 1. The model proposed in this paper has been implemented in the grinding system 9 and grinding force control experi- ments have been performed. Several grinding force models have been proposed. A good summary of these models, up to the year 1992, is given by To nsho et al. 10. In these models basically normal grinding forces are related to cutting speed, various forms of working engagement and wheel diam- eters. Recently Ludwick et al. 11, and Jenkins and Kurfess 12 suggested the model Q K p F N C0F TH V 1 where Q is material removal rate, F N is normal force, F TH is the threshold value of F N , V is the relative speed, and K p is a proportion constant. This model is a com- bination of the model proposed by Hahn and Lindsay 13, in which normal grinding force is proportional to material removal rate, and the Preston equation (see 14), 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 and Kurfess 15, 5, and Hekman and Liang 16 for grinding force estimation. A similar model was sug- gested by Kurfess, et al. 17, Whitney et al. 18 and Kurfess and Whitney 19. In a series of studies for weld bead grinding systems, they utilised the expression Q K 1 P C0K 2 2 where the power P is the product of the grinding force and the relative speed. C. H. Liu ( hence velocity V in Eq. 1 is not included in the present model. Also, in all the previous models machine compliance was not con- sidered. In this study, compliance of the grinding system shown in Fig. 1 is included in the model. 2 Grinding process model The grinding system shown in Fig. 1 is modelled after the system shown in Fig. 2. In this figure, M and k 1 represent mass and stiness of the electric hand grinder (including link and holder in Fig. 1) respectively, k 2 is stiness of the material removal process, which is de- fined as the ratio of normal grinding force to grinding depth (assumed to be a constant). The symbols m and k 3 denote mass and stiness of the work-piece, and k 4 is stiness of the force sensor. Displacements x 1 , x 2 , x 3 and x 4 are measured from static equilibrium positions; hence gravity forces may be neglected. In Fig. 2, P is the force imposed by the CNC machine centre. Also, in the present model, the ratio of the normal grinding force to the grinding depth is assumed to take a constant value k 2 , in reality, however, the relation between the grinding force and the grinding depth is expected to be nonlinear. The symbol F d is used to represent the nonlinear term (i.e. real grinding force = k 2 x 2 +F d ). Springs k 1 , k 2 , and k 3 are in series and they are equivalent to a spring with the equivalent spring con- stant K. Estimations of k 1 , k 2 , k 3 and K are shown in the appendix and it is found that the equivalent spring constant K is dominated by k 2 . Generally k 2 is much less than the stiness of the force sensor k 4 , hence one may expect that k 2 C29K. For example, the ratio k 4 /K for the current system is larger than 10 5 . With this expectation (i.e. k 4 C29K) in mind, it may be assumed that k 4 fiand x 4 fi 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 is PtF d tC0FtMx 1 3 where FtKx 1 4 is the spring force. Dierentiating both sides of Eq. 4, one may obtain FtKx 1 5 Substituting Eq. 5 into Eq. 3, one gets PtF d tC0Ft M K Ft 6 Fig. 1 Grinding system Fig. 2 Modelling of the grinding system Fig. 3 Simplified model 875 Taking the Laplace transform for both sides of Eq. 6, also assuming zero initial conditions, one may obtain PsF d sC0Fs Ms 2 K Fs 7 From this equation, the block diagram of the process model may be drawn, as shown in Fig. 4. In this diagram, the input P(s) is the force applied by CNC machine centre, the disturbance F d (s) is the non- linear grinding force defined above, the output F(s) is the force measured bytheforce sensor.Thetransferfunction of the block diagram is defined by G P (S)=F(S)/P(S). Using Eq. 7, also assuming F d (s)=0 one may obtain G P S FS PS 1 M K s 2 1 8 3 Controller design In this study, a PID controller is utilised and is rep- resented as G c SK p K i S K d S. The block diagram of the grinding system is shown in Fig. 5. In this fig- ure, F * (S) is the input force command, F(s) is the value measured by the force sensor, and e(S)=F * (s)C0F(S)is error. In order to investigate if the system may be controlled under the steady state condition, the dis- turbance F d (s) is set to zero. From Fig. 5 and Eq. 8 one may show that the system transfer function G(s)in Fig. 5 is given by Gs Fs F C3 s K d s 2 K p sK i M K s 3 K d s 2 1K p C0C1 sK i 9 In steady state, s fi 0, from Eq. 9 it is assumed that lim s!0 Fs F C3 s K i K i 1 10 This means F(s)=F * (s), hence the system may be controlled in the steady state. Since F(t)=Kx 1 (t), dierentiating both sides of this expression, one finds _ FtK_x 1 KV t 11 where V(t) is velocity of the spindle. Taking the Laplace transform of this equation, one finds SF SKSx 1 SKV S 12 Hence the term K d SF(S) in Fig. 5 may be replaced by the expression K d KV. Also, SF * (S)=0 for the condition of constant grinding force, the block diagram shown in Fig. 5 may be replaced by the diagram shown in Fig. 6. The transfer function of this diagram may be shown to be Fig. 5 Grinding system block diagram Fig. 6 Modified grinding system block diagram Fig. 4 Block diagram of the process 876 Gs Fs F C3 s K p KsK i K Ms 3 K d Ks 2 K p K K C0C1 sK i K 13 The numerator polynomial in the current transfer function is one degree less than the transfer function defined by Eq. 9, which means the current transfer function has less zero point and also less influence upon a pole. Also, in the block diagram of Fig. 5 the error e(s) is dierentiated once, implying that the induced noise will be amplified, and this does not happen in the current block diagram. Therefore in the following discussion, the block diagram shown in Fig. 6 is used. In order to determine controller gains, the first obvious point is that the denominator of Eq. 13 is a polynomial of the third order, and can be written in the form gSC17MS 3 K d KS 2 K p K K C0C1 S K i K S aS a jbS a C0jb 14 Since it is relatively dicult to analyse a third order system, the purpose now is to approximate this system with a second order one, by assuming the two complex roots are dominant roots. Requiring that percent over- shoot of the second order system does not exceed 3%, the following relation may be obtained 21 0:03 e C0pf C14 1C0f 2 p 15 which implies pf . 1C0 f 2 p 3:057, or f=0.75. As the first attempt to determine system gains, settling time T S is arbitrarily set to 0.5 s. This means 21 T S 4 fx n 0:5s 16 thus fx n =8, and x n =10.67 rad/s. The angle h (see Fig. 7) take the value cos C01 f or 41.4C176. Therefore the complex poles of the characteristic equa- tion (i.e. the equation g(S)=0, see Eq. 14) are S=C08j7.05. The idea is to keep the third pole away from these two complex poles. The value S=C015 is arbitrarily chosen, hence Eq. 14 can be written in the form gSS 3 31S 2 353:7S 1705:5 0 17 Comparing coecients of the last equations to the denominator of Eq. 13, the following relations are ob- tained K d K M 31 18a K p K K M 353:7 18b and K i K M 1705:5 18c The value of K is estimated in the appendix to be K=8.7410 3 (N/m), and mass of the hand grinder (with holder and link) is M=5 kg, substituting these values into Eqs. 18a, 18b, 18c, K d =0.0178, K d =C00.797, and K i =0.98 may be obtained. Since system gains cannot be negative, a second trial, with new values of settling time T s and natural frequency x n , is necessary. For the second trial, the settling time is assumed to be 0.25 s, i.e. T S 4= fx n 0:25 swhich means x n 21:3, and fx n 16. Following the same steps as in the first trial, the two complex poles may be determined to be S=C016j14.1. Now assuming that the third pole is at the point S=C045, then the characteristic equation is S 3 77S 2 1895S 20466 0 19 and after comparing corresponding coecients, one finds K d K M 77 20a K p K K M 1895 20b and K i K M 20466 20c which means K d =0.0443, K P =0.0891, and K i =11.762. The zero of the system is the root of the equation in the numerator of Eq. 13, i.e. the equation K p KS K i K 0 21 Substituting the above values into this equation, one may find that the zero is the point S=C0132.0. Hence the transfer function defined by Eq. 13 may be written in the form GS FS F C3 S K p K=MSK i =K P M S 3 77S 2 1895S20466 155:034 S132 S45S16j14:1S16C0j14:1 22 Note that in steady state, G(S) approaches 1 as S fi 0.Fig. 7 Root locus of the characteristic equation 877 4 Simulation results and discussions The system response may be obtained using the com- mercial software Matlab. In the simulation procedure, the spring constant K=8.7410 3 (N/m), mass of the hand grinder (with holder and link) M=5 kg, system gains are K d =0.004, K p =0.089 and K i =11.762. The sampling time is chosen to be 0.001 s. The loop transfer function of block diagram shown in Fig. 6 is LS K p KS K i K Ms 3 K d KS 2 KS 23 With the system gains just obtained, the root-locus diagram corresponding to loop transfer function 23 may be drawn and is shown in Fig. 8. The three poles of Eq. 22 are shown in this diagram. It can seen that both the zero and the third pole are much farther apart from the imaginary axis than the dominant complex roots, making them have negligible influence on the system, hence the system may be approximated by a second order one. With the gain values just obtained, the Bode diagram for 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 the process model without controllers, (i.e. Fig. 9) one may find that the resonance has been greatly reduced. Also, while the phase margin shown in Fig. 9 is approximately 180C176, which corresponds to an unstable state, the phase angle shown in Fig. 10 is approximately 45C176. Due to the fact that the grinding force control system takes step input, step responses for the grinding process with and without controller are compared. Figure 11 shows step responses without controller, and Fig. 12 shows step response with the controller just designed has Fig. 8 Root-locus diagram of open loop system Fig. 9 Bode diagram of the grinding process Fig. 10 Bode diagram of the grinding process with controller Fig. 11 Step response of the grinding process 878 been imposed. In the system without controller, grinding force oscillates with no steady state and in the system with force control; the settling time is less than 0.25 s with a percent overshoot of less than 3%. These system gains (i.e. K d =0.044, K P =0.089, and K i =11.762), have been utilised in the automatic grinding system with force control; results show that surface roughness may be re- duced 9. 5 Conclusions In this study, a grinding process model for the force control system utilises a CNC machine centre, an electric hand grinder and a force sensor has been proposed. This force control system is represented by a spring-mass system and according to estimated stiness of each component, the spring-mass system may be further simplified. A PID controller base on the simplified sys- tem has been designed. Controller gains are estimated by using the commercial software Matlab. Estimation re- sults show that a settling time of less than 0.25 s and a percent overshoot of less than 3% may be obtained. Acknowledgements The authors gratefully acknowledge that this study was supported by the National Science Council of ROC under grant no. NSC 89-2218-E032030. Appendix: Estimations of stiffness constants Stiness of hand grinder (with link and holder): k 1 In Fig. 13, segment AB represents the link (see Fig. 1) which has a length a and makes an angle h with the Z-axis. The electric hand grinder is presented by segment BC which is normal to segment AB and the length of which is b. As the normal grinding force P is applied, the corresponding displacement of member ABC (i.e. the combined parts of link, holder, and hand grinder) at the point C is DZ, which is to be estimated in two steps. First, segment AB is tightly fixed to the spindle and may be represented by a cantilever beam, as shown in Fig. 14. The angle at point B due to the force Psinh and the moment Pbcosh is h b Pbacosh E a I a C0 Pa 2 sinh 2E a I a 24 where E a and I a are modulus of elasticity and area moment of inertia of the segment AB, respectively. Secondly, segment BC is also modelled by a cantilever beam but it has an initial inclination angle h b , as shown in Fig. 15. The deflection at C of this beam due to the load is given by CC 0 Pb 3 cosh 3E b I b 25 Fig. 13 Simplified model of hand grinder (with holder) for calculating k 1 Fig. 14 Link AB is modelled by a cantilever beam Fig. 12 Step response of the grinding process with controller 879 where E b and I b denote modulus of elasticity and area moment of inertia of the segment BC respectively. The total displacement at C in the direction of Z may be approximated by the relation d c CC 0 bh b Pb 3 cosh 3E b I b Pab 2bcosh C0asinh 2E a I a 26 and thus the displacement at C in the Z direction is DZ d c cosh Pb 3 cos 2 h 3E b I b Pabcosh 2bcosh C0asinh 2E a I a 27 Link AB is made of stainless steel with the modulus of elasticity E a =210 GPa. The hand grinder is a com- bination of stainless steel and aluminum alloy. The va- lue E b =70 GPa is used and later one will see that any value between 70 Gpa (aluminum) and 210 GPa (steel) may also be used and has very little eect on the final result. Diameters of segment AB and BC are d a =0.036 m and d b =0.026 m respectively. Using the relation I pd 4 C14 64, one may determine I a = 8.2 10 C08 mandI b =2.2 10 C08 m 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 may obtain Z=1.38 10 C06 P. Then the stiness of the first spring is k 1 =P/DZ=7.25 10 5 N/m. Stiness of material removal process: k 2 Grinding force for a certain grinding depth may be obtained by using the force sensor shown in Fig. 1 (Kistler_5295A). As feed rate is 20 mm/min, rotation speed is 20,000 rpm, and tool diameter is 9.5 mm, nor- mal grinding forces are measured with various grinding depths. Average grinding forces versus grinding depths are plotted in Fig. 16. The relation is roughly linear and the slope is taken to be the stiness k 2 , hence approxi- mately k 2 =8.84 10 3 N/m. Stiness of work-piece: k 3 Stiness of the work-piece may be estimated by using the finite element method. For example, specimens used by Chen 9 have the size 40mm 40mm 15 mm, and are made of SKD61 steel. Using the commercial software I-DEAS version 7.0, the displacement at the centre as a force 100 N is applied is calculated to be 1.510 -7 m. Hence the stiness k 3 =6.67 10 8 N/m. Equivalent stiness K for the series of springs k 1 , k 2 , and k 3 The equivalent stiness K is given by 1 K 1 k 1 1 k 2 1 k 3 28 Substituting values of k 1 ,k 2 ,andk 3 into Eq. 28, it is found that K=8.74 10 3 N/m. One may notice that both k 1 , and k 3 are much larger than k 2 , hence the right- hand side of Eq. 28 and also the equivalent stiness K, are dominated by k 2 . Stiness of the force sensor: k 4 The operation manual of the force sensor (Kis- tler_5295A) gives the value k 4 =2.6 10 9 N/m. References 1. Hahn RS (1964) Controlled-force grindinga new technique for precision internal grinding. J Eng Ind 86:287293 2. Shaw MC (1996) Principles of abrasive processing. Oxford University Press, Oxford Fig. 16 Normal force versus grinding depth (tool diameter is 9.5 mm, feed rate = 20 mm/min, and rotation speed = 20,000 rpm) Fig. 15 Link BC is also modelled by a cantilever beam 880 3. Liu L, Ulrich BJ, Elbestawi MA (1990) Robotic grinding force regulation: design, implementation and benefits. In: Proceed- ings of the 1990 IEEE international conference on robotics and automation, pp 258265 4. Jenkins HE, Kurfess TR (1995) Adaptive pole-zero cancella- tion in grinding force control. IEEE Trans Contr Syst Tech 7:363370 5. Jenkins HE, Kurfess TR (1999) Adaptive process estimation for a grinding system. In: Proceeding of the ASME dynamic system and control division 57:483489 6. Jenkins HE (1996) Process estimation and adaptive control of grinding system. PhD thesis, Georgia Institute of Technology 7. Chen CHA, Due NA (1996) Development of an automatic surface finishing system (ASFS) with in-process surface topography inspection. J Mater Process Tech 62:427430 8. Hsu TC (1998) Automatic surface finishing with rough area pattern recognition. PhD thesis, University of Wisconsin- Madison 9. Chen JC (2000) Force control in grinding processes. Masters thesis, Tamkang University 10. To nsho HK, Peters J, Inasaki I, Paul T (1992) Modeling and simulation of grinding processes. Ann CIRP 41:677688 11. Ludwick SJ, Jenkins HE, Kurfess TR (1994) Determination of dynamic grinding model. Trans ASME Dyn Syst Contr 55:843 849 12. Jenkins HE, Kurfess TR (1996) Optimization of real-time multivariable estimation in grinding. Trans ASME Dyn Syst Contr 58:365370 13. Hahn RS, Lindsay RP (1971) Principles of grinding, part 1: basic relationships in precision machining. Machinery, pp 55 62 14. Brown N (1990) Optical fabrication, MISC4476, revision 1. Lawrence Livermore, Livermore 15. Jenkins HE, Kurfess TR (1997) Dynamic stiness implications for a multiaxis grinding system. J Vib Cont 3:297313 16. Hekman KA, Liang SY (1999) Feed rate optimization and depth of cut control for productivity and part parallelism in grinding. Mechatronics 9:447462 17. Kurfess TR, Whitney DE, Brown ML (1988) Verification of a dynamic grinding model. Trans ASME J Dyn Syst Meas Contr 110:403409 18. Whitney DE, Edsall AC, Todtenkopf AB, Kurfess TR, Tate AR (1990) Development