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Waviness compensation of precision machining by piezo-electric micro cutting device
International Journal of Machine Tools & Manufacture 38 (1998) 1305–1322
Waviness compensation of precision machining by piezoelectric micro cutting device Jeong-Du Kim*, Dong-Sik Kim Laboratory for Precision Machining and Machine Tools, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon, 305-701, South Korea Received 13 November 1996; in final form 12 September 1997
Abstract Recently, precision machining technology has been developed continuously in order to achieve high productivity and assured quality of precision parts in several industrial fields such as; lasers, optics, quartz vibrators, semiconductors, aircraft and artificial satellites etc. Waviness may occur on the surface of machined parts due to the table motion error and the dynamic cutting mechanism between the tool and the workpiece and may impair the form accuracy of the precision machined parts. In this research, a micro cutting device with piezo-electric actuator has been developed in order to control depth of cut precisely and compensate the waviness on the surface of the workpiece. Experiments have been carried out on a precision lathe. The characteristics of the surface profile and the cause of the waviness profile have been analyzed and waviness profiles of some cases have been compared with those of experiments. 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction Recently, precision machining technology has been developed continuously in order to make high productivity and quality assurance of the precision parts of several industrial fields such as lasers, optics, quartz vibrator, semiconductors, aircraft and artificial satellites etc. It is so difficult to achieve them by conventional machining that precision machining technology [1–3] is needed. In the production of precision parts using lapping and polishing, it is apparent that production * Corresponding author. 0890-6955/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 7 ) 0 0 0 8 0 - 1
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rate decreases and there is a deficiency of accuracy and limitation of geometry to be machined occurring due to handling work. It is also difficult to achieve accuracy by grinding because the materials — aluminum, copper, nickel and nonmetals like silicon, plastic etc. — are ductile. Recently, cutting using a single crystal diamond tool has been performed on precision lathes. A representative application of precision machining is the face cutting of magnetic disks [4] which are used as auxiliary memory devices in personal computers. As the recording density increases, the gap between the magnetic disk and the magnetic head decreases. The surface roughness might be less than about 0.01 m Ra to ensure stability and small head ride height. In conventional abrasive particle machining, productivity and repeated accuracy deteriorate and machining accuracy is also restricted because the production process mainly depends on manual work. In recent years, the magnetic disk has been machined on precision machine tools using single crystal diamond tools along the development of performance of the precision machine tools. However, waves occur periodically on the surface of the disk when face cutting of the magnetic disk is executed and the waviness detracts from the form accuracy of the magnetic disk. In this research, a micro cutting device with piezo-electric material was developed in order to control precise depth of cut, and the waviness on the surface in face cutting of Al alloys was compensated for by the piezo-electric micro cutting device. Experiments have been carried out on a precision lathe. The characteristics of the surface profile and the cause of the waviness profile have been analyzed and waviness profiles of some cases have been compared with those obtained in experiments. 2. Experimental setup The main spindle of the precision lathe has an aerostatic bearing mounted on a built-in type DC motor and operated above 7 bar air pressure. The guideway is also of the air slide type and the maximum travel of the z-axis is 200 mm. The rapid traverse speed is 3000 mm min⫺1 and feed rate is variable from 0.5 to 3000 mm min⫺1. The body of the machine is a granite bed with air mounting. Air bearing and air slider are mounted on the granite bed in order to isolate them from vibration due to the propagated vibration from the earth and to keep them in a parallel state. The z-axis of the precision lathe is made of ceramics to increase stiffness. A piezo-electric micro cutting device was developed to control submicron depth of cut. This machine is operated in a thermostatic room because machine components, particularly the air bearing spindle and the workpiece, are very sensitive to temperature in ultraprecision cutting. Figure 1 shows a schematic diagram of the waviness compensation system in precision face cutting using the piezo-electric micro cutting device. A personal computer monitors piezo-electric voltage and displacement of the micro cutting device and controls displacement of the micro cutting device with regard to the hysteresis loop of the piezo-electric actuator. Table 1 shows the specification of the experimental system. A high voltage power supply, a differential circuit for extracting piezo-electric voltage, and gap sensor amplifier are internally set in the driver of the piezo-electric micro-depth control system. It is connected to the personal computer by A/D converter and D/A converter. The actual piezo-electric voltage is extracted by a differential circuit of OP-amp. For the first time, table motion error of the x-axis of the precision lathe has been measured by
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Fig. 1. Schematic diagram of waviness compensation system in precision machining using the piezo-electric micro cutting device.
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Table 1 Experimental apparatus Machine tools
Precision CNC lathe Air bearing Air slide Granite bed
Tool Workpiece Gap sensor
Single crystal diamond tool: straight type Al alloy Eddy current type Applied Electronics corp. Model: AEC-5505 Response frequency: 20 kHz Output: 5 V (5 mV m⫺1) Resolution: 0.5 m
A/D converter
Advantech Co. Ltd Model: PCL 812PG Channels: 16 single-ended Resolution 12 bits Conversion speed: 30 kHz max. Accuracy: 0.015% of reading ± 1 bit
System control
IBM-PC 586 compatible
an eddy current type non-contact gap sensor. This error is saved in the personal computer which drives the micro cutting device along an absolute position of the x-axis table. When face cutting is conducted, displacement of the micro cutting device is appropriately controlled in order to ensure that the relative displacement between the workpiece and the tool on the micro cutting device become zero. 3. A piezo-electric micro cutting device Figure 2 shows an overview of the piezo-electric micro cutting device which is designed for mounting on the x-axis of the precision lathe. Displacement of the piezo-electric micro cutting device is measured by a conductance type gap sensor and the piezo-electric actuator is preloaded by the ball and screw bolt. The hinge is based on the parallel spring principle with a design stiffness of 50N m⫺1. The equation of the parallel spring is given [5] Kh =
8Ebt5/2 91/2L2
(1)
where Kh[N m⫺1] is hinge stiffness, E[N m⫺2] is Young’s modulus, b[mm] is width of hinge, r[m] is notch radius, t[m] is thickness between hinge holes and L[mm] is distance between hinges.
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Fig. 2. Overview of piezo-electric micro cutting device. (a)Photograph of piezo-electric micro cutting device; (b) schematic diagram of piezo-electric micro cutting device.
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Figure 3 shows a mechanical and electrical model of the piezo-electric micro cutting device. The characteristics of the piezo-electric actuator and several important variables are listed in Table 2. Generally, the piezo-electric micro actuator is modeled on the lumped second order mechanical system from the linear piezo-electric equation and Newton’s law, and on the first order electrical system from output impedance of driving voltage source and capacitance of piezo-electric actuator [6–10].
Fig. 3. Mechanical and electrical model of piezo-electric actuator. (a) Lumped mechanical model; (b) Equivalent electric circuit of voltage driving piezo-electric element.
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Table 2 Characteristics of piezo-electric actuator and parameters 10 × 10 × 18 (144 layers) 6500 635 × 10⫺12 8 5.5 × 1010 20.794 (N V⫺1) 20 (kg) 20144 (Ns m⫺1) 1.843 × 108(N m⫺1) 6.5 × 10⫺6 (F) 300 (⍀)
Size (mm) Static capacitance (nF) Piezo-electric constant (m V⫺1) Density (g cm⫺3) Elastic constant (N m⫺2) Kf MP DP KPZT Co Ro
The governing equation of the lumped second order mechanical system of the piezo-electric micro cutting device is as follows. MP
冉
冊
c33d 233 d2uP duP + D u = P + 1 + Fl + K P PZT P e dt2 dt ⑀3
(2)
where uP[m] is displacement of the piezo-electric actuator, MP[kg] is equivalent mass of piezoelectric actuator, DP[kg s⫺1] is equivalent damping coefficient of the piezo-electric actuator, KPZT[N m⫺1] is the equivalent stiffness coefficient of the piezo-electric actuator, Pe[N] is equivalent force, Fl[N] is external load, d33[C N⫺1] is piezo-electric charge constant, c33[N m⫺2] is piezoelectric stiffness constant and ⑀3[F m⫺1] is dielectric constant of the piezo-electric actuator. Some variables are represented in detail as following. MP =
c33d33A AL 2 ⑀33A , KPZT = c33A,Pe = Vn(t),Co = 2 8l l l
(3)
where [kg m⫺3] is the density of the piezo-electric actuator, A[m2] is the cross-sectional area of the piezo-electric actuator, l[m] is the length of the piezo-electric actuator and Co[F] is the capacitance of the piezo-electric actuator. The piezo-electric actuator is driven by the voltage source with output impedance, Ro and a high voltage amplifier. Piezo-electric voltage, VP is generated by the external load and affects the input signal. Time delay of voltage source, Vi and actual input voltage, Va occurs. The relationship between the actual input voltage and nominal input voltage is given by 1 Va ⬇ Vi RoCos + 1
(4)
Piezo-electric voltage generated by the external load is represented by Eq. (5). d33Ros VP = Fl RoCos + 1
(5)
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Net voltage, Vn is the sum of actual input voltage, Va and input voltage, VP of the piezoelectric actuator. Vn = Va + VP =
1 d33Ros Vi + F RoCo + 1 RoCos + 1 l
(6)
where, Va[V] is actual input voltage, Vi[V] is nominal input voltage, Vn[V] is net voltage, Vp[V] is piezo-electric voltage and Ro[⍀] is output impedance of the piezo-electric actuator. A complete transfer function, including feedforward compensator and considering force feedback of the displacement is described in Eq. (7). U(s) =
Vi(s) ⫺
Kf(Ts + 1) T 2Mps4 + (T 2DP + 2TMP)s3 + {T 2KPZT + 2TDP + MP + T 2Km(1 + 2P)}s2 + {2KPZTT + DP + TKm(2 + 3P ⫺ PKVP)}s + {KPZT + Km(1 + P ⫺ TPKVI)} T 2(1 + 2P)s2 + T{2 + P(3 ⫺ KVP)}s + {1 + P(1 ⫺ TKVI)} F (s) T 2Ms4 + (T 2DP + 2TMP)s3 + {T 2KPZT + 2TDP + MP + T 2Km(1 + 2P)}s2 + {2KPZTT + DP + TKm(2 + 3P ⫺ PKVP)}s + {KPZT + Km(1 + P ⫺ TPKVI)} l
(7)
where KVP is piezo-electric voltage feedback proportional gain, KVI is piezo-electric voltage feedback internal gain and Km[N m⫺1] is coefficient of force feedback. Equivalent force constant Kf, time constants T and P are as follows. Kf =
c33d33A 8K 2f ,T = RoCo,P = 2 l CoKPZT
(8)
The results of simulation of Eq. (7) are shown in Fig. 4. The simulation has been performed by MATLAB with simulink. An external mass located at the 3 m point from the start position of the piezo actuator is used as the external disturbance. From the root locus, the disturbance cannot affect the stability of the system but the reference input produces an unstable response for large Km values. The steady state error disappears. Experiments have been conducted in order to compare performances of position control by the piezo-electric voltage and by the displacement feedback. Step responses of the piezo-electric actuator to a 38 V step input voltage are shown in Fig. 5. Figures 5(a) and (b) show the result of the displacement feedback control case. Figures 5(c) and (d) show the results of the piezo-electric voltage feedback control case. From comparison with simulation results, the displacement feedback shows a five times longer settling time and the piezo-electric voltage feedback case three times longer. However, the transient motion of the real system is better than that of the simulation because of the simulation’s underestimate of damping coefficient from the hysteresis loop and friction of the vertical feeding unit. The late settling time of the real system is due to overestimation of the output impedance of the amplifier and capacitance of piezo-electric actuator. The response of the piezo-electric voltage feedback system is faster than that of the displacement feedback case due to the effect of input voltage feedback. A steady state error has been reduced by compensation of piezo-electric voltage with PI controller. Piezo-electric voltage feedback control is faster and more stable than that of the displacement feedback control using a gap sensor, and it does not need an expensive displacement sensor.
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Fig. 4. Simulation of piezo-electric micro cutting device (displacement feedback considered).(a) Root locus to disturbance input; (b) root locus to reference input; (c) step response with compensator and considered feedback.
4. Results and discussions Face cutting of Al alloy by a single crystal diamond tool on the precision lathe has been carried out. Figure 6 shows the comparison of waviness profiles of Al alloy between no compensation and compensation face cutting. In the compensation face cutting, constant depth of cut is given
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Fig. 5. Step response of piezo-electric micro cutting device. (a)Displacement feedback Vi = 38 V Fl = 3 N; (b) Piezoelectric voltage feedback Vi = 38 V Fl = 3 N.
by the piezo-electric micro cutting device overcoming external disturbances such as table motion error. The piezo-electric micro cutting device is controlled by the piezo-electric voltage feedback. The peak-to-valley waviness is 1.3 m when the piezo-electric micro cutting device compensation is not used. If the depth of cut is compensated for by the piezo-electric micro cutting device, the peak-to-valley waviness is decreased to 0.7 m. It is considered that the remaining waviness is
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Fig. 6. Compariness of waviness profiles for AL alloy between no compensation and compensation face cutting. (a) Waviness profile without the piezo-electric micro cutting device; (b) waviness profile after piezo-electric micro cutting device compensation.
due to the imperfect compensation of the table error, the cutting mechanism and the thermal displacement of the tool. Figure 7 shows a schematic diagram of the surface waviness profile, in some ideal cases, by piezo-electric micro cutting device. Case 1 is the initial waviness profile with a peak-to-valley of 4 m generated by the piezo-electric cutting device. The micro cutting device is driven by a personal computer and a high voltage amplifier. The depth of the micro cutting device is determined by the elongation of the piezo-electric material which is preloaded by the ball and screw and controlled by the program on the personal computer. The depth of cut given by the precision
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Fig. 7. Schematic diagram of surface waviness profile in ideal cases by piezo-electric micro cutting device. (a) Case 1: initial waviness profile; (b) case 2: depth of cut = 2 m, piezo depth = 2 m, tool path = triangular wave; (c)case 3: depth of cut = 0 m, piezo depth = 4 m, tool path = straight.
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lathe controller is 2 m and the depth of cut given by the piezo-electric micro cutting device is also 2 m in case 2. The tool path is triangular, with wavelength 2.0 mm and peak-to-valley 2 m. In case 3, the depth of cut given by the precision lathe controller is zero, the depth of cut given by the piezo-electric micro cutting device is 4 m and the tool path is straight. The theoretical peak-to-valley results are 2 m and zero in cases 2 and 3, respectively. Figure 8 shows the waviness profile and wavelength spectral density of the machined Al alloy for case 1 at feed rate 20 mm min⫺1, the rotation speed of the main spindle 800 r.p.m. with the
Fig. 8. Waviness profile and wavelength spectral density in face cutting: case 1. (a) Waviness profile; (b) wave spectral density of waviness profile, feed rate = 20 mm min ⫺ 1, cutting speed = 800 r.p.m., workpiece = Al alloy, triangular tool path, single crystal diamond tool: straight type.
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single crystal diamond tool. For a machined disk type workpiece, the surface waviness profile has been measured in eight directions and the results were similar. The main wavelength can be derived from the wave spectral density graph. Peak-to-valley of the surface waviness is 3.3 m and the wavelength, i.e. the period of the waviness, is 2.05 mm. The waviness appears to have a constant period. Figure 9 shows the waviness profile and wavelength spectral density of the machined Al alloy
Fig. 9. Waviness profile and wavelength spectral density in face cutting: case 2. (a) Waviness profile; (b) wave spectral density of waviness profile, feed rate = 20 mm min ⫺ 1, cutting speed = 800 r.p.m., workpiece = Al alloy, depth of cut = 0.002 mm, piezo depth of cut = 0.002 mm, triangular tool path, single crystal diamond tool: straight type.
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for case 2 at feed rate 20 mm min⫺1, with the rotation speed of the main spindle of 800 r.p.m. with the single crystal diamond tool. The depth of cut given by the precision lathe controller is 2 m, the depth of cut given by the piezo-electric micro cutting device is also 2 m and the tool path is triangular. Peak-to-valley of the surface waviness is 1.3 m and the wavelength, i.e., the period of the waviness is 2.05 mm. The peak-to-valley is reduced with the same wavelength of initial waviness profile as was expected by the theoretical value as shown in Fig. 7. Figure 10 shows the waviness profile and wavelength spectral density of the machined Al alloy for case 3 at feed rate 20 mm min⫺1 and rotation speed of the main spindle of 800 r.p.m. with the single crystal diamond tool. The depth of cut given by the precision lathe controller is zero, the depth of cut given by the piezo-electric micro cutting device is 4 m and the tool path is straight. Peak-to-valley of the surface waviness is 0.3 m and the wavelength, i.e. the period of the waviness is 8.33 mm. The peak-to-valley should be zero theoretically but the peak-to valley is 0.3 m due to the steady state error of the response of the piezo-electrical micro cutting device and the cutting mechanism between the tool and the workpiece. By using the piezo-electric micro cutting device, the initial waviness profile almost completely disappeared in case 3. The peak-tovalley of the surface waviness profile has been decreased from 3.3 to 0.3 m using the micro cutting device. The peak-to-valley of the waviness profile between theory and experiments has been compared as shown in Fig. 11. The experimental results tend to correspond with the expected value and the piezo-electrical micro cutting device is reliable for control of micrometer depth and compensation for the waviness profile. Figure 12 shows the wavelength of the waviness profile of the theoretical value and the experimental results. The experimental results are similar to the theoretical values in case 2 and 3 so the piezo-electrical micro cutting device is effective in compensating for the waviness profile in precision face cutting. In case 3, the ideal wavelength is infinite and the experimental result is 8.33 mm. The peak-to-valley is 0.3 m and this wave profile has a 8.33 mm wavelength due to the above reasons.
5. Conclusions
1. A micro cutting device with piezo-electric actuator was developed which was able to control depth of cut precisely and to compensate the waviness of the surface profile in face cutting. 2. It has been demonstrated that a piezo-electric voltage feedback system which has been constructed to drive the piezo-electric micro cutting device has a better step response than a displacement feedback system. 3. When the compensation system was not used, the surface maximum waviness was 3.3 m. However, the peak-to-valley of the surface waviness profile has been decreased to 0.3 m in one particular case, by using the piezo-electric micro cutting device with the piezo-electric voltage feedback 4. A theory for the operation of the actuator was proposed which was validated by experimental results of precision face cutting.
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Fig. 10. Waviness profile and wavelength spectral density in face cutting: case 3. (a) Waviness profile; (b) wave spectral density of waviness profile, feed rate = 20 mm min ⫺ 1, cutting speed = 800 r.p.m., workpiece = Al alloy, depth of cut = 0 mm, piezo depth of cut = 0.004 mm, straight tool path, single crystal diamond tool: straight type.
References [1] Y. Furukawa, N. Moronuki, Effect of material properties on ultra precise cutting process, Annals of the CIRP 37 (1988) 113–116. [2] T. Sugano, K. Takeuchi, T. Goto, Y. Yoshida, Diamond turning of an aluminum alloy for mirror, Annals of the CIRP 36 (1987) 17–20.
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Fig. 11. Comparison of peak-to-valley of waviness profile between theory and experiments.Case 1: initial waviness profile. Case 2: depth of cut 2 m, piezo depth 2 m, triangular tool path. Case 3: depth of cut 0 m, piezo depth 4 m, straight tool path.
Fig. 12. Comparison of wavelength of waviness profile between theory and experiments. Case 1: initial waviness profile. Case 2: depth of cut 2 m, piezo depth 2 m, triangular tool path. Case 3: depth of cut 0 m, piezo depth 4 m, straight tool path.
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[3] J.D. Kim, E.B. Lee, D.H. Hyun, A study on the modelling of tool motion and high-accuracy surface generation by the use of cutting force signal, Journal of Materials Processing Technology 47 (1994) 45–62. [4] Kim, J.D. and Kim, D.S., Surface characteristics of magnetic disk machined by single crystal diamond tool in ultraprecision lathe. 3rd International Conference on Precision Surface Finishing and Burr Technology, Seoul, 1994, pp. 428–441. [5] Y. Hara, A new micro-cutting device with high stiffness and resolution, Annals of the CIRP 39 (1) (1990) 375–388. [6] J.D. Kim, S.R. Nam, Development of a micro-depth control system for an ultra-precision lathe using a piezoelectric actuator, International Journal of Machine Tools and Manufacture 37 (4) (1997) 495–509. [7] T. Zhong, Z. Nakagawa, Development of a micro-displacement table for ultra-precision machining and grinding for curved surfaces by the use of it, JSPE 26 (2) (1992) 102–110. [8] H.S. Tzou, Design of a piezoelectric exciter/actuator for micro-displacement control: theory and experiment, Precision Engineering 13 (2) (1991) 104–110. [9] Lee, Seok-Koo, Motion analysis and control of translation device driven by piezoelectric actuator. M.S. thesis, Korea Advanced Institute of Science and Technology (in Korean), 1993. [10] C.V. Newcomb, Improving the linearity of piezoelectric ceramic actuators, Electronic Letters 11 (1982) 442–444.
Waviness compensation of precision machining by piezoelectric micro cutting device Jeong-Du Kim*, Dong-Sik Kim Laboratory for Precision Machining and Machine Tools, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon, 305-701, South Korea Received 13 November 1996; in final form 12 September 1997
Abstract Recently, precision machining technology has been developed continuously in order to achieve high productivity and assured quality of precision parts in several industrial fields such as; lasers, optics, quartz vibrators, semiconductors, aircraft and artificial satellites etc. Waviness may occur on the surface of machined parts due to the table motion error and the dynamic cutting mechanism between the tool and the workpiece and may impair the form accuracy of the precision machined parts. In this research, a micro cutting device with piezo-electric actuator has been developed in order to control depth of cut precisely and compensate the waviness on the surface of the workpiece. Experiments have been carried out on a precision lathe. The characteristics of the surface profile and the cause of the waviness profile have been analyzed and waviness profiles of some cases have been compared with those of experiments. 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction Recently, precision machining technology has been developed continuously in order to make high productivity and quality assurance of the precision parts of several industrial fields such as lasers, optics, quartz vibrator, semiconductors, aircraft and artificial satellites etc. It is so difficult to achieve them by conventional machining that precision machining technology [1–3] is needed. In the production of precision parts using lapping and polishing, it is apparent that production * Corresponding author. 0890-6955/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 7 ) 0 0 0 8 0 - 1
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rate decreases and there is a deficiency of accuracy and limitation of geometry to be machined occurring due to handling work. It is also difficult to achieve accuracy by grinding because the materials — aluminum, copper, nickel and nonmetals like silicon, plastic etc. — are ductile. Recently, cutting using a single crystal diamond tool has been performed on precision lathes. A representative application of precision machining is the face cutting of magnetic disks [4] which are used as auxiliary memory devices in personal computers. As the recording density increases, the gap between the magnetic disk and the magnetic head decreases. The surface roughness might be less than about 0.01 m Ra to ensure stability and small head ride height. In conventional abrasive particle machining, productivity and repeated accuracy deteriorate and machining accuracy is also restricted because the production process mainly depends on manual work. In recent years, the magnetic disk has been machined on precision machine tools using single crystal diamond tools along the development of performance of the precision machine tools. However, waves occur periodically on the surface of the disk when face cutting of the magnetic disk is executed and the waviness detracts from the form accuracy of the magnetic disk. In this research, a micro cutting device with piezo-electric material was developed in order to control precise depth of cut, and the waviness on the surface in face cutting of Al alloys was compensated for by the piezo-electric micro cutting device. Experiments have been carried out on a precision lathe. The characteristics of the surface profile and the cause of the waviness profile have been analyzed and waviness profiles of some cases have been compared with those obtained in experiments. 2. Experimental setup The main spindle of the precision lathe has an aerostatic bearing mounted on a built-in type DC motor and operated above 7 bar air pressure. The guideway is also of the air slide type and the maximum travel of the z-axis is 200 mm. The rapid traverse speed is 3000 mm min⫺1 and feed rate is variable from 0.5 to 3000 mm min⫺1. The body of the machine is a granite bed with air mounting. Air bearing and air slider are mounted on the granite bed in order to isolate them from vibration due to the propagated vibration from the earth and to keep them in a parallel state. The z-axis of the precision lathe is made of ceramics to increase stiffness. A piezo-electric micro cutting device was developed to control submicron depth of cut. This machine is operated in a thermostatic room because machine components, particularly the air bearing spindle and the workpiece, are very sensitive to temperature in ultraprecision cutting. Figure 1 shows a schematic diagram of the waviness compensation system in precision face cutting using the piezo-electric micro cutting device. A personal computer monitors piezo-electric voltage and displacement of the micro cutting device and controls displacement of the micro cutting device with regard to the hysteresis loop of the piezo-electric actuator. Table 1 shows the specification of the experimental system. A high voltage power supply, a differential circuit for extracting piezo-electric voltage, and gap sensor amplifier are internally set in the driver of the piezo-electric micro-depth control system. It is connected to the personal computer by A/D converter and D/A converter. The actual piezo-electric voltage is extracted by a differential circuit of OP-amp. For the first time, table motion error of the x-axis of the precision lathe has been measured by
J.-D. Kim, D.-S. Kim / International Journal of Machine Tools & Manufacture 38 (1998) 1305–1322
1307
Fig. 1. Schematic diagram of waviness compensation system in precision machining using the piezo-electric micro cutting device.
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Table 1 Experimental apparatus Machine tools
Precision CNC lathe Air bearing Air slide Granite bed
Tool Workpiece Gap sensor
Single crystal diamond tool: straight type Al alloy Eddy current type Applied Electronics corp. Model: AEC-5505 Response frequency: 20 kHz Output: 5 V (5 mV m⫺1) Resolution: 0.5 m
A/D converter
Advantech Co. Ltd Model: PCL 812PG Channels: 16 single-ended Resolution 12 bits Conversion speed: 30 kHz max. Accuracy: 0.015% of reading ± 1 bit
System control
IBM-PC 586 compatible
an eddy current type non-contact gap sensor. This error is saved in the personal computer which drives the micro cutting device along an absolute position of the x-axis table. When face cutting is conducted, displacement of the micro cutting device is appropriately controlled in order to ensure that the relative displacement between the workpiece and the tool on the micro cutting device become zero. 3. A piezo-electric micro cutting device Figure 2 shows an overview of the piezo-electric micro cutting device which is designed for mounting on the x-axis of the precision lathe. Displacement of the piezo-electric micro cutting device is measured by a conductance type gap sensor and the piezo-electric actuator is preloaded by the ball and screw bolt. The hinge is based on the parallel spring principle with a design stiffness of 50N m⫺1. The equation of the parallel spring is given [5] Kh =
8Ebt5/2 91/2L2
(1)
where Kh[N m⫺1] is hinge stiffness, E[N m⫺2] is Young’s modulus, b[mm] is width of hinge, r[m] is notch radius, t[m] is thickness between hinge holes and L[mm] is distance between hinges.
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Fig. 2. Overview of piezo-electric micro cutting device. (a)Photograph of piezo-electric micro cutting device; (b) schematic diagram of piezo-electric micro cutting device.
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Figure 3 shows a mechanical and electrical model of the piezo-electric micro cutting device. The characteristics of the piezo-electric actuator and several important variables are listed in Table 2. Generally, the piezo-electric micro actuator is modeled on the lumped second order mechanical system from the linear piezo-electric equation and Newton’s law, and on the first order electrical system from output impedance of driving voltage source and capacitance of piezo-electric actuator [6–10].
Fig. 3. Mechanical and electrical model of piezo-electric actuator. (a) Lumped mechanical model; (b) Equivalent electric circuit of voltage driving piezo-electric element.
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Table 2 Characteristics of piezo-electric actuator and parameters 10 × 10 × 18 (144 layers) 6500 635 × 10⫺12 8 5.5 × 1010 20.794 (N V⫺1) 20 (kg) 20144 (Ns m⫺1) 1.843 × 108(N m⫺1) 6.5 × 10⫺6 (F) 300 (⍀)
Size (mm) Static capacitance (nF) Piezo-electric constant (m V⫺1) Density (g cm⫺3) Elastic constant (N m⫺2) Kf MP DP KPZT Co Ro
The governing equation of the lumped second order mechanical system of the piezo-electric micro cutting device is as follows. MP
冉
冊
c33d 233 d2uP duP + D u = P + 1 + Fl + K P PZT P e dt2 dt ⑀3
(2)
where uP[m] is displacement of the piezo-electric actuator, MP[kg] is equivalent mass of piezoelectric actuator, DP[kg s⫺1] is equivalent damping coefficient of the piezo-electric actuator, KPZT[N m⫺1] is the equivalent stiffness coefficient of the piezo-electric actuator, Pe[N] is equivalent force, Fl[N] is external load, d33[C N⫺1] is piezo-electric charge constant, c33[N m⫺2] is piezoelectric stiffness constant and ⑀3[F m⫺1] is dielectric constant of the piezo-electric actuator. Some variables are represented in detail as following. MP =
c33d33A AL 2 ⑀33A , KPZT = c33A,Pe = Vn(t),Co = 2 8l l l
(3)
where [kg m⫺3] is the density of the piezo-electric actuator, A[m2] is the cross-sectional area of the piezo-electric actuator, l[m] is the length of the piezo-electric actuator and Co[F] is the capacitance of the piezo-electric actuator. The piezo-electric actuator is driven by the voltage source with output impedance, Ro and a high voltage amplifier. Piezo-electric voltage, VP is generated by the external load and affects the input signal. Time delay of voltage source, Vi and actual input voltage, Va occurs. The relationship between the actual input voltage and nominal input voltage is given by 1 Va ⬇ Vi RoCos + 1
(4)
Piezo-electric voltage generated by the external load is represented by Eq. (5). d33Ros VP = Fl RoCos + 1
(5)
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Net voltage, Vn is the sum of actual input voltage, Va and input voltage, VP of the piezoelectric actuator. Vn = Va + VP =
1 d33Ros Vi + F RoCo + 1 RoCos + 1 l
(6)
where, Va[V] is actual input voltage, Vi[V] is nominal input voltage, Vn[V] is net voltage, Vp[V] is piezo-electric voltage and Ro[⍀] is output impedance of the piezo-electric actuator. A complete transfer function, including feedforward compensator and considering force feedback of the displacement is described in Eq. (7). U(s) =
Vi(s) ⫺
Kf(Ts + 1) T 2Mps4 + (T 2DP + 2TMP)s3 + {T 2KPZT + 2TDP + MP + T 2Km(1 + 2P)}s2 + {2KPZTT + DP + TKm(2 + 3P ⫺ PKVP)}s + {KPZT + Km(1 + P ⫺ TPKVI)} T 2(1 + 2P)s2 + T{2 + P(3 ⫺ KVP)}s + {1 + P(1 ⫺ TKVI)} F (s) T 2Ms4 + (T 2DP + 2TMP)s3 + {T 2KPZT + 2TDP + MP + T 2Km(1 + 2P)}s2 + {2KPZTT + DP + TKm(2 + 3P ⫺ PKVP)}s + {KPZT + Km(1 + P ⫺ TPKVI)} l
(7)
where KVP is piezo-electric voltage feedback proportional gain, KVI is piezo-electric voltage feedback internal gain and Km[N m⫺1] is coefficient of force feedback. Equivalent force constant Kf, time constants T and P are as follows. Kf =
c33d33A 8K 2f ,T = RoCo,P = 2 l CoKPZT
(8)
The results of simulation of Eq. (7) are shown in Fig. 4. The simulation has been performed by MATLAB with simulink. An external mass located at the 3 m point from the start position of the piezo actuator is used as the external disturbance. From the root locus, the disturbance cannot affect the stability of the system but the reference input produces an unstable response for large Km values. The steady state error disappears. Experiments have been conducted in order to compare performances of position control by the piezo-electric voltage and by the displacement feedback. Step responses of the piezo-electric actuator to a 38 V step input voltage are shown in Fig. 5. Figures 5(a) and (b) show the result of the displacement feedback control case. Figures 5(c) and (d) show the results of the piezo-electric voltage feedback control case. From comparison with simulation results, the displacement feedback shows a five times longer settling time and the piezo-electric voltage feedback case three times longer. However, the transient motion of the real system is better than that of the simulation because of the simulation’s underestimate of damping coefficient from the hysteresis loop and friction of the vertical feeding unit. The late settling time of the real system is due to overestimation of the output impedance of the amplifier and capacitance of piezo-electric actuator. The response of the piezo-electric voltage feedback system is faster than that of the displacement feedback case due to the effect of input voltage feedback. A steady state error has been reduced by compensation of piezo-electric voltage with PI controller. Piezo-electric voltage feedback control is faster and more stable than that of the displacement feedback control using a gap sensor, and it does not need an expensive displacement sensor.
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Fig. 4. Simulation of piezo-electric micro cutting device (displacement feedback considered).(a) Root locus to disturbance input; (b) root locus to reference input; (c) step response with compensator and considered feedback.
4. Results and discussions Face cutting of Al alloy by a single crystal diamond tool on the precision lathe has been carried out. Figure 6 shows the comparison of waviness profiles of Al alloy between no compensation and compensation face cutting. In the compensation face cutting, constant depth of cut is given
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Fig. 5. Step response of piezo-electric micro cutting device. (a)Displacement feedback Vi = 38 V Fl = 3 N; (b) Piezoelectric voltage feedback Vi = 38 V Fl = 3 N.
by the piezo-electric micro cutting device overcoming external disturbances such as table motion error. The piezo-electric micro cutting device is controlled by the piezo-electric voltage feedback. The peak-to-valley waviness is 1.3 m when the piezo-electric micro cutting device compensation is not used. If the depth of cut is compensated for by the piezo-electric micro cutting device, the peak-to-valley waviness is decreased to 0.7 m. It is considered that the remaining waviness is
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Fig. 6. Compariness of waviness profiles for AL alloy between no compensation and compensation face cutting. (a) Waviness profile without the piezo-electric micro cutting device; (b) waviness profile after piezo-electric micro cutting device compensation.
due to the imperfect compensation of the table error, the cutting mechanism and the thermal displacement of the tool. Figure 7 shows a schematic diagram of the surface waviness profile, in some ideal cases, by piezo-electric micro cutting device. Case 1 is the initial waviness profile with a peak-to-valley of 4 m generated by the piezo-electric cutting device. The micro cutting device is driven by a personal computer and a high voltage amplifier. The depth of the micro cutting device is determined by the elongation of the piezo-electric material which is preloaded by the ball and screw and controlled by the program on the personal computer. The depth of cut given by the precision
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Fig. 7. Schematic diagram of surface waviness profile in ideal cases by piezo-electric micro cutting device. (a) Case 1: initial waviness profile; (b) case 2: depth of cut = 2 m, piezo depth = 2 m, tool path = triangular wave; (c)case 3: depth of cut = 0 m, piezo depth = 4 m, tool path = straight.
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lathe controller is 2 m and the depth of cut given by the piezo-electric micro cutting device is also 2 m in case 2. The tool path is triangular, with wavelength 2.0 mm and peak-to-valley 2 m. In case 3, the depth of cut given by the precision lathe controller is zero, the depth of cut given by the piezo-electric micro cutting device is 4 m and the tool path is straight. The theoretical peak-to-valley results are 2 m and zero in cases 2 and 3, respectively. Figure 8 shows the waviness profile and wavelength spectral density of the machined Al alloy for case 1 at feed rate 20 mm min⫺1, the rotation speed of the main spindle 800 r.p.m. with the
Fig. 8. Waviness profile and wavelength spectral density in face cutting: case 1. (a) Waviness profile; (b) wave spectral density of waviness profile, feed rate = 20 mm min ⫺ 1, cutting speed = 800 r.p.m., workpiece = Al alloy, triangular tool path, single crystal diamond tool: straight type.
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single crystal diamond tool. For a machined disk type workpiece, the surface waviness profile has been measured in eight directions and the results were similar. The main wavelength can be derived from the wave spectral density graph. Peak-to-valley of the surface waviness is 3.3 m and the wavelength, i.e. the period of the waviness, is 2.05 mm. The waviness appears to have a constant period. Figure 9 shows the waviness profile and wavelength spectral density of the machined Al alloy
Fig. 9. Waviness profile and wavelength spectral density in face cutting: case 2. (a) Waviness profile; (b) wave spectral density of waviness profile, feed rate = 20 mm min ⫺ 1, cutting speed = 800 r.p.m., workpiece = Al alloy, depth of cut = 0.002 mm, piezo depth of cut = 0.002 mm, triangular tool path, single crystal diamond tool: straight type.
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for case 2 at feed rate 20 mm min⫺1, with the rotation speed of the main spindle of 800 r.p.m. with the single crystal diamond tool. The depth of cut given by the precision lathe controller is 2 m, the depth of cut given by the piezo-electric micro cutting device is also 2 m and the tool path is triangular. Peak-to-valley of the surface waviness is 1.3 m and the wavelength, i.e., the period of the waviness is 2.05 mm. The peak-to-valley is reduced with the same wavelength of initial waviness profile as was expected by the theoretical value as shown in Fig. 7. Figure 10 shows the waviness profile and wavelength spectral density of the machined Al alloy for case 3 at feed rate 20 mm min⫺1 and rotation speed of the main spindle of 800 r.p.m. with the single crystal diamond tool. The depth of cut given by the precision lathe controller is zero, the depth of cut given by the piezo-electric micro cutting device is 4 m and the tool path is straight. Peak-to-valley of the surface waviness is 0.3 m and the wavelength, i.e. the period of the waviness is 8.33 mm. The peak-to-valley should be zero theoretically but the peak-to valley is 0.3 m due to the steady state error of the response of the piezo-electrical micro cutting device and the cutting mechanism between the tool and the workpiece. By using the piezo-electric micro cutting device, the initial waviness profile almost completely disappeared in case 3. The peak-tovalley of the surface waviness profile has been decreased from 3.3 to 0.3 m using the micro cutting device. The peak-to-valley of the waviness profile between theory and experiments has been compared as shown in Fig. 11. The experimental results tend to correspond with the expected value and the piezo-electrical micro cutting device is reliable for control of micrometer depth and compensation for the waviness profile. Figure 12 shows the wavelength of the waviness profile of the theoretical value and the experimental results. The experimental results are similar to the theoretical values in case 2 and 3 so the piezo-electrical micro cutting device is effective in compensating for the waviness profile in precision face cutting. In case 3, the ideal wavelength is infinite and the experimental result is 8.33 mm. The peak-to-valley is 0.3 m and this wave profile has a 8.33 mm wavelength due to the above reasons.
5. Conclusions
1. A micro cutting device with piezo-electric actuator was developed which was able to control depth of cut precisely and to compensate the waviness of the surface profile in face cutting. 2. It has been demonstrated that a piezo-electric voltage feedback system which has been constructed to drive the piezo-electric micro cutting device has a better step response than a displacement feedback system. 3. When the compensation system was not used, the surface maximum waviness was 3.3 m. However, the peak-to-valley of the surface waviness profile has been decreased to 0.3 m in one particular case, by using the piezo-electric micro cutting device with the piezo-electric voltage feedback 4. A theory for the operation of the actuator was proposed which was validated by experimental results of precision face cutting.
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Fig. 10. Waviness profile and wavelength spectral density in face cutting: case 3. (a) Waviness profile; (b) wave spectral density of waviness profile, feed rate = 20 mm min ⫺ 1, cutting speed = 800 r.p.m., workpiece = Al alloy, depth of cut = 0 mm, piezo depth of cut = 0.004 mm, straight tool path, single crystal diamond tool: straight type.
References [1] Y. Furukawa, N. Moronuki, Effect of material properties on ultra precise cutting process, Annals of the CIRP 37 (1988) 113–116. [2] T. Sugano, K. Takeuchi, T. Goto, Y. Yoshida, Diamond turning of an aluminum alloy for mirror, Annals of the CIRP 36 (1987) 17–20.
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Fig. 11. Comparison of peak-to-valley of waviness profile between theory and experiments.Case 1: initial waviness profile. Case 2: depth of cut 2 m, piezo depth 2 m, triangular tool path. Case 3: depth of cut 0 m, piezo depth 4 m, straight tool path.
Fig. 12. Comparison of wavelength of waviness profile between theory and experiments. Case 1: initial waviness profile. Case 2: depth of cut 2 m, piezo depth 2 m, triangular tool path. Case 3: depth of cut 0 m, piezo depth 4 m, straight tool path.
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[3] J.D. Kim, E.B. Lee, D.H. Hyun, A study on the modelling of tool motion and high-accuracy surface generation by the use of cutting force signal, Journal of Materials Processing Technology 47 (1994) 45–62. [4] Kim, J.D. and Kim, D.S., Surface characteristics of magnetic disk machined by single crystal diamond tool in ultraprecision lathe. 3rd International Conference on Precision Surface Finishing and Burr Technology, Seoul, 1994, pp. 428–441. [5] Y. Hara, A new micro-cutting device with high stiffness and resolution, Annals of the CIRP 39 (1) (1990) 375–388. [6] J.D. Kim, S.R. Nam, Development of a micro-depth control system for an ultra-precision lathe using a piezoelectric actuator, International Journal of Machine Tools and Manufacture 37 (4) (1997) 495–509. [7] T. Zhong, Z. Nakagawa, Development of a micro-displacement table for ultra-precision machining and grinding for curved surfaces by the use of it, JSPE 26 (2) (1992) 102–110. [8] H.S. Tzou, Design of a piezoelectric exciter/actuator for micro-displacement control: theory and experiment, Precision Engineering 13 (2) (1991) 104–110. [9] Lee, Seok-Koo, Motion analysis and control of translation device driven by piezoelectric actuator. M.S. thesis, Korea Advanced Institute of Science and Technology (in Korean), 1993. [10] C.V. Newcomb, Improving the linearity of piezoelectric ceramic actuators, Electronic Letters 11 (1982) 442–444.