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Relevant topics in wire electrical discharge machining control
Journal of Materials Processing Technology 149 (2004) 147–151
Relevant topics in wire electrical discharge machining control Friedhelm Altpeter∗ , Roberto Perez R&D Charmilles Technologies S.A., 1217 Meyrin 1, Switzerland Accepted 16 October 2003
Abstract A relationship between the dynamics of the wire electrode and the state of the art in wire electrical discharge machining (WEDM) control is established through wire modeling, listing the control issues related to WEDM and providing a catalogue of corresponding solutions. The results are to be used for identifying promising R&D directions in terms of customer convenience, and set up cost reduction by an improved process mastering. © 2004 Elsevier B.V. All rights reserved. Keywords: Wire electrical discharge machining control; Quasi-static longitudinal motion; Viscous damping
1. Introduction Compared to die sinking electrical discharge machining, WEDM is a technology with increasing economic importance in terms of units sold per year. This fact motivates the present work that consists of a survey on wire modeling, implications for control and a catalogue of solutions. The objective of the present study is the establishment of a relationship between the dynamics of the wire electrode and the state of the art in WEDM control. These results are to be used for identifying promising R&D directions in terms of customer convenience, and set up cost reduction by an improved process mastering. The state of the art in wire modeling and WEDM control is well established in scientific articles and patent publications. The contribution of the present paper is an aggregation of this information and conclusions concerning topics with relevant marginal potential for improving machining performance.
2. Wire modeling Certain roots of WEDM control issues are linked with the mechanical properties of the wire electrode. In the following, the partial differential equations are recalled describing a wire, clamped between two guides. This model can be simplified by assuming quasi-static longitudinal motion, an approach that retains the coupling between axial ∗ Corresponding author. Tel.: +41-22-783-3620; fax: +41-22-783-3790. E-mail address: [email protected] (F. Altpeter).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2003.10.033
and transversal motion. Finally, the frequency response of the transversal subsystem to a spark is illustrated to show steady-state deformation and vibration modes. The equations of motion for a moving string have been established [1], starting from kinetic and potential energies, and applying Hamilton’s variation principle. Furthermore, the additional terms, required for modeling the flexural stiffness of an Euler–Bernoulli beam, have been discussed [2,3]. The equations of motion can be found by means of Hamilton’s variation principle of virtual work. This provides for axial coordinates u and transversal coordinates w the equations of motion ρA
∂2 u ∂((∂u/∂x) + (1/2)(∂w/∂x)2 ) − EA =0 ∂x ∂t 2
∂2 w ∂(((∂u/∂x) + (1/2)(∂w/∂x)2 )(∂w/∂x)) − EA ∂x ∂t 2 ∂4 w + EI 4 = fw ∂x
(1)
ρA
(2)
where the boundary conditions for a wire clamped at x1 = 0 and x2 = l are EA
∂u (xi ) = Fai ∂x
(3)
w(xi ) = yi (t)
(4)
∂w (xi ) = 0 ∂x
(5)
This represents a wire that is clamped at both ends by the guides; the distance between the fixtures is l. The axial loads
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are denoted by Fa in N; the transversal load distribution is fw (x, t) in N/m. Wire displacement is written in local coordinates u(x) and w(x), both in m. ρ is mass per unit volume and A denotes the cross-section area of the wire, E is the modulus of elasticity, and I denotes moment of inertia. The axial (1) and transverse (2) equations of motion are nonlinearly coupled in the field equations [3]. Linearization can eliminate this coupling, but then axial load Fa does not influence transverse vibration, which is far from reality. Assumption of quasi-static longitudinal motion retains the coupling and simplifies the equations. In the quasi-static model, longitudinal waves propagate instantaneously; axial boundary force Fai acts quasi-immediately all over the wire. These assumptions yield into the axial 1 l ∂w 2 EA dx (6) fu = 2 u(l, t) + 2 0 ∂x l and the transversal fw = ρA
∂2 w ∂2 w ∂4 w − lfu (t) 2 + EI 4 2 ∂t ∂x ∂x
(7)
partial differential equation of motion, with the overall block diagram, shown in Fig. 1. Axial load at the wire ends is Fa = Fai = lfu (t). Viscous damping kv w, ˙ related to wire motion within the dielectric liquid, is taken into account through the specification of the force fw = fEDM − kv w, ˙ acting on the wire in transversal direction. When analyzing the block diagram, it appears that the forces related to axial deformation and those induced by the machining process drive the transversal motion. The workpiece is displaced relative to the wire at path feed rate r˙ influencing the EDM characteristics. The frequency response of the transversal subsystem to a spark is illustrated in Fig. 2. When analyzing the characteristics for various spark locations, along the wire, it results that locations and frequencies of maxima remain unchanged. When viscous damping is introduced, maximum amplitude of the resonance peaks is limited, providing the same time limits to geometrical accuracy that can be achieved on the workpiece for a linear cutting operation.
Fig. 2. Frequency response of the wire for a spark located at x/l = 10%.
When control is addressed, writing the system as an aggregate of interconnected subsystems, like shown in Fig. 1, suggests a passivity (dissipativity)-based stability analysis [4,5] which may lead faster to relevant results than an approach, based on the total energy. The concept of a passivity-based stability analysis for large-scale systems (like for example the electric power network of a country) that was worked out in the 1960s is that the feedback interconnection of systems that dissipate energy is stable, provided some restrictions on phase shift. In the context of the wire in EDM, these considerations request for sufficient viscous damping, provided by the dielectric liquid through the negative feedback gain kv . Otherwise, the wire position could not be stabilized locally and considerable geometrical machining errors would result. In fact, two issues are to be solved: (i) damping of wire vibrations at high frequency and (ii) slackness control at low frequency. Additional features that are required by WEDM are path feed rate control and machining monitoring. The issue of combined vibration and federate control is generally handled by a singular perturbation approach, separating fast vibrations from slow machining motions.
3. Wire vibrations handling
Fig. 1. Block diagram: coupling between axial and transversal coordinates.
Transversal wire vibrations lead to a limitation of geometrical precision, but may improve machining efficiency through homogeneous spark location distribution. Furthermore, there is an important risk with path feed rate control owing to the separation between the actuator (wire guides) and the sensor (gap width). It has been emphasized [6] that non-collocated actuator–sensor systems lead easily to an unstable closed loop. Therefore, even simple velocity feedback is difficult, as soon as required closed-loop bandwidth ωcl is larger than the first eigenfrequency ω1 of system (7): π Fa ωk = k (8) l ρA
F. Altpeter, R. Perez / Journal of Materials Processing Technology 149 (2004) 147–151
with boundary conditions (4)–(5). Because the EDM control problem implies inherently non-collocation of sensing and actuation, increased natural damping kv w ˙ through viscous dielectrics is preferable to active vibration control. Furthermore, wire preload Fa is chosen large enough to meet ωcl ω1 . Experimental validation for the values of the first eigenfrequency f1 = ω1 /2π, according to Eq. (8), has been provided recently [7]. Therein, the characteristics for various thin wires have been measured and agreement with theoretical data, corresponding to relation (8), has been observed. A comparison of thin wires for micro-machining and the standard 0.25 mm wire for roughing cut can be done on the basis of relationship (8). It results that machining settings are always chosen such that the first eigenfrequency is in the range of 1 kHz and above in any used configuration of wire diameter and preload (for thin wires 1–5 N; for standard wires about 10 N). Closed-loop bandwidth of industrial wire guide drives is rarely exceeding 100 Hz; therefore, active vibration damping is not feasible even through guide control enhanced, for example, by a notch filter. Up to now, engineers at machine manufacturers were forced to hope that viscous damping by the surrounding dielectric liquid is sufficient to master wire vibrations that reduce machining precision. No references are known to the authors that would make use, in the context of WEDM, of some additional actuator that could introduce active damping, such as known for high buildings, large telescopes or large bridges. Therefore, the control concept applied in practice is a singular perturbation approach [8] that consists of a two-time-scale approach, a fast asymptotically stable subsystem (damping of wire vibrations) and a slow manifold handled by the main controller (path feed rate control). Damping of wire vibrations is commonly guaranteed by the viscous damping, provided through the dielectric liquid. In general, deionized water is preferred in WEDM; however, it is suggested for certain brands to use oil for high-precision machining. In principle, particular additives could be considered as well, providing electro-rheologic behavior [9]. The authors are not aware that the idea has been applied to wire damping; the essential is that such additives would enable increased viscous damping in the machining area while conserving excellent fluidity within the conditioning system (pipes, filters, pumps, etc.). Nevertheless, it has also been reported that vibrations may be beneficial for machining efficiency. High-frequency (open-loop) excitation, proposed [10] for wire EDM, pumps energy selectively into a vibration harmonic of orders 5–10, which apparently improves machining conditions by mechanical arc prevention and better flushing. Therein, transversal wire vibrations modify continuously the spatial dielectric breakdown probability, which leads to desired sparking, uniformly distributed all over the wire length.
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4. Dealing with wire slackness Dealing with wire slackness is the main issue to be solved for achieving good geometrical performance with wire EDM. Today, numerous publications may be found in the literature on mastering wire slackness; it would be surprising that fundamentally new solutions are proposed during the next years. While appropriate machining setting may be sufficient for finishing and surfacing cut, roughing cut requires particular strategies for finding a good trade off between machining speed (material removal) and geometrical precision. The forces, acting on a wire in EDM, have been analyzed previously [11–13]; wire preload, plasma pressure, electromagnetism, electrostatics, flushing and damping have all been considered. Furthermore, detailed models for the plasma have been published [14]. Wire slackness results from establishing a position that equilibrates attractive (electrostatic and electromagnetic) and repulsive (plasma pressure) forces, such as illustrated in Fig. 3 for finishing cut settings. While the equilibrium for a straight wire may be achieved in finishing and surfacing with appropriate machining settings and tool offset, wire deformation is inevitable in roughing cut as soon as reasonable machining efficiency (speed) is required. Very simple experimental observation shows that flushing dominates all other causes for roughing cut; the resulting flexure is in path direction. The state of the art for dealing with wire slackness can be found in patents that belong basically to International Patent Classification B23H7. In the past, all major machine manufacturers have addressed the topic [15–22]. When tracking publications for subclasses B23H7/02, 04 and 06, a decreasing activity is observed: while 730 patents are referenced for the publication period 1990–1992 in the database [23] only 320 are found for 2000–2002. According to Eq. (7) and as stated previously [24], wire deflection is inverse proportional to axial preload Fa . Hence, increasing preload is the most basic approach to handling wire slackness, such as discussed recently [25]. The issue is to build a wire with high tensile strength (wire core) while conserving excellent conductivity in the intermediate layer
Fig. 3. Attractive and repulsive forces on the wire during finishing cut.
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(for the machining current up to 600 A) and good EDM properties (achieved by appropriate coatings and superficial layers). The concepts of speed variation as a function of path curvature, slackness recovery at block transitions, discharge energy variations as a function of path curvature, etc. have been introduced very early [15] and are now public domain. Comparable claims and extensions to variations in flushing pressure and software particularities have been proposed later [16–20], but still all using an open-loop approach. In this context, open loop means ‘based on the reference trajectory for the wire tool’; this is in contrast to the online wire slackness measurement system [21,22] that is using an orthogonal pair of position sensors. A conceptually different but also a feed forward approach is an offline correction of the wire path [26]. This solution provides a certain independence of the user from the EDM builder, which may be an advantage for research institutions. Yet, customers prefer to buy a complete solution and patches that modify the ISO codes are rarely welcomed. The major disadvantage is that a correction of the wire path cannot solve the problem of wire slackness, the geometric imperfection remains and additional finishing cuts are required to achieve desired precision.
5. Path feed rate control and machining monitoring Traditional machining, like milling or turning, and EDM differ principally in the generation of the path feed rate. Open-loop control is generally used in traditional machining: the path feed rate is a constant parameter or depends on position only. In EDM, path feed rate is calculated online, ¯ td : a closed control loop based on actual gap condition U, is formed by the servo system, the wire dynamics, the material removal, the sparking process and the path feed rate controller. It has been suggested [27] that wire control should include actions both on path feed and on pause time t0 . This control is to be based on reference and measured sparking frequency, as well as the rate of ‘bad’ sparks. The classification into ‘good’ and ‘bad’ sparks is achieved by a specialized circuitry that uses the machining signals gap voltage U¯ and ignition delay time td . The control algorithm evaluates the change in controls as a result of a fuzzy algorithm. The power consumption and short spark ratio are chosen [28] as controlled variables, acting on cutting speed and off-time t0 through an adaptive fuzzy control algorithm. The short spark ratio is evaluated by means of a discrimination system. The motivation therein to choose this combination of controller inputs, instead of generally used mean gap voltage ¯ is to use most relevant signals for sparking condition. U, In order to prevent wire breakage by exaggerated heating, it has been suggested [29] to control off-time t0 based on actual sparking frequency. The proposed algorithm is simple integral control. A standard identification algorithm with
Fig. 4. Integrated model for wire electro-discharge machining.
forgetting factor provides information on actual workpiece height. This estimation serves in a second step to determine some other machining settings. A fundamentally new concept has been suggested recently [30] with a device for counting ‘normal’ and ‘abnormal’ sparks, where a spark is considered to be abnormal if its ignition delay time td is below a certain level. Based on the machining settings, the percentage of stability (making use of the change in abnormal spark rate) and the percentage of efficiency are evaluated. These two inputs are classified into categories and entered into a decision table for providing a change in feed rate, an approach that resembles fuzzy logic. There are two novelties: (i) making use of a completely digital implementation of the control loop and (ii) providing some hints for an appropriate choice of the entries in the decision table that assure a stable control loop. The above-mentioned approaches can be assembled into an integrated model for wire electro-discharge machining, illustrated in Fig. 4. The wire EDM process is controlled within two loops: A ‘fast’ path feed control loop and a ‘slow’ monitoring loop. Relevant machining settings are the spark duration te , the off-time t0 , the ignition voltage Utd , reference mean gap voltage U¯ ref and reference ignition delay time td,ref . The ignition delay time td and the arc voltage Ute are resulting from the gap properties that depend on the gap size δ and contamination of the dielectric liquid by machining debris. The objective of the path feed control is to maintain the wire around a stable equilibrium distance where attractive and repulsive forces balance. A feedback loop is required to take into account for variations on material removal, as well as for changes of the workpiece height. Furthermore, this gap control is useful as well to increase the basin (attractive domain, according to Fig. 3) of the equilibrium distance. In industrial practice, closed-loop bandwidth of the path feed control loop is chosen around 10–100 Hz, which requires excellent servo drive performance. Issues that remain still open for path feed rate control are the handling of the process inherent noise and the separation of the ‘very fast’ wire vibrations from the ‘fast’ path
F. Altpeter, R. Perez / Journal of Materials Processing Technology 149 (2004) 147–151
feed control loop. The noise in the EDM process is related to the stochastic properties of the dielectric breakdown process; this perturbation appears as a white noise, which is amplified, by the control loop. Furthermore, a digital implementation like in Ref. [30] presents the risk that Shannon’s theorem of sampling is not verified and high-frequency noise is mapped into low frequency; the result of these phantom signals is the appearance of undesirable oscillations in the path feed rate control loop. The principal task of monitoring is reducing the risk of wire breakage, and therefore the amount of time lost for rethreading. In general the off-time t0 , mean gap voltage U¯ ref , etc. are the principal manipulated variables produced by the machining supervision algorithm; the inputs are a cluster of signals provided by the EDM monitor.
6. Conclusion A survey on wire modeling and control of WEDM has been provided above. While numerous solutions have been proposed in the past for mastering wire slackness, very little publications are dealing with issues like vibration damping and amplification of process randomness.
References [1] F.R. Archibald, A.G. Emslie, The vibration of a string having a uniform motion along its length, J. Appl. Mech. 25 (1958) 347–348. [2] C.D. Mote, A study of band saw vibrations, J. Franklin Inst. 279 (6) (1965) 430–444. [3] C.D. Rahn, C.D. Mote, Axial force stabilization of transverse vibration in pinned and clamped beams, Trans. ASME: J. Dyn. Syst. Meas. Control 118 (1996) 379–380. [4] C.A. Desoer, M. Vidyasagar, Feedback Systems: Input–Output Properties, Academic Press, New York, 1975 (replica reprint). [5] R. Sepulchre, M. Jankovic, P.V. Kokotovic, Constructive Nonlinear Control, Springer-Verlag, London, 1997. [6] B. Yang, C.D. Mote, Active vibration control of the axially moving string in the S domain, Trans. ASME: J. Appl. Mech. 58 (1991) 189–196. [7] T. Nöthe, Funkenerosive Mikrobearbeitung von Stahl und Hartmetall durch Schneiden mit dünnen Drähten, Ph.D. Thesis, Rheinisch-Westfälische Technische Hochschule Aachen, 2000. [8] P.V. Kokotovic, H.K. Khalil, J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986. [9] T. Aoyama, I. Inasaki, Application of electrorheological fluid dampers to machine tool elements, Ann. CIRP 46 (1997) 309–312. [10] Z.N. Guo, T.C. Lee, T.M. Yue, W.S. Lau, A study of ultrasonic-aided wire electrical discharge machining, J. Mater. Process. Technol. 63 (1997) 823–828. [11] R. Siegel, Funkenerosives Feinstschneiden: Verfahrenseinflüsse auf die Oberflächen- und Randzonenausbildung, Ph.D. Thesis, Rheinisch-Westfälische Technische Hochschule Aachen, 1994.
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[12] H. Obara, T. Ishizu, T. Ohsumi, Y. Iwata, Simulation of wire EDM, in: Proceedings of the 12th International Symposium for Electromachining, VDI Berichte 1405, Aachen, Germany, 1998, pp. 99–108. [13] C. Arunachalam, M. Aulia, B. Bozkurt, P.T. Eubank, Wire vibration, bowing and breakage in wire EDM, in: Proceedings of the 12th International Symposium for Electromachining, VDI Berichte 1405, Aachen, Germany, 1998, pp. 109–118. [14] P.T. Eubank, M.R. Patel, M.A. Barrufet, B. Bozkurt, Theoretical models of the electrical discharge machining process III: the variable mass, cylindrical plasma model, J. Appl. Phys. 73 (11) (1993) 7900– 7909. [15] J. Jänicke, G.-A. Marendaz, Atelier des Charmilles S.A., Electrical discharge machining by means of a wire electrode, US Patent 4 081 652 (1978). [16] H. Obara, Fanuc Ltd., Method of controlling wire-cut electric discharge contour machines errors, US Patent 4 518 842 (1985). [17] J.M. Bosson, J. Odriozola, G.-A. Marendaz, D. Blodin, Charmilles Technologies S.A., Device and process for controlling the variation of a parameter in EDM machining with an electrode wire, US Patent 5 117 082 (1992). [18] T. Magara, H. Yamada, Mitsubishi Denki Kabushiki, Kaisha, Method and apparatus for wire-cutting electrical discharge machining of a corner, US Patent 5 418 344 (1995). [19] S. Izumiya, T. Tsutsumi, I. Higashi, Y. Kita, T. Sakuragawa, Fanuc Ltd., Method for controlling a wire-cut electric discharge machine, US Patent 5 504 294 (1996). [20] Y. Ishibashi, M. Iwai, Brother Kogyo Kabushiki Kaisha, Traveling wire electric discharge machining apparatus capable of workpiece location determining and energy level adjustment, US Patent 5 573 681 (1996). [21] D.F. Dauw, I. Beltrami, High-precision wire-EDM by online wire positioning control, Ann. CIRP 43 (1994) 193–197. [22] R. Derighetti, S. Dresti, Agie S.A., Method and apparatus for wire erosion, US Patent 5 824 986 (1998). [23] Europe’s network of patent databases. http://www.espacenet.com/. [24] D.F. Dauw, I. Beltrami, High-precision wire-EDM by online wire positioning control, Ann. CIRP 43 (1994) 193–197. [25] B. Schacht, J.-P. Kruth, B. Lauwers, Design of a coating for a high tensile strength steel electrode for wire electrical discharge machining, in: Proceedings of the 36th CIRP International Seminar on Manufacturing Systems, June 2003. [26] Z. Jian, C.Y. Yu, A new control strategy for sharp corner cutting in wire EDM, in: Proceedings of the 10th International Symposium on Electromachining (ISEM X), Magdeburg, Germany, 1992, pp. 353–358. [27] M.T. Yan, Y.S. Liao, Adaptive control of WEDM process using the fuzzy control strategy, in: Proceedings of the 11th International Symposium on Electromachining, Lausanne, Switzerland, April 1995, pp. 343–352. [28] Y.S. Liao, J.C. Woo, A new fuzzy control system for the adaptive control of WEDM process, in: Proceedings of the 12th International Symposium on Electromachining, VDI Berichte 1405, Aachen, Germany, 1998, pp. 193–202. [29] K.P. Rajurkar, W.M. Wang, J.A. McGeough, WEDM identification and adaptive control for variable-height components, Ann. CIRP 43 (1994) 199–202. [30] H.-P. Li, H.-J. Chuang, T.-P. Shieh, Method and apparatus for optimal feed rate control on wire-cutting electric discharge machine WEDM, US Patent 5 689 427 (1997).
Relevant topics in wire electrical discharge machining control Friedhelm Altpeter∗ , Roberto Perez R&D Charmilles Technologies S.A., 1217 Meyrin 1, Switzerland Accepted 16 October 2003
Abstract A relationship between the dynamics of the wire electrode and the state of the art in wire electrical discharge machining (WEDM) control is established through wire modeling, listing the control issues related to WEDM and providing a catalogue of corresponding solutions. The results are to be used for identifying promising R&D directions in terms of customer convenience, and set up cost reduction by an improved process mastering. © 2004 Elsevier B.V. All rights reserved. Keywords: Wire electrical discharge machining control; Quasi-static longitudinal motion; Viscous damping
1. Introduction Compared to die sinking electrical discharge machining, WEDM is a technology with increasing economic importance in terms of units sold per year. This fact motivates the present work that consists of a survey on wire modeling, implications for control and a catalogue of solutions. The objective of the present study is the establishment of a relationship between the dynamics of the wire electrode and the state of the art in WEDM control. These results are to be used for identifying promising R&D directions in terms of customer convenience, and set up cost reduction by an improved process mastering. The state of the art in wire modeling and WEDM control is well established in scientific articles and patent publications. The contribution of the present paper is an aggregation of this information and conclusions concerning topics with relevant marginal potential for improving machining performance.
2. Wire modeling Certain roots of WEDM control issues are linked with the mechanical properties of the wire electrode. In the following, the partial differential equations are recalled describing a wire, clamped between two guides. This model can be simplified by assuming quasi-static longitudinal motion, an approach that retains the coupling between axial ∗ Corresponding author. Tel.: +41-22-783-3620; fax: +41-22-783-3790. E-mail address: [email protected] (F. Altpeter).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2003.10.033
and transversal motion. Finally, the frequency response of the transversal subsystem to a spark is illustrated to show steady-state deformation and vibration modes. The equations of motion for a moving string have been established [1], starting from kinetic and potential energies, and applying Hamilton’s variation principle. Furthermore, the additional terms, required for modeling the flexural stiffness of an Euler–Bernoulli beam, have been discussed [2,3]. The equations of motion can be found by means of Hamilton’s variation principle of virtual work. This provides for axial coordinates u and transversal coordinates w the equations of motion ρA
∂2 u ∂((∂u/∂x) + (1/2)(∂w/∂x)2 ) − EA =0 ∂x ∂t 2
∂2 w ∂(((∂u/∂x) + (1/2)(∂w/∂x)2 )(∂w/∂x)) − EA ∂x ∂t 2 ∂4 w + EI 4 = fw ∂x
(1)
ρA
(2)
where the boundary conditions for a wire clamped at x1 = 0 and x2 = l are EA
∂u (xi ) = Fai ∂x
(3)
w(xi ) = yi (t)
(4)
∂w (xi ) = 0 ∂x
(5)
This represents a wire that is clamped at both ends by the guides; the distance between the fixtures is l. The axial loads
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F. Altpeter, R. Perez / Journal of Materials Processing Technology 149 (2004) 147–151
are denoted by Fa in N; the transversal load distribution is fw (x, t) in N/m. Wire displacement is written in local coordinates u(x) and w(x), both in m. ρ is mass per unit volume and A denotes the cross-section area of the wire, E is the modulus of elasticity, and I denotes moment of inertia. The axial (1) and transverse (2) equations of motion are nonlinearly coupled in the field equations [3]. Linearization can eliminate this coupling, but then axial load Fa does not influence transverse vibration, which is far from reality. Assumption of quasi-static longitudinal motion retains the coupling and simplifies the equations. In the quasi-static model, longitudinal waves propagate instantaneously; axial boundary force Fai acts quasi-immediately all over the wire. These assumptions yield into the axial 1 l ∂w 2 EA dx (6) fu = 2 u(l, t) + 2 0 ∂x l and the transversal fw = ρA
∂2 w ∂2 w ∂4 w − lfu (t) 2 + EI 4 2 ∂t ∂x ∂x
(7)
partial differential equation of motion, with the overall block diagram, shown in Fig. 1. Axial load at the wire ends is Fa = Fai = lfu (t). Viscous damping kv w, ˙ related to wire motion within the dielectric liquid, is taken into account through the specification of the force fw = fEDM − kv w, ˙ acting on the wire in transversal direction. When analyzing the block diagram, it appears that the forces related to axial deformation and those induced by the machining process drive the transversal motion. The workpiece is displaced relative to the wire at path feed rate r˙ influencing the EDM characteristics. The frequency response of the transversal subsystem to a spark is illustrated in Fig. 2. When analyzing the characteristics for various spark locations, along the wire, it results that locations and frequencies of maxima remain unchanged. When viscous damping is introduced, maximum amplitude of the resonance peaks is limited, providing the same time limits to geometrical accuracy that can be achieved on the workpiece for a linear cutting operation.
Fig. 2. Frequency response of the wire for a spark located at x/l = 10%.
When control is addressed, writing the system as an aggregate of interconnected subsystems, like shown in Fig. 1, suggests a passivity (dissipativity)-based stability analysis [4,5] which may lead faster to relevant results than an approach, based on the total energy. The concept of a passivity-based stability analysis for large-scale systems (like for example the electric power network of a country) that was worked out in the 1960s is that the feedback interconnection of systems that dissipate energy is stable, provided some restrictions on phase shift. In the context of the wire in EDM, these considerations request for sufficient viscous damping, provided by the dielectric liquid through the negative feedback gain kv . Otherwise, the wire position could not be stabilized locally and considerable geometrical machining errors would result. In fact, two issues are to be solved: (i) damping of wire vibrations at high frequency and (ii) slackness control at low frequency. Additional features that are required by WEDM are path feed rate control and machining monitoring. The issue of combined vibration and federate control is generally handled by a singular perturbation approach, separating fast vibrations from slow machining motions.
3. Wire vibrations handling
Fig. 1. Block diagram: coupling between axial and transversal coordinates.
Transversal wire vibrations lead to a limitation of geometrical precision, but may improve machining efficiency through homogeneous spark location distribution. Furthermore, there is an important risk with path feed rate control owing to the separation between the actuator (wire guides) and the sensor (gap width). It has been emphasized [6] that non-collocated actuator–sensor systems lead easily to an unstable closed loop. Therefore, even simple velocity feedback is difficult, as soon as required closed-loop bandwidth ωcl is larger than the first eigenfrequency ω1 of system (7): π Fa ωk = k (8) l ρA
F. Altpeter, R. Perez / Journal of Materials Processing Technology 149 (2004) 147–151
with boundary conditions (4)–(5). Because the EDM control problem implies inherently non-collocation of sensing and actuation, increased natural damping kv w ˙ through viscous dielectrics is preferable to active vibration control. Furthermore, wire preload Fa is chosen large enough to meet ωcl ω1 . Experimental validation for the values of the first eigenfrequency f1 = ω1 /2π, according to Eq. (8), has been provided recently [7]. Therein, the characteristics for various thin wires have been measured and agreement with theoretical data, corresponding to relation (8), has been observed. A comparison of thin wires for micro-machining and the standard 0.25 mm wire for roughing cut can be done on the basis of relationship (8). It results that machining settings are always chosen such that the first eigenfrequency is in the range of 1 kHz and above in any used configuration of wire diameter and preload (for thin wires 1–5 N; for standard wires about 10 N). Closed-loop bandwidth of industrial wire guide drives is rarely exceeding 100 Hz; therefore, active vibration damping is not feasible even through guide control enhanced, for example, by a notch filter. Up to now, engineers at machine manufacturers were forced to hope that viscous damping by the surrounding dielectric liquid is sufficient to master wire vibrations that reduce machining precision. No references are known to the authors that would make use, in the context of WEDM, of some additional actuator that could introduce active damping, such as known for high buildings, large telescopes or large bridges. Therefore, the control concept applied in practice is a singular perturbation approach [8] that consists of a two-time-scale approach, a fast asymptotically stable subsystem (damping of wire vibrations) and a slow manifold handled by the main controller (path feed rate control). Damping of wire vibrations is commonly guaranteed by the viscous damping, provided through the dielectric liquid. In general, deionized water is preferred in WEDM; however, it is suggested for certain brands to use oil for high-precision machining. In principle, particular additives could be considered as well, providing electro-rheologic behavior [9]. The authors are not aware that the idea has been applied to wire damping; the essential is that such additives would enable increased viscous damping in the machining area while conserving excellent fluidity within the conditioning system (pipes, filters, pumps, etc.). Nevertheless, it has also been reported that vibrations may be beneficial for machining efficiency. High-frequency (open-loop) excitation, proposed [10] for wire EDM, pumps energy selectively into a vibration harmonic of orders 5–10, which apparently improves machining conditions by mechanical arc prevention and better flushing. Therein, transversal wire vibrations modify continuously the spatial dielectric breakdown probability, which leads to desired sparking, uniformly distributed all over the wire length.
149
4. Dealing with wire slackness Dealing with wire slackness is the main issue to be solved for achieving good geometrical performance with wire EDM. Today, numerous publications may be found in the literature on mastering wire slackness; it would be surprising that fundamentally new solutions are proposed during the next years. While appropriate machining setting may be sufficient for finishing and surfacing cut, roughing cut requires particular strategies for finding a good trade off between machining speed (material removal) and geometrical precision. The forces, acting on a wire in EDM, have been analyzed previously [11–13]; wire preload, plasma pressure, electromagnetism, electrostatics, flushing and damping have all been considered. Furthermore, detailed models for the plasma have been published [14]. Wire slackness results from establishing a position that equilibrates attractive (electrostatic and electromagnetic) and repulsive (plasma pressure) forces, such as illustrated in Fig. 3 for finishing cut settings. While the equilibrium for a straight wire may be achieved in finishing and surfacing with appropriate machining settings and tool offset, wire deformation is inevitable in roughing cut as soon as reasonable machining efficiency (speed) is required. Very simple experimental observation shows that flushing dominates all other causes for roughing cut; the resulting flexure is in path direction. The state of the art for dealing with wire slackness can be found in patents that belong basically to International Patent Classification B23H7. In the past, all major machine manufacturers have addressed the topic [15–22]. When tracking publications for subclasses B23H7/02, 04 and 06, a decreasing activity is observed: while 730 patents are referenced for the publication period 1990–1992 in the database [23] only 320 are found for 2000–2002. According to Eq. (7) and as stated previously [24], wire deflection is inverse proportional to axial preload Fa . Hence, increasing preload is the most basic approach to handling wire slackness, such as discussed recently [25]. The issue is to build a wire with high tensile strength (wire core) while conserving excellent conductivity in the intermediate layer
Fig. 3. Attractive and repulsive forces on the wire during finishing cut.
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(for the machining current up to 600 A) and good EDM properties (achieved by appropriate coatings and superficial layers). The concepts of speed variation as a function of path curvature, slackness recovery at block transitions, discharge energy variations as a function of path curvature, etc. have been introduced very early [15] and are now public domain. Comparable claims and extensions to variations in flushing pressure and software particularities have been proposed later [16–20], but still all using an open-loop approach. In this context, open loop means ‘based on the reference trajectory for the wire tool’; this is in contrast to the online wire slackness measurement system [21,22] that is using an orthogonal pair of position sensors. A conceptually different but also a feed forward approach is an offline correction of the wire path [26]. This solution provides a certain independence of the user from the EDM builder, which may be an advantage for research institutions. Yet, customers prefer to buy a complete solution and patches that modify the ISO codes are rarely welcomed. The major disadvantage is that a correction of the wire path cannot solve the problem of wire slackness, the geometric imperfection remains and additional finishing cuts are required to achieve desired precision.
5. Path feed rate control and machining monitoring Traditional machining, like milling or turning, and EDM differ principally in the generation of the path feed rate. Open-loop control is generally used in traditional machining: the path feed rate is a constant parameter or depends on position only. In EDM, path feed rate is calculated online, ¯ td : a closed control loop based on actual gap condition U, is formed by the servo system, the wire dynamics, the material removal, the sparking process and the path feed rate controller. It has been suggested [27] that wire control should include actions both on path feed and on pause time t0 . This control is to be based on reference and measured sparking frequency, as well as the rate of ‘bad’ sparks. The classification into ‘good’ and ‘bad’ sparks is achieved by a specialized circuitry that uses the machining signals gap voltage U¯ and ignition delay time td . The control algorithm evaluates the change in controls as a result of a fuzzy algorithm. The power consumption and short spark ratio are chosen [28] as controlled variables, acting on cutting speed and off-time t0 through an adaptive fuzzy control algorithm. The short spark ratio is evaluated by means of a discrimination system. The motivation therein to choose this combination of controller inputs, instead of generally used mean gap voltage ¯ is to use most relevant signals for sparking condition. U, In order to prevent wire breakage by exaggerated heating, it has been suggested [29] to control off-time t0 based on actual sparking frequency. The proposed algorithm is simple integral control. A standard identification algorithm with
Fig. 4. Integrated model for wire electro-discharge machining.
forgetting factor provides information on actual workpiece height. This estimation serves in a second step to determine some other machining settings. A fundamentally new concept has been suggested recently [30] with a device for counting ‘normal’ and ‘abnormal’ sparks, where a spark is considered to be abnormal if its ignition delay time td is below a certain level. Based on the machining settings, the percentage of stability (making use of the change in abnormal spark rate) and the percentage of efficiency are evaluated. These two inputs are classified into categories and entered into a decision table for providing a change in feed rate, an approach that resembles fuzzy logic. There are two novelties: (i) making use of a completely digital implementation of the control loop and (ii) providing some hints for an appropriate choice of the entries in the decision table that assure a stable control loop. The above-mentioned approaches can be assembled into an integrated model for wire electro-discharge machining, illustrated in Fig. 4. The wire EDM process is controlled within two loops: A ‘fast’ path feed control loop and a ‘slow’ monitoring loop. Relevant machining settings are the spark duration te , the off-time t0 , the ignition voltage Utd , reference mean gap voltage U¯ ref and reference ignition delay time td,ref . The ignition delay time td and the arc voltage Ute are resulting from the gap properties that depend on the gap size δ and contamination of the dielectric liquid by machining debris. The objective of the path feed control is to maintain the wire around a stable equilibrium distance where attractive and repulsive forces balance. A feedback loop is required to take into account for variations on material removal, as well as for changes of the workpiece height. Furthermore, this gap control is useful as well to increase the basin (attractive domain, according to Fig. 3) of the equilibrium distance. In industrial practice, closed-loop bandwidth of the path feed control loop is chosen around 10–100 Hz, which requires excellent servo drive performance. Issues that remain still open for path feed rate control are the handling of the process inherent noise and the separation of the ‘very fast’ wire vibrations from the ‘fast’ path
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feed control loop. The noise in the EDM process is related to the stochastic properties of the dielectric breakdown process; this perturbation appears as a white noise, which is amplified, by the control loop. Furthermore, a digital implementation like in Ref. [30] presents the risk that Shannon’s theorem of sampling is not verified and high-frequency noise is mapped into low frequency; the result of these phantom signals is the appearance of undesirable oscillations in the path feed rate control loop. The principal task of monitoring is reducing the risk of wire breakage, and therefore the amount of time lost for rethreading. In general the off-time t0 , mean gap voltage U¯ ref , etc. are the principal manipulated variables produced by the machining supervision algorithm; the inputs are a cluster of signals provided by the EDM monitor.
6. Conclusion A survey on wire modeling and control of WEDM has been provided above. While numerous solutions have been proposed in the past for mastering wire slackness, very little publications are dealing with issues like vibration damping and amplification of process randomness.
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