- Email: [email protected]
Material removal rate prediction for ultrasonic drilling of hard materials using an impact oscillator approach
9 August 1999
Physics Letters A 259 Ž1999. 91–96 www.elsevier.nlrlocaterphysleta
Material removal rate prediction for ultrasonic drilling of hard materials using an impact oscillator approach M. Wiercigroch ) , R.D. Neilson, M.A. Player Department of Engineering, King’s College, UniÕersity of Aberdeen, Aberdeen, AB24 3UE, UK Received 28 May 1998; received in revised form 11 February 1999; accepted 17 June 1999 Communicated by A.P. Fordy
Abstract It is postulated that the main mechanism of the enhancement of material removal rate ŽMRR. in ultrasonic machining is associated with high amplitudes forces generated by impacts, which act on the workpiece and help to develop micro-cracking in the cutting zone. The inherent non-linearity of the discontinuous impact process is modelled, to generate the pattern of the impact forces. A novel procedure for calculating the MRR is proposed, which for the first time explains the experimentally observed fall in MRR at higher static forces. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Ultrasonics; Impact oscillator; Machining
Recently, nonlinear dynamics approaches have increasingly been used to explain complexities occurring in manufacturing systems. Theoretical studies have been carried out in the area of ductile metal cutting Že.g. w1,2x., where periodic Žchatter. and aperiodic Žchaos. behaviour of simple models has been demonstrated. Despite the fact that strong nonlinear dependencies have been observed in cutting brittle materials, this area has been given little attention so far. For example, one of the best known anomalies in ultrasonic machining is the decrease in material removal rate for higher values of static forces, contradicting a classical perception of the efficiency of the process mechanism. A study of this phenomenon was the stimulus for the work described in this paper.
) Corresponding author. Tel. q44-1-224-272509, fax q441224-272497, e-mail: [email protected]
Ultrasonic machining ŽUSM. offers a solution to the expanding need for machining brittle materials such as semiconductors, optical glasses, and ceramics, and for increasingly complex operations to provide intricate shapes and workpiece profiles. This form of machining is non-thermal, non-chemical, and creates no change in the metallurgical, chemical or physical properties of the workpiece. As a consequence, ultrasonic machining offers virtually stress free machined surfaces. It is therefore used extensively in manufacturing hard and brittle materials that are difficult to cut by other conventional methods. The actual cutting is performed either by abrasive particles suspended in a fluid, or by a rotating diamond-plated tool. These variants are known respectively as traditional ultrasonic machining, and rotary ultrasonic machining ŽRUM.. Traditional ultrasonic machining accomplishes the removal of material by the abrading action of a grit-loaded slurry, circulating between the workpiece and a tool that is
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 4 1 6 - 8
92
M. Wiercigroch et al.r Physics Letters A 259 (1999) 91–96
vibrated at small amplitude and high frequency, typically 10–20 mm at 20–50 kHz. In a practical machine, a high-frequency power source activates a stack of magneto-strictive or piezo-electric material to generate a standing wave in the transducer, which is coupled to the tool by a mechanical transformer or horn. This motion is transmitted under light pressure to the slurry, which abrades the workpiece into a negative image of the tool form. In traditional ultrasonic machining, therefore, the workpiece shape and dimensional accuracy are directly dependent on the geometry of the tool, there is limited capacity for circulating the abrasive slurry, and there is simultaneous abrasion of both workpiece and tool. Consequently, the method suffers from relatively poor accuracy, mediocre material removal rate and substantial tool wear. These problems were largely overcome by the introduction of rotary ultrasonic machining using diamond impregnated or coated tools. Fig. 1 shows such a rotary ultrasonic machine. A piezo-electric
Fig. 1. Schematic of rotary ultrasonic machining. 1 - piezo-electric element, 2 - transducer assembly, 3 - coupler, 4 - diamond impregnatedrcoated tool, 5 - workpiece, 6 - fixture, 7 - pump, 8 tank, 9 - coolant jet.
element Ž1. built into the rotating head provides the necessary vibration. The natural frequency of the transducer assembly Ž2. and coupler is tuned to the forcing frequency, so ideally the tip of the diamondimpregnated tool Ž4. should be at an anti-node of displacement. During machining a cutting fluid Ž9. is supplied to cool the tool and remove debris from the workpiece Ž5.. This technology was developed in the early 1960s by U.K.A.E.A. in Harwell, England. Some years later quite similar methods were studied by Markov w3–5x and Petrukha w6x, but details of the methods were not revealed. Other workers carried out experimental studies on the basic characteristics of the process, for example Kubota, Tamura and Shimamura w7x. Their tests used three configurations, namely, a glass plate drilled by a stationary ultrasonic tool with a rotary table, a glass rod turned by a lathe with an ultrasonic transducer on the carriage, and a glass plate drilled by a rotating transducer head. They established the influences of working conditions such as grain size, amplitude of vibration, rotational speed and feed pressure on the material removal rate ŽMRR., which is defined as volume of material removed in a unit of time. A particular feature of these experiments is that plots of MRR versus static load presented in w5x and w7x show a maximum for a certain value of static load. Komaraiah et al. w8x also conducted experimental studies on the ultrasonic machining of different workpiece materials including glass, porcelain, ferrite and alumina, using various tool materials, in order to analyse the effects of mechanical properties of the workpiece and tool material on the surface roughness and accuracy. Their work confirmed the superiority of the rotary technique over traditional slurry-type machining. The first theoretical approach to modelling USM was put forward by Saha et al. w9x. They attempted to develop a comprehensive analytical model for the estimation of the MRR in order to make an in-depth study of the material removal process and its dependence on major influencing parameters. Satisfactory agreement was reported between theory and experiment, apparently explaining the fall in MRR for higher static loads. However, their model uses only Hertzian theory to explain the mechanism of material removal; it seems that this approach should be more
M. Wiercigroch et al.r Physics Letters A 259 (1999) 91–96
suitable for workpieces comprising ductile rather than brittle materials. Moreover, using Hertzian theory alone to explain the relationship between MRR and static force, it appears impossible to obtain a function of the form obtained from experimental test w7x. Accordingly, this paper adopts the nonlinear dynamics approach to modelling MRR for brittle materials, which is phenomenologically different to any others previously undertaken, and hopefully would be of interest to both nonlinear dynamics and applied physiscs communities. It is based on applying impacting oscillator theory w10–12x to explain the main mechanism occurring in ultrasonic drilling. In particular, we shall address the formulation of a simple model of the non-linear dynamic interactions encountered in the machine tool — ultrasonic cutting process system, which could explain the fall in the MRR for higher static loads. The basic assumption of the proposed model is that the efficiency of cutting is dependent upon both the size and frequency of the impact force during cutting. The proposed dynamical model of ultrasonic drilling is depicted in Fig. 2, where a resonant transducer assembly tuned to the ultrasonic driving frequency is represented in a simple form as a two lump mass model. It consists of a mass m1 representing the movable headstock, and the equivalent mass m 2 of the vibrating ultrasonic horn including the tool. Linear springs of stiffness k 1 and k 2 and dashpots of viscous damping c1 and c 2 connect the headstock and the ultrasonic horn and tool to the piezo-electric ultrasonic driver. This excites the system kinematically with amplitude A and frequency v . The material of the workpiece is represented by a stiffness k 3 and damper c 3 . The process is started with an initial gap g between the tool tip and the workpiece. A force f HY D is required to drive the headstock downwards, and the velocity of the head is limited by damping c 0 in the vertical drive. This headstock model is particularly appropriate for the soft hydraulic vertical drive typically used in these machines. The model represents only the dynamical part of the displacement x 5 Žcorresponding to motion of the cut face of the workpiece., and the gradual drift resulting from penetration into the material is suppressed. Furthermore, the model does not attempt to
93
Fig. 2. A dynamic model of rotary ultrasonic drilling.
represent in detail the full dynamics of the ultrasonic transducer assembly; it is sufficient, however, to describe the essential interactions of the static force and the transducer with the workpiece. The equations of motion for the system in non-dimensional form are as follows: xXX1 q 2 j 10 xX1 q 2 j 11 xX1 y 2 j 11 xX2 q x 1 y x 2 s f HY D Ž t .
Ž 1.
2 j 11 xX1 q 2 j 11 xX2 q 2 j 12 a 12
xX2 y 2 j 12 a 12
xX4
y x 1 q Ž 1 q a 12 . x 2 y a 12 x 4 j 12 a 12 a v cos Ž nt . q a 12 a sin nt XX x 4 y 2 j 21 a 21 xX2 q 2 j 21 a 21 xX4 y a 21 x 2 q a 21 x 4
Ž
.
Ž 2.
s y2 j 21 a 21 a v cos Ž nt . q a 21 a sin Ž nt . for x 4 y x 5 - g
Ž 3a .
M. Wiercigroch et al.r Physics Letters A 259 (1999) 91–96
94
xXX4 y 2 j 21 a 21 xX2 q 2 j 21 a 21 xX4 q 2 j 22 a 22 xX5 y a 21 x 2 q a 21 x 4 q a 22 x 5 for x 4 y x 5 s g
2. Having calculated the average value fAV G of the impact force, the number of excursions of the instantaneous impact force above this value, n CROSS , is counted. The number of cycles of the impact force is then defined as:
Ž 3b . for x 4 y x 5 - g
n CY C s
s y2 j 21 a 21 a v cos Ž nt . q a 21 a sin Ž nt .
2 j 22 a 22 xX5 q a 22 x 5 s 0
Ž 4a . xX5 s xX4
for x 4 y x 5 s g
Ž 4b . where d k1 k2 X 2 2 s , t s v 11 t , v 11 s , v 12 s , dt m1 m1 2 v 21 s
j 11 s
k2
, m2 c1
2 v 22 s
k3 m2
, j 10 s c2
2 v 11 m1
, c2
It is assumed that the MRR is a function of the magnitude of the impact force and its frequency w13–16x. This is consistent with a removal process in which the tool tip impacts the workpiece, making micro-cracks on its surface. Other assumptions are that the diamond is uniformly distributed on the working part of the tool, with a uniform grit size, and that the ultrasonic amplitude and frequency, and the geometry of the tool, remain constant. The relative value of MRR is estimated from the following algorithm. 1.Initially an average value fAV G of the impact force f Žt . over all trials is calculated from t2 1 m 1 fAV G s f Ž t . dt Ž 5. Ý m is1 t 2 y t 1 t 1
H
where m is the number of numerical simulations, and t 1 , t 2 are the integration limits of a single trial, chosen to provide a stable time history. Eq. Ž5. is re-written replacing the integral with a sum as follows 1 m 1 m fAV G s Ž 6. Ý Ýf m is1 n js1 i j Žusing the trapezium rule with interval Žt 2 y t 1 .rn.
2
.
Ž 7.
3. It is assumed that there is a minimum level f MIN of impact force sufficient to cause significant damage to the material being cut. 4. It is postulated that the MRR can then be estimated from the following formula MRRs
,j s , 2 v 12 m1 21 2 v 211 m1 vi j v j 22 s , ai j s , ns . 2 v 22 m 2 v 11 v 11 2 v 11 m1 c3
, j 12 s
c0
n CROSS
n CY C n2
n
Ý js0
d j)
ž / fAV G
dj
Ž 8.
where n is the number of time steps and z is an ‘accumulated damage parameter’, normally chosen by a direct comparison with the experimental results. Here, d j and d jU are respectively the impact force Žat time-step j . and the ‘damaging’ impact force d jU s d j y fAVG . The dynamical system parameters were identified to model the ultrasonic drilling machine developed at Aberdeen University w17x and the cutting material was chosen to be a float glass. The value f MIN of impact force sufficient to cause significant damage was experimentally evaluated using a specially designed load cell w18x. Because the system under investigation is discontinuous, special precautions are needed to maintain numerical accuracy at the times when the discontinuity occurs. Initially, specially designed procedures for handling motion-dependent discontinuities were used within the fourth order Runge–Kutta scheme. These procedures involve detecting discontinuities and calculating precise values of times when they occur Žsee e.g. w19,20x.. However, it was later found that for the range of parameters investigated and the dynamic responses generated, a sixth order Runge–Kutta method w21x with automatic time step adjustment similar to the Fehlberg scheme w22x is faster and able to achieve a satisfactory accuracy of solution. With the initial gap g equal to zero, the sign of displacement of the headstock is dependent upon the magnitude of the control static force f HY D . If this force is small enough the steady state displacement x 1 is
M. Wiercigroch et al.r Physics Letters A 259 (1999) 91–96
95
Fig. 3. Time history of Ža. the tool displacement and Žb. the impact force.
negative. Fig. 3Ža. and Žb. show the time histories of the x 1 displacements and the impact force, d Žt .. The role of f HY D is to supply an optimum condition to allow the biggest impacting forces. Clearly, under the conditions of Fig. 3, for instance, the light load causes the head to ‘bounce off’ the workpiece, with only intermittent and small impact forces. However, when f HY D is too big the hammering effect disappears and the workpiece is loaded with a force which possesses a large static component but negligible impact component, decreasing significantly the cutting efficiency. Fig. 4 shows the amplitude of the impact force and its distribution in the time domain for different values of f HY D . It is evident that the magnitude of the impact force and its frequency strongly depend on the static load. This point is emphasised by the graph of Fig. 5, which shows the MRR as a function of f HY D , calculated using the above algorithm and then normalised with respect to the MRR obtained without ultrasonics. In practical terms, Fig. 5 depicts the ratio between the material removal rate with ultrasonics and without for the same cutting conditions. The upper and lower envelopes of the MRR characteristic are very similar to results obtained experimentally w5,7x. The jagged form of the theoretical MRR graph appears to arise from the nonlinear dynamics of the model, but it has not been observed experimentally.
Fig. 4. Influence of the static hydraulic force f HY D on the time history of the impacting force for Ža. f HY D s 0.1, Žb. f HYD s 0.4 and Žc. f HY D s 0.8.
In other words, as the parameter f HY D is varied, the system undergoes bifurcations, and also is influenced by the cut-off value f MIN chosen to model the minimum impact force enabling microcrack propaga-
Fig. 5. The material removal rate versus static force.
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M. Wiercigroch et al.r Physics Letters A 259 (1999) 91–96
tion. Further work is being undertaken to identify fully the origin of this behaviour and to understand why it has not so far been detected experimentally. At this stage, it is worth noting that changes in the values of z in the algorithm for the MRR do not smooth the curve, supporting the conclusion that the effect is a direct result of the nonlinear dynamics. Nevertheless, the top or bottom envelope of the graph predicts the experimentally observed form of the drop in MRR for higher values of the static loading.
Acknowledgements This research was supported jointly by the University of Aberdeen Research Committee ŽGrant No. R224. and European Community Science and Technology Research Programme ŽGrant No. ERB3510PL92103.. The authors would like also to thank the anonymous reviewer for his constructive and stimulating comments.
References w1x I. Grabec, Phys. Lett. A 117 Ž1986. 384. w2x M. Wiercigroch, ASME, J. Vib. Acoust. 119 Ž1997. 468.
w3x A.I. Markov and I.D. Ustinov, Ind. Diamond Rev., March Ž1972. 97. w4x A.I. Markov, Stanki Instrument 48 Ž1977. 33 Žin Russian.. w5x A.I. Markov, Moscow, 1980 Žin Russian.. w6x P.G. Petrukha, Russ. Eng. J. L Ž1980. 71. w7x M. Kubota, J. Tamura, N. Shimamura, Precis. Eng. 11 Ž1977. 127. w8x M. Komaraiah, M.P. Mannan, P.N. Reddy-Narasimha, S. Victor, Precis. Eng. 10 Ž1988. 58. w9x J. Saha, A. Bhattacharyya and P.K. Mishra, Proc. 27th MATADOR Conference, UMIST, Manchester, April 1988, 275. w10x J.M.T. Thompson, R. Ghaffari, Phys. Lett. A 91 Ž1982. 5. w11x S.W. Shaw, J. Sound Vib. 99 Ž1985. 199. w12x M. Wiercigroch and V.T.W. Sin, J. Appl. Mech., accepted for publication. w13x Ayal de S. Jayatilawa. Fracture of Engineering, Applied Science Publishers Ltd., London, 1979. w14x J.F. Knott, Fundamentals of Fracture Mechanics, Butterworth, Norwich, 1973. w15x C.H. Yew, P.A. Taylor, Int. J. Impact Eng. 15 Ž1994. 385. w16x J.F. Boudet, S. Ciliberto, Phys. Rev. Lett. 80 Ž1998. 341. w17x H. Barber and M.A. Player, UK patent application No. 8710439, 1987. w18x M. Wiercigroch, R.D. Neilson, M.A. Player, Proc. IMechE, Part E 212 Ž1998. 263. w19x M. Wiercigroch, Dynamics of mechanical systems with discontinuities, Silesian University Press, Gliwice, 1994 Žin Polish.. w20x M. Wiercigroch, Mach. Vib. 5 Ž1996. 112. w21x J. Legras, Methodes et techniques de l’analyse numerique, Dunod, Paris Ž1971.. w22x E. Fehlberg, Computing 6 Ž1970. 61.
Physics Letters A 259 Ž1999. 91–96 www.elsevier.nlrlocaterphysleta
Material removal rate prediction for ultrasonic drilling of hard materials using an impact oscillator approach M. Wiercigroch ) , R.D. Neilson, M.A. Player Department of Engineering, King’s College, UniÕersity of Aberdeen, Aberdeen, AB24 3UE, UK Received 28 May 1998; received in revised form 11 February 1999; accepted 17 June 1999 Communicated by A.P. Fordy
Abstract It is postulated that the main mechanism of the enhancement of material removal rate ŽMRR. in ultrasonic machining is associated with high amplitudes forces generated by impacts, which act on the workpiece and help to develop micro-cracking in the cutting zone. The inherent non-linearity of the discontinuous impact process is modelled, to generate the pattern of the impact forces. A novel procedure for calculating the MRR is proposed, which for the first time explains the experimentally observed fall in MRR at higher static forces. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Ultrasonics; Impact oscillator; Machining
Recently, nonlinear dynamics approaches have increasingly been used to explain complexities occurring in manufacturing systems. Theoretical studies have been carried out in the area of ductile metal cutting Že.g. w1,2x., where periodic Žchatter. and aperiodic Žchaos. behaviour of simple models has been demonstrated. Despite the fact that strong nonlinear dependencies have been observed in cutting brittle materials, this area has been given little attention so far. For example, one of the best known anomalies in ultrasonic machining is the decrease in material removal rate for higher values of static forces, contradicting a classical perception of the efficiency of the process mechanism. A study of this phenomenon was the stimulus for the work described in this paper.
) Corresponding author. Tel. q44-1-224-272509, fax q441224-272497, e-mail: [email protected]
Ultrasonic machining ŽUSM. offers a solution to the expanding need for machining brittle materials such as semiconductors, optical glasses, and ceramics, and for increasingly complex operations to provide intricate shapes and workpiece profiles. This form of machining is non-thermal, non-chemical, and creates no change in the metallurgical, chemical or physical properties of the workpiece. As a consequence, ultrasonic machining offers virtually stress free machined surfaces. It is therefore used extensively in manufacturing hard and brittle materials that are difficult to cut by other conventional methods. The actual cutting is performed either by abrasive particles suspended in a fluid, or by a rotating diamond-plated tool. These variants are known respectively as traditional ultrasonic machining, and rotary ultrasonic machining ŽRUM.. Traditional ultrasonic machining accomplishes the removal of material by the abrading action of a grit-loaded slurry, circulating between the workpiece and a tool that is
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 4 1 6 - 8
92
M. Wiercigroch et al.r Physics Letters A 259 (1999) 91–96
vibrated at small amplitude and high frequency, typically 10–20 mm at 20–50 kHz. In a practical machine, a high-frequency power source activates a stack of magneto-strictive or piezo-electric material to generate a standing wave in the transducer, which is coupled to the tool by a mechanical transformer or horn. This motion is transmitted under light pressure to the slurry, which abrades the workpiece into a negative image of the tool form. In traditional ultrasonic machining, therefore, the workpiece shape and dimensional accuracy are directly dependent on the geometry of the tool, there is limited capacity for circulating the abrasive slurry, and there is simultaneous abrasion of both workpiece and tool. Consequently, the method suffers from relatively poor accuracy, mediocre material removal rate and substantial tool wear. These problems were largely overcome by the introduction of rotary ultrasonic machining using diamond impregnated or coated tools. Fig. 1 shows such a rotary ultrasonic machine. A piezo-electric
Fig. 1. Schematic of rotary ultrasonic machining. 1 - piezo-electric element, 2 - transducer assembly, 3 - coupler, 4 - diamond impregnatedrcoated tool, 5 - workpiece, 6 - fixture, 7 - pump, 8 tank, 9 - coolant jet.
element Ž1. built into the rotating head provides the necessary vibration. The natural frequency of the transducer assembly Ž2. and coupler is tuned to the forcing frequency, so ideally the tip of the diamondimpregnated tool Ž4. should be at an anti-node of displacement. During machining a cutting fluid Ž9. is supplied to cool the tool and remove debris from the workpiece Ž5.. This technology was developed in the early 1960s by U.K.A.E.A. in Harwell, England. Some years later quite similar methods were studied by Markov w3–5x and Petrukha w6x, but details of the methods were not revealed. Other workers carried out experimental studies on the basic characteristics of the process, for example Kubota, Tamura and Shimamura w7x. Their tests used three configurations, namely, a glass plate drilled by a stationary ultrasonic tool with a rotary table, a glass rod turned by a lathe with an ultrasonic transducer on the carriage, and a glass plate drilled by a rotating transducer head. They established the influences of working conditions such as grain size, amplitude of vibration, rotational speed and feed pressure on the material removal rate ŽMRR., which is defined as volume of material removed in a unit of time. A particular feature of these experiments is that plots of MRR versus static load presented in w5x and w7x show a maximum for a certain value of static load. Komaraiah et al. w8x also conducted experimental studies on the ultrasonic machining of different workpiece materials including glass, porcelain, ferrite and alumina, using various tool materials, in order to analyse the effects of mechanical properties of the workpiece and tool material on the surface roughness and accuracy. Their work confirmed the superiority of the rotary technique over traditional slurry-type machining. The first theoretical approach to modelling USM was put forward by Saha et al. w9x. They attempted to develop a comprehensive analytical model for the estimation of the MRR in order to make an in-depth study of the material removal process and its dependence on major influencing parameters. Satisfactory agreement was reported between theory and experiment, apparently explaining the fall in MRR for higher static loads. However, their model uses only Hertzian theory to explain the mechanism of material removal; it seems that this approach should be more
M. Wiercigroch et al.r Physics Letters A 259 (1999) 91–96
suitable for workpieces comprising ductile rather than brittle materials. Moreover, using Hertzian theory alone to explain the relationship between MRR and static force, it appears impossible to obtain a function of the form obtained from experimental test w7x. Accordingly, this paper adopts the nonlinear dynamics approach to modelling MRR for brittle materials, which is phenomenologically different to any others previously undertaken, and hopefully would be of interest to both nonlinear dynamics and applied physiscs communities. It is based on applying impacting oscillator theory w10–12x to explain the main mechanism occurring in ultrasonic drilling. In particular, we shall address the formulation of a simple model of the non-linear dynamic interactions encountered in the machine tool — ultrasonic cutting process system, which could explain the fall in the MRR for higher static loads. The basic assumption of the proposed model is that the efficiency of cutting is dependent upon both the size and frequency of the impact force during cutting. The proposed dynamical model of ultrasonic drilling is depicted in Fig. 2, where a resonant transducer assembly tuned to the ultrasonic driving frequency is represented in a simple form as a two lump mass model. It consists of a mass m1 representing the movable headstock, and the equivalent mass m 2 of the vibrating ultrasonic horn including the tool. Linear springs of stiffness k 1 and k 2 and dashpots of viscous damping c1 and c 2 connect the headstock and the ultrasonic horn and tool to the piezo-electric ultrasonic driver. This excites the system kinematically with amplitude A and frequency v . The material of the workpiece is represented by a stiffness k 3 and damper c 3 . The process is started with an initial gap g between the tool tip and the workpiece. A force f HY D is required to drive the headstock downwards, and the velocity of the head is limited by damping c 0 in the vertical drive. This headstock model is particularly appropriate for the soft hydraulic vertical drive typically used in these machines. The model represents only the dynamical part of the displacement x 5 Žcorresponding to motion of the cut face of the workpiece., and the gradual drift resulting from penetration into the material is suppressed. Furthermore, the model does not attempt to
93
Fig. 2. A dynamic model of rotary ultrasonic drilling.
represent in detail the full dynamics of the ultrasonic transducer assembly; it is sufficient, however, to describe the essential interactions of the static force and the transducer with the workpiece. The equations of motion for the system in non-dimensional form are as follows: xXX1 q 2 j 10 xX1 q 2 j 11 xX1 y 2 j 11 xX2 q x 1 y x 2 s f HY D Ž t .
Ž 1.
2 j 11 xX1 q 2 j 11 xX2 q 2 j 12 a 12
xX2 y 2 j 12 a 12
xX4
y x 1 q Ž 1 q a 12 . x 2 y a 12 x 4 j 12 a 12 a v cos Ž nt . q a 12 a sin nt XX x 4 y 2 j 21 a 21 xX2 q 2 j 21 a 21 xX4 y a 21 x 2 q a 21 x 4
Ž
.
Ž 2.
s y2 j 21 a 21 a v cos Ž nt . q a 21 a sin Ž nt . for x 4 y x 5 - g
Ž 3a .
M. Wiercigroch et al.r Physics Letters A 259 (1999) 91–96
94
xXX4 y 2 j 21 a 21 xX2 q 2 j 21 a 21 xX4 q 2 j 22 a 22 xX5 y a 21 x 2 q a 21 x 4 q a 22 x 5 for x 4 y x 5 s g
2. Having calculated the average value fAV G of the impact force, the number of excursions of the instantaneous impact force above this value, n CROSS , is counted. The number of cycles of the impact force is then defined as:
Ž 3b . for x 4 y x 5 - g
n CY C s
s y2 j 21 a 21 a v cos Ž nt . q a 21 a sin Ž nt .
2 j 22 a 22 xX5 q a 22 x 5 s 0
Ž 4a . xX5 s xX4
for x 4 y x 5 s g
Ž 4b . where d k1 k2 X 2 2 s , t s v 11 t , v 11 s , v 12 s , dt m1 m1 2 v 21 s
j 11 s
k2
, m2 c1
2 v 22 s
k3 m2
, j 10 s c2
2 v 11 m1
, c2
It is assumed that the MRR is a function of the magnitude of the impact force and its frequency w13–16x. This is consistent with a removal process in which the tool tip impacts the workpiece, making micro-cracks on its surface. Other assumptions are that the diamond is uniformly distributed on the working part of the tool, with a uniform grit size, and that the ultrasonic amplitude and frequency, and the geometry of the tool, remain constant. The relative value of MRR is estimated from the following algorithm. 1.Initially an average value fAV G of the impact force f Žt . over all trials is calculated from t2 1 m 1 fAV G s f Ž t . dt Ž 5. Ý m is1 t 2 y t 1 t 1
H
where m is the number of numerical simulations, and t 1 , t 2 are the integration limits of a single trial, chosen to provide a stable time history. Eq. Ž5. is re-written replacing the integral with a sum as follows 1 m 1 m fAV G s Ž 6. Ý Ýf m is1 n js1 i j Žusing the trapezium rule with interval Žt 2 y t 1 .rn.
2
.
Ž 7.
3. It is assumed that there is a minimum level f MIN of impact force sufficient to cause significant damage to the material being cut. 4. It is postulated that the MRR can then be estimated from the following formula MRRs
,j s , 2 v 12 m1 21 2 v 211 m1 vi j v j 22 s , ai j s , ns . 2 v 22 m 2 v 11 v 11 2 v 11 m1 c3
, j 12 s
c0
n CROSS
n CY C n2
n
Ý js0
d j)
ž / fAV G
dj
Ž 8.
where n is the number of time steps and z is an ‘accumulated damage parameter’, normally chosen by a direct comparison with the experimental results. Here, d j and d jU are respectively the impact force Žat time-step j . and the ‘damaging’ impact force d jU s d j y fAVG . The dynamical system parameters were identified to model the ultrasonic drilling machine developed at Aberdeen University w17x and the cutting material was chosen to be a float glass. The value f MIN of impact force sufficient to cause significant damage was experimentally evaluated using a specially designed load cell w18x. Because the system under investigation is discontinuous, special precautions are needed to maintain numerical accuracy at the times when the discontinuity occurs. Initially, specially designed procedures for handling motion-dependent discontinuities were used within the fourth order Runge–Kutta scheme. These procedures involve detecting discontinuities and calculating precise values of times when they occur Žsee e.g. w19,20x.. However, it was later found that for the range of parameters investigated and the dynamic responses generated, a sixth order Runge–Kutta method w21x with automatic time step adjustment similar to the Fehlberg scheme w22x is faster and able to achieve a satisfactory accuracy of solution. With the initial gap g equal to zero, the sign of displacement of the headstock is dependent upon the magnitude of the control static force f HY D . If this force is small enough the steady state displacement x 1 is
M. Wiercigroch et al.r Physics Letters A 259 (1999) 91–96
95
Fig. 3. Time history of Ža. the tool displacement and Žb. the impact force.
negative. Fig. 3Ža. and Žb. show the time histories of the x 1 displacements and the impact force, d Žt .. The role of f HY D is to supply an optimum condition to allow the biggest impacting forces. Clearly, under the conditions of Fig. 3, for instance, the light load causes the head to ‘bounce off’ the workpiece, with only intermittent and small impact forces. However, when f HY D is too big the hammering effect disappears and the workpiece is loaded with a force which possesses a large static component but negligible impact component, decreasing significantly the cutting efficiency. Fig. 4 shows the amplitude of the impact force and its distribution in the time domain for different values of f HY D . It is evident that the magnitude of the impact force and its frequency strongly depend on the static load. This point is emphasised by the graph of Fig. 5, which shows the MRR as a function of f HY D , calculated using the above algorithm and then normalised with respect to the MRR obtained without ultrasonics. In practical terms, Fig. 5 depicts the ratio between the material removal rate with ultrasonics and without for the same cutting conditions. The upper and lower envelopes of the MRR characteristic are very similar to results obtained experimentally w5,7x. The jagged form of the theoretical MRR graph appears to arise from the nonlinear dynamics of the model, but it has not been observed experimentally.
Fig. 4. Influence of the static hydraulic force f HY D on the time history of the impacting force for Ža. f HY D s 0.1, Žb. f HYD s 0.4 and Žc. f HY D s 0.8.
In other words, as the parameter f HY D is varied, the system undergoes bifurcations, and also is influenced by the cut-off value f MIN chosen to model the minimum impact force enabling microcrack propaga-
Fig. 5. The material removal rate versus static force.
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tion. Further work is being undertaken to identify fully the origin of this behaviour and to understand why it has not so far been detected experimentally. At this stage, it is worth noting that changes in the values of z in the algorithm for the MRR do not smooth the curve, supporting the conclusion that the effect is a direct result of the nonlinear dynamics. Nevertheless, the top or bottom envelope of the graph predicts the experimentally observed form of the drop in MRR for higher values of the static loading.
Acknowledgements This research was supported jointly by the University of Aberdeen Research Committee ŽGrant No. R224. and European Community Science and Technology Research Programme ŽGrant No. ERB3510PL92103.. The authors would like also to thank the anonymous reviewer for his constructive and stimulating comments.
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