9 August 1999

.Physics Letters A 259 1999 9196www.elsevier.nlrlocaterphysleta

Material removal rate prediction for ultrasonic drilling of hardmaterials using an impact oscillator approach

M. Wiercigroch ), R.D. Neilson, M.A. PlayerDepartment of Engineering, Kings College, Uniersity of Aberdeen, Aberdeen, AB24 3UE, UK

Received 28 May 1998; received in revised form 11 February 1999; accepted 17 June 1999Communicated by A.P. Fordy

Abstract

.It is postulated that the main mechanism of the enhancement of material removal rate MRR in ultrasonic machining isassociated with high amplitudes forces generated by impacts, which act on the workpiece and help to develop micro-crack-ing in the cutting zone. The inherent non-linearity of the discontinuous impact process is modelled, to generate the pattern ofthe impact forces. A novel procedure for calculating the MRR is proposed, which for the first time explains theexperimentally 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 haveincreasingly been used to explain complexities oc-curring in manufacturing systems. Theoretical stud-ies have been carried out in the area of ductile metal

w x. .cutting e.g. 1,2 , where periodic chatter and ape- .riodic chaos behaviour of simple models has been

demonstrated. Despite the fact that strong nonlineardependencies have been observed in cutting brittlematerials, this area has been given little attention sofar. For example, one of the best known anomalies inultrasonic machining is the decrease in material re-moval rate for higher values of static forces, contra-dicting a classical perception of the efficiency of theprocess mechanism. A study of this phenomenonwas the stimulus for the work described in thispaper.

) Corresponding author. Tel. q44-1-224-272509, fax q44-1224-272497, e-mail: M.Wiercigroch@eng.abdn.ac.uk

.Ultrasonic machining USM offers a solution tothe expanding need for machining brittle materialssuch as semiconductors, optical glasses, and ceram-ics, and for increasingly complex operations to pro-vide intricate shapes and workpiece profiles. Thisform of machining is non-thermal, non-chemical,and creates no change in the metallurgical, chemicalor physical properties of the workpiece. As a conse-quence, ultrasonic machining offers virtually stressfree machined surfaces. It is therefore used exten-sively in manufacturing hard and brittle materialsthat are difficult to cut by other conventional meth-ods. The actual cutting is performed either by abra-sive particles suspended in a fluid, or by a rotatingdiamond-plated tool. These variants are known re-spectively as traditional ultrasonic machining, and

.rotary ultrasonic machining RUM . Traditional ul-trasonic machining accomplishes the removal of ma-terial 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: S0375-9601 99 00416-8

( )M. Wiercigroch et al.rPhysics Letters A 259 1999 919692

vibrated at small amplitude and high frequency, typi-cally 1020 mm at 2050 kHz. In a practical ma-chine, a high-frequency power source activates astack of magneto-strictive or piezo-electric materialto generate a standing wave in the transducer, whichis coupled to the tool by a mechanical transformer orhorn. This motion is transmitted under light pressureto the slurry, which abrades the workpiece into anegative image of the tool form. In traditional ultra-sonic machining, therefore, the workpiece shape anddimensional accuracy are directly dependent on thegeometry of the tool, there is limited capacity forcirculating the abrasive slurry, and there is simulta-neous abrasion of both workpiece and tool. Conse-quently, the method suffers from relatively pooraccuracy, mediocre material removal rate and sub-stantial tool wear.

These problems were largely overcome by theintroduction of rotary ultrasonic machining usingdiamond impregnated or coated tools. Fig. 1 showssuch a rotary ultrasonic machine. A piezo-electric

Fig. 1. Schematic of rotary ultrasonic machining. 1 - piezo-electricelement, 2 - transducer assembly, 3 - coupler, 4 - diamondimpregnatedrcoated tool, 5 - workpiece, 6 - fixture, 7 - pump, 8 -tank, 9 - coolant jet.

.element 1 built into the rotating head provides thenecessary vibration. The natural frequency of the

.transducer assembly 2 and coupler is tuned to theforcing frequency, so ideally the tip of the diamond-

.impregnated tool 4 should be at an anti-node of .displacement. During machining a cutting fluid 9 is

supplied to cool the tool and remove debris from the .workpiece 5 .

This technology was developed in the early 1960sby U.K.A.E.A. in Harwell, England. Some yearslater quite similar methods were studied by Markovw x w x35 and Petrukha 6 , but details of the methodswere not revealed. Other workers carried out experi-mental studies on the basic characteristics of theprocess, for example Kubota, Tamura and Shima-

w xmura 7 . Their tests used three configurations,namely, a glass plate drilled by a stationary ultra-sonic tool with a rotary table, a glass rod turned by alathe with an ultrasonic transducer on the carriage,and a glass plate drilled by a rotating transducerhead. They established the influences of workingconditions such as grain size, amplitude of vibration,rotational speed and feed pressure on the material

.removal rate MRR , which is defined as volume ofmaterial removed in a unit of time. A particularfeature of these experiments is that plots of MRR

w x w xversus static load presented in 5 and 7 show amaximum for a certain value of static load. Komara-

w xiah et al. 8 also conducted experimental studies onthe ultrasonic machining of different workpiece ma-terials including glass, porcelain, ferrite and alumina,using various tool materials, in order to analyse theeffects of mechanical properties of the workpieceand tool material on the surface roughness and accu-racy. Their work confirmed the superiority of therotary technique over traditional slurry-type machin-ing.

The first theoretical approach to modelling USMw xwas put forward by Saha et al. 9 . They attempted to

develop a comprehensive analytical model for theestimation of the MRR in order to make an in-depthstudy of the material removal process and its depen-dence on major influencing parameters. Satisfactoryagreement was reported between theory and experi-ment, apparently explaining the fall in MRR forhigher static loads. However, their model uses onlyHertzian theory to explain the mechanism of materialremoval; it seems that this approach should be more

( )M. Wiercigroch et al.rPhysics Letters A 259 1999 9196 93

suitable for workpieces comprising ductile ratherthan brittle materials. Moreover, using Hertzian the-ory alone to explain the relationship between MRRand static force, it appears impossible to obtain afunction of the form obtained from experimental testw x7 .

Accordingly, this paper adopts the nonlinear dy-namics approach to modelling MRR for brittle mate-rials, which is phenomenologically different to anyothers previously undertaken, and hopefully wouldbe of interest to both nonlinear dynamics and appliedphysiscs communities. It is based on applying im-

w xpacting oscillator theory 1012 to explain the mainmechanism occurring in ultrasonic drilling. In partic-ular, we shall address the formulation of a simplemodel of the non-linear dynamic interactions en-countered in the machine tool ultrasonic cuttingprocess system, which could explain the fall in theMRR for higher static loads. The basic assumptionof the proposed model is that the efficiency ofcutting is dependent upon both the size and fre-quency of the impact force during cutting.

The proposed dynamical model of ultrasonicdrilling is depicted in Fig. 2, where a resonanttransducer assembly tuned to the ultrasonic drivingfrequency is represented in a simple form as a twolump mass model. It consists of a mass m represent-1ing the movable headstock, and the equivalent massm of the vibrating ultrasonic horn including the2tool. Linear springs of stiffness k and k and1 2dashpots of viscous damping c and c connect the1 2headstock and the ultrasonic horn and tool to thepiezo-electric ultrasonic driver. This excites the sys-tem kinematically with amplitude A and frequencyv. The material of the workpiece is represented by astiffness k and damper c . The process is started3 3with an initial gap g between the tool tip and theworkpiece. A force f is required to drive theHYDheadstock downwards, and the velocity of the head islimited by damping c in the vertical drive. This0headstock model is particularly appropriate for thesoft hydraulic vertical drive typically used in thesemachines.

The model represents only the dynamical part ofthe displacement x corresponding to motion of the5

.cut face of the workpiece , and the gradual driftresulting from penetration into the material is sup-pressed. Furthermore, the model does not attempt to

Fig. 2. A dynamic model of rotary ultrasonic drilling.

represent in detail the full dynamics of the ultrasonictransducer assembly; it is sufficient, however, todescribe the essential interactions of the static forceand the transducer with the workpiece.

The equations of motion for the system in non-di-mensional form are as follows:x

XX q2j xX q2j xX y2j xX qx yx1 10 1 11 1 11 2 1 2s f t 1 . .HYD

2j xX q2j xX q2j a xX y2j a xX11 1 11 2 12 12 2 12 12 4yx q 1qa x ya x j a avcos nt . .1 12 2 12 4 12 12qa a sin nt 2 . .12

xXX y2j a xX q2j a xX ya x qa x4 21 21 2 21 21 4 21 2 21 4

sy2j a avcos nt qa a sin nt . .21 21 21for x yx -g4 5

3a .

( )M. Wiercigroch et al.rPhysics Letters A 259 1999 919694

xXX y2j a xX q2j a xX q2j a xX4 21 21 2 21 21 4 22 22 5

ya x qa x qa x21 2 21 4 22 5sy2j a avcos nt qa a sin nt . .21 21 21

for x yx sg4 53b .

2j a xX qa x s0 for x yx -g22 22 5 22 5 4 54a .

xX sxX for x yx sg5 4 4 5

4b .where

d k k1 2X 2 2s , tsv t , v s , v s ,11 11 12dt m m1 1k k c2 3 02 2v s , v s , j s ,21 22 10m m 2v m2 2 11 1

c c c1 2 2j s , j s , j s ,11 12 212v m 2v m 2v m11 1 12 1 211 1

c v v3 i jj s , a s , ns .22 i j2v m v v22 2 11 11

It is assumed that the MRR is a function of themagnitude of the impact force and its frequencyw x1316 . This is consistent with a removal process inwhich the tool tip impacts the workpiece, makingmicro-cracks on its surface. Other assumptions arethat the diamond is uniformly distributed on theworking part of the tool, with a uniform grit size,and that the ultrasonic amplitude and frequency, andthe geometry of the tool, remain constant. The rela-tive value of MRR is estimated from the followingalgorithm.

1.Initially an average value f of the impactAVG .force f t over all trials is calculated from

m t1 1 2f s f t dt 5 . . HAVGm t yt t2 1 1is1

where m is the number of numerical simulations,and t , t are the integration limits of a single trial,1 2

.chosen to provide a stable time history. Eq. 5 isre-written replacing the integral with a sum as fol-lows

m m1 1f s f 6 . AVG i jm nis1 js1

.using the trapezium rule with interval t yt rn.2 1

2. Having calculated the average value f ofAVGthe impact force, the number of excursions of theinstantaneous impact force above this value, n ,CROSSis counted. The number of cycles of the impact forceis then defined as:

nCROSSn s . 7 .CYC 2

3. It is assumed that there is a minimum levelf of impact force sufficient to cause significantMINdamage to the material being cut.

4. It is postulated that the MRR can then beestimated from the following formula

n )n dCYC jMRRs d 8 . j2 /fn AVGjs0where n is the number of time steps and z is anaccumulated damage parameter, normally chosenby a direct comparison with the experimental results.

U Here, d and d are respectively the impact force atj j. Utime-step j and the damaging impact force d sj

d y f .j AVGThe dynamical system parameters were identified

to model the ultrasonic drilling machine developed atw xAberdeen University 17 and the cutting material

was chosen to be a float glass. The value f ofMINimpact force sufficient to cause significant damagewas experimentally evaluated using a specially de-

w xsigned load cell 18 . Because the system underinvestigation is discontinuous, special precautionsare needed to maintain numerical accuracy at thetimes when the discontinuity occurs. Initially, spe-cially designed procedures for handling motion-de-pendent discontinuities were used within the fourthorder RungeKutta scheme. These procedures in-volve detecting discontinuities and calculating pre-

cise values of times when they occur see e.g.w x.19,20 . However, it was later found that for therange of parameters investigated and the dynamicresponses generated, a sixth order RungeKutta

w xmethod 21 with automatic time step adjustmentw xsimilar to the Fehlberg scheme 22 is faster and able

to achieve a satisfactory accuracy of solution. Withthe initial gap g equal to zero, the sign of displace-ment of the headstock is dependent upon the magni-tude of the control static force f . If this force isHYDsmall enough the steady state displacement x is1

( )M. Wiercigroch et al.rPhysics Letters A 259 1999 9196 95

. .Fig. 3. Time history of a the tool displacement and b theimpact force.

. .negative. Fig. 3 a and b show the time histories of .the x displacements and the impact force, d t .1

The role of f is to supply an optimum condi-HYDtion to allow the biggest impacting forces. Clearly,under the conditions of Fig. 3, for instance, the lightload causes the head to bounce off the workpiece,with only intermittent and small impact forces. How-ever, when f is too big the hammering effectHYDdisappears and the workpiece is loaded with a forcewhich possesses a large static component but negli-gible impact component, decreasing significantly thecutting efficiency. Fig. 4 shows the amplitude of theimpact force and its distribution in the time domainfor different values of f . It is evident that theHYDmagnitude of the impact force and its frequencystrongly depend on the static load.

This point is emphasised by the graph of Fig. 5,which shows the MRR as a function of f , calcu-HYDlated using the above algorithm and then normalisedwith respect to the MRR obtained without ultrason-ics. In practical terms, Fig. 5 depicts the ratio be-tween the material removal rate with ultrasonics andwithout for the same cutting conditions. The upperand lower envelopes of the MRR characteristic are

w xvery similar to results obtained experimentally 5,7 .The jagged form of the theoretical MRR graph

appears to arise from the nonlinear dynamics of themodel, but it has not been observed experimentally.

Fig. 4. Influence of the static hydraulic force f on the timeHYD . .history of the impacting force for a f s0.1, b f s0.4HY D HYD

.and c f s0.8.HYD

In other words, as the parameter f is varied, theHYDsystem undergoes bifurcations, and also is influencedby the cut-off value f chosen to model theMINminimum impact force enabling microcrack propaga-

Fig. 5. The material removal rate versus static force.

( )M. Wiercigroch et al.rPhysics Letters A 259 1999 919696

tion. Further work is being undertaken to identifyfully the origin of this behaviour and to understandwhy it has not so far been detected experimentally.At this stage, it is worth noting that changes in thevalues of z in the algorithm for the MRR do notsmooth the curve, supporting the conclusion that theeffect is a direct result of the nonlinear dynamics.Nevertheless, the top or bottom envelope of thegraph predicts the experimentally observed form ofthe drop in MRR for higher values of the staticloading.

Acknowledgements

This research was supported jointly by the Uni-versity of Aberdeen Research Committee Grant

.No. R224 and European Community Science andTechnology Research Programme Grant No.

.ERB3510PL92103 . The authors would like also tothank the anonymous reviewer for his constructiveand stimulating comments.

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