a mechanistic approach to the prediction of material ... - CiteSeerX

PED-Vol. 64, MANUFACTURING SCIENCE AND ENGINEERING ASME 1993 And to appear at The ASME Transactions, Journal of Engineering for Industry

A MECHANISTIC APPROACH TO THE PREDICTION OF MATERIAL REMOVAL RATES IN ROTARY ULTRASONIC MACHINING

Z. J. Pei, D. Prabhakar, P. M. Ferreira Department of Mechanical and Industrial Engineering University of Illinois at Urbana-Champaign

M. Haselkorn Technical Center Caterpillar Inc. Peoria, Illinois

ABSTRACT ratio, high hardness, corrosion resistance, and oxidatio An approach to modeling the material removal rateresistance) (MRR) result in superior performance which in tur during rotary ultrasonic machining (RUM) of ceramics translates is to significant cost savings. proposed and applied to predicting the MRR for the case The of very properties of ceramics that make them attracti magnesia stabilized zirconia. The model, a first attempt from aatproduct performance standpoint are also responsible predicting the MRR in RUM, is based on the assumption difficulties that encountered in shaping/machining them to brittle fracture is the primary mechanism of material precise removal. size and shape (often demanded by such application To justify this assumption, a model parameter (whichPresently, models the machining cost associated with certain ceram the ratio of the fractured volume to the indented volume components of a can be as high as 90 percent of the total co single diamond particle) is shown to be invariant for (Jahanmir most et al., 1992). Additionally, the machining o machining conditions. The model is mechanistic in the shaping sense process is often responsible for strength degradat that this parameter can be observed experimentally from of the a fewceramic components. This can increase the experiments for a particular material and then used susceptibility in of a ceramic component to sudden failur prediction of MRR over a wide range of process parameters. making it necessary to use processes which involve very lo This is demonstrated for magnesia stabilized zirconia, pressures where (e.g., grinding and lapping). Such processes tend very good predications are obtained using an estimate beof extremely this slow and typically expensive. single parameter. On the basis of this model, relations The above-mentioned difficulties associated with the use between the material removal rate and the controllable advanced ceramics are addressed by two different areas machining parameters are deduced. These relationships ceramic agreeresearch: processing and machining. In machining, well with the trends observed by experimental observations is apparent that there is a crucial need for the developmen made by other investigators. processes which are capable of relatively high materia removal rates while maintaining the sub-surface damage to ceramics at an acceptable level. 1. INTRODUCTION Studies (Prabhakar, 1992; Stinton, 1988) of various mater The demand for improved product performance has removal led to theprocesses applicable to ceramics indicated that rot emergence of advanced ceramics as an important class ultrasonic of machining (RUM) has the potential for high materials in the latter half of the twentieth century. Currently, material removal rates while maintaining low cutting pressu advanced ceramics are increasingly used for commercial and resulting in little surface damage and consequently lit applications in the aerospace, automotive, electronics, strength and reduction. Further, the potential exists o cutting tools industry. The superior properties of the implementing advanced RUM on conventional machines with some ceramics (such as chemical inertness, high strength modifications. and stiffness at elevated temperatures, high strength to weight

771


Coolant In MRR (mm3/sec)

6

Vibration

Rotation

Coolant Out

Core Drill

5 4 3 2 1 0

0

5

10 15 20 25 30 35 40 Amplitude (µm) 800 rpm 1500 rpm

Workpiece

3000 rpm

Magnesia stabilized zirconia Machining pressure = 0.155 MPa FIGURE 2 INFLUENCE OF VIBRATION AMPLITUDE ON MRR

Constant Force FIGURE 1 SCHEMATIC ILLUSTRATION OF ROTARY ULTRASONIC MACHINING

influences of the ultrasonic vibration amplitude, the appli static pressure, the rotating speed, the diamond type, grit s and bond type, etc., on the material removal rate (MRR) ha been investigated experimentally (Kubota et al., 1977; Legg In rotary ultrasonic machining process, a rotating core drill 1966; Markov, 1966; Markov and Ustinov, 1972; Markov e with metal bonded diamond abrasives is ultrasonicallyal., vibrated 1977; Petrukha et al, 1971; Prabhakar et al., 1992 by means of an ultrasonic transducer while the workpiece Tyrrell, 1970). To the authors' best knowledge, no theoreti being fed towards the core drill at a constant pressure. models have been published to explain the material remov Coolant pumped through the core of the drill washes mechanism away the and predict the MRR for RUM. Attempts to swarf, prevents jamming of the drill and keeps it cool.develop This is theoretical models to predict the MRR for RUM ar illustrated in Figure 1. Light pressures and clean cuts produced desirable since they would help in understanding th by this process make it ideal for ceramic machining (Stinton, mechanism of RUM and in the optimization of parameters 1988). Experimental results (Prabhakar, 1992) haveobtain shown required performance from the process. that the machining rate obtained from rotary ultrasonic The aim of this paper is to develop a mechanistic model machining is nearly 6-10 times higher than that predict from a the MRR in rotary ultrasonic machining. The paper conventional grinding process under similar conditions, see organized into five sections. Section 1 (this section) is a Figure 2. Important parameters involved in the process include general introduction to rotary ultrasonic machining o ultrasonic vibration amplitude, static pressure or static force, Section 2 outlines the approach used for mod ceramics. rotating speed of the tool, grit size and grit number of the development, while Section 3 develops the model. Section abrasives, and the bond type. The frequency of vibrations can the experiments performed and the verification of discusses also be considered a process parameter. However, hypotheses posed in Section 3. Conclusions are drawn up experimental evidence indicates that at frequenciesSection above5 along with a discussion of future directions of work 15kHz (which is typically the case for RUM), it has no observable effects on the process. RUM has been around for more than twenty 2. years APPROACH TO MODEL DEVELOPMENT (Anonymous, 1964; Hards, 1966; Kubota et al., 1977; Legge, RUM might be considered as a combination of the ultrason 1965; Legge, 1966; Markov, 1966) and many investigators machining process and the diamond grinding process. Hen have reported their studies on it. However, these studies have there are two principal approaches to developing a model been primarily experimental with little or no attempt to predicting MRR for RUM: one considers the process as bein develop a model governing material removal mechanism. The

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predominantly ultrasonic machining and superimposes the effect of rotating motion of the tool, the other reverses the two Load primary processes. The first approach is used here. To develop the model for RUM process, the following steps are carried out: Diamond (1) Estimate δ, the depth to which the diamond particles Particle penetrates into the workpiece surface; (2) Estimate F, the contact force between the diamond d particles and the workpiece; (3) Estimate W, the volume removed by one diamond particle in a single cycle of the ultrasonic vibration; δ (4) Aggregate the effect of all the diamond particles to obtain the material removal rate of the process. Workpiece Similar steps have been followed by other investigators (Kanith et al., 1979; Shaw, 1956), in developing theoretical models to predict material removal rates for conventional ultrasonic machining. However they used different methods to FIGURE 3 SCHEMATIC ILLUSTRATION OF THE calculate δ, F and W. One of the primary differences betweenABRASIVE-WORKPIECE INTERFACE conventional ultrasonic machining and rotary ultrasonic machining is the rotary motion of the tool which makes it possible for diamond particles to hit the workpiece at equation different can be used to relate the contact force with th locations in consecutive cycles. Unlike the conventional indentation depth. Although the Hertz equations are stric ultrasonic machining, the rotary ultrasonic machining does applicable only up to the point of initial surface fractur not use an abrasive slurry. Therefore, in the absence Sheldon of rotaryand Finnie (1966) showed that "the Hertz relatio motion of the tool, the diamond particles would always hit be theused to predict the penetration of a particle into may workpiece in the same locations, making the material surface removaleven after cracking occurs" within certain ranges. rate very low. Our analysis will model the rotation of the toolindentation depth, δ, is the maximum depth to whi If the and demonstrate how it affects the MRR. This effect diamond will be particles penetrate the workpiece surface, as shown discussed in greater detail later in this paper. Figure 3, then, according to Timoshenko and Goodier (1970 δ can be calculated as follows (The Young's modulus of diamond Ed is taken to be much larger than the Young 3. DEVELOPMENT OF THE MODEL modulus of workpiece material E. So, E/Ed approaches zero.) As Sheldon and Finnie (1966) pointed out, "Before analyzing 1/3 any material removal process it is convenient to idealize the 9 (F/n)21-ν22 material's behavior as either 'ductile' or 'brittle'. δ =This ( ( ) ) (1) 1 6 d/2 E approach is clearly an oversimplification, since materials may exhibit both ductile and brittle behavior, depending on testing where, conditions, but it enables solutions to be developed which are F -- the maximum contact force between tool and workpiec useful in many practical situations." In a ductile material, N; large plastic strains precede fracture and material is removed by n -- the number of active abrasive particles across the to the displacement action of the cutting tool or abrasive face; particles. In contrast, for an ideally brittle material, no plastic d -- the diameter of the abrasive particles, mm; deformation is present and material is removed by the E -- Young's modulus of workpiece material, MPa; propagation and intersection of cracks ahead of and around the ν -- Poisson's ratio of workpiece material. cutting tool or abrasive particles. The model developed below In the right side of the above equation, all the parameters is based on several simplifications: known except F, the maximum contact force. The workpiece material is an ideally brittle material; The material is removed by Hertz fracture; The diamond abrasive particles are assumed to be rigid 3.2 Estimation of the Maximum Contact Force F spheres of the same size, and all the particles are assumed to The workpiece is fed into the tool by a constant pressure take part in cutting during each ultrasonic cycle. static force Fs . The tool is not in continuous contact with t workpiece due to its oscillatory motion. When an abrasiv particle has penetrated the workpiece surface to maximu 3.1 Estimation of δ In each ultrasonic cycle of the tool, each diamond depth, particlethe force transmitted between tool and workpiec through will make contact with the workpiece. The classical Hertz a single abrasive particle will be F/n, where F

773


30

0

t1 t2

t

Second)

y=Asin2πft

25

−6

δ

(10

Indentation Time

∆t

y A A- δ

20

Equation 4

15 10 Equation 3

5 0 0

Core drill

0.2

0.4

0.6

0.8

1

Indentation Depth/Amplitude

Mean position

0

FIGURE 5 DIFFERENCES OF INDENTATION TIME CALCULATED BY EQUATIONS (3) AND (4)

A

This has the advantage of simplicity and might be applicab when the motion of abrasives relative to the workpiece is on vertical (due to the oscillation of the tool). However , whe the tool is also rotating with respect to the workpiece (as the case with RUM), large inaccuracies in volume calculation can result, especially when δ is very small relative to A. The tip of the core drill oscillates with an amplitude A and frequency f. The motion is considered to be sinusoidal and position of the tool tip relative to its mean position may described by the following equation:

A

y

Workpiece

δ

FIGURE 4 CALCULATION OF ∆t

y = A sin (2πft) maximum contact force between tool and workpiece and n the number of active abrasive particles across the tool face. It will take the tool tip ∆t/2 to move from y = A - δ to y = A Assuming that the diamond particle is incompressible, thecan be accurately calculated using the following So, ∆t impulse for one cycle in terms of the maximum contactequation force F (see Figure 4). is:

∫ Fdt =~ F ∆t cycle

δ 1 π ∆t = 2 (t2 - t1) ={ - arcsin(1 )} 2 A πf

(2)

(4)

For different values of δ/A, ∆t is calculated by equations (3 where ∆t is the period of time during which the particle has and (4) and shown in Figure 5. It can be seen that th penetrated the workpiece surface. Some investigators (for percentage differences, specially at lower values of example, Shaw, 1956) have used a very simple relation to quite large. The MRR of rotary ultrasonic machining is mor calculate ∆t . sensitive to the value of ∆t than that of conventiona ultrasonic machining, because the distance L moved by δ 1 ∆t = 2 (3) diamond particle (due to the rotating motion of the tool) wh Α 4f penetrating the workpiece surface is also dependent on might be noted that attempts to predict MRR using where, A -- the ultrasonic vibration amplitude, mm; calculated by equation (3) exhibits a larger deviation from f -- the ultrasonic vibration frequency, Hz; experimental data than using ∆t calculated by equation (4).) δ -- the indentation depth, mm.

774


L Diamond grit d 2

d 2

δ Workpiece

d+L 2 FIGURE 6 CALCULATION OF CONTACT LENGTH L FIGURE 7 EFFECTIVE SIZE OF DIAMOND GRIT DUE TO THE ROTATING MOTION OF THE TOOL Substituting (4) into (2), we get the impulse for one cycle in terms of the maximum contact force F:

where, D -- the tool diameter, mm; cycle S -- the rotating speed, rpm; ∆t -- the period of time during which the particle has inden The impulse for one cycle in terms of the static force Fs is: into the workpiece surface, second. Substituting (4) into the above equation, we get: 1 Fs Fs = f f δ DS π L= { - arcsin(1 )} (7) 60f 2 A By equating the two impulses, we get the following relation: F π

δ

∫ Fdt =~πf {2 - arcsin(1A)}

(5)

Now, during the period of time ∆t, the penetration of th particle increases from 0 to δ and decreases to 0 while th π δ particle moves through a distance L on the surface of th ( ּ-ּarcsin(1ּ-ּ )) 2 A workpiece. As a result, the width of the "intersection crate between the workpiece and the particle will also increase fro δ and F are the only two unknown terms in equationszero (1) to and some maximum value and decrease to zero. In short, (6), and hence they can be solved for by using equations shape (1) of the intersection crater will be a part of an ellipso and (6). Or The volume of this part of the ellipsoid, or, the indentati volume, can be calculated by integration (see Figure 7). 1/3 π 91 (F/n)21-ν22 F { - arcsin(1 (( ) ) )} = πFs 2 A 1 6 d/2 E Ld δ V = π (1 + ) ( - ) δ2 (8) d 2 3 can be solved for F, the maximum contact force, by trial and where, error. δ can be obtained by substituting for F in equation (1). d -- the diameter of the spherical abrasive, mm; δ −− the depth of maximum penetration, mm; 3.3 Estimation of the Indentation Volume V L -- the distance moved by the particle during penetrati Having developed an approach to estimating the maximum into the workpiece due to the rotary motion of the tool, mm contact force and consequently the maximum depth of indentation, this sub-section deals with estimating the volume of penetration due to a single diamond particle in 3.4 a single Estimation of MRR ultrasonic cycle. Once the indentation volume per particle per cycle has be Due to the rotating motion of the tool, the diamondobtained, particle the MRR can be predicted if the relationship betwe would move a distance L while in contact with the workpiece the fractured volume and the indentation volume is know (see Figure 6). However, such relationship is not available in literature perhaps due to its complexity and the number of facto πDS influencing it. We propose a simplified possibility which L= ∆t 60 aggregates these influences into a single parameter. If W is t F=

πFs

(6)

775


Electric Motor Ultrasonic Spindle Kit

Spindle Power SupplySpeed Controller Vibrations Core Drill Workpiece

Rotary Motion Constant Pressure Feed System Feed Control Panel

Milling Machine Table

Coolant System Pump

Coolant Tank

FIGURE 8 SCHEMATIC ILLUSTRATION OF THE ROTARY ULTRASONIC MACHINING SETUP

volume of the fractured zone, it can be consideredvolume to be is usually larger than indentation volume since t proportional to V. Therefore, cracks may initiate and propagate outside the indentati volume. k1 and k 2 have the opposite effects (k1 decreasin W = kV and k2 increasing) on the value of k. The MRR is given by the product of the volume of materia where the constant of proportionality, k, could conceivably removed be by one particle, W, the frequency of vibration, f, a a function of the material properties, process parameters, the number of active diamond particles, n. Hence, the MRR probability of causing fracture, etc.. However, for V given to be by a the following equation: useful intermediate parameter in the estimation of the MRR in RUM, k would have to stay relatively constant for a MRR given= nfW = nfkV material over a wide range of process parameters. At this point we will assume that k is independent of process parameters, Substituting (8) into this equation, we get: proceed to develop the equation for MRR and then experimentally verify that the assumption on the invariance is Ld δ MRR = knfπ( 1 )+ ( - ) δ2 (9) indeed valid. d 2 3 If the assumption made above is valid then k can be considered to be a function of two primary effects, i.e., k = where δ can be obtained from equations (1) and (6). L can f(k1 , k2 ), where k1 is the probability coefficient which brings from equation (7). On the right hand of equation obtained into consideration the fact that it may take more than only one k has not been decided. As mentioned earlier, this val indentation to remove the volume W. k 2 is the volume will have to be experimentally determined. coefficient which accommodates the fact that the fractured

776


Actuating Circuit

Core Drill Workpiece

Pressure Regulator Pressure Gage AC

Die-Set Upper Plate Lower Plate Feed Direction Hydraulic Cylinder

Valve Milling Machine Table

Air/Oil Tanks

FIGURE 9 CONSTANT PRESSURE FEED SYSTEM

machine setup, in which the drill is fed down into th workpiece, in the present setup the workpiece is raised u towards the drill. Further, the feed pressure is controlle 4.1 Experimental Setup instead of the feedrate. The support fixtures aid in securing Figure 8 schematically illustrates the rotary ultrasonic lower plate to the milling machine table. The workpiece machining setup. It consists of an ultrasonic spindle clamped kit, a firmly on the upper plate. The only moving parts constant pressure feed system and a coolant system. the feed system are the upper die-set plate (work-table) and The ultrasonic spindle kit comprises of an ultrasonicworkpiece. spindle, The feed (static) pressure is preset before the st a power supply and a motor speed controller. The ultrasonic of machining operation. When the circuit is activated th spindle is mounted on a milling machine (replacing piston therises from the hydraulic cylinder raising the workpie toolhead of the milling machine). The spindle contains towards an the drill. ultrasonic transducer. The power supply converts The 50Hzexperimental setup employs the coolant system of t electrical supply to high frequency (20kHz) AC output.milling This ismachine. fed to the piezoelectric transducer located in the spindle. The transducer converts electrical input into mechanical 4.2 Design of Parametric Set of Experiments vibrations. By changing the setting of the output control of the power supply, the amplitude of the ultrasonic vibration For can the purpose of estimating k, data are used from be adjusted. The rotational motion of the tool is supplied parametric by study conducted on RUM. If k is independent the motor attached atop the spindle and different speeds machining can be parameters, as assumed in the model developme obtained by adjusting the motor speed controller. then theoretically only one experiment is needed to get Figure 9 shows a schematic illustration of the constant value. However, to verify that it is indeed independent pressure feed system. The basic purpose of the system machining is to parameters, a number of different experiments feed the workpiece towards the tool at a constant pressure. various Thecombination of machining parameters are neede system consists of a four-pillar die-set, an actuating Further, circuit, because of the large variations that usually occurs and support fixtures. The die-set consists of two plates any experiments of that involves ceramics, to get a bette which the upper plate serves as the work-table. Thisestimate, plate is a number of experiments will be needed. A moved vertically by means of a double acting hydraulic systematic way of doing this would require a proper cylinder. The cylinder is driven by an actuating circuit experimental which design. The design of experiments involved fi consists of an electronic pressure regulator, pressure gage, control and variables (machining parameters) shown in Table a three port-two way valve. Unlike a conventional drilling Therefore, for a two-level full-factorial design, at least 3 4. VERIFICATION OF THE MODEL

777

PED-Vol. 64, MANUFACTURING SCIENCE AND ENGINEERING ASME 1993 And to appear at The ASME Transactions, Journal of Engineering for Industry TABLE 1 CONTROL VARIABLES AND THEIR LEVELS IN EXPERIMENTS

10

Experimental MRR (mm

3

Control Variable Low level High Level Static Force (N) 111 356 Vibration Amplitude (mm) 0.023 0.033 Rotating Speed (rpm) 1000 3000 Abrasive Bond Type Brass-copperIron-nickel Abrasive Grit Size (mm) 0.05 0.22

8

6

4

experiments need to be performed. Each test is replicated once, bringing the total number of tests to 64. The levels shown in Table 1 represent the typical high and low settings for the process parameters. 2 The following variables were held constant during all test runs. Workpiece Material: Magnesia Stabilized Zirconia (Young's 0 modulus E = 205000MPa, Poisson's ratio ν=0.31); 0 2 4 6 8 10 12 Coolant: A water based semi-synthesis emulsifier; 2 3 nfπ(1 + L/d)(d/2 - δ/3)δ (mm /s) Abrasive: Diamond particles. In each test the time taken to drill a hole of 12.7mm in FIGURE 10 CALCULATION OF K diameter into the 6.35mm thick workpiece was recorded. The material removal rate for any machining operation is given by: the process parameters. However, we suspect that Hert fracture, as modeled above, is the dominating influence. As VolumeּofּMaterialּRemoved first approximation, we are assuming that it is independent MRR = ּTime machining parameters. The purpose of this section is t validate this claim for one material and to estimate the value Hence, for rotary ultrasonic machining, i.e., drilling operation k for the material using the data obtained from the experim using core drills, MRR is described in the previous section. For the purpose of evaluating the assumption that k sta 2 D h 2 ּ-ּD d πH relatively constant over the parameter ranges, the data from MRR = two-level, full-factorial experiment (described above) is use 4t so that all process parameters are varied over their ranges. D from all the experiments will be used to estimate k as the slo where, of the least-squares straight line (Neter et al., 1990) passi Dh -- the hole diameter, mm; through the origin and relating the observed MRR for eac Dd -- the machined core diameter, mm; d δ 2L H -- the thickness of the workpiece, mm; experiment with the corresponding )nfπ( ( - 1) + δ value d 2 3 t -- the time consumed during machining, second. forofthe For further details of the experiments, the calibration theexperiment (see Figure 10). The value of k for th overall setup, measurement of variables and results the reader isdata is 0.618. Next, to test if the value is in fac constant, referred to Prabhakar (1992) and Prabhakar et al. (1992). In for each parameter, the data will be divided into t those studies, the experimentally observed effects ofgroups, each of corresponding to the high and low level of th parameter. The parameter k will be estimated for each group the parameters on the MRR were studied. Further, an empirical a manner similar to that described above. If our assumpti models was given. In this paper, the results of the experiment that k is are used for estimating k and testing the hypothesis that it constant is true, the estimated values for each of groups will not differ significantly from the value determin does stay constant for a particular material. for the overall data. From Figure 10, it can be seen that there exists some deviation of the predicted MRR values from the experiment 4.3 Analysis of the Experimental Results The scattering of the data may be the results of th The model developed relies on the assumption that kdata. remains following. invariant across a broad range of process parameters for a particular material. Strictly speaking, k might be a function of

(

)

778

0.943

3

-2

0.575

0.587

0.597

0.639

0.703 0.523

0.6

Rotating Speed -

-1

0.6

0

Amplitude +

0.618

0.8

0.644

1

0.781

1

2

k

Estimated Cumulative Probability Converted to Standard Normal Variable


0.4

-3 -3

-2

-1

0

1

2

3

Standard Value of the Differences Between the Predicted and Experimental MRR 0.2

Grit Size +

Grit Size -

Force +

Force -

Bond Type +

Bond Type -

Rotating Speed +

Amplitude -

0

Total Data

FIGURE 11 Q-Q PLOT

(1) Large dispersion of the data may result from unbiased measurement errors and the probabilistic nature of the flaw distribution in ceramics. (2) All abrasives are assumed to take part in machining, with all the particles having the same shape and size. In reality, it is more likely that only a fraction of particles take part in machining. (3) Other assumptions previously described. Parameter Level First, a check needs to be made to test whether there exist any particular trends in the residuals, i.e., differences between the FIGURE 12 INFLUENCE OF PARAMETER LEVELS ON K MRR values predicted by the straight line and the observed values. For this a Q-Q plot (DeVor et al., 1992) is made and shown in Figure 11. Clearly, the residuals are random in nature the material removal mechanism at very low contact force and no underlying trends are detected. Next, for each Our initial analysis suggests that this could be plastic parameter, the data is separated into two groups corresponding deformation. As the contact force is reduced, the depth to the high level and low level of that parameter. Figure 12 penetration of the abrasives into the work material keep shows the values of k estimated for each of these groups. At reducing. At some contact forces, this could be close to th the 0.001 level of significance, the hypothesis that the value ductile-brittle transition for the material. (For the mater of k is not significantly different from 0.618 has not been used, this is roughly 0.1 to 0.4 microns). This suggests that rejected for all groups except the group that corresponds to the lower forces a different model might be required. low level of static force. Based on this analysis, one can state that the assumption of k being constant for a particular material is reasonable and can be applied as a first 4.4 The Influence of Different Parameters on approximation to evaluating the material removal rate for a MRR given material and set of process parameters. In the previous sections, we have developed a simplifie If k were to be thought of as the efficiency of the rotary analytical model for MRR in rotary ultrasonic machining und ultrasonic machining for a particular material, then at low the assumption that a particular model parameter, k, rema forces, even though the MRR is low, a statistically significant constant for a given material. In this section, we will use th higher efficiency is obtained. This significant change in the model to study how individual machining parameters influe "efficiency" of the process might be attributable to a the MRR and compare the trends predicted by the model w mechanism other than Hertz fracture, that begins to dominate

779


40

F/n (N)

5 4

30 20 10

3

0 8

2

6

δ (µm)

MRR (mm3/s)

6

1

2 0

0.01

0.02 0.03 0.04 Amplitude (mm)

Fs=600N Fs=400N

0.05

16

L (µm)

0 0

4

Fs=200N Fs=100N

12 8 4 0

S=1000rpm, d=0.05mm, n=100

3 MRR (mm3/s) V (10-6 mm )

5 4 3 FIGURE 13 RELATION BETWEEN AMPLITUDE AND MRR 2 1 those observed by other researchers. It must be noted here that 0 only trends can be compared, because the precise experimental conditions for the experiments (reported by other6

investigators) are not known. The model is applied to predict the relations between the MRR and the different parameters 4 for rotary ultrasonic machining of magnesia stabilized zirconia. 2 The value of k is taken as 0.618. The predicted relation between MRR and the amplitude of the 0 ultrasonic vibration has been plotted in Figure 13. Figure 14 10 20 30 40 50 shows the variation of different important components of0 equation (9) (the MRR equation) with variation in amplitude. Amplitude (µm) Specifically, the indentation force/indenter, the depth of S=1000rpm, Fs=600N, indentation and the distance moved by an indenter when in d=0.05mm, n=100 contact with the workpiece (length of contact) due to the rotational motion of the tool are shown in this figure. Finally the volume of the indentation and the MRR are also shown. FIGURE 14 INFLUENCE OF AMPLITUDE Two important effects are visible. First, the indentation force and depth increase at a decreasing rate with amplitude. Second, the length of contact decreases with increasing amplitude. This is certainly true, however, our model (which does n These two effects cause the MRR to increase at a decreasing consider wear) suggests that in addition, the proces rate, suggesting that at some amplitude the curve will flatten mechanics, explained above, is also responsible for such and possibly begin to drop. The experimental data reported by behavior. Markov and Ustinov (1972) shows that further increases The of relation between MRR and the static force is shown i ultrasonic amplitude above a certain value will result in a15. It can be seen that the predicted MRR will alway Figure reduction in the MRR. The reason for this has been explained increase with the static force. The experimental data repor as "due to an excessive increase in alternate loadingby onMarkov the and Ustinov (1972) also show that MRR first diamond grits and a weakening of the bond" and increases further with the static force until reaching a certain valu increase of ultrasonic amplitude "may result in complete andfailure then decreases with the static force. No explanation h of the diamond core bits as a result of the high cycle stresses". been given for this phenomenon by them. According to o

780


20 F/n (N)

3

1

10 5

0

100

200 300 400 Static Force (N)

A=0.035mm A=.025mm

4 2 0 8

500 L (µm)

0

15

0 6

2 δ (µm)

MRR (mm3/s)

4

A=0.015mm A=0.005mm

6 4 2 0

MRR (mm3/s) V (10-6mm3

S=1000rpm, d=0.05mm, n=100 FIGURE 15 RELATION BETWEEN STATIC FORCE AND MRR

4 3 2 1

analysis, it may also be due to the rapid wear of the core drill. 0 Over the region of comparable forces, the trends of the 3 proposed model and the experimental trends are similar. The difference at higher values is due to the fact that our model does 2 not account for such factors as tool wear. Figure 16 shows the variation of different components of the MRR equation with 1 static force. The almost linear trend is due to the fact that both the indentation depth and the contact length increase with static force causing the indentation volume to increase0 0 100 200 300 400 500 steadily. The predicted relation between MRR and the rotating speed Static Force (N) has been plotted in Figure 17. It is seen that MRR increases as S=1000rpm, A=0.025mm, the rotating speed is increased. Markov and Ustinov (1972) d=0.05mm, n=100 state that "material removal rate increases as the peripheral speed of the drill is increased". They do not report on the characteristics. Our model agrees with this qualitativeFIGURE 16 INFLUENCE OF STATIC FORCE statement. Figure 18 shows the different components of the MRR equation. From this figure, one can observe that the However, indentation force and depth are unaffected by the rotating speedthe number of indenters increases to offset th rendering MRR almost independent of which only causes a change in the length of contact.reduction As the concentration. It must, however, be pointed out that th rotating speed increases, the indentation volume changes independence is observed for the particular constants (mate proportionally and the MRR also increases. properties) chosen. A harder or softer material may result i The predicted relation between MRR and the diamond grit decrease concentration (through the grit number) has been plotted inor increase of MRR with diamond concentration. Figureּ19. Figureּ20 shows the components of the The MRRpredicted relation between MRR and the diamond gr diameter equation. It can be observed that as the grit number increases, has been plotted in Figure 21. The components the MRR the indentation force per indenter and the indentation depthequation are shown in Figure 22. It can be seen th force increases as the diameter of the diam decrease, which in turn reduces the contact length. the All indentation this particles increases. However, the indentation depth reduces causes a reduction in the indentation volume per indenter.

781


6 F/n (N)

6

4

2 1

2

Fs=600N Fs=400N

4 2 0 16

1000 2000 3000 4000 Rotating Speed (rpm)

L (µm)

0 0

4

0 6

3 δ (µm)

MRR (mm3/s)

5

Fs=200N Fs=100N

12 8 4 0


A=0.025mm, d=0.05mm, n=100 FIGURE 17 RELATION BETWEEN ROTATING SPEED AND MRR

0.6 0.4 0.2

causes the contact length to do the same. The particle diameter, however, plays a dominant role in the calculation of the 0 6 indentation volume causing it to increase which increases the MRR. No experimental data on these relations are available 4 in literature. Figures 13, 15, 17, 19 and 21 collectively suggest that 2 second order effects exist between the process variables which is in agreement with the results obtained from a full factorial 0 experiment reported by Prabhakar et al. (1992). For example, 0 the positive effect of machining pressure (or static force) on the effectiveness of increased rotating speed in improving MRR is clearly evident in Figure 17.

3 4 1 2 Rotating Speed (1000 rpm) Fs=600N, A=0.025mm, d=0.05mm, n=100

5. CONCLUSION A theoretical model to predict MRR for RUM has beenFIGURE 18 INFLUENCE OF ROTATING SPEED developed. The model is based on the assumption that the brittle fracture mechanism is the dominant mode of material case, no statistically significant difference was observed in t removal. It might be only one of the mechanisms actually estimate of the model parameter. This leads us to believe th effecting material removal during rotary ultrasonic machining spite of the assumptions that workpiece material is idea of ceramics. The model requires the estimation of ainsingle brittle and the diamond particles are rigid spheres of the sa parameter (which is dependent on the properties of the material size being oversimplifications of the actual situation, th being machined). modelthe is, at least, a good first approximation of the mater An experimental investigation was conducted to estimate removal model parameter and verify that it was, in fact, constant over mechanism. The model parameter which can b estimated for different materials can be thought of as th the entire range of machining parameters for magnesia efficiency of the RUM process for that material. stabilized zirconia. Statistical analysis of the experimental Thethe model has been used to study the influence of differe data indicated that it was reasonable to assume that machining parameter is constant. For all cases except the low static force parameters on the MRR. The trends predicted by

782


8 F/n (N)

5 4

δ (µm)

2 1

4 2

0

300

600 900 Grit Number

Fs=400N Fs=300N

2 1 0 6

1200 L (µm)

0

6

0 3

3

Fs=200N Fs=100N

S=1000rpm, A=0.025mm, d=0.05mm FIGURE 19 RELATION BETWEEN GRIT NUMBER AND MRR


MRR (mm3/s)

6

4 2 0 0.8

0.4

model are consistent with those reported in experimental 0 investigations. Mechanistic explanations for these idealized 1 trends are given in our discussion. This investigation, being the first attempt to theoretically predict the MRR in rotary ultrasonic machining, leaves a 0.5 number of avenues for follow-up work. First, models which account for plastic flow (especially at very low static forces and consequently very small indentation depths) are required. Next, a study of different materials for the estimating the 0 0 efficiency of RUM in machining them is required. MRR is often not the only consideration in process design. Tool wear and workpiece surface damage are important factors which need to be studied along with MRR.

2

4 6 8 10 12 Grit Number (100)

S=1000rpm, Fs=100N, A=0.025mm, d=0.05mm

ACKNOWLEDGMENT FIGURE 20 INFLUENCE OF GRIT NUMBER Financial assistance was provided by Caterpillar Inc., Hards, K. W., 1966, "Ultrasonic Speed Diamond Machining" through the Manufacturing Research Center of the University of Illinois, and by the National Science Foundation PYI Ceramic Award Age, Vol. 82, No. 12, pp. 34-36. Jahanmir, S., Ives, L. K., Ruff, A. W. and Peterson, M. B., (DDM - 9157191). 1992, "Ceramic Machining: Assessment of Current Practice and Research Needs in the United States", NIST Special Publication 834. REFERENCES Kanith, G. S., Nandy, A., and Singh, K, 1979, "On the Anonymous, 1964, "Ultrasonic Drilling with a Diamond Mechanics of Material Removal in Ultrasonic Machining," Impregnated Probe", Ultrasonics, Vol. 2, pp. 1-4. International Journal of Machine Tool Design and Research, DeVor, R. E., Chang, T. H., and Sutherland, J. W., 1992, Vol. 19, pp. 33-41. Statistical Quality Design and Control: Contemporary concepts and methods, Macmillan Publishing Company, New Kubota, M., Tamura, Y., and Shimamura, N., 1977, "Ultrasonic Machining with a Diamond Impregnated tool" York.

783


F/n (N)

8 6 4

δ (µm)

MRR(mm3/s)

10

2

0.2 0.3 0.4 Grit Diameter (mm)

Fs=400N Fs=300N

0.5 L (µm)

0.1

2 1 0 6

0 0

10 8 6 4 2 0 3

Fs=200N Fs=100N

4 2 0


S=1000rpm, A=0.025mm, n=100 FIGURE 21 RELATION BETWEEN GRIT DIAMETER AND MRR

2 1

Bulletin of the Japan Society of Precision Engineering, Vol.ּ11, No. 3, pp. 127-132. 0 Legge, P., 1965, "Ultrasonic Drilling of Ceramics", 3 Industrial Diamond Review, Vol. 24, No. 278, pp. 20-23. Legge, P., 1966, "Machining Without Abrasive Slurry", 2 Ultrasonics, Vol. 4, pp. 157-164. Markov, A. I., 1966, Ultrasonic Machining of Intractable 1 Materials (translated from Russian), Illife Books, London. Markov, A. I., and Ustinov, I. D., 1972, "A Study of the 0 Ultrasonic Diamond Drilling of Non-Metallic Materials", 0 0.1 0.2 0.3 0.4 0.5 Industrial Diamond Review, March, pp. 97-99. Grit Diameter (mm) Markov, A. I. et al., 1977, "Ultrasonic Drilling and Milling of Hard Non-Metallic Materials with Diamond Tools", S=1000rpm, Fs=100N, Machines and Tooling, Vol. 48, No. 9, pp. 45-47. A=0.025mm, n=100 Neter, J., Wasserman, W., and Kutner, M., 1990, Applied Linear Statistical Models: Regression, Analysis of Variance, and Experimental Designs, Irwin, Boston. FIGURE 22 INFLUENCE OF GRIT DIAMETER Petrukha, P. G. et al, 1970, "Ultrasonic Diamond Drilling of Deep Holes in Brittle Materials", Russian Engineering Sheldon, G. L., and Finnie, I., 1966, "The Mechanism of Material Removal in the Erosive Cutting of Brittle Materials" Journal, Vol. 50, No. 10, pp. 70-74. ASME Journal of Engineering for Industry, pp. 393-399. Prabhakar, D., 1992, "Machining Advanced Ceramic D. P., 1988, "Assessment of the State of the Art i Materials Using Rotary Ultrasonic Machining Process", Stinton, M.S. Machining and Surface Preparation of Ceramics", ORNL/TM Thesis, University of Illinois at Urbana-Champaign. Report Prabhakar, D., Ferreira, P. M., and Haselkorn, M., 1992, "An 10791, Oak Ridge National Laboratory, Tennessee. S., and Goodier, J. N., 1970, Theory of Experimental Investigation of Material Removal RatesTimoshenko, in Rotary Ultrasonic Machining", Transactions of the NorthElasticity, 3rd edition, McGraw-Hill Book Company, New York. American Manufacturing Research Institution of SME, Tyrrell, W. R., 1970, "Rotary Ultrasonic Machining", SME Vol.ּXX, pp. 211-218. Technical Paper MR70 - 516. Shaw, M. C., 1956, "Ultrasonic Grinding", Microtechnic, Vol. 10, No. 6, pp. 257-265.

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