Modelling of tool wear based on cutting forces in turning

Wear, 169 (1993)

25

25-32

Modelling of tool wear based on cutting forces in turning H.V. Ravindra, Mechanical

Engineering

Y.G. Srinivasa Department,

Indian

and R. Krishnamurthy Institute

of Technology,

Madras

600 036 (India)

(Received September 17, 1992; revised and accepted April 1, 1993)

Abstract Tool wear estimation

is essential for on-line process control and optimization. Wear and breakage of the tool are usually monitored by measuring force, load current, vibration, acoustic emission and temperature. These measurements are important for reliability and for the implementation of an adaptive control system. This paper proposes a development of mathematical models to describe the wear-time and wear-force relationships for turning operation. Cutting force components have been found to correlate well with progressive wear and tool failure. The results show that the ratio between force components is a better indicator of the wear process, compared with the estimate obtained using absolute values of the forces. It also eliminates variation in material properties, which was identified as a major noise source in signals measured during machining. This paper presents data on techniques based on force measurement for tool wear monitoring.

1. Introduction In-process assessment of the state of the cutting tool remains one of the major obstacles to the optimization of the metal cutting process. Hence monitoring and diagnostic systems are becoming increasingly necessary in manufacturing optimization. Tool failure is an important factor which affects productivity and manufacturing efficiency. Hence many systems have been developed to detect tool status (wear and breakage)

PI* Machine tools are required to operate at optimal efficiency. Several approaches have been used for process optimization, for example, adaptive control. Such optimization methods require estimation of process state variables such as tool wear, dimensional change and surface condition of the workpiece. To obtain a clear knowledge of methods suitable for tool wear estimation, the nature of the manufacturing system and that of signals measured from it must be known. Tool wear is influenced by the characteristics of the tool-work material pair and working conditions. Among the tool-work material characteristics, material hardness plays a significant role [2]. Variations in work material hardness (hot hardness) have been shown to be a contributing factor for stochastic variations in tool life. Further, the mean and variance of tool life are functions of cutting conditions [3]. Thus, it may be expected that signals from machining are both random and non-stationary, particularly when inhomogeneities and large-scale variations in material

0043-1648/93/$6.00

properties, such as those relating to cast iron, are present. The monitoring and estimation schemes must therefore be designed taking into account such sources of noise. The techniques for on-line tool wear sensing are usually divided into direct and indirect methods. Direct methods involve measurement of volumetric loss of material from the tool. Methods based on optical, pneumatic and irradiation techniques etc. have also been attempted. Direct methods generally tend to be off-line and cannot monitor tool failure during the process. Indirect methods involve measurement of a variable that can be related to tool wear. Hence methods based on measurement of tool forces [4,5], power, vibration [6], dynamic force [7] and acoustic emission (AE) etc. have been proposed. One of the most promising techniques for tool wear detection and breakage involves the measurement of cutting forces [8,9]. In turning operation, it is convenient to consider the tool forces as a three-component system. These are the tangential component F,, the axial (feed) component F, and the radial component F,,. Research on alternative signals for tool condition monitoring has not produced a universally accepted method. The research has tended towards multiple sensors of mainly cutting forces and AE [lO,ll]. The best results seem to be obtainable from tool forces and acoustic emission [12]. Vibration signals when used in conjunction with dynamic force seem to be a potential methodology [7]. There is no doubt that, for sophisticated signal processing, methods like AR [13] and ARMA modelling

0 1993 - Elsevier Sequoia. All rights

resewed

]14], data-dependent systems strategy [IS], and spectral analysis may be required, particularly of dynamic signals, Apart from these, second-stage processing using pattern recognition [11,16,17], fuzzy logic 1181, neural networks 1191 and group methods of data handling ]20] may produce more accurate monitoring systems. Many of the above methods are capable of handling inputs from multiple sensors and these have to be exploited. In general, these methods are computationally demanding and may not be ~plemented in small systems. The present work attempts to devise schemes for tool wear estimation in a turning process by measurement of steady state cutting force components, the dynamic component of the main cutting force and toolholder vibration in the direction of the main cutting force. A rn~~ti-sensor approach has been proposed, The turning trials have been carried out on S.G. iron - a cast iron with a steel matrix, using coated cemented carbide tools. Forces are a function of cutting conditions, properties of tool and work material, structural rigidity of the machine, status of lubrication etc. Hence the direct use of forces will require a threshold based on the working conditions. The effects of feed and speed on forces have been indicated &?I]. From such data, it will be possible to model forces as functions of cutting conditions. The effects of other variables, such as tool and work materials, status of lubrication etc., are more subtfe and complex and difficuh to quantity. The possible solution to this is to develop different models for each pair of tool-work material combination. This is complex in terms of both experimentation and modelling. Further, the effect of random variables may be substantial and can cause false alarms [22]. Modelling is required even when a simple threshold method is used for tooI failure detection. Similar modelling methods can be used to obtain more ~nfo~at~on. for example, to estimate the amount of tool wear instead of only tooli failure detection, using data from multiply sensors. Cutting force and wear models were constructed using multiple regression analysis 1231. Different methods of pre-processing the data have been tried to reduce the effects of noise.

vibration m the direction of the mam curtmg force. were measured. The surface finish of the machined surface was also measured. A shred-component p~e~o~~e~tr~~crystal type of dy-namometer (Kistler type 944lj WHS used along with charge amplifiers and analogue indicators to measure cutting force components. The dynamic characteristics of tangential force were recorded using a signal analyser SM 2701 (of Japanese rnanufa~~ur~~~ in ~onjun~t~~~~~ with the dynamometer and charge amplifier. The signal analyscr has a floppy disk drive.. where the data c‘au be stored for subsequent analysis. :2 piezoelectric at. celerometer (Bruel and Kajaer) was used to mcasurc the vibration. The vibration signals, after ampli~~ation using a charge amplifier, were also acquired using rhc signal analyser. Initial experiments were carried out to fix the Iimits of cutting conditions for steady progressive wear using a fresh cutting edge. The depth of cut was kept constant and within limits, thereby avoiding chatter. Fur&her ~xp~riI~ents were performed to obtain measilren~en~s with progressive wear. These experiments were conducted with the selcctcri cutting conditions, keeping the depth of cut constant. Depending on length of cut, ma~hiniIlg was stopped after ~cry 60 s and width of flank wear was measured Force ~onlp~~~ents were noted at two or three intermediate points between hvo wear ~lleasurements. Similarly, the dynamjcs of the main cutting force and vibration were also recorded. The set of measurements immediately prior to wear nleasurements were used for estiu~ating cutting force models. The experiment trial was iits{>carried out by varying the depth of cut.

3. Multiple regression analysis The objective of multiple regression analysis is to construct a model that e~lains, as far as possible, the variability in a dependent variabte using several independent variables. The model fit is usually a linear model, though sometimes non-linear models such as log-linear models are also constructed. The least squares estimate is the best linear unbiased estimate. It is given by

2. Experimental details

Y=a+h,X,+h,X,+

The experimental work consisted of turning S.G. cast iron using coated carbide tools. The composition of the work material, the details of the cutting tool used and cutting conditions are presented in Table 1. The machining trials were carried out on a Precision highspeed VDF lathe, with positive infinitely variable, stepless speed regulation. The three components of force, dynamic (transient) and tangential force and tool holder

where Y is the dependent variabie and X, . ..X. are the independent variables. The coefficients a, b,, .-“, b, are not known and estimates of these values must he determined from sampled data. There are cases where non-linear models may be required, where the difficulty lies in the infinite variety of non-linear functional forms. The estimated log&near model will be of the form

. . . -t&X,,,

t I)

Fi. i? Ruvindra et al. / Modeliing TABLE

1. Specification

of work material,

Composition ofwork material Work material Hardness Carbon Composition: Chromium Phosphorus Manganese Vanadium Tooling details Holder type Tool type Tool material

tool material

of tool wear bused on cutting forces in turning

and cutting conditions

Spheroidal graphite cast iron 220-240 HB 3.463% (C) 0.039% (Cr) 0.041% (PI 0.577% (Mn) 0.004% (V)

Sulphur Copper Silicon Nickel Molybdenum

Overhang Entry angle Rake angle Included angle Side clearence angle End clearence angle Nose radius

MS BNR 25 2.5 12 SNMA 12 04 08 WIDALON HK 15 13 layers of coating consisting of ALON. Tic, TIC, Ti (C, N) over a carbide substrate 26.8 mm 75” -7” 90” 7” 6” 0.8 mm

Qx+nenral conditions Sharp tool: Cutting speed Feed Depth of cut

200, 300, 400, 500 m min-’ 0.063, 0.08, 0.1, 0.2, 0.25 mm rev-’ 0.5, 1.0, 1.5, 2.0 mm

Progressive wear: Cutting speed Feed Depth of cut

300, 350, 400, 450 m min-’ 0.1 mm rev-’ 0.5, 1.0, 1.5, 2.0 mm

(S) (Cu) (Si) (Ni) (MO)

0.008% 0.101% 2.799% 0.063% 0.002%

4. Results and discussion log KCa+b,

logX,+

. . . +b,

logX,

(2)

a, b,, b,, are least squares estimates obtained from sampled data. From this we may obtain the functional form

where

y = kX,~!X&b2.. &pm

(3)

Some important statistical values used in the present work for comparing different models are multiple correlation, standard error, maximum error and F values [241. multiply correlation gives an idea of the goodness of fit or the amount of variance in the dependent variable that is explained by regression equation. The standard error gives the deviation of a dependent variable over a fit model. The maximum error indicates the maximum deviation for the sampled data. F-values gives the significance of the model or individual independent variables in explaining the dependent variable. The F-distribution can be used in conjunction with these F-values to determine the confidence level at which the hypothesis of Iack of significance can be rejected.

In this section, an attempt is made to obtain a better understanding of the signals involved in machining. Simple functional relationships between parameters have been plotted to derive a basis for more detailed analysis. The machinabili~ of an engineering material can be evaluated from the tool wear and associated mechanisms. Figure 1 gives the wear curves at three different speeds. It can be observed here that the wear curves have clearly defined regions of running-in, steady state and rapid wear. The increasing trend of wear propagation with cutting speed and occurrence of inflexion in the wear characteristics at critical speed (marked B) indicates that wear mechanisms are predominantly thermally controlled. Figure 2 shows the three components of force and flank wear for different durations of machining for one particular cutting condition. There is an increase in all components of force with progressive increase in wear; the fluctuations in force values can be attributed to random variations in material properties. Close analysis of Fig. 2 indicates a good correlation between flank wear V,, feed force F, and radial force &.

28

H.V. Ravindra et al. / Modelling of tool wear based m curt@ forces in turning

i i

X 300mlmin. 0 350m/min. 0 &OOm/min. Feed = 0 1Omm /rev Depth L 1 OOmm

A - Running in wear A-B-Region of normal

I

I

I

0

LOO

200

600 Time, set

u

I

I

800

1000

i

Time,

I

I LOO

set

5200

Fig. 3. Force ratios and wear with time

I

200

_ _ ..iL 800

LO+3

TABLE ~._I

Speed = 300mlmtn =Otmmlrev Feed Depth = l.Omm

I

600 Time,sec

I

800

: 1000

1200

Fig. 2. Forces and wear with time.

During machining with coated tools, the defo~ation of the coating will result in changes in geometry of the cutting wedge. The continuous change in wedge geometry, along with fluctuations in material properties (as in the case of S.G. cast iron) can be treated as noise sources. In order to minimize the effects of noise, it may be advantageo~ to use the ratios of force components [S]. Figure 3 gives three different ratios of forces, FJFz, Fy/Fz and Fm along with flank wear for different durations of machining. It is observed here that, although random variations are present in the

2. Model of forces as a function of cutting conditicms -.---._.-- --- ___-..- _.- _____._.____.-..._

Equation

Multiple” corr&tion

Fz = 3540.24 (feed)0,46s (speed)-“.‘“’ F$ = 313.174 (feed)c.sss (speed)a.W’ Fz = 2097.12 (feed)0,m3 (speed)-“.“XX

0.904 0.948

-.----~

0

i

JF?+ F$~Fz = Fr

____i.__._...

1200

1

A Fz

Speed = 3OOm/rnm Feed - Olmmirev Depth : 1.0 mm

X Vb Fx/Fz o FyJFz

3001

% Vb o Fx 0 Fv

i

A

Fig. 1. Flank wear with time.

300r

2501.

-.-..-~-_.--.“.-

0.9F13 ._-_.-

-

force ratios, they are much smaller compared with the overall increase in the values. A rapid increase in the force ratio is observed at the onset of rapid wear rate. These ratios can be reliably used for monitoring the onset of tool failure. Even though the effect of random variations is largely filtered out, the modelling of tooi wear, while fixing the threshold in terms of speed, feed and depth of cut, is still required. Thus there is a requirement for more ~phisti~ated methods of signal analysis, such as multiple regression analysis, pattern recognition and group methods of data handling. These methods are robust for randomvariations in the variables and are capable of integrating information, such as measured force components, speed, feed, material properties etc. Log-linear force models were constructed to estimate force compotients for nnworn cutting edges as influenced by speed and feed using the statistical computer package SPSS on a Siemens mainframe computer. The models obtained are shown in Table 2. The regression coefficient for the proposed model was above 0.9. The effect of speed appears significant only for Fx_ Despite this, the speed was not eliminated from the equations because its removal would not only give a larger bias to the

H. K R~vin~ra et al. / ~~eiling TABLE

3. Models

for flank

29

of tool wear based on cutting forces in turning

wear Multiple correlation

Model

Equation

(a)

V,= 33.34 i-0.305F;

@‘I

V, = 133.07 + O.ZOuO:, + 0.243F;

(c)

0.887

(cl

Vb = 43.83 + 20.19F; + 31.841’; + 37.8OFs Vb = - 3557.55 t 1624F: f 2065.SSF; f 2531.56F; +&347R* V,, = 155.49 + 1593.77F;, + 2167SF:, -I-2591.72FT, + 0.309RT

(9

Vb = - 3646.47 i- 739,9lF;,

0.908

Cd1

0.709

sO.207F; -I-0.173F:

0.700

+ O.l9OF;,

0.699 0.902

-I-991.87Fg f ~977.07F~ + 90.79Rf

models, but also would not truiy represent situations that are largely temperature-dependent. Based on the above, a number of models were proposed to estimate tool wear, involving different combinations of input variables and different schemes of pre-processing, which are given below.

The regression models obtained by using the above pre-processing methods are shown in Table 3. The models are referred to as (a), . . . . (f) in the table. From Table 3, it is seen that models (d), (e) and (f) give better correlation. This is due to the facts that model (d) is an improvement over model (c) as it accounts for the overall wedge effect of cutting; model (e) is an improvement over model (d) that accounts for experimental error apart from the wedge effect; and model (f) is an improvement over model (e) in that it accounts for deviations due to material properties other than the above two factors. Figure 4 illustrates the significance of model (d) compared with model (a) in estimating the tool wear. It is seen that model (d) estimates are closer to the observed values. Table 4 gives the corresponding statistics for the models in Table 3. In actual practice, all the three cutting parameters, namely v, s and a will vary. So, in order to predict tool wear under different conditions, it is essential to construct separate models accounting for the influence of all the cutting parameters along the duration of machining (7’). Hence cutting force models influenced by v, s and a are set out in Table 5 and the corresponding flank wear models V, for different combinations of forces are shown in Table 6. Model (g) is based on the cutting force transient Fd and vibration Vd, along with the cutting forces (a multi-sensory approach). It was observed that flank wear is largely influenced by

300

Speed = 3OOm/min. Feed = 0 lmm/rev. Oepth 1: 1 Omm

25Ll-

X Measured fI1 Estimated l Estimated

of 0

(mode! a) (model d)

I

I

LOO

800

12(

Time,sec

Fig. 4. Multiple TABLE Model

regression

estimates.

4. Statistics for models

of flank wear

F-values

Standard error

Maximum error

x

Y

2

Eqn.

(pm)

bmf

:z;

0.41 0.97

1.84 1.56

0.64 0.52

8.69 9.12

42.06 41.55

100.36 130.40

icf

0.26

2.02

2.02

8.60

42.16

139.23

Model

(4

F-values

Standard error

Maximum error

F,

E;

Fx

R

Ew.

b4

(w)

31.4 34.6 37.4

35.5 46.4 50.9

30.9 38.8 42.3

21.1 18.0 23.0

24.09 28.43 30.42

27.69 25.92 25.21

51.29 55.05 53.81

transients. The lower significance of vibration may be due to the fact that the flank wear was smooth and no abrasive wear was observed. Referring to Fig. 5, it is seen that most of the models fairly represent the

30

Fig. 5. Multiple regression

H.V. Ravindra et al. I Modelling of tool wear based on cutting forces

m turnrng

Fig. 7. Multiple regression

estimates.

Fig. 8. Multiple regression

estimates.

estimates.

TABLE

5. Model

of force

as a function

of speed,

feed and

depth of cut

measured value above 400 s of machining. Further, models (c) and (f) exhibit large errors during the early part of machining. In Fig. 6, even though models (c) and (f) exhibit higher error values compared with other models, they represent fairly the measured value when machining time is longer. Referring to Figs. 7 and 8, all the models represent the measured values with varying depths of cut. This indicates that the proposed models hold good at higher speeds and depths of cut. Hence, by using a coated carbide tool at a lower depth of cut, the coating peels off and the tool itself does

Equation

Multiple correlation

F, = 21.5 (feed)OJ4 (speed)0.60 (doe)“.” F,, = 1619 (feed)o,49 (speed)-“.n (dot)‘.”

0.768 0.833 0.935

F, = 657.1 (feed)a”’ (speed) -oz (doc)“.ps

not wear. The computed F-ratios given in Table 7 are much higher compared with the statistical table values at various confidence levels. The higher the F-ratios

Zi. V. Ravindra et al, / Modelling of tool wear based on cutting forces in turning TABLE

6. Models for flank wear

Model

TABLE

Equation

Multiple correlation

V, = 36.6 + 0.27F, + 0.39F, - O.O42F,+ 0.29T V,=21.5+0.38Fx,+0.17F,,+0.36Fz,+0.28T V, = - 157 +6’7.732;;,+ 30.25F,,+ 125.87Fz2-0.105T V, = - 241 f 225.8F, + 1886.6F, - 0.261ij f 0.36R f 0.25T Vb=15.4+W.64F,,-24.26F,,-8.98F,3-i-0.43R,+0.30T V, = - 913 + 534F,, + 394.1F,+ 526.89Fz i- 82.0lR,- O.MT V, = 12.79 f 0.26F, + 0.6OF, - 0.26F, + 3.26F* - O.OW,

0.913 0.921 0.954 0.965 0.916 0.946 0.920

7. F-ratio values for wear models

Confidence limit

90% 95% 99%

31

Model (a)

(b)

cc>

(d)

(e)

(ff

(8)

51.35 41.08 28.52

57.05 45.64 31.69

104.1 83.24 57.80

39.31 32.47 22.63

44.42 36.69 25.57

73.52 60.73 42.33

58.35 46.68 32.40

resemble the measured values. This is attributed to the thermal dependent of tool wear. The higher F-ratio values justify the significance of the models and usefulness in predicting tool wear. Using these models, an effective control strategy can be implemented for on-line monitoring and control during turning.

References confirm the significance of these models and justify their usefulness in tool wear predictions.

5. Conclusions Experimental work was designed to obtain data for sharp tools and different stages of flank wear. The data obtained were used to establish the effect of cutting conditions on cutting forces and tool wear. Material properties and tool wedge geometry variation have been identified as major noise sources in signals. It was observed that, although random variations were present in force ratios, they were much smaller compared with the overall increase in the values. Force models were constructed to fix the threshold in terms of speed, feed and depth of cut using multiple regression analysis. These force models were used to estimate tool wear using a variety of input variable combinations. Among the models tried, the wear models without accounting for machining time gave multiple correlation of around 0.7 and with machining time the multiple correlation was above 0.9. The improvement in the correlation value is due to the influence of timedependent degradation of the tool material on flank wear. The ones which eliminate the influence of experimental error, i.e. models based OR differences in the actual and estimated force components, as well as that based on the ratio of the same, gave better predictions of the tool wear. This can be seen in Figs. 5-8. These wear models hold good at higher cutting velocities and depths of cut, as these models closely

3 H.K. Toenshoff, J.P. Wulfsberg, H.J.J. Kats, W. Kong and C.A. Vam Lutervelt, Development and trend in monitoring and control of machining processes, CARP, 37 (2) (1988) 611-622. 2 SE. Oraby and D.R. Hayhurst, Development of models for tool wear force relationships in metal cutting, Znf. I. Me&. Sci., 33 (2) (1991) 125-138. 3 S. Rosetto and R. Levi, Fracture and wear as factors affecting stochastic tool life models and machining economics, Trans. ASME, J. Eng. IT&., May (1978) 281-286. 4 A. Pillipi and R. Ippolito, Adaptive control in turning, cutting forces and tool wear relationships for PlO, P20, P30 carbides, C1RP, 17 (1969) 377-385. 5 R. Mackinnon, G.E. Wilson andA.J. Wilkinson,Tool condition monitoring using multi-~mponent force measurement, Proc. Int. Mach. Tool Des. Con$, 1986, pp. 317-324. 6 M. Lee, C.E. Thomas and D.G. Wildes, Prospects for inprocess diagnosis of metal cutting by monitoring vibration signals, J. Met. Sci., 22 (1987) 3821-3830. 7 S.B. Rao, Tool wear monitoring through dynamics of stable turning, Trans. ASME, J. Eng. Ind., August (1986) 133-190. 8 P.M. Lister and G. Barrow, Tool condition monitoring systems, Froc. 26th IMTDR ConJ, Manchester, 17-18 September, 1986, Macmillan, pp. 271-288. 9 NH. Cook, Tool wear sensors, Wear, 62 (1988) 49-57. 10 D.A. Domfeld, Monitoring of cutting process by means of AE sensors, Proc. 3rd Int. Mach. Tool Eng. Co&, Tokyo, 1988, pp. 268-271. 11 E. Emel and E. Kannatey-Asibu, Acoustic emission and force sensor fusion for monitoring cutting process, Int. J, Me&. Sci., 31 (1989) 795-809. 12 K.F. Martin, J.A. Brandon, K.L. Grosvenor and A. Owen, A comparison of in-process tool wear measurement methods in turning, Proc. 26th IMTDR Cm$, Manchester, 17-18 September, 1986, Macmillan, pp. 289-296. 13 S.Y. Liang and D.A. Domfeld, Tool wear detection using time series analysis of acoustic emission, Trans. ASME, J. Eng. Znd., ZZO (1989) 199-205.

32

H. V. Ravindra et al. I Modelling of tool wear based on cuttingforces in turn&

14 Y. Yao, X.D. Fang and G. Amdt, Comprehensive tool wear estimation in finish-machining via multi-variate time-series analysis of 3-D cutting forces, Ann. CIRP, 39 (1) (1990) 57-60 15 SM. Pandit and S. Kashou, A data dependent systems strategy of on-line tool wear sensing, Trans. ASME, J. Eng. Ind., 104 (1982) 217-223. 16 E. Emel and E. Kannatey-Asibu, Tool failure monitoring in turning by pattern recognition analysis of AE signals, Trans. ASME, 1. Eng. Ind., 110 (1988) 137-145. 17 T.I. Liu and SM. Wu, On-line detection of drill wear, Trans. ASME, J. Eng. Ind., 112 (1990) 299-302. 18 P.G. Li and SM. Wu, Monitoring drilling wear states by a fuzzy pattern recognition technique, Trans. ASME, J. Eng. Ind., 110 (1988) 297-300. 19 S. Rangawala and D. Domfeld, Sensor integration using neural networks for intelligent tool condition monitoring, Trans. ASME, J. Eng. Znd., 112 (3) (1990) 219-228. 20 T. Uematsu and N. Mohri, Prediction and detection of cutting tool failure by modified group method of data handling, Int. J. Mach. Took Des. Rex, 26 (1) (1986) 69-80. 21 M.C. Shaw, Metal Cutting Princ@les, MIT Press, Cambridge, MA, 1984. 22 M. Raghunandan, Multi-sensory approaches to in-process monitoring of turning, M.S. Thesb, IIT, Madras, India, 1991. 23 T.H. Wonnacott and R.J. Wannacott, Regression - a second course in statistics, John Wiley, Chichester, 1981. 24 N.R. Drapper and H. Smith,Applied Regression Analysis, John Wiley, Chichester, 1966.

Fz

FXX CY F,

U s

a T

F,I JG Vt, F-ratio F,,=F,-Fx,,

FX 6

axial component of force radial component of force

Fyl=Fy-&,

F,, = FxIFm

F;, = FJF,,

F__=F;-F, F,, = FJF,

(Al) OQ)

FI,=FxJR,,

F,z=FyJRI,

F,x=FrlIRI

W)

Fz =&I&,

F,, = F,,IR,,

/;23 = Fz,lRz

(4

F, = FJR,

F2 = FJR,

F3 = FJR

CM)

R = (Fx” + [,.z + Fz2) ‘I2

iAh)

R,=(F,,2+F,,,2+FZ,2)1’2

(AT)

R? = (FxZL+ Fyz2 + Fz,2)‘n

W-9

F:,

Appendix A: Nomenclature

tangential component of force axial component of force model radial component of force model tangential component of force model cutting speed, m min ’ feed, mm rev’ depth of cut, mm machining time in seconds transient cutting force vibration in the direction of main cutting force flank wear, ym F-value(computed)/F-value (tabulated)

F;,

F: FL F,$ FZ

refer to the data collected with varying speed and feed, by keeping depth of cut constant (1 mm) are cutting force models based on the above data.