In-process identification of the milling operation

Int. J. Much. Tools Manufact. Printed in Great Britaifi

Vol. 30, No. 3, pp.325--337, 1990.

IN-PROCESS

0890--6955/9053.00 + ,00 Pergamon Press plc

IDENTIFICATION OPERATION

OF THE

MILLING

D. W. CHo* and K. F. EMANt (Received 11 September 1987; accepted in final form 8 August 1989) Abstract--The three dimensional milling operation was formulated in terms of multi-variate stochastic time series models. It was shown, both theoretically and experimentally, that this formulation can be effectively used for the identification of the cutting and structural dynamics properties directly from measured operating data.

NOMENCLATURE A(B)

Aw. A v,

p =x,y,z p =x.y ,z

B(B) B

c(/~) Cpq(l"CO),

p,q =x,y,z

O(j,o) D.(jco).

p =x,y,z

D, Du,, F(/o,) F.(jto). F,p,

q=x,y,z p=x,y,z p=x,y,z

F, Fp,.

p=x,y,z

Fo

F,,t

G(i,o) G.q(/Co),

p,q=x,y,z

H,(B) H, pq(B),

p,q=x,y,z

H,,,(B) H,,,mt(B ) ,

p.q=x,y,z

Kpq(jCo),

p,q=x,y,z

m n

N gtpq,

p,q=x,y.z

T

U(/oO u,,(i,o),

p=x,y,z

U, Uu,,

q=x,y,z

Vt Xt

Z,

autoregressive polynomial matrix discrete white noise vector component of workpiece acceleration component of spindle (tool) acceleration moving average polynomial matrix backshift operator defined as x,_k=Bkx, transfer function matrix of the cutting dynamics transfer function of the cutting dynamics between the p-component force and q-component chip thickness variation Fourier transform of the relative displacement vector between workpiece and tool Fourier transforms of the components of the relative displacement vector discrete displacement vector at time t components of the discrete displacement vector Fourier transform of the cutting force vector Fourier transforms of the cutting force components component of the cutting force at the ith cutting edge discrete force vector at time t components of the discrete cutting force vector at time t steady state cutting force vector discrete steady state cutting force vector transfer function matrix of the structure transfer functions of the structure between the p-component displacement and q-component force discrete cutting process transfer function matrix discrete transfer function of the cutting dynamics between the p-component force and the q-component chip thickness variation discrete transfer function matrix of the structure discrete transfer function of the structure between the p-component displacement and the q-component force cutting stiffness coefficient between the p-component force and q-component relative displacement moving average order autoregressive order number of discrete measurements directional coefficient of the ith cutting edge between the p-component force and q-component relative displacement time for one revolution of the cutter Fourier transform of the chip thickness variation component of the chip thickness variation discrete chip thickness variation vector components of the discrete chip thickness variation vector input-output vectors measurement vector number of teeth simultaneously in contact

*Pohang Institute of Science and Technology, Pohang, Republic of Korea. tNorthwestern University, Evanston, IL 60208, U.S.A. 30:3-A

325

326

D.W. Crto and K. F. EMAN

Zt

O.A, ~q(B) ~qk

Oi,B~ Oij(B) Oq~ "qm'

p=x,y,z

~,, I.t

q=x,y,z

discrete white noise vector autoregressive parameter matrices autoregressive polynomial autoregressive parameters moving average parameter matrices moving average polynomial moving average parameters discrete noise vector contaminating the displacement components of the discrete noise signals contaminating the displacement at time t discrete noise vector contaminating the force components of the discrete noise signal contaminating the cutting force overlap factor 1. INTRODUCTION

To ASCERTAINthe stability of a machining operation knowledge of the structural dynamic behavior of the machine tool and of the cutting process are required. The former can be relatively easily defined, however, the determination of the cutting dynamic behavior is still associated with difficulties from both the theoretical and experimental standpoints. Although relatively many theoretical studies have been done in this area, most of the work was based on the assumption of orthogonal cutting or theoretically derived equations which utilize constants evaluated from orthogonal cutting data [1,2]. Moreover, most of the available methods deal with the single point turning operation only. Cutting dynamics behavior is typically represented by the complex dynamic cutting force coefficients (DCFCs) [1,3] which characterize the relationship between the cutting force and the chip thickness variation components in the cutting process at one frequency, usually the chatter frequency, only. Attempts made in the past to identify the cutting process experimentally in the whole frequency range of interest were based on classical Fourier [4] and time series methods [5,6,7]. These attempts, successful for single point orthogonal turning operations, were later extended to the oblique turning case [8]. The main problems associated with the in-process identification of the instantaneous dynamics of a machining system are related to the time variant nature of the process and the three dimensionality of the system, which leads to complex analytical procedures. Closely associated with these two problems are the experimental difficulties. During the cutting experiments the parameters of the machining process are changing due to factors such as temperature rise, tool wear, changing dynamic receptance along the feed direction, etc. This in turn imposes the requirement that for the identification of the instantaneous dynamics relatively short data records must be used during which the system may be considered time invariant. There are few analytical tools that can properly handle the multiple input situation, i.e. the simultaneous three dimensional chip thickness variations, and the fact that the chip thickness variations and the cutting forces are correlated in the actual machining process. The identification problem, therefore in general, reduces to the identification of the system from multiple inputs and outputs. To address this problem in this paper, multi-variate, Modifed Autoregressive Moving Average Vector (MARMAV) models [5,9] will be used for the in-process identification of the cutting process and of the structure. First, the fundamentals of the proposed method will be formulated based on the three-dimensional cutting model of the milling operation. The MARMAV models that represent the cutting process and structural dynamics of the milling operation will be derived in Section 2. Section 3 describes the experimental work conducted on a milling machine, followed by the identification of the cutting and structural dynamics and the analysis and discussion of the obtained results in Section 4. Finally, conclusions are drawn in Section 5. 2. THE CUTTING PROCESS IN THE MILLING OPERATION

The dynamic machining system under both stable and chatter conditions is uniquely defined by the dynamics of the cutting process and by the structural dynamics of the

In-process Identification

327

machine tool at the cutting point. The dynamic force, represented by its components Fx, Fy, and Fz in three orthogonal directions, acts between the workpiece and the tool resulting in a relative displacement. At the cutting point the structural dynamic behavior of the machine tool relates these relative displacements and the cutting forces by nine response loci of the 3×3 square structural transfer matrix defined as:

o.u >l Dy(/,,,) / =

Gyy(jto)

Gy/jo~)l/FRO,,,) I

(1)

/ LF uo,)j or Oqt~) = GO~) FO~). The cutting process, in turn, can be characterized by the chip thickness and the directional coefficients [10]. In the general case of a multi degree of freedom milling system, the total chip thickness variation at the ith cutting edge in contact can be decomposed into three components. These component variations are caused by the relative movements in the x-, y- and z-directions which are produced by the components Fix, Fly, Fiz of the dynamic cutting force. Since the relative motion between the tool and the workpiece is the same at each cutting edge, the total chip thickness variation may be represented, based on the double modulation principle [3,11], by:

(2)

U(jo0 = (e-S
where r = T/Zc. Consequently the relationship between the force components and the components of the chip thickness variation can be expressed in the form:

Fip = Ripq gpq Uq,

p,q=x,y,z.

(3)

The directional coefficients are periodic time variant functions. In milling operations, several teeth are normally in contact with the workpiece; thus the directional coefficients actually are the sum of the coefficient values of all teeth simultaneously in contact. Considering all three components of the chip thickness variation, based on equation (3), the resultant force is given by:

Fp = KpxRpxUx + gpyRpyUy +

p=x,y,z

KpzRpzU z

(4)

where Z c

Rpq = E Ripq" i=1

Equation (4), which represents the cutting dynamics, can now be rewritten in matrix form associated with the simplified block diagram in Fig. 1 as: Oo

+

_I

[

D

Fio. 1. Block diagram of the closed loop machining system (continuous model).

328

D . W . CHo and K. F. EMAN

fCx. .>

Fy(jO,)I -- I¢~(jo,) F,(]o~)_]

LC,x(jw)

Gy(/~o) cx~(y,,,)-]-ux(j~,)] Czy~jOJ)

(5)

Uyq,,,)l

CzzO'(D)J_u~-o,)j

or

F(jo~) = C(j~o) U(jo0 where

Cpq(j(o) = Rpq gpq(]O)),

p,q = x,y,z.

In the figure, without loss of generality, the steady state cutting force was set equal to zero since only the dynamics of the process are of interest. Provided that the state variables D and F in the block diagram (Fig. 1) can be measured in a cutting experiment or during actual cutting, the complexity and discontinuity of the directional coefficients can be accounted for. If Ux, Uy, U~ and Fx, Fy, F~ are measured and suitably modeled, then the product of the directional coefficient and of the cutting stiffness coefficient can be obtained directly from the model. Regarding this product as one block in the block diagram, and referring to it as the stiffness transfer function, Cpq, between the p-force component and q-chip thickness variation (p,q=x,y,z), the unknown transfer matrices G and C could be identified if a method of modeling the operating data was available. For the formulation of a model which would facilitate the identification of G and C from discrete (digitized) force and displacement components, the effect of external process disturbances and of the measurement noise must be taken into consideration. Considering the noise in modeling and measurement, one can formulate a discrete model for the cutting dynamics in a milling operation corresponding to equation (5) as:

E,l Fy, F;,

= |Hcyx(B) LH,x(B)

Hcyy(B) Hczy(B)

Hcy~(B)|

Uyt

]

Hc=(B)J Uz,

=~ |my,

(6)

Lnz,

or

F, = Hc(B) U, + "qt. The structural transfer function can be represented in an analogous way from equation (1) as:

E°xl Dyt Dz,

= Hmyx(B) Hmzx(B)

Hmyy(B) Hmzy(B)

-mz.,lI

[G, ]

Hmyz(B)| Fy, + / ~ , / Hm=(B)J F~,

(7)

or

D, = H,n(B) F, + ~,.

The block diagram of the discrete system, equivalent to the continuous system of Fig. 1, is given in Fig. 2.

In-process Identification

+

For= 0

_ I

I

329

+

_

Ot

+

Hc (B)

H,e- J~r-I

FIG. 2. Block diagram of the machining system with noise (discrete model).

Equations (6) and (7) can now be expressed in the joint input-output model form [12] as: m(B)v t = n(n)

zt

(8)

where A(B) = Ao + A 1 B + A2

n 2+

B(B) =B1B + B2 B 2÷ . . . . . .

......

+ Apn q

+ Bqn p

and where the input-output vector v, is defined as vt = {Fxt, Fyt, Fzt, Uxt, Uyt, Uzt}T for cutting dynamics and as v, = {Uxt, Uy,, U=,, F~t, Fyt, F=,}T for structural dynamics identification, zt is a discrete white noise vector with zero mean, and Ai and B~ are autoregressive and moving average parameter matrices, respectively. This model is a six variate Modified Autoregressive Moving Average Vector Model of orders n and m denoted by MARMAV(n,rn) [13]. The relationship between equation (6) or (7) and equation (8) can be derived by first expanding the matrix equations and then by matching the corresponding elements, in (6) and (8) for the cutting dynamics and (7) and (8) for the structural dynamics, element by element. The six variate stochastic model in equation (8) allows the identification of all 18 transfer functions, Hcpq(B) and Hrapq(B), p , q = x , y , z , of the three dimensional cutting system. Because of the special structure of the M A R M A V model, it can be shown that in practical situations, each row of the transfer matrices of equations (6) and (7) can be identified independently from a single output and three inputs [12]. Considering for instance, the case in which Fxt is used as an output and /.Ix,, Uy, and Uz, as the inputs, the number of variables is reduced to four by eliminating the 2nd and 3rd rows and columns from equation (8). Equation (8) can therefore be written in expanded form by the definition of the M A R M A V models [13] as: (1)oX t =

( I ) l X t _ 1 "~- . . . . .

+

~nXt--n "~- a t + O l a t _ 1 + . . . + ~)mat_m

(9)

where Xt-k = {Xt,-k, g2t-k, g3t-k, g4t-k} T, k = 1,2, . . . , n, is the measurement vector (here {XI,, X2,, X3,, Xat}T has been used to represent the vector {F~t, Ux,, Uy,, Uz,}T for the sake of generality), at-k = {air-k, a2t-k, a3t-k, a4t-k} T, k = l , 2 . . . . . m, a discrete white noise vector, ~k = [qbqk], i,j = 1, . . . , 4, k=1,2 . . . . , n, is the autoregressive parameter matrix at lag k; ~o = [d~ijo], i,j= 1 . . . . . 4, is an upper triangular matrix with (~iiO = 1 and dhjo = 0 for i>j, and Oh = [0ijk], i,j=l . . . . , 4, k = l , 2 . . . . , m, is the moving average parameter matrix at lag k. From the first scalar equation obtained by expanding equation (9), it is possible to identify three transfer functions of the transfer matrix corresponding to the elements in the first row of the structural transfer matrix on the right hand side of equation (6) from:

330

D . W . CHO and K. F. EMAN

+L2(B) v

Xlt

_ +13(B) y

+ , l ( B ) "'2'

+,,(B) X4, + 0n(B)

+ l l ( B ) "~3'

alt

(lO)

where + u ( B ) = 1 - + 1 1 1 B - +112 B2 - . . .

+lln Bn

-

- . • • - +12nB n --. • • - - +I3,,B n +,4(B) = +,4o - + 1 4 1 B - - " • • - - +14nBn 0n(B) = 1 - 0 u l B - 0u2B 2 - • • . - 011~Bm. + 1 2 ( B ) = 4120 -- + l z t B +I3(B)

=

+130

--

+131B

The terms +q(B)/611(B) represent the discrete transfer functions between Xj, and X,-,, i.e. the x-component of the cutting force and the components of the chip thickness variation. Similarly, the other elements of the discrete transfer functions can be obtained. The only remaining problem is the estimation of the unknown autoregressive and moving average parameter matrices from the measured displacement and force components. The algorithms used for this purpose are beyond the scope of this paper, however, in principle any of the available linear or nonlinear procedures could be used [12,9]. The objective function adopted is the sum of squares of the residuals defined, based on equation (10), by: N

f(.

• •, %k,.

......

, %,,.

a 2,

• • ) =

(11)

t=n+l

while the adequacy tests are based on the F-test or some other statistical criterion. Once the parameters of the M A R M A V models in equation (9) are estimated, the discrete transfer functions are readily available since these parameters can be directly substituted into equation (10) to yield the transfer functions. The frequency response functions are in turn obtained by substituting B = e -j'oa where A is the sampling interval. Moreover, the discrete transfer functions +q(B)/+u(B) can be easily transformed into the continuous time domain by the impulse response invariant transformation [14]. 3.

EXPERIMENTAL MEASUREMENTS

In order to identify the transfer functions of the structure and of the cutting dynamics, cutting experiments on a vertical milling machine were carried out in upmilling. Three acceleration components of the workpiece fixture (Awx,Awy, Awz) and spindle housing (Asx, Asy, A~) and three orthogonal force components (F~, Fy, Fz) were measured under various cutting conditions. The milling cutter used had eight teeth which were equally spaced and three of them were engaged during cutting. Rectangular workpieces (100 mm × 150 ram) were used and replaced after each cutting experiment. The cutting speeds used in the experiments were 302, 365,445 and 545 rev min-1 with the feedrate, f, fixed at 270 mm min -1. The depth of cut, d, was varied from 0.5 mm to 5 mm in 0.5 mm increments. All nine signals from the tool dynamometer and the accelerometers were recorded in a data recorder through charge amplifiers for further analysis. These signals were filtered with analog low pass filters to eliminate the high frequency noise components and digitized using a four channel oscilloscope with a sampling interval of 0.0002 s. A schematic diagram of the experimental setup is shown in Fig. 3. 4.

IN-PROCESS I D E N T I F I C A T I O N OF T H E S T R U C T U R A L A N D C U T T I N G D Y N A M I C S

The sampled data from the experiments were used to identify the structural and cutting dynamics transfer matrices by the method outlined in the previous section.

In-process Identification

r-=--a Tr,ox!ol . accet.erome~.er~

T.., t

h

arcca:ltaerom et.er

II

-"r'LuQIuu~ j . / ~ I 1-101 0 0 0

III

lzy

\ \ S0in0leheo0 //

!

III

O' Face cutter

/Workpiece I/Workpiece fixture li Dynarnometer

IlLILl III

331

'/ --II Tope recorderI1=

1 1

I c°mp°ter I FIG. 3. Schematic diagram of the experimental set-up.

4.1. Transfer functions of the cutting dynamics The workpiece vibration combined vectorially with the spindle vibration (tool vibration) yields the relative vibration between the tool and workpiece. However, since the spindle vibrations were negligible compared to the workpiece vibrations (the standard deviations of the spindle vibrations were less than 10% of those of the workpiece fixture) they were neglected and the workpiece vibrations were considered as being equivalent to the relative vibration between tool and workpiece, i.e. Upt = A~p,

p=x,y,z. The chip thickness variations in three orthogonal directions (shown in Fig. 4) were used as inputs and one cutting force component at a time as output to identify the transfer functions that describe the cutting dynamics. The joint input output vectors were defined as (Fp,, Uxt, Uyt, Uzt)T, p=x,Y,Z. The ordinary coherence functions between the input signals were first obtained and plotted in Fig. 5. It can be noted that the coherences have values significantly larger than zero at all frequencies, indicating that the inputs are correlated. This fact confirms the earlier statement that all three inputs should be used simultaneously for model identification. After filtering with a digital filter, the input and output data obtained at a speed of 365 rev min -~ were fitted to the M A R M A V models of the form given by equation (10). As an example the identified model parameters of the MARMAV(2,1) models for the cutting dynamics in the x-, yand z-directions are given in Table 1. The identified discrete transfer functions were subsequently transformed into the continuous time domain. The magnitudes and phases of the characteristic transfer functions are shown in Fig. 6. For a rotational speed of 445 rev min-J with all other conditions unchanged, the same procedure as for 365 rev min-~ was repeated. The identified adequate model was, once again, a MARMAV(2,1) model. After transformation into the continuous time domain, the transfer functions were plotted in terms of magnitude and phase and shown in Fig. 7 for all three components of force and vibration. The fact that under both experimental conditions MARMAV(2,1) models turned out to be adequate, indicates that not only the chip thickness and penetration rate related effects but also inertial effects during chip formation play a significant role. Moreover, the transfer functions

332

D . W . C8o and K. F. EMAN

0,:31

-3.63 0.00

002

0 04

0.06 Time(sec)

0.08

0.10

X -direction 8.80 j

~

I

0.5I

-

7 000

. o02

5 0.04

8

~

0.06 Time(see)

0.08

olo

,

•

Y-direction 3.67 ]

LL.kh, o.oo

0.02

0.04

0.06

0.08

oJo

Time(see) Z - direction

FIG. 4. Chip thickness variation in the three orthogonal directions (d = 0.1 mm, f = 270 mm m i n - L n = 365 rev min-~). TABLE 1. M A R M A V (2,1) PARAMETERSFOR TESTS RUN AT 365 rev min -~ WITH d = l mm AND f=270 mm rain i Identified parameters Cutting force Fx Fy Fz

~blt, 1.578 1.598 1.691

p component cutting force, Fpvsq component chip thickness variation, Uq Cutting force and x-direction chip thickness variation

F:Ux Fy-Ux Fz-Ux

Cutting force and y-direction chip thickness variation

Fx-Uy Fy-Uy F,-Uy

Cutting force and z-direction chip thickness variation

Fx-U, Fy-Uz F:Uz

d~112 -0.616 -0.702 -0.699

Identified parameters

(bt2o

~b~2j

~b~22

- 10.312 1.076 14.564 qb~3(,

- 14.829 3.109 31.290 dh3~

7.888 0.072 - 12.322 ~bL~2

0.528 -7.768 -2.340 (b4o

5.052 - 12.259 -4.516 ~bj41

-5.777 5.052 2.102

-0.002 6.032 - 1.128

2.023 -7.199 -5.328

-5.068 6.780 0.234

1~)142

In-process Identification

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8

0

300

600

900

1200

1500

U,-Uy

I

~; o.s.

13 0

! 3O0

I

!

!

600

9OO

1200

1500

1200

1500

Uy- U~

I

°I

0

300

600 900 Frequency (Hz) U=-U=

FIG. 5. Ordinary coherence functions between the input signals.

of the cutting dynamics are significantly affected by the cutting conditions, thus, as anticipated, the transfer functions at only one condition cannot completely represent the cutting dynamics. 4.2. Transfer functions of the structure By following a similar procedure as in Section 4.1 in this section, all the elements of the structural dynamic transfer function matrix in equation (7) will be found for the same two cases as above, i.e. corresponding to cutting speeds of 365 rev min -1 and 445 rev min-~. Once again, the acceleration of the workpiece was taken for the relative vibration between tool and workpiece and used as the output while the three cutting force components were used as inputs in equation (10), i.e. the joint input output vector was defined as (Up,, Fx,, Fyt, F=,)T, p=x,y,z. This case is very similar to a multi-excitation test in experimental modal analysis [9,14,15] except that the excitation forces are cutting forces and the response is relative instead of absolute vibration. Before modeling, a digital filter with a cutoff frequency of 1500 Hz was used to eliminate the noise and reduce aliasing errors. Multi-variate models were successively

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In-process Identification

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fitted starting from a MARMAV(2,1) model to find the adequate order.,MARMAV(10,9) models were subsequently found as adequate for both cutting conditions. The Nyquist plots that represent the complete three dimensional transfer function matrix obtained from the data measured at 365 rev min -1, i.e., Gxx(jto) in equation (1), are shown in Fig. 8. The transfer function matrix shows a high degree of symmetry, with only relatively small differences in the magnitudes. The digitized signals obtained at 445 rev rain -~ were also fitted by MARMAV(10,9) models. The continuous time domain dynamic compliances transformed from the discrete ones have shown a very close agreement with those obtained at 365 rev min- 1. This was also true for all the other components of the transfer functions obtained under other experimental conditions. This suggests that the transfer functions of the structure were not significantly influenced by the cutting conditions in contrast to those of the cutting dynamics. The direct and cross transfer functions, obtained from the operating data, together with the cutting dynamics transfer functions, completely define the machining system represented by the block diagram in Fig. 1. 5. CONCLUSIONS

The three dimensional closed loop milling dynamics was formulated in the form of a multi-variate time series model whose parameters can be estimated from relative displacements and force components measured under actual cutting conditions. Since this method yields parametric transfer functions for both the cutting and structural dynamics, they can be directly used for stability analysis. Furthermore, since all variables are considered simultaneously their correlatedness is inherently accounted for by the identification procedure. REFERENCES

[1] M. M. NIGM, M. M. SADEKand S. A. TOBIAS, 'Determination of dynamic cutting coefficients from steady state cutting data', Int. J. M.T.D.R., 17, 19-37 (1977).

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In-process Identification

337

[2] T. SATAand T. INAMURA,'Analysis of three dimensional cutting dynamics', CIRP, 22, 119-120 (1973). [3] J. TLusl'v, 'Analysis of the state of research in cutting dynamics', Ann. CIRP, 27, 583-589 (1978). [4] M. M. NI~M and M. M. SADEg, 'Experimental investigation of the characteristics of the dynamic cutting process', Trans. ASME, J. Engng Ind., 99, 410-418 (1977). [5] T. Y. Am~, K. F. EMAN and S. M. Wu, 'Determination of the inner and outer modulation dynamics in orthogonal cutting', Trans. ASME, J. Engng Ind. 109, 275-280 (1987). [6] T. MORIWAKIand K. IWATA,'In-process analysis of machine tool structural dynamics and prediction of machining chatter', Trans. ASME, J. Engng Ind., 98, 301-305 (1976). [7] T. MORIWAKI,'Measurement of cutting dynamics by time series analysis technique', Ann. C1RP, 22, 117-118 (1973). [8] X. G. YANG, K. F. EMAN and S. M. Wu, 'Analysis of three dimensional cutting process dynamics', Trans. ASME, J. Engng. Ind., 107, 336-342 (1985). [9] S. M. Wu, 'Dynamic data system: a new modeling approach', Trans. ASME, J. Engng Ind., 99, 708--714 (1977). [10] H. OPITZ and F. BERNARDI,'Investigation and calculation of the chatter behavior of lathes and milling machines', Ann. CIRP XVII, 335-343 (1970). [11] J. TLusrv and F. ISMAIL,'Special aspects of chatter in milling', Trans. ASME, J. Vibr. Acoust. Stress Reliability Design, 105, 24-32 (1983). [12] D. W. CHo, 'A new multi-input modal analysis and three-dimensional cutting dynamics identification method applied to milling operations', Ph.D. Thesis, University of Wisconsin-Madison (1984). [13] S. M. PANDITand S. M. WE, 'Time Series and System Analysis with Applications'. Wiley, New York (1983). [14] D. W. CHO, K. F. EMAN and S. M. Wu, 'A new time domain multiple input modal analysis method', Trans. ASME, J. Engng Ind. 109, 377-384 (1987). [15] R. J. ALLEMANG,R. W. ROSTand D. L. BROWN, 'Multiple input-estimation of the frequency response functions', Proc. 2nd IMAC Conference, Orlando, Florida, pp. 710-719 (1984).