Receptance coupling for end mills

International Journal of Machine Tools & Manufacture 43 (2003) 889–896

Receptance coupling for end mills

Simon S. Parka, Yusuf Altintasa,∗, Mohammad Movahhedyb

a Department of Mechanical Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canadab Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

Received 6 January 2003; received in revised form 28 February 2003; accepted 6 March 2003

Abstract

Identification of chatter free cutting conditions, the chatter stability lobes, requires a measurement of the frequency responsefunction (FRF) of each tool mounted on the spindle. This paper presents a method of assembling known dynamics of the spindle–tool holder with an analytically modeled end mill using the receptance coupling technique. The classical receptance technique isenhanced by proposing a method of identifying the end mill–spindle/tool holder joint dynamics, which include both translationaland rotational degrees of freedom. The method requires measurement of FRFs with impact tests applied on the spindle–tool holderassembly and blank calibration cylinders attached to the spindle. The spindle and tool holder characteristics are completely identifiedfrom the two experiments, and used for the mathematical prediction of FRF for end mills with arbitrary dimensions. The proposedmethod is experimentally proven and verified in cutting tests. 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Receptance coupling; Frequency response function; Rotational degrees of freedom; Substructure coupling

1. Introduction

End mills are widely used in milling dies, molds, aero-space, and automotive components. If the cutting con-ditions, namely depth of cut and spindle speed, are notselected properly, milling operations may becomeunstable with severe chatter causing tool chipping, roughsurface finish, and overload on the spindle drive andbearings. Since the early 1950s, significant research hasbeen conducted in determining chatter free spindlespeeds and depths of cuts[1–3]. Regardless of the differ-ent approaches, the stability expressions require accuratemeasurements of frequency response function (FRF) atthe tip of the tool when it is attached to the tool holder–spindle assembly.

Analytical or finite element based prediction of FRFsfor the entire kinematic chain of spindle and machinetool system is not accurate, since the damping and stiff-ness at each joint are not known before the machine isbuilt. Numerical techniques are rather useful at the

∗ Corresponding author. Tel.:+1-604-822-2182; fax:+1-604-822-2403.

E-mail address: [email protected] (Y. Altintas).

0890-6955/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0890-6955(03)00088-9

design stage of the machine, but they cannot be usedeffectively for the process planning of productionmachines. The present practice in the shops is to measurethe FRFs of each end mill used in the shop using impactmodal tests with an instrumented piezoelectric forcehammer and a vibration sensor. The Experimental ModalAnalysis (EMA) through impact hammer excitationstake away significant time from the production machine.

Schmitz et al.[4,5] pointed out the importance of eli-minating time consuming and repetitive FRF tests foreach tool, and proposed a receptance coupling techniqueborrowed from structural engineering literature. Thetechnique allows coupling of analytical or experimentalFRFs of the components in obtaining the response of theassembly[6–10]. The accuracy of the receptance coup-ling technique depends on accurate identification of thejoint dynamics of the substructures at the assembly joint,and the FRFs of each substructure. Any noise in the FRFand inaccuracies in the joint dynamic parameters amplifythe errors in obtaining the FRF for the assembled tool–spindle system.

This paper presents an improved receptance couplingtechnique to minimize errors in evaluating the FRF ofthe tool–spindle system. The end mill is modeled usinga standard finite element (FE) model of a cylindrical

890 S.S. Park et al. / International Journal of Machine Tools & Manufacture 43 (2003) 889–896

Nomenclature

1 tool tip location2 joint locationA end mill substructureB spindle substructureX displacement vectorF force vectorH receptance (X /F)H2 (HA ,22 + HB ,22)x translational displacementq rotational displacementf lateral forceM momenthff x / fhfM x /MhMf q / fhMM q /M

beam, and the FRF of the spindle–tool holder system isidentified using impact modal tests at the blank cylinderfree end, which is mounted with a set length to the toolholder. Schmitz et al. [4,5] used a similar technique butconsidered only translational degree of freedom at thespindle portion of the substructure and considered thejoint between the tool and the spindle–tool holder to beflexible. The flexible joint parameters were identifiedusing iteration of the joint values until the measurementof a sample tool and model fitted. The rotational degreeof freedom (RDOF) at the assembly joint was neglectedin their study. However, the tool bends inside the toolholder, which acts as a torsional spring, hence therotational displacement of the tool at the joint cannot beneglected for accurate construction of FRF at the tool tip.On the other hand, it is rather difficult to experimentallymeasure the dynamics of angular displacements byapplying a moment and force. In this paper, an algorithmthat allows analytical extraction of rotational dynamicsat the joints from linear displacements and impact forcetests is proposed. The rotational dynamic response of thespindle–holder is identified analytically by substitutingthe direct and cross FRF measurements taken at the freeend of the assembly and joint of the blank cylinder in thereceptance coupling expressions. The model is comparedagainst the previous approach, and is shown to haveimproved accuracy on experiments carried out with vari-ous sizes of end mills.

Henceforth, the paper is organized as follows. Thereceptance coupling technique enhanced with the identi-fication of joint dynamics, namely RDOF at the rigidjoint, is presented in the next section, which is followedby experimental verification and discussions. The paperis concluded with a summary of contributions.

2. Receptance coupling of end mill with jointdynamics

The objective of this article is to analytically couplea structural dynamic model of an end mill to an exper-imentally identified dynamic model of the tool holder–spindle assembly as illustrated in Fig. 1. The structuraldynamic model, i.e. FRF, of the end mill (SubstructureA) is obtained either using the beam theory or a finiteelement method, since it is a linear structure. The FRFof the spindle assembly (Substructure B) is measuredexperimentally by inserting a blank cylinder (i.e.carbide) into the tool holder. Consider the FRF of endmill (A) at two free ends (1, 2):

�X1

XA,2� � �HA,11 HA,12

HA,21 HA,22��F1

FA,2� (1)

where X, F are the displacement and force vectorsapplied on the structure at points 1 and 2, respectively.HA,ij vectors are FRFs between points i and j. Similarly,the FRF of the spindle structure (B) at its free end (2) is:

{XB,2} � [HB,22]{FB,2} (2)

The equilibrium and compatibility conditions at the tool–spindle joint (2) provide the following boundary con-ditions:

F2 � FA,2 � FB,2

X2 � XA,2 � XB,2

(3)

which are used in coupling the spindle (B) with the free–free model of the end mill (A). By considering the com-patibility and equilibrium conditions (Eq. (3)) in FRFs(Eqs.(1) and (2)), we obtain the following equation:

891S.S. Park et al. / International Journal of Machine Tools & Manufacture 43 (2003) 889–896

Fig. 1. Spindle/tool holder and tool substructures.

X2 � HB,22FB,2 � HA,21F1 � HA,22FA,2←FA,2 � (F2�FB,2)

FB,2 � (HA,22 � HB,22)�1(HA,21F1 � HA,22F2)(4)

Letting H2 = (HA ,22 + HB ,22), the displacements X1 andX2 can be expressed as functions of FRFs and appliedforces F1 and F2 as follows:

X1 � HA,11F1 � HA,12(F2�FB,2) � HA,11F1

� HA,12F2�HA,12FB,2 � HA,11F1 � HA,12F2

�HA,12H�12 (HA,21F1 � HA,22F2) � (HA,11

�HA,12H�12 HA,21)F1 � (HA,12

�HA,12H�12 HA,22)F2

X2 � HA,21F1 � HA,22(F2�FB,2)

� HA,2F1 � HA,22F2�HA,22H�12 (HA,21F1 � HA,22F2)

� (HA,21�HA,22H�12 HA,21)F1

� (HA,22�HA,22H�12 HA,22)F2

(5)

The equations can be rearranged in a matrix form as fol-lows:

�X1

X2

� (6)

� �(HA,11�HA,12H�12 HA,21) (HA,12�HA,12H�1

2 HA,22)

(HA,21�HA,22H�12 HA,21) (HA,22�HA,22H�1

2 HA,22)��F1

F2

�The objective is to identify the FRF at the free end of thetool when it is assembled to the spindle. By substitutingH2 = (HA ,22 + HB ,22) into Eq. (6), we obtain the followingdirect and cross receptances at the tool tip:

X1

F1� HA,11�HA,12(HA,22 � HB,22)�1HA,21 � H11

X1

F2� HA,12�HA,12(HA,22 � HB,22)�1HA,22 � H12� (7)

The FRF at the tool tip is represented as a combinationof FRFs related to the tool, spindle assembly and jointparameters. HA,11, HA,12, HA,21, and HA,22 can be evalu-ated from a finite element model or an analytical beammodel of the free–free beam, which represents the endmill. Considering the assembly of the linear tool (i.e.HA,12 = HT

A,21) to the spindle, we need to know only HB,22,which is measured directly at the free end of the short-blank cylinder held in the tool holder. However, if thetool is assembled to the blank cylinder held at the toolholder only with translational degree of freedom at thejoint interface [5], the prediction does not lead to accur-ate results as experienced and shown in the next section.The motion at the joint interface consists of both trans-lational and rotational displacements, which need to beconsidered in the model.

In this paper, the translational and rotational degreesof freedom at the free ends of the tool (A1, A2), as wellas at the free end of the spindle–blank cylinder assembly(B2) are shown in Fig. 1. Each FRF now contains bothtranslation and rotational displacement elements, henceEq. (7) at the free end of the tool can be expanded as:

�x1

q1� � �h11,ff h11,fM

h11,Mf h11,MM��f1

M1�→{X1} � [H11]{F1}

�x1

q1� � �h12,ff h12,fM

h12,Mf h12,MM��f2

M2�→{X1} � [H12]{F2}

(8)

where each element in the matrix (hij) must be evaluatedfrom the receptance coupling expression (Eq. (7)) byincluding both translational (x) and rotational (q) dis-placements due to lateral force (f) and moment (M). Bysubstituting Eq. (8) into Eq. (7), the direct and crossFRFs at the tool tip, which include both the translationaland rotational degrees of freedom, can be shown as:


[H11] � �hA11,ff hA11,fM

hA11,Mf hA11,MM��hA12,ff hA12,fM

hA12,Mf hA12,MM�

[H2]�1�hA21,ff hA21,fM

hA21,Mf hA21,MM�

[H12] � �hA12,ff hA12,fM

hA12,Mf hA12,MM��hA12,ff hA12,fM

hA12,Mf hA12,MM�

[H2]�1�hA22,ff hA22,fM

hA22,Mf hA22,MM�

(9)

where

[H2]�1 � ��hA22,ff hA22,fM

hA22,Mf hA22,MM� � �hB22,ff hB22,fM

hB22,Mf hB22,MM��1

� �h2,ff h2,fM

h2,Mf h2,MM��1

�1

h22,Mf�h2,MMh2,ff

��h2,MM h2,fM

h2,Mf �h2,ff�

where h2,ij = (hA,22,if + hB22,if) for each element (i,j→f,M)in the matrix. Furthermore, h2,Mf and hT

2,fM are equal dueto reciprocity. Substituting [H2]�1 into Eq. (9) leads to:

[H11] � �hA11,ff hA11,fM

hA11,Mf hA11,MM��

1h2

2,Mf�h2,MMh2,ff

�hA12,ff hA12,fM

hA12,Mf hA12,MM� ��h2,MM h2,fM

h2,Mf �h2,ff� �hA21,ff hA21,fM

hA21,Mf hA21,MM�

[H12] � �hA12,ff hA12,fM

hA12,Mf hA12,MM��

1h2

2,Mf�h2,MMh2,ff

�hA12,ff hA12,fM

hA12,Mf hA12,MM� ��h2,MM h2,fM

h2,Mf �h2,ff� �hA22,ff hA22,fM

hA22,Mf hA22,MM�

The first elements in the matrices [H11] and [H12] are:

H11(1,1) �x1

f1� hA11,ff

�1

h22,Mf�h2,MMh2,ff

[hA21,MF(�h2,MfhA12,ff (10)

� hA12,Mfh2,ff) � hA21,ff(�h2,MfhA12,Mf

� h2,MMhA12,ff)]

H12(1,1) �x1

f2� hA12,ff

�1

h22,Mf�h2,MMh2,ff

[hA22,MF(�h2,MfhA12,ff (11)

� hA12,Mfh2,ff) � hA22,ff(�h2,MfhA12,Mf

� h2,MMhA12,ff)]

The aim is to identify H11(1,1) = x1 / f1 without measur-ing the spindle with a full length tool, but to identify it

from a constant spindle measurement, analytical modelof the free–free tool with constant assembly joint para-meters. The FRF functions of the free–free end mill canbe evaluated from the analytical method [11] or the finiteelement model of the beam, thus the elements (hA11, hA12,hA21, hA22) in substructure (A) can be evaluated using apredetermined damping ratio for the tool material. Weneed to evaluate only the FRF values at the joint (h2,Mf,h2,ff, h2,MM). The direct transfer function h2,ff = (hA,22,ff

+ hB22,ff) can be evaluated by measuring the spindle witha short blank at point 2 (hB22,ff) and hA22,ff is obtainedfrom the free–free beam model. The remaining two(h2,Mf, h2,MM) reflect the rotational degree of freedom ofthe assembly, and it is difficult to experimentally meas-ure directly at the joint. We need to apply both forceand moment at joint (2) for the experimental evaluationof the joint properties, which is not a practical exerciseon the production floor. Instead, the paper proposes apractical methodology based on the two sets of equations(Eqs. (10) and (11)) which have common FRF terms.Eqs. (10) and (11) can be rewritten as:

u � a �f(�bb � ek) � c(�be � db)

b2�dk

v � b �g(�bb � ek) � d(�be � db)

b2�dk

(12)

where

H11(1,1) � u, H12(1,1) � v, hA11,ff � a,

hA12,ff � b, hA21,ff � c, hA22,ff � d, hA12,Mf � e,

hA21,Mf � f, hA22,Mf � g, h2,ff � k, h2,Mf � b,

h2,MM � d

The equations are reduced to the following form:

(b2�dk)(u�a)�f(�bb � ek)�c(�be � db) � 0 (13)

(b2�dk)(v�b)�g(�bb � ek)�d(�be � db) (14)

� 0

Two unknowns in the above Eqs. (13) and (14) are b,d. Through the utilization of the symbolic non-linear ana-lytical toolbox [12], the two unknowns are expressed as:

b � (�kug � kfv � kag�kfb � fdb�cbg) /

(ad�ud�cb � cv)

d �1

(ab�ud�cb � cv)2 [kf 2v2 �

(2kagf � bf 2d�2kugf�ec2g � defc�2bkf 2�bfcg)

v�d2efu � d2efa � g2ka2�decga � decgu �

g2ku2 � bdgfa�2g2kua � bec2g�bdgfu�bdefc

� 2bkugf � b2kf 2�2bkagf � bg2cu

� b2fcg�b2f 2d�bg2ca]

(15)


Therefore, the rotational degrees of freedom FRFs,hB22,Mf and hB22,MM can be obtained as:

hB22,Mf � b�hA22,Mf

hB22,MM � d�hA22,MM

(16)

It is proposed that one short and one long blank cylin-der be used as calibration gauges. The short cylinder isused to identify the spindle–tool holder assemblydynamics (hB,22,ff) and the long cylinder is used to ident-ify the joint parameters (h2,Mf, h2,MM). Direct (H11(1,1)= x1 / f1) and cross (H12(1,1) = x1 / f2) transfer functionsof the system with a long cylinder can be measured byattaching the accelerometer at point (1), and applyingimpact tests at both points 1 and 2. Since the H11(1,1)and H12(1,1) are now available, the two unknown FRFparameters (h2,Mf, h2,MM) can be extracted directly fromEqs. (15) and (16), and stored as constant properties ofthe spindle–tool holder assembly. After keeping thespindle–tool holder FRF parameters (h2,Mf, h2,ff, h2,MM) ina constant database and FRFs of a free–free end mill(hA11, hA12, hA21, hA22) of any geometry identified withFE, one can evaluate the FRF of a tool–spindle assembly(x1 / f1) using Eq. (10).

3. Experimental results

In the experimental work performed, a mechanicalchuck was used on a vertical three-axis milling machine(Fadal VMC40) as shown in Fig. 1. Three carbide-cylin-drical blanks and a four fluted end mill are used to ident-ify the spindle and joint dynamic properties, as well asto test the validity of the approach, see Fig. 2. The den-sity and the modulus of the carbide blanks, which areused as end mill materials, are 14,450 kg/m3 and 5.8e11

Fig. 2. Cylindrical blanks and four fluted end mill used in identifying the spindle–tool holder joint dynamics.

N/m2, respectively. The blanks were pushed 20 mminside the collet in all cases. The proposed method isverified using a four fluted carbide end mill as well.

A short blank cylinder of 19.05 mm diameter wasinserted into the chuck through the collet. An impacthammer instrumented with a force sensor and acceler-ometer attached to the FRF measurement point is usedfor obtaining FRF experimentally. The FRF of thespindle system is measured at the short blank’s end point(hB22,ff, Eq. (9)). Fig. 3a shows the translational part ofthis response. The spindle system has three dominantmodes at 465, 617 and 812 Hz, respectively. Fig. 3bshows the FE based free–free response of the long toolsubstructure modeled using Timoshenko beam elements.The first non-rigid body mode of the tool occurs atapproximately 8400 Hz. Fig. 4 shows the comparisonbetween the experimental and predicted coupling resultswithout the RDOF FRFs (i.e. hB22,Mf and hB22,MM arezero) for the spindle with the long blank tool. The firstthree dominant modes occur very close to the modes ofthe spindle system without tool, which shows that thesemodes are clearly influenced by the spindle. The fourthmode for the long blank is at 1015 Hz, which is due tothe interaction between the tool and the spindle system.The prediction is inaccurate in Fig. 4, and some modes,which are due to rotational joint dynamics (i.e. 1015 Hz),are not even visible when the RDOF FRFs are neglected.Based on the algorithm derived in Eqs. (15) and (16),the RDOF FRFs (i.e. hB22,Mf and hB22,MM) were acquiredby a second impact test applied to the medium lengthblank inserted into the spindle. The obtained RDOFFRFs are depicted in Fig. 5. Based on Figs. 4 and 5, themagnitude of translational FRF was of the order of 10�6

m/N, while the magnitudes of RDOF FRFs, hB22,Mf =q2 /F2 or x2 /M2 and hB22,MM = q2 /M2, were of the orders


Fig. 3. Frequency responses of spindle/tool holder and long blank.(a) FRF of spindle/tool holder measured with impact tests; (b) FRF ofthe long blank identified with finite element method.

of 10�5 and 10�4 m/N, respectively. This observationhelps to understand that the RDOF FRF responses makethe largest contribution to the flexibility of the assembledstructure, and their inclusion in the coupling analysis isessential in predicting the FRF of the spindle–toolassembly accurately.

The dynamics of the spindle assembly (hB22,ff, Eq. (9))and end mill–spindle/holder joint (hB22,Mf and hB22,fM) arenow identified experimentally from the impact testsapplied on short and medium blanks, respectively. Theidentified dynamics can now be used to evaluate the FRFof any end mill attached to the spindle–holder system.Two experiments were conducted to verify the pro-posed model.

The long blank cylinder was modeled with the FEmethod, and its free–free dynamics on both ends are cal-culated. The experimentally measured and analytically

Fig. 4. Predicted and measured FRF of the long blank attached to thespindle when the rotational dynamics are neglected.

predicted FRF at the free end of the blank mathemat-ically assembled to the system with known joint dynam-ics are compared in Fig. 6. The prediction accuracy isexcellent for the first, and quite acceptable for thesecond mode.

A four fluted carbide end mill of 19.05 mm diameterwith the length sticking out 82 mm from the collet wastested on a vertical machining center, which had a taper40 spindle with 15,000 rpm spindle speed range. Thefluted section of the end mill (i.e. 40 mm) was con-sidered to be 80% of the total diameter [13] in the finiteelement model. The measured and predicted FRF of theend mill attached to the spindle are shown in Fig. 7. Theprediction accuracy is good for the first mode (465 Hz),and quite acceptable for the second dominant modearound 1600 Hz.

4. Conclusion

A receptance coupling technique is proposed to pre-dict the dynamics of end mills attached to machine toolspindles. The machine tool spindle assembly dynamicsare measured using a short (30.67 mm) blank insertedinto the tool holder. The FRF of a free–free end mill isidentified using either beam theory or the FE model. Therotational dynamics of the end mill–spindle assembly areextracted mathematically from the direct and cross FRFmeasurements applied to a medium length (68.33 mm)blank attached to the spindle. The FRF of end mills hav-ing arbitrary dimensions is predicted by coupling thereceptances of the spindle–holder and free–free end mill.The proposed method significantly improves the accu-racy of previous receptance coupling techniques whichneglected the rotational degrees of freedom at the tool


Fig. 5. Identified rotational FRFs at the free end of the short blankattached to the spindle. (a) RDOF FRF (hB22,Mf = q2 /F2 or x2 /M2); (b)RDOF FRF (hB22,MM = q2 /M2).

holder. The proposed method eliminates repetitive FRFmeasurement of each tool attached to the spindle–toolholder system, whose dynamics are measured once andstored in a process planning database. The proposed sys-tem is used successfully in identifying chatter free cut-ting conditions in milling operations.

Acknowledgements

This research was sponsored by NSERC, Pratt & Whi-tey Canada, Boeing Commercial Airplane Division,Sandvik Coromant, and Weiss Spindle GmbH.

Fig. 6. Predicted and measured FRF of the long blank attached to thespindle when the rotational dynamics are considered.

Fig. 7. Predicted and measured FRF of the four fluted end millattached to the spindle when the rotational dynamics are considered.

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Receptance coupling for end mills