Receptance coupling for tool point dynamic prediction by fixedboundaries approach

Iker Mancisidor a,n, Aitor Urkiola a, Rafael Barcena b, Jokin Munoa a, Zoltan Dombovari c,Mikel Zatarain d

a Dynamics and Control, IK4-Ideko, Arriaga Industrialdea 2, 20870 Elgoibar, Basque Country, Spainb Department of Electronic Technology, University of Basque Country (UPV/EHU), School of Industrial Technical Engineering, 3 Rafael Moreno Ave.,48013 Bilbao, Basque Country, Spainc Department of Applied Mechanics, Budapest University of Technology and Economics, Muegyetem rkp. 3, H-1111 Budapest, Hungaryd Head of Scientific Development, IK4-Ideko, Arriaga Industrialdea 2, 20870 Elgoibar, Basque Country, Spain

a r t i c l e i n f o

Article history:Received 5 September 2013Received in revised form12 December 2013Accepted 23 December 2013Available online 8 January 2014

Keywords:ChatterMillingReceptance coupling

a b s t r a c t

The material removal capability of machines is partially conditioned by self-excited vibrations, alsoknown as chatter. In order to predict chatter free machining conditions, dynamic transfer function at thetool tip is required. In many applications, such as high-speed machining (HSM), the problematic modesare related to the flexibility of the tool, and experimental calculation of the Frequency Response Function(FRF) should be obtained considering every combination of tool, toolholder and machine. Therefore, it isa time consuming process which disturbs the production. The bibliography proposes the ReceptanceCoupling Substructure Analysis (RCSA) to reduce the amount of experimental tests. In this paper, a newapproach based on the calculation of the fixed boundary dynamic behavior of the tool is proposed.Hence, the number of theoretical modes that have to be considered is low, instead of the high number ofmodes required for the models presented up today. This way, the Timoshenko beam theory can be usedto obtain a fast prediction. The accuracy of this new method has been verified experimentally fordifferent tools, toolholders and machines.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, high-speed machining has been widely appliedin aerospace industry, due to the good machinability provided byaluminum alloys. In this type of application, typically, up to 95% ofthe initial material can be removed.

Under these conditions, the main limitation in the materialremoval rate is self-excited vibrations, also known as chatter,because they cause a reduction in the surface quality and in thelifetime of mechanical elements and tools. One of the mostpopular techniques to avoid chatter is the employment of stabilitydiagrams in order to determine the best cutting conditions. Theseso-called stability lobes separate the stable and unstable regionsdepending on the spindle speed and depths of cut.

Right now, the main reference among stability models is thezero order approach, proposed by Altintas and Budak [1]. Thissemi-analytical model has been shown to be very precise, but in

case of special tool geometries and low immersion milling, theexistence of additional stability lobes related to flip bifurcationsand mode interactions were found [2]. These inaccuracies can besolved by the employment of other models, both in frequencydomain [3] and in time domain [4].

The common point of stability models is the fact that they allrequire accurate measurements of FRF at the tool tip. The problemis that the measurements have to be repeated for each combina-tion of tool/toolholder/spindle.

In order to overcome this drawback, Schmitz and Donalson [5]proposed a receptance coupling technique to predict the dynamicresponse at the tool tip. The technique allows coupling of analy-tical and/or experimental FRFs of individual components to get theresponse of the final assembly.

In this first model, two components were joined just consideringtranslational degrees of freedom, while the interface between themwas considered flexible. The method was improved introducingthe rotational degree of freedom related to bending with its jointflexibility value [6]. The mentioned joint conditions were determinedcorrelating theoretical and experimental values and updating theflexibility values to obtain the best fitting. In recent years, severalauthors have proposed improvements to this method and havestudied the influence of contact parameters [7–13].

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ijmactool

International Journal of Machine Tools & Manufacture

0890-6955/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijmachtools.2013.12.002

n Corresponding author. Tel.: þ34 943 748000; fax: þ34 943 743800.E-mail addresses: imancisidor@ideko.es (I. Mancisidor),

aurkiola@ideko.es (A. Urkiola), rafa.barcena@ehu.es (R. Barcena),jmunoa@ideko.es (J. Munoa), dombo@mm.bme.hu (Z. Dombovari),mzatarain@ideko.es (M. Zatarain).

International Journal of Machine Tools & Manufacture 78 (2014) 18–29

Park et al. [14] proposed a rigid joint where the flexibility of theconnection is taken into account by the experimental tests withblanks with different lengths. In this way, they proposed a newmethodology to identify the dynamics on the toolholder nose,based on an inverse receptance coupling. Moreover, they provedthat rotational displacement of the tool cannot be neglected foraccurate construction of FRF at the tool tip. Some years later,Namazi et al. [13] demonstrated that this rigid-connectionassumption yields an acceptable match with the experimentalresults.

By means of the RCSA, considerable time can be saved on theprediction of the FRFs at the tool tip for different tools, combiningthe theoretical response of the different tools and the experimen-tal results of the toolholder/machine assembly.

There are many ways to achieve dynamic response of the tools.Schmitz et al. [5–9] used the analytical Euler–Bernoulli beamtheory. This theory is distinguished by its simplicity and suppliesreasonable engineering approximations for many problems.Nevertheless, it tends to overestimate slightly the natural frequen-cies, principally when working with high frequencies [15]. There-fore, the prediction is better for slender beams than non-slender beams.

Timoshenko beam model overcomes these inaccuracies addingthe effects of shear and bending to Euler–Bernoulli0s theory. Onthe contrary, this beam model introduces a cut-off frequencyabove which the results are inaccurate [15]. Sometimes this cut-off frequency is very low and consequently, only an insufficientnumber of modes are available. This beam theory was introducedin an analytical model for prediction of tool point FRF by Ertürket al. [10–12] and compared with respect to the results obtained byEuler–Bernoulli0s theory [10].

In most of these works only bending modes has been studied.However, axial and torsional modes can also be problematic insome cutting operations such as drilling, plunge milling or con-ventional milling with high diameter tools [16,17]. In this way,Schmitz [18] took into account this type of modes in the applica-tion of receptance coupling method.

At last, the theoretical response of the tools can also beobtained using the Finite Element Method (FEM) [13,14,19,20].This method discretizes the system in several elements, obtainingaccurate results based on variational principles. However, bymeans of the FEM complex modeling and meshing operationscan be required, and it is difficult to obtain a quick dynamicresponse [10].

In the bibliography, all these methods are applied consideringthe tool without any kind of restraint or connection, which isknown as free–free boundary condition. Taking into account theseconditions and applying directly the free–free boundary condi-tions, a high number of analytical modes is required to obtain anaccurate description of strain distribution near the joint (typicallymore than 80) [5,6]. This way, in some cases the employment ofTimoshenko theory is limited due to the cut-off frequency.

In order to overcome this drawback, the fixed boundary approachis proposed, following the idea presented by Mancisidor et al. [24].This novel methodology calculates a free–free beam response basedon clamped-free boundary condition beam results. This way, anaccurate result can be achieved without calculating a high numberof modes and it is possible to use Timoshenko0s model.

2. Receptance coupling

In Receptance Coupling Substructure Analysis (RCSA), experi-mental or analytical FRFs of the individual components can becoupled to predict the final dynamic response of the assembly atany selected spatial coordinate.

In this paper, an analytical model of the tool is coupled to anexperimentally identified dynamic response of the toolholder–spindle assembly, as illustrated in Fig. 1. According to Park et al.[14], a rigid joint has been considered for coupling both responses.

Mathematically, the Gij matrix describes the FrequencyResponse Function of the final assembly measured in point i andthe excitation in point j (Eq. (1)).

Xi

θi

( )¼

gixjx gixjθgiθjx giθjθ

" #FjMj

( ): ð1Þ

The point FRF at the tool tip G11 is one of the main inputsto feed the stability model [1]. By means of RCSA, this mainreceptance is estimated using receptances Hij measured or calcu-lated when the two substructures are disconnected. Operatingaccording to RCSA [14] the next basic expressions will be obtained:

½G11� ¼ ½H11��½H12�ð½H22�þ½H33�Þ�1½H21�; ð2Þ

½G12� ¼ ½H12��½H12�ð½H22�þ½H33�Þ�1½H22�; ð3Þwhere

½Gij� ¼gixjx gixjθgiθjx giθjθ

" #and ½Hij� ¼

hixjx hixjθhiθjx hiθjθ

" #:

The response in the substructure A will be obtained theoreti-cally with free–free boundary conditions. The direct FRF h3x3x inthe substructure B, can be obtained experimentally by measuringthe spindle with a short blank, but the other two receptances(h3x3θ, h3θ3θ) related to the rotational degree of freedom, aredifficult to measure directly. However, following the methodologyproposed by Park et al. [14], these functions can be obtained usinga long blank, measuring the direct response in the tool tip andfinally subtracting the effect of the blank. Other two alternativeswere presented by Kumar and Schmitz [9].

To simplify the formulation proposed by Park et al. [14],the names of variables of Eqs. (2) and (3) have been changed,obtaining Eqs. (4) and (5):

g1x1x ¼ u; g1x2x ¼ v; h1x1x ¼ a; h1x2x ¼ b; h2x1x ¼ c; h2x2x ¼ d;

h2θ2x ¼ e; h2θ2θ ¼ f ; h2x2xþh3x3x ¼ g; h2θ2xþh3θ3x ¼ β;

h2θ2θþh3θ3θ ¼ δ;

u¼ aþeð�βbþegÞþcð�βeþδbÞβ2�δg

; ð4Þ

v¼ bþ f ð�βbþegÞþdð�βeþδbÞβ2�δg

: ð5Þ

Two unknowns (β and δ) are going to be considered, and theycan be solved by the utilization of the symbolic non-linear

Fig. 1. Tool and toolholder–spindle substructures.

I. Mancisidor et al. / International Journal of Machine Tools & Manufacture 78 (2014) 18–29 19

analytical toolbox:

β¼ ðgbe�egv�edbþgf u�gf aþ f cbÞðcbþdu�da�vcÞ ; ð6Þ

δ¼ �1ðcbþdu�da�vcÞ2

ð�ge2v2�ge2b2þuedbf þe2b2d�e2d2a�gf 2a2� f 2gu2þue2d2

�e2vdbþevf cbþcef ad�eb2f cþ f 2acbþe2cbd�e2dvc

�c2ef bþc2ef vþ2ge2vbþ2ugf 2a

�uf 2cb�edbf aþ2gebf a�2gevf aþ2ugevf �2uf egb�ucef dÞ:ð7Þ

After this extraction, h3x3θ and h3θ3θ receptances can beobtained:

h3x3θ ¼ β�h2x2θ ; h3θ3θ ¼ δ�h2θ2θ : ð8Þ

Furthermore, the damping of the joint interface has been takeninto account with this methodology and it is not necessary toupdate any flexibility value.

3. Analytical prediction of the tool FRF

In the RCSA method, it is necessary to obtain the response ofthe tool considering that both ends of the beam are free, which isknown as free–free boundary approach [5–14].

This section presents a new approach to calculate theseresponses based on the fixed boundary model. The proposedmodel calculates the response of a free–free beam by means ofclamped-free boundary conditions with an addition of the effect ofthe clamped point movement. Therefore, the number of modesrequired is low, just one or two for each bending plane. This way, itprovides the possibility to use Timoshenko0s beam theory solvingthe problems created by the cut-off frequency.

In this work, tools with small diameters have been employedfor HSM, as shown in Table 1. Since the stability is mainly affectedby bending modes, the effect of torsional and axial modes hasbeen neglected. However, the method can be generalized con-sidering torsional and axial receptances [18].

3.1. Mathematical development of the fixed boundaries model

The first step with the fixed boundaries model is to calculatethe modes of the beam with clamped-free boundary conditions.Hence, a good representation of the strain distribution near thejoint is provided in low frequency modes. The next step is to addthe effect of rigid movements (δ1, δ2) restrained in the clamped-free model (see Fig. 2):

xðz; tÞ ¼ ~xðz; tÞþδ1ðtÞþδ2ðtÞz; ð9Þwhere ~xðz; tÞ and xðz; tÞ are the displacement functions of theclamped-free Timoshenko beam and the free–free beam derivedby fixed boundaries. The vibratory beam can be modeled bythe following linear differential equation using spatial dis-cretizationfxg : ¼ collðxðzl; tÞÞ:½M�f€xgþ½K�fxg ¼ fFg: ð10Þ

Mathematically, Eq. (10) represents the general equation ofvibration, and the exact displacement will be approximated by acombination of rigid body displacements plus a linear combina-tion of the low frequency modes (Eq. (11)).

fxg ¼ ∑2

k ¼ 1fφkgδkþ ∑

n

l ¼ 1fϕlgql

¼ ½φ1 φ2j ϕ1 ϕ2 … ϕn �

δ1δ2q1q2…qn

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼ ½φ ϕ �δq

( ); ð11Þ

where fφkg : ¼ collðφkðzlÞÞ and fϕkg : ¼ collðϕkðzlÞÞ. In Eq. (11), thefirst term describes rigid body displacements of the part followingunitary movements (displacement δ1 and rotation δ2) of the jointand the second refers to clamped-free modal displacements ql,while n is the number of modes considered. The rigid bodymodeshapes considered here unnormalized, like: φ1ðzÞ ¼ 1 andφ2ðzÞ ¼ z, while the clamped-free modeshapes are mass normal-ized as

ϕlðzÞ ¼ cl ~ϕ lðzÞ; ð12Þwhere

cl ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρAR L0~ϕ2l ðzÞdzþρI

R L0 ~ψ 2

l ðzÞdzq and

~ψ lðzÞ ¼ ~ϕ0lðzÞþ

ω2n;lρκG

Z~ϕðzÞdz:

Above ~ϕ lðzÞ and ~ψ lðzÞ represents the unnormalized lth mode-shapes related to the displacement and cross section deflection ofthe clamped-free Timoshemko beam, where prime means deriva-tive with respect to z. Meanwhile ωn,l, ρ, A, I, κ and G are thecorresponding natural frequencies, the density of the beam mate-rial, the cross section area, the cross section inertia, theTimoshenko shear factor and the torsional Young modulus,respectively. The modeshape vectors φk and ϕk at Eq. (11) aresufficient discrete representation of the otherwise transcendentalcontinuous modeshape functions.

Table 1Parameters of the employed tools.

Tool 1 Tool 2 Tool 3 Tool 4 Tool 5 Tool 6

Material Carbide Carbide Carbide Carbide Carbide HSSE [GPa] 580 580 580 580 510 206.8Ρ [kg/m3] 14,455 14,350 14,455 14,000 13,800 7820Diameter [mm] 20 20 20 20 20 18Total length [mm] 150 145 105 104 100 123Flute length [mm] 44 36 44 44 36 63Number of flutes 3 3 3 4 3 2r [mm] 8.88 8.52 8.88 7.3 9.07 7.11a [mm] 3 2.50 3 2.7 2.3 1.9fd [mm] 4.11 3.47 4.11 3.22 2.99 3.80

Fig. 2. Addition of clamped point movements.

I. Mancisidor et al. / International Journal of Machine Tools & Manufacture 78 (2014) 18–2920

This change drives to the following equation:

φT

ϕT

" #½M�½φ ϕ �

€δ€q

( )þ

φT

ϕT

" #½K�½φ ϕ �

δq

( )¼

φT

ϕT

" #fFg: ð13Þ

If each part of this equation is analyzed a simple form of thestiffness matrix is obtained:

ð14Þ

since:

However, the mass matrix presents more complexity in someof its terms:

φT

ϕT

" #½M�½φ ϕ � ¼

φTMφ φTMϕ

ϕTMφ ϕTMϕ

" #: ð15Þ

Operating:

φTMφ¼M0 ¼mtot mtotzgmtotzg Itot

" #;

φTMϕ¼Mϕ0 ¼R L0 ρAϕ1ðzÞdz …

R L0 ρAϕnðzÞdzR L

0 ρAϕ1ðzÞzdz …R L0 ρAϕnðzÞzdz

24

35;

ϕTMϕ¼⋱ ⋮ ⋰⋯

R L0 ρAϕkðzÞϕlðzÞþρIψ kðzÞψ lðzÞdz ⋯

⋰ ⋮ ⋱

264

375

¼

1 0 … 00 1 … 0… … … …0 0 … 1

26664

37775;

where zg is the distance from the mass center of the beam. FinallyEq. (15) can be written as

ð16Þ

If in the general Eq. (13) a sinusoidal force with a certainfrequency ω is considered, the following expression is obtained:

ð17Þ

where the modal transformation matrix related to the end points(z1, z2) displacements and cross section deflections is

Finally, the damping can be added introducing a loss factor, andobtaining the final receptance ready to be introduced in the RCSA:

ð18Þ

Taking into account the numeration defined in Fig. 2:

½H� ¼

h2x2x h2x2θ h2x1x h2x1θh2θ2x h2θ2θ h2θ1x h2θ1θh1x2x h1x2θ h1x1x h1x1θh1θ2x h1θ2θ h1θ1x h1θ1θ

266664

377775:

This way, the free–free state response is obtained describingthe correct strain distribution in the joint with a low numberof modes.

3.2. Mathematical model of the tool

In HSM it is very common to use solid carbide tools or high speedsteel (HSS) tools. These tools have few flutes in order to assure a goodchip evacuation with high material removal rates. As describedbefore, in this study the analytical model is based on Timoshenkobeammodel and the tool has been divided into two beams: the flutedpart and the pure cylindrical side. Thus, different beam properties canbe introduced along different segments of the model.

One of the most important parameters of the tool is thecalculation of its section and inertia, which is hard to model dueto the complexity of the cross-section of the fluted part. In thisstudy an approach proposed by Kivanc and Budak [21] has beenused to calculate equivalent inertias and cross-sections of thefluted beam section.

They presented cross-section models of 2, 3 and 4 fluted mill,as shown in Fig. 3, and proposed to divide each cross-section inregions determined by the following variables: the position of thecenter of the arc (a), the diameter of the flute (fd) and the radius ofthe arc (r).

The model does not take into account the helix angle, so theinertias do not change over all fluted segment. For 3 and 4 flutedtools, the inertias of the two principal bending planes are equal,and hence, the bending mode frequencies can be calculatedaccurately. However, in the case of 2 fluted tools, the inertias aredifferent for each plane and in consequence, the effect of the helixangle produces differences respect to the real response. It will beanalyzed in the next section.

4. Verification of the analytical model

In this section the analytical model based on Timoshenko0stheory with fixed boundaries approach is verified. This verificationis divided in two parts. On the one hand, results obtained by theanalytical model are compared with those obtained by FEM andthe ones obtained experimentally for free–free boundaries. On theother hand, the improvement introduced by the fixed boundariesapproach is demonstrated theoretically in a simple assembly,comparing its results by the directly applied free–free boundariesapproach.

4.1. Free–free response of the tool

The target of the analytical model is to calculate the response ofthe tool with free–free boundary conditions. By means of fixed

Fig. 3. Cross-sections considered [18].

I. Mancisidor et al. / International Journal of Machine Tools & Manufacture 78 (2014) 18–29 21

boundaries approach, firstly the clamped-free tool response iscalculated using Timoshenko0s beam theory and then the effect ofrestrained movements is added, obtaining the free–free response.The model is fed by principal properties of the tool and parametersproposed by Kivanc and Budak [21], which are defined by iterativecalculations.

In this section, the results obtained by the analytical model arecompared with the experimental and FEM results (see Fig. 4). Thiscomparison is based on the frequency deviation for the fourprincipal modes (two modes for each bending plane), and thetools defined in the Table 1 have been tested.

Table 2 reports comparison results for three of these tools,showing tools with different number of flutes. It can be observedthat results obtained by the analytical model for three and fourflutes tools are very close to those obtained by FEM, and bothsimulations show an excellent correlation with the experiments.However, when the inertias of the two principal bending planesare different, as in two fluted tools, the analytical model simulatesa major deviation. In this way, the results for two flutes tool differfrom the FEM and experimental, because in the analytical modelthe helix angle is not taken into account.

The finite elements model can model the helix angle, so theresults are very good for the three cases. However, the preparationof a new model for each tool is needed and therefore, it is a timeconsuming process, with respect to the fast response offered bythe analytical model.

In this way, it is possible to say that results obtained by theanalytical model are acceptable and hence, both the analyticalmodel and the cross section proposed by Kivanc and Budak [21]are validated.

4.2. Improvement introduced by the novel analytical model

A simple case has been selected to verify the employment ofthe new fixed boundary model with receptance coupling theore-tically. The selected structure can be seen in Fig. 5, where thecharacteristics of the substructures are classified. The idea is to

obtain the response of the assembly by means of RCSA, where theA substructure0s response is calculated by the analytical model.

The results of receptance coupling using the new analyticalfixed boundary model and the usual direct free–free boundarymodel have been compared with the real theoretical response ofthe assembly. Every calculation has been based on theTimoshenko0s beam formulation.

In the substructure A, which simulates the tool, the cut-offfrequency is reached analytically for the 12th mode. Consequentlyonly 11 modes can be taken into account using Timoshenko0sbeam model. Fig. 6 shows the effect of the number of consideredmodes. The graphics on the left have been obtained by freeboundary formulation while the ones on the right are based onthe fixed boundary model.

It can be concluded that it is not possible to create an accurateresult by free–free formulation. However, when the new approachis used two modes are enough to obtain an exact receptance.Therefore, the great advantage introduced by the novel fixedboundary model proposed in this paper was shown.

Fig. 4. (a) Small crane for the free–free holding of the tool; (b) real tool; (c) finite elements model of the tool; and (d) analytical model of the tool with the necessary inputparameters.

Table 2Comparison between results obtained experimentally, by FEM and by analyticalmodel, for tools with different number of flutes.

Mode Experimental (Hz) FEM (Hz) Analytical (Hz)

Tool 1(3 flutes)

1 4925 4923 (�0.03%) 4928 (þ0.07%)2 4925 4923 (�0.03%) 4928 (þ0.07%)3 10,860 11,174 (þ2.89%) 11,178 (þ2.92%)4 10,860 11,174 (þ2.89%) 11,178 (þ2.92%)

Tool 4(4 flutes)

1 9752 9352 (�4.01%) 9649 (�1.05%)2 9764 9358 (�4.15%) 9649 (�1.17%)3 21,534 21,914 (þ1.77%) 22,473 (þ4.36%)4 21,534 21,931 (þ1.82%) 22,473 (þ4.33%)

Tool 6(2 flutes)

1 3904 3735 (�4.31%) 3659 (�6.25%)2 4260 4072 (�4.40%) 4834 (þ13.48%)3 10,690 10,377 (�2.92%) 10,406 (�2.65%)4 11,760 11,188 (�4.85%) 12,441 (þ5.79%)

I. Mancisidor et al. / International Journal of Machine Tools & Manufacture 78 (2014) 18–2922

5. RCSA experimental application

Several experimental tests have been carried out to obtain theresponse of the assembly formed by the machine–spindle, thetoolholder and the tool, following the method proposed by Parket al. [14]. This methodology has been chosen because it presentsthe possibility of characterizing the damping of the interfacebetween tool and toolholder in the experimental receptances ofthe assembly.

This way, the receptance h3x3x is measured directly using ashort blank cylinder, and a long blank cylinder is introduced toextract (Eqs. (4)–(8)) the other two responses of the machine–spindle–toolholder assembly (h3x3θ, h3θ3θ). These two blank cylin-ders are described in Table 3. The receptances of the blanks are

also calculated by the analytical model based on Timoshenko withfixed boundaries approach.

With the objective of studying several combinations, differentmachines and toolholders (Tables 4 and 5, respectively) have beentested with the tools previously reported (Table 1).

5.1. Overview of general results

The dynamic response predictions of different assemblies,combining the reported equipment, have been performed in thissection. These results are compared with the pure experimentalreceptances.

Fig. 5. Structure for the simple case.

Fig. 6. Results of RCSA: (a1) real part with usual free boundaries model; (a2) imaginary part with usual free boundaries model; (b1) real part with the new fixed boundariesmodel; and (b2) imaginary part with the new fixed boundaries model.

Table 3Parameters of the employed blanks.

Short blank Long blank

Material Carbide CarbideE [GPa] 580 580Ρ [kg/m3] 14,500 14,500Diameter [mm] 20 20Total length [mm] 50 120

Table 4Parameters of the employed machines.

Machine 1 Machine 2 Machine 3

Model DanobatDS 630

SoraluceSV 6000

Danobat FALCON500 2G

Holder HSK-100A HSK-63A HSK-63ASpindle Kessler Kessler FisherPower [kW] 35 34 35Torque [Nm] 240 102 33Speed range [rpm] 12,000 18,000 20,000

I. Mancisidor et al. / International Journal of Machine Tools & Manufacture 78 (2014) 18–29 23

In order to validate the predictions, every response has beenstudied in frequency as well as in amplitude. The two mostdominant modes have been considered for each case. This waythe difference of both the frequency and the amplitude areanalyzed in percentages.

Fig. 7 summarizes the results obtained for these combinations,in average form. The a, b and c figures shows the average resultsdepending on the influence of the slenderness of the tool, the typeof toolholder or machine employed, respectively. They are ana-lyzed in detail in the next sections.

In Fig. 7d, it can be observed that the averages of deviation ofall studies are 1.48% and 30.38% for frequency and amplitude,respectively. The influence of the deviation for the stability infrequency is more critical than the deviation in amplitude. Thefrequency determinates the most stable regions while the ampli-tude defines somehow the minimum depth of cut. Therefore, itcan be concluded that the RCSA method with the new fixedboundaries approach is an excellent method in order to predictthe tool point response.

In the following sections, the influence of each element overthe accuracy of the method is analyzed.

5.2. Influence of the slenderness of the tool

This section focuses on discussing the influence of the slender-ness of the tool. Hence, combinations of different tools areanalyzed. The tools used are classified as slender (tool 1 and tool 2)and non-slender (tool 3, tool 4, and tool 5). This way the results of thecombinations have been averaged depending on the tool employed.Ertürk et al. [22] also analyzed the effect of the slenderness of the tool,concluding that by changing the tool overhang length, its frequencycan easily be altered, modifying the tool point FRF and stability lobesdiagram of the assembly.

Table 6 shows these results in frequency and amplitude devia-tion percentages (see also Fig. 7a). In general, the results for alltools are good even though the signals are dynamically complex.However, the results with the slender tools are better than theother tools. The reason of this is that when the tool is slender, thebending mode is more dominant when it is coupled with themachine and in this way, the importance of spindle deformationand other modes decreases. Therefore, the success of the recep-tance depends mainly on the prediction of this clear mode.However, when the tool is non-slender, different modes withdifferent shapes are significant in the receptance of the system andin the cutting stability. Therefore, the receptance coupling methodachieves better results when a slender tool is used.

Table 5Parameters of the employed toolholders.

Model Holder Type

Toolholder 1 LAIP 1214121420 HSK-63A Shrink fitToolholder 2 Gühring GM 300 HSK-63A HydraulicToolholder 3 ROMH CMBH 753216 HSK-63A Collet chuckToolholder 4 LAIP 00414250606 HSK-100A HydraulicToolholder 5 Kennametal Hertel 40120 M HSK-100A Collet chuck

Fig. 7. Percentages in amplitude and frequency deviation: (a) depending on the slenderness of the tool; (b) depending on the toolholder employed; (c) depending on themachine employed; and (d) for all combinations studied.

Table 6Average of deviations of all combinations studied, classified depending on the toolslenderness.

Frequency deviation (%) Amplitude deviation (%)

Slender tools 1.37 17.99Non-slender tools 1.58 42.77

I. Mancisidor et al. / International Journal of Machine Tools & Manufacture 78 (2014) 18–2924

Fig. 8 shows an example of receptances where machine 3 andshrink fit toolholder (toolholder 1) in combination with tool 1(slender) and tool 3 (non-slender) are tested. They are comparedwith experimental measurements and it is clear that better resultsare obtained for the slender tool.

5.3. Influence of the toolholder

The objective of this section is to validate this method fordifferent types of toolholders. This way, different interfaces

between toolholder and tool have been studied, using the tool-holders described in Table 5 (see Fig. 9).

Analyzing Table 7, it is clear that proper predictions areobtained for all types of toolholders using RCSA (see also Fig.7b).Nevertheless, in this study a major deviation is obtained whenpredicting the behavior of hydraulic toolholders. This table showsthe average of all combinations depending on the toolholderemployed.

Fig. 11 shows an example of receptances where machine 2 iscombined with tool 2 (slender tool) and tool 4 (non-slender tool)

Fig. 8. Real and Imaginary parts of RCSA prediction and measured receptance, with slender and non-slender tools.

Fig. 9. Toolholder 1: shrink fit; toolholder 2: hydraulic; and toolholder 3: collet chuck.

Table 7Average of deviations of all combinations studied, classified depend-ing on the toolholder employed.

Toolholder Frequency deviation (%) Amplitude deviation (%)

Shrink fit 1.33 30.10Hydraulic 2.13 37.80Collet chuck 0.92 23.15

Fig. 10. Machine 1: DS 630; machine 2: SV-6000; and machine 3: FALCON 500 2G.

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with different toolholders. They are compared with experimentalmeasurements and it can be concluded that this method can beapplied for different type of toolholders.

However, since impact tests have been used, some non-linearities produced in the toolholder interfaces could beneglected. Hence, a more detailed study could be necessary.

Fig. 11. Real and imaginary parts of RCSA prediction and measured receptance, with different toolholders.

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5.4. Influence of the machine–spindle

Finally, the effect of the machine with different spindles hasbeen analyzed, comparing the results obtained for differentcombinations with each machine (see Table 4 and Fig. 10).

As in the case of toolholders, Table 8 shows that accuratereceptances are obtained for all machines studied, even thoughthe results for machine 1 are slightly worse (see also Fig. 7c). This

Table 8Average of deviations of all combinations studied, classified depending on themachine employed.

Frequency deviation (%) Amplitude deviation (%)

Machine 1 3.02 40.00Machine 2 1.27 26.57Machine 3 0.65 27.78

Fig. 12. Real and imaginary parts of RCSA prediction and measured receptance, with different machines.

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way, it can be concluded that the effect of the machine has noinfluence for receptance coupling prediction.

Fig. 12 shows an example of receptances where the threemachines are combined with the collet chuck toolholder anddifferent tools, and they are compared with the experimentalresults. It can be observed, that all predictions are good in spite ofthe employment of different machines.

It is noteworthy the fact that although the same tools havebeen used, different levels of responses have been obtaineddepending on machine employed. It means that different spindlesgive rise to different performance for the same tool. This fact waswell predicted by RCSA.

6. Stability prediction

In this section, the accuracy of the predictions obtained usingthe RCSA with the new approach of fixed boundaries has beenmeasured by stability lobes diagrams. On the one hand, thestability diagrams fed by RCSA responses have been comparedwith the results provided by directly measured FRF. On the otherhand, they are correlated with the experimental cutting tests.

Since slotting operations have been performed, in this studythe zero order stability model has been employed for the obtain-ment of stability lobes diagrams [1,23]. The cutting coefficients are780 N/mm2 for tangential direction, 120 N/mm2 for radial direc-tion and 184 N/mm2 for axial direction. The machine employed isthe Soraluce SV 6000 (machine 2) with the shrink fit toolholder(toolholder 1).

Fig. 13 shows the results for tool 3, considered previously as anon-slender tool. The comparison between the two theoreticalstability lobes is quite good, since the different lobe zones aresimilar. However, the experimental cutting tests differ fromthese lobes.

By means of tool 1 (slender tool) the results shown in Fig. 14 areobtained. In this case, the diagrams obtained by the RCSA responseand experimental response are almost identical. Moreover, thecutting tests also show similar results, although some differenceis observed at high speed tests. It is important to remark thatreceptances have been calculated for the machine at rest, and it ispossible that rotation of spindle and the displacements of theslides could have some influence for the cutting test.

However, the differences observed between the cutting tests andpredicted diagrams are related to the stability model, since thecorrelation between the FRFs obtained by the RCSA method isexcellent, as shown in the previous sections. One of the major causesof error can be that FRF measurement is not performed while thespindle is rotating. In this way, there could be a difference between thestatic and dynamic responses of the machine.

7. Conclusions

This paper has proposed a new analytical model based onTimoshenko beam theory with the fixed boundary approach toperform the receptance coupling. Applying this method, it is notnecessary to calculate a high number of modes, and therefore theycan be calculated analytically by the Timoshenko beam theorywithout the cut-off frequency problem. The effectiveness of themodel and the introduced improvement has been demonstrated.

This article confirms the good results obtained by the analyticalmodel with the methodology proposed by Park et al. [14] tocharacterize the experimental data. In fact, this method provides auseful tool to consider the damping of the interface between tooland toolholder without any additional parameter.

Furthermore, the influence of different parameters has beenstudied. In this way, it has been showed that the method is valid topredict dynamic parameters of assemblies with different tool-holders and machines without substantial differences amongthem. However, it has been demonstrated that the more slenderis the beam the more accurate will be the RCSA predictions.

Finally, stability diagrams have been obtained with RCSApredictions, showing that it can be an adequate method in orderto predict the stability of the system. Nevertheless, the accuracy ofthe stability model for high speeds should be improved.

Acknowledgments

This research was partially supported by the ER-2012/00019PAINT project funded by the Basque Government, through ETOR-GAI program and the Hungarian Scientic Research FoundationOTKA Grant no. K101714.

Fig. 13. Comparison between stability lobes for a non-slender tool obtained byexperimentally measured FRF and RCSA0s predicted FRF, and they correlation withexperimental cutting tests.

Fig. 14. Comparison between stability lobes for a slender tool obtained byexperimentally measured FRF and RCSA0s predicted FRF, and they correlation withexperimental cutting tests.

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