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Modeling and identification of tool holder–spindle interface dynamics
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 47 (2007) 1333–1341 www.elsevier.com/locate/ijmactool
Modeling and identification of tool holder–spindle interface dynamics Mehdi Namazia, Yusuf Altintasa,, Taro Abeb, Nimal Rajapaksea a
Manufacturing Automation Laboratory, Department of Mechanical Engineering, The University of British Columbia, 6250 Applied Science Lane, Vancouver, BC, Canada V6T1Z4 b Advanced Tools Group, R&D Department, Tsukuba Plant, Mitsubishi Materials Corporation, 1511 Furumagi, Ishige, Ibaraki 300-2795, Japan Received 23 June 2006; received in revised form 31 July 2006; accepted 1 August 2006 Available online 10 October 2006
Abstract The majority of the chatter vibrations in high-speed milling originate due to flexible connections at the tool holder–spindle, and tool–tool holder interfaces. This article presents modeling of contact stiffness and damping at the tool holder and spindle interface. The holder–spindle taper contact is modeled by uniformly distributed translational and rotational springs. The springs are identified by minimizing the error between the experimentally measured and estimated frequency response of the spindle assembly. The paper also presents identification of the spindle’s dynamic response with a holder interface, and its receptance coupling with the holder–tool stick out which is modeled by Timoshenko beam elements. The proposed methods allow prediction of frequency-response functions at the tool tip by receptance coupling of tools and holders to the spindle, as well as analyzing the influence of relative wear at the contact by removing discrete contact springs between the holder and spindle. The techniques are experimentally illustrated and their practical use in high speed milling applications is elaborated. r 2006 Elsevier Ltd. All rights reserved. Keywords: Tool holder; Spindle; Contact dynamics; Receptance; Coupling
1. Introduction High-speed machining is mainly limited by the spindle and tool holder structures, which are the most flexible parts with high natural frequencies. The identification of holder–spindle taper interface stiffness as well as their combined dynamics are important for an improved design and prediction of chatter stability lobes [1]. Rivin [2] extensively assessed the state of the art in the tooling structures’ technology, and discussed six important subjects related to tooling: the influence of machining parameters on tool life and stability; stiffness and damping of tools; tool–tool holder interfaces; modular tooling; tool–machine interfaces and tool balancing for high-speed machines. Levina [3] studied the effects of angular Corresponding author. Tel.: +1 604 822 5622; fax: +1 604 822 2403.
E-mail addresses: [email protected] (M. Namazi), [email protected] (Y. Altintas), [email protected] (N. Rajapakse). URL: http://www.mech.ubc.ca/mal. 0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.08.003
deformations in the spindle–tool holder interface on deflection at the tool tip. Levina [3] investigated the effect of drawbar force and taper tolerance on the static stiffness of the tool holder–spindle connection. Smith et al. [4] showed that increased drawbar force increases the static stiffness of the tool holder–spindle interface, at the expense of reduced damping. Predicting the dynamic response of the tool holder– spindle assembly is important in order to assess the chatter stability during machining. Schmitz and Donaldson [5], were first to propose a method for predicting the frequency-response function (FRF) at the tool tip using the receptance coupling technique. Kivanc and Budak [6] used an approach similar to that of Schmitz and Donaldson [5] in their frequency-response prediction, but the complex end-mill geometry was modeled in finite elements and equations were developed to predict the static and dynamic properties of the tools. Although the dynamics of the tool-tip FRF prediction was improved due to accurate modeling of the flute geometry, the rotational degree of freedom (RDOF) of the spindle–tool
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holder assembly were neglected, and the joint parameters were identified using the least-squares-error minimization method. Movahhedy and Gerami [7] proposed two joint models with linear and rotational springs to model the tool–tool holder connection. An optimization method based on a genetic algorithm was employed to find the parameters of the joint model. Park et al. [8] proposed an improved receptance coupling technique to identify the dynamics on the spindle. Park et al. [8] demonstrated that the spindle RDOF dynamics have to be considered in order to obtain accurate FRF predictions at the tool tip. Park included the rotational dynamics at the joint, which were indirectly identified using translational responses measured from a set of short and long blank tools. The end mill was modeled in finite elements and subtracted from the spindle–tool holder assembly by employing the inverse receptance coupling technique. This paper presents modeling and identification of tool holder–spindle interface stiffness using translational and rotational springs which were uniformly distributed at the contact zone. Later, the tool holder–spindle assembly dynamics are predicted using receptance coupling technique which includes both translational and RDOF. The stick out of the holder and tool are modeled with the finite element method, and subtracted from the system leaving only the spindle with a holder taper assembly. Arbitrary tool holders and tools are mathematically added to the spindle using the receptance coupling technique. The proposed method allows prediction of FRFs of tools
mounted on the spindle without having to repeat impact modal tests at each tool change. Henceforth, the paper is organized as follows. The identification of the tool holder–spindle taper contact stiffness is presented in Section 2 along with experimental procedures. The receptance coupling with an experimental proof is presented in Section 3. The paper is concluded in Section 4. 2. Modeling of tool holder–spindle taper connection Conical tapered connections are used as the interface between the spindle and the tool holder. The holder is pulled towards the spindle taper by a drawbar force mechanism. The friction between the holder and spindle taper interfaces determine the translational and rotational stiffness of the connection. 2.1. Modeling of contact stiffness The connection between the tapered surfaces is modeled by springs in the x and y directions that prevent the tool holder from rotating and translating inside the spindle taper. Two-degree-of-freedom Timoshenko beam elements are used to model the tool holder taper and the spindle. Each element has one translational (y) degree of freedom in the radial direction, and one RDOF (yz), and the elements are connected to each other through rotational and radial springs as shown in Fig. 1.
Fig. 1. Timoshenko beam element model of the tool holder–spindle interface with distributed contact springs.
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The translational springs are converted into rotational springs as follows. When an element of length dx along the taper is subjected to a rotation of dyz, the horizontal springs on both sides of the tool holder taper deform by an amount dxs. Since the rotational deflection of the element, dyz, is small, the distortion of the springs can be neglected and the deformation of the horizontal springs, dxs, can be expressed as Dx dyz , (1) 2 where Dx is the diameter of the element. The resisting axial forces acting on the element due to the deformation in the horizontal springs is dxs ¼
Dx dyz , (2) 2 where Kx is the stiffness of the horizontal contact spring. The resisting axial forces generate a moment about the center of the element, which prevents the element rotating: dF x ¼ K x dxs ¼ K x
Dx D2 ¼ K x x dyz 2 2 D2 ð3Þ ¼ K y dyz ; ! K y ¼ K x x , 2 where Ky is the equivalent stiffness of the rotational spring. The equivalent radial stiffness is Kr. An element of length dx and surface area dA at axial location Lx along the tool holder taper is subjected to a radial contact force Fc. The radial deformations, y, are linearly related to the contact force as follows:
dM ¼ 2 dF x
F c ¼ K r y.
Integrating over the whole contact length, the total potential energy stored in the contact springs is expressed as Z L Z L 1 1 2 ky y dA; U y ¼ ky y2 dA, (9) Uy ¼ 2 0 0 2 where L is the contact length of the tool holder with the spindle taper. The total potential energy stored in the contact springs is added to the internal virtual work as the result of bending deformations of the elastic tool holder. The internal virtual work for the Timoshenko beam undergoing radial translation (y), and rotational deformation (y) can be expressed as follows: Z L 1 Ub ¼ EI x ðy0 Þ2 þ kGAðy0 yÞ dx, (10) 0 2 where E is the Young’s modulus of elasticity and I is the area moment of inertia of the cross-section. G is the modulus of traverse elasticity, k is the cross sectional factor and A is the cross sectional area of the tool holder taper. Finally, the total internal virtual work of the spindle–tool holder interface is the sum of the three terms obtained in (9) and (10): U tot ¼ U b þ U y þ U y .
(11)
The finite element formulation of the tool holder–spindle interface from the internal virtual work is formulated, and the details can be found in any Timoshenko beam formulation or in the thesis of Namazi [9].
(4)
Similarly, the rotational deformations (y) are linearly related to the contact bending moment, Mc, as follows: M c ¼ K y y,
(5)
where Kr and Ky are the stiffness of each spring at a distance x along the contact length. The computational model of the contact-interface considers the distributed radial and rotational springs on the tapered surface of the tool holder, as shown in Fig. 1. The stiffness per unit area of the distributed radial (kr) and rotational (ky) springs are assumed to be constant, K r ¼ kr dA ¼ kr p Dx dx, K y ¼ ky dA ¼ ky p Dx dx
dA ¼ pDx dx.
ð6Þ
The resulting taper contact force and bending moment become F c ¼ ðkr dAÞ y;
1335
M c ¼ ðky dAÞ y.
(7)
The potential energy stored in the contact springs at the tool holder spindle interface as a result of radial deformations (y) and rotational deformations (y), is dU y ¼ 12 F c y ¼ 12 kr y2 dA, dU y ¼ 12 M c y ¼ 12 ky y2 dA,
ð8Þ
2.2. Experimental procedure The radial and rotational stiffness per unit area of the distributed springs, kr and ky are identified experimentally through dynamic testing. Numerical identification of contact springs were also attempted by considering contact elements in commercial finite element software, but the accuracy was inevitably questionable for a practical use in production floors [9]. The experimental setup (Fig. 2) is designed to investigate the spindle–tool holder interface dynamics without the effects of the spindle bearings and other machine elements that are inevitable on a real machine tool. The large steel block (approximately 40 kg) with the spindle taper has a load cell on the back to measure the drawbar force. The block is placed on a supporting cushion to simulate free–free boundary conditions, and impact modal tests are performed to obtain the FRF at the tool tip [10]. Both the spindle block and tool holder are modeled using Timoshenko cylindrical beam elements by including shear and rotational deformations. The equivalent circular cross-section, which has the same area and second moment of area as the cross section of the rectangular block with
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Fig. 2. Experimental setup for dynamic analysis of spindle–tool holder interface. Fig. 3. Frequency-response function of the CAT 40 taper in experimental setup—10 kN drawbar force. Table 1 Contact stiffness per unit area for CAT 40 taper determined by experimental identification—95 mm overhang length from spindle face Drawbar force (kN)
Radial stiffness per unit area, kr (N/m3)
Rotational stiffness per unit area, ky (N/rad m)
4 6 8 10
0.75 1012 0.82 1012 0.95 1012 1.09 1012
5.25 108 5.87 108 6.25 108 6.79 108
dimensions (b, h), is calculated as bh
p d 2eq ; 4
4 bh3 p d eq . 12 64
(12)
Impact hammer tests are performed on a shrink-fit tool holder without any tool in the block. The drawbar force is adjusted through the load cell, and the FRF is obtained at the tool holder tip through impact modal tests. The FRF of the finite element model is simulated by adopting the same damping ratios obtained from modal tests. By using a nonlinear least-squares curve-fitting algorithm, the two contact stiffness per unit area of the contact springs, kr and ky , are determined to match the finite element simulations with experimentally measured FRF. The identification is carried out for CAT 40, CAT 50 and the HSK A63 taper interface at 2 kN drawbar force increments until the limit load recommended by the manufacturers. A sample results for CAT 40 taper are shown in the article as an example. A CAT 40 shrink-fit tool holder without any tool, and with 95 mm shank gauge length measured from the spindle face. The contact stiffness per unit area of the CAT40 taper are listed in Table 1, and the comparison between the fitted FRFs and the experimental results for the CAT 40 taper with 10 kN drawbar force is shown in Fig. 3.
Table 2 Comparison of the contact stiffness per unit area for HSK A63 and CAT 40 taper with 10 kN drawbar force Tool holder taper
Radial stiffness per unit area, kr (N/m3)
Rotational stiffness per unit area, ky (N/rad m)
HSK-A63 CAT 40
3.87 1012 1.09 1012
2.83 109 6.79 108
The equivalent of the CAT 40 taper is the HSK 63, because they have relatively equal gauge diameters, power and torque transmission capabilities. The identified stiffness per unit area of the two tapers for 10 kN drawbar force is compared in Table 2. The radial and rotational contact stiffness per unit area of the HSK taper are about 4 times the CAT 40 taper. This is mainly due to the dual-face contact of the HSK taper design. However, the surface contact area of the CAT 40 taper is about 2 times that of the HSK 63A taper, thus making up for one half of the loss in stiffness due to the face contact. In order to investigate the effect of tool length on the contact stiffness of the holder, a short and a long, carbide blanks with 16 mm diameter are attached to the shrink CAT 50 tool holder. The drawbar force applied on the tool holder was 20 kN, and the tool holder–tool connection is assumed to be rigid. The comparison of experimental and predicted FRFs of the assembly at the free end of the blank tool using the identified holder–spindle taper contact stiffness are shown in Fig. 4 for tools with long (64 mm) and short (15 mm) overhangs. While the predicted FRF for short overhang agrees well with the measurements (Fig. 4b), the influence of long overhangs is clearly visible in Fig. 4a. Increased bending loads change the contact stiffness between the tool holder and spindle taper, which need to be included in the estimated contact stiffness.
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Fig. 5. Finite element model of spindle system on machine.
Table 3 Radial and rotational spring stiffness of CAT 40 tool holder–spindle Spring no.
Radial stiffness, Kr (N/m)
Rotational stiffness, Ky (N m/rad)
1 2 3 4 5
4.70 109 8.99 109 6.40 109 5.80 109 3.60 109
6.3 105 1.20 106 8.60 105 7.70 105 4.80 109
springs is calculated in discrete form of Eq. (6), K r ðiÞ ¼ kr p DðiÞ LðiÞ, K y ðiÞ ¼ ky p DðiÞ LðiÞ,
Fig. 4. Comparison of frequency-response functions of the CAT 50 shrink-fit holder with a long (64 mm) and short (15 mm) tool overhangs under 20 kN drawbar force: (a) 64 mm tool overhang and (b) with 15 mm tool overhang.
2.3. Prediction of tool FRFs on the machine tool spindle The contact stiffness constants for the CAT 40 taper are used to couple the tool holder to a finite element model of a spindle system developed by Cao [11] on the machine. The finite element model of the spindle on the machine tool is created using the Spindle Pros [12] finite element program, as shown in Fig. 5. First, the tool holder–spindle connection is assumed to be rigid, and in the second case the connection is assumed to be modeled by the distributed radial and rotational springs along the taper. The identified radial and rotational stiffness per unit area of the connection springs are kr ¼ 1.09 1012 (N/m3) and ky ¼ 6:79 108 (N/rad m), respectively as shown in Table 1 for 10 kN drawbar force. The tool holder taper is modeled by 4 beam elements. Five contact springs are considered between the tool holder and the spindle taper. The stiffness of the connection
ð13Þ
where Kr(i) and Ky(i) are the radial and rotational stiffness of the connection springs at node i, D(i) is the diameter of the taper element and L(i) is the tapered element length. The stiffness values for the 5 springs at a drawbar force of 10 kN are listed in Table 3. The large contact spring constants correspond to the sections of the tool holder with larger diameters. The FRF of the finite element model of the spindle–tool holder assembly is simulated at the tool tip for the following two cases: rigid tool holder–spindle connection and distributed-springs connection. The simulation results are compared with the experimental results as shown in Fig. 6. The rigid-connection assumption yields results close to those of the spring-connection, and both simulations show an acceptable match with the experiments. However, the spring-connection model is able to simulate the mode at around 2900 Hz more accurately because this mode is dominated by the tool–tool holder assembly. The static stiffness of the assembly is also accurately simulated using the spring connection. It is obtained by calculating the magnitude of the FRF at zero frequency. By comparing the FRFs shown in Fig. 6, the experimental FRF and spring connection FRF match well at the lowerfrequency range; however, the rigid connection is statically stiffer by 15%.
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Fig. 6. Frequency-response function at the tool tip.
3. Structural dynamics of the holder–spindle interface The identification of static contact stiffness between the holder and spindle is important to compare various holder shapes and contact mechanisms. However, FRF of the holder–spindle interface determines the chatter stability of the machining process, and need to be identified. Although the most accurate method is to measure the FRF at the tool tip, it is a costly process since FRF needs to be measured at each tool change. A receptance coupling technique proposed by Park and Altintas [8] is extended here to identify the rotational and translational dynamics of the spindle up to the tool holder flange, which is standard and remains unchanged as shown in Fig. 7. 3.1. Mathematical model
Fig. 7. Receptance coupling for obtaining frequency-response function at the tool tip.
point 2 are: F 2 ¼ F A;2 þ F B;2 , X 2 ¼ X A;2 þ X B;2 .
ð16Þ
By letting H 2 ¼ H A;22 þ H B;22
(17)
and substituting (17) into (15) The machine tool assembly (structure AB) is divided into two substructures. Substructure A represents the tool holder, and substructure B represents the remaining machine tool assembly up to the tool holder flange. The two structures are rigidly connected at point 2. The FRFs of substructure A at the two free ends is ( ) " # ( ) X1 H A;11 H A;12 F1 ¼ , (14) X A;2 H A;21 H A;22 F A;2 where X1 and XA,2 are displacement vectors with both translational and angular displacement components. F1 and FA,2 are force vectors containing both force and moments applied at points 1 and 2. HA,ij are FRFs between points i and j. Similarly, the FRFs of substructure B at point 2 is fX B;2 g ¼ H B;22 fF B;2 g. (15) After rigidly coupling the two structures A and B at point 2, the equilibrium and compatibility conditions at
X 2 ¼ H B;22 F B;2 ¼ H A;21 F 1 þ H A;22 ðF 2 F B;2 Þ.
(18)
By rearranging (18), the forces on structure B are: 1 H A;21 F 1 þ H A;22 F 2 F B;2 ¼ H B;22 þ H A;22 ¼ ðH 2 Þ1 H A;21 F 1 þ H A;22 F 2 .
ð19Þ
Finally the displacements at points 1 and 2 are expressed as functions of FRFs and the applied forces as follows: X 1 ¼ H A;11 F 1 þ H A;12 ðF 2 F B;2 Þ ¼ H A;11 F 1 þ H A;12 F 2 H A;12 ðH 2 Þ1 ðH A;12 F 1 þ H A;12 F 2 Þ ¼ ðH A;11 H A;12 ðH 2 Þ1 H A;21 Þ F 1 þ ðH A;12 H A;12 ðH 2 Þ1 H A;22 Þ F 2 , X 2 ¼ H A;21 F 1 þ H A;22 ðF 2 F B;2 Þ ¼ H A;21 F 1 þ H A;22 F 2
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"
H A;22 ðH 2 Þ1 ðH A;21 F 1 þ H A;22 F 2 Þ
¼ ðH A;21 H A;22 ðH 2 Þ1 H A;21 Þ F 1 þ ðH A;22 H A;22 ðH 2 Þ1 H A;22 Þ F 2 .
"
ð20Þ
Eqs. (20) are arranged in matrix form as follows: 2 H A;11 H A;12 ðH 2 Þ1 H A;21 X1 ¼ 4 X1 H A;21 H A;22 ðH 2 Þ1 H A;21 3 H A;21 H A;12 ðH 2 Þ1 H A;22 F1 , 5 F2 H A;22 H A;22 ðH 2 Þ1 H A;22
ð21Þ
hA12;fM
hA12;Mf
hA12;MM # "
h21;ff
X1 F1 X2 F1 X1 F2 X2 F2
¼ H A;21 H A;22 ðH 2 Þ1 H A;21 ¼ H 12 ,
¼ H A;21 H A;12 ðH 2 Þ1 H A;22 ¼ H 21 , ¼ H A;22 H A;22 ðH 2 Þ1 H A;22 ¼ H 22 .
ð22Þ
Each FRF contains both translational and rotational displacements, thus Eqs. (22) can be expanded as ( ) ( ) " # ( ) h11;fM h11;ff x1 f1 f1 ¼ ½H 11 ¼ , h11;Mf h11;MM y1 M1 M1 ( ) ( ) " # ( ) h12;fM h12;ff x2 f1 f1 ¼ ½H 12 ¼ , h12;Mf h12;MM y2 M1 M1 ( ) ( ) " # ( ) h21;fM h21;ff x1 f2 f2 ¼ ½H 21 ¼ , h21;Mf h21;MM y1 M2 M2 ( ) ( ) " # ( ) h22;fM h22;ff f2 x2 f2 ¼ ½H 22 ¼ . h22;Mf h22;MM y2 M2 M2 ð23Þ By substituting Eqs. (23), into Eqs. (22), direct and crosstransfer functions at points 1 and 2 with both rotational and translational degrees of freedom are found as " ½H 11 ¼
h11;ff
h11;fM
h11;Mf
h11;MM
#
" ¼
hA11;ff
hA11;fM
hA11;Mf
hA11;MM
½H 2
h21;fM
hA21;ff
#
hA21;ff
hA21;fM
hA21;Mf
hA21;MM #
hA21;fM
,
h21;Mf h21;MM hA21;Mf hA21;MM " # " # hA12;ff hA12;fM hA22;ff hA22;fM 1 ½H 2 , hA12;Mf hA12;MM hA22;Mf hA22;MM " # " # h22;fM hA22;fM h22;ff hA22;ff ½H 22 ¼ ¼ h22;Mf h22;MM hA22;Mf hA22;MM "
hA22;ff
hA22;fM
hA22;Mf
hA22;MM
#
" 1
½H 2
hA22;ff
hA22;fM
hA22;Mf
hA22;MM
# , ð24Þ
where ("
½H 2
1
¼ " ¼
hA22;ff
hA22;fM
hA22;Mf
hA22;MM #1
h22;ff
h2;fM
h2;Mf
h2;MM
#
" þ
.
hB22;ff
hB22;fM
hB22;Mf
hB22;MM
#)1
ð25Þ
The first elements in the three matrices [H11], [H12] and [H22] in Eq. (24) along with the equation for the reciprocity condition yields 4 sets of nonlinear equations:
¼ H A;11 H A;12 ðH 2 Þ1 H A;21 ¼ H 11 ,
" 1
¼
where H 2 ¼ H A;22 þ H B;22 . Eq. (21) represents the receptance coupling of the spindle with holder up to the flange and holder–tool stick out structures. The receptances of the free–free tool holder–tool assembly, HA,11, HA,12 and HA,22, are modeled using the finite element model and the receptance of the spindle at point 2, HB,22, is obtained through the inverse receptance coupling method as explained in the following. The following cross and direct receptances are obtained from Eqs. (20) and (21):
#
hA12;ff
½H 21 ¼
1339
#
1 h11;ff ¼ hA11;ff ðh2;ff h2;MM h2;fM h2;Mf Þ
ðhA12;ff h2;MM hA12;fM h2;Mf Þ hA21;ff þ þ ðhA12;fM h2;ff hA12;ff h2;fM Þ hA21;Mf , 1 h12;ff ¼ hA12;ff ðh2;ff h2;MM h2;fM h2;Mf Þ
ðhA12;ff h2;MM hA12;fM h2;Mf Þ hA21;ff þ þ ðhA12;fM h2;ff hA12;ff h2;fM Þ hA22;Mf , 1 h22;ff ¼ hA22;ff ðh2;ff h2;MM h2;fM h2;Mf Þ
ðhA22;ff h2;MM hA22;fM h2;Mf Þ hA22;ff þ þ ðhA22;fM h2;ff hA22;ff h2;fM Þ hA22;Mf , h2;fM ¼ h2;Mf . ð26Þ The four unknowns are, h2,ff, h2,fM, h2,Mf and h2,MM which are the receptances of the assembly at point 2 and need to be solved. The terms h11,ff , h12,ff ,and h22,ff are obtained by 3 impact hammer tests at points 1 and 2. The FRF’s of the free–free substructure A, are obtained through the finite element method. This system of nonlinear equations is symbolically solved in MAPLEs [13]. The translational and RDOF FRFs at point B can be obtained as: hB22;ff ¼ h2;ff hA22;ff ; hB22;fM ¼ hB22;Mf ¼ h2;fM hA22;fM ; hB22;MM ¼ h2;MM hA22;MM .
ð27Þ
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The spindle dynamics, which include the RDOF, are stored in a matrix as shown below " # hB22;ff hB22;fM H B;22 ¼ . (28) hB22;Mf hB22;MM
3.2. Experimental procedure A shrink fit HSK 63 tool holder without a tool is attached to the spindle as shown in Fig. 8. Three impact modal tests are performed at points 1 and 2: the direct FRF at point 1; h11,ff cross FRF at points 1 and 2, h12,ff and direct FRF measurement at point 2, h22,ff as shown in Fig. 8. Structure A, the stick out of the tool holder is modeled by finite element method using Timoshenko beams. The tool holder and spindle with HSK 63A taper assembly is coupled by employing Eqs. (17) and (22) as follows: H 11 ¼ ðH A;11 H A;12 ðH B;22 þ H A;22 Þ1 H A;21 Þ.
The proposed receptance coupling method is experimentally evaluated on a horizontal machining center. The Mori Seiki SH403 horizontal machining center has an HSK 63A spindle taper. The spindle with HSK 63 interface was identified using a short shrink fit holder with a gauge length of 60 mm first. The FRF of another tool holder with 140 mm gage length (Fig. 9) was estimated using the proposed receptance coupling technique. The tool–tool holder connection in the shrink-fit is modeled as a rigid connection and the fluted tool is considered to be 80% of the total shank diameter. The predicted FRF at the tool tip are compared with the experiments in both x and y directions as shown in Fig. 10, which has acceptable accuracy for use in chatter stability
(29)
The fluted section of the end mill is considered to be 80% of the total diameter in the finite element model and the tool–tool holder connection in the shrink-fit is assumed to be rigid. The tool–tool holder model is in the free–free condition as the rigid body modes play an important role in the coupling between the structures. The damping ratio used for the finite element model is assumed to be 1–3%, which was verified by several impact tests.
Fig. 9. Tool–tool holder assembly—tooling A.
Fig. 8. Inverse receptance coupling for obtaining spindle dynamics.
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2
x 10-7 Predicted Experiment
Re (Φ(ω))
(m/N)
1.5 1
0.5
1341
The overall structural dynamics of the spindle must be identified at the tool tip in order to predict chatter stability lobes. The paper presents a method, which allows identification of the holder and spindle assembly at the spindle face, and allows receptance coupling of free-free holder with a cutter. The proposed method allows prediction of FRFs at the tool tip without having to measure the FRFs at each tool–holder change.
0
Acknowledgments -0.5 -1 -1.5 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz) 0.5
x 10-7
This research is sponsored by Mitsubishi Materials, Natural Sciences and Engineering Research Council of Canada and Pratt & Whitney Canada under Industrial Virtual High Performance Machining Research Chair Grant. Mori Seiki HS403 is loaned by Machine Tool Technology Research Foundation (MTTRF). References
0
Im (Φ (ω)) (m/N)
-0.5 -1
-1.5 -2 -2.5 -3 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)
Fig. 10. Measured and predicted frequency-response functions in X direction at the tool tip on the Moriseiki SH403.
prediction methods [14,15]. The predicted FRFs are also able to model the higher frequencies more accurately, which are difficult to measure accurately via impulse modal tests due to loss of spectral strength of the hammers at high-frequency range dominated by the flexible tool modes. 4. Conclusion Tool holder–spindle interface stiffness is one of the main obstacles in achieving chatter free, high-speed machining. The holder–spindle interface stiffness depends on the drawbar force as well as total contact, which is modeled by a uniformly distributed translational and rotational springs in this paper. The influence of loosing some of the contact areas due to wear can be simulated by removing the corresponding spring elements. Furthermore, different holder design scenarios and shapes can be investigated by applying the proposed contact model.
[1] Y. Altintas, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press, Cambridge, 2000. [2] E.I. Rivin, Tooling structure: interface between cutting edge and machine tool, Annals of CIRP 49/2 (2000) 591–634. [3] Z.M. Levina, Stiffness calculations for cylindrical and taper joints, Machines and Tooling XLI (1970) 5–10. [4] S. Smith, W.R. Winfough, J. Halley, The effect of drawbar force on metal removal rate in milling, Annals of CIRP 48/1 (1998) 293–296. [5] T. Schmitz, R. Donaldson, Predicting high-speed machining dynamics by a substructure analysis, Annals of CIRP 49 (2000) 303–308. [6] E.B. Kivanc, E. Budak, Structural modeling of end mills for form error and stability analysis, International Journal of Machine Tools & Manufacture 44 (2004) 1151–1161. [7] M. Movahhedy, J. Gerami, Prediction of spindle dynamics in milling by sub-structure coupling, International Journal of Machine Tools & Manufacture 46 (2006) 243–251. [8] S.S. Park, Y. Altintas, M. Movahhedy, Receptance coupling for end mills, International Journal of Machine Tools & Manufacture 43 (2003) 889–896. [9] M. Namazi, Mechanics and dynamics of the tool holder–spindle interface, M.A.Sc. Thesis, Mechanical Engineering Department, The University of British Columbia, Vancouver, BC, Canada, 2006. [10] CutPros—Advanced Cutting Process Measurement and Simulation Software, Manufacturing Automation Laboratory, University of British Columbia. [11] Y. Cao, Y. Altintas, A general method for the modeling of spindlebearing systems, Transactions of ASME, Journal of Mechanical Design 126 (2004) 1089–1104. [12] SpindlePros—Advanced Spindle Dynamics Modeling Software, Manufacturing Automation Laboratory, University of British Columbia. [13] MAPLEs—Waterloo MAPLE Inc. [14] E. Budak, Y. Altintas, Analytical prediction of chatter stability in milling—Part I: general formulation, Journal of Dynamic Systems, Measurement, and Control, Transactions of the ASME 120 (1998) 22–30. [15] E. Budak, Y. Altintas, Analytical prediction of chatter stability in milling—Part II: application of the general formulation to common milling systems, Journal of Dynamic Systems, Measurement, and Control, Transactions of the ASME 120 (1998) 31–36.
International Journal of Machine Tools & Manufacture 47 (2007) 1333–1341 www.elsevier.com/locate/ijmactool
Modeling and identification of tool holder–spindle interface dynamics Mehdi Namazia, Yusuf Altintasa,, Taro Abeb, Nimal Rajapaksea a
Manufacturing Automation Laboratory, Department of Mechanical Engineering, The University of British Columbia, 6250 Applied Science Lane, Vancouver, BC, Canada V6T1Z4 b Advanced Tools Group, R&D Department, Tsukuba Plant, Mitsubishi Materials Corporation, 1511 Furumagi, Ishige, Ibaraki 300-2795, Japan Received 23 June 2006; received in revised form 31 July 2006; accepted 1 August 2006 Available online 10 October 2006
Abstract The majority of the chatter vibrations in high-speed milling originate due to flexible connections at the tool holder–spindle, and tool–tool holder interfaces. This article presents modeling of contact stiffness and damping at the tool holder and spindle interface. The holder–spindle taper contact is modeled by uniformly distributed translational and rotational springs. The springs are identified by minimizing the error between the experimentally measured and estimated frequency response of the spindle assembly. The paper also presents identification of the spindle’s dynamic response with a holder interface, and its receptance coupling with the holder–tool stick out which is modeled by Timoshenko beam elements. The proposed methods allow prediction of frequency-response functions at the tool tip by receptance coupling of tools and holders to the spindle, as well as analyzing the influence of relative wear at the contact by removing discrete contact springs between the holder and spindle. The techniques are experimentally illustrated and their practical use in high speed milling applications is elaborated. r 2006 Elsevier Ltd. All rights reserved. Keywords: Tool holder; Spindle; Contact dynamics; Receptance; Coupling
1. Introduction High-speed machining is mainly limited by the spindle and tool holder structures, which are the most flexible parts with high natural frequencies. The identification of holder–spindle taper interface stiffness as well as their combined dynamics are important for an improved design and prediction of chatter stability lobes [1]. Rivin [2] extensively assessed the state of the art in the tooling structures’ technology, and discussed six important subjects related to tooling: the influence of machining parameters on tool life and stability; stiffness and damping of tools; tool–tool holder interfaces; modular tooling; tool–machine interfaces and tool balancing for high-speed machines. Levina [3] studied the effects of angular Corresponding author. Tel.: +1 604 822 5622; fax: +1 604 822 2403.
E-mail addresses: [email protected] (M. Namazi), [email protected] (Y. Altintas), [email protected] (N. Rajapakse). URL: http://www.mech.ubc.ca/mal. 0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.08.003
deformations in the spindle–tool holder interface on deflection at the tool tip. Levina [3] investigated the effect of drawbar force and taper tolerance on the static stiffness of the tool holder–spindle connection. Smith et al. [4] showed that increased drawbar force increases the static stiffness of the tool holder–spindle interface, at the expense of reduced damping. Predicting the dynamic response of the tool holder– spindle assembly is important in order to assess the chatter stability during machining. Schmitz and Donaldson [5], were first to propose a method for predicting the frequency-response function (FRF) at the tool tip using the receptance coupling technique. Kivanc and Budak [6] used an approach similar to that of Schmitz and Donaldson [5] in their frequency-response prediction, but the complex end-mill geometry was modeled in finite elements and equations were developed to predict the static and dynamic properties of the tools. Although the dynamics of the tool-tip FRF prediction was improved due to accurate modeling of the flute geometry, the rotational degree of freedom (RDOF) of the spindle–tool
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holder assembly were neglected, and the joint parameters were identified using the least-squares-error minimization method. Movahhedy and Gerami [7] proposed two joint models with linear and rotational springs to model the tool–tool holder connection. An optimization method based on a genetic algorithm was employed to find the parameters of the joint model. Park et al. [8] proposed an improved receptance coupling technique to identify the dynamics on the spindle. Park et al. [8] demonstrated that the spindle RDOF dynamics have to be considered in order to obtain accurate FRF predictions at the tool tip. Park included the rotational dynamics at the joint, which were indirectly identified using translational responses measured from a set of short and long blank tools. The end mill was modeled in finite elements and subtracted from the spindle–tool holder assembly by employing the inverse receptance coupling technique. This paper presents modeling and identification of tool holder–spindle interface stiffness using translational and rotational springs which were uniformly distributed at the contact zone. Later, the tool holder–spindle assembly dynamics are predicted using receptance coupling technique which includes both translational and RDOF. The stick out of the holder and tool are modeled with the finite element method, and subtracted from the system leaving only the spindle with a holder taper assembly. Arbitrary tool holders and tools are mathematically added to the spindle using the receptance coupling technique. The proposed method allows prediction of FRFs of tools
mounted on the spindle without having to repeat impact modal tests at each tool change. Henceforth, the paper is organized as follows. The identification of the tool holder–spindle taper contact stiffness is presented in Section 2 along with experimental procedures. The receptance coupling with an experimental proof is presented in Section 3. The paper is concluded in Section 4. 2. Modeling of tool holder–spindle taper connection Conical tapered connections are used as the interface between the spindle and the tool holder. The holder is pulled towards the spindle taper by a drawbar force mechanism. The friction between the holder and spindle taper interfaces determine the translational and rotational stiffness of the connection. 2.1. Modeling of contact stiffness The connection between the tapered surfaces is modeled by springs in the x and y directions that prevent the tool holder from rotating and translating inside the spindle taper. Two-degree-of-freedom Timoshenko beam elements are used to model the tool holder taper and the spindle. Each element has one translational (y) degree of freedom in the radial direction, and one RDOF (yz), and the elements are connected to each other through rotational and radial springs as shown in Fig. 1.
Fig. 1. Timoshenko beam element model of the tool holder–spindle interface with distributed contact springs.
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The translational springs are converted into rotational springs as follows. When an element of length dx along the taper is subjected to a rotation of dyz, the horizontal springs on both sides of the tool holder taper deform by an amount dxs. Since the rotational deflection of the element, dyz, is small, the distortion of the springs can be neglected and the deformation of the horizontal springs, dxs, can be expressed as Dx dyz , (1) 2 where Dx is the diameter of the element. The resisting axial forces acting on the element due to the deformation in the horizontal springs is dxs ¼
Dx dyz , (2) 2 where Kx is the stiffness of the horizontal contact spring. The resisting axial forces generate a moment about the center of the element, which prevents the element rotating: dF x ¼ K x dxs ¼ K x
Dx D2 ¼ K x x dyz 2 2 D2 ð3Þ ¼ K y dyz ; ! K y ¼ K x x , 2 where Ky is the equivalent stiffness of the rotational spring. The equivalent radial stiffness is Kr. An element of length dx and surface area dA at axial location Lx along the tool holder taper is subjected to a radial contact force Fc. The radial deformations, y, are linearly related to the contact force as follows:
dM ¼ 2 dF x
F c ¼ K r y.
Integrating over the whole contact length, the total potential energy stored in the contact springs is expressed as Z L Z L 1 1 2 ky y dA; U y ¼ ky y2 dA, (9) Uy ¼ 2 0 0 2 where L is the contact length of the tool holder with the spindle taper. The total potential energy stored in the contact springs is added to the internal virtual work as the result of bending deformations of the elastic tool holder. The internal virtual work for the Timoshenko beam undergoing radial translation (y), and rotational deformation (y) can be expressed as follows: Z L 1 Ub ¼ EI x ðy0 Þ2 þ kGAðy0 yÞ dx, (10) 0 2 where E is the Young’s modulus of elasticity and I is the area moment of inertia of the cross-section. G is the modulus of traverse elasticity, k is the cross sectional factor and A is the cross sectional area of the tool holder taper. Finally, the total internal virtual work of the spindle–tool holder interface is the sum of the three terms obtained in (9) and (10): U tot ¼ U b þ U y þ U y .
(11)
The finite element formulation of the tool holder–spindle interface from the internal virtual work is formulated, and the details can be found in any Timoshenko beam formulation or in the thesis of Namazi [9].
(4)
Similarly, the rotational deformations (y) are linearly related to the contact bending moment, Mc, as follows: M c ¼ K y y,
(5)
where Kr and Ky are the stiffness of each spring at a distance x along the contact length. The computational model of the contact-interface considers the distributed radial and rotational springs on the tapered surface of the tool holder, as shown in Fig. 1. The stiffness per unit area of the distributed radial (kr) and rotational (ky) springs are assumed to be constant, K r ¼ kr dA ¼ kr p Dx dx, K y ¼ ky dA ¼ ky p Dx dx
dA ¼ pDx dx.
ð6Þ
The resulting taper contact force and bending moment become F c ¼ ðkr dAÞ y;
1335
M c ¼ ðky dAÞ y.
(7)
The potential energy stored in the contact springs at the tool holder spindle interface as a result of radial deformations (y) and rotational deformations (y), is dU y ¼ 12 F c y ¼ 12 kr y2 dA, dU y ¼ 12 M c y ¼ 12 ky y2 dA,
ð8Þ
2.2. Experimental procedure The radial and rotational stiffness per unit area of the distributed springs, kr and ky are identified experimentally through dynamic testing. Numerical identification of contact springs were also attempted by considering contact elements in commercial finite element software, but the accuracy was inevitably questionable for a practical use in production floors [9]. The experimental setup (Fig. 2) is designed to investigate the spindle–tool holder interface dynamics without the effects of the spindle bearings and other machine elements that are inevitable on a real machine tool. The large steel block (approximately 40 kg) with the spindle taper has a load cell on the back to measure the drawbar force. The block is placed on a supporting cushion to simulate free–free boundary conditions, and impact modal tests are performed to obtain the FRF at the tool tip [10]. Both the spindle block and tool holder are modeled using Timoshenko cylindrical beam elements by including shear and rotational deformations. The equivalent circular cross-section, which has the same area and second moment of area as the cross section of the rectangular block with
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Fig. 2. Experimental setup for dynamic analysis of spindle–tool holder interface. Fig. 3. Frequency-response function of the CAT 40 taper in experimental setup—10 kN drawbar force. Table 1 Contact stiffness per unit area for CAT 40 taper determined by experimental identification—95 mm overhang length from spindle face Drawbar force (kN)
Radial stiffness per unit area, kr (N/m3)
Rotational stiffness per unit area, ky (N/rad m)
4 6 8 10
0.75 1012 0.82 1012 0.95 1012 1.09 1012
5.25 108 5.87 108 6.25 108 6.79 108
dimensions (b, h), is calculated as bh
p d 2eq ; 4
4 bh3 p d eq . 12 64
(12)
Impact hammer tests are performed on a shrink-fit tool holder without any tool in the block. The drawbar force is adjusted through the load cell, and the FRF is obtained at the tool holder tip through impact modal tests. The FRF of the finite element model is simulated by adopting the same damping ratios obtained from modal tests. By using a nonlinear least-squares curve-fitting algorithm, the two contact stiffness per unit area of the contact springs, kr and ky , are determined to match the finite element simulations with experimentally measured FRF. The identification is carried out for CAT 40, CAT 50 and the HSK A63 taper interface at 2 kN drawbar force increments until the limit load recommended by the manufacturers. A sample results for CAT 40 taper are shown in the article as an example. A CAT 40 shrink-fit tool holder without any tool, and with 95 mm shank gauge length measured from the spindle face. The contact stiffness per unit area of the CAT40 taper are listed in Table 1, and the comparison between the fitted FRFs and the experimental results for the CAT 40 taper with 10 kN drawbar force is shown in Fig. 3.
Table 2 Comparison of the contact stiffness per unit area for HSK A63 and CAT 40 taper with 10 kN drawbar force Tool holder taper
Radial stiffness per unit area, kr (N/m3)
Rotational stiffness per unit area, ky (N/rad m)
HSK-A63 CAT 40
3.87 1012 1.09 1012
2.83 109 6.79 108
The equivalent of the CAT 40 taper is the HSK 63, because they have relatively equal gauge diameters, power and torque transmission capabilities. The identified stiffness per unit area of the two tapers for 10 kN drawbar force is compared in Table 2. The radial and rotational contact stiffness per unit area of the HSK taper are about 4 times the CAT 40 taper. This is mainly due to the dual-face contact of the HSK taper design. However, the surface contact area of the CAT 40 taper is about 2 times that of the HSK 63A taper, thus making up for one half of the loss in stiffness due to the face contact. In order to investigate the effect of tool length on the contact stiffness of the holder, a short and a long, carbide blanks with 16 mm diameter are attached to the shrink CAT 50 tool holder. The drawbar force applied on the tool holder was 20 kN, and the tool holder–tool connection is assumed to be rigid. The comparison of experimental and predicted FRFs of the assembly at the free end of the blank tool using the identified holder–spindle taper contact stiffness are shown in Fig. 4 for tools with long (64 mm) and short (15 mm) overhangs. While the predicted FRF for short overhang agrees well with the measurements (Fig. 4b), the influence of long overhangs is clearly visible in Fig. 4a. Increased bending loads change the contact stiffness between the tool holder and spindle taper, which need to be included in the estimated contact stiffness.
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Fig. 5. Finite element model of spindle system on machine.
Table 3 Radial and rotational spring stiffness of CAT 40 tool holder–spindle Spring no.
Radial stiffness, Kr (N/m)
Rotational stiffness, Ky (N m/rad)
1 2 3 4 5
4.70 109 8.99 109 6.40 109 5.80 109 3.60 109
6.3 105 1.20 106 8.60 105 7.70 105 4.80 109
springs is calculated in discrete form of Eq. (6), K r ðiÞ ¼ kr p DðiÞ LðiÞ, K y ðiÞ ¼ ky p DðiÞ LðiÞ,
Fig. 4. Comparison of frequency-response functions of the CAT 50 shrink-fit holder with a long (64 mm) and short (15 mm) tool overhangs under 20 kN drawbar force: (a) 64 mm tool overhang and (b) with 15 mm tool overhang.
2.3. Prediction of tool FRFs on the machine tool spindle The contact stiffness constants for the CAT 40 taper are used to couple the tool holder to a finite element model of a spindle system developed by Cao [11] on the machine. The finite element model of the spindle on the machine tool is created using the Spindle Pros [12] finite element program, as shown in Fig. 5. First, the tool holder–spindle connection is assumed to be rigid, and in the second case the connection is assumed to be modeled by the distributed radial and rotational springs along the taper. The identified radial and rotational stiffness per unit area of the connection springs are kr ¼ 1.09 1012 (N/m3) and ky ¼ 6:79 108 (N/rad m), respectively as shown in Table 1 for 10 kN drawbar force. The tool holder taper is modeled by 4 beam elements. Five contact springs are considered between the tool holder and the spindle taper. The stiffness of the connection
ð13Þ
where Kr(i) and Ky(i) are the radial and rotational stiffness of the connection springs at node i, D(i) is the diameter of the taper element and L(i) is the tapered element length. The stiffness values for the 5 springs at a drawbar force of 10 kN are listed in Table 3. The large contact spring constants correspond to the sections of the tool holder with larger diameters. The FRF of the finite element model of the spindle–tool holder assembly is simulated at the tool tip for the following two cases: rigid tool holder–spindle connection and distributed-springs connection. The simulation results are compared with the experimental results as shown in Fig. 6. The rigid-connection assumption yields results close to those of the spring-connection, and both simulations show an acceptable match with the experiments. However, the spring-connection model is able to simulate the mode at around 2900 Hz more accurately because this mode is dominated by the tool–tool holder assembly. The static stiffness of the assembly is also accurately simulated using the spring connection. It is obtained by calculating the magnitude of the FRF at zero frequency. By comparing the FRFs shown in Fig. 6, the experimental FRF and spring connection FRF match well at the lowerfrequency range; however, the rigid connection is statically stiffer by 15%.
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Fig. 6. Frequency-response function at the tool tip.
3. Structural dynamics of the holder–spindle interface The identification of static contact stiffness between the holder and spindle is important to compare various holder shapes and contact mechanisms. However, FRF of the holder–spindle interface determines the chatter stability of the machining process, and need to be identified. Although the most accurate method is to measure the FRF at the tool tip, it is a costly process since FRF needs to be measured at each tool change. A receptance coupling technique proposed by Park and Altintas [8] is extended here to identify the rotational and translational dynamics of the spindle up to the tool holder flange, which is standard and remains unchanged as shown in Fig. 7. 3.1. Mathematical model
Fig. 7. Receptance coupling for obtaining frequency-response function at the tool tip.
point 2 are: F 2 ¼ F A;2 þ F B;2 , X 2 ¼ X A;2 þ X B;2 .
ð16Þ
By letting H 2 ¼ H A;22 þ H B;22
(17)
and substituting (17) into (15) The machine tool assembly (structure AB) is divided into two substructures. Substructure A represents the tool holder, and substructure B represents the remaining machine tool assembly up to the tool holder flange. The two structures are rigidly connected at point 2. The FRFs of substructure A at the two free ends is ( ) " # ( ) X1 H A;11 H A;12 F1 ¼ , (14) X A;2 H A;21 H A;22 F A;2 where X1 and XA,2 are displacement vectors with both translational and angular displacement components. F1 and FA,2 are force vectors containing both force and moments applied at points 1 and 2. HA,ij are FRFs between points i and j. Similarly, the FRFs of substructure B at point 2 is fX B;2 g ¼ H B;22 fF B;2 g. (15) After rigidly coupling the two structures A and B at point 2, the equilibrium and compatibility conditions at
X 2 ¼ H B;22 F B;2 ¼ H A;21 F 1 þ H A;22 ðF 2 F B;2 Þ.
(18)
By rearranging (18), the forces on structure B are: 1 H A;21 F 1 þ H A;22 F 2 F B;2 ¼ H B;22 þ H A;22 ¼ ðH 2 Þ1 H A;21 F 1 þ H A;22 F 2 .
ð19Þ
Finally the displacements at points 1 and 2 are expressed as functions of FRFs and the applied forces as follows: X 1 ¼ H A;11 F 1 þ H A;12 ðF 2 F B;2 Þ ¼ H A;11 F 1 þ H A;12 F 2 H A;12 ðH 2 Þ1 ðH A;12 F 1 þ H A;12 F 2 Þ ¼ ðH A;11 H A;12 ðH 2 Þ1 H A;21 Þ F 1 þ ðH A;12 H A;12 ðH 2 Þ1 H A;22 Þ F 2 , X 2 ¼ H A;21 F 1 þ H A;22 ðF 2 F B;2 Þ ¼ H A;21 F 1 þ H A;22 F 2
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"
H A;22 ðH 2 Þ1 ðH A;21 F 1 þ H A;22 F 2 Þ
¼ ðH A;21 H A;22 ðH 2 Þ1 H A;21 Þ F 1 þ ðH A;22 H A;22 ðH 2 Þ1 H A;22 Þ F 2 .
"
ð20Þ
Eqs. (20) are arranged in matrix form as follows: 2 H A;11 H A;12 ðH 2 Þ1 H A;21 X1 ¼ 4 X1 H A;21 H A;22 ðH 2 Þ1 H A;21 3 H A;21 H A;12 ðH 2 Þ1 H A;22 F1 , 5 F2 H A;22 H A;22 ðH 2 Þ1 H A;22
ð21Þ
hA12;fM
hA12;Mf
hA12;MM # "
h21;ff
X1 F1 X2 F1 X1 F2 X2 F2
¼ H A;21 H A;22 ðH 2 Þ1 H A;21 ¼ H 12 ,
¼ H A;21 H A;12 ðH 2 Þ1 H A;22 ¼ H 21 , ¼ H A;22 H A;22 ðH 2 Þ1 H A;22 ¼ H 22 .
ð22Þ
Each FRF contains both translational and rotational displacements, thus Eqs. (22) can be expanded as ( ) ( ) " # ( ) h11;fM h11;ff x1 f1 f1 ¼ ½H 11 ¼ , h11;Mf h11;MM y1 M1 M1 ( ) ( ) " # ( ) h12;fM h12;ff x2 f1 f1 ¼ ½H 12 ¼ , h12;Mf h12;MM y2 M1 M1 ( ) ( ) " # ( ) h21;fM h21;ff x1 f2 f2 ¼ ½H 21 ¼ , h21;Mf h21;MM y1 M2 M2 ( ) ( ) " # ( ) h22;fM h22;ff f2 x2 f2 ¼ ½H 22 ¼ . h22;Mf h22;MM y2 M2 M2 ð23Þ By substituting Eqs. (23), into Eqs. (22), direct and crosstransfer functions at points 1 and 2 with both rotational and translational degrees of freedom are found as " ½H 11 ¼
h11;ff
h11;fM
h11;Mf
h11;MM
#
" ¼
hA11;ff
hA11;fM
hA11;Mf
hA11;MM
½H 2
h21;fM
hA21;ff
#
hA21;ff
hA21;fM
hA21;Mf
hA21;MM #
hA21;fM
,
h21;Mf h21;MM hA21;Mf hA21;MM " # " # hA12;ff hA12;fM hA22;ff hA22;fM 1 ½H 2 , hA12;Mf hA12;MM hA22;Mf hA22;MM " # " # h22;fM hA22;fM h22;ff hA22;ff ½H 22 ¼ ¼ h22;Mf h22;MM hA22;Mf hA22;MM "
hA22;ff
hA22;fM
hA22;Mf
hA22;MM
#
" 1
½H 2
hA22;ff
hA22;fM
hA22;Mf
hA22;MM
# , ð24Þ
where ("
½H 2
1
¼ " ¼
hA22;ff
hA22;fM
hA22;Mf
hA22;MM #1
h22;ff
h2;fM
h2;Mf
h2;MM
#
" þ
.
hB22;ff
hB22;fM
hB22;Mf
hB22;MM
#)1
ð25Þ
The first elements in the three matrices [H11], [H12] and [H22] in Eq. (24) along with the equation for the reciprocity condition yields 4 sets of nonlinear equations:
¼ H A;11 H A;12 ðH 2 Þ1 H A;21 ¼ H 11 ,
" 1
¼
where H 2 ¼ H A;22 þ H B;22 . Eq. (21) represents the receptance coupling of the spindle with holder up to the flange and holder–tool stick out structures. The receptances of the free–free tool holder–tool assembly, HA,11, HA,12 and HA,22, are modeled using the finite element model and the receptance of the spindle at point 2, HB,22, is obtained through the inverse receptance coupling method as explained in the following. The following cross and direct receptances are obtained from Eqs. (20) and (21):
#
hA12;ff
½H 21 ¼
1339
#
1 h11;ff ¼ hA11;ff ðh2;ff h2;MM h2;fM h2;Mf Þ
ðhA12;ff h2;MM hA12;fM h2;Mf Þ hA21;ff þ þ ðhA12;fM h2;ff hA12;ff h2;fM Þ hA21;Mf , 1 h12;ff ¼ hA12;ff ðh2;ff h2;MM h2;fM h2;Mf Þ
ðhA12;ff h2;MM hA12;fM h2;Mf Þ hA21;ff þ þ ðhA12;fM h2;ff hA12;ff h2;fM Þ hA22;Mf , 1 h22;ff ¼ hA22;ff ðh2;ff h2;MM h2;fM h2;Mf Þ
ðhA22;ff h2;MM hA22;fM h2;Mf Þ hA22;ff þ þ ðhA22;fM h2;ff hA22;ff h2;fM Þ hA22;Mf , h2;fM ¼ h2;Mf . ð26Þ The four unknowns are, h2,ff, h2,fM, h2,Mf and h2,MM which are the receptances of the assembly at point 2 and need to be solved. The terms h11,ff , h12,ff ,and h22,ff are obtained by 3 impact hammer tests at points 1 and 2. The FRF’s of the free–free substructure A, are obtained through the finite element method. This system of nonlinear equations is symbolically solved in MAPLEs [13]. The translational and RDOF FRFs at point B can be obtained as: hB22;ff ¼ h2;ff hA22;ff ; hB22;fM ¼ hB22;Mf ¼ h2;fM hA22;fM ; hB22;MM ¼ h2;MM hA22;MM .
ð27Þ
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The spindle dynamics, which include the RDOF, are stored in a matrix as shown below " # hB22;ff hB22;fM H B;22 ¼ . (28) hB22;Mf hB22;MM
3.2. Experimental procedure A shrink fit HSK 63 tool holder without a tool is attached to the spindle as shown in Fig. 8. Three impact modal tests are performed at points 1 and 2: the direct FRF at point 1; h11,ff cross FRF at points 1 and 2, h12,ff and direct FRF measurement at point 2, h22,ff as shown in Fig. 8. Structure A, the stick out of the tool holder is modeled by finite element method using Timoshenko beams. The tool holder and spindle with HSK 63A taper assembly is coupled by employing Eqs. (17) and (22) as follows: H 11 ¼ ðH A;11 H A;12 ðH B;22 þ H A;22 Þ1 H A;21 Þ.
The proposed receptance coupling method is experimentally evaluated on a horizontal machining center. The Mori Seiki SH403 horizontal machining center has an HSK 63A spindle taper. The spindle with HSK 63 interface was identified using a short shrink fit holder with a gauge length of 60 mm first. The FRF of another tool holder with 140 mm gage length (Fig. 9) was estimated using the proposed receptance coupling technique. The tool–tool holder connection in the shrink-fit is modeled as a rigid connection and the fluted tool is considered to be 80% of the total shank diameter. The predicted FRF at the tool tip are compared with the experiments in both x and y directions as shown in Fig. 10, which has acceptable accuracy for use in chatter stability
(29)
The fluted section of the end mill is considered to be 80% of the total diameter in the finite element model and the tool–tool holder connection in the shrink-fit is assumed to be rigid. The tool–tool holder model is in the free–free condition as the rigid body modes play an important role in the coupling between the structures. The damping ratio used for the finite element model is assumed to be 1–3%, which was verified by several impact tests.
Fig. 9. Tool–tool holder assembly—tooling A.
Fig. 8. Inverse receptance coupling for obtaining spindle dynamics.
ARTICLE IN PRESS M. Namazi et al. / International Journal of Machine Tools & Manufacture 47 (2007) 1333–1341
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The overall structural dynamics of the spindle must be identified at the tool tip in order to predict chatter stability lobes. The paper presents a method, which allows identification of the holder and spindle assembly at the spindle face, and allows receptance coupling of free-free holder with a cutter. The proposed method allows prediction of FRFs at the tool tip without having to measure the FRFs at each tool–holder change.
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Acknowledgments -0.5 -1 -1.5 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz) 0.5
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This research is sponsored by Mitsubishi Materials, Natural Sciences and Engineering Research Council of Canada and Pratt & Whitney Canada under Industrial Virtual High Performance Machining Research Chair Grant. Mori Seiki HS403 is loaned by Machine Tool Technology Research Foundation (MTTRF). References
0
Im (Φ (ω)) (m/N)
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Fig. 10. Measured and predicted frequency-response functions in X direction at the tool tip on the Moriseiki SH403.
prediction methods [14,15]. The predicted FRFs are also able to model the higher frequencies more accurately, which are difficult to measure accurately via impulse modal tests due to loss of spectral strength of the hammers at high-frequency range dominated by the flexible tool modes. 4. Conclusion Tool holder–spindle interface stiffness is one of the main obstacles in achieving chatter free, high-speed machining. The holder–spindle interface stiffness depends on the drawbar force as well as total contact, which is modeled by a uniformly distributed translational and rotational springs in this paper. The influence of loosing some of the contact areas due to wear can be simulated by removing the corresponding spring elements. Furthermore, different holder design scenarios and shapes can be investigated by applying the proposed contact model.
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