See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/233926065 Modeling and experimentation for threedimensional dynamics of endmills Article in International Journal of Machine Tools and Manufacture · February 2012 DOI: 10.1016/j.ijmachtools.2011.09.005 CITATIONS READS 9 75 4 authors: Bekir Bediz Uttara Kumar 18 PUBLICATIONS 57 CITATIONS 4 PUBLICATIONS 38 CITATIONS Sabanci University SEE PROFILE GE Global Research SEE PROFILE Tony L. Schmitz O. Burak Ozdoganlar 231 PUBLICATIONS 2,344 CITATIONS 118 PUBLICATIONS 1,149 CITATIONS University of North Carolina at Charlotte SEE PROFILE Carnegie Mellon University SEE PROFILE All content following this page was uploaded by Bekir Bediz on 02 February 2015. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. International Journal of Machine Tools & Manufacture 53 (2012) 39–50 Contents lists available at SciVerse ScienceDirect International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool Modeling and experimentation for three-dimensional dynamics of endmills Bekir Bediz a, Uttara Kumar b, Tony L. Schmitz c, O. Burak Ozdoganlar a,n a b c Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, Charlotte, NC 28223, USA a r t i c l e i n f o abstract Article history: Received 11 May 2011 Received in revised form 12 September 2011 Accepted 13 September 2011 Available online 22 September 2011 This paper presents a model for the three-dimensional (3D) dynamic response of endmills while considering the actual fluted cross-sectional geometry and pretwisted shape of the tools. The model is solved using the spectral-Tchebychev (ST) technique. The bending and the coupled torsional-axial behavior of four different fluted endmills is compared to finite element model (FEM) predictions and experimental results obtained using modal testing under free-free boundary conditions. For the first eight modes, including six bending and two torsional/axial modes, the difference between the 3D-ST and experimental natural frequencies is shown to be 3% or less for all four tools tested during this study. For the same modes, the 3D-ST and FEM predictions agree to better than 1%. To demonstrate its application, the 3D-ST model for the fluted section of a commercial endmill is coupled to the spindle– holder to predict the tool-point dynamics using receptance coupling substructure analysis (RCSA) with a flexible connection. The coupled model is validated through experiments. & 2011 Elsevier Ltd. All rights reserved. Keywords: Endmill dynamics Three-dimensional spectral Tchebychev Modal analysis Receptance coupling substructure analysis 1. Introduction The dynamic behavior of the tool–holder–spindle–machine assembly, as reflected at cutting points along a milling tool, often determines the achievable process efficiency and quality that can be obtained during milling. The combination of the cutting mechanics, which describe the machining forces for the selected machining conditions, tool geometry, and relative tool-workpiece motion; and the structural dynamics, which dictate the relative tool-workpiece motion at the cutting point in response to the machining forces, determine the dynamic behavior of machining operations [1–3]. The structural dynamics include the dynamic behavior of the entire structural assembly, including the tool, holder, spindle, and machine. Unless cutting conditions and tool geometry are selected carefully, the mechanics–dynamics interaction can cause unstable cutting conditions (chatter) due to the over-cutting of the surface left by one tooth of the vibrating cutter by subsequent teeth. Even under stable cutting conditions, forced vibrations of the flexible tool can lead to errors in the location of the machined surface (in milling). If the dynamic response is well-known, however, models can be applied to guide selection of cutting conditions that increase stability and reduce surface location errors. In order to accurately predict the tool–holder–spindle–machine (THSM) assembly dynamics, two fundamental capabilities are n Corresponding author. Tel.: þ1 412 268 9890; fax: þ1 412 268 3348. E-mail address: ozdoganlar@cmu.edu (O. Burak Ozdoganlar). 0890-6955/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2011.09.005 needed: (1) accurate and numerically efficient dynamic models of realistic, arbitrary tool and holder geometries that capture nonsymmetric bending and coupled torsional/axial dynamics; and (2) a means to join the experimentally determined spindle–machine dynamics to the model-based tool–holder dynamics to obtain the THSM assembly dynamics at any point along the tool [4–13]. This paper describes new results for modeling three-dimensional dynamic behavior of macro-scale milling tools using the spectral-Tchebychev1 (ST) technique while considering the actual fluted geometry. The 3D-ST model results for both bending and torsion are compared to finite element computations and experimental data obtained using impact testing with free–free boundary conditions. The 3D-ST model for the actual geometry of the fluted portion of the tool is then combined with the holder model and the measured spindle dynamics to predict the THSM assembly dynamics through receptance coupling substructure analysis (RCSA). 2. Modeling and spectral-Tchebychev solution of macroendmill dynamics This section briefly describes the derivation of the endmill dynamics model and the associated 3D-ST technique used to solve the model numerically. The approach outlined here follows 1 The name of the Russian mathematician P.L. Tchebychev can be alternatively transliterated as Chebychev or Chebyschev. 40 B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 that presented (and experimentally validated) by Filiz and Ozdoganlar [14,15]. A traditional macro-scale endmill can be represented by the diameter of the shank section (commonly referred to as the tool diameter, ds), helix angle C, the shank length Ls, the flute length Lf, number of flutes, and the cross-sectional geometry of the fluted region. Due to its simple circular cross-section, the axial, torsional, and bending deflections of the shank region are uncoupled. Thus, an appropriate one-dimensional (1D) beam models (e.g., the Timoshenko beam model) is sufficient to describe the dynamics of the shank. However, the fluted section with its pre-twisted geometry and complex cross-section causes the axial and torsional deflections to be coupled. Furthermore, unlike those of the shank portion, bending deflections of the fluted portion are not symmetric. As a result, accurately capturing the dynamic behavior of the fluted section necessitates use of 3D modeling techniques. Therefore, to simultaneously achieve numerical efficiency and modeling accuracy, a 1D model for the shank and a 3D model for the fluted section are used. The extended Hamilton’s principle is applied to obtain the integral boundary value problem for the each of the shank and fluted sections. The complete model is then obtained by combining the shank (1D) and fluted (3D) portions through component mode synthesis. given as q ni ¼ Q n q i and Z L ^ AðzÞqðzÞqðzÞ dz ¼ q T V A q^ , ð3Þ 0 where Qn is the nth derivative matrix, VA is the inner product matrix and the underline indicates the sampled functions based on Gauss–Lobatto sampling. The deflections associated with each of the six degrees-of-freedom for each of the sample points can be described in terms of q, e.g. u ¼ ½I O O O O Oq ¼ Iu q, ð4Þ cx ¼ ½O O O I O Oq ¼ Icx q: ð5Þ Substituting Eqs. (2)–(4) in Eq. (1) and using the weighted residual method, the mass, stiffness and forcing matrices can be obtained as M ¼ P T rðITu V A Iu þ ITcy V Ix Icy þ ITv V A Iv þ ITcx V Iy Icx V A þ ITw Iw þ ITcz V Ip Icz ÞP, ð6Þ T T K ¼ P T ½EQcy V Ix Qcy þ ks GðQu Icy ÞT V A ðQu Icy Þ þ EQcx V Iy Qcx T T þ ks GðQv þ Icx ÞT V A ðQv þ Icx Þ þEQw V A Qw þ GQcz V Js Qcz P, ð7Þ F ¼ PT VF q P T rðITu V A Iu þ ITcy V Ix Icy þ ITv V A Iv þ ITcx V Iy Icx 2.1. Solution of the 1D-ST for the dynamics of the shank T To insure the accuracy of the dynamic model, particularly for higher modes, the Timoshenko beam model, which includes the shear and rotary inertia effects, is used to model the 1D dynamics of the shank section [16,17]. The model includes bending, axial, and torsional motions of the shank section. Following the extended Hamilton’s principle, the integral boundary value problem for the 1D model can be expressed as Z L" @2 c @2 u @2 v @2 c rA 2 u^ þ rIx 2y c^ y þ rA 2 v^ þ rIy 2x c^ x @t @t @t @t 0 @2 w @2 cz ^ ^ þ rIp cz w 2 @t 2 @t    ^ ^ @cy @c @u @u^ ^ @cx @c y x cy c y þ EIy þ ks GA þ EIx @z @z @z @z @z @z    ^ @v @v^ @w @w ^ þ cx þc þ ks GA x þEA @z @z @z @z # Z L ^ @cz @c z ^ dz, þ GJs fF q gT fqg dz ¼ @z @z 0 þ rA ð1Þ ^ f ^ gT are the functions representing ^ f ^ T ¼ fu^ v^ w ^ f where fqg x y z the variation terms, L is the length of the beam, r is the density, A(z) is the cross-sectional area along the beam axis, E is Young’s modulus, G is the shear modulus, Ip(z) is the polar moment of inertia, Js(z) is the torsion constant, Ix(z) and Iy(z) are the second area moments, ks is the shear constant, and x and y are the angular deflections of the beam due to shear deformations. The solution of Eq. (1) is obtained using the 1D spectralTchebychev technique described by Yagci et al. [18]. The generalized coordinates are sampled in the spatial domain, and expressed as a truncated series expansion of orthogonal Tchebychev polynomials as qi F Nz X aqi ðtÞT nz 1 ðzÞ, ð2Þ nz ¼ 1 where qi is the generalized coordinate, T nz 1 are the scaled Tchebychev polynomials, Nz is the number of polynomials used in expanding the generalized coordinate, and aqi are the timedependent expansion coefficients. The derivative and inner product operations are defined in the sampled domain and can be þ ITw V A Iw þ ITcz V Ip Icz ÞRq€b PT ½EQcy V Ix Qcy T þ ks GðQu Icy ÞT V A ðQu Icy Þ þ EQcx V Iy Qcx T T þ ks GðQv þ Icx ÞT V A ðQv þ Icx Þ þEQw V A Qw þ GQcz V Js Qcz Rqb : ð8Þ 2.2. Solution of the 3D-ST for the dynamics of the fluted section The fluted section and the transition region (from the flutes to the shank sections) are modeled using 3D dynamics. For a 3D structure, axial, torsional, and bending motions are coupled. To obtain the boundary-value problem (BVP) for the 3D deflections of the fluted section, 3D linear elasticity theory is used to obtain the strain (elastic) energy. Using the extended Hamilton’s principle and applying integration by parts, the integral BVP for the fluted section can be obtained as Z t2 Z T € T fqgfqg ^ ^ þ fF q gT fqg ^ dx dy dz dt ¼ 0, ½r fqg ½BT ½BC fqg t1 Volume ð9Þ where fF q g ¼ fF x F y F z gT . In arriving at the BVP given in Eq. (9), the stress–strain (constitutive) equations were fsg ¼ fCgfeg, where C is the constitutive matrix, and the strain-displacement relations were feg ¼ fBgfqg, where B is the differential operator matrix. In order to solve the 3D fluted section problem defined by Eq. (9) using the 3D-ST method, coordinate transformations and a mapping of the complex cross-sectional flute geometry into a rectangular domain are required. The pretwist effect can be accommodated by using a local coordinate frame ðx, y, zÞ, and the complex crosssection defined by the ðx, y, zÞ domain is mapped onto a rectangular cross-section domain defined by the coordinates ðx, Z, zÞ. Combining both operations, the relation between the final coordinates ðx, Z, zÞ and the physical domain ðx,y,zÞ can be written as @ @ ¼ Jxi jx @x i @xj and @ @ ¼ J ijx x , @xi @x i i,j ¼ 1; 2,3, . . . , ð10Þ xx where J xx ij and J ij are the elements of the Jacobian matrices of the cross-section transformation and mapping, respectively. The integral BVP given in Eq. (9) is then solved using the 3D-ST technique. First, each deflection term is expressed using a three- 41 B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 dimensional Tchebychev (truncated) series expansion qi ðx, Z, z,tÞF Nx X Nz NZ X X aqi ðtÞT l1 ðxÞT m1 ðZÞT n1 ðzÞ, ð11Þ l¼1m¼1n¼1 where the T terms represent the orthogonal (scaled) Tchebychev polynomials, aqi are the polynomial coefficients, and Nj is the number of polynomials associated with the coordinate j ¼ ðx, Z, zÞ. Similar to the 1D approach, the derivative and inner product operations are defined in the sampled domain as Z LZ ð12Þ q ni,x ¼ Q xni Ii q and q q^ dx dy dz ¼ q T V xyz q^ , i 0 Area where q ni,x is the nth spatial derivative and Q xni is the nth extended i derivative matrix with respect to coordinate xi, and Vxyz is the inner product matrix. Similarly as in the 1D solution, the mass, stiffness, and force matrices can be found from Eq. (9) as M ¼ rðITu V xyz Iu þ ITv V xyz Iv þ ITw V xyz Iw Þ, ð13Þ K ¼ B T Vxyz B C , ð14Þ F ¼ V xyz F q : ð15Þ 3a 1 2b 2a 3b 4a 4b Measured Timoshenko beam model 3D-ST model Fig. 2. RCSA components with coordinates. 3b 1 3a 2.3. Combining 1D and 3D solutions Fig. 3. Component coordinates for flexible coupling. The 1D formulation for the shank section provides six deflection components (three displacements and three angles) for a given axial position. The 3D formulation for the fluted section, on the other hand, provides three deflection components for each sampling point at given three-dimensional position within the solid. To combine the 1D-ST solution for the shank with the 3D-ST solution for the fluted (and transition) regions, and thus, to form a global boundary value problem, compatibility conditions are used at the boundaries shared by the two sections. The compatibility equations relate the three displacement degrees of freedom ðu, v, wÞ from the 3D model and the six degrees of freedom ðu, v, w, cx , cy , cz Þ for the 1D model for each point ðx,yÞ within the (boundary) cross-section. Assuming small angles, this compatibility condition can be expressed mathematically as fu, v, wg ¼ fuycz , v þ xcz , w þxcy þ ycx g: ð16Þ the archived spindle–machine dynamics and this result was subsequently coupled (using a flexible connection) to the free– free response of the tool. The component coordinates are identified in Figs. 2 and 3. The receptance matrices for the assembly, Gij, and substructures, Rij, can be represented as shown in Eq. (17), where o is the frequency, Xi and xi are the assembly and substructure displacements at the coordinate location i, Yi and yi are the assembly and substructure rotations, Fj and fj are the assembly and substructure forces applied at the coordinate location j, and Mj and mj are the assembly and substructure and moments: " # 2 Xi Xi 3 " # 2 xi xi 3 mj fj hij lij Hij Lij Fj Mj 4 5 and Rij ðoÞ ¼ Gij ðoÞ ¼ ¼ Yi Yi ¼ 4 yi yi 5: nij pij N ij P ij m Fj Mj fj j ð17Þ 3. Receptance coupling substructure analysis To demonstrate the application of the 3D-ST model for obtaining the tool-point dynamics, the RCSA technique is used. Fig. 1 shows the schematic of the THSM assembly used for experimental validation. The free–free response of the fluted portion of the tool was obtained analytically by the 3D-ST technique. Using the three-component RCSA technique for tool point dynamics prediction [7], the modeled holder was coupled to Spindle-Machine Flange Holder Tool Shank Flutes Using RCSA, the assembly dynamics at coordinate 1 is obtained in two steps. First, the modeled free–free receptances of the holder-shank component are coupled to the free–free receptances of the remainder of the tool (fluted part) using the following: G11 o ¼ R11 oR12a ðoÞ½R2b2b ðoÞ þ R2a2a ðoÞ1 R2a1 ðoÞ, ð18Þ G3a1 ðoÞ ¼ R3a2b ðoÞ½R2b2b ðoÞ þR2a2a ðoÞ1 R2a1 ðoÞ, ð19Þ G3a3a ðoÞ ¼ R3a3a ðoÞR3a2b ðoÞ½R2b2b ðoÞ þ R2a2a ðoÞ1 R2b3a ðoÞ, ð20Þ G13a ðoÞ ¼ R12a ðoÞ½R2b2b ðoÞ þ R2a2a ðoÞ1 R2b3a ðoÞ: ð21Þ This coupling result represents the tool model with end coordinates 1 and 3a ðG11 ,G13a ,G3a3a ,G3a1 Þ. Second, the modeled free–free receptances of the holder with portion of the shank inside it is coupled to the spindle machine receptances: G3b3b ðoÞ ¼ R3b3b ðoÞR3b4a ðoÞ½R4b4b ðoÞ þ R4a4a ðoÞ1 R4a3b ðoÞ: 2 3 4 Fig. 1. THSM assembly and coordinates. 1 ð22Þ The free–free receptances are then considered as a component, using the receptance matrices R11, R13a, R3a3a and R3a1 . G3b3b is considered as R3b3b . Third, the holder–spindle–machine component is then flexibly coupled to the tool using translational and rotational spring constants assembled in the stiffness matrix k. 42 B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 sectioned by electrical discharge machining (EDM), and the crosssection was imaged at several locations, as shown in Fig. 4(b). The s Matlab image-processing toolbox was then used to identify the boundaries (periphery), and sample points along the periphery were extracted to define the cross-section. The RCSA equation for the flexible coupling tool point FRF [9] is provided as   1 1 G11 ðoÞ ¼ R11 ðoÞR13a ðoÞ R3b3b ðoÞ þ R3a3a ðoÞ þ R3a1 ðoÞ: ð23Þ k Identification of the stiffness matrix k is discussed in Section 7. 4.2. Experimental setups and conditions 4. Experimental methods To determine the natural frequencies of the aluminum endmills, the frequency response functions (FRFs) of the endmills were obtained by impact testing. Fig. 5 shows the experimental procedure used to obtain the bending and torsional natural frequencies of the endmill. To approximate the unconstrained boundary conditions, the endmills were placed on soft foam. The very low stiffness of the foam base relative to the endmills provided a reasonable approximation to unconstrained (free– free) boundary conditions. The FRFs were measured by exciting them using a miniature impact hammer (PCB 0841A17, sensitivity 46.95 N/V) and recording the corresponding vibration using a low mass accelerometer (PCB 352C23, sensitivity 1727 (m/V s2). To assess the effect of the miniature accelerometer in the measured dynamic response, a set of finite elements simulations were conducted. It was seen that the added mass due to accelerometer does not change the natural frequencies by more than 0.05% for the first eight natural frequencies of the tool. Therefore, the effect of the miniature accelerometer is deemed negligible. Depending on the locations of the accelerometer and hammer impact, direct and cross FRFs, Hij, were obtained, where i and j represents the measurement and force locations, respectively. For instance, in the configuration shown in Fig. 5(a), the accelerometer was placed at the shank end, and the force was applied at the same location (from the other side of the endmill). Therefore, the direct FRF at the endmill’s shank end was obtained. Fig. 5(b) shows the experimental procedure used to identify the torsional natural frequencies. The endmill was excited by applying the force to one flute in the tangential direction and the response was recorded in the same direction using an accelerometer placed on the opposite flute. In these experiments both the bending and the torsional natural frequencies were excited. The results for the first eight natural frequencies were recorded. The resonant frequencies of the solid carbide endmill span a considerably wider range than those of the aluminum endmills. Therefore, the hammer impact test described above cannot be effectively used to determine the first eight natural frequencies of endmill 4. Fig. 6(a) shows the experimental setup used for obtaining the natural frequencies of the solid carbide endmills. The endmill was suspended using flexible elastic bands from two sections along its length to approximate unconstrained boundary conditions. The dynamic excitations within a frequency range of 0–40 kHz were provided using a 3 mm by 4 mm piezoelectric element with a This section describes the experimental setup and procedures used for validating the presented 3D-ST solution. For this purpose, a set of modal tests were conducted on the four different endmills described in Table 1. These four-flute endmills each had different geometries (shank diameter, shank length, flute length, and helix angle). Endmills 1–3 were machined from aluminum blanks so that the cross-sectional geometry was explicitly known; the geometry for the commercial endmill (identified as endmill 4) was determined by sectioning and measurement. 4.1. Description of endmill geometries The accuracy, and thus effective application, of the 3D-ST solution for macro-endmills requires accurate knowledge of the geometry and material parameters of the endmills. In particular, cross-sectional geometry, twist rate, and the geometry of the transition region (from the shank to the fluted portion) must be well known. To facilitate evaluation of the 3D-ST model for wellknown geometries and for a range of geometric parameters, a set of aluminum endmills (test endmills) were fabricated using fiveaxis machining. Since the aluminum endmills were described s using a 3D solid model (SolidWorks ), their geometry was accurately defined within the accuracy of the manufacturing technique. The cross-sectional geometry of the endmills are given in Fig. 4(a). To enable further assessment of the 3D-ST solution, a commercial (solid) carbide endmill (endmill 4) was selected for testing. However, the detailed information about the cross-sectional geometry was not available. Therefore, the endmill was Table 1 Properties of the endmills used for validation experiments (all parameters except the helix angle were measured using digital calipers with a resolution of 0.01 mm). Properties Endmill 1 Endmill 2 Endmill 3 Endmill 4 Shank diameter (mm) Shank length (mm) Flute length (mm) Helix angle (mm) Material 38.20 101.35 201.14 30 Aluminum 38.17 102.98 175.23 30 Aluminum 38.20 77.66 200.44 30 Aluminum 12.66 66.40 84.41 30 Carbide 2 4 3 1 1 2 3 6 5 4 5 7 6 Fig. 4. Cross-section of (a) endmills 1–3 (coordinates are provided in mm), (b) endmill 4. 7 43 B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 Foam Base Miniature Impact Hammer Endmill Miniature Impact Hammer Endmill Accelerometer Foam Base Accelerometer Fig. 5. Impact hammer experimentation setup (a) direct FRF measurement at the endmill shank end and (b) torsional measurement. Microscope Head Piezo Tool Tip View Metal Stripes Laser Source Microscope Elastic Bands Tool Piezo Metal Stripes Laser Spot Piezo Cables Base Mount Fig. 6. Experimental setup for endmill 4 free–free boundary condition measurements. 125 mm thickness. As shown in Fig. 6(b), the piezoelectric element was glued onto the macro endmill shank. When a voltage is applied, the piezoelectric element expands and contracts along its polarization axis. A pseudo-random excitation having a range of 0–40 kHz with 140 V amplitude was supplied to the piezoelectric element. For each experiment, 50 measurements of 0.320 s each were completed and averaged to eliminate unbiased noise. The response is measured using a laser Doppler vibrometer s (LDV) system (Polytec MSA-400) with two fiber-optic laser sources. Each fiber-optic laser source splits into a pair of channels and, therefore, both relative and absolute measurements can be performed. During the experiments, the fiber-optic laser sources were fed through a microscope to reduce the laser-beam spot sizes. For the measurements presented in this study, a 5  objective is used to obtain a spot size of 2:8 mm was used. The axis of the endmill is located in the vicinity of the nodal line of the torsional motions. Therefore, measurements along the axis of the endmill do not provide accurate determination of the torsional modes. To facilitate measuring the torsional modes, metal strips (0.22 mm by 3.12 mm) were glued to the sides of the macro endmill. The response measured from the metal strips amplifies the torsional motions, thereby enabling accurate determination of torsional/axial natural frequencies. 5. Alternative modeling techniques for comparative analysis In this section, two alternative approaches for modeling the dynamics of Table 1 endmills are described. The main purpose of this study is to provide a comparative analysis of the 3D-ST technique with two other popular solution approaches. First, a finite element (FE) solution of the endmill dynamics using a commercial solver is outlined. Next, simplified 1D solution of the endmill dynamics using four different equivalent cross-section formulations is provided. 5.1. Finite element modeling (FEM) of macro-endmills Using the geometry of the four endmills, finite element (FE) models were constructed. Analysis was performed on the meshed solid model of each endmill using the commercial software ANSYSs Workbench (SOLID186 elements). Table 2 lists the material properties (Young’s modulus E, density, r, and Poisson’s ratio, n). Modal analysis was used to identify the natural frequencies. The convergence of the FE model was evaluated for all endmills based on the changes in the second torsional natural frequency to ensure that a sufficient number of elements were included. Table 3 summarizes the convergence study for the FE model of endmill 1. The convergence study was performed in a consecutive way, where the percent change in natural frequencies 44 B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 were calculated between the two consecutive FE simulations taking the finer meshed model as reference. Table 2 Endmill material properties. Endmill material E (GPa) Aluminum Carbide r 70 580 kg m3  n  2700 14 500 5.2. Equivalent diameter approach 0.33 0.24 One of the common approaches to describe the dynamics of the fluted section is to approximate it as a uniform circular crosssection beam with an equivalent diameter. Four methods were applied to calculate this diameter: (1) the cross-sectional area; (2) the area moment of inertia of the cross-section; (3) the volume; and (4) the mass. Table 3 Convergence study for endmill 1 from Table 1. Second torsional natural frequency (Hz) Number of nodes Number of elements % Change in natural frequencies with increasing number of nodes 7776.5 7756.3 7746.3 7742.4 7739.3 3056 4077 7226 20 689 22 454 578 792 1485 4488 4828 –  0.26  0.13  0.05  0.04 1. The cross-sectional area of the fluted portion, Af, was obtained from the solid model (endmills 1–3) or analysis of crosssection image (endmill 4) and the corresponding equivalent diameter, deqA , was calculated using rffiffiffiffiffiffiffiffiffi 4 Af : ð24Þ deqA ¼ p 2. The area moment of inertia for the modeled cross-section of the fluted portion, If, was determined and the equivalent diameter, deqI , was calculated using   64 If 1=4 deqI ¼ : ð25Þ Table 4 Equivalent diameter of endmills. p Endmill deqA (mm) deqI (mm) deqV (mm) deqM (mm) Aluminum Carbide 30.40 9.90 31.10 10.10 30.96 9.95 31.55 9.92 3. The volume of the fluted section, Vf, was obtained and the equivalent diameter was determined as sffiffiffiffiffiffiffiffiffiffi 4 Vf deqV ¼ : ð26Þ p Lf Table 5 Comparison of the experimental and predicted natural frequencies. Endmill 1 B11 B12 B21 B22 TA1 B31 B32 TA2 Experiment (Hz) FE (Hz) 3D-ST (Hz) 1426 1400 1404 1430 1403 1407 4059 4000 4007 4068 4002 4023 4218 4115 4091 7530 7484 7486 7543 7492 7497 7777 7739 7750 Difference (%) between ST and FEM Difference (%) between FE and experiment Difference (%) between ST and experiment 0.28  1.81  1.54 0.32  1.92  1.60 0.17  1.45  1.28 0.53  1.62  1.10  0.60  2.42  3.00 0.03  0.61  0.58 0.06  0.68  0.62 0.14  0.49  0.34 Endmill 2 B11 B12 TA1 B21 B22 TA2 B31 B32 Experiment (Hz) FE (Hz) 3D-ST (Hz) 1710 1691 1690 1717 1691 1695 4716 4606 4583 4802 4776 4772 4811 4776 4790 8558 8523 8563 8698 8673 8665 8719 8674 8680 Difference (%) between ST and FEM Difference (%) between FE and experiment Difference (%) between ST and experiment  0.06  1.12  1.18 0.22  1.47  1.26  0.51  2.32  2.81  0.08  0.55  0.63 0.30  0.72  0.43 0.47  0.40 0.06  0.09  0.28  0.37 0.07  0.52  0.45 Endmill 3 B11 B12 TA1 B21 B22 TA2 B31 B32 Experiment (Hz) FE (Hz) 3D-ST (Hz) 1634 1611 1612 1638 1611 1615 4414 4349 4315 4588 4523 4529 4596 4524 4551 8412 8286 8293 8602 8556 8561 8615 8558 8568 Difference (%) between ST and FEM Difference (%) between FE and experiment Difference (%) between ST and experiment 0.07  1.40  1.34 0.21  1.64  1.43  0.79  1.46  2.24 0.13  1.42  1.30 0.60  1.57  0.98 0.09  1.49  1.41 0.06  0.54  0.48 0.13  0.66  0.54 Endmill 4 B11 B12 B21 B22 TA1 B31 B32 TA2 Experiment (Hz) FE (Hz) 3D-ST (Hz) 2487 2463 2486 2487 2471 2495 7343 7242 7271 7343 7248 7272 11273 11365 11384 13515 13283 13337 13515 13336 13393 20071 20143 20163 Difference (%) between ST and FEM Difference (%) between FE and experiment Difference (%) between ST and experiment 0.93  0.96  0.05 0.98  0.64 0.33 0.40  1.37  0.98 0.33  1.29  0.96 0.17 0.82 0.99 0.41  1.72  1.32 0.43  1.33  0.91 0.10  0.36 0.46 45 B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 4. The mass of the endmill, m, can be expressed as the sum of the mass of the shank (first expression on the left hand side of Eq. (27)) and the mass of the fluted portion (second expression) r p 2 ds Ls 4 þr p 2 deqM 4 Lf ¼ m, ð27Þ where deqM is the equivalent diameter of the fluted portion. By weighing the endmill and substituting nominal value for the density and geometry of the endmill, the equivalent diameter was calculated according to Eq. (28). The benefit of this approach is that no tool model is required, unlike the first three methods: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u 2 u 4 m p ds Ls : ð28Þ  deqM ¼ t p Lf r 4 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 In this section, the natural frequencies obtained using modal testing, the ST method, and FE model (FEM) simulations are compared to assess the accuracy of the 3D-ST technique for modeling endmill dynamics. Table 5 provides the natural frequencies from 3D-ST, FEM, and experimentation, as well as the percent differences between the modes obtained from different methods. The labeled mode shapes (as bending or torsional/axial) are identified according to the calculated deformations. The bending modes are represented by Bij where i is the mode number and j is the principal direction of deformation. The coupled torsional–axial modes are represented by TAi where i is the mode number. For each of the endmills, the percent errors between the ST method and experimentation are seen to be less than 3.0%. This 4 11 9 10 7 8 es 6 d o 5 TM 3 4 3D-S 1 2 12 13 14 MAC 3 4 5 FE 6 7 M 8 M 9 od es 10 11 12 13 14 11 9 10 7 8 6 s e d 5 T Mo 3 4 3D-S 1 2 12 13 14 3 4 5 FE 6 M 7 M 8 od 9 es 10 11 12 13 14 11 9 10 7 8 es d o 6 5 TM 3D-S 3 4 1 2 12 13 14 12 13 14 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 2 5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 3 FE 6 M 78 M od 9 es 10 11 12 13 14 MAC 6.1. Comparison of tool modeling techniques MAC MAC Table 4 lists the equivalent diameter (inmm) calculated by the four methods for the four endmills. 6. Modal assessment 2 3 4 5 FE 6 M 78 M od 9 es 10 11 12 13 14 10 11 8 9 7 des 5 6 T Mo 3 4 3D-S 1 2 Fig. 7. Modal Assurance Criteria (MAC) analysis for (a) endmill 1; (b) endmill 2; (c) endmill 3; and (d) endmill 4. 46 Cross-sectional Area Area Moment of Inertia Volume Cross-sectional Area Area Moment of Inertia Volume Mass Mass 14 12 10 Absolute Percent Error (%) 8 6 4 2 0 1 2 3 compared to experimental results compared to FE results B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 4 5 6 7 8 10 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 14 12 8 10 6 8 4 6 4 2 2 0 1 2 3 4 5 6 7 8 0 Mode Number Fig. 8. Absolute percent errors of the approximate methods for (a) endmill 1; (b) endmill 2; (c) endmill 3; (d) endmill 4. small difference between the 3D-ST and experimental natural frequencies could be the result of differences between the modeled and actual geometry and the non-uniform/inaccurate material properties. Furthermore, it is observed that the largest errors occur in the first torsional mode, which suggests that it is the most sensitive mode to geometric uncertainties. A comparison between natural frequencies from the 3D-ST and FEM results, on the other hand, shows a difference less than 0.8% for all four endmills and modes. This good match between the 3D-ST and FEM results, which use identical geometries and material properties, also supports this hypothesis. It should be noted that the dynamic response is very sensitive to the geometry of the transition region. Therefore, it is critical to accurately capture this geometry. For a more in-depth assessment of the 3D-ST model, the modal assurance criterion (MAC) was used [20,21]. The modal assurance criterion is a measure of consistency of the modal vectors obtained through two different methods. If the MAC value for a given mode is above 0.8 (i.e., 80%), it is considered that the modes are consistent [21]. Therefore, this method was used to compare the mode shapes (other than rigid body modes) obtained from the 3D-ST method and FEM analysis. To perform the MAC analysis, the position and associated displacement at each FEM node was obtained. Second, the 3D-ST method was used to determine the displacements at the same positions for the given mode shape. The MAC value for each mode was then calculated from 31.7 43 7 12.7 29.3 126.8 0.88 24.08 25.12 39.95 47.99 Fig. 9. Holder and blank dimensions for 127.68 mm overhang length (all dimensions are in mm). Table 6 Carbide endmill/blank and holder material properties. Component E ðGPaÞ Tool Holder 580 200 r kg m3   14 500 7800 n 0.24 0.29 2 MAC ¼ 9fFA gT fF3DST gT 9 fFA gT fFA g þfF3DST gT fF3DST g , ð29Þ where FA and F3DST are the modal matrices obtained through FE analysis (ANSYSs ) and the 3D-ST method, respectively. The calculated MAC numbers are given in Fig. 7. A MAC number closer to unity indicates good agreement between the mode shapes for the two methods; the first 14 modes of each endmill were analyzed. A good match between the 3D-ST and FEM modes is B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 observed. A minimum MAC value of 0.855, 0.993, 0.966, and 0.931 and the average MAC values of 0.975, 0.995, 0.992, and 0.989 are obtained for endmills 1, 2, 3, and 4, respectively. The equivalent diameter simplification for the twisted fluted section of the tool was also evaluated. Fig. 8 provides the percent errors for each of the approximate methods with respect to the results of the FEM results for each mode. 47 The general trend of the percent errors was similar for each tool model. Comparing the different approximation methods, the most accurate equivalent diameter was obtained using the cross-sectional area and the least accurate approach was the method based on the endmill mass. Also, considering all four approximation methods for each mode, the largest errors (410%) was observed for the first torsional mode. Furthermore, the approximation errors depend strongly on the particular geometric parameters of the endmills. Therefore, these approximations should be used carefully. The 3D-ST modeling approach presented in this study avoids such errors while retaining numerical efficiency. 7. Tool point measurements and predictions V IV III II I Fig. 10. Blank-holder model components. Table 7 Component dimensions for 127.68 mm overhang length. Dimensions (mm) Component Outer diameter (d0), left Outer diameter (d0), right Inner diameter (di), left Inner diameter (di), right Length, L III II I 43 31.7 7 7 8.04 31.7 31.7 12.7 12.7 14.83 31.7 31.7 12.7 12.7 1.04 31.7 29.3 12.7 12.7 24.08 12.7 12.7 0 0 127.68 x 10−5 Real (m/N) x 10−5 0 −5 200 x Imag (m/N) IV 400 600 800 1000 1200 1400 1600 1800 2000 0 −2 −4 −6 −8 200 400 600 800 1000 1200 1400 1600 1800 2000 5 0 −5 200 10−5 Imag (m/N) Real (m/N) 5 V Tool point measurements were performed and predictions were completed on a Mikron UCP-600 Vario five-axis machining center. Predictions and measurements of the tool point FRFs are presented here for three different overhang lengths (118.7 mm, 124.78 mm, and 127.59 mm) of endmill 4 (150.8 mm total length) inserted in a steel tapered thermal shrink fit holder. The tool point FRFs were measured by exciting the tool tip with a miniature impact hammer (PCB 0841A17, sensitivity 46.95 N/V) and recording the corresponding vibration with a low mass accelerometer (PCB 352C23, sensitivity 1727 (m/V s2). Also, measurements were performed with the same shrink fit holder using a carbide blank of approximately the same length as the endmill (152.8 mm). The overhang lengths of the blank were adjusted to match the three overhang lengths of the tool. Fig. 9 shows the dimensions of the holder and blank for an overhang length of 127.68 mm. Table 6 x 2 0 −2 −4 −6 −8 200 400 Frequency (Hz) Real (m/N) 4 600 800 1000 1200 1400 1600 1800 2000 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) 10−5 400 x 10−5 2 0 −2 −4 200 400 600 800 600 800 1000 1200 1400 1600 1800 2000 Imag (m/N) x 10−5 0 −2 −4 −6 −8 200 400 1000 1200 1400 1600 1800 2000 Frequency (Hz) Fig. 11. Measured (solid line) and predicted (dotted line) tool point FRFs for the carbide tool blank with overhang lengths of: (a) 127.68 mm; (b) 124.35 mm; and (c) 118.62 mm (rigid connection). 48 B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 holder at coordinate 2 in Fig. 12. Since no connection is rigid in reality, a flexible connection between the tool and the holder was implemented. The flexible coupling of the components is carried out in two steps: (1) the spindle–machine is first rigidly coupled to the holder and the portion of the shank inside the holder; (2) the holder–spindle–machine component is then flexibly coupled to the blank outside the holder using translational and rotational spring constants assembled in the stiffness matrix k. The RCSA equation for the flexible coupling tool point FRF in case of the blank is provided in Eq. (30). The stiffness matrix [9] is given by Eq. (31), where kxf, kyf , kxm, and kym are the displacement-toforce, rotation-to-force, displacement-to-moment, and rotationto-moment stiffness values, respectively (kyf and kxm were assumed equal due to reciprocity). lists the tool and holder material properties used for modeling both the endmill and blank. The component model for one blankholder combination is shown in Fig. 10, and the corresponding dimensions are listed in Table 7. Fig. 11 shows the comparison of the measured and the predicted carbide blank tool-point FRF for three different overhang lengths. It is observed in Fig. 11 that the predicted natural frequencies are higher than those from the experiments. This is attributed to the assumption of a rigid connection between the tool and the   1 1 G11 ðoÞ ¼ R11 ðoÞR12a ðoÞ R2b2b ðoÞ þ R2a2a ðoÞ þ k 1 2a 2b Fig. 12. Component coordinates for flexible coupling in case of blank. k¼ kyf (N/rad) kym (Nm/rad) 3.05  106 3.07  106 0 Imag (m/N) Real (m/N) 0 −5 x 10−5 4 2 0 −2 −4 −6 −8 200 400 600 800 1000 1200 1400 1600 1800 kyf kym # : ð31Þ 600 800 1000 1200 1400 1600 1800 5 0 2000 −5 200 400 600 800 1000 1200 1400 1600 1800 2000 2000 x 10−5 4 2 0 −2 −4 −6 −8 200 400 600 800 1000 1200 1400 1600 1800 2000 Imag (m/N) Real (m/N) 5 400 kxm x 10−5 x 10−5 200 kxf ð30Þ To identify the stiffness matrix, one overhang length of the blank (124.35 mm) was considered and an optimization procedure based on a genetic algorithm [19] was implemented. The variables were the three stiffness values, and the objective function to be minimized is given by Eq. (32), where the difference between the imaginary parts of the measured (m) and predicted (p) tool Table 8 Stiffness matrix values. kxf (N/m) " R2a1 ðoÞ, Frequency (Hz) Frequency (Hz) Real (m/N) x 10−5 5 0 −5 200 400 600 800 1000 1200 1400 1600 1800 2000 600 800 1000 1200 1400 1600 1800 2000 Imag (m/N) −5 x 10 0 −2 −4 −6 −8 200 400 Frequency (Hz) Fig. 13. Measured (solid line) and predicted (dotted line) tool point FRFs for the carbide tool blank with overhang lengths of: (a) 127.68 mm; (b) 124.35 mm; and (c) 118.62 mm (flexible connection). 49 B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 x 10−5 5 5 Real (m/N) Real (m/N) x 10−5 0 −5 200 400 600 800 1000 1200 1400 1600 1800 0 −5 2000 200 0 −5 −10 −15 200 400 600 800 400 600 800 1400 1600 1800 1000 1200 1400 1600 1800 2000 x 10−5 Imag (m/N) Imag (m/N) x 10−5 400 600 800 0 −5 −10 −15 200 1000 1200 1400 1600 1800 2000 Frequency (Hz) 1000 1200 1400 1600 1800 2000 Frequency (Hz) Real (m/N) x 10−5 5 0 −5 200 400 600 800 600 800 1000 1200 2000 Imag (m/N) x 10−5 0 −5 −10 −15 200 400 1000 1200 1400 1600 1800 2000 Frequency (Hz) Fig. 14. Measured (solid line) and predicted (dotted line) tool point FRFs for the carbide endmill with overhang lengths of: (a) 127.59 mm; (b) 124.78 mm; and (c) 118.7 mm (flexible connection). Table 9 Percent error between measurement (M) and prediction (P) for Fig. 13. Overhang length (mm) Table 10 Percent error between measurement (M) and prediction (P) for Fig. 14. Mode 1 Mode 2 Mode 3 Mode 4 M (Hz) M (Hz) M (Hz) P (Hz) M (Hz) P (Hz) P (Hz) P (Hz) Fig. 13(a) 127.68 % error 569.5 570.7 2030  0.2  5.1 2133.5 3073 3.1 2979 3925 1.1 3880 Fig. 13(b) 124.35 % error 595.1 594.9 2045 0.0  4.8 2143 3111 0.9 3082.3 3800  2.4 3890 Fig. 13(c) 118.62 % error 644.8 643.2 2060 0.2  4.6 2155 3230  1.1 3264 3909 4087 4.4 point FRFs was squared and summed over all frequencies within the range of interest. qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi min ðImðHm ÞImðHp ÞÞ2 : ð32Þ The k matrix values obtained by this approach (see Table 8) were used to predict the tool point FRFs for other overhang lengths of the blank as well as the endmill. The comparison of the measurements and predictions for the three overhang lengths of the carbide blank is shown in Fig. 13. Note that due to the linear axis of the FRF plots in Figs. 13 and 14, the higher modes cannot be Overhang length (mm) Mode 1 Mode 2 Mode 3 Mode 4 M (Hz) P (Hz) M (Hz) M (Hz) M (Hz) Fig. 14(a) 127.59 % error 666  2.5 682.2 2105 2.5 Fig. 14(b) 124.78 % error 688.4 716  4.0 Fig. 14(c) 118.7 % error P (Hz) P (Hz) P (Hz) 2052 2864 4.1 2745.4 3750  4.5 3920 2070  0.1 2072 2944 4.2 2819 3820  2.7 3922 739.3 766.8 2090  3.7  0.2 2094 3087 3.2 2987.3 3910  0.5 3928 observed, however, comparison of four dominant modes was included in the analyses. Table 9 lists the percent errors between the measured and predicted natural frequencies for the four dominant modes within the 5000 Hz measurement bandwidth. The stiffness values given in Table 8 were used to predict the THSM assembly tool point FRFs for the carbide endmill with different overhang lengths. Fig. 14 shows the measured and predicted results. The percent errors between the predicted natural frequencies and the measured values for the four dominant modes within the 5000 Hz measurement bandwitdh were listed in Table 10 for three overhang lengths. 50 B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50 8. Conclusion This paper presented new modeling results for the threedimensional (3D) dynamic behavior of macro-scale milling tools using the spectral-Tchebychev (ST) technique. The actual complex cross-sectional geometry and the pretwisted shape of endmills were taken into account during modeling. The bending and torsional behavior of three endmills with known cross-section was modeled and verified against both finite element models and experiments (impact testing with free–free boundary conditions). Model validation was also performed for a commercial carbide endmill. The difference between the experiments and the spectral-Tchebychev method predictions was seen to be less than 3% for all the four tools for the first six bending modes and first two torsional/axial modes. The natural frequencies from the finite elements model and the 3D-ST method were seen to match with less than 1% difference. To demonstrate the application of the modeling approach, the 3D-ST model of a commercial carbide endmill was coupled to the Timoshenko beam model of a shrink fit holder and the measured spindle receptances using receptance coupling substructure analysis (RCSA). The tool point measurements and predictions were compared for three different overhang lengths. A flexible connection between the tool and the holder was implemented, where the holder–tool interface stiffness values were determined using a carbide blank and a genetic algorithm-based optimization technique. The stiffness values were used to predict the tool point FRFs of other blanks with different overhang lengths, as well as the endmill with various overhang lengths. The maximum error between the natural frequency of the tool point measurement and prediction was less than 1% for the carbide blank and less than 4% for the endmill. Acknowledgments The authors gratefully acknowledge financial support from the National Science Foundation (CMMI-0928393 and CMMI-0928211). References [1] J. Tlusty, Manufacturing Processes and Equipment, Prentice Hall, Upper Saddle River, NJ, 2000. [2] Y. Altintas, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press, Cambridge, UK, 2000. View publication stats [3] T. Schmitz, K.S. Smith, Machining Dynamics: Frequency Response to Improved Productivity, Springer, New York, NY, 2009. [4] T. Schmitz, R. 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