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Comparison of Equivalent Diameter End Mill Models for Dynamics Prediction by Receptance Coupling Substructure Analysis

Permanent Link: http://ufdc.ufl.edu/UFE0041355/00001

Material Information

Title: Comparison of Equivalent Diameter End Mill Models for Dynamics Prediction by Receptance Coupling Substructure Analysis
Physical Description: 1 online resource (77 p.)
Language: english
Creator: Kumar, Uttara
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: coupling, element, finite, frequency, milling, receptance, response
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: COMPARISON OF EQUIVALENT DIAMETER END MILL MODELS FOR DYNAMICS PREDICTION BY RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS With the increase in the spindle speeds made available by new spindle designs coupled with higher machine velocities and accelerations, the knowledge of tool-holder-spindle-machine dynamics becomes all the more important if the improved machine capabilities are to be exploited. The wider acceptance of Receptance Coupling Substructure Analysis (RCSA) in the field of high speed machining to predict the tool point dynamics emphasizes the need to develop tool-holder models that can accurately predict the dynamics of a particular machine to determine the best spindle speed and corresponding material removal rate (MRR). This thesis describes an investigation of tool models that are required as part of the RCSA tool-holder-spindle-machine coupling procedure. Finite element analysis (FEA) is used to approximate the tool's cross-sectional geometry and generate frequency response functions which represent the free-free beam's behavior. Because FEA is computationally expensive, alternative models using equivalent diameters for the fluted portion are also considered. In this work three methods are applied in order to calculate the equivalent diameter of the fluted portion of the end mill. The aim of this work is to compare these methods for equivalent diameter selection and select the best alternative. The prediction of frequency response function (FRF) using the equivalent diameter models of the fluted end mill are compared to FRF measurements obtained using impact testing.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Uttara Kumar.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041355:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041355/00001

Material Information

Title: Comparison of Equivalent Diameter End Mill Models for Dynamics Prediction by Receptance Coupling Substructure Analysis
Physical Description: 1 online resource (77 p.)
Language: english
Creator: Kumar, Uttara
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: coupling, element, finite, frequency, milling, receptance, response
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: COMPARISON OF EQUIVALENT DIAMETER END MILL MODELS FOR DYNAMICS PREDICTION BY RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS With the increase in the spindle speeds made available by new spindle designs coupled with higher machine velocities and accelerations, the knowledge of tool-holder-spindle-machine dynamics becomes all the more important if the improved machine capabilities are to be exploited. The wider acceptance of Receptance Coupling Substructure Analysis (RCSA) in the field of high speed machining to predict the tool point dynamics emphasizes the need to develop tool-holder models that can accurately predict the dynamics of a particular machine to determine the best spindle speed and corresponding material removal rate (MRR). This thesis describes an investigation of tool models that are required as part of the RCSA tool-holder-spindle-machine coupling procedure. Finite element analysis (FEA) is used to approximate the tool's cross-sectional geometry and generate frequency response functions which represent the free-free beam's behavior. Because FEA is computationally expensive, alternative models using equivalent diameters for the fluted portion are also considered. In this work three methods are applied in order to calculate the equivalent diameter of the fluted portion of the end mill. The aim of this work is to compare these methods for equivalent diameter selection and select the best alternative. The prediction of frequency response function (FRF) using the equivalent diameter models of the fluted end mill are compared to FRF measurements obtained using impact testing.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Uttara Kumar.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041355:00001


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1 COMPARISON OF EQUIVALENT DIAMETER END MILL MODELS FOR DYNAMICS PREDICTION BY RECEPT ANCE COUPLING SUBSTRUCTURE ANALYSIS By UTTARA VIJAY KUMAR A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2009

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2 2009 Uttara Vijay Kumar

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3 To my parents, Alka and Vijay Kantute

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4 ACKNOWLEDGMENTS I want to thank my parents for their immense love encouragement and for the principles and values they have insti lled in me. I would also like to thank my sister, Shweta, who has been my best friend and my brother in law Shwetal for their care and constant motivation. I extend m y sincere gratitude to my advisor Dr. Tony L. Schmitz, a mentor in every sense of the word, for his guidance and exceptional ideas throughout my research. I feel privileged to have had the opportunity to interact with two distinguished yet so humble personalities, my committee m embers Dr. Schueller and Dr. Greenslet I am grateful to Dr. Cristescu, who has been a father like figure to me. I thank my colleagues in the M achine Tool Research Center (MTRC) for their help, support and sense of humor making th e MTRC a fun place to work. I am also thankful to all my friends in Gainesville especially Ashwin Dani, for being a pillar of strength.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...............................................................................................................4 LIST OF TABLES ...........................................................................................................................7 LIST OF FIGURES .........................................................................................................................8 ABSTRACT ...................................................................................................................................11 CHAPTER 1 INTRODUCTION ..................................................................................................................13 Research Description ..............................................................................................................13 Literature Review ...................................................................................................................14 2 RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS ............................................16 Description ..............................................................................................................................16 Free Free Beam Receptances .................................................................................................18 Coupling Free Free Receptances ............................................................................................22 Spindle Machine Receptances ................................................................................................28 Collet HolderSpindle Machine Receptances .........................................................................31 3 MODELING AND ANALYSIS .............................................................................................34 End Mill Modeling .................................................................................................................34 Analysis in ANSYS Workbench ............................................................................................36 Results from ANSYS Classi c .................................................................................................39 4 COMPARISON BETWEEN MODEL PREDICTIONS AND EXPERIMENTAL RESULTS ...............................................................................................................................42 Free Free Receptances of the End Mill ..................................................................................42 Experimental Approach ...................................................................................................42 Equivalent Diameter Approach .......................................................................................43 Finite Element Analysis (FEA) Approach ......................................................................44 Comparison between the Three Approaches ..........................................................................44 Tool Point Dynamics ..............................................................................................................50 5 CONCLUSION .......................................................................................................................58 Concluding Statements ...........................................................................................................58 Future Work ............................................................................................................................58

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6 APPENDIX A PROGRAM IN ANSYS PARAMETRIC DESIGN LANGUAGE (APDL) .........................60 B MATLAB CODES .................................................................................................................64 Importing FEA Receptances ...................................................................................................64 Free Free Beam Receptances .................................................................................................67 Coupling Free Free Beam Receptances ..................................................................................70 Coupling Free Free Section to a Cantilevered Section ...........................................................72 L IST OF REFERENCES ...............................................................................................................75 BIOGRAPHICAL SKETCH .........................................................................................................77

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7 LIST OF TABLES Table page 41 Comparison of first bending mode natural frequency (free free boundary conditions) of the flu ted portion of end mill obtained by FEA and equivalent diameter approaches. .........................................................................................................................45 42 Comparison of the entire end mill (shank and fluted portion) free free first bending mode natural frequency obtained using the three different approaches. ...........................45 43 Comparison of first bending mode natural frequency between clampedfree FEA modal analysis and coupling the harmonic analysis fluted portion receptances with a clamped free Timoshenko shank model. ...........................................................................46 44 Comparison of clampedfree first bending mode natural frequency between experiment, equivalent diameters and FEA modal analysis. .............................................47 45 Comparison between the average error in natural frequency for each equivalent diameter over all overhang lengths relative to FEA/experiment. ......................................49

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8 LIST OF FIGURES Figure page 21 Tool holder spindle machine assembly and coordinates ...................................................16 22 The machine assembly is divide d into four components I IV. .........................................17 23 Components of the modeled subassembly: fluted portion (I), tool shank (II) and holder (III). .........................................................................................................................18 24 Individual components I II III with displacements and rotations at specified coordinate locations. ..........................................................................................................19 25 Subassembly composed of the fluted portion (I) and tool shank (II). The gene ralized force Q1 is applied at U1 to determine subassembly receptances G11(direct) and G3 a 1(cross). .........................................................................................................................23 26 Subassembly composed of the fluted portion (I) and tool shank (II). The generalized f orce Q3a is applied at U3a to determine assembly receptances G3 a 3 a (direct) and G13 a (cross). ................................................................................................................................25 27 Component I II (fluted portion and shank) subassembly receptances coupled with component II I (holder) receptances. ..................................................................................27 28 Standard artifact used to determine spindle machine (component IV) receptances. .........29 29 Measuremen t of H33 using impact testing. .........................................................................30 210 Measurement of H3 a 3 using to determine N33. ...................................................................30 211 Measurement of component III (holder spindle machine) receptance, H33. ......................31 212 Impact testing to measure the direct receptance of the holder spindle machine assembly, H33. ....................................................................................................................32 213 Measurement of component III (holder spindle machine) cross receptance, H33a to determine N33, L33 and P33. ................................................................................................32 214 Impact testing to measure the cross receptance of the hold er spindle machine assembly, H33a. ...................................................................................................................33 215 Impact testing to measure the tool point FRF of the tool holder spindle machine assembly, H11. ....................................................................................................................33 31 Carbide end mill. ................................................................................................................34 32 A) Section drawn for helical sweep in Pro E. B) Dimensions used to draw the section. ...............................................................................................................................35

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9 33 Rounding of the inner edge. ...............................................................................................35 34 Final model of the fluted portion of end mill. ....................................................................36 35 Local coordinate system defined o n the end mill model. ..................................................36 36 Meshing created on end mill model ...................................................................................37 37 Force applied along the positive x axis. .............................................................................38 38 Moment applied on the top face about z axis. ....................................................................38 39 Nodes selected to obtain h and l receptances. ....................................................................39 310 Numbers of selected nodes for direct receptances. ............................................................40 311 Numbers of selected nodes for cross receptances. .............................................................40 41 Experimental free free receptance by impact testing. A) Direct receptance. B) Cross receptance. .........................................................................................................................42 42 Experimental setup to measure clamped free FRF of the end mill. ..................................47 43 Percentage error (relative to experiment) for different equivalent diameters and FEA as a function of overhang length. .......................................................................................48 44 No rmalized percentage error (average FEA mean error set to zero relative to experiment) for different equivalent diameters and FEA as a function of overhang length. .................................................................................................................................49 45 Measured direct FRF ( HAA) of the collet holder spindle machine. ....................................50 46 Clamped free FRF of the modeled tool holder substructure using different equivalent diameters. The tool overhang length is 86.5 mm. The first t wo bending modes are seen for each method. ........................................................................................................51 47 Measured (denoted as Exp in the legend) and predicted tool holder spindle direct FRFs ( H11) for three different equivalent diameters. The to ol overhang length is 86.5 mm. ....................................................................................................................................51 48 Clamped free FRF of the modeled tool holder substructure using different equivalent diameters. The overhang length is 96.5 mm. .....................................................................53 49 Measured and predicted tool holder spindle direct FRFs (H11) for three different equivalent diameters. The tool overhang length is 96.5 mm. ............................................53 410 Clamped free FRF of the modeled tool holder substructure using different equivalent diameters. The overhang length is 101.5 mm. ...................................................................54

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10 411 Measured and predicted tool holder spindle direct FRFs (H11). The overhang length is 101.5 mm. .......................................................................................................................54 412 Clamped free FRF of the modeled tool holder substructure using different equivalent diameter. The overhang length is 116.5 mm. .....................................................................55 413 Measured and predicted tool holder spindle direct FRFs (H11). The overhang length is 116.5 mm. .......................................................................................................................55 414 Clamped free FRF of the modeled tool holder substructure using different equivalent diameters. The overhang length is 126.5 mm. ...................................................................56 415 Measured and predicted tool holder spindle direct FRFs (H11). The overhang length is 126.5 mm. .......................................................................................................................56 416 Clamped free FRF of the modeled tool holder substructure using different equivalent diameters. The overhang length is 136.5 mm. ...................................................................57 417 Measured and predicted tool holder spindle direct FRFs (H11). The overhang length is 136.5 mm. .......................................................................................................................57

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11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Par tial Fulfillment of the Requirements for the Degree of Master of Science COMPARISON OF EQUIVALENT DIAMETER END MILL MODELS FOR DYNAMICS PREDICTION BY RECEPT ANCE COUPLING SUBSTRUCTURE ANALYSIS By Uttara Vijay Kumar December 2009 Chair: Tony L. Schmitz Maj or: Mechanical Engineering With the increase in the spindle speeds made available by new spindle designs coupled with higher machine velocities and accelerations, the knowledge of tool holder spindle machine dynamics becomes all the more important if the improved machine capabilities are to be exploited. The wider acceptance of Receptance Coupling Substructure Analysis (RCSA) in the field of high speed machining to predict the tool point dynamics emphasizes the need to develop tool holder models that can a ccurately predict the dynamics of a particular machine to determine the best spindle speed and corresponding material removal rate (MRR). This thesis describes an investigation of tool models that are required as part of the RCSA tool holder spindle machin e coupling procedure. Finite element analysis (FEA) is used to approximate the tools cross sectional geometry and generate frequency response functions which represent the free free beams behavior. Because FEA is computationally expensive, alternative mo dels using equivalent diameters for the fluted portion are also considered. In this work three methods are applied in order to calculate the equivalent diameter of the fluted portion of the end mill. The aim of this work is to compare these methods for equivalent diameter selection and select the best alternative. The prediction of

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12 frequency response function (FRF) using the equivalent diameter models of the fluted end mill are compared to FRF measurements obtained using impact testing.

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13 CHAPTER 1 INTRODUCT ION Research D escription The process dynamics during milling can dramatically affect productivity due to unstable cutting conditions (or chatter) and forced vibrations which can cause part geometry errors (or surface location errors) [1 3]. Modeling these process dynamics requires knowledge of the structural dynamics of the cutting system, particularly the response of the tool holder spindle machine assembly (as reflected at the tool point). With the increase in the spindle speeds made available by new spindle designs coupled with higher machine velocities and accelerations, the knowledge of tool holder spindle machine dynamics becomes indispensable if the improved machine capabilities are to be exploited. Efforts are continuously being made to develop methods that accurately predict the dynamics of a particular machine to determine the best spindle speed and material removal rate (MRR) with a minimal number of tests. The prediction of tool point dynamics using Receptance Coupling Substructure Analysis (RCSA) [4] is gaining wider acceptance in the field of high speed machining. The analytical RCSA method is applied to join models of the tool and holder with a measurement of the spindle machine. This thesis describes an investigation of tool models that are required as part of the RCSA tool holder spindle machine coupling procedure. Finite element analysis (FEA) is used to approximate the tools cross sectional geometry and generate frequency response functions which represent the free free beams behavior. Bec ause FEA is computationally expensive, alternative models are also considered. In a study by Kops and Vo [5], the fluted portion was approximated as a uniform beam by determining an equivalent diameter based on FEA and beam equations were used for deflecti on calculations. Schmitz et al. also made use of the equivalent diameter approach [4, 67]. Three methods are applied in this work to calculate

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14 the equivalent diameter of the fluted portion of the end mill: 1) the area of the cross section of the end mill s fluted area; 2) the area moment of inertia of the crosssection of the end mills fluted area; and 3) and the mass of the end mill. The goal of this work is to compare these methods for equivalent diameter selection and identify the best alternative. The results of frequency response function (FRF) measurements are compared to prediction using the equivalent diameter models of the fluted end mill. Literature R eview Many attempts have been made to model the milling process and it continues to be a widely s tudied topic. The time marching numerical integration approach to model the milling process is summarized by Smith and Tlusty [8]. Related work includes the mechanistic model approach for the prediction of the force system [9]. Frequency domain solutions have been applied to determine process stability in the form of stability lobe diagrams, which identify stable and unstable cutting zones as a function of axial depth of cut and spindle speed [10]. Altintas and Budak used a Fourier series (frequency domain) approach to approximate the time varying cutting force coefficients for stability lobe diagram development [11]. A closed form, frequency domain solution for surface location error in milling was developed by Schmitz and Mann [12]. A numerical method for the stability analysis of linear time delayed system based on a semi discretization technique was developed [13]. Modeling approaches based on finite element analysis [14] and, later, time finite element analysis [15] have also been developed. In all the m odeling methods, a description of the system dynamic response comprised of the tool holder spindle machine assembly receptance is required. This response can be obtained on a caseby case basis via impact testing, where an instrumented hammer is used to ex cite the tool point and the response is measured using (typically) a low mass accelerometer. However, because each tool holder combination must be measured on each machine, the number of experiments can be

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15 excessive. Therefore, the preferred method is appl ication of an appropriate modeling approach which reduces the number of required experiments. The preference of a modeling approach led to the application of receptance coupling [16] to predict the tool point FRF. In the initial application of receptance c oupling to tool point FRF prediction, an Euler Bernoulli (E B) beam model of the overhung portion of the tool was coupled to the displacement to force receptance of the holder spindle machine [4]. The fluted portion of the tool was approximated using the e quivalent diameter approach by Kops and Vo [5]. Many improvements have been made since then to the RCSA method. Park et al. incorporated displacement to moment, rotation to force and rotationto moment receptances in the analysis [17]. The E B beam model w as replaced with the Timoshenko beam model that takes into consideration the rotational inertia and shear deformation [18]. Schmitz et al. extended the RCSA method to three components: the overhung tool (i.e., the portion outside the holder), holder and spindle machine [6]. The RCSA method was further improved by making use of FEA to estimate the stiffness and damping values at the tool shrink fit holder connection [7]. FEA was implemented to model the spindle by Eturk et al. [19]. In the most recent study, Filiz et al. applied the spectral Tchebychev technique to model the cutting tool [20].

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16 CHAPTER 2 RECEPTANCE COUPLING SUBSTRUCTURE ANALYSI S Description This chapter describes the Receptance Coupling Substructure Analysis (RCSA) approach. In RCSA the rece ptances, or frequency response functions (FRFs), of individual components are coupled analytically to predict the assembly receptances. It involves both experimental and modeled FRFs. The tool holder spindle machine assembly is divided into separate compon ents. In the second generation RCSA method, the assembly was divided into three primary components: the tool, holder and spindle machine [6]. The tool and holder were modeled, while the receptances of the spindle machine (which are difficult to model based on first principles, primarily due to the difficulty in estimating damping at interfaces) were calculated by measuring a standard artifact and using the inverse receptance coupling method [6]. This approach is depicted in Figure 21. In Figure 22, four c omponents are identified because the fluted portion of the tool is separated from the constant diameter tool shank. Figure 21. Tool holder spindle machine assembly and coordinates

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17 Figure 22. The machine assembly is divided into four components I IV. A FRF is a transfer function (expressed in frequency domain) where only the positive frequencies are considered. FRFs of a system are expressed as complex ratios of displacementto force, velocity to force or acceleration to force at the specified coordin ate locations. They describe the system natural frequencies and mode shapes. The FRFs representing displacement to force are termed receptances. The complex ratio of the displacement to the applied force is either expressed as its real and imaginary parts or in terms of its magnitude and phase. The receptance of the tool holder spindle machine assembly as reflected at the tool point is used to produce the desired stability lobe diagram. For the tool holder spindle machine RCSA model, four bending receptance s are considered for the description of each component. They are, displacement to force, i ij jx h f displacement to couple, i ij j x l m

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18 rotation to force, i ij jn f and rotation to couple i ij jp m where i and j are the coordinate locations. If i and j are equal the receptances are referred to as direct receptances; otherwise, they are denoted cross receptances. The receptances can either be measured or predicted. Due to the generally large number of tool, holder and spindle combinations in a particular facility, measuring FRFs can be very time consuming. Thus, to save time and improve efficiency, the receptances of the assembly components may be modeled using Euler Bernoulli [16] or T imoshenko beam theory [21]. Figure 23. Components of the modeled subassembly: fluted portion (I), tool shank (II) and holder (III). FreeF ree B eam R eceptances Because the spindle machine receptances are difficult to model, they are measured using a st andard holder, while the tool and holder receptances are modeled. In this research the numerical (finite) Timoshenko beam elements are used to model the four degrees of freedom (displacement and rotations at both the ends) for free free beam receptances of the tool shank

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19 and holder [6]. Both FEA and equivalent diameter Timoshenko beam models are used to describe the fluted portion of the tool. See Figure 23, where rigid connections are assumed. The individual component or substructure, receptances, Rij are defined in Equation 21: ii jj ijij ij ijij ii jjxx fm hl R np fm (21) where i x is the substructure displacement at the coordinate location i i is the substructure rotation at the coordinate location i jf is the force applied to the substructure at the coordinate location j and jm is the couple applied to the substructure at the coordinate location j Figure 24. Individual components I IIIII with displacements and rotations at specified coordinate locations. Thus for the components I, II and III, Equation s 22 to 213 describe the direct and cross receptanc es at the coordinate locations shown in Figure 24. Component I, the fluted portion of the tool, is described using Equations 22 through 25.

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20 11 11 1111 11 1111 11 11xx fm hl R np fm (2 2) 11 22 1212 12 1212 11 22 aa aa a aa aaxx fm hl R np fm (2 3) 22 22 2222 22 2222 22 22 aa aa aaaa aa aaaa aa aaxx fm hl R np fm (2 4) 22 11 2121 21 2121 22 11 aa aa a aa aaxx fm hl R np fm (2 5) Similarly, component II, the tool shank, is described by Equations 26 through 2 9. 22 22 2222 22 2222 22 22 bb bb bbbb bb bbbb bb bbxx fm hl R np fm (2 6) 22 33 2323 23 2323 22 33 bb aa baba ba baba bb aaxx fm hl R np fm (2 7) 33 33 3333 33 3333 33 33 aa aa aaaa aa aaaa aa aaxx fm hl R np fm (2 8)

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21 33 22 3232 32 3232 33 22 aa bb abab ab abab aa bbxx fm hl R np fm (2 9) The component III, holder, receptances are given by Equati ons 2 10 through 213. 33 33 3333 33 3333 33 33 bb bb bbbb bb bbbb bb bbxx fm hl R np fm (2 10) 33 44 3434 34 3434 33 44 bb aa baba ba baba bb aaxx fm hl R np fm (2 11) 44 44 4444 44 4444 44 44 aa aa aaaa aa aaaa aa aaxx fm hl R np fm (2 12) 44 33 4343 43 4343 44 33 aa bb abab ab abab aa bbxx fm hl R np fm (2 13) The r elationships between displacements/rotations and forces/couples can be written in matrix form for convenience as shown in Equations 214 to 225. 111111 111111xhlf npm or 1111uRq (2 14) Similarly, 1122 aauRq (2 15) 2222 aaaa uRq (2 16)

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22 2211 aauRq (2 17) 2222 bbbbuRq (2 18) 2233 bbaauRq (2 19) 3333 aaaa uRq (2 20) 3322 aabbuRq (2 21) 3333 bbbb uRq (2 22) 3344 bbaauRq (2 23) 4444 aaaauRq (2 24) 4433 aabbuRq (2 25) where ui and qi are the generalized displacement/rotation and the force/couple vectors, respectively. Coupling F reeF ree R eceptances The coupling of the individual component receptances is performed sequentially with component s I and II coupled first. Then, th is subassembly s receptances are coupled with component III. The component I and II subassembly receptances are determ ined using Equations 228 to 244. In order to calculate the subassembly receptances, G11 (direct) and G3a1 (cross) (Equation s 2 26 and 227 respectively), a generalized force Q1 (representing both the externally applied force and couple) is applied at co ordinate location 1(see Figure 2 5) 11 11 1111 11 1111 11 11XX FM HL G NP FM (2 26)

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23 Figure 25. Subassembly composed of the fluted portion (I) and tool shank (II). The g eneralized force Q1 is applied at U1 to determine subassembly receptances G11(direct) and G3 a 1(cross). 33 11 3131 31 3131 33 11 aa aa a aa aaXX FM HL G NP FM (2 27) The displacement equations for the substructures can be described as follows: 1111122 aauRqRq (2 28) 2222211 aaaaauRqRq (2 29) 2222 bbbbuRq (2 30) 3322 aabb uRq (231) If rigid coupling between the two components is assumed, the compatibility condition is expressed as shown in Equation 2 32. 220bauu (2 32)

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24 Th e equilibrium condition at coordinate locations 2a and 2b is given by Equation 2 33. 220baqq (2 33) At coordinat e location 1, the equilibrium condition is provided by Equation 234. 11qQ (2 34) Substituting for u2 b and u2 a in Equation 232 gives Equation 235. 222222222110babbbaaaauuRqRqRq (2 35) Equation 236 is obtained using Equations 2 33 and 2 34. 22222211 ()0 bbaabaRRqRQ (2 36) Solving for q2b gives Equation 2 37. Given that 22 abqq from Equation 233, substitution in Equation 238 gives Equation 2 39, which can the n be written as shown in Equation 2 40. 1 22222211 () bbbaaaqRRRQ (2 37) 111122 11 11 111 aaRqRq Uu G QQQ (2 38) 1 111122222211 11 1 () abbaaaRQRRRRQ G Q (2 39) 1111 1 111112222221 1111()abbaaaHL GRRRRR NP (2 40) Similarly, the cross receptances between coordinates 3 a and 1 are given by Equations 241 and 242. 1 33322322222211 31 111 1 () aaabbabbbaaa aUuRqRRRRQ G QQQ Q (2 41) 3131 1 3132222221 3131()aa aabbbaaa aaHL GRRRR NP (2 42)

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25 To determine the other two receptances of the sub assembly III, G3 a 3 a and G13a, a generalized force Q3 a is applied to U3 a as shown in Figure 2 6. Figure 26. Subassembly composed of the fluted portion (I) a nd tool shank (II). The generalized force Q3 a is applied at U3 a to determine assembly receptances G3 a 3 a (direct) and G13 a (cross). 33 33 333 33 333 33 33 aa aa aaa aa aaa aa aaXX FM HL G NP FM (2 43) 11 33 1313 13 1313 11 33 aa aa a aa aaXX FM HL G NP FM (2 44) The component displacement/rotation equations are now described by Equations 245 to 248. 1122 aa uRq (2 45) 2222 aaaauRq (2 46) 2222233 bbbbbaauRqRq (2 47)

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26 3333322 aaaaabbuRqRq (2 48) The compatibility equation remains the same as given in Equation 232. 220abuu (2 49) The equilibrium equations are: 220baqq and (250) 33 aaqQ (2 51) Substituting for u2 a and u2 b in Equation 249 gives Equation 252. 222222222330abaaabbbbaauuRqRqRq (2 52) Using Equations 250 and 251 and substituting for q2 b and q3 a in Equation 252 gives Equation 253. 22222233 ()0 bbaaabaaRRqRQ (2 53) Solving for q2 a, Equation 254 is obtained. Using Equation 220, q2 b can be written. Equation 2 55 gives the desired expression for the subassembly direct receptances. 1 22222233 () abbaabaaqRRRQ (2 54) 33333322 33 333 aaaaaabb aa aaa UuRqRq G QQQ (2 55) By substituting for q2 b in Equation 2 55 and using Equations 2 50 and 254, Equation 2 56 is obtained. Equation 2 57 gives the final expression after simplification. 1 333322222233 33 3 ()aaaabbbaabaa aa aRQRRRRQ G Q (2 56) 3333 1 333332222223 3333()aaaa aaaaabbbaaba aaaaHL GRRRRR NP (2 57) In a similar way, the cross receptances are defined. See Equations 2 58 and 259.

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27 1 122122222233 11 13 333 3 () aaabbaabaa a aaa aRqRRRRQ Uu G QQQ Q (2 58) 1313 1 1312222223 1313() aa aabbaaba aaHL GRRRR NP (2 59) The four receptances of the subassembly III are then rigidly coupled to the component III (holder) receptances. See Figure 27. A B Figure 27. Component I II (fluted portion and shank) subassembly receptances coupled with component III (holder) receptances. The subassembly III and III direct and cross receptances are obtained in exactly the sam e manner as described for component s I and II with the generalized force Q1 applied at coordinate location 1 to determine the new G11 and G4 a 1 receptances (Figure 2 7A); Q4 a is applied at coordinate location 4a to determine G4 a 4 a and G14 a (Figure 2 7B). The resulting four receptances are described by Equations 260 to 264.

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28 1111 1 111113333331 1111()abbaaa HL GRRRRR NP (2 60) 4141 1 4143333331 4141()aa aabbbaaa aaHL GRRRR NP (2 61) 4444 1 444443333334 4344()aaaa aaaaabbbaaba aaaaHL GRRRRR NP (2 63) 1414 1 1413333334 1414()aa aabbaaba aaHL GRRRR NP (2 64) Spindle M achine R eceptances As d iscussed previously, the receptances of component IV (spindle machine) are difficult to model. Therefore, these receptances are experimentally determined. Since the flange geometry is the same for all holders (to enable automatic tool changes), only the po rtion of the holder beyond the flange (towards the tool) is modeled. The flange and the holder taper (which is inserted in the spindle) are considered part of the spindle machine. To identify the spindle machine receptances, a standard artifact (i.e., a st andard tool holder with a uniform cylindrical geometry beyond the flange) is placed in the spindle and G33 is measured as shown in Figure 2 8. Using G33 and a model of the portion of the holder beyond the flange, the spindle machine receptance R4 b 4 b is cal culated. G33 is expressed in matrix form by replacing coordinate 1 by 3, 3a by 4a and 3b by 4b in Equation 2 60. See Equation 265. 3333 1 333334444443 3333()abbaaa HL GRRRRR NP (2 65) If G33 is deter mined experimentally and R33, R4 a 3, R4 a 4 a and R34a are modeled (component III in Figure 2 8), R4 b 4 b can be calculated by rewriting Equation 265 as shown in Equation 2 66. This is referred to as inverse receptance coupling.

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29 a a a a b bR R G R R R4 4 34 1 33 33 3 4 4 4 (2 66) Figure 28. Standard artifact used to determine spindle machine (component IV) receptances. The component receptance H33 from the G33 matrix was measured by impact t esting (the TXF software from MLI was used for data acquisition and signal analysis). In this method, an impact hammer is used to apply the force and an accelerometer (piezoelectric sensor) measures the accelerance (translational acceleration to force FRF) and the software is used to twice integrate the accelerance to give the required displacementto force receptance. As shown in Figure 29, the direct FRF H33 is measured with the force and the accelerometer at the same coordinate location. A second compo nent of the G33 matrix, N33, is calculated by the finite difference approach [22] as described in Equation 2 67, where the cross

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30 FRF H3 a 3 is measured as shown in Figure 2 10 and s is the distance between the direct and cross FRF measurements. 33 33 3 333333 33 33 a a a xx xx ffHH s N ffss (2 67) Figure 29. Measurement of H33 using impact testing. Figure 210. Measurement of H3 a 3 using to determine N33. Assuming reciprocity, the off diagonal terms of the G33 matrix may be taken to be equal. See Equation 268. L33 = N33 (2 68)

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31 The P33 receptance is synthesized using Equatio n 269 [ 22] 2 333 33 33 3333 333 33331 xfN P LN mfxHH (2 69) Thus, using inverse receptance coupling and the finite difference approach, the spindle machine (component IV) receptances are obtained. These receptances can then be coupled with any fluted portion tool shankholder modeled receptances to predict the tool point assembly dynamics. Collet H older S pindle M achine R eceptances In this study, a collet holder was used and the ho lder spindle machine receptances were measured as a single subassembly. Therefore, the entire tool holder spindle machine assembly was divided into only three components: the fluted portion of the tool (I), the tool shank (II) and the holder spindle machin e (III). Components I and II were modeled as described previously. The measurements for component III included the direct FRF H33 and cross FRF H3 a 3. The subassembly receptances L33, N33 and P33 were determined as described in the previous section. Figures 211 through 214 show the measurements used to determine the components of matrix G33. Figure 211. Measurement of component III (holder spindle machine) receptance, H33

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32 Figure 212. Impact testing to measure the direct receptance of the holder spindle machine assembly, H33. Figure 213. Measurement of component III (holder spindle machine) cross receptance, H33a to determine N33, L33 and P33.

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33 Figure 214. Impact testing to measure the cross receptance of the holder spindle machine assembly, H3 3 a. H33 and H33a were measured and N33, L33 and P33 were calculated using Equations 2 67 through Equation 2 69, where s = 30 mm. The modeled tool receptances were then coupled to G33. The predicted tool point frequency response was then compared to the measured tool point FRF. Figure 2 15 shows the experimental setup used to acquire the tool point FRF. Figure 215. Impact testing to measure the tool point FRF of the tool holder spindle machine assembly, H11.

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34 CHAPTER 3 MODELING AND ANALYSI S End M ill M odeling A three dimensional (3D) model of the fluted portion of the four flute, carbide end mill shown in Figure 31 was developed in ProEngineer Wildfire (Pro E). Figure 31. Carbide end mill. The model was then imported into ANSYS Workbench for the f inite element analysis. The geometry of the end mill was: 12.7 mm shank diameter, 66.15 mm shank length, 85.05 mm fluted portion length, 30 deg helix angle, and 75 mm pitch. The steps used to model the geometry of the fluted portion in ProE are detailed n ext : 1. Using the command Extrude on the top plane, sketch a circle with diameter of the shank and then extrude it to the length of the fluted portion. 2. Select Insert>>Helical Sweep>>Cut. 3. Define the attributes, Constant>> Thru axis>> Right handed>> Done 4. To spe cify the Sweep Profile, select the front plane. Draw a center axis at the center of the cylinder and, from the bottom left corner of the cylinder, draw a vertical line equal to the length of the cylinder, the arrow should be pointing upwards. 5. Specify the pitch value. 6. Define the section as shown in the Figure 32A. The section is drawn by measuring the dimensions as shown in Figure 32B. The horizontal line is the dimension on the face of the tool as indicated in the Figure 32B and the vertical line is the distance between two flutes in the front plane. The arc is tangent to the vertical and the two lines joining the arc to the vertical line represent the cutting edge. Specify the material side which should be inside the section.

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35 7. Once the first flute is created, pattern it around the central axis to obtain four flutes. 8. Use the command Round to smooth the inner edge; the radius should be equal to the inner surface of the end mill as shown in Figure 33. 9. Save the part file as the .stp file type. A B Figure 32. A) Section drawn for helical sweep in ProE. B) Dimensions used to draw the section. Figure 33. Rounding of the inner edge.

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36 Figure 34. Fina l model of the fluted portion of end mill. Analysis i n ANSYS Workbench The step file (.stp) created in Pro E was then used to perform a harmonic analysis in ANSYS workbench to calculate the direct and cross receptances of the fluted portion of the end mill. The steps are next identified : 1. Open ANSYS Workbench. Select Empty Project>> New Geometry>> File>>Import External Geometry>> Generate. 2. Click on Project>> New Simulation Figure 35. Local coordinate system defined on the end mill model.

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37 3. Model>> Geometr y>> Name of the geometry. The box on the bottom right side of the screen has an option for material type. Click on new material and specify the density, elastic (Youngs) modulus and Poissons ratio for the tool material (tungsten carbide in this case). Th e values used for this simulation are: 15000 kg/m3 density, 5.5 1011 N/m2 elastic modulus and 0.22 Poissons ratio. 4. The coordinate system should be aligned such that the x axis is along the flutes cutting edge as shown in Figure 35.A local coordinate sys tem is defined so that the x axis is along the edge of the flute. 5. For meshing, click on mesh option on the left side of the screen. Select the options for meshing. Size of the elements>> 5 104 m. This is the smallest size that can be chosen in Workbench. Right click on Mesh and click on generate mesh. The mesh generated in shown in Figure 36. Figure 36. Meshing created on end mill model 6. In order to calculate the natural frequencies, a modal analysis was performed. 7. Insert New Analysis>> Modal Analysis>> Analysis Setting. Select the number of modes. Right click on Solution>>Insert>>Deformation>>Total. Click Solve. 8. The natural frequencies of the number of modes selected can then be viewed. 9. In order to calculate the receptances, a Harmonic Analysis was pe rformed. The deformation of every node was stored in the .db file and read in ANSYS classic. New Analysis>> Harmonic Analysis>> Analysis settings. Insert the range of frequency and the

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38 number of intervals. In this analysis, a resolution of 0.1 Hz was used. Select the Full method. Specify the damping constant (0.00075 in this case). Click Yes to save the .db file. 10. Right click on Analysis>> Insert>> Force/ Moment. Select the node where the force/moment needs to be applied. To specify the magnitude of force/m oment select Components and also the coordinate system if you have specified any other than the Global Coordinates. In x / y /z component specify the force/moment (1 N along the x axis or 1 N m about negative z axis in this case) shown in Figure 3 7 and Figur e 3 8. 11. Right click on Solution and insert Frequency response>> Deformation. Select the vertex where the deformation is to be determined. In this case the vertex opposite to that where the force was applied was selected. Choose the Direction>> X Axis. Click on Solve. Figure 37. Force applied along the positive x axis. Figure 38. Moment applied on the top face about z axis.

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39 Results from ANSYS Classic The .db file saved from the harmonic analysis was imported into ANSYS Classic to read the results at p articular nodes. The direct receptances, h and l were calculated by finding the deformation in the x axis (which can be the global x axis or the x axis of the new coordinate system specified) on the node opposite to the node where force/moment was applied. To find the direct receptances, n and p, the deformations at node s 1 and 2 shown in Figures 39 and 310 w ere read. The difference between the deformations divided by the distance between the nodes gave l Similarly, for the cross receptances, the nodes at the other end of the model were selected (see Figure 3 11). Figure 39. Nodes selected to obtain h and l receptances.

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40 Figure 310. Numbers of selected nodes for direct receptances. Figure 311. Numbers of selected nodes for cross receptances.

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41 T he steps followed in ANSYS Classic are next detailed : 1. Click on File>> Change Directory>> Select the folder where the .db file is stored. 2. File>> Resume from>> Click on the .db file. 3. In order to read the direct and cross h and n receptances and calculate the direct and cross l and p receptances, a program was written in APDL (ANSYS Parametric Design Language) provided in Appendix A. The result is stored in an .inp file. To read it, click on File>> read input from>>select the .inp file. 4. The real and imaginary values for all receptances are stored in an Excel file. This file can then be imported in MATLAB to plot the free free receptances of the fluted portion and couple it with shankholder spindle machine receptances to predict the tool point frequency response function.

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42 CHAPTER 4 COMPARISON BETWEEN M ODEL PREDICTIONS AND EXPERIMENTAL RESULTS FreeFree Receptances of t he End Mill This section describes the three approaches used to obtain the freefree receptance ( h = x/f ) of the end mill. Each approach is desc ribed and the results are compared. Experimental Approach In the experimental approach, the frequency response function (FRF) of the end mill was obtained by impact testing. The free free boundary conditions were approximated by placing the end mill on a piece of foam (Figure 4 1). The very low stiffness foam base (relative to the end mill) provided adequate conditions to obtain the free free receptances of the end mill. A B Figure 41. Experimental free free receptance by impact testing. A) Direct receptance. B) Cross receptance. A low mass/metal tip (high contact stiffness) hammer was used to excite a wide range of frequencies simultaneously. The accelerometer mass was compensated to avoid the loading effects on the low mass tool. Using the mass cancellation approach [23] given by Equations 4 1 and 42, where maccelerometer is the accelerometer mass, the desired direct and cross accelerance FRFs were obtained. The direct and cross receptance FRFs were then defined by Equations 43 and 44, re spectively.

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43 11, 11, 11,A 1 m Ameasured corrected accelerometermeasuredA (4 1) 12, 12, 12, A 1 m Ameasured corrected accelerometermeasuredA (4 2) 11,corrected 11 2A H (4 3) 12,corrected 12 2A H (4 4) Equivalent Diameter Approach As mentioned previously, the equivalent diameter approach assumes a constant cross section of the fluted portion of the end mill. The equivalent diameter of the cross section was calculated in three different ways. 1. The crosssectional area of the fluted portion, Af, was modeled and the corresponding equivalent diameter, deqA, was calculated. See Equation 4 5. 4 f eqAA d (4 5) Using the software Solidworks, the cross sectional area of the modeled tool was determined [24]. The model considered in this work gave an area of 82.113 mm2. Using Equation 45, the equivalent diameter was 7.23 mm. 2. The area moment of inertia for the modeled cross section of the fluted portion, If, was determined using Solidworks and the equivalent diameter, deqI, was calculated using Equation 46 464 I f eqId (4 6) For the tested tool, the area moment of inertia was 595.25 mm4, which gave an equivalent diameter of 10.494 mm. 3. The mass of the tool can be expressed as the sum of the mass of the shank (first expression on the left hand side of Equation 47) and the mass of the fluted portion (second expression):

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44 2 2 44 eqMf ssdL dL m (4 7) where m is total mass of the tool, ds and deqM are the shank diameter and the equivalent diameter of the fluted portion, respectively, Ls and Lf are the lengths of the shank and the fluted portion, respectively, and is the tool density (carbide f or this study). By weighing the tool and substituting nominal value for the density and geometry of the tool, the equivalent diameter, deqM, was calculated according to Equation 4 8: 24 4ss eqM fdL m d L (4 8) where m = 216.5 g, Ls = 66.15 mm, Lf = 85.05, ds = 12.7 mm and = 0.015 g/mm3. Using these values, an equivalent diameter of 9.525 mm was obtained. The mass of the fluted portion, mf, can be expressed as shown in Equation 49. 2 4 eqMf fdL m (4 9) Because the mass of the fluted portion mf remains constant, the value of deqM also remains constant as it depends only on Lf and which do not change even if receptances are modeled at different overhang lengths, i.e., different values of Ls. Finite Element Analysis (FEA) Approach The freefree receptances of the fluted portion were obtained by harmonic analysis (using ANSYS Work bench as discussed in Chapter 3) and then rigidly coupled to the free free receptances of the shank using RCSA. The freefree receptances for the shank were calculated using the Timoshenko beam model [6] programmed in MATLAB. Comparison b etween t he Three Approaches The natural frequencies of the first bending mode for the fluted portion (only) of the end mill as determined by FEA and the three equivalent diameter approaches are compared in Table 41. Since it was not possible to measure the FRF of only the fluted portion, results from the experimental approach are not included.

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45 Table 4 1. Comparison of first bending mode natural frequency (free free boundary conditions) of the fluted portion of end mill obtained by FEA and equivalent diameter approaches. Approach Natural frequency (Hz) Percentage error (relative to FEA) FEA 666 2 d eqA 5230 21. 5 d eqI 7219 8. 4 d eqM 6804 2.1 Table 4 2 provides a comparison between the experimental, equivalent diameter and FEA approaches for free free receptances o f the entire end mill (tool shank coupled to the fluted portion). Table 4 2. Comparison of the entire end mill (shank and fluted portion) free free first bending mode natural frequency obtained using the three different approaches. Approach Natural frequen cy (Hz) Percentage error (relative to experiment) Experimental 2496 FEA 251 5 0.7 d eqA 164 1 34. 3 d eqI 249 3 0.1 d eqM 221 7 11.2 Using the FEA approach, the free free receptances were obtained up to a frequency of 600 Hz for the fluted portion of t he end mill. The time to compute the displacement to force receptance for a frequency range of 20 Hz with a resolution of 0.1 Hz was 20 hours, so the frequency range was limited to 600 Hz and the receptances at both the ends of the fluted portion were assu med to be the same. The significant computational expense of the FEA approach leads to the preference for an alternative, less costly approach. The first bending mode natural frequency obtained by coupling the fluted portion harmonic analysis receptances t o the clamped free Timoshenko beam model was compared to the natural frequency obtained by modal analysis in FEA for the clampedfree end mill (shown in Table 43). A long shank length was considered

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46 so that the natural frequency was less than 600Hz. The difference between the two results was caused by the assumption of equal receptances on both the ends of fluted portion. Table 4 3. Comparison of first bending mode natural frequency between clampedfree FEA modal analysis and coupling the harmonic analysi s fluted portion receptances with a clamped free Timoshenko shank model. Shank length (mm) Natural frequency (Hz) FEA modal analysis (clamped free) Fluted portion receptances (harmonic analysis) coupled to Timoshenko beam model and rigid support Percen tage error 80 467 404 13.4 90 41 6 36 7 11. 8 100 372 333 10. 3 Due to the significant computational expense associated with the harmonic analysis, only the natural frequencies obtained by FEA modal analysis (computational time was approximately two to th ree minutes) were compared to the other approaches. The first bending natural frequency obtained by modal analysis was compared to the first bending natural frequency obtained by the equivalent diameter approach (which takes just over one minute to complet e) to compare the three equivalent diameter calculation methods. Using the three equivalent diameters and the Timoshenko beam model, the clampedfree receptances were calculated by coupling the tool substructure (receptances of the fluted portion modeled using equivalent diameters coupled to the various shank lengths) to a rigid support (having zero receptances). Experiments were also performed by clamping the tool in a vise to approximate fixedfree boundary conditions (see Figure 42). Table 4 4 shows a comparison of the first clamped free bending natural frequency obtained by FEA modal analysis, the equivalent diameter methods and experiment for different overhang lengths. Figure 43 shows the percentage error relative to experimental result versus the to ol shank length for the three equivalent diameters and the FEA modal analysis results.

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47 Figure 42. Experimental setup to measure clamped free FRF of the end mill. Table 4 4. Comparison of clampedfree first bending mode natural frequency between experim ent, equivalent diameters and FEA modal analysis. Overhang length (mm) Natural frequency (Hz) Percentage e rror relative to experiment Exp d eqA d eqM d eqI FEA d eqA d eqM d eqI FEA 91.5 91 4 821 104 7 1128 1234 10.1 14.6 23. 5 35.1 9 6.5 84 8 809 1004 106 4 1156 4.6 18.4 25.5 36.4 101.5 877 795 960 1001 1082 9.3 9.4 14. 2 23.3 106.5 867 78 1 916 941 1011 10.0 5.6 8.5 16.5 111.5 83 3 76 5 872 884 944 8.2 4.7 6.1 13.3 116.5 75 6 74 8 828 830 881 1.0 9.6 9.8 16.6 121.5 71 2 730 786 779 8 23 2.6 10.5 9. 4 15.6 126.5 67 1 711 74 6 73 1 769 6.1 11.2 9.0 14.7 131.5 62 9 69 2 706 686 719 10.0 12. 3 9. 1 14.3 136.5 568 671 668 644 673 18.2 17.6 13. 4 18.4

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48 Figure 43. Percentage error (relative to experiment) for different equivalent diameters and F EA as a function of overhang length. Figure 43 shows that the first bending mode natural frequency obtained by FEA has an error of 35% relative to the experimental validation for the shortest overhang length tested. It is proposed that the large error b etween the two approaches is due to imperfect realization of the clamped boundary condition (between the tool and vise). If this clamping condition is not exactly rigid, this effectively incorporates a linear/torsional spring at the vise tool interface and would serve to lower the measured natural frequency(s). Assuming the FEA modal analysis provides the most accurate representation of the clamped free tool, the FEA and equivalent diameter approaches were compared to experiment by normalizing the mean er ror between the experimental and FEA results to zero (see Figure 4 4).

PAGE 49

49 Figure 44. Normalized percentage error (average FEA mean error set to zero relative to experiment) for different equivalent diameters and FEA as a function of overhang length. The n ew normalized natural frequency for each equivalent diameter was then compared to the FEA and experimental results. Table 4 5 shows the percentage difference between the mean error (including all overhang lengths) for the three equivalent diameter approach es and FEA. Table 4 5. Comparison between the average error in natural frequency for each equivalent diameter over all overhang lengths relative to FEA /experiment Equivalent diameters Percentage error d eqI 7.9 d eqM 9.9 d eqA 23 .0

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50 Tool Point Dynamics T he tool collet holder assembly defined in Chapter 3 was modeled for six different overhang lengths of the tool (i.e., the length of the tool beyond the free end of the collet holder). The assembly receptances were also coupled to a rigid support. The purpo se of this coupling was to identify the modes of the clamped free tool holder assembly and better understand their interactions with the spindle machine modes. The measured receptances of the holder spindle machine (not including the tool) were also plotte d. Finally, the measured and predicted receptances of the tool holder spindle machine assembly were plotted to visualize the interactions between the spindle modes and tool holder modes. The prediction of the tool point dynamics using different equivalent diameters was then compared to the experimental results. Figure 45. Measured direct FRF ( HAA) of the collet holder spindle machine.

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51 Figure 46. Clampedfree FRF of the modeled tool holder substructure using different equivalent diameters. The tool overhang length is 86.5 mm. The first two bending modes are seen for each method. Figure 47. Measured (denoted as Exp in the legend) and predicted tool holder spindle direct FRFs ( H11) for three different equivalent diameters. The tool overhang length i s 86.5 mm.

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52 In Figure 4 5, the collet holder spindle machine direct FRF (at the subassemblys free end) is displayed. Multiple modes are evident with the lowest dynamic stiffness in the frequency range near 700 Hz. By comparing Figures 4 5 and 46, the low er frequency mode in Figure 4 7 is identified as a spindle mode and the higher frequency mode as a tool holder mode. Considering the higher frequency mode, the experimental results lie between the mass equivalent diameter approach (dotted line) and the mom ent of inertia equivalent diameter approach (dash dot). The area based equivalent diameter under predicts the natural frequency of the tool holder substructure and, due to the natural frequency error, causes an incorrect interaction with the spindle machin e mode near 700 Hz. For an overhang length of 96.5 mm (Figure 49), the experimental results show improved agreement with the mass equivalent diameter approach. The area based prediction is again incorrect. In Figure 4 11, the mass equivalent diameter ap proach captures the two peaks of the higher frequency mode, which indicates an interaction between the tool holder mode and a spindle mode, and shows good agreement with the experimental results. The incorrect area based trend continues. From Figures 413, 415 and 4 17, it is seen that the reduction in natural frequency of the tool holder substructure (with increasing tool overhang length) moves the interaction from a higher frequency spindle mode to the lower frequency spindle mode (near 700 Hz). Again, the area based method offers the least prediction accuracy.

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53 Figure 48. Clampedfree FRF of the modeled tool holder substructure using different equivalent diameters. The overhang length is 96.5 mm. Figure 49. Measured and predicted tool holder spindle direct FRFs (H11) for three different equivalent diameters. The tool overhang length is 96.5 mm.

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54 Figure 410. Clampedfree FRF of the modeled tool holder substructure using different equivalent diameters. The overhang length is 101.5 mm. Figure 411. Measured and predicted tool holder spindle direct FRFs (H11). The overhang length is 101.5 mm.

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55 Figure 412. Clampedfree FRF of the modeled tool holder substructure using different equivalent diameter. The overhang length is 116.5 mm. Figure 413. M easured and predicted tool holder spindle direct FRFs (H11). The overhang length is 116.5 mm.

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56 Figure 414. Clampedfree FRF of the modeled tool holder substructure using different equivalent diameters. The overhang length is 126.5 mm. Figure 415. Measured and predicted tool holder spindle direct FRFs (H11). The overhang length is 126.5 mm.

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57 Figure 416. Clampedfree FRF of the modeled tool holder substructure using different equivalent diameters. The overhang length is 136.5 mm. Figure 417. Measured and predicted tool holder spindle direct FRFs (H11). The overhang length is 136.5 mm.

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58 CHAPTER 5 CONCLUSION Concluding Statements The comparison between the natural frequencies obtained by FEA modal analysis and equivalent diameter model results for the clamped free end mill, as well as modeled and experimental results for tool point dynamic responses (obtained using a selected collet holder spindle machine substructure), indicate that the best equivalent diameter for prediction accuracy varies with ove rhang lengths. Figure 44 shows that at lower overhang lengths, the prediction accuracy using the area moment of inertia method for equivalent diameter estimation ( deqI) is highest, but as overhang length increases the response obtained by the mass based e quivalent diameter method ( deqM) is more accurate. The results obtained using the area based equivalent diameter ( deqA) has a large percentage error with respect to FEA as compared to other methods (see Figure 4 4 and Table 4 5). The tool point FRF measure ments and predictions also mimic the nonmonotonic error behavior. The trend is further complicated by the inherent interactions between the tool holder and spindle machine modes. Though Figure 44 shows decreased percentage error for the areabased equiva lent diameter (relative to FEA) as the overhang length increases, the tool holder spindle machine assembly predictions and measurements completed here do not show good agreement for the selected overhang length range when using this method. This is partial ly the result of the complex interactions between the tool holder and spindle machine dynamics, which naturally vary from one machine to the next, even for the same tool holder geometry. Future Work Future work should include calculation of the free free receptances of the fluted portion of the tool over a range of 200 to 3000 Hz in ANSYS to compare the FEA based receptances with

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59 experimental results. Improvements in the three dimensional tool model developed in ProE will also help in accurately determini ng the equivalent diameter calculation using area and moment of inertia. Better controlled clamped free boundary conditions need to be provided in order to enable accurate validation data by experiments. Comparisons of the equivalent diameters and FEA appr oaches may also be made using a new approach in which the tool holder model is developed using the spectral Tchebychev technique to solve the Timoshenko beam equations (tool holder assembly with different geometries and material properties are modeled indi vidually and are combined analytically using component mode synthesis technique) [20]. Additional experiments and predictions should also be completed on tools with various shankto flute length ratios coupled to other spindle machine substructures and the three different equivalent diameter methods compared.

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60 APPENDIX A PROGRAM IN ANSYS PARAMETRIC D ESIGN L ANGUAGE (APDL) Calculation of h and n of beam! /post1 /PNUM, NODE, 1 m= 200 Number of intervals in the frequency range define the value of real and imag part of each node *DIM,xrd1,,m *DIM,xrd2,,m *DIM,xrc1,,m *DIM,xrc2,,m *DIM,xid1,,m *DIM,xid2,,m *DIM,xic1,,m *DIM,xic2,,m *DIM,hrd,,m *DIM,nrd,,m *DIM,hrc,,m *DIM,nrc,,m *DIM,hid,,m *DIM,nid,,m *DIM,hic,,m *DIM,nic,,m *DIM,fre,,m

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61 f= 1.0 define the force/moment applied *GET,yd1,NODE,75,LOC,Y get Y location of nodes for direct receptance *GET,yd2,NODE,1074,LOC,Y dyd=ABS(yd1yd2) distance between two nodes *G ET,yc1,NODE,1278,LOC,Y get Y location of nodes for cross receptance *GET,yc2,NODE,8904,LOC,Y dyc=ABS(yc1yc2) distance between two nodes *DO,n,1,m SET,1,n No. of substep *GET,fre(n),ACTIVE,0,SET,FREQ get each step's frequenc y SET,1,n,1,0, set the value of real part *GET,xrd1(n),NODE,75,U,X real UX of node for direct receptance *GET,xrd2(n),NODE,1074,U,X hrd(n)=xrd1(n)/f real part of direct h nrd(n)=(xrd1(n) xrd2(n))/(f*dyd) real part of direct n *GET,xrc1(n),NODE,1 278,U,X real UX of NODE for cross receptance *GET,xrc2(n),NODE,8904,U,X hrc(n)=xrc1(n)/f real part of cross h nrc(n)=(xrc1(n) xrc2(n))/(f*dyc) real part of cross n SET,1,n,1,1, set the value of imaginary part *GET,xid1(n),NODE,75,U,X imag UX of node for direct receptance *GET,xid2(n),NODE,1074,U,X hid(n)=xid1(n)/f imag part of direct h

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62 nid(n)=(xid1(n) xid2(n))/(f*dyd) imag part of direct n *GET,xic1(n),NODE,1278,U,X imag UX of node for cross receptance *GET,xic2(n),NODE,8904,U,X hic(n)=xic1(n)/f imag part of cross h nic(n)=(xic1(n) xic2(n))/(f*dyc) imag part of cross n *ENDDO *CFOPEN,H&N_by_force,xls *VWRITE, ('direct and cross recptance of H and N') *VWRITE, ('Freq',' ','HR_direct',' ','HI_direct',' ','NR_direct',' ','NI_direct',' ','HR_c ross',' ','HI_cross',' ','NR_cross',' ','NI_cross') *VWRITE,fre(1),hrd(1),hid(1),nrd(1),nid(1),hrc(1),hic(1),nrc(1),nic(1) (F5.1,' ',F20.14,' ',F20.14,' ',F20.14,' ',F20.14,' ',F20.14,' ',F20.14,' ',F20.14,' ',F20.14) *VWRITE, ('*******END********') *CFCLOS /PNUM,NODE,0 xrd1(1)= xrd2(1)= xrc1(1)= xrc2(1)=

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63 xid1(1)= xid2(1)= xic1(1)= xic2(1)= hrd(1)= nrd(1)= hrc(1)= nrc(1)= hid(1)= nid(1)= hic(1)= nic(1)= fre(1)= FINI

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64 APPENDIX B MATLAB CODES T his section includes the MATLAB programs used to import the FEA H armonic A nalysis receptances from Excel files and then couple them to Timoshenko beam model s Importing FEA Receptances % FEA.m % Uttara V Kumar % Tony Schmitz % Jun Zhang % Program to read receptances stored in Excel from harmonic % analysis in FEA f = 200.1:0.1:600; % Initializing frequency vector f = f.'; % Import the receptance of the fluted portion calculated by ANSYS real_h66 = xlsread('H&N_by_force(200-600).xls',1,'B3:B4002'); imag_h66=xlsread('H&N_by_force(200-600).xls',1,'C3:C4002'); h66=(real_h66+i*imag_h66); real_n66 = xlsread('H&N_by_force(200-600).xls',1,'D3:D4002'); imag_n66=xlsread('H&N_by_force(200-600).xls',1,'E3:E4002'); n66=(real_n66+i*imag_n66); real_l66 = xlsread('L&P_by_moment(200-600).xls',1,'B3:B4002'); imag_l66=xlsread('L&P_by_moment(200-600).xls',1,'C3:C4002'); l66=(real_l66+i*imag_l66); real_p66 = xlsread('L&P_by_moment(200-600).xls',1,'D3:D4002'); imag_p66=xlsread('L&P_by_moment(200-600).xls',1,'E3:E4002'); p66=(real_p66+i*imag_p66); real_h67 = xlsread('H&N_by_force(200-600).xls',1,'F3:F4002'); imag_h67=xlsread('H&N_by_force(200-600).xls',1,'G3:G4002'); h67=(real_h67+i*imag_h67); real_n67= xlsread('H&N_by_force(200-600).xls',1,'H3:H4002'); imag_n67=xlsread('H&N_by_force(200-600).xls',1,'I3:I4002'); n67=(real_n67+i*imag_n67);

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65 real_l67 = xlsread('L&P_by_moment(200-600).xls',1,'F3:F4002'); imag_l67=xlsread('L&P_by_moment(200-600).xls',1,'G3:G4002'); l67=(real_l67+i*imag_l67); real_p67 = xlsread('L&P_by_moment(200-600).xls',1,'H3:H4002'); imag_p67=xlsread('L&P_by_moment(200-600).xls',1,'I3:I4002'); p67=(real_p67+i*imag_p67); n66 = l66; n67 = -l67; h77 = h66; p77 = p66; h76 = h67; p76 = p67; l77 = -l66; n77 = l77; l76 = -l67; n76 = l67; % Define beam d_out = 12.7e-3; % outer diameter, m d_in = 0; % inner diameter, m lengths = 80e-3; % length of subcomponents, m Eout =5.5e11; % modulus of holder, N/m^2 Ein = 5.5e11; % modulus of tool, N/m^2 densityout = 15000; % density of holder, kg/m^3 densityin = 15000; % density of tool, kg/m^3 eta = ones(1, length(d_out))*0.0015;% structural damping Iout = pi/64*(d_out.^4-d_in.^4); % second area moment of inertia for outer diameter, m^4 Iin = pi/64*(d_in.^4); % second area moment of inertia for inner diameter, m^4 EIeq = Eout.*Iout + Ein.*Iin; % composite structural stiffness, N-m^2 EI = EIeq.*(1+i.*eta); % damped structural stiffness, N-m^2 Aout = pi/4*(d_out.^2 d_in.^2); % cross-sectional area of outer diameter, m^2 Ain = pi/4*(d_in.^2); % cross-sectional area of inner diameter, m^2

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66 m = lengths.*(Aout.*densityout + Ain.*densityin);% mass, kg % Timoshenko terms n = 50; % number of finite elements nu_out = 0.22; % Poisson's ratio of outer diameter nu_in = 0.22; % Poisson's ratio of inner diameter Gout = Eout./(2*(1+nu_out)); % shear modulus of outer diameter, N/m^2 Gin = Ein./(2*(1+nu_in)); % shear modulus of inner diameter, N/m^2 A = Aout + Ain; % composite cross-sectional area, m^2 AG = Gout.*Aout + Gin.*Ain; % produce of area and shear modulus For composite beam, N nu = (nu_out.*Aout + nu_in.*Ain)./A; % composite Poisson's ratio I = Iout + Iin; % composite second area moment of inertia, m^4 rg = (I./A).^0.5; % radius of gyration, m mpl = densityout.*Aout + densityin.*Ain; % mass per unit length, kg/m for cnt = 1:length(d_in) if (d_in(cnt) == 0) | ((d_in(cnt) ~= 0) & (Ein(cnt) ~= 0)) % solid cross-section kp(cnt) = 6*(1+nu(cnt))^2/(7+12*nu(cnt)+4*nu(cnt)^2); % shape factor else % hollow cross-section num = 6*(1 + nu_out(cnt))^2*(d_in(cnt)^2 + d_out(cnt)^2)^2; den = 7*d_in(cnt)^4 + 34*d_in(cnt)^2*d_out(cnt)^2 + 7*d_out(cnt)^4 + nu_out(cnt)*(12*d_in(cnt)^4 + 48*d_in(cnt)^2*d_out(cnt)^2 + 12*d_out(cnt)^4) + nu_out(cnt)^2*(4*d_in(cnt)^4 + 16*d_in(cnt)^2*d_out(cnt)^2 + 4*d_out(cnt)^4); kp(cnt) = num/den; end cnt end % Determine free-free receptance of beam % h = x/f % l = x/m % n = theta/f % p = theta/m [h33, l33, n33, p33, h44, l44, n44, p44, h34, l34, n34, p34, h43, l43, n43, p43] = timo_free_free(f, EI(1), lengths(1), AG(1), kp(1), rg(1), mpl(1), n);

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67 [h66, l66, n66, p66, h77, l77, n77, p77, h67, l67, n67, p67, h76, l76, n76, p76] = couple_free_free(f, h66, l66, n66,... p66, h77, l77, n77, p77, h67, l67, n67, p67, h76, l76, n76, p76, h33, l33, n33, p33, h44, l44, n44, p44, h34, l34,... n34, p34, h43, l43, n43, p43); % Rigid support receptances h55 = zeros(size(h33)); l55 = zeros(size(h33)); n55 = zeros(size(h33)); p55 = zeros(size(h33)); %Coupling free-free beam receptances to rigid wall [H11 L11 N11 P11] = Couple_forward2(f,h66, l66, n66, p66, h77, l77, n77, p77, h67, l67,... n67, p67, h76, l76, n76, p76, h55, l55, n55, p55); %Plot figure figure(1) subplot(211) plot(f, real(H11), 'm--') set(gca, 'FontSize', 12) xlim([200 600]) ylabel('Real (m/N)') hold on subplot(212) plot(f, imag(H11), 'm--') set(gca, 'FontSize', 12) xlim([200 600]) xlabel('Frequency (Hz)') ylabel('Imaginary (m/N)') hold on FreeFree Beam Receptances % timo_free_free.m % T. Schmitz (8/20/04) % This program uses n Timoshenko beam elements to determine end receptances for free-free beam. % Input variables are: f, frequency, Hz; EI, structural rigidity including hysteretic damping, N-m^2;% L, beam length, m; % density, kg/m^3; A, cross sectional area, m^2; kp, shear factor; G, shear modulus, N/m^2; % rg, radius of gyration; n, number of elements

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68 function [h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21] = timo_free_free(f, EI, L, AG, kp, rg, mpl, n); l = L/n; % length of each finite element, m phi = 12*EI/(kp*AG*l^2); % shear deformation parameter % Single element matrices for Timoshenko beam % Mass matrix Mt = mpl*l/(1+phi)^2*[(13/35+7*phi/10+phi^2/3) (11/210+11*phi/120+phi^2/24)*l (9/70+3*phi/10+phi^2/6) (13/420+3*phi/40+phi^2/24)*l; (11/210+11*phi/120+phi^2/24)*l (1/105+phi/60+phi^2/120)*l^2 (13/420+3*phi/40+phi^2/24)*l -(1/140+phi/60+phi^2/120)*l^2; (9/70+3*phi/10+phi^2/6) (13/420+3*phi/40+phi^2/24)*l (13/35+7*phi/10+phi^2/3) -(11/210+11*phi/120+phi^2/24)*l; -(13/420+3*phi/40+phi^2/24)*l -(1/140+phi/60+phi^2/120)*l^2 -(11/210+11*phi/120+phi^2/24)*l (1/105+phi/60+phi^2/120)*l^2]; Mr = mpl*l/(1+phi)^2*(rg/l)^2*[6/5 (1/10-phi/2)*l -6/5 (1/10phi/2)*l; (1/10-phi/2)*l (2/15+phi/6+phi^2/3)*l^2 -(1/10-phi/2)*l (1/30+phi/6-phi^2/6)*l^2; -6/5 -(1/10-phi/2)*l 6/5 -(1/10-phi/2)*l; (1/10-phi/2)*l -(1/30+phi/6-phi^2/6)*l^2 -(1/10-phi/2)*l (2/15+phi/6+phi^2/3)*l^2]; M = Mt + Mr; % Stiffness matrix Kb = EI/(l^3*(1+phi)^2)*[12 6*l -12 6*l; 6*l (4+2*phi+phi^2)*l^2 -6*l (2-2*phi-phi^2)*l^2; -12 -6*l 12 -6*l; 6*l (2-2*phi-phi^2)*l^2 -6*l (4+2*phi+phi^2)*l^2]; Ks = kp*AG*phi^2/(4*l*(1+phi)^2)*[4 2*l -4 2*l; 2*l l^2 -2*l l^2; -4 -2*l 4 -2*l; 2*l l^2 -2*l l^2]; K = Kb + Ks; Mtemp2 = M; Ktemp2 = K; % Build full mass and stiffness matrices for cnt = 2:n % Concatenate left element matrices with required zeros right = zeros(cnt*2, 2); bottom = zeros(2, (cnt+1)*2); % Mass matrix

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69 temp = cat(2, M, right); Mtemp1 = cat(1, temp, bottom); % Stiffness matrix temp = cat(2, K, right); Ktemp1 = cat(1, temp, bottom); % Concatenate right element matrices with required zeros left = zeros(cnt*2, 2); top = zeros(2, (cnt+1)*2); % Mass matrix temp = cat(2, left, Mtemp2); Mtemp2 = cat(1, top, temp); % Stiffness matrix temp = cat(2, left, Ktemp2); Ktemp2 = cat(1, top, temp); % Add two matrices M = Mtemp1 + Mtemp2; K = Ktemp1 + Ktemp2; end % Calculate required direct and cross receptances for ends of beam for cnt = 1:length(f) w = f(cnt)*2*pi; % frequency, rad/s D = inv(-M*w^2 + K); % dynamic matrix h11(cnt) = D(1,1); l11(cnt) = -D(1,2); n11(cnt) = -D(2,1); p11(cnt) = D(2,2); h12(cnt) = D(1,2*(n+1)-1); l12(cnt) = -D(1,2*(n+1)); n12(cnt) = -D(2,2*(n+1)-1); p12(cnt) = D(2,2*(n+1)); h21(cnt) = D(2*(n+1)-1,1); l21(cnt) = -D(2*(n+1)-1,2); n21(cnt) = -D(2*(n+1),1); p21(cnt) = D(2*(n+1),2); h22(cnt) = D(2*(n+1)-1,2*(n+1)-1); l22(cnt) = -D(2*(n+1)-1,2*(n+1)); n22(cnt) = -D(2*(n+1),2*(n+1)-1);

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70 p22(cnt) = D(2*(n+1),2*(n+1)); clear D; end Coupling Free Free Beam Receptances % T. Schmitz and Gregory Duncan % Couple_free_free.m % Function used to couple two free-free beams. function [h11, l11, n11, p11, h44, l44, n44, p44, h14, l14, n14, p14, h41, l41, n41, p41] = couple_free_free(f, h11, l11, n11, p11,... h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21, h33, l33, n33, p33, h44, l44, n44, p44, h34, l34, n34, p34,... h43, l43, n43, p43); [N col] = size(f); h11=reshape(h11,1,1,N); l11=reshape(l11,1,1,N); n11=reshape(n11,1,1,N); p11=reshape(p11,1,1,N); h12=reshape(h12,1,1,N); l12=reshape(l12,1,1,N); n12=reshape(n12,1,1,N); p12=reshape(p12,1,1,N); h21=reshape(h21,1,1,N); l21=reshape(l21,1,1,N); n21=reshape(n21,1,1,N); p21=reshape(p21,1,1,N); h22=reshape(h22,1,1,N); l22=reshape(l22,1,1,N); n22=reshape(n22,1,1,N); p22=reshape(p22,1,1,N); h33=reshape(h33,1,1,N); l33=reshape(l33,1,1,N); n33=reshape(n33,1,1,N); p33=reshape(p33,1,1,N);

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71 h34=reshape(h34,1,1,N); l34=reshape(l34,1,1,N); n34=reshape(n34,1,1,N); p34=reshape(p34,1,1,N); h43=reshape(h43,1,1,N); l43=reshape(l43,1,1,N); n43=reshape(n43,1,1,N); p43=reshape(p43,1,1,N); h44=reshape(h44,1,1,N); l44=reshape(l44,1,1,N); n44=reshape(n44,1,1,N); p44=reshape(p44,1,1,N); RS11 = [h11 l11; n11 p11]; RS12 = [h12 l12; n12 p12]; RS21 = [h21 l21; n21 p21]; RS22 = [h22 l22; n22 p22]; RS33 = [h33 l33; n33 p33]; RS34 = [h34 l34; n34 p34]; RS43 = [h43 l43; n43 p43]; RS44 = [h44 l44; n44 p44]; R11=zeros(2,2,N); R41=zeros(2,2,N); R14=zeros(2,2,N); R44=zeros(2,2,N); for n=1:N R11(:,:,n) = RS11(:,:,n) RS12(:,:,n)*((RS22(:,:,n)+RS33(:,:,n))\RS21(:,:,n)); R41(:,:,n) = RS43(:,:,n)*((RS22(:,:,n)+RS33(:,:,n))\RS21(:,:,n)); R14(:,:,n) = RS12(:,:,n)*((RS22(:,:,n)+RS33(:,:,n))\RS34(:,:,n)); R44(:,:,n) = RS44(:,:,n) RS43(:,:,n)*((RS22(:,:,n)+RS33(:,:,n))\RS34(:,:,n)); end h11 = R11(1,1,:); l11 = R11(1,2,:); n11 = R11(2,1,:); p11 = R11(2,2,:); h14 = R14(1,1,:);

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72 l14 = R14(1,2,:); n14 = R14(2,1,:); p14 = R14(2,2,:); h41 = R41(1,1,:); l41 = R41(1,2,:); n41 = R41(2,1,:); p41 = R41(2,2,:); h44 = R44(1,1,:); l44 = R44(1,2,:); n44 = R44(2,1,:); p44 = R44(2,2,:); h11=reshape(h11,1,N); l11=reshape(l11,1,N); n11=reshape(n11,1,N); p11=reshape(p11,1,N); h14=reshape(h14,1,N); l14=reshape(l14,1,N); n14=reshape(n14,1,N); p14=reshape(p14,1,N); h41=reshape(h41,1,N); l41=reshape(l41,1,N); n41=reshape(n41,1,N); p41=reshape(p41,1,N); h44=reshape(h44,1,N); l44=reshape(l44,1,N); n44=reshape(n44,1,N); p44=reshape(p44,1,N); Coupling Free Free Section t o a Cantilevered Section % Couple_forward2.m % Scott Duncan % Program to couple in a forward direction based on a free-free section attached to a cantilevered section, finding output at end of attached % free-free Coordinate system 3 -2 1 % Inputs (f, h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21)

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73 % Outputs ( H11, L11, N11, P11) function [H11, L11, N11, P11] = Couple_forward2(f, h11, l11, n11, p11, h22, l22, n22, p22, h12, l12, n12, p12, h21, l21, n21, p21, h33, l33, n33, p33) N=length(f); %%reshape vectors to assemble 3D matrices h11=reshape(h11,1,1,N); l11=reshape(l11,1,1,N); n11=reshape(n11,1,1,N); p11=reshape(p11,1,1,N); h22=reshape(h22,1,1,N); l22=reshape(l22,1,1,N); n22=reshape(n22,1,1,N); p22=reshape(p22,1,1,N); h12=reshape(h12,1,1,N); l12=reshape(l12,1,1,N); n12=reshape(n12,1,1,N); p12=reshape(p12,1,1,N); h21=reshape(h21,1,1,N); l21=reshape(l21,1,1,N); n21=reshape(n21,1,1,N); p21=reshape(p21,1,1,N); h33=reshape(h33,1,1,N); l33=reshape(l33,1,1,N); n33=reshape(n33,1,1,N); p33=reshape(p33,1,1,N); %%Assemble matrices size 2X2XN R21 = [h21 l21; n21 p21]; R12 = [h12 l12; n12 p12]; R22 = [h22 l22; n22 p22]; R11 = [h11 l11; n11 p11]; R33 = [h33 l33; n33 p33]; %%predefine G11f matrix for for loop G11f=zeros(2,2,N); for n=1:N G11(:,:,n) = R11(:,:,n) R12(:,:,n)*((R22(:,:,n)+R33(:,:,n))\R21(:,:,n));

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74 end H11 = G11(1,1,:); L11 = G11(1,2,:); N11 = G11(2,1,:); P11 = G11(2,2,:); H11=reshape(H11,1,N); L11=reshape(L11,1,N); N11=reshape(N11,1,N); P11=reshape(P11,1,N);

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75 LIST OF REFERENCES 1. J. Tlusty, Manufacturing Processes and Equipment, Prentice Hall, Upper S addle River, NJ 1999. 2. Y. Altintas, Manufacturing Automation, Cambridge University Press, UK, 2000. 3. T. L. Schmitz, K.S. Smith, Machining Dynamics: Frequency Response to Improved Productivity, Springer, NY 2009. 4. T. Schmitz R.R. Donaldson, P redicting highs peed m achining dynamics by s ubstructure a nalysis Annals of the CIRP 49 ( 1) (2000) 303308. 5. L. Kops, D. Vo, Determination of the e quivalent diameter of an e nd m ill based on its c ompliance Annals of the CIRP 39 ( 1) (1990) 93 96. 6. T. Schmitz G.S. Duncan, T hree c omponent r eceptance c oupling s ubstructure a nalysis for t ool point dynamics prediction Journal of Manufa cturing Science and Engineering 127 ( 4) (2005) 781 790. 7. T. Schmitz, K. Powell, D. Won, G.S. Duncan, W.G. Sawyer, J. Ziegert Shrink f it t ool holde r c onnection s tiffness/ damping m odeling for f requency r esponse p rediction in m illing International Journal of Machine Tools and Manufacture 47 (9) (2007) 1368 1380. 8. S. Smith, J. Tlusty, An overview of m odeling and s imulation of the m illing process, Journa l of Engineering for Industry, Transactions of the ASME 113 (1991) 169175. 9. R. E. Devor, W.A. Kline, W.J. Zdeblick, A m echanistic m odel for the force s ystem in e nd m illing with a pplication to m achining a irframe s tructures Transactions of the NAMRI/SME 18 (1980) 297303. 10. J. Tlusty, W. Zaton, F. Ismail Stability l obes in m illing Annals of the CIRP 32 ( 1) (1983) 309313. 11. Y. Altintas, E. Budak, Analytical p redictio n of s tability l obes in m illing, Annals of the CIRP 44 ( 1) (1995) 357362. 12. T. Schmitz, B. Mann, Closed f orm s olutions for s ur face l ocation e rror in m illing, International Journal of Machine Tools and Manufacture 46 (2006) 13691377. 13. T. Insperger, G Stpn, Semi discretization m ethod for delayed s ystems, International Journal for N umerical Methods i n Engineering 55 ( 5) (2002) 503518. 14. I. Deiab, S. V eldhuis, M. Dumitrescu, Dynamic m odeling of f ace m illing process i ncluding the e ffect of f ixture dynamics Transact ions of the NAMRI/SME V 30 (2002) 461468.

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76 15. B. Mann, P. Bayly, M. Davies, J. Halley, Limit c ycles, b ifurcations, and a ccuracy of the m illing process, Journal of Sound and Vibration 227 (2004) 3148. 16. R. E. D. Bishop, D.C. Johnson, The Mechanics of Vibration, Cambridge University Press, Cambridge, UK 1960. 17. S. Park, Y. Altintas, M. Movahhedy, Re ceptance c oupling for e nd m ills, International Journal of Machine Tools and Manufacture 43 (2003) 889896. 18. E. B. Kivanc E. Budak, Structural m odeling of e nd m ills for f orm e rror and s tability a nalysis International Journal of Machine Tools and Manufacture 44 (2004) 11511161. 19. A. Ertrk, H.N. zgven, E. Budak, Analytical m odeling of s pindle t ool dynamics on m achine t ools using t imoshenko beam m odel and r eceptance c oupling for the prediction of t ool point FRF International Journal of Machine Tools and Manuf acture 46 ( 15) (2006) 1901 1912. 20. S. Filiz, C. H. Cheng, K. Powell, T. Schmitz, O.B. Ozdoganlar An i mproved t ool holder m odel for RCSA t ool point f requency r esponse p rediction Precision Engineering 33 ( 1) (2009) 2636. 21. W. Weaver Jr. S. Timoshenko, D. Young, Vibration Problems in Engineering, 5th Ed., John Wiley and Sons, New York, NY, Section 5.12. 22. S. Sattinger, A m ethod for e xperimentally determining r otational m obilities of s tructures, Shock and Vibration Bulletin 50 (1980) 1727. 23. D.J. Ewins, Modal Testing: Theory, Practice and Applications, 2nd Edition, Research Studies Press, 2000. 24. G. S. Duncan, Milling d ynamics prediction and uncertainty a nalysis using r eceptance c oupling s ubstructure a nalysis, PhD Dissertation, University of Florida, 2006.

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77 BIOGRAPHICAL SKETCH Uttara Vijay Kumar was born and raised in New Delhi, the capital city of India. She received her Bachelor of Technology degree in m echanical and a utomation e ngineering from Indira Gandhi Institute of Technology, a constitue nt college of Guru Gobind Singh Indraprastha University, Delhi in May 2007. In f all 2007 she began her graduate studies at the Department of Mechanical and Aerospace Engineering University of Florida, in pursuit of her M S degree in m echanical e ngineering. In s pring 2008, she joined the Machine Tool Research Center under the guidance of Dr. Tony L. Schmitz