Mechanics and Dynamics of the Tool Holder - Spindle Interface

MECHANICS AND DYNAMICS OF THE TOOL HOLDER - SPINDLE INTERFACE

MEHDINAMAZI

B.Sc, Sharif University of Technology

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

THE FACULTY OF GRADUATE STUDIES

(Mechanical Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA

April 2006

This thesis presents a general method for identifying and modeling the tool holder- spindle interface in machine tools, using an experimental technique and the finite element method.

The spindle assembly is one of the weakest parts in the machine tool and contributes to the chatter vibrations. The unwanted vibrations lead to a poor surface finish and can damage the tool, tool holder and spindle bearings. The tool holder-spindle interface is the connection closest to the cutting, and its dynamics can affect the stability of the cutting process and the dimensional accuracy of the work-piece. In this thesis, Timoshenko beam elements are used to model the tool holder, and an experimental setup is used to identify the contact stiffness of the interface for CAT and the HSK tapers. The finite-element models of the tool holder and the spindle are coupled through a receptance coupling model. The effect of the drawbar force is investigated as the main factor affecting the dynamics of the interface. It is shown that with an increase in the drawbar force, the dynamic stiffness of the connection between the holder and spindle taper decreases and saturates after a certain force level. The dynamics of various tool holder types is also investigated in the setup as a guideline to select tool holders for low- speed and high-speed milling operations.

This thesis also presents the coupling of tool holder dynamics identified through the finite element method with the experimentally identified spindle. The structural dynamics of the spindle with a tool-holder taper is identified experimentally through an inverse receptance coupling technique. The tool holder stick-out and tool are assumed to be a lightly damped linear structure, and its analytically predicted dynamics is coupled to the spindle with the aid of a receptance coupling method. This approach greatly reduces the number of impact modal tests needed to identify the dynamics of the machine at the tool tip after each tool change. The dynamics of the machine tool and the properties of the work-piece material are used to calculate chatter stability lobes. The proposed method is applied on a horizontal machining center and verified experimentally.

ii Table of Contents

Abstract ii

Table of Contents iii

List of Tables v

List of Figures vii

Acknowledgement xi

Chapter 1 Introduction 1

Chapter 2 Literature Review 5

2.1 Modeling the Tool Holder-Spindle Interface 5 2.2 Chatter Vibrations 6 2.3 Substructure Coupling and Joint Identification 7

Chapter 3 Modeling and Identification of Tool Holder-Spindle Interface Dynamics.... 12

3.1 Overview 12 3.2 Finite-Element Model of Tapered Connections 12 3.3 Computational Model of Contact Stiffness in Tapered Connections 14 3.4 Contact Stiffness Derivation through Experimental Identification 17 3.5 Finite Element Model of the Experimental Setup 19 3.5.1 Finite Element Model of the Simulated Spindle 19 3.5.2 Finite Element Model of Tool Holder 20 3.6 Contact Stiffness Identification Results 21 3.7 Simulation Results on the Test Setup 26 3.8 Simulation Results on the Machine Tool 27 3.9 Contact Stiffness Derivation Using Contact Elements 32 3.10 Summary 39

Chapter 4 Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies.... 40

iii 4.1 Overview 40 4.2 Modal Analysis of Tool Holder-Tool Assembly on a Simulated Spindle 41 4.2.1 Modal Analysis on a Short Overhang 42 4.2.2 Modal Analysis on a Long Overhang 44 4.3 Tool Holder Selection and Tool-Tool Holder Joint Stiffness and Damping 46 4.4 The Effect of Drawbar Force Variation on the Dynamics of the Interface 50 4.5 Case Study: Modal Analysis of a Collet Tool Holder on the Machine Tool 56 4.6 Summary 60

Chapter 5 Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 61

5.1 Overview 61 5.2 Receptance Coupling 62 5.3 Inverse Receptance Coupling 64 5.4 Inverse Receptance Coupling - Experimental Procedure 67 5.5 Receptance Coupling of Arbitrary Tool-Tool Holder Assembly to the Spindle 68 5.6 Simulation and Experimental Results 69 5.7 Tool Length Tuning 77 5.7.1 Chatter-Free Machining 78 5.7.2 Optimization for Maximum Productivity 79 5.7.3 Optimization Results 81 5.8 Summary 85

Chapter 6 Conclusions 86

6.1 Conclusions 86 6.2 Future Research Directions 87

Bibliography 88

Appendix A Timoshenko Beam Element Formulations 91

A.l Beam Element Formulations 91

Appendix B Tables of Modal Parameters 95

Appendix C Inverse Receptance Coupling Solution 103

iv List of Tables

Table 3.1: Natural Frequencies of a Free-Free CAT 50 Shrink-Fit Holder 20 Table 3.2: Contact Stiffness per Unit Area for CAT 50 Taper Determined by Experimental Identification - 105 mm Overhang Length from the Spindle Face 21 Table 3.3: Contact Stiffness per Unit Area for CAT 40 Taper Determined by Experimental Identification-95 mm Overhang Length from Spindle Face 23 Table 3.4: Contact Stiffness per Unit Area for HSK A63 Taper Determined by Experimental Identification -87 mm Overhang Length from Spindle Face 25 Table 3.5: Comparison of the Contact Stiffness per unit Area for HSK A63 and CAT 40 Taper - 10 kN Drawbar Force 25 Table 3.6: Radial and Rotational Spring Stiffness of CAT 40 Tool Holder - Spindle 29 Table 3.7: Contact Stiffness Constants for CAT 50 Taper Using Contact Elements 36 Table 3.8: Contact Stiffness Constants for CAT 50 Taper for Different Overhang Lengths with 20 kN Drawbar Force 37 Table 4.1: Modal Parameters at 20 kN Drawbar Force for 64 mm Overhang and 16mm Diameter -1st Mode 46 Table 4.2: Static Stiffness and Dynamic Stiffness of the Four Types of Tool Holders - Simulation Results vs. Experimental Results 48 Table 4.3: Percentage Error in Modal Stiffness - Rigid Tool - Tool Holder Connection 49 Table 4.4: Normalized Dynamic Stiffness and Modal Damping of 4 Types of Tool Holders with 64 mm Overhang 50 Table 5.1: Constraints on Tool Overhang and Spindle Speed 83 Table 5.2: Cutting Conditions 83 Table B. 1: Modal Parameters vs. Drawbar Force for Collet Chuck - 10 mm Overhang 95 Table B.2: Modal parameters vs. Drawbar Force for Power Chuck-10 mm Overhang 96 Table B.3: Modal Parameters vs. Drawbar Force for Hydraulic Chuck - 10 mm Overhang 97 Table B.4: Modal Parameters vs. Drawbar Force for Shrink-Fit - 10 mm Overhang 98 Table B.5: Modal Parameters vs. Drawbar Force for Collet Chuck - 64 mm Overhang 99 Table B.6: Modal Parameters vs. Drawbar Force for Milling Chuck - 64 mm Overhang... 100 Table B.7: Modal Parameters vs. Drawbar Force for Hydraulic Chuck - 64 mm OverhanglOl

v Table B.8: Modal Parameters vs. Drawbar Force for Shrink-Fit - 64 mm Overhang

vi List of Figures

Figure 1.1: Spindle Assembly and Tool Holder-Spindle Interfaces 1 Figure 1.2: CAT Spindle Taper (Left) and the HSK Taper (Right) 2 Figure 1.3: Schematic of Angular Mismatch between Spindle Taper and CAT Tool- Holder Tapers 3 Figure 2.1: Chatter Vibration Mechanism in Turning [2] 6 Figure 2.2: Schmitz's Tool Holder - Spindle Assembly Model[32] 8 Figure 2.3: Tool-Tool Holder Joint Model - Rotational and Linear Joint Elements (Model A) versus Two Linear Joint Elements (Model B)[25] 9 Figure 3.1: Schematic of a Tapered Tool Holder inside the Spindle Taper 12 Figure 3.2: Timoshenko Beam Element Model of the Tool Holder-Spindle Interface with Distributed Contact Springs 13 Figure 3.3: Equivalent Rotational Springs 14 Figure 3.4 :Timoshenko Beam Element Model of Tool Holder-Spindle Connection with Distributed Rotational and Radial Springs 15 Figure 3.5: Experimental Setup for Dynamic Analysis of Spindle-Tool Holder Interface.... 18 Figure 3.6: Flowchart for Deriving the Stiffness Constants through Experimental Identification 19 Figure 3.7 : Equivalent Circular Cross-Section of Spindle Block 20 Figure 3.8: Experimental and Finite Element Model of the Free-Free Tool holder 20 Figure 3.9: Frequency Response Function of the CAT 50 Taper in Experimental Setup - 20 kN Drawbar Force 22 Figure 3.10: Frequency Response Function of the CAT 40 Taper in Experimental Setup - 10 kN Drawbar Force 23 Figure 3.11: HSK A63 Shrink-Fit Tool Holder 24 Figure 3.12 : Spindle Block - HSK A63 Tool Holder Assembly 24 Figure 3.13: Frequency Response Function of the CAT 50 Shrink-Fit Holder with a 64 mm Overhang Blank Tool in Experimental Setup - 20 kN Drawbar Force 26 Figure 3.14 : Frequency Response Function of the CAT 50 Shrink-Fit Holder with a 15 mm Overhang Blank Tool in Experimental Setup - 20 kN Drawbar Force 27

vn Figure 3.15 : Finite Element Model of Spindle System on Machine 28 Figure 3.16 : Finite Element Model of Tool Holder in Spindle Taper with Connection Springs 28 Figure 3.17: Frequency Response Function at the Tool Tip 30 Figure 3.18 : Mode Shapes Contributed by Tool - Tool Holder Assembly 31 Figure 3.19 : Schematic of Bending Deformations of the Tool Holder - Spindle Interface.. 32 Figure 3.20: 3D Finite Element Model of Tool Holder in Spindle Taper with Boundary Conditions - Model A 34 Figure 3.21: 3D Finite Element Model of Tool Holder with Rigid Spindle-Tool Holder Connection - Model C 34 Figure 3.22: The Effect of Drawbar Force on the Static Stiffness of a CAT 50 Tapered Connection - Simulation Results 35 Figure 3.23: Finite Element Model of Tool Holder in Spindle Taper using Timoshenko Beam Elements 36 Figure 3.24 : Flowchart for Obtaining the Stiffness Constants by Using Contact Elements.. 37 Figure 3.25: Frequency Response Function of the CAT 50 Taper in Experimental Setup - 20 Drawbar Force 38 Figure 4.1: Common Types of Tool Holders 41 Figure 4.2 : FRF Comparison of Different Tool Holders with Short 10 mm Overhang 42 Figure 4.3: 1st Mode of the Collet Chuck Tool Holder at 3508 Hz with 10mm Overhang.... 43 Figure 4.4: FRF Comparison of Different Tool Holders with Long 64 mm Overhang 44

Figure 4.5: 1st and 2nd Modes of the Collet Chuck Tool Holder with Long Blank Tool 45 Figure 4.6 : Dimensions of the Four Types of Tool Holders 47 Figure 4.7 : Finite Element Model of the Four Types of Tool Holders - With 16mm Diameter Tool and 64 mm Overhang 48 Figure 4.8 : The Effect of Drawbar Force on the Natural Frequency, Dynamic Stiffness and Damping of the Collet Chuck with 10 mm Blank Tool 52 Figure 4.9 : The Effect of Drawbar Force on the Natural Frequency, Dynamic Stiffness and Modal Damping of the 1st Mode of the Collet Chuck -64 mm Tool Overhang 54 Figure 4.10 : The Effect of Drawbar Force on the Natural Frequency, Dynamic Stiffness and Modal Damping of the 2nd Mode of the Collet Chuck-64 mm Overhang 55

Vlll Figure 4.11 : The Modal Analysis of the Collet Chuck on the Mori Seiki SH403 56 Figure 4.12 : Measured Transfer Function at the Tool Tip in the Y axis of the Machine 57 Figure 4.13: Stability Lobes 57 Figure 4.14 : The Dominant Mode Shapes of the Collet Chuck Tool Holder on the Machine Tool - Tool Diameter 20 mm , Tool Overhang 50 mm 58 Figure 4.15: FRF Comparison of Different HSK Tool Holders with 50 mm Overhang on Simulated Spindle Block 59 Figure 4.16: FRF Comparison of Different HSK Tool Holders with 50 mm Overhang on the Machine Tool 60 Figure 5.1: Spindle on Machine Tool with Tool holder Taper only 61 Figure 5.2: Receptance Coupling for Obtaining Frequency Response Function at the Tool Tip 62 Figure 5.3: Experimental Procedure for Decoupling and Identifying Spindle Dynamics 67 Figure 5.4: Inverse Receptance Coupling for Obtaining Spindle Dynamics 68 Figure 5.5: Finite Element Model of Tool - Tool Holder Shank 69 Figure 5.6: Shrink-Fit Holder to Identify Spindle Dynamics at the Flange on the Mori Sieki SH403 Horizontal Machining Center 69 Figure 5.7: Tool - Tool Holder Assembly - Tooling A 70 Figure 5.8: Measured and Predicted Frequency Response Functions in X Direction at the Tool Tip on the Moriseiki SH403 71 Figure 5.9: Measured and Predicted Frequency Response Functions in Y Direction at the Tool Tip on the Moriseiki SH403 72 Figure 5.10: Stability Lobes Diagram for Tooling A 73 Figure 5.11: Tool - Tool holder Assembly - Tooling B 74 Figure 5.12 : Measured and Predicted Frequency Response Functions in X Direction at the Tool Tip on the Moriseiki SH403 75 Figure 5.13 : Measured and Predicted Frequency Response Functions in Y Direction 76 Figure 5.14 : Stability Lobes Diagram - Tooling B 77 Figure 5.15 : Typical Stability Lobe for Milling 79 Figure 5.16 : Typical Stability Lobe for Milling 80 Figure 5.17 : Tool Tuning Optimization Flowchart 81

ix Figure 5.18 : Shrink-fit Tooling with 2 Fluted ,10mm Slender End mill for Machining Helicopter Gearbox 82 Figure 5.19 : Comparison between Predicted and Experimental FRFs for 45 mm Overhang 82 Figure 5.20 : Comparison between Predicted Frequency Response Functions 84 Figure 5.21 : Comparison between Tuned Tool and Short Tool Stability Lobes 84 Figure A.l: Timoshenko Beam Element 91 Acknowledgement

I would like to extend my deepest appreciation to my supervisor, Dr. Yusuf Altintas, a considerate and caring professor. His support and guidance have motivated me throughout my masters program at UBC. His great depth of knowledge helped me throughout my research, and his consideration and kindness were a source of strength during the hardest times.

I would also like to thank Dr. Nimal Rajapakse, the department head and my co- supervisor. I greatly appreciate his help during his busy schedule.

I would like to extend my gratitude to Mitsubishi Materials and their research engineer, Mr. Taro Abe who provided the setup and helped greatly in the experiments.

I also want to thank my friends at the Manufacturing Automation Laboratory (MAL) for their friendship and help. They have made this place like a home to me. In addition I would like to thank Yuzhong Cao, our knowledgeable senior PhD candidate.

I would also like thank my parents and brother, who supported me greatly over the course of my research and studies.

xi Chapter 1 Introduction

The recent trend in manufacturing has been to machine monolithic parts from blocks, bars or billets. This practice has greatly reduced assembly costs and the stress concentration effects of bolting and connecting parts. With the advent of high-speed machining, monolithic parts are machined by removing as much material as possible in the shortest time. However, operating the machine at high spindle speeds and significant depths of cut without chatter vibrations has been the challenge faced by the industry.

With major advances in the design of machine tools and cutters, the weakest part in the machine tool is the spindle-tool holder assembly. The assembly consists of a rotating spindle shaft, tool holder and drawbar mechanism, as shown in Figure 1.1.

Figure 1.1: Spindle Assembly and Tool Holder-Spindle Interfaces

Chatter stability depends on the frequency response function (FRF) of the spindle structure at the tool tip and the work-material cutting coefficients. Tool holders and the interfaces are known to be one of the relatively flexible parts in machining centers that dominate the FRF of the machine tool, and are the main source of chatter vibrations in milling operations.

1 Chapter I. Introduction 2

The tool holder is held in the spindle by the drawbar force, which is maintained through a stack of springs, and the cutting tool is gripped by various clamping mechanisms. The tool- holder taper, which acts as the adapter between the machine tool and the cutting tool, enables automatic tool change. The two most common types of tapers on machining centers are the steep 7/24, or CAT taper, and the modern hollow HSK (1/10 taper) as shown in Figure 1.2.

Figure 1.2: CAT Spindle Taper (Left) and the HSK Taper (Right)

The most common size CAT tool holders are the CAT 10, 20, 30, 40, 50, and 60. All these tool holders have the same taper angle (7/24), but differ in diameter at the gauge line. The gauge line is determined by the diameter of the tool holder at the large end of the taper. For example, the gauge diameter for a CAT 40 tool-holder is 44.45 mm. Standard HSK tool holders are the HSK No. 32, 40, 50, 63, 80, 100, 125 and 160. The taper angle on all the HSK tool holders is 1/10, but they differ in gauge diameter, e.g., the gauge diameter of the HSK 63 taper is 49.3 mm. Thus the HSK taper equivalent of the CAT 40 taper is the HSK63 since the two tool holders have relatively close gauge diameters, and similar power and torque transmission capabilities.

The dynamics of the tool holder - spindle interface depends on the drawbar force, taper tolerance, surface finish of the tapered parts and taper geometry. The drawbar force maintains the necessary contact pressure on the taper and prevents slippage against high cutting torques. The taper tolerance, the angular mismatch between the spindle taper and the tool-holder taper, has a direct impact on the stiffness of the connection. The tolerance level is rated on a scale of AT 1 to AT 9, with AT 1 being the most accurate [16]. Figure 1.3 shows the two possible cases: On the left, the tool holder taper has a steeper angle than the spindle taper, and on the right, the spindle taper is steeper than the tool-holder taper. Chapter 1. Introduction 3

Toolholder

Figure 1.3: Schematic of Angular Mismatch between Spindle Taper and CAT Tool- Holder Tapers

The shaded area in Figure 1.3 shows where contact would occur between the two surfaces in each case. When contact occurs at the front end of the taper, the stiffness of the connection is not significantly affected. However, at high loading or during milling, the front-end vibrations lead to corrosions known as fretting, which increases the run-out or unbalance at the tool tip. When the contact area is at the smaller diameter zone, there would be no contact between the two surfaces at the front end, unless the drawbar force is increased. This leads to an increased overhang and thus decreases the bending stiffness at the tool tip [30].

The taper design is an important factor in the inherent dynamic stiffness of the connection. In the CAT taper, only one contact surface, i.e., the tapered surface, exists between the spindle and the tool holder. The simultaneous contact on the face and tapered surfaces in the HSK hollow taper increases the rotational stiffness of the connection as well as improving the axial positioning accuracy of the tooling. The face contact prevents the holder from being pulled back as the spindle opens up at high speeds due to the centrifugal forces. In this research, the effect of the drawbar force on the dynamics of the tool holder- spindle connection is investigated.

In this thesis, the spindle assembly is designed and analyzed in a virtual environment at the design stage with the aid of the available finite element method, which considers the nonlinearity of the bearings. To be able to predict the frequency response function at the tool tip and the static stiffness of the whole assembly, the tool holder-spindle connection has to be taken into account. By modeling the connection, the tool holder can be coupled to the Chapter 1. Introduction 4 spindle, and virtual cutting tests can be performed on the assembly prior to the actual production of the spindle.

There are several dominant types of tool holders used in industry, with different dynamics. The most common types are the collet, hydraulic chuck, power chuck and the shrink-fit tool holder. Although some factors affecting the tool-holder dynamics have been investigated to some extent through research or experience, comparing the milling performances of different tool holders has not been conducted.

Impact modal testing methods are widely used in industry to measure the frequency response function at the tool tip. Machine tool dynamics vary after each tool change, which requires that impact modal tests on the machine be repeated whenever a new tool is used. However, the time and expertise required to perform impact tests and obtain accurate results are an issue in most manufacturing facilities. Experimental identification of the spindle dynamics by including the tool-holder taper in the spindle will make it possible to mathematically couple arbitrary shrink-fit tool holders to the spindle, and obtain the frequency response function at the tool tip. The objective of this thesis is to study the tool holder-spindle interface, and develop structural modeling and coupling techniques that would allow the dynamics at the tool tip to be predicted without conducting modal tests after each tool change.

The thesis is structured as follows: In Chapter Two, a review of the literature on identification of the tool holder - spindle interface through finite element modeling and experimental methods is presented. Substructure coupling techniques and joint identification methods employed to predict the tool-tip frequency response function are also reviewed. In Chapter Three, a general finite element model for tapered connections is presented. In Chapter Four the effects of the drawbar force are investigated experimentally on the test setup and compared to results of previous reported research. The dynamics of four common types of tool holder are studied using experimental modal analysis, and dynamic stiffness and damping ratios are compared. In Chapter Five, the receptance coupling method is employed to obtain the dynamics of the machine tool reflected at the tool-holder interface. Finally, conclusions and future work are discussed. Chapter 2 Literature Review

2.1 Modeling the Tool Holder-Spindle Interface

Rivin [29] extensively assessed the state of the art in the tooling structures' technology. He discussed six important subjects related to tooling: (1) the influence of machining parameters on tool life and stability; (2) stiffness and damping of tools; (3) tool - tool holder interfaces; (4) modular tooling; (5) tool - machine interfaces and (6) tool balancing for high speed machines. His study revealed the importance of the tooling structure for modern machining systems and documented the research and development efforts in this field over the past two decades.

Reshetov and Levina [21], [30] studied the effects of angular deformations in the spindle-tool holder interface on deflection at the tool tip. They investigated the effect of drawbar force and taper tolerance on the static stiffness of the tool holder - spindle connection. They showed that the increased drawbar force results in increased static stiffness of the connection.

Week et al [40] carried out extensive tests on spindles with CAT 50 taper tool holders and HSK tool holders. Their tests showed that the angular deflections at the tool holder- spindle interface increase linearly with the radial load at the tool tip. The increase in drawbar force results in the increased stiffness and this effect is especially more obvious when the connection is inaccurate, and is slight when the taper is precision machined.

Tsutsumi et al [30] carried out extensive finite element modeling of the 7:24 taper. Static characteristics of the taper and the effect of drawbar force on radial and axial stiffness were investigated. The effect of spindle diameter on the axial and radial stiffness of the connection was also considered in the finite element model. They showed that the axial displacement is important, since it determines the axial repeatability and thus the axial accuracy of the tool holder.

Smith et al [33] investigated the effects of drawbar force on metal removal rates in milling. Their approach was similar to the one chosen in this study. They found that

5 Chapter 2. Literature Review 6 increased drawbar force increases the static stiffness of the tool holder-spindle interface, at the expense of reduced damping. Smith et al carried out numerous static and dynamic measurements on spindle tapers with HSK and CAT tapers. However, their finite element model of the interface was not very comprehensive and included a number of trials and guesses to match the natural frequencies. The spindle-tool holder connection in their model consisted of one rotational and one translational spring, and the spring positions were adjusted along the taper to fit the natural frequencies of the model to the experiments on the set-up.

2.2 Chatter Vibrations

Chatter vibrations result from a self-excitation mechanism in the generation of chip thickness during machining operations. A simple turning operation is shown in Figure 2.1. One of the structural modes of the machine tool is initially excited by the cutting forces. A wavy surface finish left during the previous revolution is removed during the succeeding revolution, which also leaves a wavy surface owing to structural vibrations. Depending on the phase shift between the two successive waves, the maximum chip thickness may exponentially grow while oscillating at a chatter frequency that is close to, but not equal to, a dominant structural mode in the system. This phenomenon is known as regenerative chatter [2]. The growing vibrations increase the cutting forces, may chip the tool and produce a poor surface finish.

Feed direction I n

Figure 2.1: Chatter Vibration Mechanism in Turning Chapter 2. Literature Review 1

Extensive research has been conducted on establishing chatter-stability lobes [1],[7],[8],[24],[26],[36],[38]. The regeneration phenomenon was first explained by Tobias [38] and Tlusty [36], who identified the main sources of self-excitation as being associated with the structural dynamics of the machine tool and the feedback between subsequent cuts. Tlusty presented a practical stability law for orthogonal cutting systems, where the chatter- free axial depth of cut is expressed as a function of the real part of the dynamic compliance of the machine tool and work-piece. Later, Merrit [24] used feedback-control theory to develop the stability lobes proposed by Tlusty. These theories are applicable only to orthogonal cutting, where the direction of cutting force, chip thickness, and structural dynamics do not change with time. For 2D milling chatter problems, Tlusty applied his orthogonal cutting stability formulation to milling by using an average directional coefficient and an average number of flutes in cut. Opitz [26] approximated the periodic coefficients with their average values. Instead of using a one-dimensional solution for milling stability like Tlusty and Opitz , Altintas and Budak [1],[7],[8] developed a two-dimensional approach for milling stability problems. The method developed by Altintas is used in this thesis to obtain stability lobes.

Regardless of the different approaches mentioned above, the stability expression requires accurate frequency response function measurements at the tool tip. The measurements are obtained by impact testing at the tool tip, which requires time and expertise. The prediction of the tool tip frequency response function has been investigated by various researchers as explained in Section 2.3.

2.3 Substructure Coupling and Joint Identification

Predicting the dynamic response of an assembly by combining its components has been a subject of interest for a long time. The objective is to assist design engineers in optimizing components without altering the main assembly, and to avoid simulating the whole assembly, which is computationally expensive. In the receptance coupling technique, the frequency response functions of individual components are obtained analytically or experimentally and then analytically assembled to predict the FRF of the whole assembly. In the scope of this research, identifying the machine tool dynamics, including the spindle-tool holder interface, Chapter 2. Literature Review 8 will enable the coupling of shrink-fit holders to the machine tool, and selecting the optimum tooling for a specific cutting operation.

Schmitz et al [32], were first to propose a method for predicting the frequency response function at the tool tip using the receptance coupling technique. Schmitz's tool holder- spindle assembly model is shown in Figure 2.2. Structure A represents the tool holder- spindle and structure B the tool overhang. The two structures are joined through linear and

rotational springs and dampers. The stiffness and damping terms are labeled kx, he, cx, and ce respectively.

Figure 2.2: Schmitz's Tool Holder - Spindle Assembly Model1321

The frequency response function at the tool tip was predicted by coupling the structures using the receptance coupling technique. However, the rotational dynamics of the spindle- tool holder assembly, structure A, were neglected, and the joint stiffness and damping terms were identified through trial and error rather than systematically. Schmitz verified the simulation results by a tool-tuning example, where the tool length was varied and the effect was seen on the critical depth of cut. However, the experimental verifications were performed on very long tool overhangs, with length-to-diameter ratios ranging from 8:1 - 12:1 , where the tool mode is dominant and the effect of the spindle rotational dynamics, which were neglected, are not obvious.

Kivanc and Budak [19] used an approach similar to that of Schmitz [32] in their frequency response prediction, by employing the receptance coupling technique. The complex end-mill geometry was modeled in finite elements and equations were developed to predict the static and dynamic properties of the tools. Although the dynamics of the tool-tip FRF prediction was improved due to accurate modeling of the flute geometry, the rotational Chapter 2. Literature Review 9 dynamics of the spindle-tool holder assembly were neglected, and the joint parameters were identified using the least-squares-error minimization method. The experimental verifications were performed on long overhangs where the tool dynamics were dominant and the effect of the rotational dynamics was therefore not clearly shown.

Movahhedy et al [25] proposed a method for predicting spindle dynamics. Two joint models were proposed to model the tool-tool holder connection as shown in Figure 2.3 . Model A consisted of rotational and linear springs and model B consisted of two linear springs. An optimization method based on a genetic algorithm was employed to find the parameters of the joint model. The rotational dynamics of the spindle-tool holder assembly were taken into account in Movahhedy's model. However, the algorithm converged to the optimum solution within 30 iterations in about 10-20 minutes on a Pentium IV processor.

Model A Model B

Figure 2.3: Tool-Tool Holder Joint Model - Rotational and Linear Joint Elements

(Model A) versus Two Linear Joint Elements (Model B)1251

Park and Altintas [27] proposed an improved receptance coupling technique to identify the dynamics on the spindle. This technique enabled the coupling of the spindle and arbitrary tool dynamics. Park clearly demonstrated that the spindle rotational degrees of freedom (RDOF) dynamics have to be considered in order to obtain accurate frequency response function predictions at the tool tip. However, the difficulty in obtaining accurate RDOF responses has been a major issue in the literature and the subject of numerous studies. The use of laser instruments and angular transducers has been limited given their high cost. Another method used extensively in the literature is the finite difference method, in which Chapter 2. Literature Review 10 the RDOF response is obtained from the translational response of two accelerometers.[5], [13]. However, the presence of noise in the measurements has been an obstacle in employing this method effectively.

Park included the rotational dynamics at the joint, which were indirectly identified using translational responses measured from a set of short and long blank tools. The end mill was modeled in finite elements and subtracted from the spindle-tool holder assembly by employing the inverse receptance coupling technique.

In this thesis the methodology proposed by Park and Altintas is used to identify the spindle dynamics on the tool-holder flange. The dynamics of the spindle, including the rotational dynamics, are identified indirectly by performing three impact measurements on a shrink-fit holder on the machine. This method enables the coupling of the machine to arbitrary shrink-fit tool holders and tools. The end mill and the shrink-fit holder are modeled in finite elements, and the connection between the tool holder and the tool is assumed to be rigid. Some modeling of the different kinds of tool holders such as the hydraulic, collet and the power chuck was performed in order to couple them to the machine tool. However, the difficulties in identifying the joints and modeling these tool holders outweighed their benefits of including them, thus the receptance coupling technique is limited to coupling arbitrary shrink-fit holders and tool assemblies.

Tlusty et al [37] were first to introduce the notion of "tool tuning". The concept was numerically demonstrated by Tlusty, showing that it is possible to dynamically tune long slender end mills for optimal stability at a specified machining condition. Davies et al [12] experimentally demonstrated the tuning of an 11.8 mm diameter slender end mill in cutting aluminum by increasing its length from 104 mm to 118 mm. Smith et al [34] modeled the effect of tool length on the dynamic flexibility in finite elements. The highest spindle speed was set as the constraint and the tool length was adjusted within a permissible range to achieve maximum metal removal rates. Cutting tests were performed to verify the effect of tool overhang on the stability of the milling process, using four sets of tools. The general trend observed in the experiments performed by Smith was that metal removal rates decreased with increased overhang length. However, there were local increases in metal removal rates for certain tool lengths. Chapter 2. Literature Review 11

Experimental and predicted results obtained by Schmitz [14], showed interactions between the tool and the spindle-tool holder assembly similar to the vibration absorber affect. Schmitz suggested two parameters for optimizing the tool length for stable milling: (1) tool- length selection in order to increase the critical depth of cut, and (2) tool-length selection to move a high stable lobe to the top spindle speed of the machine.

The receptance coupling method proposed in this thesis is used to optimize the tool length by shifting a high stable lobe to the top spindle speed of the machine. Chapter 3 Modeling and Identification of Tool Holder-Spindle Interface Dynamics

3.1 Overview

Conical tapered connections are used as the interface between the spindle and the tool holder. The conical connection provides self centering and is also ideal for automatic tool change, since tool holders can be easily assembled and disassembled. The drawbar force holds the tool holder, and is one of the major factors affecting the dynamics and static stiffness of the connection. In this chapter, a finite element model of this type of connection is formulated. This model is applicable to both the static and dynamic analysis of the connection and the effects of the drawbar force are investigated and simulated using this model. The model can also be used to simulate the contact forces on the interface, and model the vibrations that lead to fretting and wear on the tapered surfaces.

3.2 Finite-Element Model of Tapered Connections

The tapered connection is modeled as a beam supported on an elastic foundation. The connection between the tapered surfaces is assumed to be distributed springs in the x and y directions that prevent the tool holder from rotating and translating inside the spindle taper as shown in Figure 3.1

Figure 3.1: Schematic of a Tapered Tool Holder inside the Spindle Taper

12 Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 13

Two-degree-of-freedom Timoshenko beam elements are used to model the tool holder taper and the spindle. Each element has one translational (y) degree of freedom in the radial

direction and one rotational (6>z) degree of freedom, as shown in Figure 3.2.

Figure 3.2: Timoshenko Beam Element Model of the Tool Holder-Spindle Interface with Distributed Contact Springs

To account for the rotational and translational degrees of freedom in the beam elements the distributed springs in the x and y directions are transformed into radial and rotational springs. The springs in the y direction only resist motion in the radial direction. The horizontal springs in the x direction are transformed into rotational springs in the following manner.

When an arbitrary element of length dx along the taper is subjected to a rotation of dOz, as shown in Figure 3.3, the horizontal springs on both sides of the tool holder taper deform

by an amount dxs. Since the rotational deflection of the element, d9z, is small, the distortion

of the springs can be neglected and the deformation of the horizontal springs, dxs, is:

dxs=^-d6z (3.1) where A; is the diameter of the element, as shown in Figure 3.2. Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 14

IT* \ I

\ #A/VvWVWj

Figure 3.3: Equivalent Rotational Springs

Thus, the resisting axial forces acting on the element due to the deformation in the horizontal springs is,

(3.2) dFx=Kx-^-d9z 2

where Kx is the stiffness of the horizontal contact spring.

As seen from Figure 3.3, the resisting axial forces generate a moment about the center of the element as follows:

dM = 2-dF-^ = K-^--dff (3.3)

x 2 x 2 The resisting moment, dM, prevents the element from rotating, thus acting as a rotational spring with the equivalent stiffness, Kg:

Dr (3.4) Ke ~Kx

3.3 Computational Model of Contact Stiffness in Tapered Connections

The connection between the tool-holder taper and the spindle taper is modeled with radial and rotational springs which are distributed on the tapered surface of the tool holder, as shown in Figure 3.4. The stiffness per unit area of the distributed radial and rotational springs are assumed to be constant and are respectively labeled as: the translational contact stiffness

per unit area, Kr, and the rotational contact stiffness per unit area, Kg. Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 15

Figure 3.4 :Timoshenko Beam Element Model of Tool Holder-Spindle Connection with Distributed Rotational and Radial Springs

When an arbitrary element of length dx and surface area dA at axial location Lx along the

tool holder taper is subjected to a radial contact force Fc , the radial deformations, y , are linearly related to the contact force as follows:

Fc=Kr-y (3.5)

Similarly, the rotational deformations, 9, are linearly related to the contact bending moment,

Mc, as follows :

MC=K0-O (3.6)

where Kr and Kg are the stiffness of each spring at a distance x along the contact length. Since the stiffness per unit area of the springs is constant,

Kr=Kr-dA , Ke = Kg • dA (3.7)

and the differential contact area, dA, is

dA = n • Dx • dx (3.8) Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 16

where Dx is the diameter of the taper at axial location Lx along the taper, as shown in Figure 3.4.

By substituting (3.8) into (3.7), the stiffness of the each spring at a distance x along the contact length is:

K=K-n-D-dx (3.9)

Ka =Ka-n-D-dx

By substituting (3.7) into (3.6) and (3.5),

FC={Kr-dA)-y , MC={K0 -dA)-e (3.10)

The potential energy stored in the contact springs at the tool holder spindle interface as a result of radial deformations (y), and rotational deformations (d9), is respectively,

dUy=jFc-y , dUe=jMc-6 (3.11)

By substituting (3.10) into (3.11),

2 2 dUY=YKr-y -dA , dUG^Kg-d -dA (3.12)

Integrating over the whole contact length, the total potential energy stored in the contact springs is expressed as:

U K 2dA 2 yA\- y-y UE=)\-Ke-e dA (3.13) o 2 o 2

where L is the contact length of the tool holder with the spindle taper.

The total potential energy stored in the contact springs is added to the internal virtual work as the result of bending deformations of the tool holder. The internal virtual work for the Timoshenko beam undergoing radial translation (y), and rotational deformation (9) can be expressed as follows: Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 17

(3.14)

o where E is the Young's modulus of elasticity and I is the area moment of inertia of the cross section. G is the modulus of traverse elasticity, k is the cross sectional factor and A is the cross sectional area of the tool holder taper.

Finally, the total internal virtual work of the spindle-tool holder interface is the sum of the three terms obtained in (3.13) and (3.14):

uM=ub+uy+ul e (3.15)

The finite element formulation of the tool holder-spindle interface from the internal virtual work is given in Appendix A.

The radial and rotational stiffness per unit area of the distributed springs, Kr and Kg are identified in sections 3.4 and 3.9,using two methods: (1) experimental identification using dynamic testing and (2) finite element modeling of the tool holder-spindle interface using contact elements. The experimental method is based on a dedicated experimental setup, consisting of a block with the spindle taper and a load cell to adjust the drawbar force. The finite element method is based on modeling the tool holder - spindle interface using contact elements, and is explained in Section 3.9.

3.4 Contact Stiffness Derivation through Experimental Identification

The finite element model in Section 3.2 gives a general relationship between the taper geometry and the contact stiffness of the tool holder-spindle interface. However the actual contact stiffness of the tool holder-spindle interface, which is a function of the drawbar force applied on the tool holder, is determined through experiments.

The experimental setup is shown in Figure 3.5. The setup is designed to investigate the spindle - tool holder interface dynamics without the effects of the spindle bearings and other machine elements that are inevitable on a real machine tool. The large steel block (approximately 40 kg) with the spindle taper has a load cell on the back to measure the drawbar force. The experimental setup is placed on a supporting cushion to simulate free-free Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 18 boundary conditions, and impact modal tests are performed on the setup to obtain the frequency response function at the tool tip. The transfer functions were measured with an impact hammer, an accelerometer, signal conditioner I/O box, a laptop computer with a data acquisition card and MALTF (transfer function measurement software that is a part of CUTPRO® [11]).

Load Cell

Tool Holder

Figure 3.5: Experimental Setup for Dynamic Analysis of Spindle-Tool Holder Interface

The identification procedure to obtain the relationship between the drawbar force and the translational and rotational stiffness per unit area of the contact springs at the tool holder-

spindle interface, Kr and Ko, is shown in the flowchart in Figure 3.6.

Impact hammer tests are performed on a shrink-fit tool holder without any tool in the block. The drawbar force is adjusted through the load cell, and the frequency response function is obtained at the tool holder tip. The finite element model of the setup is constructed using Timoshenko beam elements, as explained later in Section 3.5, and the frequency response function of the finite element model is simulated. By using a nonlinear least-squares curve-fitting algorithm, the two contact stiffness per unit area of the contact

springs, KR and KQ , are determined to match the experimental frequency response function (FRF) to the finite element simulations. Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 19

Select Tool Holder Taper Set Drawbar Force Perform Impact Tests on Setup Obtain FRF

Construct FE Model of Setup, and Tool Holder Taper

Calculate FRF of FE Model

Solve for Kr ,Ke using Nonlinear Least Squares to fit Simulated FRF to Experimental FRF

Figure 3.6: Flowchart for Deriving the Stiffness Constants through Experimental Identification

3.5 Finite Element Model of the Experimental Setup

In this section, the experimental setup is modeled using the finite element method. Timoshenko beam elements are used to include shear and rotational effects, and the tool holder - spindle interface is modeled using the connection springs, as explained above in Sections 3.2 and 3.3 .

3.5.1 Finite Element Model of the Simulated Spindle

The spindle block is also modeled using Timoshenko beam elements. The equivalent circular cross-section, which has the same area and second moment of area as the cross section of the block, is readily calculated as :

b n bh = ^<- , JL= .

b=160 d«,=180

h=J60

Figure 3.7 : Equivalent Circular Cross-Section of Spindle Block

3.5.2 Finite Element Model of Tool Holder

Figure 3.8: Experimental and Finite Element Model of the Free-Free Tool holder

A CAT 50 taper shrink-fit tool holder is modeled using both Timoshenko beam elements and solid cubic 3D elements in ANSYS, as shown in Figure 3.8. The natural frequencies and the bending mode shapes are compared with the experimental results in Table 3.1. The finite element model using Timoshenko beam elements shows an acceptable prediction.

Experimental Timoshenko Beam Solid Cubic 3D

1st Mode Freq. (Hz) 5380 5376 4807

2nd Mode Freq. (Hz) 9836 9990 9739

Table 3.1: Natural Frequencies of a Free-Free CAT 50 Shrink-Fit Holder Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 21

3.6 Contact Stiffness Identification Results

The radial and rotational contact stiffness per unit area of the tool holder-spindle interface are identified using the experimental method explained in Section 3.4. The experimental identification method is used to identify the stiffness constants for CAT 50, CAT 40 and the HSK A63 taper interface as functions of the drawbar force.

The drawbar force is varied within a range recommended by spindle manufacturers, and the contact stiffness is identified at 2 kN drawbar force increments. The recommended drawbar force for the CAT 50 taper is 20-25 kN, and the drawbar force for both CAT 40 and HSK 63 taper are 10-12 kN.

Nonlinear least-squares curve-fitting is employed to determine the stiffness constants in the finite element model and the modal damping in the finite element simulations is obtained from the experiments.

The radial and rotational stiffness per unit area for the CAT 50 taper are listed in Table 3.2 as functions of the drawbar force.

Drawbar Radial Stiffness per Rotational Stiffness per

Force (kN) 3 Unit Area K$ (N/rad.m) Unit Area, KR (N/m )

6 1.84xl012 1.38xl09

8 1.75xl012 1.69xl09

10 1.65xl012 1.97xl09

12 1.60xl012 2.18xlOy

14 1.59xl012 2.313xl09

16 1.62xl012 2.37xl09

18 1.61xl012 2.48xl09

20 1.64xl012 2.56xl09

Table 3.2: Contact Stiffness per Unit Area for CAT 50 Taper Determined by Experimental Identification - 105 mm Overhang Length from the Spindle Face Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 22

Figure 3.9 shows the experimental and finite element model frequency response functions for the CAT 50 tool holder taper with 20 kN drawbar force. The experiments are performed using a shrink-fit tool holder without any tool, and the tool holder shank overhang, measured from the spindle face is 108 mm.

x 10 Experiment — FEM Model- Fitted

CD CD

4000 5000 6000 7000 8000 9000

x 10

^3, -0.5 e "En E -1 3000 4000 5000 6000 7000 8000 9000 Frequency (Hz)

Figure 3.9: Frequency Response Function of the CAT 50 Taper in Experimental Setup - 20 kN Drawbar Force

The experimental identification method is also used to identify the CAT 40 taper contact stiffness per unit area. The experiments are performed on a CAT 40 shrink-fit tool holder without any tool and the tool holder shank gauge length, measured from the spindle face, is 95 mm. The contact stiffness per unit area of the CAT40 taper are listed in Table 3.3, and the comparison between the fitted frequency response functions and the experimental results for the CAT 40 taper with 10 kN drawbar force is shown in Figure 3.10. Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 23

Drawbar Radial Stiffness per Rotational Stiffness per

Force (kN) 3 Unit Area, KQ (N/rad.m) Unit Area, KR (N/m )

4 0.75xl012 5.25xl08

6 0.82xl012 5.87x10s

8 0.95xl012 6.25xl08

10 1.09xl012 6.79xl08

Table 3.3: Contact Stiffness per Unit Area for CAT 40 Taper Determined by Experimental Identification-95 mm Overhang Length from Spindle Face

CAT 40 Shrinkfit In Block without Tool

3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500

.-6 X 10

ft -0.! ( ro \ Drawbar Forde = 10 kN J i i i 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 Frequency(Hz)

Figure 3.10: Frequency Response Function of the CAT 40 Taper in Experimental Setup -10 ^Drawbar Force Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 24

The contact stiffness per unit area for the HSK A63 taper is identified using the experiments. Figure 3.11 shows the HSK A63 tool holder. The grooves and chamfers are neglected in the finite element model of the tool holder since they did not contribute to the stiffness of the system, and the shrink-fit tool holder is modeled using Timoshenko beam elements.

Figure 3.11: HSK A63 Shrink-Fit Tool Holder

As explained in Section 3.5.1, the spindle block is also modeled using Timoshenko beam elements. The HSK tool holder-spindle block assembly is shown in Figure 3.12.

mm I L-LLLLI

Figure 3.12 : Spindle Block - HSK A63 Tool Holder Assembly

Nonlinear least-squares curve-fitting is employed to determine the stiffness constants per unit area of the HSK 63A taper in the finite element model. The radial and rotational stiffness per unit area for the HSK 63A taper are listed in Table 3.3 as functions of the drawbar force. Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 25

Drawbar Radial Stiffness per Rotational Stiffness per

Force (kN) 3 Unit Area, Kg (N/rad.m) Unit Area, Kr (N/m )

4 2.86xl012 1.02xl09

6 3.33*1012 1.83xl0y

8 3.74xl012 2.51xl09

10 3.87xl012 2.83xlOy

12 3.89xl012 3.02xl09

14 4.12xl012 3.10xl09

16 4.03xl012 3.26xlOy

Table 3.4: Contact Stiffness per Unit Area for HSK A63 Taper Determined by Experimental Identification -87 mm Overhang Length from Spindle Face

The HSK taper equivalent of the CAT 40 taper is the HSK 63 since they have relatively equal gauge diameters and equal power and torque transmission capabilities. The identified stiffness per unit area of the two tapers for 10 &iV drawbar force is compared in Table 3.7.

Tool holder Radial Stiffness per Rotational Stiffness per

Taper 3 Unit Area, Kg (N/rad.m) Unit Area, Kr (N/m )

HSK -A63 3.87xl012 2.83xl09

CAT 40 1.09xl012 6.79x10s

Table 3.5: Comparison of the Contact Stiffness per unit Area for HSK A63 and CAT 40 Taper - 10 kN Drawbar Force

The radial and rotational contact stiffness per unit area of the HSK taper are about 4 times the CAT 40 taper. This is mainly due to the dual-face contact of the HSK taper design. However, the surface contact area of the CAT 40 taper is about 2 times that of the HSK 63 A taper, thus making up for one half of the loss in stiffness due to the face contact. Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 26

3.7 Simulation Results on the Test Setup

In Section 3.6, the contact stiffness per unit area for the CAT 50 taper is identified using a shrink-fit holder without any tool. The predicted contact stiffness per unit area listed in Table 3.2 are used to model the CAT 50 shrink-fit tool holder with a 16 mm diameter blank tool and 64 mm tool overhang. The drawbar force applied on the tool holder is 20 kN, and the tool holder-tool connection is assumed to be rigid. This assumption is valid since the tool holder directly clamps onto the tool, creating one solid structure. The simulation results and comparisons with the experiments are shown in Figure 3.13.

,-5 x 10 1 Experiment FEM Model s til QJ * -0.5 r rr -1 1500 2000 2500 3000 3500 4000 4500 5000 5500

x 10'

-1.5 1500 2000 2500 3000 3500 4000 4500 5000 5500 Frequency (Hz)

Figure 3.13: Frequency Response Function of the CAT 50 Shrink-Fit Holder with a 64 mm Overhang Blank Tool in Experimental Setup - 20 kN Drawbar Force

The comparisons between the simulation results and the experimental results show an acceptable match, however the error is due to the effect of the overhang on the stiffness constants. Figure 3.14 shows experimental and simulation frequency response functions on a Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 27 short 15 mm overhang. The error between the experimental and simulation results is smaller in this instance, due to the shorter overhang.

„ x 10

Experiment FEM Model 3 e CD CD ,f I f I I I I i i 1000 1500 2000 2500 3000 3500 4000

x 10

.3,

e -0.5

CD E

-1 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz)

Figure 3.14 : Frequency Response Function of the CAT 50 Shrink-Fit Holder with a 15 mm Overhang Blank Tool in Experimental Setup - 20 kN Drawbar Force

The simulation results on a long and short tool overhang clearly show the effect of the tool overhang on the stiffness of the tool holder-spindle interface. However the cost in tooling and number of experiments needed to predict the contact stiffness as a function of the drawbar force and overhang length outweigh the benefits of such a modeling effort. Thus, in this section the stiffness constants are assumed to be a function of the drawbar force only, and independent of overhang length. The effect of overhang length on the stiffness per unit area of the interface is simulated in Section 3.9.

3.8 Simulation Results on the Machine Tool

The contact stiffness constants for the CAT 40 taper are used to couple the tool holder to a finite element model of a spindle system developed by Cao [9] on the machine. The finite Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 28 element model of the spindle on the machine tool is created using the Spindle Pro ® [35] finite element program, as shown in Figure 3.15. Two cases are investigated in modeling the tool holder-spindle connection. In the first case the tool holder-spindle connection is assumed to be rigid, and in the second case the connection is assumed to be distributed radial and rotational springs along the taper. The stiffness per unit area of the distributed springs is derived using the experimental procedure explained in Sections 3.2-3.6.

Figure 3.15 : Finite Element Model of Spindle System on Machine

The finite element model of the tool holder taper using 4 beam elements is shown in Figure 3.16 .

Tool Holder -Spindle Connection Springs

Figure 3.16 : Finite Element Model of Tool Holder in Spindle Taper with Connection Springs

There are 5 connection springs between the tool-holder taper and the spindle taper. The stiffness of the connection springs is calculated using the discretized form of equation (3.9) , which is Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 29

Kr(i) = Kr-7r-D(i)-L(i)

( } Ke(i) = K8-7T-D(i)-L(i) '

where Kr(i) and Kg(i) are the radial and rotational stiffness of the connection springs at node i, D(i) is the diameter of the taper element and L(i) is the tapered element length. The drawbar force applied on the tool holder by the drawbar mechanism in the spindle is 10 kN, which was obtained from the spindle manufacturer. Thus, the radial and rotational stiffness per unit area of the connection springs are respectively:

12 3 8 Kr = 1.09xl0 (N/m ) and K9= 6.79* 10 (N/rad.m) which were identified in Section 3.6, Table 3.3. The stiffness values for the 5 springs at a drawbar force of 10 kN axe listed in Table 3.7.

Radial Stiffness, Kr Rotational Stiffness, Spring No. (N/m) Kg (N.m/rad)

1 4.70xlOy 6.3xl05

2 8.99xlOy 1.20xl06

3 6.40x109 8.60xl05

4 5.80xl09 7.70x10s

5 3.60xl09 4.80xl09

Table 3.6: Radial and Rotational Spring Stiffness of CAT 40 Tool Holder - Spindle

The FRF of the finite element model of the spindle-tool holder assembly is simulated at the tool tip for the following two cases: rigid tool holder-spindle connection and distributed- springs connection. The simulation results are compared with the experimental results as shown in Figure 3.17. Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 30

-7 x 10 6 Rigid Connection Spring Connection 5h

4h E

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)

Figure 3.17: Frequency Response Function at the Tool Tip

The rigid-connection assumption yields results close to those of the spring-connection, and both simulations show an acceptable match with the experiments. However, the spring- connection model is able to simulate the mode at around 2900 Hz more accurately because the tool-tool holder assembly is largely contributing to the bending deformations at 2900 Hz. The mode shapes related to the dominant modes of the spindle assembly, 988 Hz and 2932 Hz, are simulated using the finite element model of the machine, as shown in Figure 3.18. The bending deformations of the tool-tool holder assembly are dominant in contributing to the modes at 988 Hz and 2932 Hz . The first dominant mode at 988 Hz is similar to the first bending mode of a cantilever beam; the second dominant mode at 2932 Hz is similar to the second bending mode of a cantilever. Thus the spindle - tool holder connection dynamics has affected the frequency and the mode shapes of these dominant modes.

The static stiffness of the assembly is also accurately simulated using the spring connection. It is obtained by calculating the magnitude of the frequency response function at zero frequency. By comparing the frequency response functions shown in Figure 3.17 at the lower frequency range, the experimental FRF and spring connection FRF match perfectly; however, the rigid connection is statically stiffer by 15 %. Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 31

r, , , , • • , Spindle Head

u-»-J— • Spindle Housing

Spindle Shaft •_• • • ; • •

Tool- Tool Holder

Mode @ 988 Hz

—~—•—.———— Spindle Head

— 1 u—1 1,1 Spindle Housing

Mode @ 2932 Hz

Figure 3.18 : Mode Shapes Contributed by Tool - Tool Holder Assembly Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 32

3.9 Contact Stiffness Derivation Using Contact Elements

Bending deformations in the spindle-tool holder interface are due to large bending moments caused by the tool overhang. These bending deformations cause very non-uniform pressure distributions in the spindle-tool holder connection [30]. Figure 3.19 shows the schematic of bending deformations in the tool holder-spindle interface. The rotational deformations in the spindle - tool holder taper, 6o, cause large projected deflections at the tool tip.

N L

\ \

Figure 3.19 : Schematic of Bending Deformations of the Tool Holder - Spindle Interface

The radial deflection at the tip, y, is:

y = yr+y0+e0.L (3.18) Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 33

where yr is the radial deflection at the tip when the spindle-tool holder interface is assumed to be rigid, yo and 9o are the radial and rotational deformations in the tool holder - spindle interface, and L is the overhang length, as shown in Figure 3.19. The projected deflection at the tool tip due to the rotational deformations inside the interface is 6n.L. The projected deflections are important in the overall static and dynamic stiffness of the tool holder-spindle assembly.

As explained in Section 3.3, identifying the contact stiffness per unit area of the distributed springs at the tool holder spindle interface through experiments requires a dedicated test setup. Another proposed methodology in this thesis for identifying the contact stiffness per unit area is using 3D contact elements to model the tool holder - spindle interface in ANSYS [3]. The tool holder and a simulated spindle taper are modeled using 3D cubic solid elements.

Using ANSYS, it is not necessary to construct gap elements or contact interface objects. The contact model relies on the definition of contact pairs in master-slave relationships. During model execution in different sub-steps, the solver internally imposes reaction forces to keep the nodes on the slave surfaces from penetrating the element walls on the master surfaces. The ANSYS solver is capable of performing nonlinear simulations of the contact between deformable bodies. These simulations can also take into account friction and the geometric interferences and clearances in the order of microns. Thus, the contact stiffness depends on the coefficient of friction between the two interfaces, the elastic properties of the two bodies in contact and the boundary conditions, mostly the applied forces and displacement constraints.

Three models of the spindle - tool holder connection are taken into account and simulated. In models A and B, 3D surface contact elements are constructed on the tapered surfaces between the tool holder and the spindle. A friction coefficient of 0.2, which is typical for steel surfaces in contact is assumed between the two surfaces, and the drawbar force, Fa, is applied as an axial force on the tool holder as shown in Figure 3.20. Details from the tool holder such as the threads on the retention bolt hole and chamfers are not considered in the model, since they did not contribute to the overall stiffness of the interface. The taper mismatch between the two tapered surfaces is assumed to be zero in Model A, which results Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 34 in a perfect fit between the tool-holder taper and the spindle taper. In Model B, the tool- holder taper has a steeper angle than that of the spindle taper. The taper mismatch between the tool holder and the spindle taper is 21 angular seconds, corresponding to the AT 5 tolerance level.

Fixed Displacemnt Boundary Conditions

Figure 3.20: 3D Finite Element Model of Tool Holder in Spindle Taper with Boundary Conditions - Model A

In Model C, the spindle-tool holder connection is assumed to be rigid, and the tool holder is rigidly connected to the spindle as shown in Figure 3.21.

Fixed Displacemnt Boundary Conditions

Figure 3.21: 3D Finite Element Model of Tool Holder with Rigid Spindle-Tool Holder Connection - Model C

The static radial and rotational deflections of the tool-holder tip are simulated by applying a force at the tool-holder tip in the 3D solid model of a CAT 50 shrink-fit tool holder without any tool, as shown in Figure 3.20 and Figure 3.21. In Model A and B, fixed Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 35 displacement constraints are placed at the front end of the spindle face, shown in Figure 3.20. The drawbar force, Fd, is 20 kN, the radial load P is 1000 N and the overhang length L is 108 mm. In Model C, the fixed displacement constraints are applied on the end of the tool-holder flange as shown in Figure 3.21. The simulated tool tip deflection is 14.397 microns in Model A, 15.543 microns in Model B and 11.719 microns in Model C.

The effect of the drawbar force on the static stiffness of tool holder spindle connection is also simulated using model A and B, as shown in Figure 3.22.

4.50E+07

£• 4.00E+07

3.50E+07 CO o

CO Q. 3.00E+07 - Perfect Fit ( No Taper o .o Mlsmatch)_Model A Rigid Conection_Model 2.50E+07 C 21 Angular Sec. Taper Mismatch Model B

2.00E+07 12 16 20 24 28 32 Drawbar Force (kN)

Figure 3.22: The Effect of Drawbar Force on the Static Stiffness of a CAT 50 Tapered Connection - Simulation Results

The static stiffness of the connection increases with increased drawbar force and saturates after 12 kN in Model A. This is due to the fact that full contact has been established between the tool holder and the spindle taper, and the tool-holder spindle connection is no longer affected by the bending moment due to the radial force P. The saturation phenomenon occurs at around 24 kN for Model B, due to the angular mismatch between the two tapers which requires a larger drawbar force to establish full contact. Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 36

The comparison between the rigid connection, Model C, and Model A show a decrease of about 20 % due to the compliance of the connection.

To obtain the radial and rotational contact stiffness per unit area, Kr and Kg of the tool holder-spindle interface, the finite element model of the tool holder spindle assembly is constructed by using Timoshenko beam elements, as shown in Figure 3.23 Distributed contact springs are used to model the contact between the two interfaces, as explained in Section 3.2 and Section 3.3.

—I \ \ \ \ \ \ \ \ \ \ / 7 \ i_ ^ \ \ \ i m—n—i—n 7 / / —i.

/ / / • z ~I - \ \ \\\^ \ \ \ \

\ \ \ \

Figure 3.23: Finite Element Model of Tool Holder in Spindle Taper using Timoshenko Beam Elements

By employing a nonlinear least-squares technique the stiffness constants per unit area of the connection, which yield static deflections in the beam model equal to the 3D solid model, are determined. The identification procedure is shown in Figure 3.24.

The stiffness constants for the CAT 50 taper and overhang length of 108 mm obtained by modeling the spindle - tool holder interface using contact elements are tabulated in Table 3.7.

Drawbar Radial Stiffness per Rotational Stiffness per

3 Force (kN) Unit Area Kr (N/m ) Unit Area Ke( N/rad.m)

10 4.29xl012 4.84xl08

20 6.37xl012 6.0445 xlO8

Table 3.7: Contact Stiffness Constants for CAT 50 Taper Using Contact Elements Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 37

Select Tool Holder Taper Construct 3D FE Model Model Interface with Contact Elements

ApplyDisplacement Constraint Apply Drawbar Force Apply Radial Force

I Calculate Nodal Displacements

Construct Timoshenko Beam Model of 3D Model Model Interface with Distributed Springs Apply Radial Force

Solve for Kr ,Ke using Nonlinear Least Squares to fit 3D Model Displacements to Beam Model Displacements

Figure 3.24 : Flowchart for Obtaining the Stiffness Constants by Using Contact Elements

By modeling the spindle-tool holder interface using contact elements, it is possible to model the effect of the overhang on the stiffness of the connection. By increasing the overhang length of the tool holder to 216 mm, the stiffness constants obtained for the 20 kN drawbar force are calculated and listed in Table 3.8.

Overhang Radial Stiffness per Rotational Stiffness per

3 (mm) Unit Area Kr (N/m ) Unit Area Ke( N/rad.m)

108 6.37xl012 6.0445 xlO8

216 6.33xl012 4.24xl08

Table 3.8: Contact Stiffness Constants for CAT 50 Taper for Different Overhang Lengths with 20 kN Drawbar Force Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 38

By comparing the two stiffness constants in Table 3.8, it is observed that the overhang effect is more pronounced on the rotational stiffness per unit area of the connection. Increasing the overhang by about two times decreases the rotational stiffness, K& by about 30 %. The reason for this phenomenon is the nonlinear nature of the spindle - tool holder connection. The stiffness of the tool holder - spindle interface depends on the bending deformations in the interface, which are subject to the bending moments applied on the tool holder.

The spindle-taper and the tool holder are constructed using the Timoshenko beam elements and the radial and rotational stiffness per unit area obtained in Table 3.7 are used to construct the contact springs distributed in the tool holder-spindle interface. The frequency response simulation results are compared with the experimental results on the setup as shown in Figure 3.25.

x 10"'

Experiment FEM Model

3000 4000 5000 6000 7000 8000 9000

x 10

3000 4000 5000 6000 7000 8000 9000 Frequency (Hz)

Figure 3.25: Frequency Response Function of the CAT 50 Taper in Experimental Setup - 20 fc/V Drawbar Force Chapter 3. Modeling and Identification of Spindle - Tool Holder Interface Dynamics 39

Modal damping obtained from the experiments, is applied to the modes. The comparison between the frequency response functions shows an acceptable match, taking into account that the stiffness constants are obtained by modeling the interface using 3D solid contact elements.

3.10 Summary

A finite element model for tapered connections is developed in this chapter. The tool holder-spindle interface is modeled using distributed rotational and translational springs. The stiffness of these springs, which is related to applied drawbar force on the tool holder, is obtained using two methods: the experimental identification method and modeling the interface using contact elements. The model was experimentally verified on a dedicated test setup and an experimental spindle. Chapter 4 Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies

4.1 Overview

Tool holders are practically essential to serve as an adapter between different types of spindles and cutting tools. In modern machine tools they also allow for automatic tool change. There are several dominant types of tool holders used in the manufacturing industry with different dynamics. Although their dynamics as well as some factors affecting them have been investigated to some extent through research or experience, there is not enough research comparing the milling performance of different types of tool holders. Moreover, performing experimental modal analysis [15] on tool holders to obtain detailed information about their dynamics would be useful in selecting a specific tool holder for roughing or finish milling operations. The results could also be used to provide some directions in modeling the spindle-tool holder-tool assembly for prediction of the frequency response function at the tool tip.

The most common types of tool holders in the industry are the collet, the milling, the hydraulic chuck and the shrink-fit holder as shown in Figure 4.1. The clamping mechanism in these tool holders distinguishes their mechanical properties. In the shrink-fit the interference between tool diameter and inner shank of tool holder diameter is used to clamp the tool by thermally expanding the shank, and then letting it cool down and contract to grip the tool. In the collet and milling chuck a deformable spring element, i.e., the collet, is elastically deformed by applying an axial load to clamp the tool. In the hydraulic chuck the clamping force to grip the tool is supplied by hydraulic pressure.

Tool holders are characterized by certain mechanical and dynamic properties, including the following: (1) concentricity or run-out; (2) clamping torque; (3) accessibility to the work- piece; (4) ease of axial adjustment; and (5) dynamic stiffness. In this study, the dynamic stiffness of these tool holders is compared.

40 Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 41

Figure 4.1: Common Types of Tool Holders

The shrink-fit holder has gained popularity with the advent of high-speed machining, because of its higher static stiffness and less run-out. However the upfront cost of the shrink- fitting equipment and the tool change times involved have kept it from gaining overall popularity.

4.2 Modal Analysis of Tool Holder-Tool Assembly on a Simulated Spindle

Four types of CAT 50 taper tool holders with equal gauge lengths, measured from the tool tip to the spindle face, are compared to each other on the experimental setup. The setup is designed to investigate the spindle - tool holder interface dynamics without the effects of the spindle bearings and other machine elements, which are inevitable on a real machine tool. The experimental setup consists of a large steel block with the spindle taper, and a load cell at the back to control the drawbar force. The drawbar force is kept constant at 20 kN, and the setup is placed on a supporting cushion to simulate free - free boundary conditions. Both short and a long overhang are used, and the frequency response functions are measured at the tool tip. The short overhang is used to investigate the effect of the spindle-tool holder interface on the FRF whereas the longer overhang is used to study the effect of the tool-tool holder connection. Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 42

4.2.1 Modal Analysis on a Short Overhang

With the 16 mm diameter blank tool and a short 10 mm overhang, one dominant mode is present in the 5000-Hz range for all the tool holders, as shown in Figure 4.2.

1 Collet Hydraulic j Power Chuck ij Shrink Fit .'i

/l / 1 . JM.. • \ f 2500 300.0 1 3500 4000 4500 Frequency (Hz)

2 - J

_31 i i i I 2500 3000 3500 4000 4500 Frequency (Hz)

Figure 4.2 : FRF Comparison of Different Tool Holders with Short 10 mm Overhang Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 43

With the effect of the spindle bearings and other machine elements not present on the setup, the shrink-fit has the least dynamic stiffness compared to the other types of tool holders. The hydraulic chuck is dynamically stiffer by 2.5 times compared with the shrink-fit, due to the fact of the hydraulic chuck is more damped because of the hydraulic fluid.

Experimental modal analysis is performed on the tool holders to observe the bending mode shapes. In all cases, the mode shapes are similar. The modal analysis result for the collet chuck is shown in Figure 4.3.

Figure 4.3:1st Mode of the Collet Chuck Tool Holder at 3508 Hz with 10mm Overhang

With a short tool overhang, the effects of the tool - tool holder connection are minimal and the effects of the tool holder-spindle interface are more distinct. This is due to the fact that the bending moment experienced by the tool holder - spindle connection is larger than that of the tool -tool holder joint. The modal analysis results in Figure 4.3 show that there is bending around the stepped part at the lower end of the flange (between points 5 & 6) and between the upper end of the flange and the block (points 7 & 8). Moreover, it is not possible Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 44 to distinguish the bending between the upper end of the flange and the block due to the interface from the bending due to the elastic deformation of the tool holder inside the block.

4.2.2 Modal Analysis on a Long Overhang

With the 16 mm diameter blank tool and 64 mm overhang two dominant modes are present in the 5000-Hz frequency range for all the tool holders as shown in Figure 4.4.

x 10'

Collet Hydraulic Power Chuck Shrinkfit

T3 13 CD CO

1400 1500 1600 1700 1800 1900 2000 Frequency (Hz)

x 10

3600 3800 4000 4200 4400 4600 4800 Frequency (Hz)

Figure 4.4: FRF Comparison of Different Tool Holders with Long 64 mm Overhang Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 45

The shrink-fit tool holder has the least dynamic stiffness of the tool holder tested, while both the power chuck and hydraulic chuck are the most dynamically stiff tool holders. This is due to the high damping of the hydraulic and power chuck. The shrink-fit tool holder clamps the tool directly and thus has fewer connections than the three other types. In the collet, power and hydraulic chucks, relative motion between the parts leads to a higher damping.

The bending mode shapes for the long overhang are analyzed using experimental modal analysis. The tool holders have similar bending mode shapes and the modal analysis results for the collet chuck are shown in Figure 4.5.

Figure 4.5: 1st and 2nd Modes of the Collet Chuck Tool Holder with Long Blank Tool Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 46

As shown in Figure 4.5, both bending mode shapes are dominated by deformations in the tool-tool holder connection. For the first mode, at 1738 Hz, both the tool clamp interface at the tool holder joint (point 5) and the stepped part at the lower end of the flange, (point 9) are contributing to the bending vibrations at the tool tip. This contribution is larger for the second mode, where the tool holder - spindle interface (point 10) also contributes to the mode shape. The modal analysis results with the longer overhang clearly show the effect of the tool-tool holder joint in the bending modes of the assembly.

4.3 Tool Holder Selection and Tool-Tool Holder Joint Stiffness and Damping

The performance of the tool holders are relatively compared to each other by comparing the modal parameters obtained from the dominant mode, i.e., 1st mode, of the FRFs on the long 64mm tool overhang with 16 mm diameter. With the longer overhang the tool-tool holder joint dynamic has more of an effect on vibrations at the tool tip than the tool holder- spindle interface, as seen in Section 4.2.2. The dynamic stiffness, modal stiffness, natural frequency and modal damping of the 1st mode for the four different kinds of tool holders are listed in Table 4.1

1st Mode Tool Holder Type 2kC[N/m] c co [Hz] AfN/m] Shrink-fit 1.39E+04 7.32E-04 1693.5 9.53E+06 Collet Chuck 4.36E+04 2.06E-03 1738.0 1.06E+07 Hydraulic Chuck 5.3E+04 3.69E-03 1687.5 7.19E+06 Power Chuck 6.24E+04 3.53E-03 1607.8 8.85E+06

Table 4.1: Modal Parameters at 20 kN Drawbar Force for 64 mm Overhang and 16mm Diameter -1st Mode

In terms of tool holder selection for roughing operations where higher metal removal rates are desired, the power chuck is the most suitable, since it has a higher dynamic stiffness, i.e., 2k<^. For finish milling operations where static stiffness and accuracy are preferred the shrink-fit and the collet chuck are the most suitable, since they have the highest modal stiffness, k. However, the run-out or eccentricity on the shrink-fit tool holder is less Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 47 than on the collet, making it more suitable for finishing operations, especially in high-speed machining.

Although the distance between the gauge plane on the spindle face and the tool tip is the same for all the tool holders used in the tests, the tool holder shank diameters are different as shown in Figure 4.6. By modeling the tool holders using Timoshenko beam elements and assuming the tool holder - tool joint to be a rigid connection, the size effect factors as well as the effect of the tool holder - tool joint compliance are obtained.

J(H_.

_6.7_

-4-

Shrink Fit Collet Chuck

105 18,5 48,5 41 . ,.?f.

Hydraulic Chuck Power Chuck

Figure 4.6 : Dimensions of the Four Types of Tool Holders

The tool holders are modeled in finite elements, as shown in Figure 4.7, and the dynamic and static stiffness are compared to each other. The tool holder-spindle interface and the tool holder-tool connection are assumed to be rigid. Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 48

Shrink Fit Tool Holder Hydraulic Tool Holder

Collet Chuck Power Chuck

Figure 4.7 : Finite Element Model of the Four Types of Tool Holders - With 16mm Diameter Tool and 64 mm Overhang

The comparisons between the modal parameters from simulations and experiments are listed in Table 4.2. The modal parameters are for the first mode since it is the most dominant as previously shown in Figure 4.4. The modal damping used in the simulations is from the experiments.

Damping Simulation Experiment Tool Holder Ratio Type kevp[N/m] kslalic[N/m] kC[N/m] k.I-m[N/m] k£[N/m]

Shrink-fit 7.32E-04 1.07E+07 1.02E+04 1.396E+07 6.97E+03 9.53E+06 Collet Chuck 2.06E-03 1.41E+07 3.25E+04 1.577E+07 2.18E+04 1.06E+07 Hydraulic 3.69E-03 1.35E+07 6.75E+04 1.536E+07 2.65E+04 7.19E+06 Power Chuck 3.53E-03 1.69E+07 5.42E+04 1.83E+07 3.12E+04 8.85E+06

Table 4.2: Static Stiffness and Dynamic Stiffness of the Four Types of Tool Holders - Simulation Results vs. Experimental Results

Due to the presence of one dominant mode, the assembly is assumed to be a one-degree- of-freedom mass-spring system. Thus, the modal stiffness and the static stiffness for each of Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 49 the four tool holders are relatively close to each other. This is evident by comparing the

simulated static stiffness, kstanc, and simulated modal stiffness, ksim, listed in Table 4.2. Thus, the modal stiffness can be used as a measure for comparing the actual stiffness of the different tool holder - tool connections.

The percentage error between the experimental and simulations for modal stiffness are listed in Table 4.3. The main source of error is from the assumption that the tool - tool holder

joint is a rigid connection. By comparing the experimental modal stiffness, keXp,, with the simulation results in Table 4.2, it is shown that the simulated modal stiffness of the shrink-fit is closest to the experimental results, due to the fact that the rigid connection model is more acceptable in the shrink-fit than the other types of tool holders.

Tool Holder Type Percentage Error (%)

Shrink-fit 46

Collet Chuck 49

Hydraulic 106

Power Chuck 114

Table 4.3: Percentage Error in Modal Stiffness - Rigid Tool - Tool Holder Connection

Although the distance between the gauge plane on the spindle face and the tool tip is the same for all the tool holders used in the tests, the tool holder shank diameters are different as

shown in Figure 4.6. The normalized modal stiffness, ft normal, is the ratio of the experimental modal stiffness to the simulated static stiffness:

rnormal ; (4.1) static

where kexp is the modal stiffness obtained from the experiments and kstatic is the simulated static stiffness , listed in Table 4.2, Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 50

The normalized modal stiffness which can be used to compare the stiffness of the joint in the four types of tool holders is listed in Table 4.4. The modal stiffness of shrink-fit is higher compared to other types of tool holders. Thus for finish machining operations where accuracy and precision is needed the shrink-fit holder is the most suitable type of tool holder.

Normalized Normalized Damping Simulation Experiment Modal Dynamic Tool Holder Ratio Type Stiffness Stiffness

kstalic[N/m] keJN/m] /^normal ^normal- C

Shrink-fit 7.32E-04 1.07E+07 9.53E+06 0.89 0.065 Collet Chuck 2.06E-03 1.41E+07 1.06E+07 0.75 0.155 Hydraulic 3.69E-03 1.35E+07 7.19E+06 0.53 0.196 Power Chuck 3.53E-03 1.69E+07 8.85E+06 0.52 0.184

Table 4.4: Normalized Dynamic Stiffness and Modal Damping of 4 Types of Tool Holders with 64 mm Overhang

The power chuck has the least amount of normalized modal stiffness, although it yields a relatively high normalized dynamic stiffness due to the damping in the connection. Thus in roughing operations where maximum material removal rates are desired the hydraulic chuck and the power chuck are the most suitable types of tool holders.

The above performance comparisons of different kind of tool holders are obtained on a simulated spindle block where the effects of the spindle bearings and the machine are not present. When the tool holder is attached to the actual spindle the interactions between the tool holder dynamics and the spindle dynamics greatly affect the performance of the tool holder. The guidelines presented in this section are valid where the tool holder dynamics are dominant. Such cases arise when the tool overhang is very long or in the case of slender end mills. The comparisons between the dynamic stiffness of tool holders on a high speed machining center is performed in Section 4.5.

4.4 The Effect of Drawbar Force Variation on the Dynamics of the Interface

The effect of the drawbar force on the dynamics of the connection is investigated on four different types of tool holders with the CAT 50 taper with short and long blank tools. The Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 51 drawbar force was varied from 2 kN to 30 kN (the drawbar force recommended by drawbar manufacturers for the CAT 50 taper is 20-25 kN). With the 10 mm short tool overhang, all tool holders showed the same trend. Based on the observations in the modal analysis, Section 4.2, the effect of the tool holder - spindle interface in the bending vibrations at the tool tip is more dominant when the overhang is shorter. This is due to fact the bending moment in the tool holder- tool joint is smaller than the bending moment in the spindle-tool holder interface. The effect of the drawbar force on the natural frequency, modal stiffness and modal damping of the collet chuck with the short 10 mm tool overhang are shown in Figure 4.8.

The experimental results in Figure 4.8 indicate that with the increase of the drawbar force the natural frequency of the dominant mode increases up to 15 kN and then does not change much if the drawbar force is increased beyond 15 kN. The dynamic stiffness decreases as the drawbar force is increased and as can be seen from Figure 4.8 , this is due to the fact that the modal damping is decreasing due to the established contact in the connection which leads to less play in the joint. Although the increase in the drawbar force increases the stiffness of the connection [33], it has an adverse effect on the damping of the connection. As a result the dynamic stiffness, which is the product of modal damping and modal stiffness decreases with the increasing drawbar force.

It is shown that after 15 kN drawbar force, the modal parameters do not change much. The reason is that full contact has been established between the tapered parts and thus the stiffness and damping of the connection do not change much after 15 kN drawbar force. Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 52

3550

10 15 20 25 35 Drawbar Force(kN)

x 10

10 15 20 25 Drawbar Force(kN)

x 10'

10 15 20 25 30 35 Drawbar Force(kN)

Figure 4.8 : The Effect of Drawbar Force on the Natural Frequency, Dynamic Stiffness and Damping of the Collet Chuck with 10 mm Blank Tool Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 53

The effect of the drawbar force on the dynamics of the tool holder with a long 64 mm overhang on the same collet chuck is also investigated. With the longer overhang two dominant modes are present in the 5000 - Hz frequency range, and the change in the modal parameters of these two modes are considered, and the experimental results are shown in Figure 4.9 and Figure 4.10.

The effect of the drawbar force on the natural frequency of the first mode is about 1 % which is almost negligible. This is due to the fact that the natural frequency is a function of the modal stiffness and modal mass which does not change much for the first mode as the drawbar force is increased. This due to the fact that tool holder - spindle interface dynamics do not contribute much to the first bending mode as seen in Section 4.2. However the dynamic stiffness and the modal damping ratio decrease by about two to three times, since the drawbar force has a much stronger effect on the damping of the connection rather than the stiffness.

The effect of the drawbar force on the natural frequency of the second mode is more pronounced as expected, an increase of about 5 %. The reason for this is due to the fact the spindle - tool holder interface contribute to the bending vibrations of the second mode ,and tool - tool holder joint dynamics contribute to the first mode of the bending vibrations at the too tip ,as seen in Section 4.2.2.

The contact between the tool holder taper and spindle taper is maintained through the application of the drawbar force and as the drawbar force is increased from 2 - 15 kN the contact surface increases.The dynamic stiffness and modal damping of both modes decrease with the increase in the drawbar force and almost saturate after the drawbar force is increased to about 15 kN. It is observed that there is no significant change in the dynamic stiffness, natural frequency, and damping ratio when the drawbar force is increased beyond 15 kN. The reason for this phenomenon is that after 15 kN full contact is established between the two surfaces, and thus the increase in the drawbar force beyond 15 kN has no significant effect on the stiffness and damping of the interface. Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 54

1745 i N X

c 1740 -

cr 0 @ CD -0 m 1735 p—er 3 / CO / 1730

x 10 •c? 12

10 15 20 25 35 Drawbar Force(kN)

x 10'

10 15 20 25 30 Drawbar Force(kN)

Figure 4.9 : The Effect of Drawbar Force on the Natural Frequency, Dynamic Stiffness and Modal Damping of the 1st Mode of the Collet Chuck -64 mm Tool Overhang Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 55

Figure 4.10 : The Effect of Drawbar Force on the Natural Frequency, Dynamic Stiffness and Modal Damping of the 2nd Mode of the Collet Chuck-64 mm Overhang Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 56

It is concluded that a lower drawbar force is suitable for roughing operations were maximum material removal rates are desired and thus a higher dynamic stiffness leads to a higher stability margin. For finishing operations where accuracy is the main objective a higher drawbar force can be selected, since it leads to higher stiffness in the tool holder spindle interface. It is also concluded that there is no significant change in the dynamic stiffness of the connection above 15 kN drawbar force for the CAT 50 taper due to the fact that full contact is established between the tool holder taper and the spindle taper at this drawbar force.

4.5 Case Study: Modal Analysis of a Collet Tool Holder on the Machine Tool

The results obtained so far are on a simulated spindle block where the effects of spindle bearings and other machine elements are not present. To better investigate the mode shapes of the tool holder and the tool - tool holder joint dynamics on the actual machine tool, modal analysis on a collet chuck is performed on a horizontal machining center, Mori Seiki SH 403. The collet is tightened to the recommended torque by the manufacturer which is 55 Nm. The measurement points are selected along the tool holder and shown in Figure 4.11.

145

Figure 4.11: The Modal Analysis of the Collet Chuck on the Mori Seiki SH403 Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 57

The measured frequency response function of the machine at the tool tip in the Y direction is shown in Figure 4.12. Nine dominant modes are present within the 5000-Hz frequency range. The stability lobes for a work-piece material of aluminum alloy 7050 are calculated and shown in Figure 4.13 .

x 10

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)

Figure 4.12 : Measured Transfer Function at the Tool Tip in the Y axis of the Machine.

Mori Seiki SH403 / Mitsubishi 4 Flute Endmill D20 H50 / AL7050 16 864 - 877 Hz 14

£ 12

3 o 10

CL 8 Q 6

4 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 Spindle Speed [rpm]

Figure 4.13: Stability Lobes Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 58

The chatter frequencies above the 10000 RPM spindle speed are close to the natural frequencies of the second, third, sixth, eight and ninth mode as shown in Figure 4.13 . These mode shapes are plotted in Figure 4.14 . It is observed that for the two higher modes at 3148 Hz and 4374 Hz, tool- tool holder vibrations are dominant and for the three lower frequency modes at 534 Hz, 872 Hz and 1394 Hz spindle vibrations are contributing to the vibrations at the tool tip.

Mode 8: 3148 Hz Mode 9: 4374 Hz

Figure 4.14 : The Dominant Mode Shapes of the Collet Chuck Tool Holder on the Machine Tool - Tool Diameter 20 mm , Tool Overhang 50 mm

Full slotting milling tests are performed to verify the stability lobes as shown in Figure 4.13. When the cutting conditions are selected at 12000 RPM and 8 mm depth of cut the chatter vibrations occur at 1303 Hz , which is close to the mode at 1394 Hz where the spindle Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 59 vibrations are dominant. When the cutting conditions are selected at 14000 RPM and 10 mm depth of cut, the chatter vibrations occurred at 3175 Hz where the tool holder - tool joint vibrations are dominant.

By comparing the frequency response functions of the different kind of HSK tool holders on the simulated spindle block , where the effect of spindle dynamics are not present, it is observed that the shrink-fit has the least dynamic stiffness and the collet chuck has the highest dynamic stiffness as shown in Figure 4.15.

x 10"6

6 Collet Hydraulic _ 5 • Power Chuck Shrinkfit 4 1e

CD T3 3

CD 2

2000 2500 3000 0 Frequency (Hz) 1500

Figure 4.15: FRF Comparison of Different HSK Tool Holders with 50 mm Overhang on Simulated Spindle Block

However with the 50 mm overhang and 20 mm diameter tool, the spindle dynamics are still dominant and when coupled to the tool holders and the results are totally reversed as shown in Figure 4.16.It is observed that in this case the collet chuck and the shrink-fit are the most dynamically stiff, and the hydraulic chuck is the least stiff in terms of chatter stability. However when the tool overhang is very long and the tool - tool holder dynamics become dominant, the results obtained on the simulated spindle block can be used to select the tool holder for the cutting operations. Chapter 4. Experimental Analysis of the Dynamics of Tool Holder-Tool Assemblies 60

-7 x 10 3

Collet z 2.5 Hydraulic Power Chuck Shrinkfit

0L 500 1000 2500 3000 Frequency (Hz)

Figure 4.16: FRF Comparison of Different HSK Tool Holders with 50 mm Overhang on the Machine Tool

4.6 Summary

An experimental study on the various types of tool holders further highlights the importance of the tool holder - tool joint dynamics. Modal analysis of the tool holders in the experimental spindle block show two dominant mode shapes coming from the tool and the tool holder interface. The effect of the drawbar force on the overall dynamics of the tool holders shows that with the increase of the drawbar force the dynamic stiffness of the connection decreases, and this is more pronounced on the second mode. The effect of the drawbar force on the first mode depends on the coupling between the spindle - tool holder connection and the tool- tool holder connection. It is shown that the collet chuck was more influenced by the drawbar force affect than the shrink-fit. Furthermore modal analysis of the collet chuck on a high speed machining center demonstrates the tool- tool holder connection as contributing to the higher frequency vibrations. Chapter 5 Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics

5.1 Overview

The frequency response function of the machine tool at the tool tip along with the work- piece cutting coefficients are extremely important for the simulation of cutting forces and obtaining chatter stability lobes [1],[2]. The accuracy of the frequency response function measurement determines the ability of the process planner and the machinist to maximize the productivity of the cutting operation without chatter vibrations. However the frequency response function at the tool tip varies after each tool change and the upfront cost in performing frequency response function measurements in terms of hardware and the expertise to obtain accurate measurements is an issue in most manufacturing facilities. In this chapter a method to obtain the rotational and translational dynamics of the spindle up to the tool holder flange, as shown in Figure 5.1, is proposed. The spindle assembly up to the tool holder flange remains unchanged after each tool change, thus making it possible to rigidly couple a shrink-fit tool holder with arbitrary dimensions, and cutting tool. The proposed method is an extension of the work done by Park et al [27 ].

Machine Column

Spindle Toolholder Flange

Figure 5.1: Spindle on Machine Tool with Tool holder Taper only

61 Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 62

Furthermore the frequency response functions are used to select the optimum tooling for a specific cutting operation. In this specific industrial application known as tool tuning, the tool length is adjusted to shift the stable pockets of the stability lobe to the maximum spindle speed of the machine.

5.2 Receptance Coupling

The machine tool assembly (Structure AB) is divided into two substructures as shown in Figure 5.2. Substructure A represents the tool holder and substructure B represents the remaining machine tool assembly up to the tool holder flange which is standard in all tool holders for a specific spindle taper. The two structures are rigidly connected at point 2.

Figure 5.2: Receptance Coupling for Obtaining Frequency Response Function at the Tool Tip Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 63

The frequency response functions of substructure A at the two free ends which relates the displacements to the applied forces is :

H HA,\\ A,n

\XA,2 H A,2\ ^ A,22 where X] and XA,2 are displacement vectors with both translational and angular displacement components. Fj and FA,2 are force vectors containing both force and moments applied at points 1 and 2. HAJJ are frequency response functions between points i and j

Similarly the frequency response functions of substructure B at point 2 is:

{X^)=[HB22[{FB2} (5.2)

After rigidly coupling the two structures A and B at point 2, the equilibrium and compatibility conditions at point 2 are:

F = F +F 1 2 1 A,2 B,2 (5.3)

X2 ~ X A,2 ~ XB,2

By letting:

H2=HAa2+HBt22 (5.4)

And substituting (5.4) into (5.2):

X2 = HB,22 - FB,2 = HA,2\ ' F\ + HA,22 " (F2 ~ FB,2 ) (5.5)

By rearranging (5.5), the forces on structure B are:

i FB,2 ={HB,22 +HA,22Y(HAai-F, +HA22-F2)=(H2)-{HAa, • F, +HAa2-F2) (5.6)

Finally the displacements at points 1 and 2 are expressed as functions of frequency response functions and the applied forces as follows:

X\ = HA,U - ^1 + #4,12 • (^2 _ FB,2 )

= HAM -Fx +HAA2 -F2 -HAX2 -{H2Y \HA2l -Fx +HAa2 • F2) (5.7)

= (HA,l 1 _ HA,\2 ' (H2 ) ' • HA,2l )' F\ + (HA,12 ~ HA,\2 ' (H2 ) ' - H A,22 )' F2 Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 64

F F ^2 ~ #,4,21 ' Fi + #A,22 ' ( 2 B,2 )

F F H F = #A,2\ • \ + #,4,22 - 2 ~ A,22 " (#2 ) ' * (#,4,21 ' ^1 + #,4,22 " 2 ) (5.8)

H H H F = (#4,21 _ #.4,22 - ( 2 ) ' - #.4,21 )" ^1 + (#.4,22 ~ A,22 ' ( 2 ) ' " #,4,22 )" 2

Equations (5.7) and (5.8) are arranged in matrix form as follows:

[#,4,11 -#4,12 '(#2) ' '#,4,21 [#.4,21 ~#W,12 '(#2) ' •#/!,: (5.9) H I#.4,2;! _#/f,22 '(#2) "#,4,21 )t #,4,22 ~ A,22 " (#2 ) ' " HA,

where H2 = HA,22 +HBj2.

Equation (5.9) is the standard receptance coupling equation for the coupling of two

structures. The receptances of the free - free tool holder - tool assembly, HA.H , #,4,72 and

HA,22, are modeled using the finite element model and the receptance of the spindle at point

2, HB,22, is obtained through the inverse receptance coupling method explained in Section 5.3.

5.3 Inverse Receptance Coupling

The inverse receptance coupling method allows the identification of the spindle assembly at the tool holder flange, point 2, with both rotational and translation degrees of freedom responses.

By applying a force at point 1 and measuring displacements at points 1 and 2, the force

at point 2 is zero , F2=0 , and the following cross and direct receptances are obtained from equations (5.7) and (5.8):

H { A,n ~HA,\2 "(#2) ' •#/f,2i)-#i F 11 (5.10)

Similarly by applying a force at point 2 and measuring displacements at points 1 and 2, the force at point 1 is zero, Fi-0, and the cross and direct receptances are obtained from equations (5.7) and (5.8): Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 65

~ {HA,2\ ~ HA,\2 " (#2 ) ' * #,4,22 ) = H2 (5.11) X, = (#,4,22 _ #,4,22 - (#2 ) ' • #,4,22 ) = #:

Each frequency response function contains both translational and rotational displacements, thus equations (5.10) and (5.11) can be expanded as :

h h H\\,fM If.] h h "\ \,MM _ (5.12) ' h h \X2\ = [Hn\r n\2,MM 1M,

' h ^2\,fM 21, ff 0, h ^2\,MM _ (5.13) ^22,ff h

0, h h Jl22,Mf

where each element in the matrix, hy, is evaluated from the receptance coupling equations, (5.10) and (5.11), by including both rotational and translational displacements due to forces and moments.

By substituting equations (5.12) and (5.13) into equations (5.10) and (5.11) , direct and cross transfer functions at point 1 and 2 with both rotational and translational degrees of freedom can be shown as :

ft- h h h h h A\\,ff A\\,fM ^A\2,ff ^A\2,JM H H llff A2\,ff A2\,fM [#nl = [HA h h h h h h '1 \,MM n n n A\\,Mf A\ \,MM A\2,Mf A\2,MM nA2\,Mf nA2\,MM l2\,ff 21, JM lA2l,ff A2\,jM lA\2,ff lA\2,fM

l l h h h h Â22,ff hA22jM 2\,Mf 2\MM n n n ,l A2\,Mf A2\,MM A\2,Mf A\2,MM h h 22,ff l22,jM hA22,ff hA22.JM hA22,ff hA22.JM nA22,Mf nA22,MM [#22 ] = i [*.l l h h h h h2,Mf 22,MM n n n n A22,Mf A22,MM A22,Mf A22,MM Â22,ff Â22,JM h h n n (5.14) A22,Mf A22,MM

where : Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 66

-1

h h h h l n n B22,ff B22.JM 2,fM A22,ff A22,jU '2,ff (5.15) tor- h h h h h h nA22,Mf nA22,MM nB22,Mf nB22,MM n2,Mf n2,MM

The first elements in the three matrices [HiffHn] and [H22] in (5.14) along with the equation for the reciprocity condition yields 4 sets of nonlinear equations:

1 h -h 1 1 ~\WnA\2,ff ' ^2,MM ^A\2,fM ' ^2,Mf/'^A2l,ff n n Ik ll,ff AU,ff ,l 2,MM I*-,2,fM ft* ' fir,2,Mf, \,ff J

+ {Â\2,JM ' ^2,ff Âl2,ff ' ^2,/M )' Â2\,Mf ] 1 h = h 77 7 7 \[(^/d2,# ' ^2,MM Â\2,fM ' ^2,Mf)' Â22,ff \n2,ff ' n2,MM ~ n2,JM ' n2,Mf I

^ {Â\2,/M ' ^2,ff Â\2,ff ' ^2,/M )' Â22,Mf ] 1 ^22,# — Â22,ff 77 7 ~T 7 \ \&A22,ff ' ^2,MM Â22,JM ' ^2,Mf)' Â22,ff + • • • n h V*2,# " "2,MJW — 2,JM ' 2,Mf )

(h-A22,JM ' ^2,ff ~ ^A22,ff ' ^2,JM )' ^A22,Mf ]

^2,fM = ^2,Mf

(5.16)

The four unknowns are, h2jf, h2,jM, h2,Mf and ^ZMM which are the receptances of the assembly at point 2 and need to be solved.

The terms on the left side of the first 3 equations in (5.16), hu,jj, hn,ff,and h22,jf are obtained by 3 impact hammer tests at points 1 and 2, as explained in Section 0 . The FRF's of the free-free substructure A, are obtained through the finite element method. This system of nonlinear equations is symbolically solved in MAPLE® [23] and is given in Appendix C.

Therefore the translational and rotational degrees of freedom FRFs at point B can be obtained as :

^B22,ff = ^2,ff ~ ^A22,ff

^B22,/M = ^B22,Mf ~ ^2,JM ~ ^A22,fM (5.17)

^B22,MM = ^2,MM ~ ^A22,MM

As it will be explained later in Section 5.5, the spindle dynamics are used to couple any arbitrary tool/ tool holder assembly to the spindle. The spindle dynamics which include the rotational degrees of freedom are stored in a matrix as shown below: Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 67

h h nB22,ff nB22,JM " B.22 ~ (5.18) h h nB22,Mf nB22,MM

The above receptance matrix is symmetric due to the reciprocity and linearity of the structure.

5.4 Inverse Receptance Coupling - Experimental Procedure

It is proposed that a shrink-fit holder without any cutting tool is used as a reference gauge to identify the spindle dynamics at point 2. Three impact modal tests are performed at points 1 and 2: The direct transfer function at point 1; hu,ff cross transfer function at points 1 and 2, hn,ff and direct transfer function measurement at point 2, ti22,ff as shown in Figure 5.3.

hnjf hii.u

ZEE

Tool Holder Flange Gauge\—y-j^ Line Accelerometer

Impact Hammer

Figure 5.3: Experimental Procedure for Decoupling and Identifying Spindle Dynamics

The inverse receptance coupling procedure is shown in Figure 5.4. The assumption is that after each tool change the machine dynamics remain unchanged up to point 2 on the machine tool. The uncommon part which is the tool holder shank is analytically modeled Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 68 using the finite element method and is decoupled and the receptance at point 2 on the spindle is obtained by solving the system of nonlinear equations in (5.16).

Di Structure AB Structure A Structure B

Figure 5.4: Inverse Receptance Coupling for Obtaining Spindle Dynamics

5.5 Receptance Coupling of Arbitrary Tool-Tool Holder Assembly to the Spindle

To couple the tool holder tool assembly to the machine tool, the coupling procedure is performed by employing (5.4) and (5.10) as follows:

H H +H l H H\\ =(HA,U ~ A,\2 ( B,22 A,22)~ ' A,2\) (5.19)

The frequency response functions at the two free ends of the tool holder (points 1 and 2) are obtained analytically by modeling the tool holder in finite elements using Timoshenko beam elements as shown in Figure 5.5. The fluted section of the end mill is considered to be 80% of the total diameter in the finite element model [20] and the tool-tool holder connection in the shrink-fit is assumed to be rigid. The tool - tool holder model is in the free-free Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 69 condition as the rigid body modes play an important role in the coupling between the structures. The damping ratio used for the finite element model is assumed to be 1-3 % which was verified by several impact tests.

%1,

Figure 5.5: Finite Element Model of Tool - Tool Holder Shank

5.6 Simulation and Experimental Results

The proposed receptance coupling method is experimentally evaluated on a horizontal machining center. The Mori Sieki SH403 horizontal machining center has an HSK 63A spindle taper. A shrink-fit holder without any tool is used as the reference gauge to obtain the dynamics at the flange as shown in Figure 5.6.

Figure 5.6: Shrink-Fit Holder to Identify Spindle Dynamics at the Flange on the Mori Sieki SH403 Horizontal Machining Center

The spindle dynamics are obtained through the inverse receptance coupling technique explained in Section 5.3. Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 70

19.05

i i

Figure 5.7: Tool - Tool Holder Assembly - Tooling A

The spindle dynamics can be used to obtain the frequency response function at the tool tip for any shrink-fit holder - tool assembly. The tool / tool holder assembly, tooling A, with a gauge length of 140 mm which is shown in Figure 5.7 is mathematically coupled to the spindle by employing the receptance coupling method explained in Section 5.5. The tool - tool holder connection in the shrink-fit is modeled as a rigid connection and the fluted tool is considered to be 80% of the total shank diameter. The predicted frequency response function at the tool tip is compared with the experiments in both x and y directions as shown in, Figure 5.8 and Figure 5.9. Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 71

Figure 5.8: Measured and Predicted Frequency Response Functions in X Direction at the Tool Tip on the Moriseiki SH403 Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 72

x 10

2.5 r Predicted Experiment

I -1.51 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)

x 10

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)

Figure 5.9: Measured and Predicted Frequency Response Functions in Y Direction at the Tool Tip on the Moriseiki SH403 Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 73

The comparisons between the predicted and the measured frequency response functions at the tool tip are in good agreement as shown in Figure 5.8 and Figure 5.9. The predicted frequency response functions are also able to model the higher frequencies more accurately. The hammer force input spectrum dropped drastically above the 2500-Hz frequency range, thus making it impossible to measure the higher frequency modes accurately. Chatter stability lobes are constructed using frequency response functions in the feed direction (X) and normal direction of cutting (Y) on the machine as shown in Figure 5.10. The comparison between the stability lobes, obtained by using the predicted frequency response functions and the measured FRFs, show an acceptable match. The critical depth of cut using the predicted measurements is however more conservative due to the fact that in the measurements the dominant modes are dynamically stiffer than the FRF predictions.

• Predicted Experiment 10

CL CD Q

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Spindle Speed (RPM) x 10

Figure 5.10: Stability Lobes Diagram for Tooling A

The effect of tool change on the frequency response function at the too tip is also investigated, and a different tooling assembly, tooling B , shown in Figure 5.11 is coupled to the spindle. The tool holder shank is shorter compared to the first tooling assembly and the overall gauge length is 112 mm. Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 74

Figure 5.11: Tool - Tool holder Assembly - Tooling B

The comparisons between the measured and predicted frequency response functions are shown in Figure 5.12 and Figure 5.13 . The chatter stability lobe in Figure 5.14 shows an increase in the axial depth of cut of about 30 % compared to the previous tooling. The comparisons between the experimental and the predictions employing the receptance coupling method show a very close agreement, due to the accurate prediction of the frequency response functions. The slight shift in the stable pockets is due to the fact that the predicted natural frequency is higher than the measurements. Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 75

x 10 15 Predicted Experiment

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)

x 10

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency(Hz)

Figure 5.12 : Measured and Predicted Frequency Response Functions in X Direction at the Tool Tip on the Moriseiki SH403 Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 76

x 10

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)

x 10

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)

Figure 5.13 : Measured and Predicted Frequency Response Functions in Y Direction Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 11

0I i i i i i i i i l 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 Spindle Speed (RPM) x 10

Figure 5.14 : Stability Lobes Diagram - Tooling B

The experiments clearly demonstrate the ability of the receptance coupling method to couple both long and short tool overhangs, with different tool holder geometries. In the next section, the obtained spindle dynamics are used to predict the FRF at the tool tip for a 10 mm diameter end mill in a shrink-fit tool holder. The tool length is tuned to optimize the material removal rate for machining a gearbox cover.

5.7 Tool Length Tuning

The effect of the tool overhang on the stability of the cutting operation has been investigated by various researchers as explained in Chapter 2. The tool overhang directly affects the frequency response function and the location of the stability lobes. Thus by varying the tool overhang the lobes can be shifted .The objective is to match the stable zones on the stability lobe to the spindle speed where the machine has optimum performance. For aluminum machining, this generally translates into running the machine at the maximum spindle speed without chatter vibrations and saturating the power and torque limits of the machine tool. Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 78

Once the spindle dynamics or machine tool dynamics are experimentally identified, the tool holder - tool assembly is modeled in finite elements and coupled to the spindle, as explained in Section 5.5. The tool overhang can be varied in finite elements and the frequency response function at the tool tip can be predicted for each case.

In order to apply the optimization technique to the tool tuning problem, objectives and design variables must first be identified. Schmitz [32] suggested two objectives for optimizing the tool length for stable milling: (1) tool length selection in order to increase the critical depth of cut, and (2) tool length selection to move a high stable lobe to the top spindle speed of the machine. In this thesis the objective is to select a tool length to maximize productivity, or the product of spindle speed and depth of cut, by shifting a high stable lobe to the maximum spindle speed of the machine.

The parameters used in tool tuning are the tool overhang and the spindle speed. There are constraints on the minimum and maximum tool overhang. The minimum tool overhang is selected by the NC programmer depending on the geometry of the tool holder and the final part. There are also limits on the torque and the power of the machine tool.

5.7.1 Chatter-Free Machining

Productivity in high speed machining of aluminum is often limited by chatter vibrations. The chatter vibrations depend on the cutting conditions and the dynamic property of the machine at the tool tip. The cutting conditions include tool geometry, work piece material, machining method, spindle speed and depth of cut. In order to select optimum cutting conditions, chatter stability lobes are developed. Figure 5.15 shows a typical stability lobe diagram for a milling operation. If the cutting conditions, i.e. spindle speed and depth of cut, are selected under the lobe, the process is stable .Otherwise, it will be unstable. For example at point B, with a lower spindle speed and depth of cut compared to point A, the cutting is unstable. However, at point A the cutting operation is stable and productivity has increased significantly. Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 79

0 2000 4000 6000 8000 10000 Spindle Speed [rpm]

Figure 5.15 : Typical Stability Lobe for Milling

Once the cutting conditions are selected by the process planner, the stability lobes are affected by the dynamic properties at the tool tip. Therefore the dynamics of the machine can be tuned by adjusting the tool overhang.

5.7.2 Optimization for Maximum Productivity

The objective function is the product of spindle speed and depth of cut, which is a measure of the material removal rate, and is defined as:

Maximize: fobj =n-a max (5.20)

where n is the spindle speed and amax is the maximum depth of cut corresponding to the selected spindle speed.

The cutting conditions, namely the depth of cut and the spindle speed, must be selected under the stability lobes to avoid chatter vibrations. The locations of the stability pocket depend on the natural frequencies of the machine tool, while the allowable depth of cut is determined by the dynamic stiffness of each mode at the tool tip. The proposed optimization method tunes the machine tool modes by varying the tool overhang in such a way that chatter free stability pockets are created at the maximum spindle speed of the machine. An example

is shown in Figure 5.16. The maximum spindle speed, n max, of the machine is 20000 rpm. Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 80

However the most stable zone is located above the top speed. By varying the tool overhang the stable zone can be shifted further to the left to match the top spindle speed on the machine.

5000 10000 15000 20000 25000 30000 Spindle Speed ( RPM )

Figure 5.16 : Typical Stability Lobe for Milling

The design variables are the tool overhang, / overhang, and the spindle speed, n. The constraints on the design variables are:

n I min ^ I overhang ^ I max , min It < il max (5.21)

Where / mi„ and / max are the minimum and maximum tool overhang, n max is the maximum

spindle speed on the machine tool and n min is the minimum speed depending on the range of

the search. A smaller n min, results in a wider search field. The tool overhang is iteratively varied by: hverhang W ~ ^min + (5.22) Where k is the iteration number and Al is the tool overhang increment. Practical tool overhang increments can be set to 1 or 2 mm. Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 81

Since there are only two design variables with explicit constraints, a linear search algorithm is employed to find the optimum tool overhang. The optimization procedure is shown in Figure 5.17.

Set Design variables / overhang , H

Objective function fuhJ(n,l overhang) Constraints Lin, Lax

Number of iteration k=0

k=k+l I overhang(k) = lmin+k.AI

Couple Finite Element Model of Tool / Tool Holder Assembly to Spindle (Receptance Coupling)

Calculate FRF and chatter stability lobes,

Search for Maximum /ohj(k) = Yl . Qmax

in Range f fimm , Ylmax] , noplim.— Yl

Optimum Overhang: / Optimum Spindle Speed: n <>/,nm jYes

/ opt int — I overhatig(k)

Figure 5.17 : Tool Tuning Optimization Flowchart

5.7.3 Optimization Results

The tooling assembly, consisting of a 10 mm, 2-fluted end mill and a shrink-fit holder shown in Figure 5.18,used in machining a helicopter gearbox cover, is tuned using the optimization method explained in Section 5.7.2 .The machine tool dynamics are identified using the inverse receptance decoupling technique explained in Section 5.6. Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 82

1 k ir MO i i >« •

Figure 5.18 : Shrink-fit Tooling with 2 Fluted ,10mm Slender End mill for Machining Helicopter Gearbox

The comparison between the predicted and experimental FRFs for the initial 45 mm overhang show an acceptable match, as plotted in Figure 5.19. Although the predictions are 20 % stiffer than the actual measurements on the machine, the split mode and the natural frequencies are well predicted.

x 10" 2.5

Predicted Experiment

£ 1.5

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)

Figure 5.19 : Comparison between Predicted and Experimental FRFs for 45 mm Overhang Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 83

The constraints for the minimum tool overhang and spindle speed are listed in Table 5.1.

1 min 45 (mm)

I max 55 (mm)

ft max 15000 (RPM)

ft min 12000 (RPM)

Table 5.1: Constraints on Tool Overhang and Spindle Speed

The required cutting conditions are listed in Table 5.2.

Cutter Diameter 10 (mm)

Number of Flutes 2

Workpiece Material AL 7050-T6

Feed Rate 0.1 (mm/flute)

Milling Type Full Slotting

Table 5.2: Cutting Conditions

Figure 5.20 shows the predicted frequency response function for the minimum and optimum tool overhang by using the receptance coupling method. It is clear that by increasing the tool overhang the dynamic stiffness and the natural frequency at the tool tip decreases. The split mode present in the case of the minimum tool overhang has also been attenuated by increasing the overhang. Figure 5.21 shows the comparisons between the two stability lobes. The decrease in the natural frequency shifts the stable pocket to match the maximum spindle speed of the machine. By increasing the tool overhang by 4 mm the productivity has increased by almost 2 times. The optimum spindle speed for maximum productivity is 14950 RPM.

Although in most cases the minimum tool overhang is still the optimum length for maximum productivity, there are cases present where the longer overhang can lead to a higher productivity as shown in the above example. 4000 6000 8000 10000 12000 15000 18000 20000 Spindle Speed ( RPM )

Figure 5.21 : Comparison between Tuned Tool and Short Tool Stability Lobes Chapter 5. Receptance Coupling of the Spindle and Arbitrary Tool Holder Dynamics 85

5.8 Summary

In this chapter, the inverse receptance coupling method is used to obtain the dynamics of the machine tool spindle including the tool holder flange. After each tool change, the dynamics of the machine up to the tool holder flange which is standard on all tool holders for a specific taper remains unchanged. A shrink-fit holder without any tool is used to identify the dynamics, and three impact modal tests are performed on the machine tool. The finite element model of the tool holder shank is mathematically subtracted from the machine tool. The identified spindle dynamics is used to couple any arbitrary shrink-fit tooling to the machine. This methodology greatly reduces the number of impact modal tests needed to identify the frequency response function at the tool tip. The receptance coupling method is also used to select the optimum tool overhang for a specific cutting operation. By varying the tool overhang, the higher stable lobes are shifted to match the maximum spindle speed on the machine, thus maximizing the productivity. Chapter 6 Conclusions

6.1 Conclusions

In this thesis the finite element model of tool holder and the spindle taper are developed using Timoshenko beam elements, and the spindle - tool holder interface is modeled using distributed rotational and translational springs. The contact stiffness per unit area of the distributed springs is derived by using two methodologies: the experimental method using a dedicated setup, and the finite element method, in which the interface is modeled using 3D contact elements. In both methods, the contact stiffness per unit area is calculated for a certain drawbar force. The finite element model is experimentally verified by coupling a CAT 40 shrink-fit tool holder to an experimentally identified spindle. It is found that the effect of the tool holder interface is reflected on to the higher frequency modes and the static stiffness at the tool tip. Furthermore the effect of the drawbar force on the dynamic stiffness of the spindle - tool holder interface is experimentally investigated on the setup. It is observed that with increased drawbar force, the dynamic stiffness of the interface decreases due to the decrease in damping. However, there is no significant change in the dynamic stiffness and damping after a certain drawbar force level, 15 kN for the CAT 50 taper. This is due to the fact that full contact has been established between the spindle taper and the tool holder. The tool - tool holder connection dynamics are also investigated for four different types of tool holders on the setup. Four different types of tool holders having the same overhang length are selected for comparison. The mode shapes relating to the bending of the tool - tool holder interface are obtained by using the experimental modal analysis technique on the setup, and their dynamic stiffness are compared to each other as a measure of chatter resistance. The comparisons are used as a guideline to select a tool holder for a specific milling application. Finally, the spindle dynamics on the machine tool are identified, using the inverse receptance coupling technique. The FRF at the tool tip varies after each tool change and time

86 Chapter 6. Conclusions 87 consuming measurements must be repeated. An inverse receptance coupling technique is developed to predict the rotational and translational spindle dynamics. The identified dynamics is used to couple any arbitrary shrink-fit tooling to the spindle. The receptance coupling method greatly reduces the number of impact modal tests needed to obtain the frequency response function of the machine. Furthermore, the mathematical model of the machine is used in an industrial application known as tool tuning to select the optimum tool overhang for a specified cutting operation.

6.2 Future Research Directions

The tool holder spindle connection is vital in determining the overall dynamics of the machine tool and the accuracy of the machined part since 15-20% of the deflection at the tool tip is contributed by the tool holder-spindle interface. The finite element model of the tool holder - spindle interface presented in this thesis uses the assumptions that the radial and rotational contact stiffness per unit area are independent of the overhang length of the tool holder. The model can be extended to include the effect of taper tolerance and the overhang length using set-ups with specified taper tolerances. Furthermore the spindle identification on the machine, which was obtained using the inverse receptance coupling technique, has to be combined with the finite element model of the interface. The effect of the drawbar force and the taper tolerances as well as the overhang effect on the interface can be taken into account when using the receptance coupling method.

The selection of tool holders for a specific cutting operation in terms of dynamic stiffness needs to be studied in more extent, and modeling the tool holder - tool connection in different kind of tool holders can be taken into account. Bibliography

[I] Altintas, Y., Budak, E., 1995 "Analytical Prediction of Stability Lobes in Milling", Annals ofCIRP, Vol. 44, pp.357-362. [2] Altintas, Y., 2000, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press [3] ANSYS9 [4] Aoyama,T., Inasaki , I. ,2001, "Performances of HSK Tool Interfaces under High Rotational Speed ", Annals ofCIRP, Vol. 50/1, pp.281-284. [5] Avitabile,P.,2002, "Twenty Years of Structural Dynamics Modification", Proc. Of 20th IMAC, Los Angeles, USA, pp.356-372. [6] Bathe , K.J, 1996 , Finite Element Procedures , Prentice Hall. [7] Budak, E., and Altintas, Y., 1998, "Analytical prediction of chatter stability in milling -Part I: general formulation", Journal of Dynamic Systems, Measurement, and Control, Transactions of the ASME, Vol.120, pp.22-30. [8] Budak, E., and Altintas, Y., 1998, "Analytical prediction of chatter stability in milling -Part II: application of the general formulation to common milling systems", Journal of Dynamic Systems, Measurement, and Control, Transactions of the ASME, Vol.120, pp.31-36. [9] Cao, Y., Altintas, Y., 2004, "A General Method for the Modeling of Spindle- Bearing Systems", Transactions of ASME, Journal of Mechanical Design, Vol. 126 ,pp. 1089-1104. [10] Cowper G.R., 1966, "The Shear Coefficient in Timoshenko Beam Theory", Transactions of ASME, Journal of Applied Mechanics, pp. 335 -339. [II] Cut Pro ® - Milling Module . Manufacturing Automation Laboratory , University of British Columbia. [12] Davies, M., Dutterer, B., Pratt, J.R., Schaut, A.J. 1998"On the Dynamics of High- Speed Milling with Long ,Slender End Mills", Annals ofCIRP, Vol. 47, pp.55-60. [13] Durate ,M.L.M, Ewins ,D.J. ,2000 "Rotational Degrees of Freedom For Structural Coupling Analysis Via Finite Difference Technique with Residual Compensation", Mechanical Systems and Signal Processing, Vol. 14, pp.205-212 [14] Duncan , G.S, Tummond ,M.F. , T.,L. Schmitz , 2005 "An Investigation of the Dynamic Absorber Effect in High-Speed Machining", International Journal of Machine Tools & Manufacture, Vol. 45, pp.497-507 [15] Ewins, D.J., 1984 , Modal Testing : Theory and Practice, Research Studies Press Ltd. [16] International Standard Organization 1947 , Standard Taper Rate Tolerances

88 Bibliography 89

[17] Jacobs, T.P., 1999, M.A.Sc Thesis "Characterization of the Machine Tool Spindle to Tool holder Connection ",University of North Carolina at Charlotte , Department of Mechanical Engineering. [18] Kim , T.R. , Wu , S.M., 1989 "Identification of Joint Parameters for a Taper Joint "Journal of Engineering for Industry, Transactions of the ASME , Vol. Ill pp.282- 287. [19] Kivanc, E.B., Budak , E.. 2004 "Structural Modeling of End Mills for Form Error and Stability Analysis", International Journal of Machine Tools & Manufacture, Vol. 44, pp.1151-1161 [20] Kops, L. Vo.D.T. , 1990, "Determination of Equivalent Diameter of an End Mill based on its Compliance", Annals of CIRP, Vol. 39, pp.93-96. [21] Levina, Z.M., "Stiffness Calculations for Cylindrical and Taper Joints", Machines and Tooling, Vol. XLI, pp.5-10. [22] Levina, Z.M., et al ,'Taper Connection Stiffness", Machines and Tooling, Vol. XLIVNo. 10, pp.21 -26. [23] MAPLE® - Waterloo MAPLE Inc. [24] Merritt, H. E., 1965, "Theory of self-excited machine tool chatter", ASME Journal of Engineering for Industry, Vol.87, pp.447-454. [25] Movahhedy, M.,Gerami, J. 2006 "Prediction of Spindle Dynamics in Milling by Sub-structure Coupling", International Journal of Machine Tools & Manufacture, Vol. 46, pp.243-251 [26] Opitz, FL, and Bernardi, F., 1970, "Investigation and calculation of the chatter behavior of lathes and milling machines", Annals of CIRP, Vol.18, pp.335-343. [27] Park, S.S, Altintas, Y., Movahhedy, M., 2003 "Receptance Coupling for End Mills", International Journal of Machine Tools & Manufacture, Vol. 43, pp.889-896 [28] Park, S.S, 2003, Ph.D. Thesis: High Frequency Bandwidth Cutting Force Measurements in Milling Using The Spindle Integrated Force Sensor System, University of British Columbia , Department of Mechanical Engineering ,Vancouver, Canada. [29] Rivin, E.I., 2000, "Tooling Structure: Interface between Cutting Edge and Machine Tool", Annals of CIRP, Vol. 49/2, pp.591-634. [30] Rivin, E.I., 1999, Stiffness and Damping in Mechanical Design, Marcel Dekker Inc. [31] Rivin, E.I., 1993, "Influence of Tool holder Interfaces on Tooling Performance", Transactions of NAMRI/SME, Vol. XXI, pp.173-179. [32] Schmitz, T., Donaldson, R., 2000 "Predicting High -Speed Machining Dynamics by A Substructure Analysis", Annals of CIRP, Vol. 49, pp.303-308. [33] Smith, S., Jacobs, T.P., Halley, J., 1999, "The Effect of Tool Length on Stable Metal Removal Rate in High Speed Milling", Annals of CIRP, Vol. 47/1, pp.307- 310. Bibliography 90

[34] Smith, S., Winfough, W.R., Halley, J., 1998, "The Effect of Drawbar Force on Metal Removal Rate in Milling", Annals ofCIRP, Vol. 48/1, pp.293-296. [35] Spindle Pro ® . Manufacturing Automation Laboratory , University of British Columbia. [36] Tlusty, J., and Polacek, M., 1963, "The stability of machine tools against self- excited vibrations in machining", International Research in Production Engineering, ASME, pp.465-474. [37] Tlusty , J.,Smith, S.,Winfough, 1996,"Techniques for the Use of Long Slender End Mills in High Speed Milling", Annals ofCIRP, Vol. 45, pp.393-396. [38] Tobias, S. A., 1965, Machine Tool Vibrations, John Wiley, New York. [39] Wang , J.H. , Horng , S.B., 1994 "Investigation of the Tool Holder System with a Taper Angle 7:24 ", InternationalJournal of Machine Tools and Manufacture , Vol. 34,No.8 , pp.1163-1176. [40] Week , M. , Schubert , I., 1994 " Final Report on the Research Project Interface Machine/Tool: Testing and Optimization ", WZL Laboratory for Machine Tools and Applied Economics, Aachen. Appendix A Timoshenko Beam Element Formulations

A.l Beam Element Formulations

x,u

Vi v2

Figure A.l: Timoshenko Beam Element

A general 2D Timoshenko beam element is formulated here however the developed model contains translation in the x direction and rotation about the y axis. Figure A.l shows a two-noded Timoshenko beam element of length Lsim- u, v and w indicate the displacements

in the radial x, radial y and axial directions, while 6X, dy and dz represent the rotations around the x, y and z (torsional) axes, respectively. The subscripts 1 and 2 respectively denote displacements and/or rotations belonging to node 1 and node 2. £ is the non-dimensional position of the point P%, measured from node 1. It assumes a minimum value of 0 when P$ is located at node 1, and a maximum value of 1 when it is located at node 2.

The stiffness and mass matrices of the Timoshenko beam element, K~Eim and Msim, are given respectively in Eq.(A.l) and (A.2).

91 Appendix A.Timoshenko Beam Element Formulation 92

k? 0 0 0 0 -K 0 kxs 0 0 0 0 0 0 0 0 (A.1) 0 0 0 ^3 0 K 0 -K 0 ^3 0 0 0 K

0 0 -k2 0 0 0 0 0

0 ^*3 0 0 0 0 Jc-^ 0 0 0 0 K 0 ~kl 0 0 0 0 0 0 0 0 -K 0 0 0 0

mi m,

0 0 m2

x 0 -ml 0 m s ml 0 0 0

0 0 0 0 0 w4 PA Elm (A.2) -'Elm 0 0 0

0 0 m7 0 0 0 0 0 m2

0 0 0 0 0 -W3 0 W5

The expressions for the elements of the stiffness and mass matrices are given in Eq.(A.3) and (A.4), respectively.

, _ 12£/ „. 6EI jt = £/m U _ A-ElmE . Eim,ii

k2— , 4,ma+,) ^Elm (A.3) G Elmji ElmM K = hlm,zz ,(4 +

N,«/ 3

3 2 'u/4 £ - l—^ £ —^£ l + O. 2 2

(l + Oj)LE (A.8)

(1 + *j)L*

N22 = Ngz2 = £

Just as in the previous case, the subscript j represents either the x or y-axis. Appendix B Tables of Modal Parameters

Drawbar 1st Mode ForcefkN] kC[N/m] co [Hz] k[N/m] 2 8.28E+05 5.79E-03 3253.4 1.43E+08 4 6.45E+05 4.97E-03 3356.9 1.30E+08 6 4.26E+05 3.32E-03 3423.8 1.28E+08 8 3.75E+05 2.95E-03 3454.1 1.27E+08 10 3.34E+05 2.61E-03 3479.5 1.28E+08 12 3.28E+05 2.57E-03 3483.8 1.28E+08 14 3.10E+05 2.43E-03 3490.3 1.28E+08 16 2.92E+05 2.31E-03 3493.8 1.26E+08 18 2.89E+05 2.27E-03 3497.2 1.28E+08 20 2.70E+05 2.13E-03 3504.9 1.27E+08 22 2.60E+05 2.03E-03 3508.4 1.28E+08 24 2.48E+05 1.95E-03 3511.0 1.27E+08 26 2.39E+05 1.88E-03 3513.9 1.27E+08 28 2.30E+05 1.81E-03 3517.5 1.27E+08 30 2.21E+05 1.74E-03 3519.6 1.27E+08

Table B.l: Modal Parameters vs. Drawbar Force for Collet Chuck -10 mm Overhang

Common experimental conditions:

Tightening torque: 30 Nm

Spindle taper: CAT50 Tool Holder: BT50-CTM16-105 (Showa Tool)

Tool: q> 16 carbide test bar (OAL 60 mm) Tool overhang from holder end: 10 mm

Distance between gauge plane and holder end: 105 mm

Hammer: Kistler 9722A2000 with steel tip Accelerometer: Kistler 8778A500

Lubrication of spindle taper and tool clamp: dry

95 Appendix B. Tables of Modal Parameters 96

Drawbar 1st Mode ForcefkN] k£[N/m] co [Hz] k[N/m] 2 1.98E+06 9.23E-03 3092.4 2.15E+08 4 1.50E+06 7.12E-03 3210.8 2.11E+08 6 1.32E+06 6.55E-03 3290.0 2.02E+08 8 1.29E+06 6.63E-03 3317.4 1.94E+08 !0 1.13E+06 6.19E-03 3347.0 1.83E+08 12 1.17E+06 6.24E-03 3355.7 1.88E+08 14 1.10E+06 5.83E-03 3375.3 1.89E+08 16 1.01E+06 5.35E-03 3388.3 1.88E+08 18 9.20E+05 4.98E-03 3398.0 1.85E+08 20 9.16E+05 4.85E-03 3410.7 1.89E+08 22 8.77E+05 4.57E-03 3416.7 1.92E+08 24 8.14E+05 4.43E-03 3422.7 1.84E+08 26 7.68E+05 4.20E-03 3430.6 1.83E+08 28 7.62E+05 4.17E-03 3434.6 1.83E+08 30 7.56E+05 4.07E-03 3439.3 1.86E+08

Table B.2: Modal parameters vs. Drawbar Force for Power Chuck-10 mm Overhang

Common experimental conditions:

Tightening torque: (until the end of the screw, according to instruction manual)

Spindle taper: CAT 50 Tool Holder: BT50-HPC16-105A (Showa Tool)

Tool: (#16 carbide test bar (OAL 60 mm) Tool overhang from holder end: 10 mm

Distance between gauge plane and holder end: 105 mm

Hammer: Kistler 9722A2000 with steel tip Accelerometer: Kistler 8778A500

Lubrication of spindle taper and tool clamp: dry (all surfaces wiped with a dry cloth) Appendix B. Tables of Modal Parameters

Drawbar 1st Mode ForcefkN] kc;[N/m] co [Hz] k[N/m]

2 1.92E+06 1.30E-02 3480.9 1.48E+08 4 1.41E+06 1.03E-02 3639.1 1.37E+08 6 1.26E+06 9.30E-03 3718.8 1.36E+08 8 1.16E+06 8.50E-03 3759.2 1.36E+08 10 1.07E+06 7.94E-03 3779.9 1.35E+08 12 9.89E+05 7.44E-03 3798.6 1.33E+08 14 9.56E+05 7.11E-03 3815.7 1.35E+08 16 8.98E+05 6.77E-03 3825.9 1.33E+08 18 8.57E+05 6.54E-03 3838.0 1.31E+08 20 8.38E+05 6.39E-03 3843.3 1.31E+08 22 8.06E+05 6.12E-03 3853.8 1.32E+08 24 7.87E+05 5.98E-03 3858.1 1.32E+08 26 7.67E+05 5.82E-03 3866.0 1.32E+08 28 7.40E+05 5.62E-03 3870.7 1.32E+08 30 7.18E+05 5.53E-03 3872.9 1.30E+08

Table B.3: Modal Parameters vs. Drawbar Force for Hydraulic Chuck -10 mm Overhang

Common experimental conditions:

Tightening torque: (until the end of the screw, according to instruction manual)

Spindle taper: CAT50 Tool Holder: BBT50-HDC16L-105 (Daishowa Seiki)

Tool: ^16 carbide test bar (OAL 60 mm) Tool overhang from holder end: 10 mm

Distance between gauge plane and holder end: 105 mm

Hammer: Kistler 9722A2000 with steel tip Accelerometer: Kistler 8778A500

Lubrication of spindle taper and tool clamp: dry Appendix B. Tables of Modal Parameters 98

Drawbar 1st Mode ForcefkN] kt;[N/m] co [Hz] k[N/m] 2 8.37E+05 7.68E-03 3089.6 1.09E+08 4 6.56E+05 5.98E-03 3161.5 1.10E+08 6 5.07E+05 4.59E-03 3213.3 1.11E+08 8 3.20E+05 3.40E-03 3246.3 9.41E+07 10 2.67E+05 2.97E-03 3272.0 8.98E+07 12 2.65E+05 2.90E-03 3280.7 9.15E+07 14 2.67E+05 2.88E-03 3288.6 9.28E+07 16 2.38E+05 2.64E-03 3302.1 9.02E+07 18 2.28E+05 2.54E-03 3305.4 8.96E+07 20 2.00E+05 2.25E-03 3312.5 8.90E+07 22 1.88E+05 2.14E-03 3317.3 8.79E+07 24 1.77E+05 2.04E-03 3320.1 8.67E+07 26 1.74E+05 1.98E-03 3322.4 8.81E+07 28 1.63E+05 1.87E-03 3328.6 8.70E+07 30 1.61E+05 1.82E-03 3331.1 8.84E+07

Table B.4: Modal Parameters vs. Drawbar Force for Shrink-Fit -10 mm Overhang

Common experimental conditions:

Spindle taper: CAT 50 Tool Holder: BT50-SF16-105-N (Showa Tool)

Measurement point: Tool tip Tool: \6 carbide test bar (OAL 60 mm)

Tool overhang from holder end: 10 mm

Distance between gauge plane and holder end: 105 mm

Hammer: Kistler 9722A2000 with steel tip Accelerometer: Kistler 8778A500

Lubrication of spindle taper and tool clamp: dry Appendix B. Tables of Modal Parameters 99

st Drawbar 1 Mode 2nd Mode force [kN] kt,[N/m] co [Hz] k[N/m] kc;[N/m] co [Hz] k[N/m]

2 2.48E+04 2.46E-03 1717.1 1.01E+07 1.13E+06 7.19E-03 3913.4 1.57E+08

4 2.98E+04 2.47E-03 1726.8 1.21E+07 1.81E+06 1.03E-02 4008.5 1.76E+08

6 4.24E+04 3.50E-03 1735.0 1.21E+07 1.61E+06 7.84E-03 4078.7 2.05E+08

8 5.44E+04 5.85E-03 1730.4 9.29E+06 1.17E+06 5.86E-03 4116.8 2.00E+08

10 3.58E+04 3.45E-03 1732.3 1.04E+07 9.11E+05 4.96E-03 4148.7 1.84E+08

12 3.02E+04 3.05E-03 1735.2 9.91E+06 8.57E+05 4.84E-03 4163.7 1.77E+08

14 3.18E+04 3.24E-03 1735.2 9.81E+06 8.12E+05 4.52E-03 4178.7 1.80E+08

16 2.68E+04 2.43E-03 1735.9 1.10E+07 8.50E+05 4.37E-03 4186.9 1.94E+08

18 2.31E+04 2.17E-03 1737.1 1.07E+07 7.67E+05 4.16E-03 4195.5 1.84E+08

20 2.18E+04 2.06E-03 1738.0 1.06E+07 7.10E+05 4.02E-03 4202.3 1.77E+08

22 2.20E+04 2.08E-03 1738.2 1.06E+07 7.15E+05 3.90E-03 4206.4 1.83E+08

24 2.05E+04 2.08E-03 1739.1 9.87E+06 6.82E+05 3.91E-03 4211.7 1.74E+08

26 2.08E+04 2.05E-03 1739.3 1.01E+07 6.78E+05 3.84E-03 4214.8 1.77E+08

28 1.92E+04 2.02E-03 1739.9 9.54E+06 6.48E+05 3.76E-03 4219.0 1.72E+08

30 2.22E+04 2.03E-03 1739.8 1.09E+07 7.29E+05 3.78E-03 4221.2 1.93E+08

Table B.5: Modal Parameters vs. Drawbar Force for Collet Chuck - 64 mm Overhang

Common experimental conditions:

Tightening torque: 30 Nm

Spindle taper: CAT 50 Tool Holder: BT50-CTM16-105 (Showa Tool)

Measurement point: Tool tip Tool: cM6 carbide test bar (OAL 125 mm)

Tool overhang from holder end: 64 mm

Distance between gauge plane and holder end: 105 mm

Hammer: Dytran 5800SL with extender mass Accelerometer: Kistler 8778A500

Lubrication of spindle taper and tool clamp: dry Appendix B. Tables of Modal Parameters 100

st Drawbar 1 Mode 2nd Mode

force [kN] kc;[N/m] co [Hz] k[N/m] k^[N/rn] co [Hz] k[N/m]

2 8.58E+04 8.50E-03 1582.9 1.01E+07 2.65E+06 1.38E-02 3514.9 1.93E+08

4 4.78E+04 4.15E-03 1590.1 1.15E+07 2.16E+06 1.13E-02 3650.0 1.90E+08

6 4.32E+04 3.94E-03 1596.3 1.10E+07 1.84E+06 9.62E-03 3727.3 1.92E+08

8 3.89E+04 3.85E-03 1599.5 1.01E+07 1.57E+06 8.63E-03 3769.0 1.81E+08

10 3.46E+04 3.84E-03 1601.8 9.01E+06 1.30E+06 7.55E-03 3801.9 1.72E+08

12 3.53E+04 3.80E-03 1603.6 9.27E+06 1.16E+06 6.90E-03 3827.6 1.67E+08

14 3.65E+04 3.72E-03 1605.0 9.81E+06 1.22E+06 6.70E-03 3844.5 1.83E+08

16 3.68E+04 3.68E-03 1606.1 1.00E+07 1.27E+06 6.75E-03 3858.4 1.89E+08

18 3.31E+04 3.57E-03 1607.2 9.26E+06 1.02E+06 5.84E-03 3875.3 1.75E+08

20 3.12E+04 3.53E-03 1607.8 8.85E+06 9.97E+05 5.98E-03 3884.0 1.67E+08

22 3.06E+04 3.53E-03 1608.4 8.67E+06 9.23E+05 5.71E-03 3894.7 1.62E+08

24 3.13E+04 3.53E-03 1608.9 8.87E+06 9.84E+05 5.66E-03 3901.1 1.74E+08

26 3.07E+04 3.50E-03 1609.6 8.78E+06 9.71E+05 5.50E-03 3910.0 1.76E+08

28 2.91E+04 3.49E-03 1609.9 8.34E+06 9.27E+05 5.62E-03 3915.6 1.65E+08

30 2.83E+04 3.45E-03 1610.4 8.20E+06 8.77E+05 5.46E-03 3922.1 1.60E+08

Table B.6: Modal Parameters vs. Drawbar Force for Milling Chuck - 64 mm Overhang

Common experimental conditions:

Tightening torque: (until the end of the screw, according to instruction manual)

Spindle taper: CAT 50 Tool Holder: BT50-HPC16-105A (Showa Tool)

Measurement point: tip Tool: 016 carbide test bar (OAL 125 mm)

Tool overhang from holder end: 64 mm

Distance between gauge plane and holder end: 105 mm

Hammer: Dytran 5800SL with extender mass Accelerometer: Kistler 8778A500

Lubrication of spindle taper and tool clamp: dry Appendix B. Tables of Modal Parameters 101

st nd Drawbar 1 Mode 2 Mode

force [kN] k£[N/m] co [Hz] k[N/m] kC[N/m] co [Hz] k[N/m]

2 3.59E+04 4.18E-03 1665.8 8.59E+06 4.50E+06 2.27E-02 3962.0 1.98E+08

4 4.97E+04 6.56E-03 1669.7 7.57E+06 3.43E+06 1.47E-02 4132.7 2.33E+08

6 2.89E+04 3.38E-03 1676.6 8.54E+06 3.08E+06 1.38E-02 4221.3 2.23E+08

8 2.78E+04 3.30E-03 1678.7 8.43E+06 2.81E+06 1.30E-02 4273.6 2.16E+08

10 2.61E+04 3.14E-03 1681.1 8.31E+06 2.53E+06 1.16E-02 4313.0 2.17E+08

12 2.58E+04 3.04E-03 1683.2 8.47E+06 2.53E+06 1.11E-02 4343.7 2.27E+08

14 2.58E+04 3.19E-03 1685.0 8.10E+06 2.34E+06 1.05E-02 4368.8 2.22E+08

16 2.56E+04 3.33E-03 1686.4 7.70E+06 2.02E+06 1.03E-02 4386.4 1.97E+08

18 2.81E+04 3.60E-03 1687.0 7.81E+06 2.10E+06 9.70E-03 4399.9 2.17E+08

20 2.65E+04 3.69E-03 1687.5 7.19E+06 1.79E+06 9.21E-03 4413.8 1.94E+08

22 2.81E+04 3.77E-03 1688.0 7.44E+06 1.77E+06 8.31E-03 4425.4 2.13E+08

24 2.81E+04 3.77E-03 1688.2 7.44E+06 1.73E+06 8.55E-03 4434.3 2.02E+08

26 2.77E+04 3.81E-03 1688.7 7.26E+06 1.71E+06 8.40E-03 4443.2 2.03E+08

28 2.76E+04 3.83E-03 1688.7 7.20E+06 1.59E+06 7.91E-03 4448.4 2.02E+08

30 2.81E+04 3.87E-03 1689.3 7.27E+06 1.62E+06 7.58E-03 4456.2 2.14E+08

Table B.7: Modal Parameters vs. Drawbar Force for Hydraulic Chuck - 64 mm Overhang

Common experimental conditions:

Tightening torque: (until the end of the screw, according to instruction manual)

Spindle taper: CAT 50 Tool Holder: BBT50-HDC16L-105 (Daishowa Seiki)

Measurement point: Tool Tip Tool: \6 carbide test bar (OAL 125 mm)

Tool overhang from holder end: 64 mm

Distance between gauge plane and holder end: 105 mm

Hammer: Dytran 5800SL with extender mass Accelerometer: Kistler 8778A500

Lubrication of spindle taper and tool clamp: dry Appendix B. Tables of Modal Parameters 102

sl Drawbar l Mode 2nd Mode

force[kN] kc;[N/m] co [Hz] k[N/m] kC[N/m] co [Hz] k[N/m]

2 1.48E+04 1.34E-03 1665.4 1.10E+07 7.01E+05 5.04E-03 4159.5 1.39E+08

4 1.14E+04 1.09E-03 1676.0 1.05E+07 6.38E+05 5.47E-03 4272.0 1.17E+08

6 9.54E+03 9.69E-04 1681.4 9.84E+06 4.56E+05 3.79E-03 4334.1 1.20E+08

8 9.12E+03 8.87E-04 1685.7 1.03E+07 4.52E+05 3.74E-03 4377.4 1.21E+08

10 8.89E+03 8.39E-04 1687.4 1.06E+07 4.32E+05 3.55E-03 4398.2 1.22E+08

12 7.85E+03 7.98E-04 1689.3 9.84E+06 3.63E+05 3.14E-03 4419.7 1.16E+08

14 8.38E+03 7.84E-04 1690.7 1.07E+07 3.64E+05 2.76E-03 4434.8 1.32E+08

16 7.81E+03 7.62E-04 1691.7 1.03E+07 2.99E+05 2.33E-03 4445.9 1.28E+08

18 7.41E+03 7.50E-04 1692.8 9.88E+06 2.51E+05 2.06E-03 4457.1 1.22E+08

20 6.97E+03 7.32E-04 1693.5 9.53E+06 2.29E+05 1.96E-03 4464.2 1.17E+08

22 7.20E+03 7.28E-04 1694.3 9.88E+06 2.26E+05 1.83E-03 4473.2 1.24E+08

24 6.99E+03 7.18E-04 1694.8 9.73E+06 2.12E+05 1.76E-03 4477.7 1.21E+08

26 6.76E+03 7.16E-04 1695.2 9.44E+06 2.08E+05 1.69E-03 4482.4 1.23E+08

28 6.62E+03 7.08E-04 1695.6 9.34E+06 1.95E+05 1.64E-03 4486.9 1.19E+08

30 6.74E+03 6.98E-04 1695.9 9.65E+06 1.92E+05 1.59E-03 4490.2 1.21E+08

Table B.8: Modal Parameters vs. Drawbar Force for Shrink-Fit - 64 mm Overhang

Common experimental conditions:

Spindle taper: CAT 50 Tool Holder: BT50-SF16-105-N (Showa Tool)

Measurement point: Tool tip Tool: 016 carbide test bar (OAL 115 mm)

Tool overhang from holder end: 64 mm

Distance between gauge plane and holder end: 105 mm

Hammer: Dytran 5800SL with extender mass Accelerometer: Kistler 8778A500

Lubrication of spindle taper and tool clamp: dry Appendix C Inverse Receptance Coupling Solution

The symbolic solution to the system of equations (5.16) is given below in (C.3), and the variables in these solutions are defined in (C.l) and (C.2).

hnff=u, hl2iff=v, h22Jf=w (C.l)

a = n c ^A\\,jf~ >hAUjrr b ^ Ai\,ff~ >hA22,jf~d (C.2) nA\2,JM ~ ei nA2\,Mf =f ^A22,Mf =& >nA22,JM =K

Finally, the solution obtained by using MAPLE is:

, _ {d2fev + d2age-d2uge + dbkug-dfebw-dbkag-dcgbe-kcbgv+cb2gk + cbwge)

_ [d3e(a-u)+cb2(gw+ dk) + ced2(v-b)- fb2dw-cbdv(k +g) + d2bk(u-a) + fbd2v\

Kff - -p

2 2 2 2 2 2 h2MM -{je d + dg ea + dbkf^-dkfbe+dej^g-dfe w-dge c-dg eu-dgfbe-

- bg2ka - kvgce-kvgfb + kbug2 + kb2gf + kcbge+ ge2cw)/ R

(C.3) where the common denominator term in the solution is :

P = {-ugbw- fb2w + cb2k + Jbwv-cewv + cbew-2cbkv+cv2k-

+ d2ae - d2ue - dfv2 +dvak-duvk-dagv+dcev + dubk- (C4)

- dcbe - dabk + duew- daew+ dfbv + dugv + agbw)

103