CHATTER ANALYSIS OF MACEUNE TOOL

SYSTEMS IN TURMNG PROCESSES

Zhanchen Wang

-4 thesis submitted in conformity with the requirements

for the Degree of Doctor of Philosophy in

Graduate Department of Mechanka1 and indusmal Engineering

University of Toronto

O Copyright by Zhanchen Wang 200 1

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Chatter Analysis of Machine Tool Systems in Turning Processes

Ph.D. Thesis, 2001

Zhanchen Wang

Department of Mechanical and Industrial Engineering

University of Toronto

Abstract

Chatter of a machine tool system consisting of a flexible workpiece and a cutting twI flexibly

mounted on a guided bed in turning process is investigated in this thesis. Chatter onset conditions

are accurately determined using combinations of the finite element method, the modal analysis

method, and the Laplace transform technique. Stability charts separating stable and unstable

cutting operations are determined using the Nyquist criteria and provided for conventional lathes

and workpieces of various shapes subjected to different boundary constraints simulating the

effects of chuck and center. These charts can be conveniently used in industry to select an

appropriate set of cutting parameters for a chatter-ftee turning operation.

Four dpamical models are proposed in this thesis to handie chatter of machine tool systems. The

frrst mode1 is applicable to machine tool systerns having workpieces of very large stifhess

compared to the cutting tooI mounting stiffiiess. The second mode1 is developed for workpiece

of reiativeiy srna11 stiflhess. The third model handles general machine tool systems in which

motions of the workpiece and cutting tool structure are truIy coupled. The fourth mode1 is

applicabie to tuming processes involving the novel use of two cutting tools.

in chatter andysis, the tool structure is considered as a mass-spring-darnper system having two

degrees of fieedom; the workpiece is considered a spinning beam structure whose displacement-

smin relationships obey the Timoshenko theory. The finite element method and Lagrange

equations are employed to formulate the system equations of motion for the workpiece. In al1

cases studied, the cutting force may be applied at any locations along the workpiece.

The procedure for determining the chatter onset conditions of machine too1 systems represented

by the four dynamical models is programrned into a computer code written in the Matlab

Ianguage. Chatter-free cutting conditions may now be easily established for any machine tooI

system in tuniing process by simply providing a few input parameters and running the computer

program.

The author would like to express his sincere gratitude and appreciation to his supervisors, Dr. W.

L. Cleghom of the University of Toronto and Dr. S. D. Yu of Ryerson University, for their

invaluable inspiration and guidance thmughout this thesis work. Without their advice,

encouragement and support, this work could not have been a reality.

Particular thanks are due to Prof. M. A, Elbestawi, Dr. R T e k and Mr. G. Quintero, Deparunent

of Mechanical Engineering at McMaster University, for their helpful suggestions

He would Iike to thank Ms, B. Fung, Mr. L. Roosman and Mr. D. Esdaile, in the Department of

Mechanical and industrial Engineering at the University of Toronto for their help. He gratefully

acknowiedges the financia1 support fiom the N a m l Sciences and Engineering Research Council

of Canada.

Most imponantly, the author wouId like to thank his parents, his brother and sistets for their

meat suppon throughout his academic years, -

Finally, fie would like to express tiis specid thanks to his wife, HuiIing, and his daughter, Annie,

for their support, understanding and patience.

1 Introduction ............................................................................................................................... I ........................................................................................................................ 1 .1 Background 1

1.2 Lirerature Review ................................................................................................................ 9

1.2.1 Causesofchatter .......... ... ........................................................................................ 9 ............................................................................................ 12.2 Models of cutting force. 12

................................................................................. 1 - 2 2 1 GeneraI cutting force models 12

.......................................................................................... 1.2.2.2 Ploughing force model I f

............................................................................................................ 1.2.3 Chatter mode1 18

............................................................................ 1 .2.3. 1 Charter of cutting tool structure 1 S

1.2.3.2 Chatter of workpiece ............................................................................................. 20

3 3 ........................................................................................................... 1.3 Overview of Thesis --

2 Cbatter of Tool Structure Incorporating the Effect of

Plougbing Force ........................................ .................................................... ............. 2 5 2.1 Introduction ....................................................................................................................... 35

vi

................... Chatter Mode1 of TooI Structure .. ............................................................ 26

.............................................................................................................. Stability Analysis 31

Numerical Examples ......................................................................................................... 33

Summary ........................................................................................................................... 41

3 Free Vibration of Spinning Stepped-Shaft Workpiece. ....................................................... 42 introduction ................................................................................................................... 43

Governing Equations of Motion for Rotating .............................................................. 35

Stepped Shaft .................................................................................................................... 45

Modal Analysis ................................................................................................................ 55

Convergence Tests and Numerical Results ....................................................................... 58

........................................................................................................................... Summary 69

................................................... 4 Stability Analy sis of Spinning Stepped-Shaft Workpiece 71 ....................................................................................................................... 4.1 htroductioa 71

4.2 Chatter Mode1 ............................................................................................................... 72

4.2.1 Stepped workpiece mode1 ........................................................................................... 77

4.3 Cutting Force Mode1 .......................................................................................................... 74

4.4 StabiIity Analysis ............................................................................................................... 77

4.41 Modal analysis method ................................................................................................ 77

............................................................................... 4.4.2 Direct Lapiace rransfonn method 80

................................................... 4.4.3 Cumng force at an arbitrary location on workpiece 82

4.5 Numericd Examples .......................................................................................................... 88

vii

4.5.1 Effects of mode order on stability ....................................................................... 89

4.5.2 Cornparisons of the modal analysis method and direct Laplace

transfom method ...................................................................................................... 94 . .

4.5.3 Effects of damping factor on stability ........................................................................ 95

4.5.4 Cornparison of present work and literature ............................................................ 97

........... 4.5.5 Simulation results of the dyaarnic system with typical boundary conditions 100

4.6 Summary .......................................................................................................................... 115

5 Chatter Model and Stability Analysis of Coupled System Consisting of Spinnimg

Stepped-Sbaft Workpiece and Tool Structure .............................................................. 116 5.1 Introduction ...................................................................................................................... 116

5.2 Chatter Mode1 of the Coupled System ............................................................................. 117

5.3 Cutting Force ............................................................................................................. 119

5 -4 Equations of Motion for the Coupled Systern and Stability Analysis ............................. 120

5.5 Numerical Simulations ..................................................................................................... 124

5.6 Conclusions .................................................................................................................... 146

6 Chaiter of Spinning Stepped-Shaft Workpiece with Two Cutting Tools ........................ 148 ......................................................................................... 6.1 Introduction 148

6.2 Equations of Motion and Cutting Forces ....................................................................... 149

........................................................................................................ 6.3 Stability Analysis 151

............................................................... 6.3.1 Cutting force applied at nodes of elements 152

........................................... 6.3.2 Stability analysis with arbitrary cutting force locations 162

....................................................................................................... 6.6 Numerical Examples 176

viii

6.7 Summaxy ......................................................................................................................... 190

7 Conclusions and Recomrnendations .................................................................................. 191 ...................................................................................................... 7.1 Conclusions ......... .. 191

................................................................................. 7.3 Recommendations for Future Work 194

Bibliography .............................................................................................................................. 195

Roman Characters

cross-section area of beam

~ ~ N t ~ t , = (Y}: [D,](x),

pararneter in cutting force component in the y direction

parameter in cutting force component in the 3 direction

constant, = (Y}: [D, ](X},

cutting process pararneter related to velocity in the y direction

cutting process parameter related to velocity in the t direction

ratio of cutting stifiess, = k, 1 k,

viscous damping factor

cutting process pararneter related to cutting speed in the y direction

cutting process parameter related to cuning speed in the z direction

element gyroscopic matrix

element damping matrix

depth of cut

normal cutting depth

geometric mat& for a two-node Timoshenko beam element

modulus of elasticity

ploughing force

proportionaIity constant in cutting force mode1

cutting force in the x direction

cutting force in the y direction

Laplace transfonn of F'

Laplace transfonn of F,.

cutting force component of cutting tool 1 in the x direction

cutting force component of cutting tool 1 in the y direction

cutting force component of cutting tool2 in the x direction

cutting force component of cutting tool2 in the y direction

i-th generaiized coordinate

shear modulus

global gyroscopic matrix

second moment of area

identity mamx of dimensions n x n

cutting force coefficient in the x direction

cutting force coefficient in the y direction

static cutting stiffiiess of m u l g tool 1 in the x direction

static cutting stifhess of cutting tool 1 in the y direction

static cutting stiüness of cutting tool2 in the x direction

static cutting stifbess of cutting tm13 in they direction

global stiffriess matrix

gIobaI stifiess matrix used in iiee vibration anaiysis

element stifhess matrix used in fiee vibration anaiysis

length of a finite beam element

total length of workpiece

global mass matrix

eiement mass matrix in fiee vibration analysis

global mass matrix before modification in free vibration analysis

number of degrees of freedom

node number at cutting point

order of mode truncation

global displacement vector of dimensions nx 1

element nodal displacement vector

global force vector of dimensions nxl

global force vector before modification

element force vector

Laplace transform of ~ ( t )

random cutting force disturbance in the y direction

random cutting force disturbance in the y direction

shape fùnction of Timoshenko beam element

time deIay

kinetic energy of an element

transformation matrix

tool structure dispIacement at cutting point

workpiece displacernent at cutting point

xii

instantaneous depth of cut

instantaneous depth of cut for cutting tooI 1

instantaneous depth of cut for cutting tool2

nominal depth of cut

lateral displacement of workpiece in the x direction

lateral displacements of workpiece in y direction

instantaneous depth of cut in the x direction

displacement of cutting 1001 in the x direction

displacement of workpiece in the x direction

Rayleigh dissipation function

Laplace transform of u,

Laplace aansform of u,

total volume of the ploughed workpiece material

strain energy due to deformation of an element

cutting speed

strain energy in the xo: plane

strain energy in theyoz plane

superscnpt representing workpiece

i-th eigenvector in state space

element [n + 8(n, - 1) + i] of the eigenvalue vector @} conesponding to

cutting point

Y.,

state vector of adjoint system

j* eigenvector of adjoint system

dement In c 8(n, - 1) t 11 of cigmveaor {Y},

eiement E + 8(n, - i) c 51 of eigenvector [F},

coordinate of cutting point

Laplace transform of nodal displacement vector q

Greek Characters

a shear angle in the xoz plane

4. shear angIe in the yot plane

f i effective clearance angle

KI clearance angle

6 Dirac d e i function

- : damping ratio

Il0 constant used in ploughing force modei

'II constant used in ploughing force mode1

K shear correction factor

4 ith eigenvalue of original system

4 Jh eigenvalue of adjoint system

P overlap factor of successive cuts

PC proportionality constant

xiv

local coordinate of a beam element

mass density of beam material

ultimate shear stress of workpiece material

shear angle

bending angle in the xoz plane

bending angle in the yoz plane

spin rate of workpiece

Table 3.1

Table 3.2

Table 3.3

Table 3.4

Table 3.5

Table 3.6

Table 3.7

Table 3.8

Table 4.1

Table 4.2

Table 1.3

Table 4.4

Cornparison of At-Rest Nahuai Frequencies (radk) for a Uniform Beam with

Clamped-Free Boundary Conditions ........................................................................ 6 1

Comparison of At-Rest Natural Frequencies (rad/s) for a Stepped Beam with

Free-Free Boundary Conditions ............................................................................... 62

Cornparisons of Natural Frequencies (radk) of Spinning Uniform Bearn with ....... 63

Natural Fequencies of Stepped Shaft with Clamped-Free Bundary Cnditions ........ 64

Natural Frequencies of Stepped Shaft with Hinged-Hinged Boundary

Conditions ...................... .,. ................................................................................. 65

Natural Frequencies of Stepped Shafl with Clamped-Hinged Boundary

................................................................................................................. Conditions 66

Natural Fequencies of Stepped Shaft with Free-Free Boundary Conditions .........,.. 67

Natural Frequencies of Spinning Stepped Shaft with

Two Rotational Springs at One .................................................................................. 68

Comparison of Predicted Chatter Onset Locations with the Experirnental

Results (Lu, 1990) for Uniform Workpiece ............................................................ 99

Natural Frequencies of the Uniform and Stepped Workpieces with One End

Supported by a Chuck and the Other Hinged (radls) ............................................ t O3

Nature Frequencies of the Uniform and Stepped Workpieces with One End

Supponed by a Chuck and the Other Free (radk) ................................................... 107

Nature Frequencies of the Unifom Workpieces Supported by Chuck, Tailstock and

................................................................................................. Steady Rest (rad/s) 1 10

xvi

Table 4.5 Nature Frequencies of the Uniform and Stepped Workpieces Supported by Two

Centers (radls) ................................................................................................ 1 13

Table 5.1 Natural Frequencies of the First Four Modes for Uniform and Stepped

Workpieces for Case 1 (radis) .......... .. . .... .... .- ............... . . . . . . ................ . 128

xvii

Fig . 1.1 Workpiece deflections before and after onset of chatter (Lu. 1990) ............................. 7

Fig . 1.2 Wavy surface of a shafi afier chatter (Chinacescu, 1990) ........................................... 3

Fig . 1.3 Wavy surface of a holiow cyiinder workpiece after chatter (Stephen, 1999) ............... 3

Fig . 1.4 A lathe (Boothroyd and Knight. 1989) ...................................................................... 4

Fig . 1.5 Cylindrical turning on a lathe (Boothroyd and Knight, 1989) ...................................... 5

Fig . 1.6 Contact region and cutting force ................................................................................... 7

Fig . 1.7 Regenetative effect ..................,............................................................................... . 10 Fig . 1.8 Cutting with overlap region ..............................,.................................................... . I I

Fig . 1.9 Cutting without overlap region .................................................................................. 11

Fig . 1.1 O Tenns used for cutting process (Boothro yd and Knight. 1 989) .................................. 13

Fig . 1.1 1 Orthogonal cutting process (Trent, 1991) .............. ,.., ............... ,., . 1 1 Fig . 1.12 Illustration of curting force components (Trent, 1991) ........................................... 15

Fig . 1 -13 Feedback loop of a typical machining system ......................................................... 19

Fie . 1.14 Mass-spring-darnper workpiece mode1 ................................................................... 70

Fig . 2.1 Chatter mode1 of tool smicture ................ ,. ......... ,.. ..,...... 10

Fig . 7.2 Schematic diagram of tooI penetration ........................................................................ Z S

Fig . 2.3 Effect of tool vibration on tool penetration ..................................... ... ..... 28

7; .......................................................................... Fig . 1.3 Nyquist plot of function [G(jo)H(i@]

Fig . 2.5 Cornparison of the stabiIity charts between the present work

.... and Liu (1990) ..........................,................................................*............................ 36

xviii

Fig . 2.6 Cornparison of the chatter fiequency between the present work and Liu (1 990) ........ 36

Fig- 2.7 Stability curve for tool structure with ploughmg force (case 1) .................................. 37

Fig . 2.8 Chatter frequency on the stability threshold with ploughing force (case 1) ................ 37 . .

Fig . 2.9 Effect of ploughing force on stability ........................................................................ 38

Fig . 2- 10 Effects of poughig force on chatter fiequency ...................................................... 38

Fig . 2.11 Effect of cutting force term BZ on stability .................................................................. 39

Fig . 2.13 Effect of cutting force term C, on stability ................................................................. 40

Fig . 2.14 Effect of cutting force term C= on chatter fiequency ................................................ 40

..................................................................... Fig . 3.1 A spinning stepped shaft 46

Fig . 3.2 A Cinite beam element .................................................................................................. 46

Fig . 3.3 Unifom shafl and its boundary conditions used in Table 3.1 ..................................... 60

Fig . 3.4 Stepped shaA and its boundary conditions used in Table 3.2 ...................................... 61

Fig . 3.5 Unifonn shaft and its boundary conditions used in Table 3.3 ..................................... 62

Fig . 3.6 Stepped shaft and its boundary conditions used in Table 3.4 ................. .... ........ 63

Fig . 3.7 Stepped shaft and its boundary conditions used in Table 3.5 ...................................... 64

Fig . 3.8 Stepped shaft and its boundary conditions used in Table 3.6 ...................................... 65

Fig . 3-9 Stepped shaft and its boundary conditions used in Table 3.7 ...................................... 66

Fig . 3.10 S tepped shafi and its boundary conditions used in Table 3.8 ...................................... 67

..... Fig . 3.1 1 Stepped shaft used to investigate effects of spring constant on natural fiequency 68

Fig . 3 -12 Effects of rotational spring stiffhess on natural fiequencies ..................... .. ............. 69

Fig . 4.1

Fig . 4.2

Fig . 4.3

Fig . 4.4

Fig . 4.5

Fig . 4.6

Fig . 4.7

Fig- 4.8

Fig . 4.9

xix

Stepped workpiece mounted on a lathe ....................................................................... 72

Chatter mode1 of stepped-shaft workpiece .................................................................. 73

Feedback loop of cutting process ................................................................................ 75 . .

Schematic view of a cutting process ............................................................................ 76

........... Cutting force at an arbitrary point and its equivalent nodal force components 83

Dimensions (unit: mm) and boundary conditions of the uniform shaft ...................... 90

Nyguist plots ofk, ~ ( s , r,)h e-" - 1) using the first three modes ............................. 90 Comparisons of Nyquist plots of the h c t i o n [k,H(jw, zc)(eyw~l)J using

three different modal tmcation schemes ..................................................................... 91

Comparisons of Nyquist plots function [kxH(jo7 zc)(ejor-l)] near the first natural

fiequency ............... ,. ................................................................................................. 91

Fig . 4-10 Stability charts of stepped workpiece near the first and second

natural fiequencies ...................................................................................................... 92

Fig . 4.1 1 Chatter fiequencies near the first and second natural frequencies

of stepped workpiece ................................................................................................... 93

Fig . 4.12 Cornparison of the Nyquist plots for the modal analysis method and

the direct Laplace tranfonn method ............................................................................. 95

Fig . 4.13 Stabiiity chart of uniform workpiece 1 ........................................................................ 96

Fig . 4.14 Chatter frequencies of spinning unifonn workpiece 1 ................................................ 97

Fig . 4-15 Cntical relative tool position along the workpiece fiom the chuck ............................ 96

Fig . 4.16 Dimensions (unit: mm) and boundary conditions of the unifonn and

.................................................................................. stepped workpieces for Case A 103

Fig. 4.17 Stability region of the stepped workpiece for Case A ...................... .... ............ 104

Fig . 4.1 8 Stability threshold of workpieces for Case A ........................................................ 104

Fig . 4.19 Chatter fieyencies of uniforrn and stepped workpieces for Case A ........................ 105

Fig . 4.20 Stability region of workpieces for Case A ................................................................. 105

Fig . 4.21 Dimensions (unit: mm) and boundary conditions of the uniform and

stepped workpieces for Case B .................................................................... IO6

Fig . 4.22 Stability chart of the uniform and stepped workpieces for Case B ........................... 107

Fig . 4.23 Chatter frequencies of workpieces for Case B ..................................................... 108

....................................................................... Fig . 4.24 Workpiece supported by a steady rest 109

Fig . 4.25 Dimensions (unit: mm) and boundary conditions of the workpieces for Case C ...... 109

........................ .............. Fig . 4.26 Effects of steady rest on the stability of workpiece .. 110 Fig . 4.27 Effects of steady rest on the chatter fiequencies of workpiece ................................. 111

Fig . 4.28 Workpiece supported by a dog and two centers ........................................................ I I2

Fig . 4.29 Dimensions (unit: mm) and boundary conditions of the workpieces

supported by a dog ..................................................................................... LI3

Fig . 4.30 Stability charts of workpiece supported by a dog .................................................... Il3

Fig . 4.3 1 Chatter fiequencies of workpiece supported by a dog ......................................... 114

Fig . 5.1 Coupied system consisting of workpiece and tool structure ..................................... 117

Fig . 5.2 Dimensions (unit:rnm) and geometry of workpieces for Case 1 ............................... 129

........... Fig . 5.3 Stability charts of the coupled systern with a unifotm workpiece for Case I 130

Fig . 5.5 Chatter fiequencies of the uniform and stepped workpieces for Case 1 ................... 131

Fig- 5.6 Cornparison of the stability charts for a uniform and stepped workpieces

for Case 1 ................................................................................................................... l3l

.......................... Fig . 5 . 7 Stability charts of the uniform and stepped workpieces for Case 1 132

Fig . 5.8

Fig . 5.9

Fig . 5.10

Fig . 5.1 1

Fig . 5.12 Fig . 5.13

Fig . 5-14

Fig . 5.15

Fig . 5-16

Fig . 5.17

Fig . 5.18

Fig 5-19

Fig . 5.20

Fig . 5.21

Fig . 5.22

Fig . 5.23

Fig . 5.24

Fig . 535

Fig . 5.26

Fig- 5.27

Cornparison between the stability charts of the workpiece and the

coupled systems for Case 1 ....................................................................................... 133

Dimensions (unit: mm) and geometry of workpieces for Case 2 ............................. 133

Stability chart of uniform workpiece 1 for Case 2 .................................................. 134

Stability charts of workpieces for Case 2 ................................................................ 134

Chatter fiequencies of workpieces for Case 2 ........................................................ 135

Effects of tool structure on the stability threshold of the stepped workpiece

for Case 2 ................................................................................................................. 135

Effects of tool structure on the chatter fiequencies of the stepped workpiece for

Case 2 ....................................................................................................................... 136

Dimensions (unit: mm) and geometry of workpieces for Case 3 ............................ 137

Stability chart of uniform workpiece 1 for Case 3 .................................................. 138

Chatter fiequencies of workpieces for Case 3 ....................................................... 138

Stability charts of workpieces for Case 3 ................................................................ 139

Effects of tool structure on stability of stepped workpiece for Case 3 .................... 139

Effects of tool structure on chatter fiequencies of stepped workpiece for Case 3 .. 140

Dimensions (unit: mm) and geometry of workpieces with steady rest for Case 4. . 141

Effects of steady rest on the stability thresholds for Case 4 .................................... 142

.................................... Effects of steady rest on the chatter fiequencies for Case 4 142

Effects of tooI structure on the stability thresholds for Case 4 ................... ............. 143

Effects of tool structure on the chatter fiequencies for Case 4 ................................ 143

.......................... The stepped workpiece and boundary conditions used for Case 5 144

............ Effects of rotational SDME constant on the stability thresholds for Case 5 144

Fig . 5.28

Fig . 5.29 Fig . 5.30

Fig . 6.1

Fig . 6.2

Fig . 6.3

Fig . 6.4

Fig . 6.5

Effects of rotational spring constant on the chatter fiequacies for Case 5 ............ 145

Effects of rotational spring constant on the stability thresholds for Case 5 ............ 145

Effects of rotational spring constant an the cbatta fiequencies for Case 5 ............ 146

Workpiecc systern with cutting forces at two locations ........................................... 149

Cutting forces at nodes of eiements ........................................................................ 152

............................................................. Cutting forces at arbitrary cutting locations 163

Nyquist plot of [kXH@, I, )(e9"tl)] for the uniform workpiece

......................................................................................... with only one cutting tool 180

Nyquist plot of k , , ~ , (s) for cutting 1001 1 of the uniform woikpise

with two cutting tools ................................................................................................ 180

Fig . 6.6 Effects of cutting tool2 on the Nyquist plot of k , , ~ , (s) for the

uniform workpiece with two cutting tools (2,=200 mm) ................... .. ................. 181

Fig- 6.7 Cornparison of Nyquist plot of k , , ~ , (s) for cutting tool 1

..... and Nyquist plot of k,, H , (s) for cunuig tool 2 .............. ... ................. t SZ

Fig . 6.8 Stability charts of cutting two tool 1 and cuning toor 2 for the unifonn workpiece 183

Fig . 6.9 Two tools Chatter fiequencies of cuîting tool 1 and cutting tool ............................. 183

Fig . 6.10 Stability charts of cutting tool 1 and cutting tool2 for stepped workpiece 1 .......... 184

Fig . 6.1 i Stability charts of cutting tu01 I and cutting tool2 for stepped workpiece 2 ........... 164

.............. Fig . 6.1 2 Chatter fkequencies of cutting tool 1 for uniforrn and stepped workpieces 1 SS

............. Fig . 6.13 StabiIity charts of cutting tool s for uniforni and stepped workpieces .. .... .. 182

................... .................. . Fig 6.14 Chatter fiequencies of cutMg tool 1 for al1 workpieces .. 186

....................... . Fig 6-15 Workpiece system with two cuffing tooIs appiied on opposite sides 186

Fig. 6.16

Fig. 6.17

Fig. 6.18

Fig. 6.19

Fig. 6.20

Fig. 6.21

xxiii

Nyquist plot of k,,~,(s)for cuning tool 1 of the uniforni workpiece .................. 187

Nyquist plots of k , ~ , ( . s ) and kGq H~ (s)for cuîting tao1 1 and cutting tool2 .... 187

Stability charts of the uniform workpiece for cutting tool 1

(&, =ZOO mm, Q = 400 mm) .................................... .. ................................... 188

Chatter frequencies of the uniform workpiece for cutting tool 1

(z,~ =ZOO mm, ,72 = 400 mm) ......................................... 188

Cornparison of the stability charts of the unifom workpiece for cutting tool 1

(with or without cutting tool 21.. . .-. . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... . . . . . . . . .. 189 Cornparison of chatter fiequemies of the unifonn workpiece for cutting tool 1

(with or without cutting tool2).. .. . .. . . .. .. .. ..... . . . .. . . . .. . . .. . . . .. . . .. . ..... . .... . ... +. . ... 189

Chapter 1

Introduction

1.1 Background

Metal cutting procas is ofien accompanied by a violent vibration between workpiece and cutting

tool. This type of vibration is called chatter. Chatter may generate high pitch noise, cause poor

surface finish, tool Wear, tooI fiacture and darnage to the machine tool system. To avoid chatter,

metal removal rate has CO be reduced CO maintain chatter-fiee operation. The detrimental effects

of chatter on product qualit., machine tool, and production rate make the analysis of chatter an

essential activity in machinhg process.

One important aspect of chatter is that it occm suddenly. Mer the onset, it almost irnmediatel-

develops into fuIl scde, disastrous vibration as shown in Fig. 1.1. As a result, significant

damage may occur before any action may be taken, Figures 1.2 and 1.3 illustrate the effect of

chatter on workpiece surface finish.

Fig. 1.1: Workpiece deflections before and after onset of chatter (Lu, 1990)

The main objective of a chatter analysis is to predict its onset conditions. The onset of chatter is

the rnovement at which amplitude of vibration of workpiece with respect to the tool starts to rise

rapidly during normal cutting. This increase in amplitude occurs because the energy supplied to

the cutting system surpasses the energy dissipated by the system. If the dissipated energy is

greater than the suppiied energy, the system is said to be stabIe. Otherwise, when the dissipated

energy is less than the supplied energy, the system is unstable. From the point of view of energ!

transfer in the cutting systern, the onset of chatter can be regarded as the stability threshold of tht

system in which the energy supplied to the system is equal to the energy dissipated by the‘

system in this thesis. only the stability threshold in turning process is of interest.

Fig. 1.2: Wavy surface of a workpiece after chatter (Chiriacescu, 1990)

Fig. 1.3: Wavy surface of a holIow cylinder workpiece after chatter (Stephen, 1999)

Headstock (containing main spindle)

Fig. 1.4: A lathe (Boothroyd and Knight, 1989)

Tuming is one of the most cornmon machining operations in industry. In a tumïng process, it

workpiece rotates about its longitudinal axis on a machine tool cailed a lathe, as shown in Fi-.

1.4. The workpiece is supported by a chuck at one end and by a tailstock at the other. A cutting

tool mounted on the lathe is fed dong the workpiece axis to remove material and produce thc

required shape. The principal surface machined is concentric with the axis of the workpiece. A

schematic diagram of a turning process is shown in Fig. 1.5.

In a tuming process, there are several parameters that d e h e the cutting conditions. They are

cutting speed, feed rate, and cutting depth. Cutting speed is the rate at which the uncut surface of

the workpiece passes the cutting edge of the tool. Feed rate is the distance moved by the cutting

tool in the longitudinal direction in each revolution. Cutting depth is the thickness of the metal

removed in the radial direction.

Work surface

/ Transien t surface < M ~ h i n e d surface

- . . *.- -

~ontinuous Feed motion

Fig. 1.5: CyIindrical turning on a lathe (Boothroyd and Knight, 1989)

Cutting force is an important quantity in machining. It determines machine power requirements

and bearing loads. It also causes deflections of workpiece, cutting tool, and machine-tool

structure. Its magnitude is influenced by the cutting conditions, geometry and material of the

cutting tool, continuous or intermittent cutting, usage of cutting fluià, and workpiece material. A

reliable mathematical model of cutting force is needed for stability analysis.

Most of the cutting force models assume that the cutting tool edge is sharp. in reality, tool edge

is not perfectly sharp. As shown in Fig, 1.6, the tool edge has a mal1 radius. There is a contact

rirea between the workpiece and tool. Neither the force acting on the tooi edge nor the force that

may act on the tool flank contributes to removal of the chip. The resuitant of these disnibuted

forces is referred to as the p loughg force&. incorporation the ploughing force in the cutting

force model will improve the accuracy of stability analysis.

In addition to cutting force models, a mathematical model of the machine tool system is needed.

Depending on the stifhess of the workpiece and suppott of the tool structure, there are three

types of dynarnic modek If the stifmess of the workpiece is much Iarger than that of the support

of the tool structure, the defotmation of the workpiece may be ignored, and only the vibration of

the tooI structure is considered. If the stiffness of the tool structure is much larger than that of the

workpiece, only the vibration of îhe workpiece is considered. The third type is a coupled system

consisting of both tool structure and workpiece. In this case, the values of the stiffhess of the tool

structure and workpiece are comparable.

chiptool interface

Fig. 1.6: Contact region and cutting force

Chatter is affected by the cutting conditions, type of cutting tooI, and material of workpiece. It is

also affected by the variation of the tool position along a workpiece when the flexibility of the

workpiece is considered. Under the same cutting conditions, when the cutting tool moves from

the chuck to the tailstock along the workpiece, chatter may occur when the cool passes a critical

position.

Chatter is a challenging research subject in metal cutting field. Aithough significant progress

has been made, most of the existing cutting force models are either too complicated to be

applicable to the stability analysis, or too simpiified to incorporate the ploughing force acting on

the tool edge and the workpiece-tool intetface region. At low cutting speed or small chip

thickness, the pIoughing force is very large and cannot be neglected. Work on the application of

cutting force models including ploughing force is limiteci.

In the Iiterature, chatter analysis was conducted only for workpiece of uniform cross-section. in

practice, the cross-section of a workpiece is ofien non-uniform. Stepped-shah, for example, are

commonly encountered in a tuniing process. With today's increasing requirernents for hi&

quality and high productivity in manufachring operations, the stabili ty analysis for non-uni form

workpiece becomes especially hpoaant.

To increase chip load, muitiple cutnng tools may aiso be used in industry. When cutting long

and slender workpieces, two cutting tools may be used to increase production rate or to reduce

deformation of the workpiece. in this case, a more complicated model may be required. Chatter

mode1 about this type of m i n g process has not been found in literame.

The objectives of this thesis are to (a) develop a cbatter model for the tool structure including the

pIoughing force, (b) develop a finite eternent chatter model for stepped-sh& workpiece, (c)

develop a chatter mode1 for a coupled system consisting of a stepped-sh& workpiece and a tool

structure, (d) develop a chatter model for a workpiece system with two cutting tools, (e) perform

stability anaIysis of tuming process, and (f) provide guidehes for selections of cutting

conditions.

Literature Review

1.2.1 Causes of chatter

The machine, cutting tool, and workpiece foxm a complicated dynamic. Under certain conditions,

severe vibration of the system may occur. The vibration may be divided into the three types, fiee

vibration, forced vibration, and self-excited vibration (Boothroyd and Knight, 1989; Astkhov,

1999; Welboum and Smith, 1970; Olgac and Hosek, 1997). The self-excited vibration is also

called chatter in study of machine tool dynamics.

Free vibration results fiom impulses transferred to structure through its foundation. The structure

wiIl vibrate in its natural modes until the damping causes the motion to die out. Forced vibration

results fiom periodic forces within the system such as unbalanced rotating masses, or transmitted

through the foundations fiom nearby machinery. The causes and control of fiee and forced

vibrations are well understood and the sources of vibration can be removed or avoided during

operation of the machine.

Chatter or self-excited vibration occurs oniy during material removing process. It is complex in

mechanism and difficult to control (Tobias, 1965; and Welbourn and Smith, 1965; Bao et aI.,

1994). Many researchers investigated the causes of chatter (Cook, 1959; Andrew and Tobias,

1961). The main cause is the regenerative effect in the cutting process (Tobias, 1965). Some

extemal perturbations or a hard spot in the workpiece material causes initial variation in cutting

force and results in vibration of the dynamic system. The vibration leaves a wavy tool path on

the workpiece surface as shown in Fig. 1.7, The wavy surface will affect subsequent chip

removal load. As the cutting toat removes material 5om this surface, the unevenness in the chip

will result in vibration. If the magnitude of subsequent vibration decreases, the cutting process is

stable. However, under certain conditions, the magnitude will increase and chatter will occur.

ïhis phenornenon is called regmerative chatter. Kato and Mami (1974) investigated the cause of

charter due to workpiece deflection. They performed cutting tests on mild steel and cast ion.

tool

wovy surface cut on previous tool pass

wavy surface cut on current tool pass

1 I

Fig. 1.7: Regenerative effect

Workpiece ,Feed ,

Feed direction

Fig. 1.8: Cutting with overiap region

Workpiece

No overlap region - Feed direction Fig. 1.9: Cutting without overlap region

h a tuniing process, the tool moves dong the workpiece in the axial direction during successive

revolutions. The current revolution of cut may overlap part of the surface left on the previous cut

in the feed direction as show in Fig. 1.8. The portion of overlapping between successive cuts

depends on the feed rate. An overlap factor is used to account for effect. The overlap factor is

zero if the previously machined surface does not affect the present cut for a very large feed as

shown in Fig. 1.9. The overlap factor is bounded between zero and unity in a tuniing process.

1.2.2 Models of cutting force

1.2.2.1 General cutting force models

Cutting force acting on the tool has been the subject of metal cutting research for decades (Kegg,

1965; Lauderbaugh and Larson, 1990; Zhang, 1991; Stakhov and Viktor, L999). It is affected by

many parameters such as the feed rate, depth of cut, cutting speed, angie of approach, rake angle,

and hardness of workpiece material (Bayoumi et al., 1994; Hine, 1971). The terminology

comrnonly used in this thesis is illustrated in Fig. 1.10.

Researchers have been trying to establish a relationship among these factors to model the cutting

force. The early work by Merchant (1944, 1945) has been a foundation used by many other

researchers in the modelling of cutting force (Stephenson and Agapiou, 1996; DeMies, 1992).

Merchant's model is based on the concept of a steady process in which a chip of metai is

produced by shearing a strip of uncut metal continuousIy and unifonniy, and the defoxmation of

Undeformed chip thickness

Chip Tool

Cu tt ing Tool flank C

Clearance crevice

New workpiece surface

Positive rake angle

4 - \ Negative rake angle X 1

Clearance Angle

Fig. 1.10: Terms used for cutring process (Boothroyd and Knight, 1989)

the chip takes place dong a shear plane. As shown in Fig. 1.1 1, the uncut metal comes up to the

tooI to be sheared, and it leaves parailel to the face of the tool with a new thickness. The width of

the chip is assumed to be constant throughout the process, and neither face of the metal being cut

is supported. The face of the tool is perpendicular to the plane of cutting. This kind of cutting is

called orthogonal cutting as shown in Fig. 1.1 1.

Fig. 1.1 1 : Oahogunai cutting process (Trent, 199 1)

Cutthg forces are measured in three directions as shown in Fig. 1.12. The component of the

force acting on the rake face of the tool, normal to the cutîhg edge, is the main cutting force. The

force component, acting in the radial direction, tending to push the tool away fiom the

workpiece, is called the radiai force. The third component is acting on the tool in the horizontai

direction, paralle! to the direction of feed, is referred to as the feed force (Trent, 1991; Oxley,

1989).

Main cutting force

Radial forcé

Fig. 1.12: Iilustration of cutting force components (Trent, 199 1)

Many researchers (Oxley et ai., 1974) attempted to improve the models developed by Mechant,

Lee and Shaf5er. They included sophisticated mathematical formulation of the fictionai behavior

on the tool rake face, high strain-rate and work hardening of the workpiece material, and high

temperature. Endres et ai. (1995% 1995b) developed a cutting force model incorporating

parameters of tool geometry. Lee and Shfler (1952) developed a more sophisticated model by

introducing plasticity of the workpiece materid into the solution.

Wu and Liu (1985a) proposed an improved model to determine the dynarnic cutting force

components from cutting tests. ïhey assurned that the mean hictional coefficient fluctuates as a

result of the variation in the relative velocity on the tool-chip interface. Based on the model of

Wu and Liu (1985a), Minis and Tembo (1993) provided a cutting force model accounting for

changes in the inner and outer chip surface shape.

In chatter analysis, the cutting force is usually assurned to be proportional to the cross-section

area of chip for steady state cutting. The simplest expression is that the cutting force is

proportional to the instantaneous depth and cutting width. Many other cutting force models can

be found in the review literature (Mohamed, 1994; Shawky, 1996). They are valid for a specific

cutting tool and workpiece material. In recent years, some tesearchers have developed cutting

force models for fuIly developed chatter (Stephen, l999,l998a, 1998b; Stephen and Kalmar-

Nagy, 1997; Johnson, 1996). They reported ihat after chatter occurs, the nonlinear factors of

cutting prevent the amplitude of vibration h m going to infïnity (Hwang et al., 1997).

1.2.2.2 Ploughing force model

The ploughing force generated as the cutting tool penetrates into the workpiece material is a part

of the total cutting force. The p lougbg force exists in metal cutting process because (1) the

actual cutting edge of the tool is not perfectly sharp and has a srna11 radius; and (2) the built-up-

edge developed in fiont of the tool faces f o m a larger effective edge radius. As a result, the

built-up edge cannot move upward to become part of the chip, instead is extmded and pressed

under the tool. The ploughing force is known to contribute to the cutting process damping and

hence to the stability of machine tools (Wu 1988, 1989; Elanayer and Shin, 1996; Shawky and

Elbestawi, 1997).

Wu (1988, 1989) developed a comprehensive ploughng force mode1 based on the principles of

cutting mechanics. It takes into account the fluctuations of the mean fiictional coefficient on the

tool-chip interface, as well as the variation of the normal hydrostatic pressure distribution and the

shear flow stress along the primary plastic deformation zone. Results predicted using this model

show a good agreement with the experimentally deterrnined cntical width of cut.

Elansysr and Shin (1996) developed a general experimental procedure for the separation of

ploughing force fiom shearing force on the shear plane. Shawky and Elbestawi (1997)

decomposed the ploughing forces into static and dynamic components. in their model, damping

is predicted by tracking the dynamic ploughed volume resulting fiom the interactions with

rnachhed surface undulations. Waldor et al. (1998) developed a slip-line field to model the

ploughing force. The resulting force measurernents match predictions using the slip-line field

model.

Although there is some research on cutting force modelling incorporating the pIoughing force,

the work on the application of these cutting force models in the stability analysis is very limited.

In this thesis, the effects of the plùughing force on chatter are considered.

1.2.3 Chatter rnodel

1.2.3.1 Chatter of cutting tool structure

The cutting tool is mounted on a turret or caniage as shown in Fig. 1.4. When chatter occurs,

both the cutting tool and the workpiece may viirate. If the ngidity of the workpiece is much

Iarger than that of the support of the tool structure, then only the vibration of the tool structure is

considered. In this case, the dimensional accuracy and surface finish of a machined component

depend on the dynamic properties of the cutting tool structure.

Many analytical and expenmental studies were conducted to understand the chatter of tool

structure (Subramaniane et ai., 1976; Thompson, 1988; Tlusty et al., 1974). in the works of Doi

and Kato (1956), Tlusty (1963) and Tobias (1965), the tool structure was modelled as a

concentrated mus; the support of the tool structure was modelled as translationai springs and

dampers. Memtt (1965) later expressed the chatter rnodel in the fonn of a feedback loop shown

in Fig. 1.13. Analysis of this loop using feedback control theory yields a straightforward method

of determinhg the stability lirnit for a machine tool system.

Displacement disturbance

Force

Fig. 1.13: Feedback loop of a typical machining system

/ 4 Displicement 1 1 I 1 I t 1 I Cutting prtwiess L 1 I 1 I

The mass-spring-damper system mode1 was used by many investigators such as Wu and Liu

(1985b), Minis et al. (1990), Hwang et al. (1997), Saravanja-Fabris and D'souza (1974). The

main difference among them is the cutting force models. Some researchers are interested in the

stability Iimit (Mani et al., 1983% f983b, 1988a, 1988b, 1988~; Masory and Koren, 1985; Minis

et ai., 1Ç90a, 1990b, 1990c), the others are interested in the tully developed chatter (Hwang et

al., 1997; Stephen, 1989). Their work helps to understand the mechanism of chatter.

1 I 1 1 1 I I

I I I I I a 1 ' I I 1

- Machine-tool

1 I 1 l Cutting force I 1 1 1 I L----------------------------------------------------------------------L

' structural dynamics

Endres et al. (1990), Sahay and Dubay (1991) and Marui et al. (1995) modelled the tool as a

cantilever beam. In the work of Marui et al., the regenerative chaiter is investigated

experimentally- These models are valid for cutting tool with long tool shank. In industry, the

rigidity of the tool is normally very high and the length of tool shank is very short. Therefore,

these modeis are rarely used in industry, Therefore, in this thesis, only the mas-spring-damper

system is used for tool structure.

1.2.3.2 Cbatter of workpiece

The workpiece is held by a chuck at one end, and is ffee or supported by a tailstock at the other

end. In the 1980s, the workpiece was usually modelled (Kaneko et al., 1984; Klamecki, 1989) as

a lumped m a s with springs and darnpers attached to it as shown in Fig. 1.14.

Fig. 1.14: Mass-spring-damper workpiece mode1

Because a workpiece is constrained by the chuck and tailstock, a realistic description of

boundary conditions is critical for an accurate prediction of the onset of chatter. Lu and

KIamecki (1990) modelled a slender workpiece as a uniform Euler beam with hvo types of

boundary conditions. In their work, the chuck is considered as a rotational spring attached to a

hinge at one end of the workpiece; the tailstock is considered as a translationai spring attached to

the other end. The cutting force is considered to be proportionai to the instantaneous depth of cut.

The Euler beam theory was afso used by Jeu and Magrab (1996) in their stability analysis of

uniforrn workpiece. Shawky and EIbestawi (1998) modelled the uniform workpiece as an Euler

bearn in their control systern for workpiece accuracy in unifonri shaft turning.

Critical cuaing conditions are usualiy illustrated in a stability chart. Various methodoiogies have

been used to obtain the stability chart, Lin (1990) investigated the stability of a lurnped mass

system using an analytical method. Lin separates the characteristic equation of this dynarnic

system into a reai part and an irnaginary part in the frequency domain, The chatter fiequemies

and critical cuning conditions may be obtained anaiyticaliy or numerically.

Many researchers have used the gain-phase plot to obtain the stability chart. Intersection of the

dynaniic cornpliance with the points on the critical ioci gives harmonic solutions of the

characteristic equaiion, which define the b o u n d d of stability. Another method to obtain the

stabirity chart is the gain-factor method. Chen et al. (1994% 1994b) and Wang et ai. (1999a,

1999b) used the gain-factor mehod to obtain the criticd cutting conditions h m the intersection

of the non-zero term of the characteristic equation with the real axis- The advantage of this

method is that the criticd cuthng conditions and the chatter firequencies can be obtained dwctly.

Up to now, the workpiece is modelled in the literature either as a iumped mass, which is difficult

to descnbe the true behavior of îhe workpiece and to incorporate the realistic boundary

conditions; or as a unifonn beam, which is valid only for uniforni cross-section workpiece. In the

chatter model of a workpiece, incorporation of the effects of rotary inertia and transverse shear

deformations is necessary for wider application. Unfortunattiy, important factors such as

spinning, shear deformation and non-unifonn cross-section are not considered in the literature-

Overview of Thesis

Prediction of the onset of chatter for tool structure and workpiece in the tuming process is the

focus of the remaining chapters, In Chapter 2, the effect of ploughing force on the stability of a

system consisting of a tool structure and a rigid workpiece is investigated. Numerical results are

given to illustrate the effects of different cutting force models on the stability limit. Comparison

of the present work with that of other researcher is given for a well-accepted cutting force model.

Chapter 3 presents a chatter mode1 for spinning stepped-shaft workpiece. A Thoshenko beam

elemenr is used for fiee vibration analysis of a spinning shaft. Effects of spring constants on

natural fiequencies of the dynamic system are investigated. Convergence tests of naturai

Eiequencies are carried out, and cornparisons of the present work with that of others are made.

Chapter 4 presents a stability analysis of spinning stepped-shaft workpiece. Based on the work of

Chapter 3, the modal analysis technique and the direct Laplace tranform methcd are employed to

obtain the characteristic equation of the dynamic system. Four types of boundary conditions of

the workpiece are investigated. Exarnples are given to illustrate how the criticaI curves

separating stable and unstable motion of the dynamic system are obtained. The effects of the

boundary conditions on stability of the dynamic system are also examined. Comparison of the

modaI analysis technique and direct Laplace transform method is given. The predicted chatter

onset conditions using the present method are compared with the theoretical results and

experimental data obtained by other researchers for uniform workpiece.

Chapter 5 focuses on the chatter analysis of coupled system consisting of cool structure and

stepped-shafi workpiece. The goveming equations of motion for the spinning workpiece are

derived using the Lagrange equations. Modal analysis technique is used to obtain the

characteristic equation of the dynamic system. Numencal results are given to iIIustrate the

procedure of stability analysis. Effects of the vibration of the tool structure on the overall

stabiiity limit are also presented.

In Chapter 6, a chatter mode1 and stability analysis of a workpiece system with two cutting too1s

are presented. This modei may be used in a fast cutting process to increase the productivity. The

two cutting tools are applied independently at arbitrary locations dong the workpiece. The case

that the two cutting tools are ananged on the opposite sides of a workpiece is investigated.

In Chapter 7, a summary of the thesis work is given. Some conclusions fiom this thesis work are

drawn. Recomrnendations for future work are presented.

Chapter 2

Chatter of Tool Structure Incorporating the

Effect of Ploughing Force

2.1 Introduction

In this Chapter, the stability analysis of tuming process is performed based on a cutting force

mode1 that inciudes the effect of ploughing force. The Laplace transform is employed to obtain

the characteristic equation of the dynamic system. Nyquist stability criterion is employed to

determine the stability limit of the tool structure. The stability curves that separate stable and

unstable cutting conditions are piotted. The effect of different cutting force parameters on the

stability is ïnvestigated. The gain-factor method used to obtain the criticai cutting conditions is

compared with other method in the literature.

2.2 Chatter Mode1 of Tool Structure

A two-dimensional dynamic cutting mode1 is shown in Fig. 2.1 (Wang et al., 1999b), where the

tool is removing an uncut chip with a wavy top surface.

workpiece l

Fig. 2.1 : Chatter mode1 of tool structure

The equations of motion can be written as (Wu, 1985b, 1989; Shawky and Elbestawi, 1997)

where m is the equivalent mass of tool structure; c, and c, are the damping factors in the z and y

directions, respectively; and ky are the equivalent stiffness constants; Fz and Fy are the r and y

components of the cutting force on the tool rake; f, and f , are the z and y components of

ploughing force on the tool nose region resisting the peneûation of tool.

According to Wu (1985a, 1985b), F= and Fy are

where to is the time delay, ro = 2xlR; R is the spin rate of workpiece; Vo is the cutting speed; r i s

the ultimate shear stress of workpiece material; d is depth of cut; A, A,. C, Cy, B: and B,. are

dynamic coefficients related to the fictional behavior during cutting process. The mechanism of

ploughing force is illustrated in Figs. 2.2 and 2.3.

The two pbughing force components are

whereLp and & are proportionality constants; V is the total volume of the ploughed worh~iece

material.

Fig. 2.2: Schematic diagram of tool penetration

Fig. 2.3: Effect of tool vibration on tooI penetration

From Wu (1 989), the effective volume of ploughed material, as shown in Fig. 2.3, is

V= d (shaded area)

where d is the depth of cut. The effective volume is

1 V = d h ( & g =d(cotYe --cot'y. tany, 7-

O 2 1 where

where yo is the clearance angle; 4 is the shear angle; B, qr are constants; do is the normal cutting

depth and y, is recognized as the effective ctearance angle. Utilizing the following Taylor senes

1 * 1 3 coty,--cot-y, tany, =-- 2 ( 2 2tanyo btantan?yO 4 t )

Equation (2.7) may be written as

For small vibration, u(t) , u(t - t , ) and ~ ( r ) are mail values. Higher order t m s are neglected.

The constant term is also neglected, since it does not play a role in the vibration. Therefore,

where

Substituting the above reiations into Eqs. (2.1) and (2.2), we obtain

where - ( t ) and rv(t) are random cutting force disturbances in the z and y directions.

respectiveiy.

2.3 Stability Analysis

The goveming equations can be reduced to an equivalent single degree-of-iieedorn system.

Taking Laplace ûansform on both sides of Eq. (2.1 1) with

Equation (2.1 1) becomes

~ 2 ~ ( s ) + 2 ~ ~ m s ~ ( s ) + ~ n ' ~ ( s ) = ~ ( ~ ) ( e - " - 1 ) ( ~ , + ~ , ~ ) + ~ ( ~ ) D , ~ + ~ ( ~ ) / n i (2.16)

where

3 2

The tram fer function between r, (t) anddt) is defined as the ratio of ~ ( s ) and .(S), thus

From Eq. (2.161, we obtain

The characteristic equation of the systern is

or

where

2.4 Numerical Examples

Numencd simulations are carried out to hvestigate the effect of ploughing force on stability of

tool structure. The effects of different cutting force models on stability are also investigated. The

present work is compared with the work of Liu (1990) for a cumng force model.

Substituting s = jo into Eq. (2.20), where w is chatter fiequency, we will examine the

encirclement of the point (l+jO) by the ~ ( j o ) ~ ( j u ) loci. The presence of the time-delay term

in the characteristic equation leads to mdtiple intersections of the plot with the real axis as

shown in Figure 2.4. Let P be the coordinate of the right-most intersection point between the

Nyquist contour of the t e r m ~ ( j o ) H b ) and the real a i s . The closed-loop system is stable if P

> 1, unstable if P c 1, and criticai if P = 1, With a specific cutting width of d,, the right-most

intersection point of the open-loop locus is at P. A gain factor h = 1 / P can thus be obtained. It

is concluded that the criticai cutting depth is d =d,h , where the open-loop locus will pass

through point (l+jO). With this approach, the limit of critical cutting width can be determined.

The above mentioned method is the gain-factor method used in this thesis. This method is

compared with the work of Liu (1990) for a given tool structure system. The given equation of

motion of the dynamic system is

where damping ratio < = 0.05; natural fkequency a, = 600 rad/s; and cutting force constant k, = 60000. The stability curve of this system is show in Fig. 2.5, and the cnticd chatter

frequency cume is shawn in Fig. 2.6. It cm be found that the results of ihe present work and the

results of Liu (1 990) are identicat

The vaiues of the physical and dimensional properties have been based, largely, on the machine

tooi structure used by Wu (1985) and his experimental results. The effective mas m = 74 kg,

damping ratio 6 = 0.05, natural fiequency o, = 600 radis, and .È = 200 Mpa, A, = 3.0, BS = 0.82

sech, C, = 0.3 seclm. The data related to the ploughing forces are f, = 4.1 x 1 6 ~/mm', &I =

20', = 3', = 0.0046 mm, q, = 0.005, do = 0. 15 rndrev. For srnail vibration, steady shear

angle &, is used as the dynamic shear angle #.

The effect of different cutting force parameters on the chatter stability is also studied here, as

illustrated in Figs, 2.7-2.14. In case 1, the ploughing force is considered; in case 2, the ploughing

force is neglected; in case 3, the ploughing force is neglected and the force term B, = 0; and in

case 4, the ploughing force is neglected, the force term Bz = O and C, = 0.

As show in Fig. 2.7, case 1, the stability of the system is hfluenced by the tirne delay. The

critical cutting depth decreases as the time deIay increases; and the w-idth decreases as the time

delay increases. The cutting width behaves like a periodic function of time delay, The shapes of

the stability lobes look similar to each other. Figure 2.8 shows that the corresponding chatter

fiequency occurs at the stabifity threshold increases as spin rate ïncreases within each stabibty

lobe.

In the very low rotational speed region, the ploughing force

rises sharply at low cutting speed and increases gradudly to

plays a dominant d e . The curve

approach an asymptotic borderline

at high cutting, as shown in case 2, Fig. 2.9. In the high rotationai speed region, the effect of

ploughing force is very srnail and can be neglected. Figure 2.10 shows the chatter Erequencies of

case 1 and case 2. As the spin rate increases, their difference becomes smdler and smaller.

When the velocity term parameter Bz=O, the stability curve becomes Iower as shown in case 3,

Fig. 2.1 1. Figure 2.12 shows the effect of ploughing force on chatter (case 1 and case 3). If the

cutting speed influence is aiso neglected, i.e., C,=O, the borderline of the stability curve becorne a

horizontal line as show in Fig, 2.13. These results are in good agreement with the resuIts

obtained by other researchers (Wu, 1985; Lin, 1990). Figure 2.14 illustrates the effect of C, on

chatter fiequencies (case 1 and case 4).

Red part of [@)H(@)]

Fig. 2.4: Nyquist plot of function [G(jo~)H(jw)]

- Presmt work

* Liu ( I W O )

Spin rate (rad/s)

Fig. 2.5: Comparison of the stability charts between the present work and Liu (1990)

1000 i

Present work 9 5 0 ~ -

9001 * Liu (1990)

Spin rate (radis)

Fig. 2.6: Comparison of the chatter frequency between the present work and Liu (1990)

2.5' O 10 20 30 40 50 60 70

Spin rate (rad/s)

Fig. 2.7: Stability curve for tool structure with ploughing force (case 1 )

Spin rate (radis)

Fig. 2.8: Chatter fiequency on the stability threshold with ploughing force (case 1)

- With ploughing force

------ No ploughing force

25' 1 O 1 O 20 JO 40 50 60 70

Spin rate (radh)

Fig. 2.9: Efféct of ploughing force on stability

- With poughmg force

Spin rate ( d s )

Fig. 2.10: Eff~ects of poughig force on chatter fiequency

Spin rate (cad/s)

Fig. 2.1 1 : Effect of cutting force term Bz on stability

- 670 i Case I

Case 3

" .- 40 45 50 55 60 65 70

Spin rate (radls)

Fig. 2.12: Effect of ploughing force on chatter fiequency

I 4.q -

Case i Case 4

Spin rate (radfs)

Fig. 2.13: Effect of cutting force t m C, on stability

- 670t - Case 1

Case 4

Spin rate (rad/s)

Fig. 2.14: Effet of cutting force tm Cz on chatter frequency

2.5 Summary

A cutting force madel of machine too! structure is incorporating the effect of ploughing force

was used for stability analysis of tuming process. The Laplace transform was imptoyed to

identib the characteristic roots of the dynamical system. The stability of the system was

determined by Nyquist criterion.

The effects of different cutting force parameters on stability were also investigated. The results

obtained h m the simuiations indicate that the ploughing force play as a role of damper in

chatter of tool structure. At very low cutting speed, the ploughing force has a significant effect

on the stability; at hi& cutting speed, the effect of ploughing force on the stability is negligible-

Chapter 3

Free Vibration of Spinning Stepped Workpiece

Free vibration analysis of a rotating stepped shafi or workpiece is the foundation of stability

analysis of workpiece system. In this Chapter, Eree lateral vibration of stepped shafts iç

investigated using the Tirnoshenko beam theory and the h i t e element method. Beam finite

elements having two nodes and 16 degrees of fieedom are employed to mode1 flexural vibration

of a stepped shaft for four field variables - two lateral displacements and two bending angles.

Witfiin each uniform segment, the stepped shaft is modeled as a substructure for which a system

of equations of motion may be easily fomiated using the Galerkin method. The global

equations of motion for the entire stepped shaft are subsequently formulated by enforcing the

displacement continuity and force equilibriurn conditions across the interface between two

adjacent substructures. The second order governing differential equations for a non self-adjoint

dynamic system are then reduced to the quivalent f h t order differential equations for which

eigenvalue problern is fonnulated and sotved using the Matlab program. Values of naturaI

tiequencies are in excellent agreement with those available in the Iiterature. Effects of rotational

springs attached to the end of a stepped shafl, used to simulate non-classical boundas.

constraints of chuck on a work piece in a typical tuniing process, are also investigated. The bi-

orthogonal conditions for modal vectors, which are very usefiil in chatter analysis during tumir~g

processes, are given in this Chapter.

Introduction

The dynamic behavior of a spinning beam type structure is of great interest to researchers during

the past few decades. Rotating structures such as the workpiece machined using a lathe, the shaft

in a turbine unit, and the spindle in a milling machine can be modeled as such a system. For

slender structures, the Euler-Bernoulli beam theory may be used. However, if length of a rotating

beam is not significantly larger than its cross sectionai dimensions, effects of shear deformation

and rotary inertia must be considered, The Tioshenko beam theory may be used to handle

flexural vibration problern of spinning structures. In engineering practice, stepped workpieces

are commonly encountered. Because the chatter frequency of a workpiece is closely related to

its fundamentai natural fiequencies and vibration modes, it is necessary to perform fiee vibration

analysis of spinning stepped workpieces.

Extensive work has been done for uniform or linearly taperd shafts using theoretical or numerical

methods. Early investigations of non-spinning shafts can be found in the work of Anderson

(1953), Dolph (1954) and Carr (1970). For spinning system, Lee et al. (1988) proposed a closed-

form theoretical solution for natural fiequencies and mode shapes of a spinning Rayleigh beam.

They also appiied the Galerkin method to analyze the forced response of an undamped

gyroscopic systern. Chen et al. (19944 1994b) presented a study of spindle system in metal

manufacturing machinery using Rayleigh beam theory. In their work, Gaierkin's method is used

to calculate the natural kquencies. Katz et al. (1988) used the Euler-Bernoulli, Rayleigh and

Timoshenko beam theories to mode1 the rotating shaft. Using integral transform technique, they

developed the frequency equation for simply supported beams.

Zu and Han (1992) carried out fiee vibration analysis of a spinning Timoshenko beam with

hinged-hinged, clamped-clamped, hinged-fke, clamped-fiee, free-free and clarnped-hinged

boundary conditions. Using an analyticai method, they caiculated the natural fiequencies and

normal modes, Furthemore, they extended their work to forced vibration of spinning

Timoshenko beams with the six combinations of classical boundary conditions. A solution of the

problem is achieved by fomulating the spinning Timoshenko beam as a non-self-adjoint system

(Zu and Han, 1994). Melanson (1996) perfomed free viiration and stability analysis of spinning

uniform Timoshenko shah with extemd and internai damping under general boundary

conditions.

Numerical methods such as the finite element method have been applied to analyze the vibration

of spinning and non-spiming beams. A number of non-spinning Timoshenko beam elements are

proposed in the literature @avis et al., 1972; Thomas et ai., 1973; Dawe, 1978). The main

difference among them is the nurnber of degrees of fieedom to describe the Timoshenko beam

element. The use of finite element method for simulation of rotor systems has received

considerable attention within the past three decades. Nelson (1980) used a two-node, eighr-

degree-of-freedom Timoshenko beam element to incorporate the gyroscopic effect. Rouch and

Kao (1979) deveIoped a linearly tapered Timoshenko beam elernent with 12 degrees of fieedom-

Wu et aI. (1997) formulated a linearly tapered Timoshenko beam with eight degrees of fieedom.

Yu and Cleghorn (2000) calculated the naturai frequencies of stepped shah using a three-node,

24-degree-of-fieedom Timoshenko beam element.

In this cbapter, a 16-degree-of-fieedom Timoshenko beam element is used to determine the

natural fiequencies and mode shapes of spinning stepped workpieces. Displacement continuity

and force equilibrium conditions are applied at the interface of two elements in assembling

global equations of motion. Natural fiequencies of stepped workpieces are obtained and

convergence tests are perfonned, The results are compared with îhose obtained using ANSYS

and those of other reseatchers.

3.2 Governing Equations of Motion for a Spinning

Stepped Workpiece

The stepped shaft is modelled as a stepped Timostienko beam. Various boundary conditions

may be appIied to the beam. Fig. 3.1 shows a stepped shaft with two rotationai springs attached

to one end and two translational springs attached to the 0 t h Using the finite dement method

(Zienkiewicz, 1989), the stepped shafi may be divided into a number of unifonn beam elements

as shown in Fig. 3.2.

Fig. 3.1: A spinning stepped shaft

Fig, 3.2: A f i t e beam eIement

For a differential Timoshenko beam element, the relationships among the bending angles,

tramverse displacements and shear angles are

au au x-=#, +4, ' -=4 ,+P, a.? a.?

where u, and u, are lateral displacements of the beam in the x and y directions, respectively; 4x

and @,, are bending angles in the xoz and yoz planes, respectively; and are shearangles in the

xoz and yoz pianes, respectively, Assurning the lateral displacements and bending angles Vary

cubicdly dong the z-axis, the field displacements within an element may be written as

where

where 5 is the local coordinate; q is the nodal displacement vector, the shape functionS(t) and

the geometric rnatrix D, for the two-node element are

and the element nodai displacement sub-vectors are

and

According to Timoshenko beam theory and rotor dynamics, the kinetic energy and strain energy

of an element are

The Rayleigh dissipation function associated with viscous damping force is

where p is the mass density of the beam material; G is the shear moduius; E is the modulus of

ehsticity; A is the cross-section area; I is the second moment of area; K is the shear correction

factor; cd is the equivalent viscous damping coefficient; Q is the spin rate of the workpiece. From

Eq. (3.2), 1', V and UL can be expressed in terms of nodal displacements. Substituting Eqs. (3.1)

and (3.2) into Eq. (3.8) , we obtain the total strain energy

where

The kinetic energy cm be expressed as

w here

4 L A, = I[N]~[N)~Z, Ao, = ~(N]'[N')~z

O O

1.

A,, = S[N']~[N~~Z, A,, = l~N1]TIN'& O O

For Iineariy viscous damping, the Rayleigh dissipation function becomes

Finaily, we arrive at

where

The Lagrange equation for an element rnay be written as

where Le = T- P; Q, is the element force vector. Substituting Eqs. (3.13)-(3.15) into Eq. (3.16),

the equations of motion for an unconstained element can be obtained. The governing equations

of motion of a beam element may be written as

(3.17)

where

Equation (3.17) may be rewritten as

where

For convenience, a transformation maûix [T.] is introduced, and Eq. (3.18) may be rewritten as

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 l 0 0 0 0 0 0 0 0 C

O O O O O O O l O O O O O O O C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 C

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -

Equation (3.19) may be written as

where

The global equations of motion for the system are denved by assembling the equations of motion

for ail elements, and the final form is

where

Equation (3.21) can be rewritten as

where Mg is the gIobal mass matrix; G, is the global gyroscopic matrix; Kg is the global stifhess

rnamx; q, is the global nodal displacement vector.

To satisfy the continuity conditions of displacement and force equilibriurn across the interface

where the two elements of different cross-sectional areas are joined, the Lagrange rnultipIier

method (Tabarrok and Etirnrott. 1990) is used to assemble the global equations of motion for a

stepped shah. The displacement continuity and force equilibrium conditions may be written in

tems of the nodal displacements of the two beam elements as

4 where a =-, =- IL , and subscripts L and R indicate the lefi and the right rides of the joint 4 IR

of the two elements. The final equations of motion of the workpiece may be written in terms of

the modified dispiacement vector as

[ M I ij + [ G I ~ + [ K ] ~ = O (3 -24) where q is the modified global displacement vector of dimensions nx 1, and n is the total nurnber

of degrees of fieedom.

3.3 Modal Analysis

Modal anaiysis is carried out to determine naturai frequencies and mode shapes of the second-

order dynamic system govemed by Eq. (3.24). The second-order dynamic system can be reduccd

to an equivalent first-order system through the following transformation (Merovitz, 1997)

Equation (3.24) can be written as

where

where 1 is the identity matrix. Because G' = -G , the spinning woricpiece system is non-self-

adjoint (Katz et al., 1988; Zu and Hm, 1994). To solve the eigenvalue problem of the non-self

adjoint system, we can define the adjoint system as

where subscnpt n represent the adjoint system; Y is the state vector of the adjoint systern, and

If Ai and 2; are the i" a n d r eigenvalua of the original and adjoint systems, the eigenvalue

problems associated with these two dynamic systems are

where positive values of i irnplicitly indicate the forward precession; negative values of i

correspond to backward precession. Eigenvalues Ai and A; satisfy

Applying mode superposition, the response may be written as

i-ln

where (xb is the i'h eigenvector of the rystem; and g.(t) is the i-th generalized coordinate. The biorthogonal relationships are derived to be

where ai and bi are constants; (Y} is the,& eignivector of the adjoint system. Multiplying Eq.

(3.35) on the left by (Fr, we obtain

Equation (3.35) is a set of decoupled first-order ordinary differential equations.

3.4 Convergence Tests and Numerical Results

Two cases are investigated for non-spinning and spinning shafts. For non-spinning shafts, a

convergence test waç performed. Cornparisons with the results obtained using ANSYS" were

made. For a spinning shaft, numerical results of a uniform beam are presented, and are compared

with the results in the Iiterature. The effects of boundary conditions on the natural frequencies

are examined.

To test the convergence of at-rest naturai fiequencies cdculated using the present method. a

uniform beam and a stepped beam with three uniforni segments are selected. Values of material

properties of the bearns are

In order to compare with the results of others, the damping coefficient cd is selected to be zero in

al1 cases. The geomem of the shaft and its boundary conditions used for each example are al1

shown in Figs. 3.3 - 3.11, Tables 3.1 and 3.2 are the convergence test results and comparisons

with ANSYS. Table 3.1 presents the results of the uniform beam. It can be seen that the present

work requires oniy f 5 elements to achieve convergence to the hrst digit d e r the decima1 point

for the first four vibration modes. Table 3.2 presents the nanird fiequencies of the stepped shaft

with ftee-free boundary conditions obtained using ANSYS with 150 elements and the present

work with 15 elements. The differences in namral fkquencies for the first five modes are less

than 0-Olpercent.

Table 3.3 presents the comparisons of the natural fiequencies of the unifom beam obtained

using the present method and the exact an