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 Title Page
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 The transfer functions and the...
 Background and development of the...
 Derivation improvement and vailidity...
 Discussion of the impact excitatio...
 Characterization of the white sundstrand...
 Comments and recommendation for...
 Reference
 Biographical sketch
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Title: Theoretical and practical background for the characterization of CNC machining centers
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    Abstract
        Page v
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    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
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    The transfer functions and the methods for computing chatter limits and errors of accuracy
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    Background and development of the cutting tests
        Page 35
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    Derivation improvement and vailidity of the simple formulae for chatter limits in milling
        Page 72
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    Discussion of the impact excitatio technique
        Page 87
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    Characterization of the white sundstrand series 20 omnimil machining center
        Page 98
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    Comments and recommendation for future research
        Page 116
        Page 117
        Page 118
        Page 119
    Reference
        Page 120
        Page 121
    Biographical sketch
        Page 122
        Page 123
        Page 124
    Copyright
        Copyright
Full Text












THEORETICAL AND PRACTICAL BACKGROUND FOR THE
CHARACTERIZATION OF CNC MACHINING CENTERS











By

CARLOS A. ZAMUDIO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1991















ACKNOWLEDGEMENTS


The author wishes to express his appreciation to all who

in one form or another contributed to the completion of this

work. He is very grateful to Dr. J. Tlusty for giving him the

opportunity to work with him in the Machine Tool Laboratory.

His advice and always accurate assessments were invaluable

ingredients for the completion of this dissertation. The

author also wishes to express his appreciation to the present

and past members of the Machine Tool Laboratory with whom he

shared the many experiences which constitute his graduate

education. Specifically, the author wishes to thank Dr. K. S.

Smith for his assistance throughout this work. He is also

specially thankful to Dr. T. Delio, C. Vierck, W. Winfough, R.

L. Wells and M. Nallakatla. Finally the author is most

grateful to his wife, Ana, for her constant unconditional

support.

This work was partially sponsored by the National Science

Foundation, grant, "Programming for Quality in Milling."

















TABLE OF CONTENTS


ACKNOWLEDGEMENTS . . . . . . . ... ... ii

ABSTRACT . . . . . . . .. .. ..... V

CHAPTERS

1 INTRODUCTION . . . . . . . .. ... 1

The Need for CNC Machine Tool Characterization . 1
Limitations on the Cutting Performance . . . 2
Scope of the Dissertation . . . . . . 3

2 THE TRANSFER FUNCTIONS AND THE METHODS FOR COMPUTING
CHATTER LIMITS AND ERRORS OF ACCURACY . . . 7


The Critical Parameters . . . . . .
The Cutting Conditions . . . . .
Material Cutting Stiffness . . . .
Structural Parameters . . . . . .
Obtaining the Dynamic Characteristics . .
Obtaining Modal Parameters . . . . .
Computation of Chatter Limits . . . .
The Time Domain Chatter Simulation Routines
Other Approaches for the Computation of
Limits. . . . . . . .
Computing Errors Induced by the Cutting Force


. . 7
7
. . 7
. . 9
. . 9
. 12
. 15
. 18
. 20
Chatter
. 25
. 32


3 BACKGROUND AND DEVELOPMENT OF THE CUTTING TESTS . 35

Machine Tool Ranges . . . . . ..... 35
Selection of Conditions for the Cutting Tests . 36
Selection of Tools and Tool Holders ...... 36
Selection of the Spindle Speed . . . ... 39
Selection of Workpiece Material . . . 49
Positioning and Clamping of the Workpiece . . 49
Procedure for the Cutting Test for Chatter Limits 50
The Regular Procedure . . . . . ... .55
The Reverse Procedure . . . . . .. .55
Procedure for the Cutting Test for Errors Induced by
the Cutting Force . . . . . . . . 58
End Milling . . . . . . . . ... 60
Face Milling . . . . . . . ... 64


iii









4 DERIVATION IMPROVEMENT AND VALIDITY OF THE SIMPLE
FORMULAE FOR CHATTER LIMITS IN MILLING . . ... .72

The Oversimplification of Terms . . .. . . 72
The Average Tooth Position . . . . . ... 72
The Orientation Factors . . . . . ... 73
The Slotting Case . . . . . .. . . 75
Corrective Factors . . . . . .. . . 78
Other Immersions . . . . . . . . . 81
Corrective Factors for the Simple Formula for Slottinq4

5 DISCUSSION ON THE IMPACT EXCITATION TECHNIQUE . 87

The Impact Excitation Technique . . . ... .87
The Effect of the Tip Mass . . . . . .. .88
Effect on the Dynamic Flexibility . . . ... .90
How to Avoid Hammer Induced Errors . . . ... .92
A Three Degree of Freedom Illustrative Example . 92

6 CHARACTERIZATION OF THE WHITE SUNDSTRAND SERIES 20
OMNIMIL MACHINING CENTER . . . . . . . 98

Description of the Machine . .. . . . . 98
Dynamic Characteristics of the Machine . . .. 101
The Complete Set of the Characterization ... 101
Computed Chatter Limits . . . . ... .108
Measured Chatter Limits ........... 109
Computed Cutting Force Induced Errors .... 114
Measured Cutting Force Induced Errors .... 114

7 COMMENTS AND RECOMMENDATIONS FOR FURTHER RESEARCH 116

Comments on the Reasoning Behind This Work . 116
The B5 TC54 Committee Standard. . . . .. .116
Programming for Quality in Milling . . ... .117
Recommendations for Further Research ...... 118

REFERENCES . . . . . . . . .... . .120

BIOGRAPHICAL SKETCH . . . . . . . .... .122















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THEORETICAL AND PRACTICAL BACKGROUND FOR THE
CHARACTERIZATION OF CNC MACHINING CENTERS

By

Carlos A. Zamudio

August 1991

Chairman: Dr. J. Tlusty
Major Department: Mechanical Engineering

The greater demand on the cutting capabilities of machine

tools has prompted the need for the characterization of their

cutting performance as a method of measuring and comparing the

machines' cutting capabilities. This dissertation describes

the development of methods for the computation and measurement

of these capabilities. Previous attempts to generate a general

methodology for the characterization of the machine tools have

failed, in part, because the conditions at which the tests

should be performed were not systematically selected. By using

up-to-date knowledge of the machining process, conditions for

the testing have been properly selected.

The basic knowledge required for the characterization of

the cutting performance of a machine tool is the dynamic

characteristic of the machine tool-tool-workpiece system which

can be obtained by measuring transfer functions. Once the

dynamic characteristics of the machine tool are known, it is









possible to compute the chatter limitations and to estimate

the errors of accuracy produced by the cutting force.

The methods developed for the computation of chatter limits

include the use of very accurate time domain simulations of

the milling process. Similar simulations are used to determine

the cutting force induced errors in the finishing cuts.

Cutting test procedures are proposed as an alternative to

find the chatter limits and errors of accuracy. These cutting

tests must be carried out at specific conditions where the

results are not affected by any of the spindle speed dependent

phenomena. An important question with regards to the cutting

tests, which have been resolved in this work, is the selection

of tools and tool holders to be used.

The cutting tests for both chatter limits and errors of

accuracy required the development of clever procedures which

simplify the otherwise lengthy processes. These procedures and

the philosophical reasons behind their development are

discussed in detail. The results of the proposed cutting tests

together with the results of the computational methods are

presented and compared for a particular machine.















CHAPTER 1
INTRODUCTION

The Need for CNC Machine Tool Characterization

The need for machine tool characterization has been

recognized for many years and many attempts have been launched

at the national [1-3] and international [4-6] levels to

standardize this procedure. However, the manner in which

machines are specified has not changed through the years and

has not kept up with either the technical advances in machine

tool design or with the greater knowledge of the machining

processes. Machine tools now are rather costly and the demands

on their metal removal ability and accuracy of machining have

increased considerably. The need for precise and more advanced

methods of specification has become urgent.

The American Society of Mechanical Engineers B5 TC54

Committee has recently proposed a draft standard B5.54 titled

"Methods for Performance and Evaluation of Computer

Numerically Controlled Machining Centers" [7]. This standard

clarifies the performance evaluation of machining centers and

at the same time facilitates the performance comparison

between machines by unifying terminology, generally

classifying machines, and treating environmental effects.

Section 7 of the ASME B5.54 standard, which has been prepared

and developed in the Machine Tool Laboratory at the University

of Florida, deals solely with the cutting performance tests.
1







2

The basic theoretical and practical background required for

the development of these tests is presented and discussed in

the following chapters. This work, which includes the author's

original contributions, is the subject of this dissertation.



Limitations on the Cutting Performance

The objective of this work is to provide a way in which to

define and measure the capabilities of the machine tools with

reference to their cutting performance. From this point of

view the cutting process limitations may arise from either the

spindle drive limitations or from vibrational problems, mainly

chatter and errors of accuracy, which are dependent on the

dynamic characteristics of the machine. A more formal listing

of these limitations is shown below:

a) The drive is not able to deliver the necessary torque and

the machine stalls.

b) The process becomes unstable and chatter occurs. Chatter

has detrimental effects on both tool life and the quality

of the finished part.

c) The errors induced by the cutting force exceed permissible

levels. This means that tool deflections produced errors of

position and/or profile that are not acceptable.

These limitations must be evaluated in well prepared tests

under carefully studied conditions. Establishing the

capabilities of a machine tool in terms of the above

limitations is considered its characterization. This work







3

concentrates on the computation and measurement of chatter

limits and errors produced by the cutting force.



Scope of the Dissertation

The characterization of machine tools includes the

computation and/or measurement of chatter limits and the

computation and/or measurement of errors of accuracy. The

results of the characterization are presented in tabular form

and can be used to represent the cutting capabilities of the

machine and to compare the cutting performance of different

machines. It can also be used optimally by the part programmer

to select cutting parameters which will result in stable

cutting in the roughing cuts and in minimized errors in the

finishing cuts. This dissertation presents the methods for the

determination of chatter limits and errors of accuracy by

computational techniques and by well developed cutting tests.

These two methods are used to complement each other. The

measurement of transfer functions and computation of chatter

limits is cross checked by cutting tests for a selection of

tools. The cutting tests require the usage of the machine

tools for long periods of time in which the machine will not

be involved in part production. The transfer functions, on the

other hand, are measured quickly and the machine can be back

in production in a few hours. It is recommended therefore that

the bulk of the characterization be done by computational

methods and that the results be cross checked by cutting at a

reduced selection of tools and cutting conditions.







4

The theory of chatter in machining has been well understood

for many years and various methods for computing chatter

limits have been developed. End and face milling are selected

for the characterization of machining centers because they are

the most demanding of the machining operations. For these

operations the most precise and reliable method for the

computation of chatter limits is the time domain simulation of

the milling process. This method together with alternative

methods of computing chatter limits are discussed in Chapter

2. Alternatively the chatter limits can be obtained by cutting

tests. The specific conditions for these tests are presented

and recommended procedures for these tests are proposed in

Chapter 3.

The determination of errors induced by the cutting force

(errors of accuracy) is also considered in this work. However,

the current state of the art is such that the computational

methods for errors in finishing cuts are not yet

satisfactorily developed. In end milling it is possible to use

simulations to compute the errors generated in roughing

passes, but this technique does not work fully for fine

finishing passes. In face milling the computation of errors

has not yet been attempted. A discussion on the current state

of the methods for computation of errors is included in

Chapter 2 but no computational tool is presented for the

determination of errors in finishing cuts. It is possible,

however, to determine these errors by performing cutting tests

for errors of accuracy. The methodology for the determination







5

of these errors through well organized cutting tests is

discussed and corresponding tests are proposed in Chapter 3.

Sometimes it is advantageous to have a simple way in which

to approximate chatter limits. The "simple formula" can be

used in such occasions or when other tools to compute the

chatter limits are unavailable. The simple formula is studied

and further improvements are presented in Chapter 4.

The computation of chatter limits requires the measurement

of the dynamic characteristics of the machine tool (transfer

functions) and the extraction of the modal parameters from

these measurements. A phenomenon that may distort the shape of

the transfer function has been found when making such

measurements made with the impact hammer excitation technique

in systems with small reflected masses. This distortion is

traced to the physical characteristics of the impact hammer.

This occurrence is presented and studied for the impact

excitation technique in Chapter 5. Most impact hammers are

designed so that this problem does not arise. It is important,

however, to understand hammer behavior in measuring systems

with small reflected masses.

An example characterization of a machine tool is included

in this work. A White Sundstrand S-20 Omnimil machining center

is characterized for illustrative purposes. The results

obtained through computation and measurement are presented in

tabular and graphical forms in Chapter 6.







6

Finally, in Chapter 7 comments on the usage of the results

produced by the characterization of the machine tools are made

together with recommendations for further research.















CHAPTER 2
THE TRANSFER FUNCTIONS AND THE METHODS FOR COMPUTING
CHATTER LIMITS AND ERRORS OF PROFILE

The Critical Parameters

The parameters involved in the chatter process can be

separated into three groups: the cutting conditions, the

material properties and the machine tool structural

properties. Figure 1 shows these parameters for both the face

and the end milling processes. The effects of these parameters

on the chatter limit have been studied by many authors

including Tlusty [5,8-12] and are listed below for

completeness.



The Cutting Conditions

Feed per tooth (chip load) c does not affect the limit of

stability; however if the cut is unstable, the saturation

amplitude of the vibration is heavily dependent on the value

of c. The higher the chip load the higher the amplitude of

vibration.

Axial depth of cut b is the most directly influential of

all cutting conditions. The cutting force is proportional to

b for any particular state of the vibration. The depth of cut

is the best expression for the gain in the chatter self-































FACE MILLING


FulI Immersion


n Crpm) is spindle speed,

v Is the peripheral velocity
Ccutting speed),

C is the chip load
Cfeed/tooth),

i is the radial immersion
Ca/d)

a Is the radial depth of cut


b Is the axial depth of cut


m Is the number of teeth


Figure 1 Schematic of the milling process.


END
MILLING


Up-mi li ng







9

excitation process. Increasing b will always convert the

process from stable to unstable.

Number of teeth m has an effect similar to the axial

depth of cut because increasing the number of teeth increases

the cumulative axial depth of cut which leads to instability.

Radial depth of cut a (or in a better way the ratio i=a/d,

where d is the tool diameter) has a strong influence on

chatter. Increasing i may lead to instability because it

increases the average number of teeth cutting simultaneously.

Cutting speed v (or spindle speed n) has a rather complex

effect on the stability of the milling process. The

understanding of this effect is fundamental to the formulation

of the standard cutting tests. This effect comprises the

"process damping" at the lower end and the "stability lobes"

at the upper ranges. This significant parameter will be

discussed later in this chapter.



Material Cutting Stiffness

The cutting stiffness of the material Ks has a direct

effect on the chatter limit very similar to the effect of the

axial depth of cut. The product KsOb is the gain of the self-

excitation mechanism.



Structural Parameters

The effect of the machine tool structure is represented by

the dynamic flexibilities contained in the "Oriented Transfer







10

Function." This is explained in section 4.2 of this

chapter.

The two sides of the system of the chatter vibrations, the

cutting conditions and the structural dynamics, interact in

such a way that higher dynamic stiffness leads to stable

machining under higher metal removal rate per unit feed rate.

Tlusty and Smith [9] presented and clarified the parameters

affecting the accuracy of the milling operation. They

demonstrated the effect of the dynamics of the machine tool on

accuracy and demonstrated the effect of helical teeth in end

milling. The effect of the cutting force on accuracy is

explained with the help of Figure 2 where an end mill with

helical flutes is shown. The periodic cutting force produces

a vibration which is reproduced in the surface as the teeth

pass. The surface S is produced only when the cutting edge is

at the line of engagement A-A. Because of the helix angle the

cutting edge travels axially on this line (from point 1 to 6

in the figure) imprinting the deflections as the tool rotates.

The cutting force and the vibrations which develop from it

repeat in a synchronized manner every tooth period thus

producing the same deflections over and over again. The

resulting surface has errors which are consistent throughout

it. A time domain simulation has been developed to predict the

errors induced by the cutting force in end milling. This

simulation uses the dynamics of the machine tool to find the

vibration which is excited by the cutting force. The phase












b)





c)
y'


-\1 \ 1 I \ / \ .


9\_/1


A\
Fav
V


time


n- Y


a yov


A time


Figure 2 Errors in end milling, a) Tool. b) Force. c) Vibration.
d) Surface profile.







12

angle 4 between the vibration and the force depends on the

spindle speed and the dynamics of the system. The result of

this simulation is the profile generated by the cutter in the

line A-A.



Obtaining the Dynamic Characteristics

The dynamic characteristics of a machine tool are found by

measuring a set of Relative Transfer Functions Gx, Gy and G,.

A Relative Transfer Function (RTF) in the X direction

Gx=X(()/Fx(o) represents the frequency domain relationship

between the force acting between the tool and workpiece and

the vibration as measured between the tool and the workpiece

in the X direction. The same is analogous for the RTFs in the

other directions. The Direct Relative Transfer Function (DRTF)

is a RTF which is excited and measured at the same location

and in the same direction. Commonly, the DRTF is measured at

the tip of the tool because this is the location where the

cutting force will be acting. The DRTFs are taken in the three

principal directions (X, Y and Z axes) of the machine tool. In

most instances the modes are located in these directions due

of the symmetry of the design of the machine tool structure.

In general, the DRTFs should be complemented with cross

directional RTFs (where the system is excited in one direction

and its response measured in another) to check for modes whose

directions are not aligned with the principal axes.

For the measurement of RTFs various makes of Fourier

Analyzers are available. They measure and process an exciting







13

force and the system response to it. A practical way in which

to measure the RTFs is to excite the machine tool with a

shaker (using white noise) or using an instrumented impact

hammer and measure the motion response of the system using a

displacement or acceleration measuring device. The impact

hammers are instrumented and permit the measurement of the

impact force directly. One disadvantage of using the shaker

excitation technique is that the shaker needs to be positioned

between the tool and the workpiece. This is cumbersome

especially because the shaker and the displacement transducer

must be repositioned for each direction to be measured. Also,

the mass of the shaker and of the force transducer used to

measure the force produced by the shaker may affect the

measured results. These disadvantages are avoided in the

impact excitation technique where it is relatively easier to

locate the transducer used to measure the system response. The

mass of this transducer and the mass of the hammer are

selected so as not to affect the measured results. For these

practical reasons the impact excitation technique was selected

as the method to be used for the measurement of the RTFs

throughout this work.

Both methods (the shaker and the impact hammer), however,

present a problem when measuring machine tool configurations

with a small reflected mass at the tip of the tool (typical of

small end mills). The shaker membrane and the force rod affect

the reflected mass of the system yielding RTFs which are in

error. The impact excitation technique requires that the mass







14

of the hammer be relatively smaller than the reflected mass of

the system being measured. To design an impact hammer

acceptable for the measurement of small reflected mass systems

the total mass of the hammer must be reduced. This results in

a design where the mass in front of the piezoelectric crystal

is similar to the mass behind it. This has the effect of

making the hammer behave as a two degree of freedom system

with the crystal acting as a spring between the mass in front

of it and the mass behind it. This crystal produces a measured

impact force different from the actual impact force. This

phenomenon results in measured RTFs which have an extraneous

phase shift and in many cases introduce errors in the measured

dynamics of individual modes. This aspect is interesting

enough so that the author has made a special study of it which

is presented in Chapter 5.

Many machine tools have bearing configurations which are

preloaded or which make use of constant preload mechanisms

that use spring or pneumatic loading. This provides the

spindle/bearing configurations with dynamic characteristics

which change little throughout the spindle speed range

including standstill. Therefore, it is possible in these cases

to measure the RTFs with the spindle stationary. Sometimes,

however, it is necessary to measure the RTF while the spindle

is running. It is good practice to check for strong non-

linearities of the system. This can be done by measuring the

RTFs using different amounts of force and checking for

sizeable changes in the measured RTFs. A linear system will







15

produce identical RTFs for all levels of force provided that

enough energy excites the prominent modes. In the case where

strong non-linearities are found it is necessary remeasure the

transfer functions of the system while applying a preload

which simulates the DC component of the cutting force. The

preload must be applied carefully so as not to affect the

dynamic characteristics of the system. This can be

accomplished by using a long piano wire arrangement which will

not change the system stiffness significantly.

In some instances the process of measuring RTFs can be

simplified by disregarding a given direction or part of the

overall structure. These simplifications should be made very

carefully as they can result in gross errors. In milling, for

example, the modes in the axial (Z axis) direction are

sometimes significantly stiffer than the modes in the other

directions. It is possible in these cases to disregard the

dynamic activity in the axial direction. In other instances

the workpiece might be significantly stiffer than the tool

making it possible to disregard the workpiece dynamics without

producing significant errors. These simplifications should not

be applied generally and should be studied for each

configuration before they are made.



Obtaining Modal Parameters

Transfer Functions can be plotted or displayed in several

formats. The most common forms in which these are presented

include the phase-magnitude, complex plane, DB or log real-







16

imaginary, and linear real-imaginary. Through this work the

linear real-imaginary format is used to present transfer

functions. Figure 3 (a and b) shows a typical set of DRTFs

measured using an impact hammer and a capacitance probe for

the X and Y directions. These DRTFs are plotted in units of

micrometers per Newton versus frequency.

Once the RTFs are measured, they are used to extract the

dynamic parameters for the most significant (most flexible)

modes in each direction which can later be used for the

computation of chatter limits and errors induced by the

cutting force. The parameters of interest for each protuberant

mode are the modal mass mq, modal damping ratio (q, modal

stiffness kq, and its frequency fn. There are many curve-

fitting methods available for the extraction of modal

parameters [13,14] from the measured transfer function. Some

are computer based iterative subroutines which fit the

individual transfer functions to a selected number of modes.

Other methods present the measured and fitted RTFs after each

iteration in a graphical representation and allow the user to

vary the parameters until the curve is fitted satisfactorily.

Sometimes, however, it is useful to have a method for

obtaining approximate modal parameters which does not require

iterative computer based routines. This can be accomplished in

cases where the modes are well separated by fitting one mode

at a time. Figure 4 shows a single degree of freedom RTF and

an approximated relationship between the shape of the real and
















5. E-B6

, m/M
G. ..'"

2.5E-86



a. BE+BB



-2.5E-B6



-5.BE-06





B.BEtBB

C, n/N

-2. 5E-B6



-5. BE-B6



-7. 5E-86



-1.BE--85S


R- V d-ti DT











a Ze8 480 688 sea ile
Frequency. H.



!.cer dI..ctrc ____________ IRTF




Y 1


288 400 688
Frequency. Hz


888


1088


(b)



Figure 3. Direct Relative Transfer Function. a) X direction;

b) Y direction.


-----------;- --*-*** -- ________ ; ------



Frequency. Nx



Imag lnaer X direct lIon _RTF


















a 208 488 6EBB 808 18
Frequency. Hz


5. BE-86

C. rm/

2. 5E-B6







-2. 5E-86



-5.8 E-86






B. BE+BB

G, m/H

-2. 5E-86



-S5. E-6B



-7.5E-B6



-1. BE-B5


Real X Il.ntionL


IIRTF


"-- Y ~----'---


"""







18

imaginary curves and the modal parameters to be extracted.

This method is very useful in situations where quick

approximated results are required. Table 1 shows the

parameters extracted for the set of transfer functions

presented in Figure 3 by using the manual curve fitting

technique and compares them to the actual values. It is seen

in this example that the manual curve fitting technique gives

reasonable results for the estimation of modal parameters from

transfer functions where the modes are well separated. Note

that the parameter extracted from the two first modes (which

are relatively close to each other) yielded the highest

errors.



Table 1. Dynamic parameters extracted from Figure 3.

Curve Fitted Actual

Mode m,Kg k,N/m C m,Kg k,N/m C

1 (X) 1.39 1.56e6 .045 1.2 1.35e6 .05
2 (X) 0.62 2.06e6 .040 0.80 2.75e6 .030
3 (X) 0.34 6.86e6 .041 0.35 7.20e6 .040
4 (Y) 0.91 1.92e6 .038 1.00 2.10e6 .035
5 (Y) 0.62 5.80e6 .042 0.67 6.21e6 .040



Computation of Chatter Limits

The theory of chatter limits has been explored in depth by

many researchers [6,8,12]. The theoretical background has been

developed during the last 40 years. The best and most reliable

way to compute chatter limits is the time domain simulation of













Re[G]max


4ke
V If_ Re[G]mi
I mm


f


Im[G]








Figure 4








Figure 4


f-=fn


SIm[G]min
L-mmn


Simple curve fitting of transfer
obtain modal parameters.


functions to


Re[G]

] i







20

the milling process. In its current stage the simulation,

which is based on the regeneration of waviness model, includes

the process damping effect and other non-linearities like the

teeth jumping out of contact [15-16]. A more complex

formulation for the theoretical computation of chatter limits

has been presented by Minis et al [12]. This formulation has

significant value but its complexity makes it impractical to

use in comparison with the much easier simulation routines.

Simpler close formula methods for the approximation of chatter

limits are also available though not as accurate as the

simulation. The "simple formula," lobing diagram, and Minis'

formulation are discussed later in this chapter.



The Time Domain Chatter Simulation Routines

The mechanics of the time domain simulation were presented

by Tlusty and Ismail [17] and Smith and Tlusty [16]. The

algorithm is based on the regeneration of waviness criteria

where the waviness imprinted in the surface as the teeth pass

combines with the motion of the cutting tooth to produce chip

thickness variations. This chip thickness variation results in

a variational cutting force which feeds the vibration.

Whether the vibrations grow or die out depends on whether the

system is in an unstable or stable condition. The conditions

of stability are dictated by the parameters described earlier.

These parameters can be separated into four groups:

a) cutter geometry. These are parameters describing the cutter

geometry, such as tool diameter and number of teeth.







21

b) system dynamics. These include the modal parameters of the

system as described earlier.

c) material properties. These are mainly the cutting stiffness

of the material and the process damping coefficient.

d) cutting geometry. These include the spindle speed, the

axial and radial immersion, the chip load, the direction of

feed, etc.

All these parameters are used to simulate the milling

process. The information is read and processed resulting in

time and frequency domain records of the cutting force and the

vibration of the system. These are examined by the user to

determine whether the system is stable or unstable. Figure 5

presents the partial output of such a simulation program. Here

the time and frequency domain records of the vibration in the

X direction for a 0.75 radial immersion cut with a six fluted

end mill are presented for stable and unstable cases. Chatter

is most easily recognized by a large spectral line in the

frequency domain which is not a harmonic of the spindle runout

frequency. The severity of the vibration in the time domain

plot can also be used to distinguish chatter.

It is possible to find the chatter limit for a given radial

immersion by running the simulation program many times at

different axial depths of cut. This method, however, is not

very efficient. A better way to determine the chatter limits

is to produced a composite peak-to-peak diagram of the cutting

force or the system vibration or both. The idea of the





















Stable Cut


5bm


O- m






- 50,um i
0 Time (s) .2


O/m


o Frequency


100m -





lim
-400/m ,
o Time (s)


Unstable
-i 1 00um


e


0 Frequency (Hz)


Figure 5 Time and frequency domain records of stable and
unstable cuts produced by simulation.


tooth


(Hz)


1250


Cut


chatter


1250







23

composite p-t-p diagram and its algorithm was developed and

proposed by Smith in 1987 [18]. The simulation program is run

for a matrix of spindle speeds and axial immersions, and the

peak-to-peak value of the time domain force and vibration

signals are extracted and saved. The p-t-p values are then

plotted as shown in Figure 6 where the p-t-p vibration (top)

and force (bottom) in the X direction are plotted versus

spindle speed for lines of constant axial immersion. The axial

immersion in this case is increased from 0.5 mm to 3.5 mm in

steps of 0.125 mm. As the axial immersion is increased some

regions grow faster than others. The regions where the p-t-p

values are relatively smaller are the stable regions and the

regions where they increase considerably faster are the

unstable regions. It is possible to determine the chatter

limit at a given spindle speed by finding the lowest axial

immersion at which the p-t-p values start to increase rapidly.

Since the simulation program was not efficient at the time

this technique was developed it took a considerable amount of

time to construct a composite diagram (more than 24 hours for

a 4000 point diagram using a VAX 750 computer). Since then,

the author has improved the efficiency of the algorithm, the

simulation, and the peak search routine. At the present

stage it is possible to compute a 6000 point p-t-p diagram in

fewer than 3 hours using a fast personal computer.

A piece of information which is dropped when using the

composite peak-to-peak diagram is the frequency of chatter. In









Bmin =.5 iMax = 1.75 Binc = .125 mm 24


5588


588 1588 2588 3588 4588
RPI


688




458




388




158



8 -
588





Figure 6


1588 2588 3588 4588 5588
RPH




Composite peak-to-peak diagram for a slotting cut.
a) Vibration. b)Force.


X Vibration (a)







25

situations where the frequency of chatter is required (to

determine which mode is responsible for the chatter, for

example) the simulation program is run for the chattering

conditions already determined through the p-t-p diagram.

Alternatively, future versions of the composite p-t-p diagram

program should include frequency domain information.



Other Approaches for the Computation of Chatter Limits

The theory behind the simple formulation of chatter in

milling was derived from the chatter theory developed for

turning. In stable steady state turning the cutting force is

constant in both magnitude and direction. A simple plunging

case is presented in Figure 7. This figure also shows the

geometric construction of the directional orientation

factors px and p, [17]. These factors represent the

activity of the modes in the X and Y directions respectively.

To compute these factors the cutting force is first

projected into the directions of the two RTFs (these produces

the X and Y vibrations) and then they are projected in the

direction normal to the cut. The orientation factors are

computed by



9x=cos(aC-P)cosax
(1)
Sy=cos(y-p)cosay


and used to compute the oriented Transfer Function from












Plunge


Cutting


in Turning


WORK PIECE


N 3K-


ly


Figure 7 Derivation of the "simple formula" for turning.










Goriented= x*Gx+ y*Gy (2)



Once the Oriented Transfer Function (OTF) is obtained it is

possible to compute the limit of stability as presented by

Tlusty [8] as

-1
b- (3)
S2KRe [Goriented] min



where bcr is the limiting chip width, Ks is the specific

stiffness of the material and Re[Goriented]min is the minimum

value of the real OTF. This equation has been used very

successfully for many years.

In milling, however, the cutting force is not constant.

This is illustrated in Figure 8 where the turning operation is

compared with the milling operation. Here it is shown how the

force varies in direction and magnitude as the cutter rotates

for the milling case. In an effort to produce a similar

"simple formula" as in the turning case the "average tooth

position" ave is defined as the average position of the teeth

in the cut and the "average number of teeth in the cut" mave is

defined as one half of the number of teeth in the cutter times

the radial immersion as a ratio of the tool diameter. That is




Saventry- exit +ntry
2 (4)

mave= M'i









Turning


Milling





v n v



N N



/////Workpiec Workpiece






N b)














Figure 8 Orientation of the cutting force. a) Turning
b) Milling.









where Dentry and exit are the entry and exit angles of

engagement of the cut, i is the radial immersion as a ratio of

the tool diameter and m is the total number of edges in the

cutter. Then the orientation factor can be found for the

average tooth position using equation 1 and the critical limit

of stability in equation 3 is divided by the average number of

teeth as follows:

-1
b (5)
r 2KRe [ Goriented] minave



This equation is known as the "simple formula" for the

computation of the critical chatter limits. This formulation

is found to produce erroneous results in many situations and

especially in the slotting case. A discussion, further study

and improvement of the simple formula for slotting is

presented in Chapter 4.

In order to include the effect of the spindle speed in

computing the chatter limits it is necessary to generate

"lobing diagrams." Tlusty et al [11] and Tlusty and Ismail

[19] developed a way in which to generate the lobing diagram

from a model similar to the "simple formula." The "lobing

diagram," however, accounts for the effect that the spindle

speed has on the phase in which the vibrating tooth encounters

the surface waviness. The number of waves between two

subsequent teeth is found from










=INT( 6 0 f
(6)
e =[ 60"*f 360


where n is the spindle speed in RPM, m is the number of teeth

of the cutter, f is the frequency of the mode in Hz, N is the

integer number of waves between the subsequent teeth and e is

the phase (in degrees) between them. The top of Figure 9 shows

the effect that the spindle speed has on the number of waves

between two subsequent teeth as well on the phase between

them. The phase E has an effect on the chip thickness

variation which affects stability of the process. This can be

seen in Figure 9 where three cases with different values of e

are illustrated. It is seen that the chip thickness variation

is different for all the cases even if the amplitude of

vibration is the same. The last case is the 0 degree phase

situation where there is no chip thickness variation. This is

known as the "miracle speed" at which the stability is

improved significantly. This frequency domain solution results

in a mapping of the limiting chip width versus spindle speed.

The lobing diagram is made up of lobes whose envelope

separates the stable region from the unstable region. The

lobing diagram contains similar information as the composite

p-t-p diagram. In the bottom of Figure 9 a lobing diagram for

a single DOF system is shown. Here the effect of the lobes is

seen at higher speeds where the "miracle speed" effect is










The Effect of the Spindle Speed on Stability










The Effect of the Phase
o)


-- n


I n=fn
Spindle Speed, rev/sec


The Effect of the Number of Waves
b)






Figure 9 The effect of the spindle speed. a) Phase effect.
b) Lobing diagram and the effect of the number of
waves (lobes).


E
E



0

c-


b

bcr







32

stronger. The lobing diagram has been used in turning to

determine the stability pockets and the limiting chip widths

very successfully. In milling, however, errors occur because

this method is based on the same simplifications which were

used in the development of the simple formula. For this reason

it is preferable to use the composite p-t-p diagrams for

milling whenever possible.

Recently, Minis et al [12] have presented a complete

formulation which does not require any simplifications for the

computation of the chatter limits in milling. This

formulation, which is very complex, includes the effect of

non-linearities (like the tool jumping out of the cut) and

process damping. This formulation has only been presented for

a model with a single degree of freedom in each direction. It

is possible to use this formulation to compute accurate

chatter limits but the practicality of its use is not obvious.

This is specially true in the presence of well developed

simulation programs which are available.



Computing Errors Induced by the Cutting Force

The techniques for the computation of errors are not in

such an advanced stage as the methods used to compute chatter

limits. The errors in the end milling case can be well

approximated by a non-regenerative simulation of the process.

By non-regenerative it is meant that the waviness imprinted in

the surface of the workpiece as the chip is removed is assumed

small and its effect on the cutting force is not accounted







33

for. The modal information of the system is included in the

simulation. The author has developed a program to simulate

errors in end milling with helical teeth based on the

algorithm presented by Tlusty [20]. Figure 10 presents the

computed profile errors for radial immersions for increasing

values of radial immersion computed with the simulation

program. This program is very good for radial immersions over

10% of the diameter. For smaller radial immersion the program

does not reproduce the cutting process accurately enough and

the results do not represent the actual situation. The current

state of the simulation of errors in end milling, as reported

by some authors [21] suggests that it will soon be possible to

compute the errors produced by small radial immersion cuts.

The computation of errors in face milling has not yet been

studied. It will be possible to develop analogous simulation

programs to predict these errors. This task is now being

undertaken at the Machine Tool Laboratory.



























tILSIM/P Ver i..0


Profile Display Facility


Com: Accuracy Errors
Morn. Factor: 02487

Defl 1: .1
Def l 2: .2
iBefl 3: .3
Defl 4: .4
Defl 5: .5
Defl 6: .6
Defl 7: .7
Defl 0: .8
Def1 9: .9
Defl 18: 1.


Figure 10. Errors of accuracy in end milling computed through
simulation for increasing radial immersions.















CHAPTER 3

BACKGROUND AND DEVELOPMENT OF THE CUTTING TESTS



Machine Tool Ranges

The state of the technology is such that it is not

necessary to divide the machine tools in ranges. The attitude

of machine tool users and builders, however, is such that it

demands general guidelines based in simple formulations. The

machines are grouped in ranges which have somehow similar

dynamic characteristics so that general recommendations can be

made. Following this principle machining centers have been

divided into ranges according to their torque capabilities.

Table 2 lists these ranges.




Table 2. Machining center ranges.

CNC Machining Center Range Maximum Torque (Nem)
A. Very Light < 10
B. Light 10 25
C. Medium Light 25 65
D. Medium 65 150
E. Medium Heavy 150 375
F. Heavy > 375









Selection of Conditions for the Cutting Tests

The conditions at which to perform the cutting tests have

to be carefully selected. These include between others the

selection of tools and tool holders as well as the selection

of the spindle speed. The philosophy used in the selection of

tools and spindle speeds will be discussed in detail in the

following paragraphs. Other conditions which are not as

critical for the tests but are important enough to be

discussed are the material, geometry and location of the

workpiece and the tool motion procedures used to alleviate the

tests. Previous attempts to create rules for the

characterization of machine tools failed because the

conditions listed above where not given careful consideration.



Selection of Tools and Tool Holders

The tools must be selected to assure that the system being

tested is the machining center and not the tool itself. All

machining centers tested with the same very flexible tool will

have a most flexible mode at about the same frequency with

about the same dynamic flexibility. This mode is close to a

pure "tool mode" where most of the system's flexibility comes

from the tool itself. Thus comparing the performance of the

machines using this tool would have no value because all

machines will perform similarly. This illustrates why the tool

selection process requires knowledge of not only the transfer

function but also of the mode shapes of the spindle / holder

/ tool configurations. A good tool for testing is such a tool







37

where the most flexible mode has significant spindle

participation. This means that the spindle should participate

as much as the tool in the most prominent mode. Figure 11

shows the mode shapes of a spindle with two different tools.

The first case corresponds to an end mill 19 mm in diameter

and 75 mm long. Here the most flexible mode (2144 Hz) has a

mode shape where the tool participates by itself. The second

case corresponds to a tool 50 mm in diameter and 75 mm long.

Here the most flexible mode (838 Hz) shows significant spindle

participation. This latter tool will be a good tool to use in

testing the machining center.

Another important consideration is the selection of the

tool holders. The holders should be selected as to give the

highest stiffness and the shortest overhang. Using these

criteria the V-flange holder is selected for face mills and

the V-flange set screw type holder is selected for end mills.

The set screw is selected over the collet type holder because

it is stiffer.

Considering all the above the tools to be used in the

cutting performance test have been selected as follows:

a) Face mills. Uniform pitch face mills with negative square

carbide inserts will be used (with negative/ negative

geometry, 0.5 mm nose radius and small lead angle). Because

many face milling operations require the use of extensions

holders the face mills will be use with and without

extension (see Figure 12).



























S934Hz, 304kN/mm






1014 hZ, 62.7kN/mm






1596 Hz, 126kN/mm






S 2144 Hz, 9.4kN/mm
* ------


838 Hz, 59kN/mm






1547 Hz, 584kN/mm







2429 Hz, 234kN/mm






3193 Hz, 247kN/mm




b)


Figure 11 Mode shapes. a) 19 mm by 75 mm. b) 50 mm by 75 mm.







39

b) End mills. These are HSS or solid carbide end mills. These

should have continuous smooth edges with uniform pitch. End

mills will be used with set screw holders. If available

short overhang holders (like shrink fit type) could be

used. These are depicted in Figure 13.

c) Shell end mills. These cutters commonly have bodies that

are integral with the holder as shown in Figure 14. The

carbide inserts are set to resemble a flute. When possible

the tool should be selected such that the inserts fill the

cutting edge completely. Cutters with gaps in the cutting

edges may be used but the limiting axial immersion will be

reported as the product of the actual chatter limits times

the average filling factor (See Figure 14).

Using these criteria Table 3 has been created where the

tools recommended for the cutting test are listed for the

different machining center ranges. The transfer functions and

mode shapes required for the proper selection of these tools

have been gathered or generated from the extensive work

performed at the University of Florida's Machine Tool

Laboratory during the past six years.



Selection of the Spindle Speed

Ideally the spindle speed to be used in the cutting test

can be obtained from the computed p-t-p diagrams which are

generated for the computation of the chatter limits. The speed

should be selected to test at the most unstable condition to

obtain the critical limit of stability. Conversely it is









Face Mi s


Regular face mill


Extension ength, L


Face mill


with extension


Figure 12 Face mills, a) Standard holder. b) Extension
holder.









End Mills


[-- overhang

M50 Neldon holder


Special
Short overhang holder
(shrink fitted)


Figure 13 End mills, a) Standard holder. b) Shrink fit.









End mill with


Overlap


inserts


I


type


Gap type



Figure 14 Integral shell end mills, a) Full edge. b) Gap
type.


bl
p b









Table 3. Selected tools for cutting test.

Range Standard Extension End Mill End Mill
Face Mill Face Mill Dia(mm)x W/Inserts
Dia(mm)x Dia(mm)x Length Dia(mm)x
Length(mm) Length(mm) (mm) Length(mm)
No. Teeth No. Teeth No. Flutes No.Flutes
A N/A N/A 12.5x30 N/A
2
B 50x0 N/A 19x50 N/A
4 4
C 75x0 75x50 25x75 N/A
6 6 4
D 100x0 100x100 38x75 50x75
8 8 6 4
E 100x0 100x125 50x75 50x100
8 8 6 4
F 150x0 150x150 50x125 67.5x150
12 12 6 6



possible to also test at the most advantageous speed to obtain

a range for the chatter limits.

It is also possible that the p-t-p diagrams are not

available. In this case it is necessary to at least know the

frequency of the most prominent modes. In a general case the

process of selecting the spindle speed at which to perform the

cutting tests requires the satisfaction of three conditions.

The first condition to be considered for the selection of

speeds is tool life criteria. The maximum speed is set by the

empirical tool life data. A maximum surface speed of 120 m/min

is selected for cutting steel 4340 with coated carbide

inserts. The corresponding maximum spindle speed for cutting

steel according to tool life criteria is found by










nt(rpm)=1000 120(m/mi) (7)
x -d(mm)



where d is the tool diameter in meters and nt is the spindle

speed in rpm. The tool life criteria does not apply to

machining aluminum. HSS and Carbide tools can machine this

material at very high speeds with little wear.

The two other conditions which must be met are related to

the dynamic characteristics of the machine tool with its

different tool configurations. The first criteria which is

considered is the lobing effect due to the relationship

between natural frequencies of the system and spindle speed.

The effect of the stability lobes may pollute the results. The

test must be carried at such a spindle speed where the lobing

effect is minimized. From chatter theory it is known that the

first stable region lies at spindle speeds at which the tooth

frequency matches the chatter frequency. Other regions of

stability lie at integer divisions of the first stable speed.

If there is a first stable region at 12000 rpm there will be

a stable region at half this speed (6000 rpm) another one at

one third (4000 rpm) etc. The stability pockets become smaller

as the speed decreases because the lobes get tighter together.

Figure 15 shows a typical lobing diagram derived for a single

mode. It is possible to see the large pockets of stability to

the right of the plot. It is also possible to see these

pockets shrink in size as the speed decreases. The speed at

which the first lobe occurs is found from









60f, (Hz)
n, (rpm) (8)
m



where ni is the speed in rpm and fn is the natural

frequency of the system in Hz. Other stability pockets are

found from

60*fn(Hz)
n (rpm) =6f (Hz) (9)
i'm



where ni is the spindle speed for the ith region of stability.

Below the third pocket, the stability varies little. The

maximum spindle speed to avoid significant lobing effect is

set to a speed 80% of one third of the first lobe

(10) 0.8 -n (rpm)
nm (rpm) -
3



Depending on the tool life criteria nmx or nt will limit the

maximum speed for the tests.

The last condition which must be met for the proper

selection of the spindle speed is the avoidance of process

damping. The test must be carried a such speed where the

effect of process damping is minimized. As the spindle speeds

are lowered the effect of process damping is first seen when

the ratio of surface speed v (m/sec) over natural frequency fn

(Hz) is 0.0015 (m) for both steel and aluminum. This

correspond to a wavelength A of 1.5 mm in the waviness left in













Lobing Diagram






Process
damping



(9






cr

CF


Lobing Effect


n min n max


ni- 60f
m


Spindle Speed, rpm


Figure 15 Lobing diagram for a SDOF system.









the workpiece surface. This relationship is expressed as


S(mM) = 1000-v(m/sec) _*'d(mm) 'nmi (rpm)
(mm)--- -1.5 (11)
fn (Hz) 60 fn (Hz)


where v(m/sec)=r*d(mm) n(rpm)/1000. Solving for nmin and

multiplying all constants together the equation becomes

29*fn(Hz)
in (rpm) 29f(Hz) (12)
d(mm)


The value nmi is the minimum value at which the chatter test

should be performed.

The three speeds nt, nmx, and nmin determine the range of

speeds at which the chatter test should be conducted. To test

for the critical chatter limits it is necessary to test at

speeds where the system is most unstable. Depending on how

much knowledge of the machine dynamics is available the

spindle speeds for chatter testing are selected using the

following rules:

a) Rule 1. If the lobing diagrams or the composite peak-to-

peak diagrams are available the highest speed in the range

nmin to nmx (or nt if cutting steel) such that the cut will

be at a critical condition. This will produce the critical

chatter limits. It is possible to repeat the test at such

a speed where the system is most stable to establish the

range between the critical chatter limits and the limits at

the most stable speeds.

b) Rule 2. If only the frequency and dynamic flexibility of

the most pronounced mode in each direction is available the







48

test is performed at the speed nmx as determined from the

formulation presented earlier in this section. If nmax is

greater than nt then the test is performed at the speed

nt.

c) Rule 3. Rules 1 and 2 apply for most situations. There are

cases where the dynamics of the system are so complex that

the test speed can only be determined from the peak-to-peak

diagrams. This is increasingly important in systems with

separate modes of about the same flexibility. To illustrate

lets assume a system with a strong mode at about 70 Hz and

another strong mode at 450 Hz. This system yields

conflicting speed ranges depending on which mode is used to

compute nmx and nmin. To avoid the lobing effect in the 70

Hz mode it is necessary to cut at a speed at which the 450

Hz mode is in the process damping area. This problem is

partially solved by using the peak-to-peak diagram to

select the speed. The test should be carried at the most

unstable speed in the range nmin (for the 70 Hz mode) to nMx

(for the 450 Hz mode). It is also possible to test in the

most stable regions to determine the chatter limit range.

d) Rule 4. If no data is available the test is performed at

default speeds which have been derived from typical

frequencies and dynamic flexibilities for the different

configurations of all the machine ranges. Table 4 lists the

default spindle speeds to be used in the testing. The

speeds in Table 4 apply only to the tools listed in Table

3.









Table 4


49

Default cutting test speeds. Numbers in
parenthesis refer to short overhang tool holders.


Selection of Work~iece Material


The materials selected are 4340 steel for the face milling

operations and aluminum 7075-T6 for HSS and solid carbide end

mills. For end milling with the integral shell end mills 4340

steel will be used. The chip loads selected are 0.15 mm per

tooth for the machining of steel and 0.1 mm for the machining

of aluminum.



Positioning and Clamping of the Workpiece

The workpieces used for the chatter and surface error

cutting tests are rectangular in geometry. They must be

clamped as close to the center of the work area as possible.

For horizontal machining centers the workpiece should be

clamped at the center of the table. For horizontal machining


Range Face Extension End Mill End Mill
Mill Face Mill (Aluminum) W/Inserts
(Steel) (Steel) rpm (Steel)
rpm rpm (Short) rpm
A N/A N/A 11200 N/A
(12800)
B 763 N/A 2800 N/A
(5200)
C 510 510 2100 N/A
(5600)
D 380 380 1600 760
(2200)
E 380 380 1600 760
(2100)
F 250 250 2400 570
(2700)







50

centers the workpiece should be clamped in an angle plate at

such a height so that the center workpiece is close to the mid

point of the Y axis travel. The Z axis cutting plane should be

close to the mid point of the Z axis travel.



Procedures for the Cutting Test for Chatter Limits

The material, tool, and spindle speed are selected using

the criteria presented above. The task is then to generate the

mapping of the limiting axial and radial depths of cut as

proposed earlier in section 1 of Chapter 2. The test procedure

must then allow for the variation of either the radial or the

axial depths. That is, once the axial depth of cut is selected

the radial immersion is increased in the cut until chatter is

produced or once the radial immersion is selected the axial

depth of cut is increased until chatter is produced.

Originally the test was proposed so that the radial

immersion was increased through the cut until chatter was

produced. Then the axial depth of cut was increased and the

test repeated. This test procedure was run for various tools

and machines an seemed very efficient. Either the rectangular

workpiece was clamped at an angle or it was pre-machined into

a wedge shape as shown in Figure 16. One problem with this

test is that results are found for radial and axial depths

which are scattered and thus direct comparison between the

results found for different machines would be difficult. This

means that the chatter limits are found for a given set of

radial immersions in one machine and for a different set in















1.1d


0.12d! j


a) Block
Workpiece


)y Down Mill



Up Milling






Y
x


---- A


b) Pre-Machined
Workpiece


Figure 16 Chatter test by increasing radial immersion.


ing


0.125d


2.25d



F-


0.125d







52

another one. This is illustrated in the following example. For

a given tool in machine 1 the chatter limits in the X

direction are found for 0.3, 0.7 and 0.9 radial immersion up

milling and 0.35, and 0.8 radial immersion down milling. For

a second machine using the same tool cutting in the same

direction the chatter limits are found for 0.2, 0.85 radial

immersion up milling and 0.15, 0.75 and 0.98 radial immersion

down milling. The data obtained through the cutting test as

described above is not directly comparable between the

machines.

Another version of this test was to first select a set of

radial immersions at which to test. Once the radial immersion

is selected the axial depth of cut is increased through the

cut by using a wedge-like workpiece until chatter was

produced. This test solved the problem presented above but

failed because the axial depth of cut varied from the entry to

the exit angles of engagement. This was very significant in

slotting with face mills which have large diameter where the

axial depth of cut at the entry and exit of the cut were

significantly smaller than the axial depth at the center of

the cut.

From these experiences it was decided that the test should

be performed for five selected radial immersions and the axial

depth of cut should be increased by steps. Figure 17 shows the

geometry of the workpieces selected as well as the locations

at which the workpiece should be clamped for both horizontal

and vertical spindle machines. The chatter test will be







53

carried out in the X and Y directions of cut. The radial

immersions selected for the testing are 0.25 up and down

milling, 0.5 up and down milling and full radial immersion.

Before the test is started the available torque and power

at the test speed are calculated for the selected immersions.

The maximum axial depth of cut due to torque limitations is

computed for each case using the nominal torque T from

b amM) = 2000 7~T(Nm)
b (mm)- (13)
l a (mm) -c (mm) -.mK, (N/mm2)

where a is the radial immersion, c is the feed per tooth, m is

the number of teeth and Ks is the cutting stiffness of the

material. During the cutting test the axial depth of cut

should not surpass 90% of the values found through equation 12

to avoid stalling the spindle drive.

The initial axial depths of cut for each case are computed

as one half the limiting axial depth of cut as found through

the simulation. If the simulation results are not available

the "simple formula" may be used to get a good estimate from

which to start.

The test procedure is the same for both end and face

milling. The difference being that the face milling test

requires extra machining to clean the workpiece after each set

of tests (see Figure 18 and 19). The following procedure is

repeated for the 5 cases (0.25 up, 0.25 down, 0.5 up, 0.5

down, and 1.0) in each direction (X and Y 10 cases per

tool).












Chatter Test Workpiece


Face Mills


h=0.75d

End Mills


-=d h=75mm
K- w=d





hw=100mm




Horizontal Spindle

S]"-, I ~i Vertical Spindle


Figure 17 Workpiece geometry and clamping.









The Regular Procedure

A given radial immersion and mode of cut (up or down

milling) are selected. The computed chatter limit is used to

determine a initial depth of cut b=bcompud/2. The tool path is

programmed to cut in a loop starting at b, axial depth of cut

(see Figure 18). The spindle speed and feed rate are used in

accordance to the rules presented earlier. The length of cut

L=0.75d (where d is the diameter of the tool) is programmed

for the first pass. If chatter occurs the reverse procedure is

used, otherwise the tool is returned to the initial position.

The axial depth of cut is increased by bI. The length of cut

is set to L=AL+L. The tool will cut at and axial depth of bi

for the first 0.75d part of the cut and 2bi for the rest of

the cut. The value of AL is set at 0.25mm for all tools. If

chatter occurs stop the test and mark this axial depth of cut

as the chatter limit, other wise return to the starting point

increase the axial depth by bI to 3bI and cut a length L+2AL.

This procedure is repeated until chatter is obtained or the

maximum allowed axial depth due to torque limitations is

reached. In this last case the chatter limit will be mark as

blim>bTtim



The Reverse Procedure

If chatter occurred in the first cut the axial depth is

reduced by an amount b1/4. The tool pass is programmed to cut

a length L+AL. If chatter reoccurred the tool is returned to








Face Milling
Slotting, i=1.0


AL=0.5d
b2
t t _b___


Down milling 1/2 immersion, i=0.5





Up milling 1/2 immersion, i=0.5


/)t


Down milling 1/4 immersion, i=0.25.


Up milling 1/4 immersion, 1=0.25.


Figure 18 Chatter test procedure for face mills.












End lingg


Down


3 i=0.25


Figure 19 Chatter test procedure for end mills.


i=0.5


Slot
i=1.0 1


Up


,i= c







58

the initial position the axial depth is further decreased by

another b,/4 and the pass programmed to cut a length L+2AL.

This procedure is repeated until chatter is not encountered or

until the axial depth of cut is equal to b,/4. In this last

case the chatter limit will be stored as bim
the chatter limit is marked at the last cut at which chatter

was observed.

The chatter limits found through the cutting tests are

tabulated for each tool. For each combination of radial

immersion and direction of cut a chatter limit (bim) and a

frequency of chatter is listed. These tables can be used to

verify the validity of the computed chatter limits which are

reported in similar tables. The chatter limits found through

both the simulation and the cutting test are also plotted in

the form presented in Figure 20. These knowledge can then be

made accessible to the part programmer and the machine

operator as a guide to the capabilities of the machine from

the point of view of chatter and torque overload avoidance.



Procedure for the Cutting Test for Errors Induced by the

Cutting Force

The cutting test for errors induced by the cutting force

underwent a similar development period as the chatter test.

Originally it was though that the force induced errors in end

milling could be evaluated by the "dip" test. Here the feed is

stopped in the middle of a finishing cut so that the tool

deflections can relax generating a dip. Using the bottom of
















30
blim (mm)


6
ah. (mm


Evaluation of the Chatter Test
100mm-8 insert Face Mill (X dir)
Machining 4340 Steel


18

12





0 0.25 0.5 1.0 0.5 0.25 (
Up Milling Down milling
Radial Immersion (a/d)


oo0


al m
0/-


.U-tu ------ -^ ------------------- -------------------------------------

360

240


0 0.25 0.5 1.0 0.5 0.25
Up Milling Down milling
Radial Immersion (a/d)


Figure 20 Results from the chatter test.


_ b _ _


T(N/m)







60

the dip as an ideal reference surface from which to compare

the rest of the cut would provide a measure of the errors. A

series of tests were performed to establish the validity of

this theory. After careful study it was found that the bottom

of the dip does not represent the ideal surface and the "dip"

test had to be discarded. As a result two new test procedures

were developed by the author; one for end milling and one for

face milling. In this tests the ideal location of the surface

is inferred from reference surfaces which are machined with

very small immersions so that the errors are minimized.

The tools, spindle speeds, materials and feed rates are the

same ones used for the chatter test. Since the aim of the test

is to measure the errors imprinted in the workpiece surface it

is critical that the cuts made do not chatter.



End Milling

The first step in this test is to determine the chatter

limit at a radial depth of cut of 2 mm. The procedure

described in the chatter test can be used to accomplish this.

Once this chatter limit is determined the axial depth of cut

for the test is set to 80% of the limit or 50 mm whichever is

smaller.

Figure 21 shows the geometry of the test. The workpiece is

prepared by machining two incline surfaces. The geometry of

the prepared part is described by the lengths P, R, and h. The

values of these dimensions are chosen so that P=125 mm,

R=1.2-d, and h=1.6 mm as shown in Figure 21.







61

The workpiece is prepared with two inclined surfaces to

allow testing of the two modes (up and down milling) of

cutting in the same test for an axis. Using Figure 21 as

reference the test is described for the X direction. The left

most inclined surface is machined in the +X direction down

milling. Conversely the right most inclined surface is cut in

the -X direction up milling. The tool is set so that its

center is one diameter to the left of the workpiece and so

that the maximum radial immersion is 1.5 mm. This is

accomplished by setting the tool 1.5 mm below the top surface

or much more easily by programming the workpiece preparation

and the test in the same program. This way the location of the

tool with respect to all surfaces is known. The tool is fed in

the +X direction until the center of the tool coincides with

the mid point of the workpiece. Because of the incline

geometry the tool will meet the workpiece about 7.8 mm after

its center has passed the left corner of the workpiece. Once

this cut is completed the tool is retrieved from the cut and

positioned so that its center is one diameter to the right of

the workpiece and so that the maximum radial depth of cut is

again 1.5 mm (same level as before). The tool is fed in the -X

until the center of the tool coincides with the center of the

workpiece. The machined surface by both the up and down

milling cuts is shown as line A-A in the figure. At this point

the tool is retrieved from the cut and repositioned to the

original position to the left of the workpiece. The tool is
















End Milling


1.5mm


50/zm


before
machining


after
machining


Figure 21 Procedure for the test of errors in end milling.







63

retrieved in the Z axis direction until the axial depth of cut

is reduced to 1.5 mm. Then the tool is moved 50 microns below

the original cut to generate a reference surface. The tool is

fed in the +X direction until the whole workpiece is machined.

The reference surface is marked as line B-B in Figure 21. The

errors are measured at locations where the cut had radial

immersions of 50 microns, 500 microns and 1.5 mm. This are

shown as lines T1 to T6 in the figure. Because of the

variation of radial immersion through the cut the location of

the center of the tool (position at which the surface was

produced) is found from the point in which the immersion is

measured by


1 (mm) = (a (mm) -d(mm) -a (mm)2) (14)

where a is the given radial immersion and d is the tool

diameter. This is better illustrated in Figure 22. The

positions to be traced are computed from the slope of the

incline and adjusted by the distance 1. These dimensions are

given for the different tool diameters in Table 5. The

locations are given referred to the corners of the workpiece

as shown in Figure 21.

Table 5. Location of measurements for the end milling test.

d(mm) en(mm) Tl,T6(mm) T2,T5(mm) T3,T4(mm)
12.5 7.8 10.9 44.4 120.9
19. 7.8 10.7 43.9 119.9
25. 7.8 10.6 43.4 119.1
37.5 7.8 10.3 42.6 117.7
50. 7.8 10.1 42.0 116.5







64

The results of the test are evaluated by tracing the

profile of the surface at the positions Tl to T6. The

instrument used is mounted in the headstock and the traces are

made using the machine axes to move the tracing device over

the surface. The traces are made from the tip of the cut up to

and including the reference surface.

Once the trace is plotted the ideal surface is positioned

50 microns above the reference surface. The error are measured

in reference to this surface. The errors are evaluated as the

maximum error emx, which is positive if the tool did not

remove enough material or negative if it remove to much

material, and the error range Ae. Figure 23 shows a typical

trace made with a capacitance probe based tracer. The

reference surface, the ideal surface and the errors are marked

in this figure for illustration.



Face Milling

The workpiece is a block of 3.5*d length and one diameter

wide. A step as shown in Figure 24 can be pre-machined but is

much simpler to include this pass as part of the test program.

The test is performed first in full immersion and then 0.5

radial immersion both in up and down milling modes. The axial

depth of cut is established from the chatter limit test from

B=0.333*bcr. This value is then used to program the passes.

First the step is created by machining at an axial depth of B

a length of 2-d. The tool is retrieved axially and the second











End Millina


Computation of the location of the center of the tool I

as a function of radial immersion


Figure 22 Locating the position for tracing a given immersion.







66

pass is run at a depth B lower than the original cut for the

entire workpiece length. In this way the cut will have an

axial depth of B during the first 2.d length of cut and 2*B

during the rest of the cut. In the 0.5 immersion case this

procedure is repeated for the up and down milling modes. Then

the tool is retrieved axially and repositioned to perform the

reference cuts which are made 50 microns below the machined

surface (see Figure 24).

To obtain the errors traces are made across the workpiece

at locations Tl and T2 (see Figure 24). Locations T1 and T2

are 0.55*d and 2.05'd from the side of the workpiece where the

tool entered the cut. Figure 25 show the resulting traces for

the a typical full immersion case while Figure 26 show similar

plots for the 0.5 radial immersion cases. In both cases the

reference surfaces are used to find the ideal surface. Because

the tool is not perfectly perpendicular to the cutting plane

the reference surfaces are at different levels. This is

clearly seen in the bottom right of Figure 24. The ideal

surface is inferred from the reference surfaces. To properly

position the ideal surface in the slotting case the reference

surfaces are first transposed and then connected with a

straight line. The ideal surface is parallel to this line an

is located 50 microns below it. The 0.5 radial immersion case

requires of more manipulation. First the reference surface are

transposed and connected through a straight line. The angle of

this line represents the non-perpendicularity of the tool. The










Tracer


Ref
Sur


Y-Up.


a= 1.6mm


12.5 25.
Distance,


37.5


Ideal Surface


Minimum

Maximum

Reference


Error

Error T
50Am
IL


mm.


Figure 23 Surface trace and the evaluation of the errors.


800
Error
ptm.


750



700



650


I I 1


Ae eMAX


J^- _^-;












Face Milling


Slotting


Material removed by the
first pass to create a
step

* i I //x // 7//


--------
3.5d


I- I- ii
7--
Reference Surface
50 microns below cut


Down milling 1/2 imerssion

Reference surface








Reference surface

Up milling 1/2 imerssion


Figure 24 Procedure for the test of errors in face milling.


After
machining


2B


*V


_


I







69

reference line is bisected and a the two halves are

transposed. The right side of the bisected line is copied at

the same level in the left side and conversely the left side

of the line is copied to the right. The ideal surfaces are

positioned 50 microns below this transposed lines. The maximum

errors emx and the error ranges Ae are found with reference to

this surfaces. This is clearly illustrated in Figures 25 and

26.










Fcce Milling


Tracer


600


500



400



300


200


Slottina Cut, 1 mm Axial Depth of Cut


0 25 50 75


Position, (mm)


Figure 25 Surface trace for the slotting case.










Face Milling


Tracer


600


500



400



300


200


Half ;mmersion Up and Down Milling


0 25 50 75


Position, (mm)


Figure 26 Surface trace for the half immersion cases.















CHAPTER 4

DERIVATION, IMPROVEMENT AND VALIDITY

OF THE SIMPLE FORMULAE FOR CHATTER LIMITS IN MILLING



The Oversimplification of Terms

The simple approach to the derivation of the chatter limits

has been briefly discussed in CHAPTER 2. A more detailed

analysis of the simplified computation of chatter limits in

slotting follows. First it is necessary to point out the

characteristics which were simplified to derive the simple

formula for milling. The first simplification is the use of a

single average tooth which is positioned at the geometric

average of all the teeth in the cut. This works reasonably

well in some cases (0.25 to 0.75 radial immersion with similar

X and Y RTFs) and results in gross errors in others. In the

slotting case the simple formula gives results which may be

off by more than 300%. The other simplification made is the

usage of an average number of teeth in the cut.



The Average Tooth Position

The average tooth position tave as proposed by Tlusty [10]

is found by


av- Yexit-Yentry (15)
av 2 Yentry (15)







73

where Yentry and exit are the entry and exit angles of

engagement. A half immersion up milling cut in the X direction

has an average tooth position of 45 degrees according to the

formula above. This is illustrated in Figure 27. Here the

average tooth position for the down milling case is also

shown. The slotting case which is shown at the bottom of this

figure has an average tooth position at 90 degrees which is

the direction of the feed.



The Orientation Factors

Continuing with the half immersion case and considering

both up and down milling the orientation factors are computed

for the average tooth position. The way in which these factors

are computed is well established [8,10]. Figure 27 also shows

the geometric construction for the orientation factors for the

half immersion up and down milling. The angle 3 between the

normal and the force is assumed to be 70 degrees as proposed

by Tlusty [8]. The angles ax and a are measured from the

normal to the cut (clockwise) to the corresponding axis. For

half immersion up milling using the original average tooth

position the angles are ax=45 and ay=135. For the down milling

case these values are ax=135 and 0y=45. The values of pxu, gyu

and Axd', gyd are computed from the equations

1x=cos (cx) -cos (p-ax) (16)
py=-cos (ay) *cos (p-cay)

These values are found to be Lxu=0.64, tyU=-0.30 and xd=-0.30,

Ay=0.64.











Orientation Factors


xu F/


Down Milling


,Uxu =0.64 /ZXD =-0.30
/u,=-0.30 /-D =0.64


/-=x 0.34
/=0


Figure 27 Average tooth position and the orientation factors for
various immersions.


Milling









The Slotting Case

In the simplified formulae the slotting case is selected to

find the chatter limit. Chatter limits for other immersions

are found by dividing the chatter limit for the slotting case

by the immersion i (=a/d, ratio of the diameter). The slotting

case is a special case since the average tooth position is

s=ot=90. As explained before, this has the effect of

disregarding the Y direction mode as the orientation factor in

this direction is y =0. The directional factor in the X

direction is x,=0.34. These factors are an oversimplification

which results in errors which are significant. Nevertheless

the equation for chatter limits can be rewritten from equation

3 in Chapter 2 for slotting as

-1
cr, slot m (17)
2 ,*x' Ks*Re[G] (in
2

It is also possible to replace Re[G]min by H/2, where H is the

dynamic flexibility in Mm/N of the most prominent mode in the

X direction. Using ,~=0.34 and assuming a steel workpiece

(Ks=2000N/mm2) the last equation can be written as


bcrsot3 (18)



Similarly for aluminum (Ks=750)

8
bcr, sot= (19)
Exm


and for cast iron (Ks=1500)










bcr, slo:c= ( 20)



In these equations H is in im/N, and bcrst in mm.

In reality the regeneration occurs throughout the cut

involving the modes in both directions. To include some of

this effect into the chatter limit calculation for the

slotting case, let us assume a cutting case with four fluted

cutter. Assume an average cutting position as shown in Figure

28 where the two teeth are in contact with average tooth

positions similar to the half immersion up (tooth 1) and down

(tooth 2) milling cases described earlier. Without regarding

of the effect of one tooth to the other and just adding the

orientation factors (Ax=xu+Axd' ly=lyu +yd) the cumulative

orientation factors are found such that x=gty=0.34. This

provides a situation in which the dynamics of both directions

are incorporated. In this derivation the effect of the two

teeth were included. So the simplified formula has to be

rewritten so as to be divided by half the number of teeth in

the cut. If the two transfer functions are similar the chatter

limit will be the same using either formula. Otherwise the

oriented transfer function could be computed using x,= y=0.34

and the limit computed from

-1
bcr, slot m (21)
2 "K.Re [Goriented] min' -

In this formulation the effect of one tooth on the other is

completely disregarded. This new approach will require the

















Up milling tooth







Down milling tooth


-x =-Ux + -XD = 0.34

,U= U +/YD =0.34


Figure 28 Average teeth (two teeth) position for slotting.







78

computation of the oriented transfer function which cannot be

done manually. A program has been created to generate the

oriented transfer function for slotting using the directional

factors computed from the two teeth approach and to compute

the chatter limit.



Corrective Factors

Correction factors can be applied to the simplified

approach described above to account for the differences in the

X and Y transfer functions. A simulation program was used to

generate the limits of stability for a SDOF slotting case for

different levels of non-symmetry. The dynamic parameters were

set for the X mode. The Y mode damping and mass were set to

the same values as the X mode and its stiffness was varied.

This has the effect of changing the frequency of the mode as

well as its dynamic flexibility. These parameters are listed

in Table 6.


Table 6. Parameters for the example case used to derive
corrective factors.

X mode Y mode
Mass, mi 3.9 Kg 3.9 Kg
Damping Ratio, Ci 0.05 0.05
Stiffness, ki 42.0.106N/m 0.1 10 k]


The results from the simple formula are affected by the X

direction only. The chatter limit was found at bc=2.26 mm

using the simple formula. A cast iron workpiece was used. The







79

results from the simulations and the new simplified approach

are presented in Table 7.


Table 7.


Results from simulation of the slotting case with
varying Y mode parameters.


chat b ,or (mm)
k/k, f/f br.,. (mm) (Hz) ( Crn)
10.0 3.20 3.85 "530 4.64 (3.08)
4.00 2.00 3.05 "530 4.89 (3.27)
1.23 1.10 1.15 ~530 3.33 (1.12)
1.02 1.01 0.70 "530 2.30 (0.75)
1.00 1.00 0.70 "530 2.26 (0.75)
0.98 0.99 0.70 "530 2.28 (0.75)
0.83 0.91 0.80 "530 2.65 (.89)
0.44 0.66 1.35 "355 2.42 (1.21)
0.25 0.50 1.00 -265 1.26 (0.84)
0.11 0.33 0.40 "177 0.56 (0.37)



The results are disturbing since the simple formula and the

simulation results are off by more than 250% in some cases. In

the case where the modes are symmetrical (stiffness ratio

ky/kx=1.0) the chatter limit found by simulation is 1/3 of the

chatter limit computed by the simple formula. The relationship

between the new approach and the simulation can be described

in simple terms. It was noticed that when the ratio k/kx is

over 4 the chatter limit found through simulation was closer

to 2/3 the one found through the simple approach. For cases

where the stiffness of the Y mode was between 1.5 and 4 times

more stiffer than the X mode it was noticed that the stability

limit found by the simple approach was twice the one found by







80

simulation. For cases where the stiffness ratio was between

0.75 and 0.25 it was observed that the chatter limit found by

the simple approach was twice the one found by the simulation.

For cases where the stiffness ratio was below 0.25 it is

observed that the chatter limit found by the simple approach

was 1.5 times the one found by the simulation. Corrective

factors which account for the observation made above were

calculated and are listed in Table 8.


Table 8. Corrective factors for the simple approach.

Stiffness Corrective
Ratio Factor, K,
4 to 10 0.667
1.5 to 4 0.5
0.75 to 1.5 0.333
0.25 to 0.75 0.5
0.1 to 0.25 0.667



The simple approach formula adjusted by the corrective

factors is re-written as


bcr,slot- m (22)
2 -Ks Re [Goriented] min '



The corrective factors have been included in the program used

to generate the oriented transfer function and from it the

chatter limits are derived. This program is significantly

faster to estimate the chatter limit and frequency than the

simulation but is not as accurate. For situations where time







81

is a constraint or the simulation is not available this

program can be used to estimate the chatter limits. The

results from the previous example are improved tremendously by

applying the corrective factors. The results are listed in

parenthesis in Table 7. Figure 29 shows a graphical comparison

the results from simulation to the results from the other

methods. The two teeth average position approach including the

corrective factors follows the simulation very closely.

The selection of the stiffness ratio over the frequency

ratio as the discriminator for the application of the

corrective factors was made because it is easier to separate

the regions for which the different values of Kc would be

applied. A similar corrective technique could be used with the

frequency ratio as the discriminator. The approach was derived

assuming that the masses of the two modes are comparable.



Other Immersions

A common way to estimate the limit of stability in

immersions other than full immersion is by dividing the

chatter limits computed for the slotting criteria by the

radial immersion


b bcrslot (23)
crl- i



where i is the radial immersion as a ratio of the diameter

(i=0.5 for half immersion for example). It is useful to get an

idea of the chatter limit but is very inaccurate because the


















Comparison of methods for SDOF in each direction with similar masses.


\ Simulation
.................... ..ach ................................................. ............




-- Corrected


Simple Formula
-.................. r................. ..................t ................


18.8


4.8 6.8
Stiffness Ratio (Hy/Kx)


Figure 29


Comparison between the simple
and the new approach with and
factors.


formula, simulation
without corrective


5.8
Axial
Depth of
Cut (nm)
4.8


3.0


2.8


1.8


8.8







83

relationship between the limiting axial and radial depths of

cut is not that simple. Also, it is wrong that the limits

found for up or down modes of milling are the same using this

approach. A better approach is to use the directional factors

for the given immersion to compute the oriented transfer

function and use the formula

-1
bCr' 2- m*i (24)
2 Ks Re [ Gi, oriented] min 2




This formula is good for middle range immersions. The term

(m-i)/2 is an estimate of the average number of teeth in the

cut. It was discussed earlier how the average tooth position

formulation starts to produce severe errors as the immersion

approaches slotting. It also fails for very low immersions

because the cutting force formulation becomes very complex.

The symmetry of the modes has an effect similar to the one

found in the slotting case. A program has been written to

compute the oriented transfer function, compute the stiffness

ratio of the two most flexible modes, discriminate between up

and down milling cuts, compute the chatter limit using the

formula for bcr,i presented above, and apply corrective factors

in the same manner as the slotting case. A database of

corrective factors has to be developed for this approach to

work in general situations. This work concentrates in 0.25 and

0.5 immersions in which the X and Y modes participate roughly

in similar ratios all these cases can be paraphrased as either

up or down milling. General corrective factors have been







84

developed for the two cases (up and down milling) and are

listed in Table 9.


Table 9. Corrective factors for up and down milling cases.

Stiffness K Ko
Ratio Up milling Down milling
1.2 to 10 0.45 1.33
0.9 to 1.2 1.0 1.0
0.25 to 0.9 0.667 0.25



Using these corrective factors the equation for the chatter

limit is written as


cri m (25)
2 KRe [Gi,,oriented] min2



where j (=u or d) stands for either up or down milling. It is

important to emphasize that this equation is an approximation

which is only valid for cuts between 0.25 and 0.5 radial

immersion. Using the program to get these results is much

faster than running the simulation to establish the chatter

limit or even more faster than generating the corresponding

composite peak-to-peak diagram. Note that the program outputs

the chatter limit together with the expected frequency of

chatter.



Corrective Factors for the Simple Formula for Slotting

The techniques described above require the use of computers

and need programs to be evaluated. It is of valued use to have







85

a method of computing chatter limits directly from a formula.

Recent discussions have led to the acceptance of a formula

which accounts for the most flexible mode in the direction of

feed and uses corrective factors based on the stiffness ratio

in a similar way as the techniques described above. The

corrective factors, which were developed in the same manner as

before, are listed in Table 10.


Table 10. Corrective factors for the simple formula.

k /k, K,
4 10 1.5
2-4 1.0
1.5 2 0.667
0.74 1.5 0.4
0.25 0.75 0.5



Note that this formulation is not as accurate as the

approach presented before where the oriented transfer function

is used. In that case the effect of the two modes are

accounted for. Still the simple formulae can be re-written as

3*K
br sot= (26)
bcr,sot- Hx m


for steel, as


b 8,"K (27)
bcr,stot- C (27)Hxm


for aluminum and as







86

4- KC
bcr,stot (28)
m


for cast iron.

The corrective factor will reduce the discrepancies between

the computed and actual limit of stability. A better

discriminating technique for the application of corrective

factors which would account for the stiffness and mass (or

frequency) ratios could be developed but for estimation

purposes the formulae presented should suffice.















CHAPTER 5

DISCUSSION ON THE IMPACT EXCITATION TECHNIQUE



The Impact Excitation Technique

The need for obtaining the dynamic parameters of a

spindle/tool system (the transfer functions) is obvious in the

characterization of machine tools. Ideally the impact

excitation technique can be used to generate transfer

functions from which the dynamic parameters of the system may

be extracted. Commonly, the real and imaginary parts of the

transfer functions are used to extract these parameters. An

unexpected phenomenon has been observed where the phase of the

transfer functions changes throughout the frequency range.

This phase change was never studied by the author mainly

because it was possible to extract the dynamic parameters from

the ill transfer functions by assuming that the peak-to-peak

value of a mode (dynamic flexibility) did not change with the

change in phase nor did the frequency. After several

experiments where the measurements made with several hammers

did not match, it was decided that the phenomena of phase

shifting required explanation. It was found that the source of

the phase change was a capacitance probe transducer which had

been used for the measurements. A side effect of this study

resulted in the detail analysis of the impact hammer itself.









88

A set of simulation programs were developed to study the

different theories and to establish the actual reason for the

phase shifting. It was found that the hammer can be a source

of errors if it is not properly designed. A rather simplified

explanations follows.



The Effect of the Tip Mass

The force measured using an impact hammer is the force

acting on the crystal. Normally, it is accepted that the force

at the tip of the hammer is closely measured by the

piezoelectric crystal:

Ftip=Fcrystal (29)

This assumption might not be valid for the smaller hammers

(which are required to measure end mills with small reflected

masses) mainly because the mass of the mass in front of the

crystal becomes a significant part of the total mass of the

hammer. In these cases the hammer acts as a two degree of

freedom system with the crystal acting as a spring between the

mass in front of it and the mass behind it. This phenomena

introduces an inertial force such that


Ftip=Fcrystal +tip'tip (30)

the last term being the inertial term. The notation mip and

xtip refers to the mass in front of the hammer and its motion.

Note that the measured force is different from the actual

force by the inertial term. It has been shown through












Force Spectrun

. .... ..... ..........-..............................****




-- - - -- ! --


100888


2888


3888


4888


> Ilagnitude

..................................


1888


2888
Freq, Hz


3888


4888


Original TF >


Real


3888


> Inaqinarv


3888


Figure 30 Measured impact force and the corresponding
transfer function. a) Impact force spectrum. b)
Response spectrum. c) Real RTF. d) Imaginary RTF.


1.88

8.75

8.58

8.25

8.88


8.8E-87

6.8E-87

4.8E-87

2.8E-87

8.8E+88


4.8E-87

2.8E-87

8.8E+88

-2.8E-87

-4.8E-87


(d)


1888


2888


8.8E+88

-2.8E-87

-4.8E-87

-6.8E-87

-8.8E-87


.............. .... ..-...- .................................................................

Y:'----------------r.
..~........ .. .. .... .......... .. .. .............. ....... ......... .. .. .... .. .

.... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... .... ... ... ... ... ... ... .... .. ... ... ... ... ... ... ...


4888


1888


2888
Freq, Hz


4888


........................... .............

............................ ...... -----------------r----- ----------- .....







90

simulation that the inertial term is responsible for a phase

shift similar to the one presented above. Figure 30 shows the

ideal case for obtaining the transfer function of a system in

the real and imaginary form. The top of the figure shows a

typical impact force. Below the corresponding impact force

spectrum is shown. The bottom of this figure shows the ideal

real and imaginary transfer functions.

In reality the impact force read by the crystal is

different than this ideal condition as explained above. Figure

31 shows the impact force as seen by the crystal and at the

tip of the hammer. This forces, which were obtained by

simulation, are obviously not the same. The difference between

the two forces is also plotted in this figure. This force

difference is equal to the inertial force at the tip of the

hammer. The inertial force is shown in the bottom of the

figure. Note that the two last terms are identical. This

asserts that the crystal reads the impact force minus the

inertial force of the tip. This force difference is big in

cases where the tip mass is significant.



Effect on The Dynamic Flexibility

The simulations also showed that the inertial term also

affects the measured dynamic flexibility. The difference

between measured and actual flexibility was noticed to be as

high as 80%. This distorting effect is frequency dependent and

is present even in cases where no noticeable phase shifting is

seen. If the inertial term could be corrected, the phase














e .................................................... .--------------........ .........................


.............. ... ..................................... ....................................


680

458

388

158

8



688

458

388

158

8





1088

8

-188

-288

-3088



388

288

188

8

-1880


8888


8.88805


8.8885


Time Donain


0. 8088
(d)


8.8888


0.8885


0.0005


8.0818


8.8818


8.8815


0.8818 8.8815
Time, sees


0.8820 8.0025

Tip Impact Force


8.8828


8.8825


Force Difference


8.0015 8.8828 8.8825

Inertial Force: Xdd2*M2


0.8018 8.8815
Time, sees


8.0028


8.8825


Figure 31 Impact force. a)Force measured by the crystal.
b)Force at the tip. c)Force difference. d)Inertial
force at the tip.


8.1












8. I


..... ...................................................................... .........................




.. .. .. .. .... ..... ... .... ............. ....... ... ..... .. ........... ............. ..


8888
(b)


..... .... .... ... ...... .

... .. ... ... .. ... ... .. ... .. ... .. ... .. ... .. ..... .. ... .. .. ... .. ... .. ... .. ... .. ... .. ... .. ..

........ ........... ....................... ...... ................................. .


Time Domain


Crystal Impact Force


.. I I -







92

shifting and flexibility distortions could be eliminated.

Ideally, if the mass of the tip was known and its acceleration

measured, the force could be corrected. In reality the

acceleration of the tip is practically inaccessible.



How to Avoid Hammer Induced Errors

The most practical approach to avoid problems in the impact

excitation technique is to select or design a hammer which has

most of its mass behind the piezoelectric crystal. In

actuality, the availability of hammers that could be used to

measure tools with small reflecting mass is good but adding

mass to the back of the hammer is a good practice.



A Three Degree of Freedom Illustrative Example

An example is included here to illustrate the effect that

the hammer design infringes on the measured impact force. A 3

DOF system is selected to show the phase change effect. The 3

DOF system is defined in modal coordinates q, as shown in

Table 11.



Table 11. Structure modal parameters.

Mode 1 Mode 2 Mode 3
mq (Kg) 5.5 0.33 0.23
Cq 0.056 0.021 0.015
kq (N/m) 7.44e7 6.25e7 9.01e7
H (pm/N) 0.120 0.381 0.370







93

Two hammers are used to illustrate the effect of the

relative mass of the tip. The hammers are modelled as two DOF

systems in local coordinated xi as listed in Table 12.


Table 12. Hammer characteristics.

Hammer 1 Hammer 2
m2 (Kg) 0.05 0.05
C2 0.04 0.04
k2 (N/m) 4.0e6 1.6e7
m3 (Kg) 0.05 0.15
C3 0.04 0.04
k3 (N/m) 8.0e6 3.2e7


Where k2 refers to the hammer tip/tool contact stiffness and

k3 refers to the crystal stiffness. The mass m2 is then the

mass in front of the crystal and m3 is the mass behind it.

These masses suggest a hammer with a large mass in front of

the crystal. Figure 32 shows the schematic of the model used

in the simulation of the impact excitation technique. The

block diagram and listing of this program is included in the

Appendix. A simulation of the impact excitation technique

including the system response was run with these parameters

and the results are presented in Figure 33 for the first

hammer and 34 for the second one. The top of these figures

show the impact force spectrum as seen by the crystal. Below

it the transfer function in the magnitude, real, and imaginary

forms are plotted. The results are compared to the ideal

transfer function presented in Figure 30. The phase change

effect is clearly seen in the transfer function generated by















Hammer ,
amCell k3 c3

Tip m2 X2
k2 c2

[Mq] x l
Structure _


Figure 32 Schematic of the impact model.




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