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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