Designs of Damped Toolholders for Increased
Cutting Performance
by
Andreas Athanassopoulos
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 2000
© Andreas Athanassopoulos, MM. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
....................
Department ot Mechanical Engineering
May 18, 2000
A uthor ...........
C ertified by ....
.
.............................
Samir Nayfeh
Assistant Professor
Thesis Supervisor
Accepted by ..........
Ain A. Sonin
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
SEP 2 0 2000
LIBRARIES
Designs of Damped Toolholders for Increased Cutting
Performance
by
Andreas Athanassopoulos
Submitted to the Department of Mechanical Engineering
on May 18, 2000, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
The goal of this research project is the design and implementation of methods that
result in higher metal removal rate and improved surface quality cut by long overghang
tools. Damping treatments such as a squeeze-film damped tool and a viscoelastic
collar ring are examined. Analytical methods are developped in order to model the
dynamic behanior of the tools. Experiments are performed to determine the increase
in damping of the new designs. The results of metal cutting tests show increases up to
300% in cutting speed. A complete redesign of the widely used R8 collet is performed,
attempting to provide higher damping when long overhang tools are used.
Thesis Supervisor: Samir Nayfeh
Title: Assistant Professor
2
Acknowledgments
I am extremely grateful to my thesis advisor, friend and motivator during the last two
years, Professor Samir Nayfeh. His contribution to my transformation from college
student to graduate research assistant has helped me in various ways, that extend
beyond the purposes of my thesis. The most important thing is gaining the confidence
to believe in myself, the desire to create and the motivation to improve as individual.
I honestly wish that every graduate student's advisor was as talented and gifted as
mine.
My guiding force during all my educational career has been to make my parents
and my brother proud of me. I would like to thank them from the bottom of my
heart.
I would also like to thank:
My colleagues and friends in the MESO Lab, especially Kripa Varanasi and Paolo
Morfino, who were always there for me when I needed help. I wish good luck to both
of them.
My roommates Dimitris Georgakopoulos, David Rossow and friend Anna Vorrias
for providing me with the necessary means of transportation when I needed to work
late at school.
My friends at the MIT Machine Shop, Mark Belanger, Jerry Wentworth and Dave
Dow for bearing with me and my crazy requests. "Hey Mark..how long do you think
it's going to take for me to do this?"
Professor Chun for trusting me with a Teaching Assistant position at a time when
funding for my research was uncertain.
3
Contents
1
2
1.1
High Speed M achining . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.2
W hy Higher Speed?
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3
The M ain Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
14
Types of Machine Tool Vibration
2.1
2.2
Chatter occurence and suppression
...................
Self-Excited Vibration (Chatter) .................
16
2.1.2
Regenerative Chatter . . . . . . . . . . . . . . . . . . . . . . .
17
2.1.3
The Negative Effects of Chatter . . . . . . . . . . . . . . . . .
20
Available Methods for Chatter Reduction . . . . . . . . . . . . . . . .
21
Products for chatter reduction . . . . . . . . . . . . . . . . . .
22
29
Theory and Model Implementation
3.1
3.2
3.3
15
2.1.1
2.2.1
3
11
Introduction
Comparison of viscous and hysteretic damping . . . . . . . . . . . . .
30
3.1.1
Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.1.2
Hysteretic Damping
. . . . . . . . . . . . . . . . . . . . . . .
32
Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.2.1
Tool and Sleeve Mass element [M]et . . . . . . . . . . . . . . .
36
3.2.2
Tool and Sleeve Stiffness element [K]ei . . . . . . . . . . . . .
37
3.2.3
Tool and Sleeve Damping element [C]ei . . . . . . . . . . . . .
38
3.2.4
Boundary Condition Adjustment
. . . . . . . . . . . . . . . .
39
Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . .
39
4
4
3.4
Dynamic Response Measurement
. . . . . . . . . . . . . . . . . . . .
40
3.5
Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
42
Fluid Damped Tooling
4.1
Analytic approach and modelling . . . . . . . . . . . . . . . . . . . .
43
4.2
Preliminary Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.3
Testing on the Cincinnati Vertical Milling Machine
. . . . . . . . . .
47
4.3.1
Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.3.2
First Testing on the Horizontal Milling Machine . . . . . . . .
50
4.3.3
Modal Analysis on the foam base . . . . . . . . . . . . . . . .
51
4.3.4
Main experiment . . . . . . . . . . . . . . . . . . . . . . . . .
52
55
5 Redesign of the R8 Collet
5.1
Project Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.2
Damping using Viscoelastic Material
. . . . . . . . . . . . . . . . . .
58
5.3
Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.4
The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.4.1
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
5.4.2
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
5.4.3
Final Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
5.5
Model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
5.6
Choosing the type of viscoelastic material
. . . . . . . . . . . . . . .
65
5.7
Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
Belleville Springs . . . . . . . . . . . . . . . . . . . . . . . . .
72
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5.8.1
Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5.8.2
First Prototype . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Second Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.9.1
Modal testing . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5.9.2
Cutting tests
. . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.7.1
5.8
5.9
5
6
83
Viscoelastically Damped Tools
83
6.1
The tool ......................
6.2
Experimental Apparatus Description
. .
84
Set up . . . . . . . . . . . . . . .
85
Concept 1-Steel clamps . . . . . . . . . .
85
6.2.1
6.3
6.4
6.3.1
Description
. . . . . . . . . . . .
85
6.3.2
D ata . . . . . . . . . . . . . . . .
87
Concept 2- Collar Ring . . . . . . . . . .
91
. . . . . . . . . . . .
91
6.4.1
Description
6.4.2
Adjustments to the Experimental Apparatus
91
6.4.3
Set up . . . . . . . . . . . . . . .
92
6.4.4
D ata . . . . . . . . . . . . . . . .
93
96
7 Conclusions
7.1
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
7.2
Recommendations for future work . . . . . . . . . . . . . . . . . . . .
96
A APPENDIX A: Finite Element Code
6
98
List of Figures
2-1
Closed loop system of machine tool structure and cutting process
. .
17
2-2
Regenerative Chatter due to the cutting of an undulated surface . . .
19
3-1
Single degree of freedom system with viscous damping
. . . . . . . .
30
3-2
Single degree of freedom system with hysteretic damping . . . . . . .
33
3-3
Positive sense of beam displacements and forces . . . . . . . . . . . .
35
3-4
Assembling the system mass matrix from the individual element mass
m atrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-5
Assembling the system stiffness matrix from the individual element
stiffness m atrices
3-6
37
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Assembling the system damping matrix from the individual element
m ass m atrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
4-1
The Q-Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4-2
Q-Tool Model . . . . . . . . . . . . . . . . . . . . . .. . .
4-3
Model for damping with viscous fluid . . . . . . . . . . . . . . . . . .
45
4-4
Loss Factor as a function of length ratio, for different viscosities . . .
46
4-5
Loss Factor as a function of viscosity and length ratio . . . . . . . . .
47
4-6
Frequency Response with different viscosities . . . . . . . . . . . . . .
54
4-7
Damping as a function of viscosity
. . . . . . . . . . . . . . . . . . .
54
5-1
New Collet . . . . . . . . . . . . . . . . . . . . . .
5-2
Collet Assembly . . . . . . . . . . . . . . . . . . . .
. .
58
5-3
Model for damping with viscoelastic material . . . . . . . . . . . . . .
60
7
.. .
44
. . . . .
..
. ..
57
. . ..
.. .
5-4
Variation of w2rq for Cantilever case, as a function of Length ratio and
Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
5-5
Cantilever case: Frequency as a function of Length ratio and Stiffness
62
5-6
Cantilever case: w
as a function of Length ratio and Stiffness . . . .
63
5-7
Model A: Viscoelastic material behind the second fixed point . . . . .
64
5-8
Variation of w 2
for Model A(Visco all the way to the back), as a
function of Length ratio and Stiffness . . . . . . . . . . . . . . . . . .
65
Model A: w 2 as a function of the length ratio and stiffness . . . . . .
66
5-10 Model A: Viscoelastic behind the second fixed point . . . . . . . . . .
67
5-11 Model B: Viscoelastic material, only between fixed points . . . . . . .
68
5-9
5-12 Variation of W227 for Model B (visco between fixed points), as a function
of Length ratio and Stiffness . . . . . . . . . . . . . . . . . . . . . . .
69
5-13 Model B: w 2 as a function of the length ratio and stiffness . . . . . .
70
5-14 Shape functions for viscoelastic between nodes 3 and 4, with zero and
infinite stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
5-15 Final Model: Fixed point in the back-Viscoelastic material only at the
front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
5-16 Variation of w 2 j for the final design, as a function of Legth ratio and
Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-17 Final Model:w 2
73
. . . .
74
5-18 Comparison of maximum damping for each model . . . . . . . . . . .
75
5-19 Comparison of maximum loss factors . . . . . . . . . . . . . . . . . .
76
5-20 Model for deflection-force calculation . . . . . . . . . . . . . . . . . .
77
5-21 Force analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5-22 Belleville Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
as a function of the length ratio and stiffness
5-23 Comparison of Frequency response for new collet design and the regural
R oyal R 8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5-24 Comparison of frequency response for second prototype and the regural
Royal R 8
5-25 Regular R8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
8
5-26 New R8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 82
...............................
84
6-1
The Gun-Drill .......
6-2
Steel Clamps Assembly . . . . . . . . . . . . . . . . . . . . . . . . . .
86
6-3
Frequency response using the steel clamps, with the gundrill pinned .
89
6-4
Time response variation from no adjustment(top) to steel clamps(middle)
and steel clamps tightened (bottom) , with the gundrill pinned . . . .
90
6-5
Collar Ring Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
6-6
Frequency response of gundrill pinned using the Collar Ring . . . . .
94
6-7
Frequency response of gundrill free, using the Collar Ring . . . . . . .
95
9
List of Tables
4.1
V iscosities used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
5.1
M odel Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.2
Belleville Spring Characteristics . . . . . . . . . . . . . . . . . . . . .
72
5.3
Belleville Springs measured expansion
. . . . . . . . . . . . . . . . .
73
5.4
First Prototype Data
. . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.5
Second Prototype Data
. . . . . . . . . . . . . . . . . . . . . . . . .
79
6.1
Comparison of the modes measured to the modes of a cantilever beam
85
6.2
Damping and Frequency with and without viscoelastic
. . . . . . . .
88
6.3
Damping and Frequency for gundrill-pinned
. . . . . . . . . . . . . .
94
6.4
Damping and Frequency for Gundrill-free . . . . . . . . . . . . . . . .
95
10
Chapter 1
Introduction
1.1
High Speed Machining
Metal removal rates are faster today than ever before. What was considered high
speed machining just a few years ago is regarded as conventional today. Many factors
are driving shops to faster metal cutting rates and higher rates of productivity. The
need to put more work across machine tools has shops looking constantly to improve
metal cutting processes. As a result, cost effective, agile and high quality production
are three major issues in high speed manufacturing. The demand for higher part
surface quality and metal removal rates have lead to the slow but continuous increase
of spindle speeds in high speed machining.
As a result of the continuous effort in this area, it is expected that high speed
machining will show considerable growth in the coming years. Machine tool industries will have to develop products that will be able to satisfy the higher customer
expectations for faster cutting speeds and better cutting performance. These include
better and more capable machine tools and CNC processors that allow the machine
to accurately cut at increasingly higher speeds and feeds.
11
1.2
Why Higher Speed?
Implementing higher speed machining in a shop has many benefits, some obvious but
others less so. Obviously, making parts faster helps satisfy customers' demands for
quicker deliveries despite shorter lead times. Applications such as rapid prototyping
and mold making, in which smooth surface finishes, complicated geometries and fast
cycle times are required, take advantage of the benefits of high speed machining.
There are also benefits derived from increased tool life. It may seem paradoxical, but
machining at high speed with the right tooling matched to the application can reduce
tool wear because of the diminished cutting forces at high speed.
The decrease of cutting forces at very high speeds can help a shop manufacture
more accurate parts with better surface finishes. A less obvious benefit of high speed
machining for shops moving in that direction is derived from the exercise of implementing it. Learning to do the things necessary for successful high speed machining
can simultaneously elevate other facets of an enterprise to equivalent levels of productivity.
The major improvement is due to the reduction of cutting forces in the increased
number of tooth passes to remove a given amount of material. Much of the chip
formation mechanics at high speed involve heat. At high speeds, most of the heat
generated in deformation cannot diffuse away from the shear zone. Higher temperature at the primary shear zone helps speed up the plastic deformation process that
results in a chip being formed. Because of the increased rate of plastic flow, high
speed cutting experiences a decrease in the cutting force needed to remove a chip.
Apart from the need for high speed machining, the major driving force for this
project is the need for deep cavity machining. Conventional toolholders do not provide
the required stiffness and damping characteristics for deep cavity machining. Small
tool radii and long cutter bodies are needed to access these surfaces. As a result, due
to the increased tool overhang, cutting force variation is initiated when these kind of
tools are used.
12
1.3
The Main Problem
The main problem in deep cavity machining is the initiation of chatter when modes
of vibration of the cutting tool are excited. Chatter is manifested by strong vibration
between the cutting tool and the workpiece. It can result in poor surface finish and
accelerated tool wear. Extensive research has been performed in the recent years to
the problems of chatter in machining and measures of stability improvements have
been explored. The main reason for chatter occurence is low damping in the machine
tool system which leaves a major part of the transmitted energy unabsorbed.
Most of the solutions available for chatter supression do not account for economic
issues. As the need for high speed machining increases, low cost and effective solutions
are sought.
This research project aims at developping effective methods to improve current
toolholding mechanisms and to increase damping in long overhang tools. A complete
redesign of the widely used R8 collet is performed.
The goal is to achieve stiffer
positioning of the collet in the spindle and of the tool in the collet, through the use of
a second hard point in the back, and higher damping through the use of viscoelastic
material. Finite element models are developped to examine the different available
configurations and to measure the damping effect in each one. Cutting tests are
performed to qualitetively determine the improvement in surface finish.
Two types of tools are examined: a viscous-damped tool (of the type developped
by Rohatgi, 1999) where energy is dissipated in a thin layer of viscous fluid that
surrounds the tool shank, a viscoelastically damped tool. Finite-element modelling
provides insight on the dynamic behavior of the tools. Modal analysis is performed
to determine their dynamic behavior and measure the increase in damping. In both
cases the increase in damping is substantial.
Before presenting the research part of the project, it is useful to provide the basic
background on the types of machine tool vibration, the currently available methods
for chatter reduction and the theory behind damping.
13
Chapter 2
Types of Machine Tool Vibration
Machine Tools are subject to three types of vibration: free, forced and self excited
vibrations. Free vibrations occur when the stable system is displaced from its equilibrium position and allowed to vibrate freely. In this case the system will vibrate
and evenually return to its original position in a manner dictated by it structural
characteristics.
When a dynamic exciting force is applied to the system, forced vibration occurs.
These forces are commonly induced by one of the following three sources:
1. Alternating cutting forces such as those induced by inhomogeneities in the workpiece material (hard spots, cast surfaces, etc), cutting forces periodically varying
due to changes in the chip cross section.
2. Internal sources of vibrations, such as disturbances in the workpiece and cutting
tool drives (caused by worn components, defects in gears, instability of the
spindle,etc), out of balance forces (masses in the spindle or transmission).
3. External disturbances transmitted by the machine foundation
Self-excited vibration or chatter is initiated by variations in the cutting forces
(caused by changes in the cutting velocity or chip cross section), built-up edge, and
regenerative effects.
14
Chatter depends on the design and configuration of both the machine and tooling
structures, on workpiece and cutting tool materials, and on machining regimes. The
stiffness of the tool, spindle, workpiece and fixture are very important factors. The
cutting stiffness of the workpiece material is also an important factor; for example
steel has a greater tendency than aluminum to chatter. The chatter resistance of a
machine tool is often expressed in terms of the maximum allowable width of cut bum.
2.1
Chatter occurence and suppression
Forced vibration may be easily identified during the development stage or final inspection of a machine tool and can be easily reduced or eliminated. On the other
hand, chatter occurence may not be easily detected during the runoff stage, unless
the machine tool is thoroughly tested. In addition, since it is typically a nonlinear
phenomenon, chatter may occur only under specific cutting conditions and may appear sporadically. As a result, the elimination of chatter in a particular machining
process can be very tedious and can be accomplished only by reducing the production rate. The prediction of chatter and the determination of stability involves the
understanding of:
a. The dynamic structural response characteristics of the system
b. The cutting stiffness, cutting force, and chip thickness.
Finite element analysis (FEA) and dynamic testing are used to study the vibration
and chatter problems during.the design stage and runoff of equipment.
As mentioned before, chatter initiation is due to the fact that the damping of
the machine tool system is not sufficient to absorb the portion of the cutting energy
transmitted to the system. The practical significance of the chatter depends on the
type of operation, whether it is a finishing or roughing operation, the surface finish
requirements, and tool wear characteristics.
Chatter becomes more significant as
cutting speeds increase since the forces excited approach the natural frequencies of
the system. It is often difficult to overcome chatter, but progress can be made through
the proper selection of cutting conditions, better design of the machine tool structure
15
and spindle, and improved vibration isolation.
Chatter suppression typically referrs to increasing the machining process stability.
In practice the stability problem (chatter suppression) and the surface finish problem are coupled because the occurence of vibration and especially chatter reduces
machined surface quality.
Two approaches may be taken to solving chatter problems. The first is to choose
or change cutting conditions such as the cutting speed, feed, tool geomatry, ,etc, to
optimize the metal removal rate, while operating in a stable regime. The second is to
analyze the dynamic characteristics of the machining system to determine the stable
operating range, and suggest improvements to the system design which can extend
this range(Tlusty, 1985).
2.1.1
Self-Excited Vibration (Chatter)
Self-excited vibration occurs because the dynamic cutting process forms a closed-loop
system. Disturbances in the system are fed back into the system and may result in
instability. Self-excited vibrations do not result directlly from external forces, but
draw energy from the cutting process itself.
The characteristic features of self-excited vibrations are:
1. The amplitude increases with time, until a stable limiting value is attained
2. The frequency of vibration equals a natural frequency of the system
3. The energy supporting the vibration is obtained from an internal source.
When the dynamic cutting force is out of phase with the instantaneous relative
movement between the tool and the workpiece, this leads to the development of selfexcited vibration. This type of instability is called regenerative chatter because the
vibration reproduces itself in subsequent revolutions through the generation of the
waviness.
A second type of self-excited vibration is the nonregenerative chatter
which occurs without undulation.
16
2.1.2
Regenerative Chatter
In this section, we provide an overview of the theory of regenerative chatter following
teh presentation by Tlusty(, 1985). The dynamic machining process can be represented as a closed loop system by the block diagram in the figure below.
CUTTING PROCESS
MACHINE TOOL STRUCTURE
Figure 2-1: Closed loop system of machine tool structure and cutting process
Dynamic fluctuations of the cutting force and tool position relative to the workpiece occur in all machining processes because workpiece and tool are not infinitely
stiff. This relative motion leaves an undulation of amplitude yi on the machined
surface. Tlusty (1985) performed a simple analysis that assumes that the dynamic
cutting force is proportional to the undeformed chip thickness. The vibration of the
tool in the direction normal to the cut surface during the ith cut is:
yj = Y sin(wt) = Xi cos(a) sin(wt)
(2.1)
The direction of the principal vibration X in the figure below, forms an angle a
with the normal to the machines surface Y. The chip thickness variation due to the
surface wave Yj_1 produced by subsequent cuts depends on the phase lag Ewith the
surface wave Y left by the previous revolution. The number of waves between cuts
is:
17
n +-
E
27r
where nr
-f
(2.2)
N
is the largest possible integer (number of whole waves) such that E/27r < 1
and E is the phase of inner modulation yj to the outer modulation yi-1. The maximum
chip thickness variation occurs at c = 180'.
Based on the regenerative chatter theory, the state of the dynamic cutting process
is described by:
>1
=
Yi-1
unstable
(2.3)
=1 at stability limit
stable
<1
The relationship between the amplitudes for the ith and (i - 1)th cuts is:
I
-G(w)
Xi
Yi
Yi-1
Xi-
(2.4)
G(w) + 1/kdb
1
Regenerative instability wil occur when the vibratory motion increases with time,
in which case the magnitude of the above equation is greater than 1 for some frequencies. The stability limit is obtained when the magnitude of yi/yi_1 is set to 1
ImG(w) = 0
and
ReG(w)=
-1
2kd b
(2.5)
where kd is the specific dynamic stiffness, which is assumed to be a material constant.
Therefore, the limit width of cut is given by:
bum
=
-1
2kdRe[G(w)]
(2.6)
However, if the phase shift E (eq 2.2) is left free to accept any value, then equation
2.6 does not determine a specific value of bum and chatter may occur at various
frequencies, resulting in many values of bum(King, 1985). One of them is the minimun
value that represents the limit of stability:
-1
(blim)min = 22 kdRe[G
(w) ]m,,
(2.7)
In order to illustrate all this. let us consider the system depicted in figure 2-2. The
systems vibrates in the X direction, has damping c and stiffness k and the tool has
18
mass m. The direct transfer function of this system, denoted as Gd is the ratio of the
vibration in X over the component of force F acting in the direction X, F , between
the tool and the workpiece.
x
Too I
F
Yi
WORKP IECE
Figure 2-2: Regenerative Chatter due to the cutting of an undulated surface
We have that:
X(w)
Ga =nF-(w)
1/kw2
w2
2+
2jww(8
(2.8)
where wn is the natural frequency of the system:
k
k
m
-
(2.9)
and the damping ratio ( is:
c
(2.10)
2 Vdm
The minimum Gd,min, i.e. the maximum negative point of ReG is given by:
Re(Gd) =
-1
4k(1()
(2.11)
at Wmin = wn(1 + ().
When interpreting the model results, we want to get a feel of how bum behaves
and how it is related to the damping and stiffness of the system. To achieve this
we know that Re(G) depends on the product k(, and that ( is analogous to the loss
factor rq. From equation 2.9 we see that W2 is analogous to the stiffness k. Putting
19
everything together, gives us the term w r, which incorporates both the stiffness and
the damping characteristics of the system. The goal in each of the designs presented
is to maximize this factor, and therefore achieving maximum damping at the highest
stiffness.
To help understand how vibration at the tool tip can lead to chatter, the process
of self-excitation and regenerative chatter is represented by the block diagram of the
closed loop system shown in the figure below. Under certain conditions, the next
pass of the vibrating tool can align with the rough surface just cut (yi-i) to cause
variations in the chip thickness (regenerative feedback).
2.1.3
The Negative Effects of Chatter
1. Low quality parts:
The vibrations that result in chatter cause the cutting tool tip to entirely lose
contact with the workpiece, and then dig deep in it. Apart from the high pitched
noise that is generated, this results in poor surface quality since the depth of
cut deviates from the required tolerance limits. In some cases this may lead to
damage of the part.
2. Increased spindle wear:
Most of the commonly used machine tools are not designed for high amplitude vibrations on a continuous basis. Chatter can result in damage of the
spindle and linear axis bearing surfaces, which result in early failure and high
depreciation costs.
3. Increased tool wear:
As a result of the impact between the tool and the workpiece, the cutting tool
is also damaged which results in early tool wear. This leads to increase of the
tool change rate and tool consumption costs.
4. Lower output:
Since chatter does allow products to meet the required specifications, a decrease
20
in the amount of vibration during operation must be obtained. The simplest
way to attain this is to decrease the metal removal rates. This leads to an
increase in the machining time required for each part.
As a result of the longer machining time, the manufacturing costs rise, causing
an increase in the selling price of the product. This may very easily cause reduced
revenues and market share, which hurt the state of the business and any considerations
for future projects. Consequently the manufacturer will have to reduce profits by
chosing a smaller margin to maintain a target selling price.
2.2
Available Methods for Chatter Reduction
Most of the universal available methods available for chatter reduction do not modify
the machine tool structure in any way and have limited applicability.
1. Damping materials:
Machinists can effectively dampen the machine tool system through the use of
visco-elastic materials, mounted on certain components of the system. However,
eliminating the problematic mode is usually a harder task.
2. Reducing tool overhang:
The amplitude of vibration can be reduced as the overhang length decreases due
to the increased tool stiffness. At conventional speeds the input force spectrum
is dominated by low frequencies. In this case, the static compliance of the system
dominates the system's response, rather than the resonant peak response of its
natural modal frequencies.
Although minimizing tool overhang may seem like the obvious thing to do, its
implications are very important. The overall length between the spindle gauge
line and the tool tip should be kept to a minimum for a number of reasons.
Stiffness drops quickly as tool length increases.
stifness of a cylindrical beam are:
21
The bending and torsional
Kbending
L
Ktorsion ~
(2.12)
(2.13)
where D and L are the beams diameter and length respectively. As we can see,
the longer the tool, the less stiff it will be. So, if for example the length of the
tool-holder is doubled, its bending stiffness will drop by a factor of eight, while
its torsional stiffness will drop by 50% or more. According to these relationships, the increases in tool length can be compensated by using larger diameter
tools. However, larger diameters can cause other problems such as introducing
imbalance and, most important, restricting clearance to small cavities. Another negative effect of using long tools is the increased runout, which results in
decreased accuracy of parts produced, induced vibration and accelerated tool
wear.
3. Reducing the cutting tool rake angle:
By reducing the rake angle, i.e. the angle of inclination between the leading
edge of the cutting tool and the part being cut, smaller reaction forces are
generated when a chip is sheared from the workpiece. Also the variations in
cutting force for a given amplitude of vibration decrease with smaller rake angle.
As a consequence, it is general practice to grind tools to small rake angles.
4. Radiusing the tool:
A radiused tool does not penetrate the workpiece as readily as a sharp point.
Thus, the deflection at the tool tip for given variation of the cutting force, and
thereby the stored energy in the cantilever tool that causes chatter, is reduced.
2.2.1
Products for chatter reduction
In the case where the universal approaches to chatter reduction are not sufficient,
certain products are available to increase the machine tool system's resistance to
22
chatter. However, most of these solutions have high purchase and implementation
costs associated with them.
A machine tool designed for high speed machining
requires not only a high speed spindle, but the capability of operating at high feed
rates with high accuracy. The build up of inaccuracy in the machine tool socket and
the toolholder taper results in radial eccentricity (runout) of any spindle mounted
tool, the magnitude of which will also depend on the tool length. High runout at
high rotational speed will lead to unbalance, vibration, reduced tool life and poor
surface finish.
Several toolholder designs provide taper/face contact. All of them are more expensive than the conventional ISO taper, and require new spindles, since they are not
compatible with the standard 7/24 toolholders.
Conventional Tool holder (ISO Taper)
Tapered tool holders are located in the spindle through the mating of two tapers.
The standard 7/24 tool/spindle interface is dimensioned so that there is a guaranteed clearance between the face of the spindle and the toolholder flange. The main
advantage of the 7/24 taper connection is that it is not self-locking and is secured by
tightening the toolholder taper in the tapered hole of the spindle, thus allowing fast
connections and disconnections. Another advantage is its simple design, requiring
only one dimension, the taper angle, to be machined with a high degree of precision. As a result, the majority of machining center spindles and the spindles of many
other machine tools have 7/24 tapered holes, and there is a huge inventory of such
toolholders.
However, the conventional taper interface has many serious shortcomings. It is
not ideal when used at high rotational speeds, even when a high tolerance taper
is specified. A number of disadvantages are: large mass and size, lack of rigidity
of the machine tool socket, which may distort under high centrifugal forces at high
rotational speeds, and poor repeatability at tool changing.
The 7/24 taper must
satisfy two parameters simultaneously: precision location of the toolholder relative
to the spindle, and clamping for adequate rigidity to the connection. Radial location
23
is not adequate since standard tolerances result in clearance between the back part
of the taper and the spindle hole. Moreover, the absence of face contact between
toolholder and spindle leads to micro motions between the male and female tapers
within the clearance, and thus to fretting and wear of the spindle at high cutting
forces.
Stiffness and accuracy of axial positioning are especially important for high speed
spindles. As spindle speed increases, the spindle shaft tends to expand due to the
centrifugal force and thermal effects. This causes the tapered tool holder to be drawn
further into the spindle due to the retention force applied by the drawbar (Arnone,
Miles, 1998). Thus, the stiffness of the interface between the tool and the spindle
is reduced, making the steep tapered too holders more susceptible to chatter. The
lower stiffness limits the use of aggressive cutting parameters, and increases the inaccuracy when working with longer tools. Expansion of the spindle may also lead to
deterioration of the toolholder balance.
HSK
The HSK interface system (DIN 69893)is the most recent design, jointly developped
by German machine-tool builders, cutting tool manufacturers (including Guhring)
and end users. It is retained in the spindle by use of grippers that sit inside a hollow
cup behind the gauge line of the tooling (see picture). It uses 1:10 taper, is short
in comparison with the 7/24 taper , and has a thin-walled design. As a result, the
application of high axial force causes simultaneous contact between the taper and
face surfaces, partly due to the shrinking of the taper (Toolholder/Spindle interfaces
for CNC Machine Tools). With increasing spindle speed, centrifugal forces cause the
grippers to expand within the toolholder, pressing it tightly up against the interior
of the spindle shaft. This results in greater stiffness in the interface. The system
operates with a modified drawbar that connects to the taper through a wedge-like
connection, which applies higher load at increasing cutting speeds.
The increased stiffness of the HSK tooling system provides a higher degree of
accuracy, compared to the conventional toolholder. When drawn into the spindle, it
24
provides a simultaneous fit on both the spindle nose (via a flange) and the spindle
taper. This insures high repeatability when inserted and removed from the spindle.
Although HSK has many benefits, there are some major disadvantages associated
with it. Testing has indicated that the small size HSK systems are more susceptible
to chatter than the ISO taper tooling. This is probably due to the longer extension
from the gauge line required to support a given tool length when using HSK, since
the hollow cup behind the gauge line does not allow using collets or other clamping
mechanisms. In contrast with the ISO taper, where tooling can be placed partially
behind the gauge line in the body of the taper itself, in HSK tooling collets must
reside entirely in front of the gauge line. This increases the effective length of the tool
assembly, which results in decreased stiffness, since stiffness in bending decreases as
the cube of the length and increases with the diameter to the fourth power (Arnone,
Miles, 1998). When larger HSK toolholders are used, the diameter of the holder body
grows so that its stiffness overcomes the adverse effects of the longer overall length.
3EI
kT=L
64
One of the principal drawbacks encountered during the adoption of the HSK
system is its sensitivity to the presence of chips or contaminants. Lack of proper
cleaning can result in the presence of chips in the spindle nose, which prevents the
HSK toolholder from being located properly in the spindle.
As mentioned before, the HSK toolholder is not compatible with existing spindles and toolholders. Its high fabrication accuracy and intricate design make HSK
spindles and toolholders very expensive (toolholders are 1.5-2 times more expensive
than 7/24s). The drawbar is also more complicated and expensive, since it needs a
mechanism for kicking out the toolholder (the 1:10 taper self locks in the spindle).
25
Heat Shrink Tool Holding
The concept of induction heating the cutting tool and shrink-fitting into a conventional holder was recently introduced to industry. In this case the toolholder is completely solid, with a precision bore to accept the tool. At room temperature, the
holder bore is smaller than the tool shank diameter that it will accept. The toolholder is then heated using an induction heater, causing it to expand, which allows
the tool to be inserted in it. As the holder cools, very high pressures are exerted to
the tool, clamping it tightly in place.
Heat-shrink tool holders provide excellent results in toolholding rigidity and machining quality. The concentricity of this system is consistently 0.0002 in., or better.
Heat shrink toolholders can be perfectly symmetric. No set screws are needed to
clamp the tool, making it possible to manufacture them to a very low level of imbalance. Balance pockets are machined in the drive keys on all V-Flange tooling eliminating the imbalance inherent in the V-Flange design (R.C. Dewes, D.K.Aspinwall,
M.L.H.Wise, 1994). This results in balanced chip loads, better finishes, and increased
speeds and feed rates.
Another advantage is that the shank of the tool is gripped over its entire circumference along the length of the bore. This is not the case with milling chucks and
hydraulic chucks which leave an area of up to 0.25 in from the end of the toolholder
with no force applied to the tool. Due to its extreme rigidity and stiffness, the heatshrink system has proved to extend tool life. It is also a simple and fast method
of changing reclamping tooling. The average induction cycle takes seven seconds to
expand the bore, letting the operator remove the dull tool and insert the new tool.
Since the induction system localizes the heat to the clamping area only, a cycle time
of less than 10 sec minimizes conduction of heat through the toolholder. The entire
process takes less than 30 sec.
The performance of the heat-shrink tool holder depends greatly upon the accuracy
of the tool shank and the operating temperature during machining. Since thermal
contraction is used to secure the tools, further sensitivity to elevated operating tem-
26
peratures should be expected. As temperatures increase, the amount of clamping
pressure can decrease.
Hydraulic Toolholders
While these type of toolholders have been available for a long time, only in the last
few years have they become as high quality and low maintenance as they are today.
Hydraulic toolholders use steel's ability to stretch and compress within its elastic
limit without failure for thousands of times. An oil chamber surrounds an expansion
sleeve, and a set screw is turned to force the piston and seal against the hydraulic fluid
(see figure). This forces the fluid into the chamber, compressing the expansion sleeve
under extremely high pressure (around 20,000 psi). The pressure distributes evenly
over the entire circumference, so that the sleeve expands evenly and concentrically
over its length around the tool shaft. The uniform pressure clamps the tool, and
full contact is achieved over the length of the engagement. Releasing the hydraulic
pressure with a turn of the screw returns the sleeve to its original size.
Hydraulic toolholding minimizes runout of the tool relative to the toolholder.
Tests showed a runout of less than 0.00012" TIR measured a distance two and a
half times the cutter diameter from the end of the holder (Arnone, Miles, 1998) .
Thus a more uniform chip load on the cutting tool is achieved, which extends tool
life and improves surface finish. In comparison, the best conventional collets will
provide a runout of approximately 0.0003" TIR, while less accurate collet designs
exhibit runout values of 0.0005" at the toolholder nose. Some concentricity is lost
when intermediate slotted sleeves are used between the tool and the expansion chuck.
Overall, the combination of accuracy and stiffness obtained with hydraulic toolholders
cannot be matched by conventional means.
Vibration damping in these toolholders comes from the hydraulic fluid within the
chuck acting as a natural dampening agent and impact cushion. The damping contributes to longer tool life and improved surface finish. Tool life may go up three or
four times over less precise mechanical holders, according to hydraulic chuck manufacturers. The damping effect of the clamping system prevents microcracking of
27
the tool cutting edge caused by vibration, which is present in a mechanical clamping
system. The lack of runout makes the tool cut more evenly, which means less wear.
The principal drawback of hydraulic toolholders is their cost. A CAT40 expansion
chuck can cost as much as five times more than a conventional chuck. This difference
in cost makes it not practical for most shops to exclusively use hydraulic toolholders. In difficult, though, machining operations such as hard die milling, where the
cutters are highly susceptible to breaking due to unbalanced cutting loads, hydraulic
toolholders are recommended for use in all process steps for optimal results.
28
Chapter 3
Theory and Model Implementation
Damping is usually determined under conditions of cyclic oscillations. Thus we take
x = X sin(wt -
#)
for the steady state displacement and ± = wX cos(wt -
#)
for
the steady state velocity. Depending on the type of damping present, the forcedisplacement curve may differ greatly. In all cases however the force-displacement
curve will enclose an area, referred to as the hysteresis loop, that is proportional to
the energy dissipated per cycle due to the damping force Fd. The energy dissipated
is given by:
Wd = fFddx
(3.1)
In the case of a spring-mass system with viscous damping, the damping force is given
by Fd = cz. Using the relations w,, =
Wd
k/m and c = 2(/kn we have:
= 2(7rkX
2
(3.2)
where ( is the damping ratio of the system. One of the most commonly used measures
of damping is the loss factor 71 defined as the ratio of the energy loss per radian Wd/27
divided by the peak potential or strain energy U:
Wd
27rU
29
(3.3)
Comparison of viscous and hysteretic damping
3.1
3.1.1
Viscous Damping
Consider the simple oscillator shown in figure 4-1, consisting of a mass M attached
to a spring with stiffness K, with damping being represented classically by a viscous
dashpot, so that the damping is proportional to velocity. With an excitation force
F(t) applied to the mass, the system will respond with a displacement x(t), considered
positive in the upward direction.
M
K
C
Figure 3-1: Single degree of freedom system with viscous damping
Applying force balance using Newton's second law, and letting the excitation force
be a steady harmonic function F(t) = Fcos(wt) yields the differential equation:
(3.4)
Mx(t) + Cx(t) + Kx(t) = Fcos(wt)
The complementary solution of this equation is of the form:
xc = e-(,,nt(Cl sin(wdt) +
and the damped natural frequency
C2
(3.5)
cos(wAt))
of the system is
(
This is the
transient response of the system, which consists of oscillation at the damped natural
frequency whose amplitude decays as e-Cwnt. A particular solution x, of equation 4.5
Wd
30
wd
-
is any function x(t) that satisfies the equation. We have that:
.)
Fcos(wt 2
2
= (K - Mw ) + w C
2 2
where
q$=tan-l[
W
(K - MW2)
So, the complete solution of equation 4.5 is:
x = xC + x, = e -t(Cisin(wdt) + C 2 cos(wdt)) + Acos(wt - #)
(3.7)
(3.8)
with
A =
A=
F
(K- Mw 2 ) 2 + w 2 C 2
(3.9)
In other words, the response of this single degree of freedom system is the sum of a
transient oscillation with natural frequency Wd and amplitude that depends on initial
conditions and decays over time, and a steady state oscillation at the frequency w
of the exciting force and a phase angle 0 lagging the excitation. If the damping is
larger, the transient response quickly dies away. The steady state response, however,
remains for as long as the excitation is present and provides insight into the system
behavior when a harmonically varying force is applied to the resonant structure. As
the frequency increases, the inertia term -Mw 2 x steadily becomes larger until it is
equal to the stiffness Kx. At that point resonance occurs, and the stiffness and
inertia forces cancel out, leaving damping as the only was to limit the amplitude
of vibration(Ahid D. Nashif, David I.G. Jones, John 0. Henderson, 1985).
At
frequencies far above resonance the inertia term dominates and the response becomes
very small, and is out of phase with the excitation (0 ~_180'). At low frequencies of
excitation the motion of the system is dominated by the stiffness K, and the response
of the mass is in phase with the excitation (# c 0). Three important frequencies can
be identified for the viscous case:
1. The undamped natural frequency wn =
4.6.
31
K/M, by setting C = 0 in equation
2. The damped natural frequency from equation 4.6:
Wd =
Wn, 1
)2
-
(3.10)
3. The resonant frequency for which I xp/F I is a maximum, from equation 4.8
w
W,
(1
(2)
(3.11)
The energy dissipated per cycle in this system is given by:
Wd
=
j
Fddx = 7rCwA 2
(3.12)
and therefore q = 2(.
3.1.2
Hysteretic Damping
The effect of damping on the steady state response of the system is very important for
many engineering problems. Hysteretic damping is often utilized in the calculation
of steady response. One of the major advantages of using hysteretic damping is the
possibility of utilizing the correspondence principle in complicated elastic analyses,
where a complex number can be substituted for the real value of the modulus to account for damping. The principal difference between viscous and hysteretic damping
is that for the viscous system the energy dissipated per cycle depends linearly on the
frequency of oscillation, whereas for the hysteretic case it is independent of frequency.
Hysteretic damping is only valid for steady harmonic motion.
Now, If we let the viscous damping coefficient of equation 4.7 be
C= K
(3.13)
then the particular solution (equation4.7) becomes:
x = XP = Bcos(wt - 4)
(3.14)
where:
F
B =
V(K
- MW2)2+ K2rq2
32
(3.15)
I
F( )
X
M
k:k ( I+I h)
L
Figure 3-2: Single degree of freedom system with hysteretic damping
and
4=tan-'[
with xc set to zero.
x = Re[Bei(wt-)]
,
Setting ei(wt-)
=
K
(K - MW2)
(3.16)
4)
+ i sin(wt - q), we have that
cos(wt -
which leads to:
X =
Re[Biwe(iWt-)] = iwx
(3.17)
So, the system can be written in the form:
-w 2 Mx + K(1 + iq)x = F
(3.18)
where K* = K (1 +iq) is the complex stiffness of the spring, in which case it represents
both stiffness and damping. Thus the above first degree of freedom system can be
represented using K* = E*S/L , where E* = E(1 + iq) is the complex Young's
modulus of the system, q is the loss factor of the spring material, S is the cross
sectional area, and L the undeformed length of the spring. E, K and q are assumed
to be constant over a limited frequency range.
The energy dissipated per cycle in
this system is given by:
Wd
= 7rKnB 2
(3.19)
In order to find the equivalent viscous damping model for a hysteretic system, we
have that at the point where w = w, (resonance) the damping forces are equal.
33
Substituting w. = VK/M in equation 4.15 we have that
rq = 2(
3.2
(3.20)
Finite Element Method
The equations of motion for the tool system can be discretized to yield the following
matrix form:
[M]X + [C]X + [K]X = F
(3.21)
where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix
and F is the vector of the input forces corresponding to the degrees of freedom of
X. The finite element method is used to formulate the mass, damping and stiffness
matrix. We employ Euler beam elements to model the dynamic behavior of the tool
and sleeve. One-dimensional Euler beam elements have 4 degrees of freedom: each
of the element's two ends or nodes has a lateral displacement v and a rotation 0,
resulting in four coordinates, v 1 , v 2 , 01,
02.
The stiffness matrix as well as the mass
matrix can be obtained from the potential and kinetic energy, provided the shape
functions of the beam are known.
34
F 1 , vi
F2, V2
1
2
A
M2 ,
M1 , 1
02
EI
Figure 3-3: Positive sense of beam displacements and forces
We express the deflection as:
v(x) = Pi + P2 + P3
where
=
2
(3.22)
+ P4
x/l and pi = constants.
Differentiating the slope equation gives:
10(x) = P2 + 2p3 + p4
(3.23)
2
The boundary conditions can be expressed in the following matrix equation:
V1
1
0 0 0
Pi
191
0
1
0 0
Pi
V2
1
1
1 1
P2
102
0
1
2 3
P2
Substituting pi = v, and P2
=
101,
(3.24)
we can solve the last two rows of the matrix for
p3 and p4 . Then the desired inverse matrix becomes:
Pi
1
0
0
0
VI
Pi
0
1
0
0
101
V2
102
P2
-3
-2
3
-1
P2
2
1
-2
1
(3.25)
This equation can be used to determine the pi corresponding to the case where each
of the displacements is equal to unity, while all others are equal to zero(Thompson,
1986). For example, when v 1 (x) = 1,with all other displacements equal to zero, the
first column gives :
35
Pi = 1, P2 = 0, P3 = -2, P4 = 2
Substituting into the deflection equation gives the shape function of the firstmode:
01(x) = 1 - 3
2
(3.26)
+ 2 3
In the same manner we obtain the following beam shape functions:
# 1 (x) = 1-3
02(x)
=
l - 21
= 3 22
q3(x)
2
+2
(3.27)
3
2 + l3
(3.28)
3
(3.29)
=-W(+ W(3
#4 (x)
(3.30)
By considering the displacement in general to be the superposition of the four
shape functions we have:
y(x) =
#1 V1 +
0201 + 03v 2 +
#4 02
=
q$ 1q1 + q 2
2
+
#3q1
+
#4 q4
(3.31)
where qi are the end displacements.
3.2.1
Tool and Sleeve Mass element [M]ei
To determine the generalized mass, the preceding equation is substituted into the
kinetic energy equation:
1
T = IJy2 mdx =
2- Zj~d
#J#
mdx = I
f
2
miEEidj
(3.32)
iij
Thus the generalized mass mij, which forms the elements of the mass matrix, is equal
to
mj =
#5oq$j~
mdx
The nodal displecement vector of an element Xl is given by:
X 1 T = [vI 01 v2 02 ]
36
(3.33)
The element mass matrix for the uniform beam element is given by:
FMl
-
156 221
54
-131
412
131
-3l2
pA
420
156 -221
412
where 1 is the element length, A is the cross sectional area and p is the material
density. The Global mass matrix can be assembled by superimposing the element
stiffness matrix,as shown below for the case when three elements are used.
I
[
[M]e
Figure 3-4: Assembling the system mass matrix from the individual element mass
matrices
3.2.2
Tool and Sleeve Stiffness element [K]ei
The nodal displacement vector of an element Xl is given by
Xe1T = [vI
01
V2
62 1
The element stiffness matrix is given by
61
-12
61
412
-61
212
12
-61
12
El
[K~el =----
412
37
where I is the element length and I is the cross sectional moment of inertia of a solid
circular beam, given as a function of the beam's diameter
rD4
64
(3.34)
The Global stiffness matrix can be assembled by superimposing the element stiffness
matrix as shown below.
I
[Ke=
[K]=
Figure 3-5: Assembling the system stiffness matrix from the individual element stiffness matrices
3.2.3
Tool and Sleeve Damping element [C]ej
The nodal displecement vector of an element K1 is given by:
XeT =
[V1
01 V 2 02]
The element damping matrix is given by:
156 221
412
[C]e = Coefficient
54
-131
131
-312
156 -221
412
where the coefficient depends on whether viscoelastic material is used, or damping is
imposed through viscous fluid. In the latter case, the system has to be converted into
a first order system. The Global damping matrix can be assembled by superimposing
the element stiffenss matrix as shown below for the case when three elements are
used.
38
[C] =
Figure 3-6: Assembling the system damping matrix from the individual element mass
matrices
3.2.4
Boundary Condition Adjustment
In modelling the dynamic behavior of a long overhang tool, the tool nodes held in
the collet or toolholder are regarded as fixed or ground. Therefore all 4 degrees of
freedom at the grounded nodes are set to zero. The deflection at the tool tip is mostly
determined by the tool flexibility and only slightly affected by the preload of the tool
holder. Grounding a node is reflected by cancelling the corresponding two rows and
columns in the system stiffness, mass, and damping matrices.
3.3
Model Implementation
In order for it to be used as a helpful design tool, the model has to be flexible, compatible with PC hardware commonly available in today's engineering environment,
and cost very little to set-up and run. Thus, modelling of the designs involving both
viscous fluid and viscoelastic material, was done using MATLAB (Mathworks, 1999).
Frequency-domain analysis allows the designer to generate a complete frequency response for each tool design by evaluating the system behavior at different frequencies. Together with an eigenvalue and frequency-domain analysis enable the designer
to compare the static and dynamic stiffness of various designs by easily finding the
natural frequency corresponding to the first mode of vibration and then accurately
determining the amplitude of the response at that frequency. Using the MATLAM
39
Script, the designer can code automated loops that vary various design parameters
in increments and find the system's optimal output performance. MATLAB is available at relatively low cost compared to more sophisticated analysis software such as
ANSYS and COSMOS. Although these software packages are easier to set up, they
do not allow for automated adjust-and-search parameter optimization. Also, due to
the generality of the underlying code, they require more processing time to evaluate
the performance of each design.
With this formulation, the MATLAB functions returning the natural frequencies
and loss factor qj of the tool system can be applied to the various designs. A sample
of the code used to model the cases where either viscoelastic material or viscous fluid
are used as the damping media, are included in Appendix A.
3.4
Dynamic Response Measurement
Impact tests are a relatively simple way to measure the dynamic response of a machine
tool structure. Reducing the amplitude of the force-to-deflection transfer function of
the structure should, according to metal cutting theory, increase the stable operating
range of metal removal.
During these experiments, the transfer function (ratio of magnitude of output
deflection to the magnitude of the input force) is measured using the HP 3567A
Signal analyzer over the frequency span of interest. The force is applied using a
modally tuned hammer (PC13), while the deflection is measured using an accelerometer (PB14) mounted at the tool tip. The hammer produces a voltage proportional
to the force applied at the surface. The deflection output is measured by a single
axis accelerometer, mounted at the tool tip. The accelerometer produces a voltage
signal proportional by Cacc to the acceleration produced at the tip. To deterimne
the deflection, the acceleration has to be integrated twice. The deflection amplitude
A(w) is a function of frequency and is calculated from the accelerometer voltage Vacc
by:
40
A(w) = C2 Vacc
(3.35)
with w being the frequency in radians pes second.
Since only the first mode is of interest, the frequency span is set to 800Hz, while a
resolution of 800 lines is used, giving a frequency resolution of 1 Hz. The signal analyzer automatically calculates the natural frequency and the corresponding damping
ratio associated with the recorded transfer function. Each trial consisted of averaging
10 measurements of the transfer function in order to ensure a smoother and more accurate recording of the tool's dynamic behavior. Apart from the frequency response,
the time response is also recorded to ensure that the system behaves as expected, and
also to provide insight on how fast the vibration dies off.
3.5
Performance
The performance of the concepts tested is measured by the damping ratio (, which
gives a good indication of the damping in the system. The dynamic stiffness of the
tool is the inverse of the force to deflection transfer function and is therefore frequency
dependent. It is the most important factor in relation to chatter. As a function of
static stiffness K, dynamic stiffness Kd can be represented by
Q = Kd
41
2(1
(3.36)
Chapter 4
Fluid Damped Tooling
A squeeze film damped tool, Q - toolTM, (Gaurav Rohatgi, 1998), is used to measure
the damping imposed in relation to the viscosity of the fluid in the damping layer.
Several different experimental setups are examined in order to determine the testing
conditions that would provide the most effective collection of data. Emphasis is given
on the repeatability of data acquisition. The main part of the experiment is performed
on a horizontal milling machine.
B
CLEARANCE
0 14
15:1
025. 5
SEAL 1NG R IN
SECTION
-HOLES
B-B
THROUGH '4HICH DAMPING FLUID !S INJECTED
Figure 4-1: The Q-Tool
According to the results, the highest damping was attained using oil of viscosity
(p) 0.28 Nsec/m
2 .
42
4.1
Analytic approach and modelling
Since viscous fluid is used as the damping medium, the following equation of motion
is used to represent the system:
4y
2Y dx
+ t2
+Y
pA
-
a2y
a2 2 dx
19t
(4.1)
=0
substituting
(4.2)
y = #(A cos wt + B sin wt)
leads to:
EI
x* = x/L
Using
+ yiw# - pAw 2 0
and
#$*= #/L
a40
pAL 4 w2
El
ax4
=
0
(4.3)
leads to:
7iwL 4
(4.4)
EI
With damping imposed through a viscous fluid, the second order system:
[M]X + [C]X + [K]X = F
is transformed into a first order system by defining a vector of state variables
X
Now the first order system of equations becomes:
(4.5)
- = [A]X + [B]_F
In terms of the system's mass, stiffness and damping matrices, and the displacement
vector X the above equation becomes:
x
[
x1
0
[']
[M]' [-K] [M]~ 1[-C]
1
i
[I
43
x
0
L0
[M]
1 [-I]
[
0
01
]
F
(4.6)
where F is the vector of the input forces and moments. In this analysis, F is zero
since the cutting force at the tool tip is not accounted for.
The eigenvalues and eigenvectors are now calculated from the matrix A. In order to
obtain insight into the behavior of the system, the dynamic behavior of the following
system is analyzed:
K2
KI
M2
MI1
C
Figure 4-2: Q-Tool Model
M,
X..
0 M2
X2
+
X.
0
C
+
=,X
0 (4.7)
-K1
X2
44
K1 + K2
X
The following figure depicts the analytic model used to model the tool-sleeve
model, when viscous fluid is used as the damping medium. Each line represents
an element, represented by a Mass,Stiffness and Damping Matrix. The number of
elements used in the code can be arbitrarily chosen. The more elements used, the
more accurate the results. Using around 50 elements is adequate. The bullets between
the elements represent the nodes ( Number of nodes = Number of elements + 1).
The tool is modelled as a cantilever beam, where the deflection and the slope at
the first node of the damping layer are fixed, ie set to zero. The length of the damping
layer is varied from zero to about 1/3 the length of the tool.
2
1
3
4
.Viscous fluid
7
6
5
-
8
10
9
11
12
Free
.
Figure 4-3: Model for damping with viscous fluid
As the following graph shows, the damping increases as the length of the viscous layer
increases.
According to the model, maximum loss factor is achieved at around P = 0.35
Nsec/m 2 . However, during the experiment it is impractical to use a fluid with that
high of a viscosity. Instead the maximum viscosity used is /y = 0.28 Nsec/m 2 . As
we can from the figure, after the maximum damping point, further increase of the
viscosity results in lower loss factors. Also, maximum damping is reached at the
point where the length of the viscous layer is about 1/3 of the tool length. At higher
viscosities, as expected, maximum damping is reached at lower length ratios.
4.2
Preliminary Testing
The first part of the experiment involves determining the settings on the Signal Analyzer that would be used during the actual data acquisition. In this way a first feel
45
13
10 0
0.35
0.42
0.52
0.370.29
0.22
0.49
Higher viscosity
LL 10
Cu
(0
0
i0-2
0.15
0.2
0.3
0.25
0.35
Length Ratio L(damped)/(Lfree)
Figure 4-4: Loss Factor as a function of length ratio, for different viscosities
of the dynamic behavior of the tool and the required analysis is obtained. First the
tool is put in the tool holder, without any fluid in the squeeze film. Next, we place
teh toolholder on a base made of foam, which allows the tool to be modelled as a
free-free beam.
As an initial rough test water is inserted into the sleeve using a 10 ml plastic
syringe without a needle. Leakage occures from both the front seal ring and from
all three screws that cover the holes from which the fluid is being poured in. To
account for this, an o-ring is tightened in between the front seal ring and the tool
shank while Teflon tape is used to cover the screws. After these adjustments there is
no fluid leaking out. Modal analysis of the tool with the o-ring and the Teflon tape
on the screws, without any fluid in, shows that there is a minor increase in damping,
probably caused by the addition of the o-ring.
46
0
-1
... .
-
....
-2
t3
5
-6 . . . . .
-7
. . . ..
. .. .
....
0.5
0.45
0.4
0.35
2
0.25
Length ratio
0
2
4
14
6
Viscosity
Figure 4-5: Loss Factor as a function of viscosity and length ratio
4.3
Testing on the Cincinnati Vertical Milling Machine
4.3.1
Set up
The first measurements on the tool are performed on the Cincinnati Milacron vertical
milling machine, in order to analyze the dynamic behavior of the tool during real life
usage. As a first step, the tool is placed on the machine, without any damping fluid
in it, to determine what the experimental set up should be. A different tool holder
is used, and the tool is tucked all the way in it. The milling machine is then turned
on, the settings on the machine are adjusted, and the tool is placed. During this
experiment the tool holder is taken out of the machine each time the tool has to be
removed in order to change the damping fluid. After placing the tool holder back in,
the emergency button is pushed and the machine is turned off, in order to decrease the
amount of vibration caused by the milling machine running, that results in unclear
47
data.
During the first trials, the tool is placed at a 900, measured from the front of
the machine. The excitation hits with the modally tuned hammer are from the right
side, facing the tool. The first trials show two peaks on the frequency response, and
as a result it isn't clear which modes the analyzer is peaking. To account for this
several different tool orientations are examined, in order to minimize the effect of
the machine on the tool vibration, i.e. to isolate, if possible, the mode of the tool.
Changing the orientation of the tool relative to the machine is done simply by turning
the tool holder. The orientation that results in the least machine-tool interaction is
hitting the tool from the front side of the machine.
After several trials, each of which includs ten averages, it is unclear whether the
results are repeatable. In other words, not the same frequency and damping of the tool
are obtained each time, although the orientation remains constant. This discrepancy
is probably due to the fact that the point of the excitation hits is not consistent, since
it is impossible to hit at the same place on the tip of the tool each time. Moreover,
the accelerometer being mounted straight on the tool tip, is at an angle relative to
the point of excitation. To account for this, two bases are built (see picture), flat on
one side, curved on the inside. The inner radius is the same as the tool radius to
insure maximum surface contact with the tool. This way, the accelerometer is placed
on a flat surface, while the hammer tip provides the excitation on the exact opposite
side.
First Measurements
Testing of various damping fluids is performed, in order to determine which one has
the best effect on increasing damping. Thus, water-soluble oil is diluted with different
ratios of water to obtain the desired viscosity each time. A range of viscosity (p) from
0.00745 to 0.02809 Nsec/m
2
is initially obtained.
Water (p = 0.001Nsec/m 2 ) is poured in the tool sleeve.
All three screws are
covered with Teflon tape, and two of them are screwed in.
The tool is held up
vertically with the tip facing down, and fluid is pumped in through the syringe from
48
one of the three holes, at a slow rate to decrease the amount of bubbles developing
in the sleeve. The tool is then tilted and held horizontally while fluid is still pumped
in, in order to insure that the sleeve is filled. Finally when the sleeve is full and fluid
starts coming out, the third screw is tightened in.
Next, the tool is placed in the tool holder and back in the machine, at the same
orientation as before. An expected increase in damping is recorded. However trying
more viscosities, after cleaning the inside of the tool each time with pressurized air,
shows that the results obtained from the modal analysis are not reasonable, since
the shape of the frequency response is still not repeatable, i.e. the peaks are not at
the same range of the response. Thus, the expected increase in damping cannot be
identified. Possible causes for this discrepancy result from:
1. The orientation of the tool within the tool holder,
2. The vibration caused by the turning on/off the machine each time to place the
tool,
3. The removal and adding of new Teflon tape on the screws after each trial
4. The development of bubbles in the inside of the sleeve, while pumping the fluid
in.
To account for the first cause, an orientation is marked on the tool and the tool
holder. The vibration caused by turning on the machine is let to die off by waiting
for 2-3 minutes after the tool holder is tucked in. The change of the Teflon tape
Replacing the plastic syringe used with
is not regarded as an important factor.
a glass syringe with a needle, results in a more controllable pump of fluid in the
sleeve, that significantly decreases the development of bubbles in the inside. Also the
accelerometer cable is mounted on the tool to account for the frequency relief.
Yet, even after these adjustments, no reasonable conclusions can be drawn. In
the few times when the measurements are consistent, the viscosity used varies from
0.02357 to 0.054 Nsec/m 2 . During these measurements, the increase in the viscosity
of the damping fluid results in an increase in damping. However, due to the lack
49
of repeatability in the preceding tests, it is decided to perform the experiment on a
horizontal milling machine with larger stiffness.
4.3.2
First Testing on the Horizontal Milling Machine
Set up
The experimental set up is moved to a horizontal milling machine. A much heavier
tool holder is used this time, in order to increase the amount of stiffness. The tool
is mounted all the way in the tool holder. In this machine, there is no need to take
the tool holder out each time the damping fluid has to be changed. This makes the
data acquisition easier and, most of all, decreases the amount of inconsistency in the
measurements. In order to have the same orientation of the tool each time, a flat
piece of metal is placed on the top base where the excitation is applied, and the tool
position is adjusted until the metal piece is parallel to the surface of the machine
using a height gage.
Measurements
It is clear from the first trials that this experimental set up significantly increases the
amount of repeatability in the measurements compared to the previous tests on the
Cincinnati machine. Due to the increased stiffness of the machine, the peaks observed
in the frequency response curves are sharper. However, the tool holder is not as stiff
as expected which causes other modes of the machine to be excited when the tip of
the tool is hit with the hammer. After taking several measurements while applying
the hits at different points on the tool, and on the tool holder, the mode of the tool
is identified.
After trying the same range of viscosity, while following the same procedure as
before, the effects of increasing the viscosity remain unclear. There is no noticeable
change in the frequency response when the tool has no fluid in compared to when
a relatively high viscosity fluid is tried. The increased stiffness of the machine and
the insufficient stiffness of the tool holder are thought at the moment to be the main
50
factors responsible for this outcome. Thus, it is decided to use the experimental set
up of the preliminary tests, in order to determine the amount of interaction between
the tool holder and the tool.
4.3.3
Modal Analysis on the foam base
Set up
The tool holder used in the horizontal milling machine together with the tool in it
were placed on the foam base without any damping fluid in the tool. The purpose
of the following test is to determine the mode shape of the system (tool holder-tool)
by measuring the magnitude at evenly spaced points and plotting the results. By
determining the change in magnitude measured on the tool and on the tool holder,
the interaction between the two is determined. It is expected that if the tool holder is
stiff enough, the magnitude corresponding to a point on the tool holder will be about
1/30 of the magnitude on the tool. The tests results show a 1/13 change, which
proves that the tool holder used is not stiff enough.
However, the weight of the tool holder is expected to make the change in damping apparent, if the tests are performed on the foam base. Thus, following the same
procedure as before, different viscosity fluids are injected in the tool and the corresponding amount of damping is determined. During these experiments the fluid is
pumped through one hole, until there is fluid coming out of the other two, while the
tool is kept up, with the tip facing down. Then fluid is also injected through the
other two holes, to insure that there are no gaps in the sleeve. This procedure proves
very efficient when high viscosity fluids are tried.
The results obtained are consistent and repeatable. It is surprising to see that
damping is still increasing when a significantly higher viscosity (p = 0.134Nsec/m 2 )
of the water soluble oil undiluted is used. This fact suggests that the viscosities used
during the experiment in the horizontal milling machine are not high enough in order
for the effect on damping to be identified. Thus, it is decided to conduct a second
experiment on the horizontal milling machine, using higher viscosities this time.
51
4.3.4
Main experiment
The main part of the experiment from which results are obtained and conclusions are
drawn is performed in the horizontal milling machine.
Set up
The same set up is used as in the previous experiment on the horizontal milling
machine. The orientation of the tool relative to the tool holder is marked, and kept
constant during all trials. The tool is positioned so that the top surface where the
excitation occurres is always parallel to the surface of the machine. The Teflon tape
on the screws is changed and the inside of the tool is cleaned with high pressure air
after each run.
To get the desirable viscosity water-soluble oil was mixed with water. The ratios
and the differenet types of oil used are the following:
Oil to Water
Viscosity (p Nsec/m 2 )
18 : 20
0.06516
25 : 20
0.07625
35 : 20
0.08720
Type of Oil
Viscosity (p Nsec/m
Water Soluble
0.136
Way
0.110
Hydraulic
0.250
Cylinder
0.280
2
)
Table 4.1: Viscosities used
As higher viscosities are used, it becomes harder for the fluid to be injected in
the squeeze film. Cylinder oil is the highest viscosity oil available, and using even
higher viscosity make it extremely difficult to pump fluid in the sleeve, without leaving
any gaps inside, in order to achieve maximum damping. Also a further increase in
viscosity will eventually have a negative effect on damping.
52
Damping as a function of Viscosity
(() Q
Viscosity (At Nsec/m 2)
Frequency (Hz)
0
798
17.821e-3
28.06
0.065
801
19.761e-3
25.30
0.076
802
20.964e-
3
23.85
0.087
803
21.895e-
3
22.84
0.115
804
22.350e-
3
22.37
0.136
804
23.620e-3
21.16
0.250
806
24.392e-3
20.49
0.280
810
26.462e
3
18.89
Damping Ratio
(1/(2())
Results
According to the results, the highest damping is achieved using the highest viscosity
oil, the cylinder oil. As the graph shows, damping is increased when larger viscosity
of oil is used.
However, the increase in damping is not linear; it gets smaller as
higher viscosity is being used. Although the theory predicts that eventually a further
increase in viscosity will start having a negative effect on damping, the turnaround
point corresponding to the maximum viscosity cannot be reached, since it is extremely
hard to pump higher viscosity oil into the sleeve.
Although the Q-tool provides increased damping during deep cavity machining
its high cost and need for nonconventional tool-holding mechanisms point towards a
more widely used and cheaper toolholder. As a result, a complete redesign of the
common R8 collet is performed and discussed in the next chapter.
53
I
Frequency Response with different viscosities
58
I
I
I
I
780
790
800
Frequency (Hz)
810
56-
54-
52-
C
50-
0)
CIS
48-
46
44-
770
760
820
830
840
Figure 4-6: Frequency Response with different viscosities
0.03
0.025[
Cal
CL
E
0.02
A Al
0
0.05
0.1
0.25
0.2
0.15
Viscosity (mu)
0.3
0.35
Figure 4-7: Damping as a function of viscosity
54
Chapter 5
Redesign of the R8 Collet
5.1
Project Overview
This project involves a complete redesign of the widely used R8 collet. The goal is
to improve on the major drawbacks of the standard R8 collet. The new design aims
to accomplish:
1. Better clamping of the tool in the collet, with higher stiffness and damping
2. Stiffer placement of the collet in the spindle
In the widely used R8 collet the tool is being held only at the front portion of the
collet. This results in poor clamping force and low stiffness of the structure when
longer tools are used, which consequently increases runout of the tool. Thus accurate
deep cavity machining is not possible using the standard R8 collet.
The new collet is based on the regular R8 collet, but instead consists of four pieces:
the collet, the clamp, the Belleville springs and the washer.
Starting from a stock 10mm R8 collet, the rear insert, on which the drawbar is
originally threaded onto is machined out, leaving 0.120 in. wall thickness. The inside
of the collet is then machined, creating the taper in the back. Three evenly spaced
slots are created on the outer of the collet. Finally three openings in the back side
are machined to allow for the insertion of the clamp.
55
The clamp, as the name suggests, is used to clamp the back of the tool in the
collet. It consists of three jaws that fit inside the taper.
The purpose of the Bellevile springs is to increase the force required to pull the
clamp. This way premature clamping of the back of the tool, before the front hard
point is achieved, is avoided. The washer is added to the second prototype to provide
a larger surface on which the springs are pushing on and to avoid snapping of the
springs off the back of the collet.
According to the new design, the tool is being clamped at two points: At the front
of the collet as before, and at the very back. The goal is to achieve higher rigidity
and stiffness of the tool during machining. In order to clamp the rear portion of the
tool, a separate insert piece is used: the clamp. The clamp is inserted through the
rear opening of the collet. The drawbar is threaded onto the clamp. As the clamp
is being pulled by the drawbar, the rear taper pushes the three jaws of the clamp
inwards, grabbing the back of the tool (see figure). At the same time, the use of
the taper causes the outer portion of the collet to expand slightly, securing the collet
within the spindle.
In order to make sure that there are enough threads on the drawbar to pull the
clamp the following procedure is followed: A long overhang tool in inserted all the
way in the regural 10mm R8 collet until it touches the insert onto which the drawbar
is threaded onto. As the collet is being pulled in the spindle, the drawbar pushes the
tool out of the collet. At the point where a secure clamp is achieved, a mark is set
on the tool. The collet is then taken out of the spindle, and the tool is inserted in it,
until the marked point. The distance from the back of the collet, where the drawbar
comes in , to the back ofthe tool is measured. For the EZ-Trak Bridgeport milling
machine this distance is 0.925". The same value is being used in the new design.
The major issue during the redesign of the R8 collet is ensuring that the tool is
clamped at the front (as in the regular R8 collet) before the clamp is pulled in the
taper causing its jaws to grab the tool in the back. This is achieved through the use
of Belleville springs in the back of the collet (see figure). In order to increase the force
needed to deflect the springs, which would cause movement of the clamp backwards,
56
resulting in an early grabbing of the tool in the back, the springs are placed in parellel
format. As the drawbar pulls the clamp against the springs, the collet is being pulled
in the spindle, and the hard point in the front is achieved. Continuing pulling on the
drawbar causes the springs to deflect, which results in the clamp jaws grabbing on
the back of the tool.
F-TOOL
TA PER-
DRAWBAR
CLAMP JAWS
RING (placed behind the Belleville springs)
Figure .5-1: New Collet
In order to increase the amount of damping in the system, viscoelastic material
is used at the front part of the collet. The viscoelastic material is placed behind the
hard point in the front, and all through the area of the collet that clamps the tool. A
thin shim (0.001 in. thick) surrounds the viscoelastic material, to avoid wear during
the insertion of the tool. In the first prototype, the front diameter is turned down by
0.005 in., in order to ensure that the viscoelastic material is preloaded as the front
part of the collet is squeezed in the taper, and the tool is being clamped.
57
TAPER-\
/-C LAMP
-%LLEVILL[ SPR INGS
R NG
TOOL
DRAWBAR
SECTION
D-D
Figure 5-2: Collet Assembly
5.2
Damping using Viscoelastic Material
In the case where viscoelastic material is used as the damping medium, the damping imposed is hysterestic, where k* represents both the standard spring k and the
damping term.
k* = k(1 +ir7)
(5.1)
As a result, in order to obtain the natural frequencies for each mode, the Damping
Matrix D is added to the Stiffness matrix K:
Kf =D+K
and using x = e"
t
(5.2)
the eigenvalues and eigenvectors are obtained for the second order
system:
MW 2 X =KfX
58
(5.3)
5.3
Dimensional Analysis
The tool-sleeve system, for the case where viscoelastic material is used is represented
using the following equation of motion:
EI a4dx4
19X
+ Kydx - pA 0t2dx
2
= 0
(5.4)
-
substituting
y = q(A cos wt + B sinwt)
(5.5)
leads to:
EI a0+
Ox 4
Using
KO - pAw 2 # =0
(5.6)
x* = X/L and 0* = qO/L leads to:
qa# w2 pAL 4
aX4~ EI
KL4
EI
where K is the complex stiffness of the viscoelastic material.
59
(5.7)
The model
5.4
The basic model used to provide insight into the dynamic behavior of the system is
the cantilever case depicted below. Both the displacement and slope are fixed at the
rear support. Several alterations of this model are examined in order to see which
configuration provides maximum damping. In these alterations, a second point is
added in the front, where only the deflection is set to zero.
1
2
3
4
5
8
7
6
10
9
11
12
Free
Viscoelastic
Figure 5-3: Model for damping with viscoelastic material
At first, the cantilever case is examined, in order to verify that the model is
behaving correctly. In the case where we only have a fixed point (both deflection
and slope fixed), ie the cantilever case, maximum damping is always achieved at the
maximum length of the viscoelastic material. The stifness (K) corresponding to the
maximum damping (7 = 0.183) is 2.424e3 (1 +
j).
At that point the maximum value
for w 2 q is achieved. Increasing the length ratio, as expected, results in a higher first
mode frequency.
The two configurations that are initially examined are:
1. Model A: Having viscoelastic material all the way to the back of the collet,
behind the second fixed point.
2. Model B: Having viscoelastic material only between the fixed points.
The variation of the w2n as a function of the stiffness and the length ratio is
plotted for each of the three models.
60
13
I
5 .5
I
I
I
I
I
I
I
I
I
14
16
18
5K=2.42446e3 (1+i)
4.5-
4.5
-K=
7.017e3(1 +i)
43.53K= 2.031 e4 (1 +i)
cm P"
2.5 -
2-
K=2.894e2(1+i)
1 --
0.5
--
0
0
2
4
6
8
10
12
Length Ratio
20
Figure 5-4: Variation of w2 rq for Cantilever case, as a function of Length ratio and
Stiffness
5.4.1
Model A
According to model A, viscoelastic material is placed throughout the inside of the
collet, all the way to the back. In this case the all the portion of the tool that is
inserted in the collet comes in contact with the viscoelastic material. The first fixed
point is at the front of the collet, as usual. The position of the second fixed point is
altered from close to the front, to all the way to the back. The term length ratio is
used to determine the fraction of the length between the two fixed points, over the
length that is free in the back. In this case the length ratio is varied from 0.001 to 20.
As we can see from the figure below, maximum w2 q/ is achieved at a stiffness (K) of
around 7e 3 (1 + i). As we further increase the stiffness, damping begins to decrease.
The maximum frequency is at the maximum length ratio, i.e. when the rear fixed
point is all the way in the back.
61
Cantilever
2.2\
m1.6
1.4
1.2
15
10
5
20
30
S10
length visco/f ree
length
0
-30
-20
1
Stiff ness
Figure 5-5: Cantilever case: Frequency as a function of Length ratio and Stiffness
62
5
0-
-5
N-10-
S
-15 . .. . .
-20
-
-25
--
3
10
2
0
Length Ratio
15
20
0
1
-5
-10
-5
Stiffness (log)
Figure 5-6: Cantilever case: w2 as a function of Length ratio and Stiffness
5.4.2
Model B
In model B, viscoelastic material is placed only between the two fixed points. When
the position of the rear fixed point is varied, increasing the length ratio, the area on
which viscoelastic material is applied also increases. The behavior of this model is
3
similar to model A. Maximum value for w2 17 is reached at K = 7e (1+ j). As stifness
increases, damping decreases. As expected, the amount of damping imposed and the
overall behavior of the model is mostly affected by the position of the rear fixed point.
5.4.3
Final Model
The final design is a combination of the two previous models: the second fixed point
is all the way in the back, while the viscoelastic material is placed only through the
63
1
5
4
3
2
7
6
8
'
11
12
Free
-- Viscoelastic .
11
10
9
if ree
12
Figure 5-7: Model A: Viscoelastic material behind the second fixed point
front portion of the tool. Since there is no need to use the middle portion of the collet
to impose damping on the tool, the next question is where should the second fixed
point be placed. Keeping the length of the viscoelastic layer constant, the position
of the rear fixed point is varied. According to the model, damping increases as the
rear fixed point is moved further in the back of the collet. Thus maximum damping
is achieved when the fixed point is all the way to the back.
It is interesting to see that at the same K where maximun damping is achieved
at the previous models, the final model provides higher damping (0.082 compared to
0.070), at higher stiffness as indicated by the 2 71 value (4.3 compared to 3.6 and 3.4
for the other models)
Having viscoelastic material all the way to the back of the collet, apart from being
hard to place, does not provide any increase in the amount of damping imposed. In
fact, for small length ratios, we get higher damping at the same stiffness, compared
with models A and B. Increasing the length ratio does not have significant impact in
the increase of damping.
5.5
Model results
The results from the three models are summarized in table 5.1
For a typical R8 collet, if the tool is inserted all the way in the back, and viscoelastic material is placed in the length corresponding to the tapered portion of the
collet, the length ratio is about 2/5.
64
13
5.5
5
3
K=7.017e (1 +i)
4.5
4
_ K=5.878e (1+i)
4
3.5
3
S
K=2.031 eK4 (1+i)
2.5
2
2.424e 3(1 +i)
-K=
1.5
-2
1
-
K=8.377e (1 +i)
0.5
01
0
2
4
16
14
12
10
8
6
18
20
Length Ratio
Figure 5-8: Variation of w2 q for Model A(Visco all the way to the back), as a function
of Length ratio and Stiffness
5.6
Choosing the type of viscoelastic material
In order to determine the type of viscoelastic material to be used in the front portion
of the collet, the following approach is used:
Fn
=
/27r
2r
o-r cos Od +
Tr
sin Od9
(5.8)
using:
6 cos 9
C-
(5.9)
6 sin 0
t
(5.10)
t
7
o- = Ec
65
(5.11)
5-0-5
-10
-15,
-20>
4
-
-
-
3
-10
-
2
1
5
20
-5
-10
Length Ratio
0
-15
Stiffness (log)
Figure 5-9: Model A: w2 as a function of the length ratio and stiffness
T
=Gy
(5.12)
we have
47rrEJ
3t
Fn
Since F,
=
K6 and using G
=
(5.13)
3t
yields:
rE(+i)
K( + i)
3t
(5.14)
where E is the complex stiffness of the viscoelastic material. Using K = 3.04e 4 (1
+j),
yields a thickness of about t = 0.50mm. The type of viscoelastic material that best
combines the required features is EAR C-1002.
66
2.25
. . -.-
-.
2.2...
2.15
-
2.1
- -
2.05
..
n
in
-
0r
S2
1.95
--
1.9
-.
1.85
15
.. . .
-..
10-
5-1
Lenthraio*L2L10
-
5
-30
Length ratIo" L2/L1
-
30
10
-20
0
1
20
10
Stiffness (K)
Figure 5-10: Model A: Viscoelastic behind the second fixed point
5.7
Force Analysis
The force analysis is one of the most important aspects of the design process, since
it provides necessary information about the forces needed to obtain the required
deflections. Tools such as MATLAB(Mathworks 1999) and Pro-E are utilized for the
moment of inertia calculations and for the variation of the critical parameters during
the design optimization. There are three parts of the new collet design on which the
imposed forces are analyzed:
1. The front of the collet, where the tool is clamped
2. The clamp, where the back of the tool is clamped
3. The rear part of the collet, which expands against the spindle wall.
It is very critical that the front of the collet deflects first, providing a hard point
on the tool, while pre-loading the viscoelastic material. The clamp and the rear part
67
1
2
30
.-
4
5
.
60
s
7.
-
8.
-
-Viscoelastic
1
10
-
11
0
12
Free
12
'
9.
ifree
Figure 5-11: Model B: Viscoelastic material, only between fixed points
Model
Max Damping (rq)
A
0.070
B
Stiffness (K)
Length Ratio g
3.6
7.01e3(1 + j)
Max
0.070
3.4
7.01e 3 (1 + j)
Max
final
0.088
4.3
3.04e 4 (1 + j)
0.495
Cantilever
0.062
4.7
3.04e 4 (1 +
0.677
w27I
j)
Table 5.1: Model Results
of the collet should bend afterwards, providing the second hard point on the tool,
while securing the collet in the spindle.
As a first step, the force-deflection relationship is determined for each section.
a2y
Ox
2
Fx+M
EI
(5.15)
where M = Fl, and using the boundary conditions we have:
Mx 2
Fx3
= 6E1+ 2E1I
(5.16)
Equating the deflection and the slope at 11 to solve for the constants for the second
section yields the following formula for the deflection at the front of the collet:
Fx3
Yfrontcolet = 6E12
Flx2
2EI2
Fl2
2E
1
SF11
E±1
Fl 1
3E 12
1
(5.17)
li)
with x ~ I
When calculating the deflection formula for the clamp and the rear of the collet, it
is assumed that the sections from 11 to 1 do not bend due to their increased thickness.
68
13
5.5
5
5I
I
I
I
I
I
I
I
I
4
4.5
5 4.5-
3
K=7.017e (1+i)
4
K=5.878e (1+i)
3.532.5K=2.031e (1+i)
2K=2.424e (1+A)
1.5(1
/-K=8.376e2
0.5
+i)
--
0-
0
0.5
1
1.5
2
2.5
3.5
3
5
Length ratio
Figure 5-12: Variation of w2 for Model B (visco between fixed points), as a function
of Length ratio and Stiffness
Thus the deflection at point 1 is computed by finding the deflection at 11 and adding
a term that multiplies the slope at 11 with the remaining length (1 - 1i):
21+(2
1++
y = 6EI
2EI
2EI
EI)(l-l1)
EI
(5.18)
The next step involved calculating the moment of inertias about the center of
mass for the different cross sections. Since all the cross are round of similar shapes
the parallel axis theorem is used in cylindrical coordinates:
Icm + Ay2m
Y2 dA
= j
(5.19)
'a
where y = r sin(6) and dA = rdrd6. When calculating the moment of inertia of the
tapered section of the collet, the average of the moment of inertias is used, since the
thickness varies linearly. Pro-E is used to confirm that the calculations are correct.
69
5
0-5
-10
-15,
-20,-
3
-25
3
15
2
10
2
-10
Length Ratio
0
-15
Stiffness (log)
Figure 5-13: Model B: w2rq as a function of the length ratio and stiffness
The following numbers are calculated for the different cross sections:
Front of the Collet
Icml = 6.05871e- 10mm 4
Icm2 =
1.3366e- 9mm 4
Rear of the Collet
11 mm 4
Icm = 9.6852e
Clamp
4
icm = 2.7695e-mm
In order to confirm that the front of the collet deflects first when the collet is
pulled into the spindle by the drawbar, the force exerted by the drawbar to cause
the required deflection is calculated for each of the three sections. The following
procedure is followed:
1. Choose the necessary deflection (overestimate)
70
-0.05-0.05
-Zero
z
o
IInfinite Stiffness
-
0
Stiffness
-0.1
-0.15-
-0.2-
-0.25
0
2
4
6
NODES
8
10
12
Figure 5-14: Shape functions for viscoelastic between nodes 3 and 4, with zero and
infinite stiffness
2. Calculate the required force F
3. Compute the normal Force N by multiplying with the corresponding cosine.
4. Obtain the friction between the interface, using a coefficient of friction,
[L
= 0.74
for steel against steel : F, = 0.74N
5. Multiply by the cosine of the corresponding angle to obtain the axial force
exerted by the drawbar Fd.
The calculated Fd is a function of the following factors:
1. The moment of inertia, which varies with the cross section's thickness
2. The lengths 11 and 1
3. The corresponding angle
71
1
10
9
8
7
6
5
4
3
2
12
11
Free
-- Viscoelastic
if
'2
ree
Figure 5-15: Final Model: Fixed point in the back-Viscoelastic material only at the
front
All of these parameters are varied when designing the rear part of the collet and
the clamp. In designing the front portion of the collet, where viscoelastic material is
used, only I1 was varied since the tapered section is standard for the R8 collets. The
result of the force analysis is having the drawbar force needed to preload the front
of the collet be about four times smaller than the Fd needed for the clamp and the
rear part of the collet. However, the average drawbar pulling force is in the range of
40KN, ie much greater than the clamp can handle by itself.
5.7.1
Belleville Springs
In order to increase the force required to pull the clamp, a series of belleville springs
are used, in a parallel stack. The force absorbed by each spring is calculated using
the following formula:
F=
H
4E ST 3
KD [1+(T
T
1 2 K1D2
S H
f)(T
T )(T
H
J)]
2T
(5.20)
where pu is Poisson's ratio, K1 is a constant and E is the Young's modulus.
E (MPa)
pu
S (mm)
T (mm)
H (mm)
K1
De (mm)
Force/spring (N)
207e3
0.3
0.90
1.15
2.05
0.57
21.08
10,970
Table 5.2: Belleville Spring Characteristics
Thus in order to ensure that the front hard point closes first, four to five springs are
placed in series, resulting in required force in the order of 40 to 50 KN. The Belleville
72
13
5.5
-----
K= 5.878e 4(1 +i)
5K= 2.031e 4(1+i)
4.5K= 7.017e (1+i)
4 3.5 3 -2.5 -K=2.424e 3 (1 +i)-
2-.
1.5 -
K=8.377e (1 +i)
0.5
0
-
01
2
3
4
5
Length Ratio
Figure 5-16: Variation of w2r for the final design, as a function of Legth ratio and
Stiffness
springs used are obtained from Key Bellevilles Inc., part number K0750 - C - 028.
The outer diameter expansion is measured for both the original diameter and the case
where the diameter is turned down to 0.78".
Original Diameter
Deformed Diameter
Expansion
0.820
0.826
0.006
0.780
0.791
0.011
Table 5.3: Belleville Springs measured expansion
73
5
-10
-
-153
-220
2
Lengh Raio
-
0
Length Ratio
-15
-10
10
15
2
-5
Stiffness (log)
Figure 5-17: Final Model:w 2 as a function of the length ratio and stiffness
5.8
5.8.1
Experiment
Set up
The tool used during the experiment is a 10mm x 202mm ballnose carbide endmill.
All the data acquisition is performed on a Bridgeport vertical milling machine. A
torque-wrench is used to measure the amount of torque exerted on the drawbar,
and therefore the pulling force on the collet. This way, repeatability in the data is
obtained. The torque exerted on the drawbar is set at 220 in.lbs.
Two similar small aluminum pieces, with opposed curved and flat surfaces are used
in order to mount the accelerometer and on the front tip of the tool, and to provide
excitation with the modally tuned hammer on the other side. This set ensures that
the accelerometer is in the same direction as the force impact, and minimizes the
excitation of unwanted modes during the experiment.
The regular 10mm R8 collet is compared with the new design with and without
74
I
I
0.08
I
I
I
I
I
I
14
16
18
Final Model-fixed point in the back
0.07-
0.06o,0.05 E
CO
Q*Q.4
-
Viscoelastic all the way to the back
0 4
0.030.02Viscoelastic material only between fixed points
0.01 -
0
0
2
4
6
8
12
10
Length Ratio 12 /11
20
Figure 5-18: Comparison of maximum damping for each model
viscoelastic material.
5.8.2
First Prototype
The first prototype is shown below. It is made of a 3/8" R8 collet, with its diameter
turned down to 0.399", compared to the tool diameter of 10mm = 0.394" in order
to have the viscoelastic preloaded before the hard point on the front. The Belleville
washers are also turned down by 0.005" to accomondate for the expansion against
the collet inner surface.
During these initial measurements three belleville springs are used. As the frequency response comparison shows, the new collet design has higher damping (lower
peak) and is a little bit stiffer (higher frequency) than the regural R8 collet.
Unfortunately, after several trials, the expansion of the springs becomes larger
than expected causing the large hoop stresses at the back of the collet. This in
combination with the fact that the amount of material surrounding the springs is
small eventually causes fracture at the thinest portion, where the keyway slot exists.
75
10-1
Final Model:
SMax(h)=0.088
K=3.04 (1 +j)e4
0
2
Models A, B
Max(h)= 0.070
K=6.21 (1+j)e3
0
10
20-
10
3
10
4
56
10
10
Stiffness
7
10
10
8
Figure 5-19: Comparison of maximum loss factors
The expansion of the springs results in an increase of the outer diameter of the
collet, which consequently makes the removal of the collet from the spinde a difficult
procedure. This is mainly due to the fact that these tests are performed with only
three springs, which makes it easier for the springs to plastically deform. Also, the
amount that the washers are turned down proves insufficient to account for their
expansion.
Furthermore, as it turns out, the area where the clamp grabs the tool in not
adequate. This, in combination with the existence of a chamfer at the back of the
tool, sometimes leads to insufficient clamping at the back. As a result the jaws of the
clamp yield, when the clamp is pulled by the drawbar in the taper.
76
F
11
Figure 5-20: Model for deflection-force calculation
Frequency (Hz)
Collet
Loss factor (q)
New R8
12.04e-
3
651
Regural R8
5.072e-
3
639
Table 5.4: First Prototype Data
5.9
Second Prototype
In order to improve on the major defects of the first design, a second prototype is
designed and built. This time, the goal is to decrease the circumferencial stresses
in the back.
Thus more material is added in the area of the collet surrounding
the springs. The guiding slot in the back is flattened to minimize the local stress
concentration.
The three slots in the back are smaller, which leads to a smaller
deflection against the spindle wall. Moreover, the taper is moved 0.090" to the front,
which gives space for five belleville springs. The springs are turned down 0.040"
in diameter, to ensure that their expansion would not cause contact with the inner
surface of the collet. A ring is placed behind the washers to increase the area on
which the springs bear on. The diameter of this ring is almost the same as the inner
collet diameter, in order to provide a tight fit. The length of the clamp is increased,
which allows for a larger grabbing area on the tool. Furthermore, this time a 10mm
collet is used where the front diameter stays as is, to ensure that the hard point in
the front comes before the one in the back.
77
N I
N
)
72 d3
F1
SP 1NDLE-
COL LET
C LA MP
Fd
Fn
LI,)
En/
y2
Figure 5-21: Force analysis
5.9.1
Modal testing
During the modal testing the regular R8 collet is compared with two alterations of
the new design: Having the brass shim covering the viscoelastic material extending
all the way to the edge, allowing an easier insertion of the tool, or not extending it,
thus increasing the amount of preload on the viscoelastic material.
According to the results, a 70% increase in damping is reached when the new
collet is tested with a higher preload on the viscoelastic material. On the other case,
the increase in damping is 50%, but at a smaller decrease in the natural frequency
measured.
In order to find out which configuration actually produces better results cutting
tests are performed.
78
De
H
S
3 3ELLEVILLE SPRINGS IN PARALLElI
Figure 5-22: Belleville Spring
Collet
Loss factor (77)
New R8 (with extended shim)
8.961e-
3
650
New R8 (without extended shim)
9.983e- 3
615
New R8 (without shim)
6.983e- 3
660
Regural R8
6.587e-3
658
Frequency (Hz)
Table 5.5: Second Prototype Data
5.9.2
Cutting tests
All the cutting tests are performed on the Bridgeport EZ Trak vertical milling machine. A cutting test consists if one pass on the edge of an aluminum bar stock at
a feed rate of 4 ic./min, at 2500 rpm. After experimenting using both collets at different cutting conditions, the depth of cut is set at 0.250in. in the z direction and
0.020in. in the y. Going deeper at one pass results in increased chatter, not allowing
the difference in damping betwen the two collets to be observed.
When the new collet is tried on the milling machine, the tool does not run true.
The viscoelastic material and shim are taken out of the collet, and the tool is tested
again. This time the tool runs
The first cut each time is taken with the regural R8 collet, at the same tool
overhang as in the new collet. The tool is then placed on the new collet, the origins
of the x,y and z directions are reset, and the cut is performed at the opposite side of
the stock.
79
10-2
Old
10~3 [
new
a)
70
-/
0)
10-
10-
I
I
10 0
200
300
I
I
500
400
Frequency Hz
II
600
700
800
Figure 5-23: Comparison of Frequency response for new collet design and the regural
Royal R8
The difference in a cut of 0.0250" deep (x) and 0.250" (y) using the old and new
R8 collets is shown below:
The new collet design results in less chatter during all depths of cuts up to 0.0250
in. After that point both collets behave in similar ways.
80
I
SI
I
I
I
I
.
I
New with scheme
extending
Regural
New with not
-
extended scheme
103-
CD
CO
10~4 --
300
350
400
450
500
550
600
650
700
750
800
Frequency (Hz)
Figure 5-24: Comparison of frequency response for second prototype and the regural
Royal R8
81
-
s
W4
.161-
-.
-4-'-'N
-
-
4i~iSW
Figure .5-25: Regural R8
Figure 5-26: New R8
82
L
aN
Chapter 6
Viscoelastically Damped Tools
Methods for incorporating viscoelastic material into teh shnk of a gundrill are tested.
The type of viscoelastic material used is EAR - C1002, with a thickness of 0.015 in.
Due to its high loss factor this material is commonly used in damping machine tool
structures, since higher loss factor results in increased damping. Since this material
is resilient, the designs can achieve the proper combination of stiffness and damping.
Using a thinner layer of material results in making the damper stiffer.
6.1
The tool
A v-groove is forged into the tool to allow the coolant to flow out of the drilled hole.
Forging the v-groove changes the shape of the upper part of the tool cross section
from circular to elliptical. The steel tool tip is brazed on the tool, while the tool is
brazed on the shank.
The main problem during the usage of this type of gundrill is the presence of
chatter. This has a significant effect on guiding the tool tip, especially on curved
surfaces. The two concepts tested aim at reducing the amount of tool vibration by
minimizing the deflection in the portion of the tool next to the shank. At the same
time, vibration energy is absorbed through the use of a damping medium (viscoelastic
material).
In order to test the new concepts, the tool diameter is turned down to 0.361"
83
from 0.381", at a length of 0.500" from the shank. Thus, in that portion, the shape
of the tool cross section becomes circular. This allows the rings used in the different
damper configurations to fit on the tool body without sliding.
A
0 .275
023
y-
7/
0 .226
0.303-
77
/77/
A.586
A
1.015
10.720
SECTION
2.636
A-A
Figure 6-1: The Gun-Drill
6.2
Experimental Apparatus Description
Our measurements on dynamic behavior of the gundrill with and without the use
of the dampers are not performed during actual operation of the tool. Rather, an
experimental apparatus is used, that has four components:
1. An aluminum vice, where the gundrill is positioned (see picture),
2. The gundrill,
3. Two C-clamps,
4. A granite table.
84
Mode
Frequency(Hz)
Ratio
Compare to
1
52
1
1
2
328
6.307
6.266
3
1020
18.61
17.547
4
1928
36.07
34.386
Table 6.1: Comparison of the modes measured to the modes of a cantilever beam
6.2.1
Set up
As a first part of the experiment, the accuracy of the experimental apparatus is
tested. The tool is placed on the vice without having its tip inserted in the aluminum
piece, i.e. as a cantilever beam. The dynamic response of the tool shows a pretty
close agreement between the modes recorded and the modes of a cantilever beam (see
table 6.1). This ensures that clamping of the tool shank on the testing fixture is stiff
enough, and thus, it can safely replicate the use of a collet or other type of tool holder
for the purpose of these experiments.
The aluminum piece is then attached to the vice and with the tool tip inserted in
the hole, the tool can be approximately modeled as fixed-pinned beam. The EI/(pA)
ratio of the tool is compared for the two cases (fixed-free and fixed-pinned), after
measuring the frequency of the first mode. In the case of a cantilever the ratio is
0.0477, whereas in the fixed-pinned beam it is 0.0536. This difference is expected,
since there are micro motions of the tool tip in the drilled hole, which slightly deviate
from the model of the tool as a fixed-pinned beam.
6.3
Concept 1-Steel clamps
6.3.1
Description
The first concept consists of a set of two tapered steel clamps, placed on the tool
body in contact with the shank of the tool (see figure 6 - 2). The clamp that is fixed
85
on the tool is denoted as "the ring". It is placed on the machined portion of the tool,
next to the braze of the tool with the shank. The inner radius of the clamp is larger
than the tool radius to account for the fillet caused by the braze on the side of the
tool shaft. The outer radius of the clamp is a little bit larger than the radius of the
tool shaft, so that the clamp contacts with the side of the tool holder (collar-clamp),
when the tool is tucked all the way in it. The ring and the clamp mate with each
other through two cylindrical tapers of 26.60. Two 2-56 screws are placed (one at
each side) of both the ring and the clamp in order to adjust the ring on the tool, and
the clamp on the ring.
RING
VISCOELASTIC MATERIAL
_GUNDR ILL
CL AMP-
Figure 6-2: Steel Clamps Assembly
A thin layer of viscoelastic material is added on the tapered surfaces of the clamp
and the ring, and between the sides of the clamp and the side of the collar-clamp on
the vice. Having the ring fixed onto the tool as the screws on the clamp are tightened,
a force is exerted on both the ring pushing downwards on the taper, and on the side
wall of the collar-clamp. Due to the fact that the taper angle is less than 450 measured
from the tool, most of the force is concentrated on the side of the collar-clamp. Thus,
a possible deflection of the tool body pushes the ring against the clamp, which then
exerts a force on the side of the collar-clamp (fixture). This causes the tool deflection
to be fought back, through the interaction of the steel clamps with the tool holder
86
(collar-clamp). Due to the dampening exerted by the viscoelastic material, the time
required for the vibration on the tool to die off is decreased (see time response graphs).
Although this design results in a significant increase in damping, its major drawback is the low amount of dynamic stiffness provided by the assembly. The three
components, tool, ring/clamp, tool holder, interact only through the viscoelastic material, while the metal to metal contact area is very small. Moreover using screws to
adjust the clamping force, which affects the amount of damping imposed, results in
a nonuniform preload both radially and axially. Also, with this configuration, it is
not easy for the user to adjust the damper, in order to obtain the desirable dynamic
behavior.
6.3.2
Data
The amount of damping measured (() and the amplification factor
adjustments, are ( = 3.735e-
3
and
Q=
(Q),
without any
133.87, respectively , with the first mode at
404 Hz.
The measurements taken with the damping fixture added on the tool show a very
noticeable increase in damping. Damping values (() vary from 6.474e
3
to 8.271e-,
depending on the adjustment of the screws on the clamp. Dynamic stiffness also
increases, as
Q
decreases. As a result of the increase in the dynamic stiffness the
frequency also increases, from 404 Hz (undamped) to 420 Hz when damped.
Measurements are also taken without the presence of viscoelastic material. This
time, the increase in damping is smaller than before.
Also the natural frequency
increases to 414 Hz. Thus the increase in the dynamic stiffness is smaller compared
to when viscoelastic material is used.
87
With Viscoelastic
Q
State
Frequency (Hz)
Damping Ratio (()
Undamped
404
3.735e-3
Damped 1
414
6.474e-
3
77.23
Damped 2
416
7.543e- 3
66.28
Damped 3
420
8.271e- 3
60.45
Damped 4
420
8.269e-
3
60.47
(1/(2())
133.87
Without Viscoelastic
Q
State
Frequency (Hz)
Damping Ratio (()
Undamped
404
3.735e- 3
133.87
Damped 1
413
4.589e- 3
108.9
Damped 2
415
4.641e~ 3
107.7
Damped 3
417
5.078e- 3
98.46
Damped 4
417
4.870e- 3
102.67
(1/(2())
Table 6.2: Damping and Frequency with and without viscoelastic
88
70
65 F
60
55
50
C4
45
40
351-
30
L
380
390
400
410
Frequency (Hz)
420
430
440
Figure 6-3: Frequency response using the steel clamps, with the gundrill pinned
89
Undamped
0.08
0.06
-
0.04
0.02.
0
-0.02
-
-0.04
-0.06
-0.08
-0.1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.3
0.35
0.4
0.45
0.5
0.3
0.35
0.4
0.45
0.5
time (msec)
Damped 4
0.04
0.06 0.04
0.02-
-0.,02-
-0.04
-0.06-
-0.08
-0.1
0
0.05
0.1
0.15
0.2
0.25
time (msec)
Damped 5
0.08
0.06
0.04
0.02
C
0
-0.02
-0.04
-0.06
-0.08
-0.1L
0
0.05
0.1
0.15
0.2
0.25
time (msec)
Figure 6-4: Time response variation from no adjustment(top) to steel clamps(middle)
and steel clamps tightened (bottom) , with the gundrill pinned
90
6.4
6.4.1
Concept 2- Collar Ring
Description
The second concept is built on the previous one. Emphasis is now given to increasing the rigidity and stiffness of the assembly, while increasing damping. Preload is
adjustable and uniform without deforming the tool. Furthermore, the new design is
more practical and aesthetically acceptable, since it is adapted on the gundrill, as an
extension of the tool shank.
The new damper consists of a set of three rings: the outer, the inner, and the
clamping ring (see figure 6-3). A thin layer of viscoelastic material is attached to the
tapered surfaces of the inner ring. The outer ring is threaded onto the tool shank,
providing a solid clamp between the tool and the damper. From the outside, the
outer ring is an extension of the tool shank, with a slightly smaller diameter for
fitting purposes. Its interior is designed so that both the inner and the clamping ring
fit when using different thicknesses of viscoelastic material. The inner ring is a split,
two sided taper, of different angles. The angle of the taper that is in contact with the
outer ring is less steep than the angle of the clamping ring taper, which results in a
gradual increase in damping during the preload of the clamping ring. The inner ring
is mounted on the short segment of the gundrill that is turned down to the diameter
of the ring. The clamping ring is threaded onto the outer ring, while its taper pushes
against the outer taper of the inner ring. This way, the desired pre-load corresponding
to the desired dynamic behavior can be attained. Damping is expected to be initiated
as the inner ring is pushed against the outer and the clamping ring.
6.4.2
Adjustments to the Experimental Apparatus
The experimental apparatus in these tests is improved compared to the previous one.
As before, the same vice is used to obtain a reasonably close simulation of the use
of the gundrill in a real life situation. However, using C-clamps to attach the vice
on a rigid body (granite table) can result in non-repeatable measurements since the
91
OUTER RING
INNER RING
CLAMPING RING
7-
A
GUNDRILL
VISCOELASTIC MATERIAL
Figure 6-5: Collar Ring Assembly
dynamic behavior of the whole system (gundrill, vice, granite table) is very sensitive
to the position and adjustment of the C-clamps. The vice can easily bend when the
C-clamps are used to attach it to the granite table. Bending of the vice causes the
tool to bend, which consequently results in unclear data since the tool tip bounces
in the work piece. Although careful adjustments of the C-clamps can sometimes
result in clear measurements, increased repeatability cannot be attained when the
C-clamps are readjusted after each trial, especially when the tool has to be taken out
of the vice. Therefore attaching the vice on a steel beam is a necessary adjustment in
the apparatus, which results in increasing the stiffness of the testing apparatus, and
produces cleaner data with less variation between measurements.
6.4.3
Set up
Viscoelastic material (EAR - C1002) of 0.015 in. thickness is used, since it provides
the best combination of damping and stiffness and fits nicely into the dampers' package. It is carefully glued to the angled surfaces of the inner ring. With the outer ring
adjusted on the tool shank, the inner ring is inserted in the outer ring, until the two
tapers mate. The clamping ring is threaded on the outer ring, until its taper mates
with the inner ring's. The viscoelastic material is pre-loaded using a spanner-wrench
92
to adjust the clamping ring.
During the test the clamping ring is tightened between trials, by twisting the
spanner-wrench 900 during the first trials, and then 450 for stiffer adjustment. This
shows the effect that the amount of pre-load has on the stiffness and damping of the
tool. As before, when the tool is tested with its tip inserted into the work piece as
if a hole was being drilled, a set of rubber straps is used to pre-load the work piece
against the tool tip, before clamping the work piece to the apparatus.
Due to the increased stiffness and rigidity of this assembly, attention is given to the
excitation of vibration on the tool during the measurements. The frequency response
of the tool is very sensitive to the amount of force that initiates the vibration. The
clearest data result when small hits are applied on the tool.
The tests show the influence of the damper on the dynamic behavior of the gundrill
in both the case where the tip of the tool is inside a work piece and in the case where
the tip is free. Although the actual numbers for ( recorded are smaller than measured
with the ones with the steel clamps, this design is stiffer and allows uniform and
easily adjustable pre-load. The change of the experimental apparatus during the two
experiments is the factor mainly responsible for the difference in damping measured
between the two concepts.
6.4.4
Data
According to the measurements taken, damping rises from ( = 3.389e-
3
without the
damper to ( = 5.189e- 3 at the optimum pre-load. If the pre-load is increased beyond
this point, damping decreases as the natural frequency continues to rise. This can
be seen by the shifting to the right of the resonant peaks on the frequency response
graphs, in both the cases (pinned and cantilever). As expected, the static stiffness of
the Collar Ring design is higher than in the Steel Clamps. A higher static stiffness
yields better part accuracy and surface finish.
With proper adjustment of the clamping ring, improved dynamic behavior of the
gundrill can be attained. This will result in better performance of the tool during
drilling operations, since chatter will be lower.
93
Damping Ratio (()
Q
State
Frequency (Hz)
Undamped
405
3.389e-
3
147.54
Damped 1
406
4.882e-3
102.41
Damped 2
408
5.189e-3
96.36
Damped 3
409
4.626e-
3
108.08
Damped 4
410
4.388e-3
113.95
(1/(2())
Table 6.3: Damping and Frequency for gundrill-pinned
Frequency Response of Pinned
85
80-
_75-
0
65-/
60
385
390
395
400
410
405
Frequency (Hz)
415
420
425
430
Figure 6-6: Frequency response of gundrill pinned using the Collar Ring
94
Q
Damping Ratio (()
State
Frequency (Hz)
Undamped
50.875
1.376e-
3
363.37
Damped 1
51.250
1.821e-
3
274.57
Damped 2
51.250
2.077e-3
240.73
Damped 3
51.250
2.294e-3
217.96
Damped 4
51.750
1.909e-
3
261.91
Damped 5
51.750
1.845e-3
271.00
(1/(2())
Table 6.4: Damping and Frequency for Gundrill-free
Frequency Response Cantilever
80
70 F
60
8 50
40 -
30 -
20'
48f
49
II
II
50
51
Frequency (Hz)
52
53
54
Figure 6-7: Frequency response of gundrill free, using the Collar Ring
95
Chapter 7
Conclusions
7.1
Results
Improved surface finish and increased metal removal rates can be achieved through
the use of squeeze-film damped and viscoelastically damped tools. The results show
increases in damping ranging from 20% to 100%. In the case of the Q-ToolTM, this
allowed a 300% increase in cutting speed.
The first cutting tests performed with the new R8 collet showed improved surface
finishes, compared to the regular R8. Dynamic response tests revealed increases in
damping up to 70%, at about the same frequency as the normal R8 collet. The
improvements in surface finish are not visible at depths of cut exceeding 0.030in.
This project also involved the development of analytical models that provide insight in the dynamic behavior of the systems examined. Such models are very useful
when potential designs are examined, and help optimize the system's performance.
7.2
Recommendations for future work
Although the new R8 collet design gave promising signs on the cutting performance
using long overhang tools, there is plenty of space for improvement. The major issue
is achiving increased damping without sacrifising stiffness. The development of more
detailed models could help in optimizing the use of viscoelastic material.
96
Measurement of the cutting forces between the tool and the workpiece would also
provide useful information, as well as surface finish measurements on the samples.
Finally, developping a collet that provides increased cutting performance, without
being hard and expensive to manufacture can be a really interesting project.
97
Appendix A
APPENDIX A: Finite Element
Code
98
%Non Dimensionalized
%Collet for fixed point in the back-viscoelastic in the front.
%nodes=total number of nodes for entire beam
%d=Total number of nodes for damping Matrix
%dl=Number of nodes for damping Matrix for portion between fixed points
%h=collet4(41,16,7,1.3015*(l+j)*10^4)
%hmax=0.08759902479025
function [blim, h, f requency,V, eigenvalues,Dl,D,Di,M, K, ld1, ld2, lf ree]
collet4(nodes,d,dl,K)
k=K;
%lnd=l/L;
=
n=nodes;
%Length of the beam
Length=4;
%Nondimensionalized length
L=1;
%Diameter of the Beam
D=1;
R=D/2;
%Nondimensional Length of the element, n=number of nodes
1 = L /(n-1);
%Calculate number of nodes for portion
%between fixed points, using the length ratio and the number of nodes with viscoelastic
%dl = ((lratio*d)+l)/(l+lratio);
%length of dl, portion between fixed points
ldl
=
(d1-l)*l;
%number of nodes free in the back
d2=d-dl+l;
%length of d2, portion between left free and first fixed point
ld2 = (d2)*l;
%Free length
= (n-d)*l;
lfree
%Nondimensionalized 1
%lnd = lv/L;
p=1;
A=1;
m=p*A;
%Modulus of Elasticity
E=1;
%Moment of Inertia
I=1;
%Coefficient
Coef=K* (L^4) / (E*I);
%Local Stiffness Matrix
6*1 -12
* [12
= E*I / (lA3)
6*1;...
2*(12);...
-6*l
4*(1^2)
-12 -6*l
12 -6*1; ...
4*(lA2)];
-6*l
6*1 2*(lA2)
Kl
6*1
%Local Mass Matrix
Ml =
(m*l /
420)
*
[156 22*1 54 -13*1;...
22*1
4*(lA2)
13*1 -3*(lA2);...
54 13*1 156 -22*1;...
-13*1
-3* (lA2)
-22*1
4* (lA2)]
%Local Damping matrix
Dl = (Coef *1/420) * [156 22*1 54 -13*1;...
22*1 4*(lA2)
13*1
-3*(lA2);...
54 13*1 156 -22*1;...
-13*1
-3*
(l^2)
-22*1
4* (l^2);
%Global Stiffness Matrix
K=zeros(2*n, 2*n);
for i=1:n-1
for x=1:4
for y=1:4
K(1+2* (i-1) +x-1, 1+2* (i-1) +y-1)
+ K1 (x,y)
=
K(1+2* (i-1) +x-1, 1+2* (i-1) +y-1)
=
M(1+2* (i-1) +x-1, 1+2* (i-1) +y-1) + Ml (x,y)
end
end
end
%Global Mass Matrix
M=zeros(2*n, 2*n);
for i=1:n-1
for x=1:4
for y=1:4
M(1+2* (i-1)+x-1, 1+2* (i-1) +y-1)
end
end
end
%Global Damping Matrix
%Nodes for elements with viscoelastic material on = d
D=zeros (2*n, 2*n);
for i=d2:d-1
for x=1:4
for y=1: 4
= D(+2*(i-)+x-1,D+2*(i-)+y--)
D(1+2*(i-1)+x-1,+
end
end
end
Mi=M;
Ki=K;
Di=D;
K(2*d-1,:)=[];
K(:,2*d-1)=[]
K(1, :)= [];
K(:,1)= [];
M (2*d-1,:=]
+ Dl(x,y);
D (2*d-1, :))=[] ;
D (: ,2*d-1)=[
D(1, :)= [] ;
D(:,1)= [];
;
%Add Stiffness and Damping Matrix
Kf=K+D;
%Get eigenvalues
[V,eigenvalues]
= eig
(Kf,M);
omega=diag(eigenvalues);
w=sqrt (omega);
W=min(w);
frequency=abs (W);
wn=frequency^2;
%frequency = abs( (W)
h=(imag(W^2))
blim=h*wn;
/
*
(sqrt(E*I/(p*A*(LengthA4)))));
(real(WA2));
%Qtool
%First Order Finite Element model with viscosity as a variable
function [h] = fedim(nodes,d,mu)
%d=number of nodes for damping Matrix
n=nodes;
%viscosity
%Length of the beam
L=1;
%Diameter of the Beam
D=1;
%Length of the element, n=number of nodes
1= L /(n-1);
%Length of viscoelastic layer
lv=l*(d-1);
%Mass per unit length m p*A
m=1;
%Clearance of damping layer
h=7.62e-5;
%Modulus of Elasticity
E=1;
%Moment of Inertia
Inertia=1;
%Coefficient for Damping Matrix
Coef=mu * (L^4)/
(E*Inertia);
%Local Stiffness Matrix
6*1;...
6*1 -12
Kl = E*Inertia / (lA3) * [12
2*(lA2);...
6*1 4*(lA2) -6*1
12 -6*1;...
-12 -6*l
4* (lA2)];
6*1 2*(l^A2) -6*l
%Local Mass Matrix
Ml = (m*l / 420) * [156 22*1 54 -13*l;...
13*1 -3*(1A2); ...
22*1 4* (lA2)
54 13*1 156 -22*1;...
-22*1 4*(lA2)];
-13*l -3*(lA2)
%Local Damping matrix
Dl = (Coef * 1/420) * [156 22*1 54 -13*l;...
22*1 4*(lA2) 13*1 -3*(1A2);...
54 13*1 156 -22*1;...
-13*1 -3* (l^ 2) -22*1 4* (l^ 2)];
%Global Stiffness Matrix
K=zeros(2*n, 2*n);
for i=1:n-1
for x=1:4
for y=1:4
K(1+2*(i-i)+x-1,1+2*(i-i)+y-1)
= K(1+2*(i-i)+x-1,1+2*(i-l)+y-1)
+ Kl(x,y);
end
end
end
%Global Mass Matrix
M=zeros(2*n, 2*n);
for i=l:n-1
for x=1:4
for y=1:4
M(1+2*(i-i)+x-1,1+2*(i-i)+y-l) = M(1+2*(i-1)+x-1,1+2*(i-l)+y-l)
+
l(x,y);
end
end
end
%Global Damping Matrix
D=zeros(2*n, 2*n);
for i=l:d-1
for x=1:4
for y=1:4
D(1+2*(i-i)+x -1,1+2*(i-1)+y-1)
end
end
=
D(1+2*(i-1)+x-1,1+2*(i-i)+y-1)
end
Mi=M;
Ki=K;
Di=D;
%for Cantilever fix node 1.Always start from the one from the right.
K(2,
K(:,
K (1,
K (:,
M(2,
M
M(1,
M
[]
[II
[]
;
;
;
[1;
[] ;
[1;
I:]
D(2, :)=[ ];
D (: , 2)=[3;
D (1,: )=[;
D (:, 1)=[
%Now the third row and third column of the original global are number 1.
%Convert to 1st order system.
%Compose matrix A
H = inv(M)*(-l*K);
F = inv(M)*(-l*D);
A = zeros(2*(2*n-2),2*(2*n-2));
g=1;
h=(2*n)-2+1;
while
( g <=
A(g,h)=1;
g=g+l;
h=h+l;
end
(2*n)-2 & h <= 2*((2*n)-2)
+ Dl(x,y);
x= (2*n) -2+1;
y=1;
y<= (2*n)-2
while(
while
( x<= 2*((2*n) -2)
A(x,y)
= H(x-((2*n)-2)
y );
,
x=x+1;
end
y=y+l;
x= (2*n) -2+1;
end
x=(2*n)-2+1;
y= (2*n) -2+1;
while(
y<= 2*((2*n)-2)
while
( x<= 2*((2*n)-2))
A(x,y)
= F(
x-((2*n)-2)
,
y-((2*n)-2)
);
x=x+l;
end
y=y+l;
x= (2*n) -2+1;
end
%---------------------------------------------
eigen = eig(A);
a=all(imag(eigen));
if
(a==O)
I=find(imag(eigen));
e=eigen(I);
w=(min(e));
else
e=eigen;
w= (min(eigen));
end
el=e (size (e, 1))
e2=e (size (e, 1) -1);
e3=e(size(e,1) -2);
e4=e(size(e,1) -3);
e5=e(size(e,l)-4);
e6=e (size (e,1) -5);
e7=e (size (e,1) -6);
e8=e (size (e,1) -7);
%First Mode Frequency in Radians per sec
h=(imag(w^2)) / (real(w^2));
BIBLIOGRAPHY
1. Ahid D. Nashif, David I.G. Jones, John 0. Henderson ,1985, "Vibration Damping" Wiley Interscience
2. McConnell G. Kenneth, 1995, "Vibration Testing", John Wiley and Sons
3. King, R. Ed , 1985, "Handbook of High Speed Machining Technology", New
York, Chapman and Hall
4. Tlusty, J. 1985, "Machine Dynamics.
Handbook of High Speed Machining
Technology" R. King. New York, Chapman and Hall
5. Tlusty, J.,Smith S., 1997, "Current Trends in High Speed Machining" Journal
of Manufacturing Science and Engineering 119 (November)
6. Tobias, S.A., 1965. Machine Tool Vibration. New York, John Wiley and Sons,
Inc.
7. Arnone, Miles, 1998, "High Performance Machining", Hanser Gardner, Cincinnati, Ohio
8. Thomson, W. T., 1988, Theory of Vibration with Applications, Third Edition,
Prentice Hall, New Jersey
9. R.C. Dewes, D.K.Aspinwall, M.L.H.Wise,1994, "High Speed Machining-Cutting
Tools and Machine Requirements"
10. SAE International," Spring Design Manual" AE-11, 1990
11. www (1999). http://toolingsystems.com/univ/shrinker.ueshrink.htm, Tooling
Systems Division
12. www (1999). http:www.hsk.com/toolhold.htm, Diebold Goldring Tooling
99