DEPARTMENT OF MANUFACTURING ENGINEERING
AND MANAGEMENT
ENHANCEMENT AND VERIFICATION
OF A CUTTING FORCE MODEL FOR
MICRO CUTTING
MASTER THESIS
Jan Slunský
2007
Preface
The work has been carried out from September 2006 to March 2007 at
Department of Manufacturing Engineering and Management, Technical
University of Denmark, under supervision of Assistant professor Giuliano
Bissaco, Ph.D.
I wish to thank my supervisor Assistant professor Giuliano Bissaco, Ph.D.
for his suggestions and many inspiring discussions. I would like to thank also
Professor Hans Nørgaard Hansen for giving me the chance to carry out the
master thesis at DTU.
Renè Sobecki is acknowledged for the discussions and assistance
concerning the metrological aspects of this work.
Special thanks are for Lars Peter Holmbæk and Peter Sanderhoff for the
help with the milling and turning experiments carried out at DTU.
A very special thanks belongs to Alessandro whom I shared the office with
during all these months. He was always helpful and it was not just the work
cooperation, but also a lot of fun and nice moments I will always remember.
Thanks to every single friend that I have met in Denmark, for the
important and funny life experience.
Special thanks to Jana and David for their support in the critical moments.
They always exhilarated me and helped me, when I needed it the most.
Finally, I would like to thank my family for the constant support in all my
life, giving me the opportunity to study and for going along with all my wishes
and necessity. I am really thankful for all the comprehension and support that
they always gave me in these years of study.
Lyngby March 30th 2007
Jan Slunský
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Enhancement and verification of a cutting
force model for micro cutting
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Abstract
This thesis is aimed at developing the mathematical model for cutting force
prediction in micro end-milling operation. The development of quantitatively reliable
predictive models is driven by the need to optimize the economic performance of
machining operations and particularly by specific demands in micro-cutting operations
corresponding to the tool deflections and its breakage.
After the introduction to the world of micro-manufacturing, the modelling of
cutting processes is shortly described. The unified mechanics of cutting approach to
the prediction of forces in milling operations is adopted and extended with numerical
calculation of the chip flow angle instead of using the Stabler rule. Also tool run-out is
considered and implemented into the model.
The unified mechanics of cutting approach introduces the edge force
coefficients in addition to the cutting force coefficients. This method eliminates the
need for the experimental calibration of each tool geometry and relies just on an
experimentally determined orthogonal cutting data, which are obtained in the turning
experiments of the aluminium 6082 T6 and the carbon steel UHB 11.
The mathematical model with the cutting quantities implemented from the
turning experiments is evaluated in terms of micro milling experiments on
conventional CNC milling machine with attached high speed spindle. A method for the
axial depth of cut control is required by such a machine, when using the end mills of
200µm, and therefore it is shortly described here.
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Table of content
1
INTRODUCTION
1.1
Micro-manufacturing scenario
8
1.2
Materials for Micro-manufacturing
9
1.3 Micro-manufacturing Processes
1.3.1 Subtractive processes
1.3.2 Near-net-shape processes
1.3.3 Additive processes
1.3.4 LIGA
10
10
12
13
14
1.4 Examples of present-day microproducts
1.4.1 Automotive applications
1.4.2 Microfluidic devices and systems
1.4.3 Smart Shoe
15
15
16
19
1.5 Micromilling
1.5.1 Cutting Tools
1.5.2 Machine Tools
1.5.3 Micro-cutting process
20
20
20
21
1.6
The need for cutting force prediction
24
1.7
Aim of the work
24
1.8
Organization of the work
24
2
MODELS FOR CUTTING PROCESSES
2.1
Overview on historical development of models for machining processes
2.2 Models for cutting force prediction
2.2.1 Classification of cutting force models
2.2.2 Unified Mechanics of Cutting approach
3
7
MATHEMATICAL MODEL FOR END-MILLING OPERATIONS THROUGH
THE UNIFIED MECHANICS OF CUTTING APPROACH
25
26
28
28
29
32
3.1 End-Milling Force Model
3.1.1 Prediction of Milling Force Coefficients from an Oblique Cutting Model
33
34
3.2 Model improvements
3.2.1 Prediction of chip flow angle – removing the Stabler rule
3.2.2 Run-out
35
35
36
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EXPERIMENTAL WORK – TURNING EXPERIMENTS
4.1
Introduction to the turning experiments
42
4.2
Experimental Setup
43
4.3 Calibration Procedure for 3D Turning Dynamometer
4.3.1 Calibration of the dynamometer
4.3.2 Data analysis
46
48
49
4.4
Experimental Plan
52
4.5
Experimental Procedure
54
4.6
Measurements
55
5
EXPERIMENTAL WORK – MICRO MILLING EXPERIMENTS
58
5.1
Introduction to the micro milling experiments
59
5.2
Experimental setup
59
5.3
Experimental plan
62
5.4
Thermal distortions
64
5.5 Experimental procedure
5.5.1 Run-out measurement procedure
5.5.2 Tool length correction procedure – the compensation of axial depth of cut errors
5.5.3 Machining procedure
6
RESULTS – TURNING EXPERIMENTS
65
65
66
66
68
6.1 Data from the turning experiments
6.1.1 Chip thickness
6.1.2 Forces analysis
69
69
70
6.2
75
7
8
41
Implementation of the parameters into the model
RESULTS – MICRO MILLING EXPERIMENTS & MODEL VALIDATION
77
7.1
The effect of Stabler rule removal
78
7.2
The effect of the tool run-out
79
7.3
Validation of the model through the micro milling experiments
80
CONCLUSIONS
85
Jan Slunský
Enhancement and verification of a cutting
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APPENDIX A – DATA FROM TURNING EXPERIMENTS
OF THE ALUMINIUM 6082 T6
APPENDIX B – DATA FROM TURNING EXPERIMENTS
OF THE STEEL UHB 11
APPENDIX C – MATLAB SCRIPT
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List of symbols
A
A, B, C, D
ae
ap
b
c
CPe, CQe
[mm2]
[-]
[mm]
[mm]
[mm]
d
[mm]
dFt, dFr, dFa
[N]
dr
f
FPc, FQc, FRc
[mm]
[mmmin-1]
[N]
FPe, FQe, FRe
[N]
FX, FY, FZ
h
hc
hmin
hlim
i
j
k
Ktc, Krc, Kac
[N]
[µm]
[µm]
[µm]
[µm]
[deg]
[-]
[-]
[-]
Kte, Kre, Kae
min, max
N
q
r
R
Re
rt
st
tj
UX, UY, UZ
x, y, z
z(a), z(b)
[-]
[mm]
[-]
[mm]
[mm]
[mm]
[µm]
[-]
[mmtooth-1]
[mm]
[V]
α
αn
β
βn
φ
φn
[deg]
[deg]
[deg]
[deg]
[deg]
[deg]
[-]
[mm]
Area of cut
Parameters
Radial depth of cut (step over)
Axial depth of cut
Width of cut
Constant
Edge forces per unit active cutting edge
length
Offset of the tool axis from the axis of
rotation
Tangential, radial and axial elemental forces
acting on an edge segment
Radius deviation
Feed
Force components due to the shearing and
friction process
Edge force components due to rubbing and
ploughing process
Force in the X, Y and Z direction
Uncut chip thickness
Chip thickness
Minimum chip thickness
Limit chip thickness
Angle of obliquity, inclination angle
Tooth number
Slope of line
Tangential, radial and axial cutting
coefficients
Tangential, radial and axial edge coefficients
Minimum and maximum measured deviation
Number of teeth of cutter
Actual tool radius in zero z-level
Actual tool radius
Nominal tool radius
Cutting edge radius
Chip thickness ratio
Feed per tooth
Uncut chip thickness for tooth j
Voltages in the X, Y and Z direction
Cartesian coordinate system
Z-level, distance from tool tip to the
measured level
Rake angle
Normal rake angle
Friction angle
Normal friction angle
Shear angle
Normal shear angle
Jan Slunský
γ, υ
ηc
λ
θ
τ
ω
Enhancement and verification of a cutting
force model for micro cutting
[deg]
[deg]
[deg]
[deg]
[MPa]
[degs-1]
Auxiliary angles
Chip flow angle
Displacement angle
Immersion (or cutter orientation) angle
Shear stress on the shear plane
Angular rate
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Enhancement and verification of a cutting
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Chapter
1
Introduction
Page 7
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1.1 Micro-manufacturing scenario
The miniaturization, design flexibility, reduced energy consumption, and high
accuracy of products is demanded for various industries. The explosion of microscale
product development in consumer markets including healthcare, communications,
and electronics, to name a few, has potential to significantly improve quality of life and
personal well being. In the healthcare field alone, advances in cardiovascular system
remediation and noninvasive surgery have been tremendous. Some examples of
microproducts are shown in chapter 1.4.
Micro-manufacturing is an important new technology [1], because:
• it is an enabling technology for the widespread exploitation of nanoscience
and nanotechnology developments – it bridges the gap between the nanoand macro-worlds;
• it is a disruptive technology that will completely change our thinking as to
how, when, and where products will be manufactured – e.g., on-site, ondemand in the hospital operating room etc.;
• it is a transforming technology that will redistribute manufacturing capability
from the hands of a few to the hands of many – micro-manufacturing
becomes a cottage industry;
• it is a strategic technology that will enhance the competitive advantage of
any country – reduced capital investment, reduced space and energy costs,
increased portability, increased productivity.
Countries with traditional strengths in manufacturing, such as Japan or
Germany, have continued to invest heavily in recent years in micro-manufacturing
R&D for several reasons. First, the demand from the global market for ever-smaller
parts and systems at reasonable cost and superior performance is strong. This
demand tends to drive the high-end research. Second, the prospects of
multidisciplinary research are causing companies increasingly to blend material
science, biology, chemistry, physics, and engineering to speed up technology
innovation and thereby new applications based on micro-technology. Third, strong
government actions in the form of national R&D initiatives have resulted in more
effective R&D infrastructures that are conductive to advanced research and education
[1].
In this thesis, the term micro-manufacturing refers to the creation of highprecision three-dimensional products using variety of materials and possessing
features with sizes ranging from a few microns to a few hundreds microns. While
microscale technologies are well established in the semiconductor and
microelectronics fields, the same cannot be said for manufacturing products involving
complex 3D geometry and high accuracies in a range of non-silicon materials.
Development of manufacturing processes like wet etching, LIGA, etc., have
taken place in the last three decades as a result of a huge interest in
microelectromechanical systems (MEMS) [2]. Such technologies suffer from several
limitations, in particular, MEMS-based methods are basically planar 2,5 dimensional
processes, with relative accuracies of the order of 10-1 to 10-3, directed toward silicon
and silicon-like materials. In contrast, many emerging miniaturization applications
require wide range of materials for parts that have three-dimensional free-form
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
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surfaces with relative accuracies in the 10-3 to 10-5 range. Processes such as
machining and others discussed in this chapter can meet these needs, but there are a
number of scientific and technological challenges that must be overcome.
Relation to existing technologies
100
10-3
CONVENTIONAL & ULTRA –
PRECISION MANUFACTURING
Tolerance ∼ Object size
Object Size [mm]
103
MICRO/MESO-SCALE
MANUFACTURING
MEMS
NEMS
10-6
Object < Atom size
-9
10
100
10-1
10-2
10-3
10-4
10-5
10-6
Relative Accuracy (Feature Tolerance/Object Size)
Fig. 1: Micro-manufacturing domain [1]
1.2 Materials for Micro-manufacturing
Current manufacturing techniques are producing three-dimensional
microstructures by shaping or structuring various materials like metals, silicon,
plastics, etc. The choice of material is determined by the function and conditions of
micro-component application and also by the material ability to be manufactured.
Products of micro-fabrication have size scales ranging from about thousand
micrometers down to hundreds of nanometers. Material structure and the size scale
impacts mechanical properties of final product. Therefore, the development of unique
materials for micro-manufacturing would be needed. The economic driving force for
the development of new materials can be the demand for significant amount of
material quantities in macro-manufacturing, but in micro-production, where the
volume production is measured only in cubic centimetres, this does not exist [1].
Most of materials are characterized by presence of grains with size ranging
from a few micrometers to approximately 100µm. As the part dimensions decreases
bellow this range, grains must either be removed or refined. If removed, the resulting
part becomes a single crystal with, for the most of materials, anisotropic mechanical
and physical properties. In the second case, reduction in grain size results in
significant property changes.
For polymers, precise modeling of the specific volume of material is required for
prediction of the dimensional stability and accuracy of injection-molded products.
A detailed survey on different materials could be found in [1].
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1.3 Micro-manufacturing Processes
Manufacturing process converts raw material into desired part. The
manufacturing chain does not include just fabrication processes, but also assembling,
handling, packaging and quality control of the products.
At present, there are many different technologies that can be used for
fabrication of microstructures. This chapter attempts to give an overview of existing
main technologies for the fabrication of truly 3D microcomponents, which are required
to develop functional micromechanisms.
The technologies presented here are categorized into four main types –
subtractive, near-net-shape, additive and a special class – LIGA processes. Some of
the processes are downscaled versions of existing traditional manufacturing
technologies, and others are innovative methods using various physical and chemical
effects [1].
1.3.1 Subtractive processes
The material subtractive processes include energy-beam micromachining
(laser, electron, focus ion beam), electro-physical and chemical processes (electrochemical machining – ECM, and electro-discharge machining – EDM), ultrasonic
micromachining, and mechanical micromachining (turning, milling, drilling, grinding).
Energy-beam Machining
Laser milling is a new manufacturing technology suitable for machining a wide
range of materials. The beam from a pulsed or continuous laser source can be
focused on a solid material to cause sufficient heating to give surface evaporation –
such a process is called ablation and is particularly appropriate for hard materials that
can not be machined by conventional techniques [3]. Laser beams are also used to
join components. The types of lasers currently being used in micro-manufacturing
include excimer lasers, diode lasers, copper vapour lasers, solid state lasers, and
CO2 lasers.
Ion and electron beam machining processes are able to produce fine structures
with extremely fine details, but the removal rates of these technologies are very low.
Electro-Chemical Machining (ECM)
Micro ECM is accurate, highly repeatable process with rapid machining times
that produce final shape of the surface by controlled dissolution. The tool never
touches the workpiece and is not consumed in the process [1].
Electro-Discharge Machining (EDM)
Electro-discharge machining is based on the erosion of the material to be
machined by means of a controlled electric discharge between an electrode and the
material. The machinable materials are usually electrically conductive, but also
semiconductor materials, such as silicon, can be machined.
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Precision microholes can be fabricated with high precision accuracy (radial
deviation of 0.5µm or less) and with surface roughness smaller than 0.1µm. Holes
can usually be machined to a depth of 3 to 5 times that of the hole diameter [4].
In order to drill microholes, microelectrodes must be fabricated first. This
process is carried out using a wire as a secondary electrode and reversing the
polarity (this method is known as wire electrode discharge grinding – WEDG). The
wire travels along a wire guide and moves to the position which determines the
required diameter of the shaft. The principle of WEDG is used also in ultrasonic
machining – see the Fig. 3. With this method shafts as thin as 5µm in diameter can be
machined. Shafts can be machined to lengths up to 10 to 15 times their diameter.
Micro EDM makes it possible to machine more complex shapes by
implementing multi-axis motion mechanisms for the electrodes.
Ultrasonic Machining
Micro ultrasonic machining (micro-USM) is able to make almost any three
dimensional microstructure with high aspect ratio on most of materials – particularly
on brittle materials. Ultrasonic machining uses a tool ultrasonic vibration with
combination of favourable abrasive slurry to create accurate cavities of any shape
through the impact grinding of fine grains. The machining process is nonthermal,
nonchemical, nonelectrical and thus produces high quality surface finish.
However, the conventional USM is not capable for instance of drilling of
microholes smaller than 100µm for lack of corresponding co-axial microtools. To this
point a new machine exploiting WEDG, EDM and USM functions itself has been
developed [5].Such a micromachining system is shown in Fig. 2.
Fig. 2: Configuration of micromanufacturing system. A: processing circuits for
WEDG/EDM; B: driving and positioning unit; C: electronic oscillation generator; D:
electronic weight display; M1, M2 and M3: motors for x-, y and z-axes movement; M4:
motor for c-axis rotation; T: transducer; W: WEDG unit; AEU: assembly/EDM/USM
module; F: force sensor; h: horn; t: microtool. [5]
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The micro-USM is mainly divided into two stages as shown in Fig. 3. The
WEDG/EDM combination is used to generate co-axial microtools first with which
micro-USM of brittle materials is carried out. In this way, microholes as small as 15µm
in diameter can be fabricated.
Fig. 3: Micro-USM procedure [5]
Mechanical Machining
These processes, where the tools are in direct contact with workpiece, are able
to machine 2D and 3D microstructures in a variety of materials with higher removal
rate than the others.
Micromachining processes as grinding, drilling, turning and milling are
mechanical processes, which are not just downscaled versions of the existing
convenctional processes – as showed the study [6]. For example, the reduced tool
stiffness of micro drills caused by changes of geometry requires an adjustment of the
complete drilling process, or that the kinematics of grinding process, as well as the
process strategy, while grinding brittle materials with diamond tools, has a significant
influence on the machined surface quality and chipping of the edges.
This thesis is focused on the micromilling area, therefore more detailed aspects
of micromilling process will be discussed in the following chapter: 1.5 Micromilling
1.3.2 Near-net-shape processes
In the near-net-shape processes replication is realized by mechanical force,
solidification or polymerisation. Casting is not discussed.
Micro-moulding processes
Micro moulding of thermoplastic polymers is one of the most promising
fabrication techniques for micro devices. Moreover, thermoplastic materials are a very
large class, which allows one to find a suitable polymer for nearly every application.
There are five processes which are employed for micro moulding of thermoplastic
polymers [7]: Injection moulding, reaction injection moulding, hot embossing, injection
compression moulding and thermoforming. The most common are injection moulding
and embossing processes. Microinjection moulding is suitable for medium and large
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scaled production, whereas hot embossing shows advantages in small series
fabrication.
Injection moulding is the process of forcing melted polymer into a mould cavity.
The mould cavity equipped with a micro structured tool (mould insert) is heated up,
after injection the viscous plastic is cooled down, the part can be ejected.
On the other hand in hot embossing process a micro-structured tool with
thermoplastic material is inserted into the moulding machine (the working chamber
can be evacuated) and pressed with high force. The mould insert is filled by the
plastic material which replicates the microstructures in detail. The setup is cooled
down and the plastic part is withdrawn out.
Fig. 4: Hot embossing process [8]
Micro-forming processes
Micro-forming is an appropriate technology for use in manufacturing microparts,
in particular for bulk production. As with material-removal processes, micro-forming,
which is based on plastic deformation, requires a research and development. Flow
stress, anisotropy, ductility and forming limit, forming forces, spring back, and
tribology are some of many factors that need further study for increasing the
performance of micro-forming processes [1]. Micro-extrusion is both fast and suitable
for mass production. It is a bulk deformation process that produces less waste as
compared to machining.
1.3.3 Additive processes
To create 3D structures, additive processes usually involve depositing of the
material in different layers or using a low-energy laser to crosslink a thin layer of liquid
photosensitive polymer resin.
Micro-SLA (micro-stereolithography)
A real 3D micro-stereolithography process called the “integrated harden
polymer process” was first developed by Professor Ikuta’s group at Nagoya University
in Japan in 1992. An extremely thin (5um in 1992 to 200nm today) layer could be
created by this process, where the polymer is cured below the polymer/atmosphere
interface by an ultraviolet beam.
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Integral micro-SLA process, in which a complete layer can be built using a
single light exposure, has been realized at the Swiss Federal Institute of Technology
in Lausanne (EPFL). The principle of integral micro-SLA apparatus is described in
Fig. 5. Slices of a 3D computer-aided design (CAD) model are converted into bitmap
files and used to drive a dynamic pattern generator, which in turn shapes a light
beam. The beam is focused on the surface of a photopolymerizable liquid, which
results in a selective solidification of the irradiated areas and creates a thin layer of
polymer of the required shape. A complete layer is built using a single exposure. A
typical exposure time of one second per layer is needed with this setup. Typical
fabrication speeds of 1–1.5 mm per hour can be obtained in the vertical direction,
which corresponds to the superimposition of 200 to 300 layers per hour [1].
Fig. 5: Integral micro-SLA principle and products [1]
Metal Jet
In contrast to various additive processes developed in recent years, where the
material is limited to silicon or plastics, the authors of [9] developed a method to form
micro models by a metal jet. This method is similar to an ink jet printing system. A thin
layer is first formed by molten metal drops ejected through a nozzle, and a second
layer is formed on the first layer. The repetition of this process produces threedimensional models. The diameters of hemispherical metal dots adhered on a base
plate were about 400µm.
Fig. 6: The principle of modelling by metal jet nozzle [9]
1.3.4 LIGA
This technology (named LIGA from the German words ‘Lithographie,
Galvanoformung, Abformung’ meaning Lithography, Electroplating, and Moulding in
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English) has been developed for fabricating highly 3D microstructures with submicron
tolerances from metals, metal alloys, polymers and ceramic materials.
The first step of the process is always an x-ray or UV lithography generating a
microstructure with high depth-to-width aspect ratio. The following steps depend on
the specifications of the product and are usually microelectroforming and
micromoulding processes which use the lithographed microstructures as a threedimensional template [4]. Ceramic microstructures can also be manufactured by filling
the plastic mould with ceramic slurry. The plastic is then removed during the sintering
process of the ceramic.
Fig. 7: LIGA principle (ankaweb.fzk.de)
1.4 Examples of present-day microproducts
After previous short introduction of key technologies for the fabrication of
microproducts, several examples of present day application are presented in this
chapter.
1.4.1 Automotive applications
Due to the large number of units needed, automotive applications are expected
to be one of the biggest market segments of microsystem technology. The available
space in cars decreases, whereas the number of additional functions is growing
rapidly because of more stringent safety, environmental and economic demands. To
overcome this difficulty, microsystem technology has to be applied instead of the
conventional techniques [10].
An example is a micromechanical gyroscope needed in active chassis
development as well as for inertial navigation systems. Fig. 8 shows such a
gyroscope that measures the angular rate with resolution of 0,002degs-1 at a range of
100degs-1. Due to the Coriolis force a rotation of the sensor about its longitudinal axis
causes a vibration of the sensing tuning fork with amplitude proportional to the
angular rate [11].
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Fig. 8: Schematic drawing of a quartz micromechanical gyroscope [11]
Another key component is the monolithic accelerometer (Fig. 9), which has
been developed for airbag release and other automotive applications.
Fig. 9: Schematic drawing of a surface – micromachined accelerometer [10]
1.4.2 Microfluidic devices and systems
An attractive field is the use of microsystems for chemical analysis and for
accurate delivery of small amounts of liquids and gases. The possible application of
this technology is in industrial process control, biotechnology, environmental control,
medical applications [10] and we can find it even in the office. Microfluidic devices are
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used for example as inkjet printing heads. Another big application area for
microluidics is called Lab-on-a-chip systems and includes [12]:
•
•
•
•
•
•
DNA analysis and genomics,
microreactors,
cell based systems,
clinical diagnostics,
liquid chromatography,
etc.
Fig. 10: The processes of both traditional chemical labs and lab-on-a-chip devices
The benefits of using microfluidic devices for laboratory applications include [12]:
•
•
•
•
•
•
•
•
•
•
reduction in sample volume and reagent usage,
improved resolution of separations,
ability to run reaction and analysis processes faster,
ability to run processes in parallel,
improved control of mixing and heating of fluids,
rapid mass transfer as a result of high surface area to volume ratios
Improved integration of process steps, for example reactions and
separations,
development of new and improved detection methods,
simpler and cheaper disposable devices,
access to a wide range of fluidic geometries.
SMOCA project
An example of prototype development of a microfluidic system is the SMOCA
project carried out at DTU. During the three-week course “Micro Mechanical Systems
Design and Manufacture” the complete product, from design idea to the making of a
functional prototype, was developed. The task was to manufacture a micro size coffee
maker as the product.
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Fig. 11: Final design of the SMOCA USB coffee maker on the left and the disposable
device after successful test on the right
The coffee maker was considered as a device, where boiling water and steam
(heated up inside the device) runs or is pressed through grounded coffee grains and
then filtered into a separate container for the finished coffee. Just heating up of
finished cold coffee was not considered as a proper solution. From a mechanical
point of view there were requirements for the temperature/heating of water, pressure
in the system, filtering and for the materials used in the device. The product ended up
being a combination of a reusable USB power-unit and a disposable coffee-making
unit. This design is easy to manufacture and it is possible to automate after some
minor adjustments of the design.
Fig. 12: Process chain
Several micro-manufacturing techniques have been used in the production of
the prototype. The device was laser-welded from two plastic parts (the bulk and the
lid), which were manufactured by hot-embossing. Heating element in form of resistor
with aluminium foil was implemented in the lid. Aluminium moulds for hot-embossing
process were successfully produced by micromilling using 1mm and 2mm end mills.
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Considering that the complete design of the device and its fabrication including
several processes was done in three weeks proved used technologies to be very fast,
versatile and relatively easy to apply.
1.4.3 Smart Shoe
The main part of the new ‘smart’ Adidas running shoe [13] is a simple but very
attractive sensor system. The system consists of a microcontroller unit, a motorcontrolled lead screw, a Hall sensor and a specially designed elastic cushioning
element. The purpose of such a design was to allow the shoe to adapt to the new
conditions, for example when a runner leaves the road and goes onto a trail or grass.
The system maintains the desired cushioning level by measuring the compression of
the shoe and automatically adjusting it on the fly.
The main task was to figure out a way to accurately measure the amount of
compression in the midsole cushioning – virtually in real time. Just 25ms from the
instant a runner’s foot hits the ground, the cushioning will achieve maximum
compression. A Hall sensor on top of the cushioning and a magnet located below are
used. The voltage output from this contactless sensing device accurately tracks the
changes in magnetic flux density based on its distance from the magnet. The sensor
is not only able to measure the distance that the cushion element has compressed, it
also measures the time to achieve full compression – allowing the software to identify
particular surface conditions. To adjust compressive forces on the cushioning
element, a motor drives a lead screw that expands or contracts a cable. This alters
the space available for the movement of the cushioning element.
Fig. 13: Anatomy of a Smart Shoe [13]
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 20
1.5 Micromilling
Micro-milling is one of the mechanical cutting processes, which is very flexible
for machining three-dimensional geometries. Even though the mechanical micromachining process may not be capable of obtaining the smallest feature sizes like in
stereolitography, it is very important in bridging the macro and micro domains for
making functional components [14]. Masuzawa (2000) [16] concluded that the size of
micro-features currently being requested is 100µm, with micro-machining technology
in the research stage being able to create 5µm features. In the near future, requested
feature sizes are expected to be reduced to 50µm, with a research capability of 2µm
features.
The motivation for micro-mechanical cutting stems from translation of the
knowledge obtained from macro-machining. There are a number of issues in
microscale machining, that are different from macroscale machining, influencing the
mechanism of the process and resulting in changes in the chip-formation process,
cutting forces, vibrations and process stability, and the generation and subsequent
character of the resulting machined surface [2].
Several issues concerning micromilling process, machine tools, and cutting
tools are discussed in this chapter.
1.5.1 Cutting Tools
The surface quality and feature size of the microstructures are depended on
precision cutting tools and machine tools. Tungsten carbide cutting tools are generally
used due to their hardness over a broad range of temperatures. Diamond tools have
a limited ability to machine ferrous materials, because the high chemical affinity
between diamond and ferrous materials.
Commercially available micro-end mills with the flutes fabricated by grinding can
be as small as 50µm. Micro-tools of less than 50µm need a zero helix angle to
improve their rigidity and to reduce the limitations of fabrication techniques [14].
Onikura at al. produced an 11µm diameter micro-carbide tool by ultrasonic vibration
grinding. Sandia Labs has developed a 25µm diameter tool with five cutting edges
using focused ion beam machining. Fang et al. investigated various tool geometries
and they found that semi-circular-based mills are better than triangular of the
conventional two fluted end-mills. They also concluded that when there is zero helix
angle on the tool, poor chip evacuation may result in poor surface finish.
Fig. 14: The first three photos show end mills produced in Sandia’s Lab, the other two
photos show 18-micron-wide turning tool [17]
1.5.2 Machine Tools
Properties of the machine tools, including their overall accuracy and their
dynamic performance, have influence on the size and quality of micro-products. Ultraprecision machine tools and micro-machine tools are used for production of micro
components. Ultra-precision machine tools have the advantage of high rigidity,
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 21
damping and precision control by sensors and actuators comparing to micro-machine
tools. On the other hand, the large machine tools and precisely controlled machining
environment may add very high costs for the fabrication of miniature components
[14].
Precision machine tools
Since the tool diameter is very small in micro-machining applications, the
rotation speed of the spindle should be very high to maintain productivity and reach
required cutting speed. Electric motors with hybrid-angular contact bearings able to
run 80000rpm are used, when the torque requirements are high. Air bearing spindles
with air turbines produce very low torque, but they are able to exceed 200000rpm
[14].
Linear drive motors and a control system are commonly used in ultra-precision
machine tools in order to achieve high accuracies of fabricated structures. The typical
accuracy of machine tool using linear drive systems is usually ±1µm.
Micro-machine tools
Machine tools gain several benefits from their miniaturization such as reduction
in energy, materials and space. As they require smaller amounts of materials more
expensive materials with better mechanical properties can be used for the
construction. Due to smaller mass comparing to conventional macro-machines,
micro-machines have higher natural frequencies implicating a wide range of spindle
speeds to fabricate components without regenerative chatter instability [14].
The portability of such system is beneficial – the small size of the machine
allows for their deployment to any place. As the micro-machine tools can be exited by
external disturbances, it requires vibration isolation to achieve desired tolerances.
Another challenge is the development of accurate sensors and actuators, which must
be small enough for implementation within the machines. Micro-machines usually use
piezoelectric or voice coil actuators in order to achieve sub-micrometer accuracies.
They use high-speed air bearing spindles [14].
1.5.3 Micro-cutting process
The principles of micro-machining are similar to those of conventional cutting
operations, but some characteristics are different due to the significant size reduction.
Structure dimensions
In [6] experiments were carried out using two edged end mills made of fine
grain tungsten carbide and single edged mils made of monocrystalline diamond. The
work was focused on the achievable minimal structure dimensions and the effect of
tool wear.
With the tungsten carbide end mills, walls with a minimum thickness of 18µm
and an aspect ratio of 1:24 could be machined without any damages (Fig. 15). The
stiffness of the walls was increased by the curves of the structure. When milling
straight-lined structures, a minimum wall thickness of 30µm could be attained. A feed
rate smaller than 1µm per tooth was not suitable (the tools did not cut the material
accurately anymore). With monocrystalline diamond tools, walls with a thickness of
only 8µm and an aspect ratio of 1:13 could be machined with very small feed rates of
about 0,5µm per tooth.
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
material: CuZn39Pb2 (Ms58)
tool:
milling cutter ∅0,5 mm VHM
coolant: emulsion 3%
st = 0,002 mm/tooth
Page 22
CuZn39Pb2 (Ms58)
milling cutter ∅0,27 mm MKD
emulsion 3%
st = 0,0005 mm/tooth
Fig. 15: Machining with tungsten carbide and diamond tools – possible structure
dimensions [6]
Tool wear
The wear of the micro-mills in the area of the tool edges leads to a change in
the shape of machined microstructure.
process: micro milling
f = 600 mm/min
ap = 0,5 mm
ae = 0,05 mm
Tool edge radius Re
material: AlCu4MgSi
tool:
micro end mill ∅0,5 mm
two-edged cutter, HM
coolant: emulsion 3%
Fig. 16: Wear of the tool edge during micro-milling of aluminium [6]
Jan Slunský
Enhancement and verification of a cutting
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Page 23
During micro-milling of aluminium [6] the radius of the tool edge increased fast
from Re=1µm at the beginning up to 8µm after a cutting distance of 4m. After this, the
wear increased slower to a value of about 12µm after a cutting distance of 14m. Fig.
16 shows, that the tool wear does not have a significant influence on the surface
roughness.
Chip formation
Chip formation is a dynamic process that is nonlinear in nature. Understanding
micro-chip formation is important in an accurate prediction of cutting forces. In macromilling, the feed per tooth (i.e. uncut chip thickness) is generally larger than the
cutting tool edge radius. Therefore, macro-chip formation models are based on the
assumption that the edge is perfectly sharp and the cutting tools completely remove
the surface of the workpiece and generate the chips. In contrast to the conventional
sharp-edge cutting model, in micro-cutting operations the edge radius of the tool is
often larger than the chip thickness. The small depth of cut due to the small feed rate
and relatively large edge radius of the tool cause a large negative rake angle. This
phenomenon causes ploughing, a rough surface and elastic recovery of the
workpiece [14].
Fig. 17 explains the chip formation with respect to chip thickness. When the
uncut chip thickness is less than a critical minimum chip thickness elastic deformation
occurs and the cutter does not remove any workpiece material (a). As the uncut chip
thickness approaches to minimum chip thickness, chips are formed by shearing of the
workpiece, with some elastic deformation still occurring. As a result, the removed
depth of the workpiece is less than the desired depth (b). When the uncut chip
thickness increases beyond the minimum chip thickness, the elastic deformation
phenomena decreases significantly and the entire depth of cut is removed as a chip
(c).
h < hmin
h ≅ hmin
h > hmin
Fig. 17: The effect of the minimum chip thickness (Re – cutting edge radius; h – uncut
(undeformed) chip thickness; hmin – minimum chip thickness) [14]
The minimum chip thickness depends on the relationship between the tool
radius, the material of the workpiece and the cutting fluid. Son at al. [15] determined
the minimum chip thickness as a function of tool edge radius and the friction
coefficient. The analytical formulation of the minimum chip thickness can be
approximated as

 π β 
h min = R e ⋅  1 − cos −  
 4 2 

(1)
In [14] other approaches for estimating minimum chip thickness are discussed.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 24
1.6 The need for cutting force prediction
The motivation for high-speed micro-machining is the reduction of production
time in creating complex 3D shapes by maximizing material removal rates. It is
important to understand the relationship between tool and workpiece in order to
produce desired micro-component. One of the biggest challenges in micro-cutting
operations is maintaining machining forces below a critical limit to prevent excessive
wear, breakage and deflection of the tool and thereby maintain the desired accuracy
at the machined features while improving productivity.
Because micro-end mills are very small tools, it is very difficult to detect damage
to cutting edges and broken shafts during cutting process. Cutting force measurement
is the most effective method for monitoring tool conditions since it provides higher
signal-to-noise ratios [14]. If the tool is broken, it is nearly impossible to restart an
interrupted machining process by aligning the tool to the workpiece, because the
feature sizes are very small. Also, special instruments may be needed to observe
damage to a workpiece by a broken tool, if any.
Well developed cutting force model with precise force prediction is able to help
operators choose the right cutting parameters, the calculated deflection of the tool
can be used for modification of the tool path in cutting operation in order to increase
the accuracy of machined part and the tool breakage can also be preceded.
1.7 Aim of the work
An overview of the present technologies for fabricating micro-components with
a more detailed description of micro-milling process was given in the previous
paragraphs. It follows from the previous paragraphs that the cutting force prediction in
micro-cutting processes is a very important issue. A model for cutting force prediction
in micromilling is reported in [18]. Its prediction capabilities were verified through
upscaled experiments due to the unavailability of a device for cutting force detection
in micromilling. Furthermore, in such a model the calculation of the force acting on the
tool does not take into account the tool run out and the deviations from the Stabler
rule. Therefore, building on that work the aim of this Thesis is the improvement of the
model by removing the assumption and to verify it through comparison with cutting
force measurements.
1.8 Organization of the work
The thesis is organized in 6 chapters. The present chapter is an introduction to
the subject. Models for cutting processes are discussed in the following chapter.
Chapter 3 presents the mathematical model for cutting force prediction in micro end
milling operations. Chapter 4 deals with turning experiments for obtaining the material
data for the mathematical model and chapter 5 is dedicated to milling experiments for
later verification of predicted forces. Data from turning experiments are analysed and
subsequently implemented into the model in chapter 6. Finally, chapter 7 presents the
results of experimental work with conclusions and further suggestions.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Chapter
2
Models for Cutting Processes
Page 25
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 26
2.1 Overview on historical development of models for
machining processes
There are different objects of modelling in machining, for instance tool life,
power, forces, temperatures etc. An overview given here is general and adopted from
[19], while the following subchapter is focused on modelling of cutting forces.
The machining process came into use in industry in the late 1700’s at the very
beginning of the Industrial Revolution. For another 200 years there were no
mathematical expressions (models) able to describe the mechanics of the machining
process. Such a understanding of process is essential for determining right machining
parameters (such as cutting speed, feed rate, depth of cut etc.) maintaining high
productivity and cost-effectivity of the machining process in practice.
Finally, during the 20th Century, such models have been developed in three
main stages:
1. Empirical modelling – beginning in the early 1900s.
2. Science based modelling – beginning in the 1940s.
3. Computer-based modelling – beginning in the 1970s.
Each of these three stages was introduced by a key event. The first one was
connected with Taylor’s research and development of the method for estimating tool
life in machining. The second stage started with Merchant’s mechanics-based
analysis of the forces acting between workpiece, cutting tool and chip. The last stage
was initiated by development of digital computer technology and its application. All
three stages coexist and synergize together today.
From today’s point of view the research and modelling of machining is very
important to the industry for its economic impact. Machining is by far the most widely
used machine-performed process for manufacturing of mechanical parts in industrial
countries. The cost of machining amounts more than 15 percent of the total value of
all products produced by their entire manufacturing industry.
Empirical modelling
Sometimes it is difficult or impossible to develop a mathematical model that
explains a situation. However, if data exists, we can often extract an empirical model,
which consists of a function that fits the data. The graph of the function goes through
the data points approximately. We can use such a model to predict behaviour inside
explored range where data do not exist, but we can not use it to explain a system. It
means the model is not truly predictive and is limited to conditions of experiments
where the range of parameters to obtain the original data was used.
The person who initiated successful empirical modelling of the machining
process was F W Taylor, as he launched a whole-factory research programme in
1880. The programme was carried out at the Midvale Steel Works company and
lasted 26 years. The primary aim of his work was to establish models that could help
choosing proper tool with cutting speed and feed in machining operations. He
produced a whole series of empirical models, where the best known and still used
today is the equation for tool life:
v c ⋅ Tn = c
where T represents a tool life, vc stays for cutting speed, n and c are constants.
(2)
Enhancement and verification of a cutting
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Jan Slunský
Page 27
Since the publication of Taylor’s work, interest in the development of empirical
modelling of the machining processes strongly increased. The development
continued in almost the whole industry and also spread to universities.
Then, the science-based modelling as a new approach to the modelling of
machining began to evolve in 1940’s.
Science based modelling
Science-based modelling of machining uses natural sciences (especially the
science of physics) to establish predictive models. These models are independent of
empirical information and can be used for engineering calculations of machining
process characteristics. Development of such models depends on the knowledge and
understanding of investigated process and therefore on the empirical modelling.
Research by Ernst and Merchant at Cincinnati Milling Company is a good example.
Ernst was interested in the process of chip formation – the mechanism of
removing metal from a workpiece by a cutting tool. To investigate this, he studied the
action of chip formation during cutting using microscope, taking high-speed motion
pictures and making photomicrographs of sections through chips still attached to
workpieces. As a result of such empirical research he came to the concept of the
“shear plane” in chip formation.
As Merchant joined Ernst’s staff, he was asked to undertake research on the
mechanism of chip formation and particularly on the mechanism of the sliding friction
between the chip and the cutting tool in chip formation. He applied the science of the
mechanics of solid bodies to the “shear plane” concept and arrived at the model of
the equilibrium force system acting in the chip-tool-workpiece system shown in Fig.
18.
hc
α
FS
h
φ
FP
β-α
NS
FQ
R
FC
β
NC
α
Fig. 18: Condensed Merchant’s circle diagram showing relationships between various
components of the forces [20].
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 28
By mathematical description of such a system it became the first sciencebased, predictive model. This model allows calculation of the friction force acting
between chip and tool, shear stress at shear plane etc., starting from the knowledge
of the total force acting on the tool (or workpiece).
Publication of this showed a new approach of the machining process – the
science-based, predictive modelling. This discovery initiated a new era in metal
cutting research.
Computer-based modelling
Introduction of digital computer technology had tremendous impact in all sectors
and fields of human activities. In machining, this technology showed its potential by
application to digital control of machine tools in the 1960’s. Later it proved to be a
potent tool for modelling with its powerful capability for simulation of processes, while
it combines both empirical and science-based models.
As the computing power of the technique increased, it could create dynamic
models enabling simulation of the actual processes in machining operations.
Integration of these dynamic models with all of the rest of systems acting in
machining and full online access to the manufacturing system database greatly
enhanced the accuracy and the speed of modelling of machining operations.
Such a complex integration enabled the development of fully automated
systems of machine tools (known as “flexible manufacturing systems”), which are
able to run autonomously by avoiding or correcting process errors or failures.
Today, the synergy of the three types of modelling has resulted in rapid
development of modelling capability, which any one of the three types would be
capable of on its own.
2.2 Models for cutting force prediction
The need and importance of reliable predictive models for optimizing the
economic performance of machining operations, as well as for design of machine
tools and cutting tools that enhance this economic performance has long been
recognized by researches and engineers in industry. For this purposes, reliable
quantitative predictions of the forces, power, chip control, tool-life, and surface
roughness, which are highly dependent on the cutting conditions, are required.
For over a hundred years a wide variety of modelling approaches have been
developed for scientific description of the cutting process as well as for predictive
purposes. Such attempts have been greatly aided by the introduction of computer
technology in more recent years. A number of promising predictive performance
models based on “empirical” and “mechanics of cutting” approaches have been
identified [22].
2.2.1 Classification of cutting force models
A large number of cutting force models is available in the literature, but it is not
a purpose of this work to review the single models – only a broad classification is
given.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 29
Empirical approach
Empirical models for cutting force prediction correspond to the empirical
modelling described in previous chapter. This approach has often been used to
establish tool-life, forces, and power equations as a function of the operation
variables such as the feed, speed, and depth of cut for the major machining
operations. In these models the cutting force components are usually calculated from
the cutting force coefficients and the chip load. The cutting force coefficients are
established from experimental tests for a given machining operation, tool geometry
and workpiece material combination.
Since the empirical approach involves considerable testing with a high number
of variables to be considered, which is time consuming and expensive, there has
been a tendency to develop alternative methods. Such a new method can be the
mechanics of cutting approach.
Mechanics of cutting approach
These models are based on the “shear plane” concept and belong to the
section of science-based modelling in previous chapter. The cutting forces are
calculated from stresses and strains within the work material and at the tool chip
interface which are dependent on the geometrical configuration of the cutting tool and
workpiece material properties. “Unified Mechanics of Cutting” approach, proposed by
Armarego [21], for modelling the forces in cutting operations is one of the promising
methods and will be further described in the following chapter.
Numerical modelling
The Finite Element Method is the numerical method used for simulation of
machining operations. Although the simulations are able to produce local time varying
stresses, strains, temperatures and other parameters very precisely, a complete
description of the work material constitutive behaviour and the choice of friction laws
is required. Unfortunately the material constitutive equations for the range of
parameters occurring in cutting are not available.
2.2.2 Unified Mechanics of Cutting approach
This approach, incorporating “edge forces”, is based on the realization that a
number of the conventional practical machining operations such as drilling, milling,
and turning involve cutting on a single major cutting edge with wide but thin areas of
cut akin to those for the “classical” orthogonal and oblique cutting operations [21]
shown in Fig. 19. Except the “classical” orthogonal and oblique cutting operations,
where the forces act on a single cutting edge, there are also “generalized” orthogonal
and oblique cutting operations involving cutting on a multi edge.
In the first stage of development, predictive mechanics of cutting analyses for
the “classical” orthogonal and oblique cutting operations have been developed and
experimentally verified [22]. Extensive experimental testing has shown a new process
– rubbing and ploughing phenomena at the cutting edge resulting in “edge” forces in
addition to the forces due to shearing and tool-chip friction. Then, the both “classical”
operations have been modified and represented by the “cutting” force components
due to the shearing and friction processes FPc, FQc and FRc plus the “edge” force
components due to rubbing and ploughing processes FPe, FQe and FRe.
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
vch
Chip
α
Page 30
αn
Tool
vch
Pn ≡ PnG
Pn ≡ PnG
ηc Chip flow
angle
Chip
i
h
Tool
FR
FP
Inclination
angle
Workpiece
FQ
i
h
FP
90°
FQ
b
b
Workpiece
Fig. 19: “Classical” orthogonal and oblique cutting processes [21]
According to Armarego, the total force components FP, FQ and FR are therefore
represented by these equations:
FP = FPc + FPe =
τ ⋅ A  cos(βn − α n ) + tan i ⋅ tan ηc ⋅ sin βn
sin φ  cos 2 (φ + β − α ) + tan 2 η ⋅ sin2 β
n
n
n
c
n

FQ = FQc + FQe =

sin(βn − α n )
τ⋅A


2
sin φ ⋅ cos i cos (φ + β − α ) + tan 2 η ⋅ sin 2 β
n
n
n
c
n

FR = FRc + FRe =
τ ⋅ A  cos(βn − α n ) ⋅ tan i + tan ηc ⋅ sin βn
sin φ  cos 2 (φ + β − α ) + tan 2 η ⋅ sin 2 β
n
n
n
c
n


+F
 Pe


+F
 Qe

(3)

+F
 Re

where A is the area of cut given by the uncut chip thickness h and width of cut b:
A = h⋅b
(4)
and the edge force components are given by
FPe = C Pe ⋅ cos i ⋅ b
FQe = C Qe ⋅ b
(5)
FRe = C Pe ⋅ sin i ⋅ b
where CPe and CQe are the edge forces per unit active cutting edge length.
Based on the oblique cutting deformation geometry and the “collinearity
conditions” between the shear force and shear velocity in the shear zone as well as
the friction force and chip velocity in the rake face the following relationships apply:
tan φ n =
rt ⋅ cos α n
1 − rt ⋅ sin α n
(6)
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
tan β n = tan β ⋅ cos η c
tan(φ n + β n ) =
tan i ⋅ cos α n
tan η c − sin α n ⋅ tan i
Page 31
(7)
(8)
In order to quantitatively predict the forces it is necessary to know the cutting
conditions (b, h), the tool geometry (i, αn), the basic cutting quantities rt, β and τ as
well as the edge force coefficients for the given tool-workpiece combination.
It was shown, that the edge force coefficients and the basic cutting quantities rt,
β and τ are not affected by inclination angle i, and therefore they could be obtained
from the simpler orthogonal cutting tests. As a consequence the database found from
the tests can be considered as a “generic” and together with the mechanics of cutting
analyses for the variety of machining operations such as turning, milling and drilling is
able to form predictive cutting models without the need to run any tests for these
operations.
In the following chapter a method for predicting the milling force components in
three Cartesian directions from the unified mechanics of cutting approach is
described. Improvements to the model, consisting of removing the Stabler rule and
considering tool run-out, are implemented.
“Stabler rule or chip flow law” assumes that the chip flow angle is equal to the
angle of obliquity. In terms of milling operations it means the chip flow angle ηc is
approximately equal to the inclination angle (helix angle of the tool) i for variety of tool
and work materials, rake angles, and speeds.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Chapter
Page 32
3
Mathematical Model for End-milling
Operations through the Unified Mechanics
of Cutting Approach
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 33
Traditionally the forces in milling operations have been established by empirical
approaches. In this way, the coefficients for relating the force to chip load are
established from milling tests for a given cutter geometry and tool-workpiece material
combination. A prohibitive amount of testing is necessary as each cutter geometry
and material combination require a set of milling tests.
An alternative “unified mechanics of cutting” approach is based on the modified
mechanics of cutting analysis, incorporating the edge forces. This approach
mathematically relates the milling operation to the “classical” oblique cutting. The
values of basic cutting quantities are obtained from “classical” orthogonal cutting tests
[22]. The general description of this approach has been done in the previous chapter
and its application in micro end-milling operation, based on Armarego’s proposal, will
be introduced here. In the first step the system is considered as ideal, which means a
cutter is rigid with zero eccentricity in the cutter axis of rotation. The improvements of
the model will follow.
3.1 End-Milling Force Model
The strategy for predicting the instantaneous force components is based on thin
shear zone mechanics of cutting analyses and orthogonal cutting databases. The
elemental force components are obtained from the classical oblique cutting analysis
by partitioning the cutter teeth into a series of axial elemental oblique cutting tools.
The fundamental tool geometry, cut thickness and width of cut is determined for each
tooth element actively engaged in cutting. The elemental components are resolved in
the practical milling force directions and the force component contributions of all the
elements on each tooth are summed.
Down-milling operation
y
Engagemen
t
x
y
z
y
x
x
(a)
(b)
Fig. 20: Geometry of milling process (a), components of milling forces (b) [23]
The general geometry of the end milling operation and coordinate system of
axes is shown in Fig. 20 (a). The coordinate system corresponds to the coordinate
system of machine tool used later in micro-milling experiments. The elemental
tangential, radial, and axial cutting forces acting on flute j of an “ideal” system (no tool
deflections and run out) are shown in Fig. 20 (b) and given by:
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
[
]
[
]
Page 34
dFtj (θ, z ) = K te + K tc ⋅ t j (θ, z ) ⋅ dz
dFrj (θ, z ) = K te + K rc ⋅ t j (θ, z ) ⋅ dz
[
(9)
]
dFaj (θ, z ) = K ae + K ac ⋅ t j (θ, z ) ⋅ dz
where hj(θ,z) is uncut chip thickness:
t j (θ, z ) = s t ⋅ sin θ
(10)
θ is the cutter orientation (or immersion) angle measured clockwise from the negative
x axis to a reference flute j = 0 and st is the feed per tooth. On flute j, a differential
cutting edge element at the axial location z has an immersion angle θj(z):
θ j (z ) = θ + j ⋅ θ p − k i ⋅ z
(11)
where θp is the flute angular pitch given on cutter, N represents the number of teeth.
θp =
2⋅π
N
(12)
At an axial depth z, the angular helix lag of the differential element on tooth j from the
leading point on the tooth’s cutting edge (z=0) is kiz where
ki =
tan i
R
(13)
while i and R are the helix angle and the cutter radius, respectively [23].
The cutting forces in equation (9) are given by the edge force component
represented by Kte, Kre and Kae, and a cutting component represented by Ktc, Krc and
Kac. The force coefficients Kte, Kre, Kae, Ktc, Krc and Kac can be predicted from the
oblique cutting analysis and the basic cutting quantities from the orthogonal cutting
data. The experimental procedure for obtaining these data is described in chapter 4,
while data analysis and the implementation into the model are given in chapter 5.
3.1.1 Prediction of Milling Force Coefficients from an Oblique Cutting
Model
In order to predict the milling force coefficients, the relevant equations from the
oblique cutting model were established by Armarego [23]. The cutting action of the
helical teeth at the periphery of the end mill cutter can be represented as an oblique
cutting process with an angle of inclination equal to the helix angle i and a normal
angle αn related by this equation:
tan α n = tan α r ⋅ cos i
(14)
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 35
The milling force coefficients can be expressed as follows:
K tc =
τ cos(β n − α n ) + tan i ⋅ tan η c ⋅ sin β n
⋅
sin φ
c
K rc =
sin(β n − α n )
τ
⋅
sin φ ⋅ cos i
c
K ac =
τ cos(β n − α n ) ⋅ tan i + tan ηc ⋅ sin β n
⋅
sin φ
c
(15)
where
c = cos 2 (φ n + β n − α n ) + tan 2 η c ⋅ sin 2 β n
(16)
while the normal friction angle βn is given by equation (7) and normal shear angle φn is
given by equation (6). The milling coefficients can be evaluated by combining
equations (6), (7), (8), (15), and (16), if the elemental tool geometry αn and i, the work
material stress τ and the basic cutting quantities rt, β and ηc are known. Estimation of
the quantities is presented in chapter 5.
3.2 Model improvements
In the first version of the model [18], the chip flow angle was approximated by
the Stabler rule (ηc = i). In the following text, the influence of the Stabler rule is
discussed including the solution of its removal and also the implementation of tool
run-out effect into the model is proposed there. The cutter is still considered rigid.
3.2.1 Prediction of chip flow angle – removing the Stabler rule
The oblique tool geometry was first rigorously analyzed by Stabler (1951), who
stated the widely accepted “Stabler rule or chip flow law” which assumes that the chip
flow angle is equal to the angle of obliquity, that means the angle ηc is approximately
equal to the inclination angle i for variety of tool and work materials, rake angles, and
speeds [24].
The chip flow angle can be calculated by numerical solution of equations (6), (7)
and (8) for known values of αn, β, i and rt. By combining these equations the following
expression for the chip flow angle ηc is obtained:
A ⋅ sin ηc + B ⋅ cos η c + C ⋅ tan η c = D
(17)
where
A = (1 − rt ⋅ sin α n ) ⋅ tan β
B = (rt − sin α n ) ⋅ tan β ⋅ tan i
C = rt ⋅ cos α n
D = cos α n ⋅ tan i
(18)
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 36
In order to simulate the effect of using the Stabler rule, the chip flow equation
(17) was solved in the Matlab software for constant rake angle and friction angle while
varying the chip thickness and angle of obliquity – see the Graph 1.
Graph 1: Variation of chip flow angle with chip thickness and angle of obliquity
From the Graph 1 is apparent, that the Stabler rule has a strong influence on
calculation of chip flow angle. Nevertheless the calculation is made for constant
friction angle and rake angle, while in micro end-milling operations these parameters
are changing with respect to the actual tool geometry, therefore the influence of
Stabler rule has not to be so evident. The effect of Stalber rule in the numerical
calculation of friction angle is discussed in chapter 7.
3.2.2 Run-out
Tool run-out and unbalance is usually a minor problem in macro-machining
operations. However, the problem is severely amplified when the diameter of the tool
decreases and spindle speed increases significantly. Tool run-out is caused by
misalignment of the axis of symmetry between the tool and the tool holder or spindle
[14]. Therefore a novel tool holder design with active or passive control using
actuators deserves more study, while it may compensate any unbalance and could
improve micro-tool run-out.
An attempt of the tool run-out implementation into the force prediction model is
given here, while the run-out parameters as well as the micro milling force
components are measured in the micro-milling experiments (Fig. 40, page 62) in
order to prove the correctness of the proposed solution.
Tool run-out parameters
The deviation of tool diameter was measured in two z-levels according to Fig.
21 – (2). After the maximum value was found, the angle λ from the measuring point E
to the reference point F (the reference flute tip) was measured. This was done in both
z(a) and z(b) levels.
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 37
R
min
Rcosλ
G
max
F
Rsinλ
λ
r
H
E
reference
point
(1)
(2)
Fig. 21: (1) – the geometrical relations for any z-level and (2) – measurement of the
run-out
In the derivation procedure presented here, the tool is considered as a cylinder
with the radius R, since the deviation d represents distance between tool axis and
spindle axis G and is not dependent on the radius where the measurement was
performed. The actual tool radius r is therefore caused by this deviation.
The value R constitutes the radius of the tool in zero z-level. The d value is
given by maximum and the minimum deviation:
d=
max − min
2
(19)
From Fig. 21 – (1) is evident that the Pythagorean Theorem is valid for the
triangle FGH:
r 2 = (d + R ⋅ cos λ ) + (R ⋅ sin λ )
2
2
r 2 = d 2 + 2 ⋅ d ⋅ R ⋅ cos λ + R 2 ⋅ cos 2 λ + R 2 ⋅ sin 2 λ
r = d 2 + 2 ⋅ d ⋅ cos λ ⋅ R + R 2
(20)
Equation (20) is valid in the both measured z-levels, therefore it can be written:
r (a ) = d(a ) + 2 ⋅ d(a ) ⋅ cos λ(a ) ⋅ R + R 2
(21)
r (b ) = d(b ) + 2 ⋅ d(b ) ⋅ cos λ(b ) ⋅ R + R 2
(22)
2
2
As the tool is considered as a cylinder, the distance r in any z-level has a linear
tendency for the reference point and therefore it can be written:
r = k⋅z+q
(23)
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 38
where k is the slope of the line:
k=
r (a ) − r (b )
z(a ) − z(b )
(24)
and q parameter can be found from following expression:
r (a ) = k ⋅ z(a ) + q → q = r (a ) − k ⋅ z(a )
q = r (a ) −
r (a ) − r (b )
⋅ z(a )
z(a ) − z(b )
(25)
By combining equations (23), (24) and (25) the following dependency is
obtained:
r=
r (a ) − r (b )
⋅ (z − z(a )) + r (a )
z(a ) − z(b )
(26)
As the distance r for desired z-level is dependent only on the tool radius R,
equations (21), (22) and (26) can be extended for any flute j, where n is the number of
flutes, with influence of helix angle i:
2 ⋅ π ⋅ ( j − 1)


2
r j (a ) = d(a ) + 2 ⋅ d(a ) ⋅ R ⋅ cos λ (a ) +
− z ⋅ tan i  + R 2
n


(27)
2 ⋅ π ⋅ ( j − 1)


2
r j (b ) = d(b ) + 2 ⋅ d(b ) ⋅ R ⋅ cos λ (b ) +
− z ⋅ tan i  + R 2
n


(28)
rj =
r j (a ) − r j (b )
z(a ) − z(b )
⋅ (z − z(a )) + r j (a )
(29)
For further application in the model, the distance rj from flute j to the centre of
rotation can be expressed as sum of the tool radius R and deviation drj:
r j = R + dr j → dr j = r j − R
(30)
The rj value in equation (30) represents the actual tool radius, characterized by
the deviation drj from the nominal tool radius R, for given flute j in desired z-level. The
effect of the helix angle is taken into account. The parameters d(a), d(b), λ(a), λ(b),
z(a) and z(b) are measured in the milling experiments.
Uncut chip thickness due to the run-out
In first proposal of the model the uncut chip thickness was calculated from
equation (10) considering the rigid tool with zero run-out. However, when the tool is
affected by run-out, the effect of previous tooth has to be taken into account. Such a
situation is shown in Fig. 22 and the uncut chip thickness is expressed by following
equation:
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
t j (θ, z ) = s t ⋅ sin θ + dr j −
Page 39
dr j−1
(31)
cos γ
st
1 - actual path of the tooth j-1
j-1
j
2 - nominal path of the tooth j-1
3 - nominal path of the tooth j
4 - actual path of the tooth j
θ
R
1
γ
2
drj
3
drj-1
stsinθ
4
Fig. 22: Uncut chip thickness affected by run-out
Tooth j with positive deviation drj, which means the actual radius of the tool r is
higher than the nominal radius R, performs cutting. The uncut chip thickness
calculated form equation (30) is affected by the cut of previous tooth j-1 with negative
deviation drj-1, which means the actual radius of the tool is lower than the nominal
one. The relationship for the γ angle was determined from Fig. 23.
2
j
j-1
R
Rcosθ
θ
γ
υ
k
l
st
Rsinθ
Fig. 23: Geometrical relations for determination of γ angle (a part of Fig. 22)
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 40
For the triangle jkl stands:
θ+γ+υ+
π
=π
2
(32)
Then the υ angle is expressed by:
tgυ =
R ⋅ cos θ
s t + R ⋅ sin θ
(33)
By combining equation (32) and (33) the following dependency is obtained:
γ=
 R ⋅ cos θ 
π

− θ − arctg
2
 s t + R ⋅ sin θ 
(34)
Equations (27) – (31) and (34) for chip thickness due to the run-out calculation
were implemented into the model (see the Matlab script in Appendix C) with the
results discussed in chapter 6. The option to calculate chip flow angle using Stabler
rule or by numerical solution of equations (6) – (8) is included too.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Chapter
4
Experimental Work
Turning Experiments
Page 41
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 42
A mathematical model for calculation of force components in end milling
operations was presented in chapter 3. It required determination of the edge
coefficients Kie and relations for chip thickness ratio rt, shear stress τ and friction at
the rake face β for different work materials. Orthogonal turning experiments were
performed in order to obtain the material characteristics, while milling experiments
were required for the model validation.
Differently from [18] where the experiments were carried out on a powder
metallurgy martensitic stainless steel (quenched and tempered before machining to a
hardness of 58 HRC), other two different materials were used (carbon steel UHB11
and Aluminium 6082 T6) in order to evaluate and improve proposed model. As the
material was changed, other conditions were kept the same.
Since different equipment and approaches were used, the experimental work is
divided into two chapters – this chapter is dealing with turning experiments while the
second one is dedicated to milling experiments. As the data from turning experiments
were processed, the results were subsequently implemented into the model. Then the
forces calculated by the model were compared with the data obtained in milling
experiments.
4.1 Introduction to the turning experiments
The aim of the turning experiments was to obtain the coefficients and relations
for the mathematical model described in chapter 3. The chip thickness and forces on
the tool were measured with equipment described below. All the tests were performed
under orthogonal cutting conditions with a tube turning operation. Different
combinations of feed (uncut chip thickness) and rake angle were used, while the
cutting speed was remaining constant in all experiments.
Reason for using different rake angles
The influence of rounded edge of tool in microcutting has been explained in
chapter 1.5.3. As the edge radius of the tool is often relatively larger than the chip
thickness, chip sliding occurs along the rounded tool edge and it results in large
negative rake angle.
αreal
a) h>hlim
b) h<hlim
Fig. 24: The effect of rounded tool edge for different uncut chip thickness
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 43
To simulate such conditions, similar tests were made in a scaled up
configuration by means of a turning operation. Fig. 24 shows the effect of rounded
tool edge for different values of uncut chip thickness. The tool with edge radius of
50µm is set in both cases with nominal rake angle of -5°.
In Fig. 24 a) the uncut chip thickness is higher than hlim and chip slides along
the rake face. This is common in conventional cutting operations.
In Fig. 24 b), the uncut chip thickness is lower than hlim, the chip does not touch
the rake face and escapes under higher negative rake angle αreal. The situation in the
Fig. 24 b) could be compared to the setup of tool with nominal rake angle of -30°
(dashed line) and forces acting in these two geometrical configurations should be
theoretically the same. In order to prove this assumption, three different values of
rake angle were used in experiments with combination of several feed rates.
The following relations for α and hlim were derived:
sin α =
R e − h lim
h
= 1 − lim
Re
Re
h lim = (1 − sin α ) ⋅ R e
(35)
(36)
4.2 Experimental Setup
The experimental equipment used for the turning experiments is shown in Fig.
25 described in this section.
Machine Tool
For turning experiments a VDF Boehringer PNE 480 L CNC lathe was used.
This machine tool has power of 50kW, maximum chucking force of 60kN and
maximum workpiece size of 480mm in diameter and 1000mm in length. The lathe
with its high chucking force generates vibrations, which are transmitted to the
workpiece. To achieve acceptable accuracy, this represents a limitation of the
minimum chip thickness.
Cooling
A 7% oil emulsion was used in turning experiments. Subsequently, the same
emulsion was used in milling experiments for later comparison of results. The
emulsion was applied to the rake face from top – it is shown in Fig. 25.
Dynamometer
A three components dynamometer with four piezoelectric transducers Kistler
9251A was used for the force acquisitions. The force measuring range is ±10kN in the
cutting direction (Power force), ±5kN in the reaction direction (Reaction force) and
±3.5kN in the feed direction (Thrust force) [25]. The calibration was done before the
turning tests and is described in paragraph 4.3.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 44
turret
dynamometer
tool holder
insert
spindle
workpiece
Fig. 25: Dynamometer, tool holder (set to α = -40°) with insert and workpiece on
machine tool
Workpiece
Because the tube turning operation was needed, a number of workpieces with
two ribs on each side were machined from the bars of two different materials – carbon
steel UHB11 and aluminium 6082 T6. The shape of workpiece with its dimensions is
shown in Fig. 26.
Fig. 26: The shape of workpiece
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 45
Uddeholm’s tool steel UHB 11 is an easily machinable carbon steel characterize
by good machinability and good mechanical strength with hardness of 200 HB. The
chemical composition is: 0,46% C; 0,2% Si; 0,7% Mn.
Aluminum 6082 T6 has this chemical composition: 95,2 – 98,3% Al; 0,6 – 1,2%
Mg; 0,4 – 1% Mn; 0,7 – 1,3% Si; max 0,1% Cu; max 0,5% Fe; max 0,1% Ti; max
0,2% Zn.
Tool Holder
In order to perform orthogonal cutting with different values of the nominal rake
angle, the tool holder developed in [18] was used. Its construction is based on
commercially available tool holder type PTGNL 1616H 16 ISO 5608, which shaft was
reduced and inserted in rectangular hole on a cylindrical bush in eccentric position
with appropriate inclination angles in order to compensate the inclination angle and
cutting edge angle provided by the tool holder. The bush was mounted on a larger
holder provided with cylindrical housing and interface for the dynamometer. This
configuration maintained the position of cutting edge of mounted insert on the axis of
the bush. As the bush is rotated around its axis, the cutting edge is always aligned
with the workpiece in radial direction, which is the requirement for orthogonal cutting.
The nominal rake angle is thus limited only by the insert geometry and for the insert
used in the turning experiments it could be changed from -5° to -90°. The tool holder
is shown in Fig. 27.
Fig. 27: Tool holder (α = -40° setup)
Inserts
Flat coated carbide inserts type TNMA 16 04 04-KR manufactured by Sandvik
Coromant were used for cutting tests. Its geometry is characterized by absence of
chip breakers (a chip breaker influence the contact length and cutting forces) and 90°
angle between the rake and flank faces It is important support against the cutting
forces when using high negative rake angles. The edge radius of the insert was
measured – see the chapter 4.6.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 46
4.3 Calibration Procedure for 3D Turning Dynamometer
The 3D turning dynamometer is highly reliable instrument which gives the
possibility to perform high accuracy force measurements. A dynamometer of this kind
was available at IPL.
Static calibration of the dynamometer is important process before using this
equipment for measuring forces in turning experiments. The calibration has to be
performed in order to obtain the matrix for transforming the output voltage from
charge amplifiers to the cutting forces. The procedure is described in following
chapters.
X – Thrust force
Y – Reaction force
Z – Power force
5
3
6
+Y
7
+X
2
+Z
4
1
Fig. 28: 3D turning dynamometer (1 – main part; 2 – upper plate; 3 – top part; 4 –
lateral lid; 5 – electrical box; 6 – testing bar; 7 – dedicated device for calibration
procedure)
Due to the problems during first attempt of calibration procedure (one or two
force components went suddenly to maximum values) and because of impossibility to
obtain repeatable results, it was decided to dismount the electric box of the
dynamometer to check the connections. The loose connections were found and
consequently all cables going from charge amplifiers and the output cable with its
protection have been replaced. This procedure decreased the drift from the
dynamometer (it could be seen from the graphs below) and it also solved major
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 47
problems with voltage-drops. Therefore and also due to the time constraint it was
decided not to open main part of dynamometer with piezoelectric cells.
longtime test (OLD)
(no load, position in Z-axis calibration)
0,025
0,02
0,015
Voltage [V]
0,01
0,005
0
-0,005
-0,01
-0,015
0
0,2
0,4
0,6
0,8
Time [hours]
1
Ux [V]
1,2
Uy [V]
Uz [V]
Graph 2: Drift components before revision
longtime test (NEW)
(no load, position in Z-axis calibration)
0,025
0,02
0,015
Voltage [V]
0,01
0,005
0
-0,005
-0,01
-0,015
0
0,2
0,4
0,6
0,8
Time [hours]
Graph 3: Drift components after revision
1
Ux [V]
1,2
Uy [V]
Uz [V]
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
1
2
Page 48
3
Fig. 29: View into the electrical box of dynamometer
(1 – connections to the piezoelectric cells; 2 – charge amplifier; 3 – output connections)
4.3.1 Calibration of the dynamometer
The calibration has to be performed with the acquisition board and setup (it
could be called “measurement system”), which is used for data acquisition during the
turning experiments. Measurement system consists of dynamometer, power supply
box and PC with acquisition board manufactured by National Instruments. The
procedure presented here is based on [26] and slightly modified. The loads were
applied by means of a pres AMSTLER.
Before calibration it was necessary to do the following:
•
adjust the zero point of the indicator of the testing machine for the load
range 0 – 10 kN;
•
turn on the power supplier box to operate mode for one hour in order to
warm it up (this procedure lowers the noise and drift components).
The load ranges and the increments of calibration are presented in Table 1.
Table 1: Load range
Load direction
Power force – FZ
Thrust force – FX
Reaction force – FY
Maximum load [kg]
Increment of load [kg]
300
20
1kg = 9,80665 N
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 49
The calibration procedure was carried out in the following steps:
•
fix the dynamometer on its dedicated device and fasten it through the
five M8 screws with a torque spanner settled at 35Nm;
•
align the bar, used as tool, with direction of the calibration;
•
reset the power supply box and switch it to operate;
•
load with desired sequence of loads, then unload;
•
repeat this procedure three times;
•
align the dynamometer on other direction and acquire data using the
same steps.
Fig. 30: Dynamometer positioned for calibration in –X direction on the Amstler
machine
4.3.2 Data analysis
Data were analysed with Regression Analysis Tool in Microsoft Excel by using
the least squares method to fit a line through a set of points.
Drift components, as could be seen in Graph 3, were lower as 4µVs-1.
Graph 4, Graph 5 and Graph 6 shows the calibration curves in each direction
with the cross talk signals. The corresponding coefficients for each direction were
calculated as average value from three calibrations and they are shown in Table 2.
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 50
Dynamometer calibration
-X direction (1)
1
Uy = 0,04893x - 0,02078
2
R = 0,94910
0
0
0,5
1
1,5
2
2,5
-1
3
Uz = 0,00999x + 0,00226
2
R = 0,98981
Voltage [V]
-2
-3
-4
-5
Ux = -1,97867x + 0,00958
2
R = 0,99975
-6
-7
Force [kN]
Ux [V]
Uy [V]
Uz [V]
Graph 4: Calibration curves on –X direction (1. calibration)
Dynamometer calibration
Y direction (1)
7
Uy = 2,04852x - 0,01197
2
R = 0,99987
6
5
Voltage [V]
4
3
2
1
Ux = -0,06188x + 0,01207
2
R = 0,89308
Uz = 0,02540x + 0,00094
2
R = 0,97627
0
0
0,5
1
1,5
2
2,5
3
-1
Force [kN]
Graph 5: Calibration curves on Y direction (1. calibration)
Ux [V]
Uy [V]
Uz [V]
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 51
Dynamometer calibration
Z direction (1)
7
6
5
Voltage [V]
4
Uz = 1,03238x - 0,01128
2
R = 0,99984
3
2
1
Uy = 0,04989x + 0,00071
2
R = 0,99557
Ux = -0,05671x + 0,00953
2
R = 0,99075
0
0
0,5
1
1,5
2
2,5
3
-1
Force [kN]
Ux [V]
Uy [V]
Uz [V]
Graph 6: Calibration curves on Z direction (1. calibration)
Table 2: Calibration coefficients
-X
Y
Z
Applied load in axis
Calibration 1
output
2
channel coefficient
R
x
y
z
x
y
z
x
y
z
-1,97867
0,04893
0,00999
-0,06188
2,04852
0,0254
-0,05671
0,04989
1,03238
0,99975
0,9491
0,98981
0,89308
0,99987
0,97627
0,99075
0,99557
0,99984
Calibration 2
2
coefficient
R
Calibration 3
2
coefficient
R
-1,97769
0,04914
0,00985
-0,0995
2,05275
0,0273
-0,05782
0,04864
1,03325
-1,98308
0,04892
0,00967
-0,12733
2,04547
0,02896
-0,05653
0,04824
1,03249
0,99979
0,9512
0,98374
0,95275
0,99982
0,98668
0,99278
0,99419
0,99985
0,99976
0,94831
0,98466
0,9729
0,99985
0,98981
0,98955
0,99045
0,99988
Average
Coefficient
-1,97981
0,049
0,00984
-0,09624
2,04891
0,02722
-0,05702
0,04892
1,03271
According with previous table, the calibration matrix M(mij), where i index
represents the direction of the load and j index represents the measured voltage, has
following expression:
m xx

M = m xy
m xz

m yx
m yy
m yz
m zx   1,97981 − 0,09624 − 0,05702 
 

m zy  =  − 0,04900 2,04891
0,04892 
m zz   − 0,00984 0,02722
1,03271 
(37)
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 52
Consequently, the output voltages acquired are linked with the values of the
load through the following equation:
UFx   1,97981 − 0,09624 − 0,05702  Fx 

 
  
0,04892  ⋅ Fy 
UFy  = − 0,04900 2,04891
UFz  − 0,00984 0,02722
1,03271  Fz 
[V ] = V ⋅ kN −1 ⋅ [kN]
[
(38)
]
Multiplying the above presented formula with inversion matrix M-1 on the left we
get the following equation for determining the force components from measured
voltages:
0,02682  UFx 
Fx   0,50581 0,02340
  
 

Fy  = 0,01199 0,48893 − 0,02250  ⋅ UFy 
Fz  0,00450 − 0,01266 0,96917  UFz 
[kN] = kN ⋅ V −1 ⋅ [V ]
[
(39)
]
4.4 Experimental Plan
As mentioned in the beginning of the chapter, turning experiments were needed
for estimating the edge coefficients and other dependencies related to the current
material.
The experimental plan was based on a full factorial design with two variables –
the nominal rake angle in the range of three levels and the feed (uncut chip thickness)
in range of seven levels – see the first proposal in Table 3. The cutting speed of
30mmin-1 was remaining constant. The chip thickness and cutting force components
were measured. Each feed – nominal rake angle combination test was performed
three times and the centre point of experimental design was repeated five times in
order to test the repeatability of the measurements. Some feed – nominal rake angle
combination tests were performed more than three times, mainly as preliminary tests
or due to difficulties during experiments. While machining UHB11 and also aluminium
with the nominal rake angle set to -53° the inserts were breaking even in low feed
rates.
Table 3: Layout of the experimental design – first proposal
Feed
5 µm
8 µm
10µm
25µm
50µm
Angle
-5°
-30°
-53°
Centre
point
75µm
100µm
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 53
Insert breakage
The reasons for cutting insert breakage could be explained as following. First,
the work material is very soft and the rib of the workpiece is deformed more than cut.
The rib becomes wider (Fig. 31), the width of the cut is increasing and subsequently
the forces acting on the tool are higher.
Fig. 31: New aluminium workpiece on the left and the same workpiece after
experiments using rake angle of -53° on the right
Second, the force composition is different in cutting soft material comparing to
hardened tool steel. The thrust force is increasing with increasing negative rake angle
and is higher than the power force, which is decreasing for higher negative rake angle
values in all hardened tool steel experiments [18]. When cutting the UHB11 steel and
the aluminium, the both force components are increasing for higher negative rake
angles values. Furthermore, for rake angle setup of -40° the forces on UHB11 are
much higher than on hardened tool steel cut with rake angle of -53°, while other
cutting parameters are kept same.
Thrust force FX
Power force FZ
vc
Fig. 32: Forces acting on the tool
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 54
Because of insert breakage it was decided to lower inclination angle to -40°, but
still for this lowered rake angle the inserts were breaking when using values of feed
rate higher than 50µmrev-1. As the feed rates of 75µm and 100µm could not be used,
the feed rate of 18µm was added in order to have enough points for estimation of
force-curve. The final proposal for experiments is shown in Table 4, where the values
of real rake angle (according to the explanation on the page 43) for each combination
of feed – tool holder angle are shown. A total of 145 tests were performed including
the unusable tests where the insert was broken.
Table 4: Layout of the experimental design – final proposal
Feed 5µm
8µm 10µm 18µm 25µm 50µm
Tool holder angle
Real rake angle
-5°
-64
-57
-53
-30°
-64
-57
-53
-40°
-64
-57
-53
75µm 100µm
-30
-5
-5
-5
-30
-30
-30
-30
-40
-40
-40
4.5 Experimental Procedure
Experiments were performed with equipment described above. The
dynamometer with tool holder and insert was mounted on the turret. For the single
test, a new cutting edge was used in order to minimize the edge wear and the build
up edge. Therefore one insert could be used only for six tests. The round bar was
clamped into the jaws, workpiece was mounted on and tightened with a screw nut.
The run out of the mounted workpiece was measured by a dial gauge – see the Fig.
33. If the run out was higher than 40µm, the position of workpiece was carefully
adjusted with hammer hits.
Fig. 33: Measurement of the run out with a dial gauge
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 55
Before a test run, a steel plate for chips collection was placed on the bottom of
the lathe just under the worpiece and force acquisition was started. A feed was
activated then for a few seconds.
After a test run, the chips were collected from the steel plate and placed into the
labelled plastic bag. The plate was cleaned afterwards to avoid mixing chips from
different test runs.
The chip measurement is described in next chapter. The analysis and results of
the experiments are shown in chapter 5.
4.6 Measurements
Before starting the tests, the cutting inserts were measured in order to confirm
the geometry specification. The specification of the inserts could be found in the
manufacturer’s catalogue [27]. The most important parameter was the edge radius,
which was measured by Stylus profilometer Tylor Hobson RTH. For each insert the
average with standard deviation was computed from 40 edge profile measurements.
A profile of the edge is shown in Fig. 34 and the results of measurements are
presented in Table 5.
Fig. 34: Measurement of insert profile in SPIP software
Table 5: Edge radius measurement
Measurement of the cutting insert edge radius [µm]
Insert no.
1
2
3
4
5
6
AVG
Average
51,91 53,57 52,64 52,16 52,29 53,16 52,62
value
Standard
3,26
2,17
2,86
3,04
2,20
2,42
deviation
STD
0,58
While making the different tests, measurements of the forces on the tool where
taken by using the dynamometer. This was calibrated and attached to the machine
used for cutting the pieces. The power and the thrust forces obtained in experiments
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 56
were analysed with two different approaches. The force analysis is presented in
chapter 6.1.2.
TESA
module
Calibration
plate
Inductive
probe
Fig. 35: TESA device with inductive displacement sensor
The rest of important parameters are connected with the chips. As chip
thickness varied over a range of more than one order of magnitude, two different
devices had to be used for measurements. Thicker chips were measured by using an
inductive displacement sensor with a resolution of 1µm. A measured chip was placed
between two spherical tips, which are ensuring contact in only one point, so the chip
curvature influence was avoided. Such a device shown in Fig. 35, it was calibrated by
a 150µm thin steel plate, which is in range of measured chips. An optical coordinate
measuring machine (CMM) shown in Fig. 36 was used for measuring thinner chips
produced in experiments using feed rate lower than 25µm.
Fig. 36: DeMeet – an optical coordinate measuring machine
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 57
The measurements referred to chips are more complicated and the accuracy is
not as good as in the other ones. When evaluating the thickness, the different
sections of he chips should be taken into account. Some of them were rectangular
and could be considered constant, but other ones had thinner section on the sides
and the ticker one in the centre – see the Fig. 37. Therefore, five different chips from
each experiment were measured and the standard deviation was evaluated. Higher
number of measurements would be appropriate, but due to the number of tests this
was not possible in given time. The results of chip measurements are presented in
data analysis - chapter 6.1.1 and in Appendix A and B.
Fig. 37: The chips of aluminium produced during the lathe experiments
(test number 63 on the left and number 23 on the right)
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Chapter
5
Experimental Work
Micro Milling Experiments
Page 58
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 59
Micro milling experiments were performed in order to verify the functionality and
usability of dynamometer device developed in [30] and to obtain the data for
validating the forces predicted by the mathematical model. The equipment used for
the tests with the experimental procedure is described in this chapter.
5.1 Introduction to the micro milling experiments
Since dedicated ultra-precise machine tools, which are appropriate for micro
milling operations by means of their high accuracy, are very expensive, the solution of
a conventional milling machine with attached high speed spindle was adopted here. It
has, of course, its limitations especially in the positioning accuracy.
As mentioned in the first chapter, variation of the chip load with dimensional and
geometrical accuracy of the workpiece is important issue in micro-milling operations.
These deviations are caused by differences between the nominal and actual process
due to the stiffness of the machine tool structure, tool deflections and tool wear, the
performance of feed drives and thermal deformations of the machine tool, workpiece
and tool [18]. The variability of feed per tooth and radial depth of cut affects just the
process performance, but an error on the axial depth of cut control can lead to the
tool breakage and therefore it is the most important parameter to control.
The static stiffness of the conventional machine tool structure is not critical
since the forces acting in the micro milling process are very low. Tool deflections and
tool wear were discussed in chapter 1. Therefore these issues will not be considered
here anymore.
The performance of feed drives is given for used machine tool. It has an
influence on the feed and displacement of the tool contributing to the chip load
variation and to the final accuracy of the workpiece. The most critical parameter for
the small diameter tools is the axial depth of cut, since it may cause their breakage.
Thermal deformations of machine tool structure are the largest source of
positioning errors in micro milling operations. The main sources of heat generation
are the feed drives of the machine tool and the bearings in the spindle system. As the
material removal rate is very low, a little heat is generated and therefore the thermal
deformation of the workpiece and the tool can be neglected, especially when the
cutting fluid is applied. Nevertheless, due to the construction of the dynamometer
device, the negative effect of the workpiece thermal dilatation was observed during
force measurements.
The set-up with improved machining procedure compensating the machine tool
deformation, which can result in depth of cut errors, is presented in following
paragraphs.
5.2 Experimental setup
The milling experiments were carried out on a 3 axis vertical milling machine
with the high speed spindle plugged in by the taper fit. The devices for tool length
measurement and run-out measurements were placed on the machine table. The
forces were measured by a single piezoelectric sensor with the workpiece mounted
on it. Dedicated fluid circuit was set up for thermal stabilization of the machine tool.
The set-up is shown in Fig. 38.
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 60
Main spindle
High speed spindle
Dedicated fluid circuit
Tool length measurement
device
+Z
+Y
+X
Dynamometer device
Run-out measurement device
Fig. 38: The set-up for milling experiments
Machine tool
The machine tool used was a conventional vertical milling machine
CINCINNATI MILACRON SABRE 750 with 3 CNC controlled axes. The X and Y
horizontal movements belong to the machine table, while the Z movement is
controlled by the machine head. An AC servomotor is used for driving each axis with
positioning resolution of 1µm and repeatability of 4µm. The main characteristics of
machine tool are given in Table 6.
Table 6: Cincinnati Milacron Sabre 750 characteristic
Characteristic
Unit
Magnitude
X/Y/Z axis
[mm]
762/381/508
Speed range
[rpm]
60÷8000
Main drive
[kW]
11,2
Programmable feed rates (X,Y,Z)
[mmin-1]
15
Rapid traverse rates (X,Y,Z)
[mmin-1]
15
High speed attached spindle
As the main spindle of machine tool was capable of max 8000rpm, the high
speed spindle HES-BT 40 H made by Japanese manufacturer NSK Nakanishi was
attached in order to achieve a reasonable cutting speed with micro end mills. The
power is supplied to the brushless motor in the spindle by an external control unit,
which also maintains the air cooling required by the motor. The spindle is equipped
with ultra precision ceramic bearings reducing the heat generation by friction and
achieving high rotational accuracy. A tool has to be mounted manually by use of two
spanners.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 61
Table 7: NSK Nakanishi HES-BT40 H spindle characteristics [28]
Spindle type
HES-BT40 H
Control unit
NE52-500
Maximum speed
50000 rpm
Max. output power
195 W
Max. torque
0,06 Nm (for 0÷30000 rpm)
Motor cooling
Air cooling 2÷6 bar
Bearing lubrication
Life lubricated
Cutting fluid
A 7% oil emulsion, the same as in the turning experiments, was used in micro
milling experiments. The first experiments made with 1mm end mill showed that the
best result, according to force acquisition, was given when the fluid was applied
before each experiment and the surface of workpiece remained wet – see Fig. 40.
Tools
The tools used were PVD coated two-flute flat end mills with diameters of
0,2mm, 0,6mm, 1mm and 6mm made by Japanese manufacturer OSG.
Table 8: Cutting parameters for carbon steel milling operations [29]
n [rpm]
f [mmmin-1]
Tool
D x L1 [mm]
WX EDS
6 x 13
4450
310
WX-LN-EDS
1x6
22000÷29000
700÷1300
WX-LN-EDS
0,6 x 4
24000÷28000
250÷500
WX-LN-EDS
0,2 x 1
28000÷32000
300÷500
ap [mm]
3,000
0,070
0,043
0,014
Fig. 39: OSG WX-LN-EDS 2 flutes end mill tool, micro grain carbide material, TiAlN
coating, tolerance for outer diameter: 0÷0,015mm, cutting length: 1,5 x diameter
Dynamometer device
The dynamometer device has been developed in [30], where the specifications
including calibration procedure are available. The device consists of the workpiece
connected by a bolt through the piezoelectric sensor (type 9251A), acting as a
measuring element, to the base plate. The setup is shown in Fig. 40. The signals from
piezoelectric sensor were amplified by the charge amplifiers and brought through the
acquisition board to the computer with Labview software, where the forces acting on
the workpiece were recorded.
The charge amplifiers used were made by Kistler – two of them (type 5015)
were measuring in X and Y direction, while the third one (type 5001) was used for Z
direction. The acquisition board and the software used was the same as in the case
of turning experiments.
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 62
Tool length measurement device
An inductive displacement sensor, for estimating the tool length, with resolution
of 0,1µm and contact force of 0,6N was placed on the machine table with its axis in
vertical direction. A probe, having a 10mm in diameter flat surface, made from
sintered carbide was mounted on the sensor. The centre point of the probe in XY
plane with the Z level indicating the zero position of the probe was set as a reference
system in the machine tool. Two thermocouples type K were also a part of measuring
chain. The first one was measuring the environmental temperature and the second
was measuring the temperature of high speed spindle.
Run-out measurement device
For purpose of run-out measurements, an inductive displacement sensor, same
as for the tool length measurement, was placed on the machine table with its axis in
horizontal direction. A reference system in the machine tool computer was set to
identify the change in run-out of the tool. In Y direction the zero point was the position
of the spindle axis, the zero point in Z direction was set to the same level as the tool
length, so the distance from the end of the tool to the point of run-out measurement
was known. The disc with angular scale was placed on the tool in order to estimate
the angular position of the cutting edge during measurements. Such a device is
shown in Fig. 40.
(1)
(2)
Thermocouple
Measuring disc
Inductive probe
a
b
c
Fig. 40: (1) – Micro end milling experiment in progress, a-workpiece, b-piezoelectric
sensor, c-base plate; (2) – Equipment for run out measurement
5.3 Experimental plan
Recommended cutting parameters by the tool manufacturer are reported in
Table 8. Despite these recommendations, various cutting parameters were needed in
order to have enough data for further comparison, therefore the modular approach
was used for micro end-mill tools (WX-LN-EDS). Using this approach, the feed per
tooth st, the radial depth of cut ae and the axial depth of cut ap are kept in desired ratio
to the tool diameter – it is the kind of “scaling of the cutting process”. The modular
approach ratios used for the experiments are reported in Table 9. Due to the time
constrains the full factorial design (performing the experiments for all combinations of
ap, ae and st) was used just for 0,6mm end mill experiments carried out on the steel. In
order to validate the force prediction model, the rotational speed was selected for
each tool diameter to the value corresponding to cutting speed of 30mmin-1 used in
the turning experiments.
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Table 9: Modular approach ratios
ap
0,075
0,050
D
ae
1,0
0,5
D
st
0,020
0,015
D
Page 63
0,025
0,1
0,010
UHB 11 steel, 1mm end mill
First of all, the response of the dynamometer device had to be validated. It was
done in milling experiments of the UHB11 steel with 1mm end mill, because the
cutting speeds were too low when the lower rotational speeds were used for smaller
tool diameters. Various values of radial depth of cut and rotational speed were
selected, while depth of cut and feed per tooth was kept the same. Various methods
of cutting fluid application and the process repeatability tests were performed also.
UHB 11 steel, 0,6mm end mill
For the experiments with 0,6mm end mill the rotational speed of 16000rpm was
set and full factorial design was used.
UHB 11 steel, 0,2mm end mill
The rotation speed of 30000rpm representing the cutting speed of 19mmin-1
was set, because the force acquisition had a bad resolution and it was not possible to
use the results of experiments for later analysis when using the speed of 50000rpm.
Also the tool breakage was observed, therefore the 20% feed override was employed
and the depth of cut was decreased.
6082 T6 aluminium, 0,6mm end mill
Since a new calibration of dynamometer device for the aluminium workpiece
was necessary, only a few experiments with 0,6mm end mill were carried out due to
the time constrains.
Table 10: Cutting parameters used in the milling experiments
Tool D
n
ap
Workpiece
lubrication
[mm]
[rpm]
[mm]
50000
1,0
various
0,05
30000
10000
0,045
wet
Steel
0,6
16000
0,030
surface
0,015
Aluminium
0,2
wet
surface
30000
0,005
0,6
wet
surface
16000
0,045
ae
[%]
100
50
10
100
50
10
100
50
10
100
50
10
f
[mmmin-1]
1500
900
300
384
288
192
180
384
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 64
5.4 Thermal distortions
As mentioned in the beginning of this chapter, thermal distortion is the most
serious source of machining errors. Reduction of the power loss due to internal heat
sources and their insulation, temperature control of machine tools or different
compensation methods could solve this problem [18]. Indirect compensation
approach was used in the experiments, since it is relatively easy to apply comparing
to direct compensation methods, which require periodic measurement of machine
distortion during machining.
The indirect compensation approach is based on measuring the distance
between spindle and machine table while the temperature of the environment and the
temperature at different places in the machine structure is measured. Performing
these measurements over a wide range of temperature changes generates sufficient
input data for the mathematical description of the machine tool thermal behaviour.
This is often called “teaching phase”. Provided data are then usually fitted by a curve
using regression analysis. The curve (a mathematical function) is later used for
estimating the position error if the temperature in different points is known.
The main sources of heat generation in the set-up used for micro milling
experiments were the feed drives of the machine tool, the bearings and the motor in
the high speed spindle and the pump of the machine tool maintaining the circulation
of the cutting fluid. Therefore the thermal behaviour analysis of the system was
performed in [30] with these results:
•
Thermal behaviour of the high speed spindle with the specific steady state
temperatures for different rotational speeds is shown in Graph 7. The curves
were obtained using a single point thermocouple type K placed on the
spindle (Fig. 40) and connected through the acquisition board with PC.
Acquisition was performed for the room temperature of 21° C.
•
The high speed spindle length variation as the function of difference
between room and spindle temperature for different rotational speeds is
shown in Graph 8, for the room temperature of 21° C .
50000rpm
40000rpm
30000rpm
20000rpm
10000rpm
Graph 7: Spindle heating-up curves for different rotational speeds [30]
Enhancement and verification of a cutting
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Jan Slunský
Page 65
a)
b)
c)
Graph 8: Calibration curves for different rotational speeds [30]
Presented diagrams were needed for later tool length correction, as thermal
expansion of the high spindle could not be neglected. Setup for thermal behaviour
analysis was exactly the same as that one used for micro milling experiments.
Additional equipment – dedicated fluid circuit
Since the cutting fluid was needed to be applied in micro milling experiments, its
influence on thermal deformation of machine tool had to be considered. Due to the
heat transfer from the pump to the fluid, the fluid temperature increased with time and
the machine structure was deformed subsequently. In order to reach thermal
equilibrium between the machine tool and the fluid in shorter time, dedicated fluid
circuit wetting the machine column was established.
5.5 Experimental procedure
Experimental procedure was performed with the equipment described above.
Due to the thermal distortions of machine tool, the simplified machining procedure,
taking into account an elongation of high speed spindle, was adopted from [18]. Runout of a micro end mill was measured just after mounting the tool in the spindle before
starting machining procedure. Best set up of dynamometer device parameters is also
briefly presented here.
5.5.1 Run-out measurement procedure
Measurements of a tool run-out were performed in order to obtain input
parameters for the model and subsequently to compare measured and predicted
forces. The deviation of tool diameter was measured in two z-levels according to Fig.
21 (2) on page 37. After the maximum and minimum deviation was found, the angle λ
from the measuring point of maximum deviation to the reference point (the reference
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 66
flute tip) was taken for each z-level in clockwise direction. The device for run-out
measurement is shown in Fig. 40 on page 62.
5.5.2 Tool length correction procedure – the compensation of axial depth of cut
errors
The tool length measurement device is shown in Fig. 41. It was needed for
setting the tool length required by the machine tool computer. First of all, surface of
the probe was scanned by spherical tip in several points to find the contact point. The
contact point was considered as a highest point of the surface, where the tool length
could be measured. Measuring the tool length in such a point avoids errors due to the
surface inclination, as can be seen in Fig. 41 – (1): If the tip of 1mm end mill is placed
in the contact point as well as the tip of 6mm end mill, the tool length is set correctly.
But when the position of the 1mm tool is different from the contact point, the
positioning error occurs.
(2)
(1)
Fig. 41: (1) – The contact point for measurement of the tool length; (2) – Tool length
measurement device: scanning for a location of the contact point
5.5.3 Machining procedure
After considering the issues discussed above, a machining procedure adopted
from [18] was applied here. The procedure is divided here to the tool change
procedure and the test run procedure. The tool change procedure was used when the
tool or rotational speed, influencing steady state temperature and therefore the
elongation of the high speed spindle, had to be changed. The test run procedure had
to be used for every single experiment, which was always performed in +Y direction
of the machine tool.
The best performance and result of acquisition was obtained for following setup,
detaily described and discussed in [30]:
•
amplifiers in DC long measuring mode,
•
acquisition frequency of 30kHz,
•
no kind of filters applied during acquisition,
•
surface of the workpiece just covered with a layer of fluid, the fluid did
not flow directly on the workpiece during an experiment.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 67
Tool change procedure
1. Machine warm up for 1 hour, with no load, with cutting fluid on, at a rotational
speed of 6000rpm with a 6mm end mill mounted in the spindle. The cutting fluid
was maintained flowing permanently.
2. Tool length presetting on the inductive probe, with the cooling air off, the spindle
turned off, and recording the spindle temperature at the moment of presetting.
3. Storing the tool length value in the machine tool control, corrected for the
temperature difference between point 2 and the steady state temperature for
6000rpm.
4. Spindle warm up with 6000rpm to the steady state temperature.
5. Facing operation, afterwards changing the tool to the micro end mill.
6. Tool length presetting on the inductive probe in the same contact point as for the
6mm tool, with the cooling air off, the spindle turned off, and recording the
spindle temperature at the moment of presetting.
7. Storing the tool length value in the machine tool control, corrected for the
temperature difference between point 6 and the steady state temperature for
desired rotational speed.
8. Spindle warm up with a selected rotational speed to the steady state
temperature.
9. Performing the micro milling operation; if another micro end mill is needed to be
used, repeating the procedure from the point 6.
Test run procedure
1. Adjust the flow of cutting fluid on the workpiece, turn it off or leave the surface
wet according to demand.
2. Start a program in machine tool computer with desired cutting parameters.
3. Just before engagement of a tool to the workpiece turn the amplifiers on, with
desired measuring range and start the data acquisition in Labview program.
4. After an experiment, stop the program, stop the acquisition, check acquired
data.
Modified test run procedure
During micro milling experiments, an error of axial depth of cut causing a tool
breakage was observed, although the thermal analysis was performed and all
parameters were carefully selected. Since it was not possible to interrupt current
experiments, the modified procedure was accepted. It consists of performing one or
two additional passes over the workpiece, until the tool touched the surface. When
the contact between tool and workpiece was observed, the next pass of the tool was
recorded.
After experiments an additional thermal analysis was performed. It was
discovered, that the pump maintaining circulation of cutting fluid was not working
properly and extremely increased the fluid temperature. This contributed to
continuous machine tool structure deformation, which was even after 1 hour of
machine warm up higher than 1,3µm per hour [30].
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Chapter
6
Results
Turning Experiments
Page 68
Enhancement and verification of a cutting
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Jan Slunský
Page 69
This chapter deals with analysis of data obtained in the turning experiments and
their implementation into the model. The comparison of predicted and acquired forces
is given in the last chapter.
6.1 Data from the turning experiments
Turning experiments were performed with two work materials – the steel UHB11
and the aluminium 6082 T6. The results needed for implementation into the model
are presented here, but only detailed description of data treatment for aluminium is
made, as the procedure is the same for both work materials. All necessary data are
available in Appendix A and B.
6.1.1 Chip thickness
The results from chip measurements are reported in Appendix A for the
aluminium and in Appendix B for the steel. For each test, the average chip thickness
is represented by mean value of 5 measurements. The chip thickness ratio diagrams
(Graph 9, Graph 10 and Graph 11), which are later used in the model for calculation
of the shear angle in oblique cutting, are estimated from the average chip thickness
values for each tool holder angle. The points have been fitted with a power law in
each diagram.
Considering the chip thickness ratio and the nominal rake as a variable, the
least square fitting gives the following expressions:
rt = (0,0015 ⋅ α n + 0,2283 ) ⋅ h 0,0014⋅αn + 0,1151 for the aluminium,
(40)
rt = (0,0044 ⋅ α n + 0,2868 ) ⋅ h −0,0074⋅αn +0,0145 for the steel,
(41)
where h is the uncut chip thickness and αn is the normal rake angle. These equations
imply that the chip thickness ratio strongly varies with the uncut chip thickness, while
the influence of the rake angle is quite small.
Chip thickness ratio
Aluminium -5°
Chip thickness ratio h/hc
0,4
0,35
y = 0,2182x0,1145
R2 = 0,9283
0,3
0,25
0,2
0,15
0
20
40
60
80
100
120
Uncut chip thickness [µm]
Graph 9: Chip thickness ratio in the aluminium for a nominal rake angle of -5°
Enhancement and verification of a cutting
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Jan Slunský
Page 70
Chip thickness ratio
Aluminium -30°
Chip thickness ratio h/hc
0,4
0,35
y = 0,1928x0,0532
R 2 = 0,5056
0,3
0,25
0,2
0,15
0
20
40
60
80
100
120
Uncut chip thickness [µm]
Graph 10: Chip thickness ratio in the aluminium for a nominal rake angle of -30°
Chip thickness ratio
Aluminium -40°
Chip thickness ratio h/hc
0,4
0,35
0,3
0,25
y = 0,162x0,0764
R 2 = 0,4272
0,2
0,15
0
20
40
60
80
100
120
Uncut chip thickness [µm]
Graph 11: Chip thickness ratio in the aluminium for a nominal rake angle of -40°
6.1.2 Forces analysis
Two approaches have been used for determining the force components. The
example of test run number 20 in Fig. 42 shows the power and thrust force (the FX
force is negative here, because of the reference system of dynamometer, but for the
calculations the values are positive) for an uncut chip thickness of 50µm and a
nominal rake angle of -5°.
“First rotation” approach
The uncut chip thickness should be reached after one full rotation from the
moment of the first contact between tool and workpiece. In this example, for given
cutting parameters, one rotation takes approximately 0,44 seconds and just in this
moment the force components are estimated.
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 71
“Plateau” approach
As we can see in Fig. 42, the force components continue increasing after the
full first rotation. This could be attributed to the development of wear and build-upedge at the cutting edge. Therefore the cutting forces were estimated after they
reached a stable level. The both approaches are implemented in the model for later
comparison.
250
FZ = 192 N
200
150
FZ = 139 N
Force [N]
100
50
0
8,5
0,44s
9,5
Time [s]
10,5
-50
-100
FX = -125 N
FX = -173 N
-150
-200
-250
Fx [N]
Fy [N]
Fz [N]
Fig. 42: Two approaches for determining the cutting force components (test no. 20)
Table 11: Estimated cutting force components (test no. 20)
Fx [N]
Fz [N]
“First rotation” approach
-125
139
“Plateau” approach
-173
192
Graph 12 and Graph 14 reports the estimated forces by the “first rotation”
approach, while Graph 13 and Graph 15 are estimated by the “plateau” approach.
From the graphs is seen the effect of nominal rake angle on the measured forces
below a certain value of the feed. As the forces at zero cut thickness obtained for the
different nominal rake angles should be the same, the tendency of measured forces
cannot be presented only by straight lines over the whole range of feeds. Therefore
for a feed bellow 25µm/rev the experimental points are approximated by exponential
function.
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 72
Fx Aluminum - first rotation
Fx in Aluminium – first rotation
1000
-5°
-30°
-40°
900
800
force [N]
700
600
500
400
300
200
100
0
0
10
20
30
40
50
60
70
80
90
100
feed [µm]
Graph 12: Measured thrust force values in the aluminium, “first rotation”
FxinAluminum
- plato
Fx
Aluminium
– plateau
1000
-5°
-30°
-40°
900
800
force [N]
700
600
500
400
300
200
100
0
0
10
20
30
40
50
60
70
80
feed [µm]
Graph 13: Measured thrust force values in the aluminium, “plateau”
90
100
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 73
- first
rotation
FzFzinAluminum
Aluminium
– first
rotation
1000
-5°
-30°
-40°
900
800
force [N]
700
600
500
400
300
200
100
0
0
10
20
30
40
50
60
70
80
90
100
feed [µm]
Graph 14: Measured power force values in the aluminium, “first rotation”
Fz Aluminum - plato
Fz in Aluminium – plateau
1000
-5°
-30°
-40°
900
800
force [N]
700
600
500
400
300
200
100
0
0
10
20
30
40
50
60
70
80
90
100
feed [µm]
Graph 15: Measured power force values in the aluminium, “plateau”
Once the mean values of the forces and chip thickness ratios have been
determined, the data treatment can start. In the first step the edge force components
FXe and FZe are determined from the intercepts of the force – feed (uncut chip
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 74
thickness) function with y-axis, it means the value of the force at zero uncut chip
thickness. These values, presented in the Table 12) were found in diagrams for steel
and aluminium, and for the both approaches. The edge force coefficients are
represented by edge force components per unit cut width:
K re =
FXe
b
(42)
K te =
FZe
b
The value of Kae coefficient is usually very small and can be taken as zero,
although alternative estimates have sometimes have been used – e.g. by Armarego
and Deshpande (1993):
K ae = K te ⋅ sin i
(43)
After that, the forces due to the cutting FXc and FZc are found by subtracting the
edge force components from the mean values of FX and FZ forces.
FXc = FX − FXe
(44)
FZc = FZ − FZe
The orthogonal cutting parameters are found than from estimated force
components due to the cutting using these equations:
tan φ =
τ=
rt ⋅ cos α
1 − rt ⋅ sin α
(FZc ⋅ cos φ − FXc ⋅ sin φ) ⋅ sin φ
tan β =
(45)
b⋅t
FXc + FZc ⋅ tan α
FZc − FXc ⋅ tan α
Table 12: Edge forces and coefficients
First rotation
FXe [N]
28
FZe [N]
22
Steel
Kre [N/mm]
14
Kte [N/mm]
11
Kae [N/mm]
5,5
FXe [N]
26
FZe [N]
13
Aluminium Kre [N/mm]
13
Kte [N/mm]
6,5
Kae [N/mm]
3,25
Plateau
45
36
22,5
18
9
72
54
36
27
13,5
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 75
6.2 Implementation of the parameters into the model
The parameters obtained from the orthogonal tests need to be implemented into
the model with several considerations stated in this chapter.
The mean value of shear stress and the edge coefficients are implemented
directly into the model. The shear angle is calculated directly for actual rake angle
and chip thickness ratio, which is computed in the model for actual uncut chip
thickness and actual rake angle from equation (40) or (41).
The calculation of the friction angle is a bit more complicated. The friction angle
β (equation (45)) takes into account the actual rake angle, which is different from the
nominal one when the uncut chip thickness is under hlim – equation (36). This situation
is shown in Graph 16 – the friction angle is approximately constant above hlim, while it
strongly depends on the uncut chip thickness at values below hlim. Thus, the
relationship between friction angle and uncut chip thickness is represented by two
different equations – one is valid above hlim and the other below hlim.
The procedure is same for the both work materials and also for the both
approaches. The relationships implemented in the model are presented in Table 13.
First rotation
Plateau
Aluminium
Table 13: Relationships implemented in the model
Shear stress
[MPa]
τ = 276,0 ± 17,1
Friction angle for h<hlim
[deg]
β = −0,0163 ⋅ h 2 + 2,0244 ⋅ h − 27
Friction angle for h>hlim
[deg]
β = −0,0219 ⋅ h + 34,61
Shear stress
[MPa]
τ = 288,7 ± 12,4
Friction angle for h<hlim
[deg]
β = −0,0186 ⋅ h 2 + 2,4640 ⋅ h − 47
Friction angle for h>hlim
[deg]
β = −0,0319 ⋅ h + 28,15
Plateau
Steel
First rotation
Chip thickness ratio
[-]
rt = (0,0015 ⋅ α n + 0,2283 ) ⋅ h 0,0014⋅αn + 0,1151
Shear stress
[MPa]
τ = 406,6 ± 36,0
Friction angle for h<hlim
[deg]
β = −0,0197 ⋅ h 2 + 2,4070 ⋅ h − 44
Friction angle for h>hlim
[deg]
β = −0,0030 ⋅ h + 27,19
Shear stress
[MPa]
τ = 649,7 ± 12,1
Friction angle for h<hlim
[deg]
β = −0,0181 ⋅ h 2 + 2,2666 ⋅ h − 44
Friction angle for h>hlim
[deg]
β = −0,0070 ⋅ h + 24,334
Chip thickness ratio
[-]
rt = (0,0044 ⋅ α n + 0,2868 ) ⋅ h −0,0074⋅αn +0,0145
Enhancement and verification of a cutting
force model for micro cutting
Jan Slunský
Page 76
Friction angle in Aluminium - plateau
40
30
friction angle [deg]
20
10
0
0
10
20
30
40
50
60
70
80
90
100
-10
hlim for nominal rake angle of -5°
-20
-30
-40
-50
uncut chip thickness [µm]
Graph 16: Friction angle calculated from the orthogonal data (experiments with
nominal rake angle set to -5°) considering the actu al rake angle
Considering that the parameters for the model are calculated from the
equations (45) based on the ideal tool geometry, the relationships from Table 13
could be applied only to cutting edges having exactly the same geometrical and
dimensional characteristics as those used in the orthogonal experiments. Therefore
the improvement of the model taking into account the actual geometry, and assuming
that the absolute dimensions are not responsible for the effects of the actual process
geometry, was proposed by the author [18]. In his proposal the dependencies of the
actual rake angle, chip thickness ratio and friction angle on the uncut chip thickness
are expressed in a parametric form using the ratio h/Re as a parameter, and
implemented in the calculation of the shear angle, friction angle and eventually the
cutting coefficients Kic.
The Matlab script of the model is available in Appendix C with explanation of the
procedures, cycles and algorithms.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Chapter
7
Results
Micro Milling Experiments
&
Model Validation
Page 77
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 78
After the experimental data from orthogonal cutting tests were processed and
implemented into the model, the program was tested against the forces obtained from
the end-milling experiments. Before starting the comparison of predicted and
measured forces, the program simulation was performed in order to determine the
effect of Stabler rule and the functionality of the run-out calculation. The diagrams
presented here show the forces acting on the workpiece.
7.1 The effect of Stabler rule removal
In Graph 1 on the page 36 it was shown, that the numerical calculation, used
instead of the Stabler rule, has strong influence on estimation of the chip flow angle.
Nevertheless the calculation was made for constant friction angle and rake angle,
while in the micro end-milling operation the parameters of cutting process are
changing during the cutter engagement with the variation of uncut chip thickness. As
the influence of Stabler rule was not so apparent, the simulation of a slot milling
operation with a 0,6mm end mill was performed in the Matlab software resulting in
following graph:
Graph 17: Predicted forces using the Stabler rule or the numerical calculation of the
chip flow angle
The simulation has proved negligible effect of the Stabler rule on the force
prediction. As shown in Graph 17, the mostly influenced force component is the one
in axial direction. If the Stabler rule is not used, the chip flow angle has to be
computed for each differential cutting edge element by the numerical solution of the
system of nonlinear equations, and it is very time consuming. Therefore the force
diagrams presented here are modelled through the use of the Stabler rule.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 79
7.2 The effect of the tool run-out
The tool run-out calculation has been implemented in the model and the
example of its simulation in Matlab software is shown for following parameters: milling
of the UHB 11 steel with 0,6mm end-mill. The tool is engaging into the workpiece at
50% of its diameter with depth of cut of 0,045mm. Further process parameters are
stated in Graph 18. The axis of the tool is displaced from the axis of the rotation by
2µm in the direction of the first flute with the zero inclination angle. The dashed lines
in the diagram represent the forces caused by ideal tool, while the solid lines illustrate
the tool with the run-out.
Graph 18: Predicted forces caused by the tool with zero run-out and by the tool with
the run-out of 2µm
It is important to note, that the diagram above represents the conditions, where
the tool is considered rigid with no deflections. Though the forces are relatively small
in the micro milling, their effect on the small tool diameter is not negligible. In fact, if
the tool run-out is not too big, it can be compensated by the tool deflection, which is
caused by the forces acting during cutting operation. It was proved in micro milling
experiments performed here, that the effect of tool run-out was not observed on the
acquired curves, although the run-out of the tool (in the range of the simulated run-out
above) was measured before the experiments.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 80
7.3 Validation of the model through the micro milling
experiments
In the following diagrams, the comparison between measured and predicted
milling forces is shown for three values of radial depth of cut: 10%, 50% and 100%
engagement. The axial depth of cut and the feed rate were constant at 0,045mm and
384mmmin-1, respectively. Helical end mill with diameter of 0,6mm and normal angle
of 0° was used for the both work materials. The rot ational speed was set to
16000rpm, which corresponds to cutting speed of 30mmin-1 used for obtaining the
data for the model in the orthogonal tests.
The predicted and measured forces on the aluminium will be discussed first.
Since the aluminium is highly heat conductive material, the cutting process had a
negative effect on the force acquisition in the axial Z-direction. The drift of the Fz
force component shown in Graph 19 was caused by the combination of the
dimensional expansion of the material and the specific construction of the
dynamometer device. Therefore the z-force component was not taken into account for
the aluminium.
Graph 19: The effect of the aluminium workpiece expansion due to the heat transfer
Two approaches have been used for the analysis of the forces obtained in the
turning experiments and therefore the model had to be tested. The first rotation
approach has been identified as the best-fit for the force prediction on the aluminium,
while the plateau approach gave too high values for the both force components
comparing to the experimental data. In the diagrams bellow, the measured forces are
displayed together with the forces predicted by the first approach. A good
correspondence in the shape and also in the peak values is visible.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 81
Graph 20: Predicted and measured forces on aluminium, slot milling
Graph 21: Predicted and measured forces on aluminium, 50% engagement
The closeness of the predicted and measured forces is decreasing for lower
radial engagement values, but is still good enough. The lower exactness can be
caused by higher noises visible in the diagrams.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 82
Graph 22: Predicted and measured forces on aluminium, 10% engagement
From the previous graphs is apparent, that the prediction of force components
in the milling of the aluminium works quite well. The prediction for the steel in for
selected cutting parameters shows also good agreement, but the result is not as
potent as in the case of aluminium. For the steel, the best-fit of the force prediction
was achieved with the plateau approach.
Graph 23: Predicted and measured forces on steel, slot milling
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 83
Graph 24: Predicted and measured forces on steel, 50% engagement
Graph 25: Predicted and measured forces on steel, 10% engagement
In the last two diagrams, the predicted force acts in smaller time range than the
acquired one. If we look closer on Graph 24 for the 50% of the radial engagement,
the time interval of acting force should be the equal to the time interval of the zero
force, as the model predicts. But the acquired forces act in a longer time interval, that
means the bigger engagement probably caused by a positioning error during the
milling experiment. The same event occurred in Graph 25.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 84
Although the prediction for the steel showed relatively good correspondence to
the measurements for selected parameters, in the other tests with different
parameters and tools the results were worse. In some cases the acquired force was
twice the predicted one. But it is important to note, that the noises in force
measurements were much higher for the steel comparing to the aluminium and may
affect the results. Some of the nonconforming predictions are shown in following
diagrams, but even here the shape of the curves is predicted well.
Graph 26: Predicted and measured forces on steel, 50% engagement
Graph 27: Predicted and measured forces on steel, 100% engagement
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Chapter
8
Conclusions
Page 85
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 86
Micro fabrication technologies are able to produce high-accuracy miniaturized
components, demanded for various industries, such as biomedical, aerospace,
automotive, electronics etc. Micro milling is the one method for creating miniature
devices and components. In combination with replication techniques it has the strong
advantages for the mass production of micro products. In order to manage and
optimize the economic performance of the milling operation, the cutting force
predictions are needed.
Therefore, the unified mechanics of cutting approach to the prediction of forces
in milling operations was adopted and extended here. The method involves the
establishment of the database of basic cutting quantities from set of orthogonal
cutting tests at various cutting conditions. The cutting quantities were obtained in the
turning experiments of the aluminium 6082 T6 and the carbon steel UHB 11 and
subsequently implemented into the model.
The model was experimentally verified by comparing the predicted and
measured milling force components. It was shown, that the predicted force
components from the model closely agrees with experimentally measured forces for
the aluminium 6082 T6, when the first rotation approach is used. The plateau
approach used for prediction on the steel UHB 11 has also good response, if the
cutting speed used is close to 30mmin-1, but for other conditions the response is not
as good as it should be. For the quantitative analysis of the predicted forces, more
micro milling experiments should be performed. The inaccuracy of predicted forces
and its non-correspondence to the measured forces does not have to be caused just
by the operation of the model, but also the inaccuracy of measurements and the
precision of the machine tool used in micro milling experiments is significant and
should be considered.
The negligible effect of the Stabler rule was shown, therefore the numerical
calculation, which is very time consuming, does not have to be used for the
estimation of the chip flow angle. It also appeared, that the small run-out of micro end
mills does not have an influence on cutting forces, because it is compensated by the
tool deflections, which are becoming very important in micro milling operations.
The issues in the previous paragraphs lead to the suggestions for the future
work. As a lot of parameters for the model are estimated with the standard deviation,
they could be implemented directly into the model and the force components could be
predicted not just in the absolute values, but also with the deviations. Also tool
deflections are very important, since they significantly influence the accuracy of the
machined part. Therefore the implementation of the tool deflection in the model would
be needed.
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 87
References
[1]
World Technology Evaluation Center (WTEC), Inc. – Ehmann, K.F. at al.,
International Assessment of Research and Development in Micromanufacturing.
2005.
[2]
Liu, X. et al., The Mechanics of Machining at the Microscale: Assessment of the
Current State of the Science. Journal of Manufacturing Science and
Engineering, Transactions of the ASME. 2004, vol. 126, no. 2, p 666-678.
[3]
Pham, D.T. et al., Laser Milling as a "rapid" Micromanufacturing Process.
Proceedings of the Institution of Mechanical Engineers, Part B: Journal of
Engineering Manufacture. 2004, vol. 218, no. 3, p 1-7.
[4]
Dario, P. et al., Non-traditional Technologies for Microfabrication. Journal of
Micromechanics and Microengineering. 1995, vol. 5, no. 4, p 64-71.
[5]
Sun, X.-Q., Masuzawa, T. and Fujino, M., Micro Ultrasonic Machining and Selfaligned Multilayer Machining/Assembly Technologies for 3D Micromachines.
Micro Electro Mechanical Systems, 1996, MEMS "96, Proceedings. "An
Investigation of Micro Structures, Sensors, Actuators, Machines and Systems".
IEEE, The Ninth Annual International Workshop on. 1996, no. 5, p 312-317.
[6]
Denkena, B. et al., Micro-machining Processes for Microsystem Technology.
Microsystem Technologies. 2006, vol. 12, no. 16, p 659-664.
[7]
Journal of Micromechanics and Microengineering, PII: S0960-1317(04)68657-8
[8]
Hot Embossing for Micro-Optical Components, POLYMICRO Newsletter
01/2003, available on the World Wide Web: http://www.polymicrocc.com/site/pdf/POLYMICRO-tech_Hot-embossing.pdf
[9]
Yamaguchi, K. et al., Generation of Three-dimensional Micro Structure using
Metal Jet. Precision Engineering. 2000, vol. 24, no. 8, p 2-8.
[10] Buttgenbach, S., Spotlights on Recent Developments in Microsystem
Technology. Design Automation Conference, 1996, with EURO-VHDL "96 and
Exhibition, Proceedings EURO-DAC "96, European. 1996, no. 10, p 274-279.
[11] Baker, G.N., Quartz Rate Sensor from Innovation to Application. Proc. Gyro
Technology Symposium, Stuttgart, 1992.
[12] http://www.dolomite-centre.com/
[13] Engineering Feat (Cover Story), available on the World Wide Web:
http://www.designnews.com/index.asp?layout=article&articleid=CA452888
[14] Chae, J., Park, S.S. and Freiheit, T., Investigation of Micro-cutting Operations.
International Journal of Machine Tools and Manufacture. 2006, vol. 46, no. 9, p
313-332.
[15] Son S.M., Lim H.S., Ahn J.H., Effects of the Friction Coefficient on the Minimum
Cutting Thickness in Micro Cutting. International Journal of Machine Tools and
Manufacture. 2005, vol. 45, p 529-535.
[16] Masuzawa, T., State of the Art of Micromachining. Annals of CIRP. 2000, vol.
49, no. 2, p 473-488.
[17] MMS Online, Machining Under The Microscope, available on the World Wide
Web: http://www.mmsonline.com/articles/010504.html
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 88
[18] Bissacco, G., Surface Generation and Optimization in Micromilling. Technical
University of Denmark. 2004, MM 04.67.
[19] Merchant, M.E., An Interpretive Look at 20th Century Research on Modeling of
Machining. Proceedings of the CIRP International Workshop on Modeling of
Machining Operations. 1998, p 27-31.
[20] Komanduri, R., In Memoriam: M. Eugene Merchant. Transactions of the ASME B - Journal of Manufacturing Science and Engineering. 2006, vol. 128, no. 18, p
1034.
[21] Armarego, E.J.A., A Generic Mechanics of Cutting Approach to Predictive
Technological Performance Modelling of the Wide Spectrum of Machining
Operations. Machining Science and Technology. 1998, vol. 2, no. 2, p 191-211.
[22] Armarego, E.J.A., Unified-generalized Mechanics of Cutting Approach - a Step
Towards a House of Predictive Performance Models for Machining Operations.
Machining Science and Technology. 2000, vol. 4, no. 3, p 319-362.
[23] Budak, E., Altintas, Y. and Armarego, E.J.A., Prediction of milling force
coefficients from orthogonal cutting data. Journal of Manufacturing Science and
Engineering, Transactions of the ASME. 1996, vol. 118, p 216-224.
[24] Shaw, M.C., Metal Cutting Principles. Oxford University Press, New York, 1989.
[25] Hoffmann, J., Pedersen, K.B., Design and Calibration of Dynamometer for CNC
Lathe. 14th NAMRE. 1984, p 183-188.
[26] Axinte, D.A., Belluci, W., Calibration Procedure for 3D Turning Dynamometer.
Technical University of Denmark. 1999, MM 99.47.
[27] AB Sandvik Coromant, Main catalogue 2006, available on the World Wide Web:
http://www2.coromant.sandvik.com/coromant/pdf/Metalworking_Products_061/
main_1.pdf
[28] NSK Nakanishi Inc, Precision & reliability, Micro grinders, motors & spindles,
March 2002 edition
[29] OSG WX21 Carbide End Mill Series ultra WX, Product Information,
http://www.cresttech.com.au/OSG.html
[30] Baú, A., Design and construction of a dynamometer for cutting force
measurement in micromilling. 2007
Jan Slunský
Enhancement and verification of a cutting
force model for micro cutting
Page 89
APPENDIX A – DATA FROM TURNING EXPERIMENTS OF THE
ALUMINIUM 6082 T6
APPENDIX B – DATA FROM TURNING EXPERIMENTS OF THE
STEEL UHB 11
APPENDIX C – MATLAB SCRIPT
Appendix A
number
feed
-5°
1
7
14
2
8
16
19
3
15
20
4
10
17
9
13
21
6
11
22
5
12
18
100
100
100
75
75
75
75
50
50
50
25
25
25
10
10
10
8
8
8
5
5
5
chip thickness
AVG
STD
270,80
5,46
274,80
7,08
276,00
9,10
204,00
8,20
213,80
16,89
204,20
4,53
206,00
9,34
143,80
4,12
152,40
5,12
146,40
3,56
75,80
2,32
81,00
6,63
83,40
3,01
33,20
1,17
33,40
2,06
36,20
0,75
28,20
1,60
29,40
1,85
28,00
1,41
18,80
1,72
20,8
1,6
19,20
1,72
-30°
Data from turning experiments of the aluminum 6082 T6
49
59
63
72
50
60
64
51
57
61
52
56
65
69
73
74
53
58
100
100
100
100
75
75
75
50
50
50
25
25
25
25
25
25
10
10
478,8
431,20
497,0
392,80
309,20
313,80
326,60
226,60
215,40
195,20
109,00
107,6
112,60
111,80
97,60
99,80
47,20
NA
25,6
17,59
18,9
35,79
17,63
24,34
38,21
5,20
20,40
31,51
12,62
3,1
2,24
4,45
1,85
6,18
1,72
NA
forces - first rotation
Fx
Fz
211,17
263,30
204,74
255,40
203,25
236,71
172,63
207,90
174,3
199,3
NA
NA
157,04
193,75
140,50
145,89
124,26
128,37
125,42
138,98
89,65
78,32
88,22
83,42
83,16
68,04
55,53
29,98
45,90
31,20
53,50
35,39
40,45
28,17
32,86
22,75
44,51
25,79
33,13
20,08
32,1
15,3
32,40
21,09
376,95
385,02
412,57
376,39
323,48
332,33
333,87
207,58
196,62
239,67
123,94
111,2
128,87
118,37
130,91
118,15
55,88
44,35
343,35
333,24
337,70
352,55
242,51
240,16
237,44
155,81
153,78
170,03
87,68
87,4
80,42
86,53
89,33
80,44
29,83
30,60
forces - plato
Fx
Fz
234,46
289,39
240,07
285,08
258,35
296,45
193,42
236,26
211,0
245,6
NA
NA
209,77
244,34
157,89
175,54
153,96
181,12
172,88
192,29
101,44
105,04
118,11
120,90
113,44
116,70
84,69
74,20
80,88
70,64
76,55
65,73
73,70
59,70
77,51
67,36
66,09
58,33
56,50
46,61
69,6
50,5
66,41
50,60
NA
682,16
700,35
695,66
491,79
516,63
511,50
361,62
331,11
364,58
225,28
196,2
243,71
205,25
248,62
219,13
120,91
NA
NA
532,89
537,46
536,90
406,42
412,29
412,74
299,26
275,10
296,35
180,45
161,8
189,81
167,48
194,65
173,13
95,87
NA
number
feed
-30°
66
70
54
67
71
55
62
68
10
10
8
8
8
5
5
5
chip thickness
AVG
STD
44,20
1,72
43,60
1,85
35,00
1,79
38,60
1,02
37,20
1,17
23,60
1,20
26,20
1,72
24,80
1,47
-40°
144
145
129
134
139
130
135
140
131
136
141
132
137
142
133
138
143
50
50
25
25
25
18
18
18
10
10
10
8
8
8
5
5
5
245,20
231,80
112,20
118,20
126,00
81,60
88,60
94,40
55,00
50,20
48,40
43,20
41,00
38,80
28,20
31,80
25,60
Aluminum
15,63
13,50
3,76
11,39
21,23
2,73
3,26
2,42
3,63
2,32
2,65
2,86
2,28
2,14
2,93
5,00
2,33
FXe
FZe
forces - first rotation
Fx
Fz
40,39
20,76
54,82
25,56
40,20
29,39
38,61
17,91
41,37
25,94
38,75
21,09
35,33
19,05
33,49
17,30
367,88
357,79
140,90
137,68
145,64
97,62
97,54
83,18
74,09
57,17
60,22
36,27
44,32
54,28
26,31
31,38
35,36
Edge Forces
First rotation
26,2
13,2
240,87
234,38
98,21
98,20
107,32
53,15
62,16
48,05
35,10
31,16
24,84
27,70
27,18
23,74
18,28
15,17
15,63
Plato
71,8
53,6
forces - plato
Fx
Fz
129,62
97,31
129,01
93,24
103,02
81,81
105,86
83,23
108,10
88,66
81,75
63,53
101,48
71,87
87,89
62,45
510,87
561,14
255,99
265,75
282,50
240,47
234,29
227,99
149,09
144,83
155,39
122,37
124,34
124,01
94,51
97,25
95,05
363,00
396,31
195,80
203,70
209,10
179,86
177,70
166,35
110,66
106,85
112,77
93,89
98,80
97,12
68,05
68,85
69,41
Chip thickness ratio
Aluminium -5°
Chip thickness ratio h/hc
0,4
0,35
y = 0,2182x 0,1145
R2 = 0,9283
0,3
0,25
0,2
0,15
0
20
40
60
80
100
120
Uncut chip thickness [µm ]
Chip thickness ratio
Alum inium -30°
Chip thickness ratio h/hc
0,4
0,35
y = 0,1928x 0,0532
R2 = 0,5056
0,3
0,25
0,2
0,15
0
20
40
60
80
100
120
80
100
120
Uncut chip thickness [µm ]
Chip thickness ratio
Alum inium -40°
Chip thickness ratio h/hc
0,4
0,35
0,3
0,25
y = 0,162x 0,0764
R2 = 0,4272
0,2
0,15
0
20
40
60
Uncut chip thickness [µm ]
Fx Aluminum - plato
Fx Aluminum - first rotation
-5°
-30°
1000
-40°
900
900
800
800
700
700
600
600
force [N]
force [N]
1000
500
400
-30°
400
300
200
200
100
100
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
feed [µm]
1000
-5°
-30°
50
60
70
80
90
100
70
80
90
100
feed [µm]
Fz Aluminum - first rotation
Fz Aluminum - plato
1000
-40°
900
900
800
800
700
700
600
600
force [N]
force [N]
-40°
500
300
0
-5°
500
400
-30°
-40°
500
400
300
300
200
200
100
100
0
-5°
0
0
10
20
30
40
50
feed [µm]
60
70
80
90
100
0
10
20
30
40
50
feed [µm]
60
Appendix B
number
feed
-5°
23
33
46
24
34
37
47
25
35
40
48
26
31
36
41
44
27
30
38
45
29
39
43
28
32
42
100
100
100
75
75
75
75
50
50
50
50
25
25
25
25
25
10
10
10
10
8
8
8
5
5
5
chip thickness
AVG
STD
301,00
10,73
293,60
10,89
295,00
15,40
229,60
8,19
228,00
8,72
220,00
5,90
227,40
8,89
150,60
10,38
158,00
7,04
151,80
12,81
147,40
9,22
NA
NA
84,20
5,71
79,00
9,96
75,64
4,15
75,58
3,19
31,04
2,04
34,68
1,72
33,30
1,00
30,78
1,04
26,98
0,75
26,12
1,05
27,20
0,77
19,20
1,15
18,06
0,67
17,92
1,05
-30°
Data from turning experiments of the steel UHB11
75
99
100
101
83
85
90
76
77
86
91
78
84
87
100
100
100
100
75
75
75
50
50
50
50
25
25
25
NA
240,20
247,40
238,40
195,00
186,60
181,60
NA
129,00
133,40
124,20
71,00
73,34
69,68
NA
10,93
10,63
16,01
6,84
7,12
2,73
NA
3,63
5,68
6,37
2,60
5,83
5,15
forces - first rotation
Fx
Fz
254,92
374,68
251,96
388,24
245,35
362,76
184,31
270,51
202,66
300,84
179,92
269,50
190,92
278,27
NA
NA
157,42
235,58
109,25
166,56
144,23
196,88
NA
NA
90,57
136,05
94,59
142,05
98,68
121,90
92,61
122,94
68,14
34,18
41,31
61,19
49,97
65,96
43,79
46,82
32,83
45,45
30,37
54,15
31,44
50,19
31,38
34,59
25,06
39,67
36,18
27,21
586,21
555,84
525,63
547,51
450,92
424,51
449,84
NA
295,27
276,23
248,19
167,60
172,36
171,13
631,42
639,69
661,95
600,20
419,62
445,20
408,88
NA
314,63
316,70
252,83
140,64
167,83
144,89
forces - plato
Fx
Fz
343,09
569,15
356,03
574,40
334,42
561,01
257,54
430,53
264,35
444,90
268,69
446,87
269,61
446,24
NA
NA
212,80
319,58
199,53
315,04
187,28
302,58
NA
NA
130,21
189,38
120,30
189,32
132,41
195,57
129,16
195,64
70,99
93,28
67,44
94,01
69,19
98,83
64,34
95,63
64,72
84,92
54,31
78,18
60,24
77,42
51,85
59,03
47,08
57,97
47,21
60,74
510,36
506,70
526,85
516,94
375,74
398,99
317,09
NA
273,81
266,29
230,75
142,27
156,92
166,13
679,23
661,06
686,90
670,04
533,56
530,30
482,31
NA
349,86
362,41
295,63
188,73
184,71
213,91
number
feed
-30°
92
94
98
79
82
93
95
80
88
96
81
89
97
25
25
25
10
10
10
10
8
8
8
5
5
5
chip thickness
AVG
STD
NA
NA
67,44
5,13
67,80
5,11
33,50
1,39
35,72
2,40
38,72
2,18
33,08
3,01
31,36
1,40
30,02
1,71
33,34
1,18
23,28
1,78
22,46
1,02
25,74
1,56
-40°
126
127
111
116
121
112
117
122
113
118
123
115
119
124
114
120
125
50
50
25
25
25
18
18
18
10
10
10
8
8
8
5
5
5
155,40
148,00
83,46
79,90
86,70
71,72
69,56
65,60
41,56
48,10
43,82
36,80
41,00
42,84
31,82
29,62
33,06
Steel
9,69
5,73
6,74
5,05
6,49
4,97
4,38
3,77
6,66
2,11
2,16
3,19
2,95
2,55
2,96
2,11
3,08
FXe
FZe
forces - first rotation
Fx
Fz
176,12
163,25
171,81
159,81
181,78
154,20
72,01
56,21
55,39
51,23
80,84
58,77
88,35
64,12
59,09
48,83
48,30
35,77
57,63
46,33
35,53
26,95
46,87
31,31
33,01
28,10
497,40
462,91
213,18
269,65
212,66
158,10
146,24
137,00
100,75
78,82
85,54
52,34
78,41
62,29
32,79
44,84
39,99
Edge Forces
First rotation
28,5
22,6
432,27
541,56
197,88
226,03
213,80
107,29
118,32
135,64
86,69
91,84
80,60
41,86
39,79
27,16
21,49
18,44
36,13
Plato
45,2
35,9
forces - plato
Fx
Fz
176,21
188,72
141,87
169,27
159,84
189,66
67,54
83,29
69,51
75,69
90,24
92,70
73,55
63,10
72,53
68,04
67,27
74,81
62,32
57,56
63,12
53,76
58,28
42,14
49,65
34,09
469,64
436,34
227,82
208,92
210,87
172,80
174,40
187,77
103,14
107,34
107,54
80,01
89,36
80,66
62,16
61,31
55,15
468,40
469,59
239,57
213,59
218,52
159,73
141,09
195,37
82,23
90,46
93,14
70,98
74,34
63,75
44,38
40,80
49,19
Chip thickness ratio
Steel -5°
0,5
Chip thickness ratio h/hc
0,45
y = 0,2612x 0,0579
R2 = 0,696
0,4
0,35
0,3
0,25
0,2
0,15
0
20
40
60
80
100
120
Uncut chip thickness [µm ]
Chip thickness ratio
Steel -30°
0,5
Chip thickness ratio h/hc
0,45
0,4
0,35
y = 0,1653x 0,2136
R2 = 0,8927
0,3
0,25
0,2
0,15
0
20
40
60
80
100
120
80
100
120
Uncut chip thickness [µm ]
Chip thickness ratio
Steel -40°
0,5
Chip thickness ratio h/hc
0,45
0,4
y = 0,1004x 0,3262
R2 = 0,9195
0,35
0,3
0,25
0,2
0,15
0
20
40
60
Uncut chip thickness [µm ]
Fx Steel - first rotation
-5°
-30°
1000
-40°
900
900
800
800
700
700
600
600
force [N]
force [N]
1000
Fx Steel - plato
500
400
-30°
400
300
200
200
100
100
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
feed [µm]
1000
-5°
-30°
50
60
70
80
90
100
70
80
90
100
feed [µm]
Fz Steel - first rotation
Fz Steel - plato
1000
-40°
900
900
800
800
700
700
600
600
force [N]
force [N]
-40°
500
300
0
-5°
500
400
-30°
-40°
500
400
300
300
200
200
100
100
0
-5°
0
0
10
20
30
40
50
feed [µm]
60
70
80
90
100
0
10
20
30
40
50
feed [µm]
60
Appendix C
Matlab Script – Main Program
%
%
%
%
%
%
%
%
%
Calculation of forces for flat end milling (down milling)
References according to Altintas CHANGED!!! - machining in Y axis
Valid for any number of cutting edge
Two different material with two different approaches
- recommended to use the first rotation approach
Calculation using Stabler rule or numerical solution of betan, eta
The run-out calculation valid for any number of flutes
- but considering just pass of the previous one
clear; % clean previous variables => free memory
clc;
% clean screen
fprintf('\n');
% Input geometrical parameters - flat end mill
R=input(' End mill radius [mm] = ');
flutes=input(' number of flutes = ');
io=input(' helix angle [deg] = ');
io=io*pi/180;
% helix angle to [rad]
rake=input(' rake angle on cutting edges [deg] = ');
rake=rake*pi/180;
% rake angle to [rad]
re=input(' cutting edge radius [microns] = ');
fprintf('\n');
fprintf('\n');
fprintf('\n');
fprintf('\n');
fprintf('\n');
% Input conditions of engagement between tool and material
ap=input(' axial depth of cut [mm] = ');
fprintf('\n');
disp '
1: feed [mm/min]';
disp '
2: feed per tooth [mm/tooth]';
fprintf('\n');
feed_selection=input (' Your choice? ');
fprintf('\n');
if feed_selection == 1 % input feed [mm/min]
rpm=input(' rotational speed [rpm] = ');
fprintf('\n');
feed=input(' feed [mm/min] = ');
fprintf('\n');
st=feed/(rpm*flutes);
% convert to [mm/tooth]
else
% input feed per tooth [mm/tooth]
st=input(' feed per tooth [mm/tooth] = ');
fprintf('\n');
end
ae=input(' step over [mm] = ');
fprintf('\n');
1
% Input material characteristics
disp ' Please select workpiece material - approach '
disp '
1: steel UHB11 - first rotation approach ';
disp '
2: steel UHB11 - plateau approach ';
disp '
3: aluminium 6082T6 - first rotation approach ';
disp '
4: aluminium 6082T6 - plateau approach ';
fprintf('\n');
material=input(' Your choice? '); fprintf('\n');
% Imput parameters for axial and angular discretization
n=input(' number of axial slices = '); fprintf('\n');
m=input(' number of angular intervals = '); fprintf('\n');
% Other calculation options
stabler=input(' Use Stabler rule? [y/n] ... ', 's'); fprintf('\n');
% if not, program will calculate eta numerically
runout=input(' Consider tool run-out? [y/n] ... ', 's'); fprintf('\n');
if runout == 'y'
z_level(1)=input(' z(a) [mm] = '); fprintf('\n');
d(1)=input(' d(a) [microns] = '); fprintf('\n');
lambda(1)=input(' lambda(a) [deg] = '); fprintf('\n');
z_level(2)=input(' z(b) [mm] = '); fprintf('\n');
d(2)=input(' d(b) [microns] = '); fprintf('\n');
lambda(2)=input(' lambda(b) [deg] = '); fprintf('\n');
else
z_level=[2 1];
d=[0 0];
lambda=[0 0];
end
lambda=lambda*pi/180;
% convert to [rad]
d=d/1000;
% convert to [mm]
% Calculate derivation parameters
Dzeta=ap/n;
DFi=2*pi/m;
pitch=2*pi/flutes;
2
% Initialization of cycle in Fi (discretization of the rotation (different instants))
Fi=0;
% Condition for determination of angular angle
F=[Fi;zeros(3,1)];
% F = [Fi 0 0 0]
while Fi<=2*pi
Fz=zeros(3,1);
% Fz = [0 0 0]
for j=1:flutes
% Start the cycle for different flutes
if j == 1
previous = flutes;
else
previous = j-1;
end
for i=1:n
(matrix 3x1)
%
%
%
%
%
% Start the cycle in z (axial discretization (the same moment))
z=Dzeta*(i-0.5);
teta=acos(1-ae/R);
Fi0=pi-teta;
Psi=z/R*tan(io);
FiP=Fi-Psi-(j-1)*pitch;
% Position of the point P (the central point of element DS)
if (FiP>=Fi0 & FiP<=pi) | (FiP>Fi0-2*pi & FiP<=-pi)
DS=Dzeta/cos(io);
3
% Engagement condition
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% run-out calculation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for k = 1:2
r(j,k)
= ( d(k)^2 + 2*d(k)*R*cos(lambda(k)+
(j-1)*pitch-z*tan(io)) + R^2 )^(1/2);
r(previous,k) = ( d(k)^2 + 2*d(k)*R*cos(lambda(k)+(previous-1)*pitch-z*tan(io)) + R^2 )^(1/2);
end
dr(j)
= ( r(j,1)
+ (z-z_level(1))*((r(j,1)
-r(j,2))
/(z_level(1)-z_level(2))) ) - R;
dr(previous) = ( r(previous,1) + (z-z_level(1))*((r(previous,1)-r(previous,2))/(z_level(1)-z_level(2))) ) - R;
gama = pi/2 - FiP - atan((R*cos(FiP))/(st+R*sin(FiP)));
if (dr(j)-dr(previous)) > st
% if this condition is true, the previous tooth is not cutting => st=2*st
h=2*st*sin(FiP);
else
h=st*sin(FiP)+dr(j)-dr(previous)/cos(gama); % else the previous tooth path has to be considered
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if h < 0
h=0;
E = [0 0 0];
C = [0,0,0];
else
[rc,tau,rake_true,beta,E]=material_calc(material,rake,re,h);
fin=atan(rc*cos(rake_true)/(1-rc*sin(rake_true)));
% sub_function for materials
% Calculation of the shear angle
if stabler == 'n'
% numerical calculation of betan, eta (removing Stabler rule)
[betan,eta] = runsolve(beta,fin,rake_true,io);
%sub_function for numerical calculation
else
% using Stabler rule
betan=atan(tan(beta)*cos(io));
eta=io;
end
% Calculate the coefficients Kic
c=sqrt((cos(fin+betan-rake_true))^2+(tan(eta))^2*(sin(betan))^2);
Krc=tau/(sin(fin)*cos(io))*sin(betan-rake_true)/c;
Ktc=tau/sin(fin)*(cos(betan-rake_true)+tan(eta)*tan(io)*sin(betan))/c;
Kac=tau/sin(fin)*(cos(betan-rake_true)*tan(io)-tan(eta)*sin(betan))/c;
C=[Krc,Ktc,Kac];
% Insert the Kic to the vector C
end
4
% Calculate the elementary forces in local coordination system
DFl=E'*DS+C'*h*Dzeta;
% vector 3x1 l=local
A=[ cos(FiP),-sin(FiP), 0
%
-sin(FiP),-cos(FiP), 0
% Calculate the rotation matrix
0,
0,-1]; %
DF=A*DFl;
Fz=Fz+DF;
% Calculate the components in X, Y, Z
% Updating the total force
else
end % End the cycle if (FiP>=Fi0 & FiP<=pi) | (FiP>Fi0-2*pi & FiP<=-pi)
end % End the cycle for in z
end % End the cycle for in flutes
Fz=-Fz;
% forces acting on workpiece
Fz1=[Fi*180/pi;Fz]; % Introduction of a new vector for Fi, Fx, Fy, Fz
% Insert Fz as a new row to F
v=size(F);
F(:,v(2)+1)=Fz1;
Fi=Fi+DFi;
end % End the cycle while in Fi
F(:,1)=[];
% Plot the curve
plot(F(1,:),F(2,:),'b',F(1,:),F(3,:),'r',F(1,:),F(4,:),'g');
legend('Fx','Fy','Fz');
title('Cutting forces diagram','FontWeight','bold');
xlabel('Rotation angle [deg]','FontWeight','bold');
ylabel('Cutting forces [N]','FontWeight','bold');
axis([Fi0*180/pi-20,380,1.5*min(min(F(2:4,:))),1.5*max(max(F(2:4,:)))]);
grid;
5
% Save the file in ASCII format
directory='d:\';
filename=input('filename = ', 's');
path=strcat(directory,filename);
fprintf('\n');
fid=fopen(path,'w');
fprintf(fid,'\n\n');
l1='First column = angle (deg)';
l2='Second column = Fx (N)';
l3='Third column = Fy (N)';
l4='Fourth column = Fz (N)';
fprintf(fid,'%s\n%s\n%s\n%s\n',l1,l2,l3,l4);
fprintf(fid,'\n\n');
G=F';
u=size(G);
for i=1:u(1)
D=G(i,:);
fprintf(fid,'%6f\t',D);
fprintf(fid,'\n');
end
fclose(fid);
% end of program
6
Matlab Script – “runsolve” function
function [betan,eta] = runsolve(beta,fin,rake_true,io)
% numerical solution of 2 nonlinear equations to find betan = x(1) and eta = x(2)
x0 = [beta,io];
options = optimset('Display','off');
[x,fval]=fsolve(@objfun,x0,options);
function f = objfun(x)
f(1) = tan(x(1))-tan(beta)*cos(x(2)) ;
f(2) = tan(fin+x(1))-cos(rake_true)*tan(io)/(tan(x(2))-sin(rake_true)*tan(io)) ;
end
betan = x(1);
eta = x(2);
end
7
Matlab Script – “material_calc” function
function [rc,tau,rake_true,beta,E] = material_calc(material,rake,re,h)
% this function returns values of chip thickness ratio, shear stress, actual rake angle, friction angle and edge
coefficients for given material, rake angle, cutting edge radius and uncut chip thickness values
hlim=re*(1+sin(rake))/1000;
x=1000*h/re*50;
% relative chip thickness (h[mm],re[um])
if material == 1 | material == 2
ah= 0.0044*(180/pi)*rake+0.2868;
bh=-0.0074*(180/pi)*rake+0.0145;
% steel
% coefficient for the expression of rc
% coefficient for the expression of rc
if material == 1
tau=407;
% [MPa]
E=[14 11 5.5];
if h<=hlim % Cycle if h<=hlim
rake_true=asin(1000*h/re-1);
beta=-0.0197*x^2+2.407*x-44;
else
% steel - first rotation approach
%[rad]
%[deg]
% Cycle if h>hlim
rake_true=rake;
beta=-0.003*x+27.191;
end
else
% steel - plateau approach
tau=650;
E=[22.5 18 9];
if h<=hlim % Cycle if h<=hlim
rake_true=asin(1000*h/re-1);
beta=-0.0181*x^2+2.266*x-44;
else
% Cycle if h>hlim
rake_true=rake;
beta=-0.007*x+24.334;
end
end
8
else
% aluminium
ah=0.0015*(180/pi)*rake+0.2283;
bh=0.0014*(180/pi)*rake+0.1151;
% coefficient for the expression of rc
% coefficient for the expression of rc
if material == 3
tau=276;
E=[13 6.5 3.25];
% aluminium - first rotation approach
if h<=hlim % Cycle if h<=hlim
rake_true=asin(1000*h/re-1);
beta=-0.0163*x^2+2.0244*x-27;
else
% Cycle if h>hlim
rake_true=rake;
beta=-0.0219*x+34.61;
end
else
% aluminium - plateau approach
tau=289;
E=[36 27 13.5];
if h<=hlim % Cycle if h<=hlim
rake_true=asin(1000*h/re-1);
beta=-0.0186*x^2+2.464*x-47;
else
% Cycle if h>hlim
rake_true=rake;
beta=-0.0319*x+28.154;
end
end
end
rc=ah*x^bh;
% Calculate the chip thickness ratio
beta=beta*pi/180; % beta to [rad]
9