in v en t i o ns

Article
Modeling and Analysis of Contact Conditions during
NC-Form Grinding of Cutting Edges
Eckart Uhlmann and Joachim Bruckhof f *
Institute for Machine T ools and Factory Management, T echnical University Berlin, 10623 Berlin, Germany;
[email protected]
* Correspondence: bruckhof [email protected] ; T el.: +49-30-314-23473
Received: 31 May 2017; Accepted: 29 June 2017; Published: 5 July 2017
Abstract:
Due to incr easing demands on cutting tools, cutting edge preparation is of high priority
because of its influence on the tool life. Curr ent cutting edge preparation pr ocesses are mostly
limited to generating simple r oundings on the cutting edge. Multi-axis high pr ecision form grinding
pr ocesses offer gr eat potential to generate defined cutting edge microgeometries. Knowledge about
the r elation between grinding strategy and material r emoval rate can achieve impr oved work r esults
with r egard to higher pr ecision of shape and dimensional accuracy as well as enhanced cutting edge
quality . Ther efore, a kinematic-geometric model was developed in or der to analyze the complex
contact conditions during grinding cutting edge micr ogeometries by using a simulation appr oach
based on the intersection of geometric bodies. The subsequent grinding tests largely validated the
utilized simulation appr oach.
Keywords: NC-form grinding; cutting edge; contact conditions
1. Introduction
Incr easing demands on cutting tools with regar d to quality and economy have led to constant
pr ogress in tool development and tool pr oduction. The cutting edge in particular is attracting much
inter est as it is an essential tool element for the cutting pr ocess. The pr operties of the cutting edge
influence the pr ocess characteristics and the work r esults in machining with defined cutting edges [
1
,
2
].
The pr operties of the cutting edge can be divided into physical and chemical characteristics as well as
into the geometry and surface topography of the cutting edge. The macro and micr o geometrical design
of the cutting edge af fects the chip formation and the chip removal [
3
,
4
]. Investigations focusing on the
ef fect of the cutting edge shape on tool wear in turning experiments showed that cutting edges r ounded
by brushing have a higher stability and ther efor e do not tend to have cutting edge br eakouts [
5
]. Further
studies about dif ferent methods for cutting edge pr eparation of micro mills made of tungsten carbide
showed that magnetic abrasive finishing and drag finishing r esult in favorable wear behavior because
of homogeneous cutting edge pr ofiles [
6
]. The most common production pr ocesses for cutting edge
pr eparation are magnetic abrasive finishing, abrasive blasting, laser machining, brushing, abrasive
flow machining, vibratory finishing and drag finishing. These pr oduction pr ocesses usually generate
simple cutting edge r oundings. Until now , systematic studies on the influence of differ ent defined
cutting edge micr ogeometries were not possible, as all common cutting edge pr eparation processes
ar e not able to produce complex and exact defined forms. Developing manufacturing pr ocesses to
pr oduce defined and complex cutting geometries in the industrial and scientific environment can
incr ease the performance of cutting pr ocesses. Multi-axis high pr ecision form grinding processes of fer
gr eat potential to generate defined cutting edge microgeometries.
Due to its positive har dness and toughness characteristics, cemented carbide is often used as a
material in cutting tools [
7
]. Most cemented carbides consist of a cobalt matrix and the hard material
Inventions 2017 , 2 , 13; doi:10.3390/inventions2030013 www .mdpi.com/journal/inventions

Inventions 2017 , 2 , 13 2 of 8
tungsten carbide. The cutting material is produced in a sinter pr ocess. The grain size of the sintered
powder , as well as the mixing ratio, influences the properties of the material. Due to the high hardness
of tungsten carbide of appr oximately V icker Har dness 2000 HV0.05, economical machining of cemented
carbide can be achieved by grinding pr ocesses only [ 8 ].
NC (Numerical Contr ol)-form grinding processes with mounted points ar e used for
manufacturing-fr ee forms on difficult-to-machine materials. Examples of these ar e the machining
of optical glasses and ceramic implants or prostheses for dental and medical applications.
The main challenges in NC-form grinding are the positional accuracy and the stif fness of the
machine axes due to the simultaneous axis movement. Furthermor e, the complex tool paths
mostly r equire NC-pr ogramming with CAD/CAM (Computer-aided Design/Computer -aided
Manufacturing)technology . T ool paths are graphically planned on CAD models and converted into
an appr opriate NC-Code by post-processors.
Dif ferent form grinding strategies lead to dif fer ent courses of chip cross-sectional ar eas and thus
dif ferent material r emoval rates. As a result, dif ferent pr ocess forces occur that could lead to varying
tool deflections and defects of the cutting edge. Knowledge about the r elation between grinding
strategy and material r emoval rate can lead to impr oved work r esults in the form of higher pr eciseness
of shape and dimensional accuracy as well as enhanced cutting edge quality . CAD/CAM-programs do
not pr ovide a pr oper calculation of chip cr oss-sectional ar eas and thus material r emoval rates. Hence,
it is necessary to develop suitable simulation models.
T o enable a r eproducible manufacturing of cutting edge geometries, dif fer ent strategies for form
grinding of cutting edge micr ogeometries, with regar d to material removal rates by using permeation
simulations, ar e examined. In grinding tests, the spindle power for the dif ferent grinding strategies is
compar ed with the course of the graphs of the material removal rate. The aim of this analysis is to
assess these dif ferent form grinding strategies with r egard to material r emoval rates to incr ease the
pr ocess performance and work results.
2. T est Conditions
The pr ogram to simulate the permeation of the tool and the workpiece was developed with the
pr ogram software MA TLAB, The MathW orks Inc., Natick, MA, USA. The form grinding pr ocesses
of the cutting edge micr o geometries were executed on a five-axis-machining center RXP600DSH of
Röders GmbH, Soltau, Germany . Besides thr ee linear axes ( X , Y , Z ), this machine is equipped with two
r otatory axes ( A , C ). Furthermore, the integrated grinding spindle has a maximum power of 8.5 kW
to enable a maximum rotation speed of 60,000 rpm. T o dr ess the grinding tools, a dr essing spindle
with a sinter ed diamond form r oller is integrated. T o pr oduce the requir ed sharpness of the mounted
points, a sharpening block is used and an automated sharpening process using the thr ough-feed
method was performed. Moreover , the grinding machine enables the determination of the geometry
and the r eference of the part with the help of a 3D-pr obe. The workpieces were cutting inserts made
of cemented carbide of the type SNMA120408. As cooling lubricant, grinding oil of MKU-Chemie
GmbH, Rödermark, Germany , with a viscosity of 7.6 mm
2
/s at 40
◦
C was used. Mor eover , metal
bonded diamond mounted points with a grain size of D20 wer e utilized. Only the use of diamond
tools allows an economic grinding processing of cemented carbides. T o r educe tool wear and thus
dimensional deviations on the workpiece, metal bond tools wer e used. An important advantage of
metal bonds is the shape stability of the tool pr ofiles [
9
]. In order to gain pr ocess for ces, conventional
for ce measurement platforms cannot be satisfactorily used when grinding with 5-axis kinematics
because of r elocations of the self weight. Rotating multicomponent dynamometers could be suitable
solutions but the non-contact signal transmission caused by the design principle can be used only
at maximum spindle speeds of appr oximately 12,000 rpm. Ther efore, the performance data of the
grinding spindle was r ecorded and used to validate the simulation r esults. T o verify the accuracy of
this measur ement method, the performance data was compared to for ce values recor ded by a force
measur ement platform during a peripheral longitudinal grinding process, as shown in Figur e 1 .

Inventions 2017 , 2 , 13 3 of 8
Inventions 2017 , 2 , 13 3 of 8

Figure 1. Comparison of ca lculated mechan ical power P c with d i rectly m e asu r ed ele c t r ic spindl e
power P .
It is sh own th at, wi th th e u s ed pr ocess parameters, both f o rce pr og r e ssions, fro m depths of cu t of
ap p r oxim at el y 200 µm , show a good correl a t i on. Duri ng m a nufacturing, cutt ing edge
microgeomet r ies wit h much smaller material removal ra t e s t h an in Figure 1 will be realized. To create
equally reliable state m ents between spindle power da ta and process fo rces, the material removal rate
was increased by offsetting the targe t geometr y in the workpie c e, Fi gure 2.

Figure 2. Offse t of n o m i n a l g e om etry to increase material removal rates.
Due to the rel a ti vel y l o ng tool pa ths of the mounted poi n t wi th a consta nt depth of cut on the
rake and flank face, thes e constant v a lue s of sp in d l e power can be seen as t h e refe rence durin g
grind i ng o f the cutting ed ge. There f ore , it is po ssi b l e to i d enti fy the a c tua l m a chi n i n g of the cutti ng
edge. The difference betw een the spind l e power m i n i mum and m a xim u m is sh own in the further
graphs as a p e rcentage inc r ease of spin dle perform a nce in comp arison to the minimum .
3. Kinem a tic-Geom etr i c M o del
3. 1. C u tti ng E dge Mi crogeo metri es a n d To ol Ty pes
In the investigation s , two different cutting edge geo m etries were considere d : an ide a l rad i us
and a wat e r f al l ra di us. Bo t h m i crogeo m e t r ies c a n be cha r a c terized usi n g the f o rm-f a c tor method.
The f o rm f a ctor is the rel a ti onshi p between the cut t i n g edge segment on the ra ke f a ce S γ and t h e
cutti ng edge segment on the f l a n k f a ce S α . The ideal rad i us h a s a form factor κ = 1. There f or e, the
cut t i ng edg e segm ent s on fl ank and r a k e fac e s, as wel l a s the cutti ng edge ra di us, ha ve the sa m e

Figure 1.
Co mpa r is o n of c alc u la t ed m ech a ni c al p owe r P
c
wi th di r ec tl y mea s ur ed e lec t ri c s pi ndl e p ow e r P .
It is shown that, with the used pr ocess parameters, both for ce pr ogr essions, from depths of cut of
appr oximately 200
µ
m, show a good correlation. During manufacturing, cutting edge microgeometries
with much smaller material r emoval rates than in Figur e 1 will be realize d. T o cr eate equally reliable
statements between spindle power data and process for ces, the material removal rate was incr eased by
of fsetting the target geometry in the workpiece, Figur e 2 .
Inventions 2017 , 2 , 13 3 of 8

Figure 1. Comparison of ca lculated mechan ical power P c with d i rectly m e asu r ed ele c t r ic spindl e
power P .
It is sh own th at, wi th th e u s ed pr ocess parameters, both f o rce pr og r e ssions, fro m depths of cu t of
ap p r oxim at el y 200 µm , show a good correl a t i on. Duri ng m a nufacturing, cutt ing edge
microgeomet r ies wit h much smaller material removal ra t e s t h an in Figure 1 will be realized. To create
equally reliable state m ents between spindle power da ta and process fo rces, the material removal rate
was increased by offsetting the targe t geometr y in the workpie c e, Fi gure 2.

Figure 2. Offse t of n o m i n a l g e om etry to increase material removal rates.
Due to the rel a ti vel y l o ng tool pa ths of the mounted poi n t wi th a consta nt depth of cut on the
rake and flank face, thes e constant v a lue s of sp in d l e power can be seen as t h e refe rence durin g
grind i ng o f the cutting ed ge. There f ore , it is po ssi b l e to i d enti fy the a c tua l m a chi n i n g of the cutti ng
edge. The difference betw een the spind l e power m i n i mum and m a xim u m is sh own in the further
graphs as a p e rcentage inc r ease of spin dle perform a nce in comp arison to the minimum .
3. Kinem a tic-Geom etr i c M o del
3. 1. C u tti ng E dge Mi crogeo metri es a n d To ol Ty pes
In the investigation s , two different cutting edge geo m etries were considere d : an ide a l rad i us
and a wat e r f al l ra di us. Bo t h m i crogeo m e t r ies c a n be cha r a c terized usi n g the f o rm-f a c tor method.
The f o rm f a ctor is the rel a ti onshi p between the cut t i n g edge segment on the ra ke f a ce S γ and t h e
cutti ng edge segment on the f l a n k f a ce S α . The ideal rad i us h a s a form factor κ = 1. There f or e, the
cut t i ng edg e segm ent s on fl ank and r a k e fac e s, as wel l a s the cutti ng edge ra di us, ha ve the sa m e

Figure 2. Offset of nominal geometry to increase material r emoval rates.
Due to the r elatively long tool paths of the mounted point with a constant depth of cut on the rake
and flank face, these constant values of spindle power can be seen as the refer ence during grinding
of the cutting edge. Ther efore, it is possible to identify the actual machining of the cutting edge.
The dif ference between the spindle power minimum and maximum is shown in the further graphs as
a per centage increase of spindle performance in comparison to the minimum.
3. Kinematic-Geometric Model
3.1. Cutting Edge Microgeometries and T ool T ypes
In the investigations, two dif ferent cutting edge geometries wer e considered: an ideal radius and
a waterfall radius. Both microgeometries can be characterized using the form-factor method. The form
factor is the relationship between the cutting edge segment on the rake face S
γ
and the cutting edge
segment on the flank face S
α
. The ideal radius has a form factor
κ
= 1. Ther efore, the cutting edge

Inventions 2017 , 2 , 13 4 of 8
segments on flank and rake faces, as well as the cutting edge radius, have the same value. The waterfall
radius is characterized by a form factor gr eater than one, in this case κ = 1.5, Figure 3 .
Inventions 2017 , 2 , 13 4 of 8
v a l u e. The w a t e rf al l r a d i u s i s ch ar act e r i ze d b y a for m fact or gre a t e r t h an one , in t h is ca se κ = 1. 5,
Fig u re 3.

Figure 3. Form-factor method for idea l radius and waterfall r a dius.
For both cutti ng edge mi crogeome tries, different gr in ding str a tegie s were ex amined, as shown
in F i gure 4. Grinding str a tegy I and II differ w i th re ga rd to the acti ve tool shape. In Stra tegy I, the
sphere-shape d fo rm o f the cy linde r b a ll end mounte d point is used to cut orthogonally ab ove the
cutting edge , referr ed to as the course of the corner radius r ε . Stra tegy II ha s the sa me tool ki nema ti cs
but the cylin dric al p a rt of the mounted point is uti lized for the c u tting proces s. In Strategy III, the
ball -sh a ped p a rt o f t h e mo unt e d point is gu ided abov e the c u tting edge at an an gle of 45°, re ferred to
as the co urse of the corner rad i us. For m a chin in g the waterfall geo m etries w i th a form factor κ = 1. 5 ,
the gri n di ng stra tegies I a n d II ma tch with th e strateg i es of the ideal grin ding rad i us.

Figure 4. Di ffer e nt gri n d i ng s t rategi es for N C -form gri n d i ng c u tti ng ed ge mi c r ogeometri e s .
3 . 2 . Mod e lling of th e Permeation
The aim o f the deve lope d process simulation is t o i d enti fy the ma teria l remova l ra tes of
different gr in ding str a tegies and c u tting edge mi cro g eometries fo r one tool p a th movement. Thus,
knowled g e about force s involved and therefore abo u t tool defl ecti on ca n be genera ted to ana l yz e
different grin ding strateg i es. Th e ki nema ti c-geometri cal model si mula tes the penetra t i o n of two
geometric a l bodies. The workpiece (c utting inse rt ) i s def i ned as a cy lin dri c al s u rf ace a n d t h e
mount e d poi n t is al so de fi ned as a cy li ndric a l s u rf ac e if t h e re qu i r ed t ool t y pe is cy lindr ic al ; if t h e
req u ired tool type is spherical, a n a ppropria t e pa rtial sur f ac e is created, Fig u re 5. The geo m etric
objects are d e scribed as p o int c l oud s . To red u ce the computi n g ti me, th e progra m minimiz e s the
poi n t cl ouds t o the poi n ts of the conta c t i ng a r eas of b o th geometri c objects.

Figure 3. Form-factor method for ideal radius and waterfall radius.
For both cutting edge micr ogeometries, dif ferent grinding st rategies were examined, as shown
in Figur e 4 . Grinding strategy I and II differ with r egard to the active tool shape. In Strategy I,
the spher e-shaped form of the cylinder ball end mounted point is used to cut orthogonally above
the cutting edge, r eferred to as the course of the corner radius r
ε
. Strategy II has the same tool
kinematics but the cylindrical part of the mounted point is utilized for the cutting pr ocess. In Strategy
III, the ball-shaped part of the mounted point is guided above the cutting edge at an angle of 45
◦
,
r eferred to as the course of the corner radius. For machining the waterfall geometries with a form
factor κ = 1.5, the grinding strategies I and II match with the strategies of the ideal grinding radius.
Inventions 2017 , 2 , 13 4 of 8
v a l u e. The w a t e rf al l r a d i u s i s ch ar act e r i ze d b y a for m fact or gre a t e r t h an one , in t h is ca se κ = 1. 5,
Fig u re 3.

Figure 3. Form-factor method for idea l radius and waterfall r a dius.
For both cutti ng edge mi crogeome tries, different gr in ding str a tegie s were ex amined, as shown
in F i gure 4. Grinding str a tegy I and II differ w i th re ga rd to the acti ve tool shape. In Stra tegy I, the
sphere-shape d fo rm o f the cy linde r b a ll end mounte d point is used to cut orthogonally ab ove the
cutting edge , referr ed to as the course of the corner radius r ε . Stra tegy II ha s the sa me tool ki nema ti cs
but the cylin dric al p a rt of the mounted point is uti lized for the c u tting proces s. In Strategy III, the
ball -sh a ped p a rt o f t h e mo unt e d point is gu ided abov e the c u tting edge at an an gle of 45°, re ferred to
as the co urse of the corner rad i us. For m a chin in g the waterfall geo m etries w i th a form factor κ = 1. 5 ,
the gri n di ng stra tegies I a n d II ma tch with th e strateg i es of the ideal grin ding rad i us.

Figure 4. Di ffer e nt gri n d i ng s t rategi es for N C -form gri n d i ng c u tti ng ed ge mi c r ogeometri e s .
3 . 2 . Mod e lling of th e Permeation
The aim o f the deve lope d process simulation is t o i d enti fy the ma teria l remova l ra tes of
different gr in ding str a tegies and c u tting edge mi cro g eometries fo r one tool p a th movement. Thus,
knowled g e about force s involved and therefore abo u t tool defl ecti on ca n be genera ted to ana l yz e
different grin ding strateg i es. Th e ki nema ti c-geometri cal model si mula tes the penetra t i o n of two
geometric a l bodies. The workpiece (c utting inse rt ) i s def i ned as a cy lin dri c al s u rf ace a n d t h e
mount e d poi n t is al so de fi ned as a cy li ndric a l s u rf ac e if t h e re qu i r ed t ool t y pe is cy lindr ic al ; if t h e
req u ired tool type is spherical, a n a ppropria t e pa rtial sur f ac e is created, Fig u re 5. The geo m etric
objects are d e scribed as p o int c l oud s . To red u ce the computi n g ti me, th e progra m minimiz e s the
poi n t cl ouds t o the poi n ts of the conta c t i ng a r eas of b o th geometri c objects.

Figure 4. Different grinding strategies for NC-form grinding cutting edge micr ogeometries.
3.2. Modelling of the Permeation
The aim of the developed pr ocess simulation is to identify the material removal rates of dif ferent
grinding strategies and cutting edge micr ogeometries for one tool path movement. Thus, knowledge
about for ces involved and therefor e about tool deflection can be generated to analyze differ ent grinding
strategies. The kinematic-geometrical model simulates the penetration of two geometrical bodies.
The workpiece (cutting insert) is defined as a cylindrical surface and the mounted point is also defined
as a cylindrical surface if the r equir ed tool type is cylindrical; if the requir ed tool type is spherical,
an appr opriate partial surface is created, Figur e 5 . The geometric objects are described as point clouds.

Inventions 2017 , 2 , 13 5 of 8
T o reduce the computing time, the pr ogram minimizes the point clouds to the points of the contacting
ar eas of both geometric objects.
Inventions 2017 , 2 , 13 5 of 8

Figure 5. Sche matic represen tation of the to ol and work piece areas rele va nt for simulate d ca lculati o n.
The target ge ometry of the respective c u tting ed g e microgeomet r y is precisely position ed as a
numeric a l ap proximated c u rve in the workpiece p o int clo u d. T h is curve is the re ference for the
ra di al posi t i on of the most externa l point of the tool poi n t cl oud. For ea ch ca lcula t i o n step, the tool
point clo u d i s pl ace d at t h e fo llow i ng point of t h e cutting edge curve, from t h e flank face (ang le
position α = 0) to the ra ke fa ce ( a ngl e posi ti on α = 90°). Ther efore , t h e rotation o f the approximated
tool around t h e approx im ated c u tting edge en ables the determin ation of the ex act se ction p l anes. In
Fig u re 6, the green cycles represent the red u ced t ool poi n t cl oud a n d the blue a r ea represents the
reduced wor k piece po int c l oud .

Figure 6. Intersection of too l a n d workpiece point clouds in one position o n one tool path.
The red curv e depi cts the cutti ng curve of the cu tting edge tar g et geometry. Th e point distan ce
o f b o t h p o i n t c l o u d s i s 0 . 1 µ m t o a c h i e v e h i g h c a l c ul a t i o n a c curacy. For ea ch posi ti on of the tool
poi n t cl oud, t h e i n tersecti on poi n ts are p r ojected on a pl a n e. Af terwa r ds, the a r ea wi thi n the externa l
points c a n be determin ed which is e q uivalent to the section p l an e between the t ool and work piece,
Fig u re 7.
Bef o re running the si mula ti on, the boundary co n d it ions h a ve t o be de fine d. These inc l u d e
geometry dat a o f the c u tting ed ge m i cr ogeometry an d the tool. Further si m u lati on da ta is t h e f e ed
speed and t h e reso lut i on o f geomet r i c o b ject s inf l u e n c ing t h e acc u racy of t h e ca lcu l at ion s .

Figure 5.
Schematic repr esentation of the tool and workpiece areas r elevant for simulated calculation.
The tar get geometry of the respective cutting edge micr ogeometry is precisely positioned as
a numerical appr oximated curve in the workpiece point cloud. This curve is the refer ence for the radial
position of the most external point of the tool point cloud. For each calculation step, the tool point
cloud is placed at the following point of the cutting edge curve, fr om the flank face (angle position
α
= 0) to the rake face (angle position
α
= 90
◦
). Ther efor e, the rotation of the appr oximated tool around
the appr oximated cutting edge enables the determination of the exact section planes. In Figure 6 ,
the gr een cycles repr esent the reduced tool point cloud and the blue ar ea repr esents the r educed
workpiece point cloud.
Inventions 2017 , 2 , 13 5 of 8

Figure 5. Sche matic represen tation of the to ol and work piece areas rele va nt for simulate d ca lculati o n.
The target ge ometry of the respective c u tting ed g e microgeomet r y is precisely position ed as a
numeric a l ap proximated c u rve in the workpiece p o int clo u d. T h is curve is the re ference for the
ra di al posi t i on of the most externa l point of the tool poi n t cl oud. For ea ch ca lcula t i o n step, the tool
point clo u d i s pl ace d at t h e fo llow i ng point of t h e cutting edge curve, from t h e flank face (ang le
position α = 0) to the ra ke fa ce ( a ngl e posi ti on α = 90°). Ther efore , t h e rotation o f the approximated
tool around t h e approx im ated c u tting edge en ables the determin ation of the ex act se ction p l anes. In
Fig u re 6, the green cycles represent the red u ced t ool poi n t cl oud a n d the blue a r ea represents the
reduced wor k piece po int c l oud .

Figure 6. Intersection of too l a n d workpiece point clouds in one position o n one tool path.
The red curv e depi cts the cutti ng curve of the cu tting edge tar g et geometry. Th e point distan ce
o f b o t h p o i n t c l o u d s i s 0 . 1 µ m t o a c h i e v e h i g h c a l c ul a t i o n a c curacy. For ea ch posi ti on of the tool
poi n t cl oud, t h e i n tersecti on poi n ts are p r ojected on a pl a n e. Af terwa r ds, the a r ea wi thi n the externa l
points c a n be determin ed which is e q uivalent to the section p l an e between the t ool and work piece,
Fig u re 7.
Bef o re running the si mula ti on, the boundary co n d it ions h a ve t o be de fine d. These inc l u d e
geometry dat a o f the c u tting ed ge m i cr ogeometry an d the tool. Further si m u lati on da ta is t h e f e ed
speed and t h e reso lut i on o f geomet r i c o b ject s inf l u e n c ing t h e acc u racy of t h e ca lcu l at ion s .

Figure 6. Intersection of tool and workpiece point clouds in one position on one tool path.
The r ed curve depicts the cutting curve of the cutting edge tar get geometry . The point distance of
both point clouds is 0.1
µ
m to achieve high calculation accuracy . For each position of the tool point
cloud, the intersection points ar e pr ojected on a plane. Afterwar ds, the ar ea within the external points
can be determined which is equivalent to the section plane between the tool and workpiece, Figure 7 .

Inventions 2017 , 2 , 13 6 of 8
Inventions 2017 , 2 , 13 6 of 8

Figure 7. Generating the cut surface by lo ca ting the outer intersections.
4. R e su lts
4. 1. Si mul a ti o n R esul t s
The resul t s of the si mul a ti on show the dif f e rent ma teria l removal ra t e Q w c u rve pr ogressions fo r
each gr indin g str a tegy , as shown in F i gure 8. The lowest max i m u m m a terial removal r a te Q w,max is
achieve d by g r indin g str a te gy II.

Figure 8. Simulation results for differe nt form grinding strategies.
The ma ximum va lue is loca te d at an angl e p o s i t i o n α of appro x imately 20°. Therefo r e, the
ma ximum ma teria l remov a l ra te Q w,max of gri n di ng stra tegy I was determi n ed a t the st a r t a n gle
posi ti on of the mounted p o i n t, ref e rred to a s the course of the rounde d c u tting edge radius r β . It i s

Figure 7. Generating the cut surface by locating the outer intersections.
Befor e running the simulation, the boundary conditions have to be defined. These include
geometry data of the cutting edge micr ogeometry and the tool. Further simulation data is the feed
speed and the r esolution of geometric objects influencing the accuracy of the calculations.
4. Results
4.1. Simulation Results
The r esults of the simulation show the differ ent material removal rate Q
w
curve pr ogressions for
each grinding strategy , as shown in Figur e 8 . The lowest maximum material removal rate Q
w ,max
is
achieved by grinding strategy II.
Inventions 2017 , 2 , 13 6 of 8

Figure 7. Generating the cut surface by lo ca ting the outer intersections.
4. R e su lts
4. 1. Si mul a ti o n R esul t s
The resul t s of the si mul a ti on show the dif f e rent ma teria l removal ra t e Q w c u rve pr ogressions fo r
each gr indin g str a tegy , as shown in F i gure 8. The lowest max i m u m m a terial removal r a te Q w,max is
achieve d by g r indin g str a te gy II.

Figure 8. Simulation results for differe nt form grinding strategies.
The ma ximum va lue is loca te d at an angl e p o s i t i o n α of appro x imately 20°. Therefo r e, the
ma ximum ma teria l remov a l ra te Q w,max of gri n di ng stra tegy I was determi n ed a t the st a r t a n gle
posi ti on of the mounted p o i n t, ref e rred to a s the course of the rounde d c u tting edge radius r β . It i s

Figure 8. Simulation results for differ ent form grinding strategies.

Inventions 2017 , 2 , 13 7 of 8
Th e ma xi mum v al ue i s loc at ed a t an an gl e po sit io n
α
of a pp r ox im ate ly 2 0
◦
. T he r ef or e, t he m ax im um
ma te ri al r em ov al r at e Q
w ,m ax
of g ri nd ing s tr at egy I w as de te rm ine d at t he s tar t an gl e pos it io n of th e
mo un te d poi nt , r ef er r ed t o as th e co ur se of t he r ou n de d cut ti ng e dge r ad iu s r
β
. It i s th e hi ghe st a ch iev ed
va lu e. Fo l lo wi ng t he fu rt he r ang le p os iti on s, th e c ur ve i s r eg r es si ve an d ha s lo w val ue s fr o m an gle
po si ti on
α
= 16
◦
. T he c our se o f gr ind in g st rat eg y II I is si mil ar b ut h as a hi ghe r pe ak t han g ri nd ing s tra te gy
II . Th er ef or e, i t d oe s not a pp ea r adv an ta geo us fo r pr oc e ss f or ce s an d co ns equ en tl y for t ool d ef le cti on s.
He nc e, f urt he r co nsi de ra tio ns r el a te t o gr in din g st ra teg ie s I an d II. Th e co ur ses o f th e di ffe r e nt g rin din g
st ra te gie s I an d II f or gr ind in g wa ter fa ll r adi i wi th a f orm f ac to r
κ
= 1. 5 di ff er on ly sl ig ht ly fr o m th e
co ur se s of th e eq ui val ent g r in di ng s tra te gi es fo r th e id eal r ad ii , Fi gur e 8 . W it h an a ng le p osi ti on o f
α
= 70
◦
,
al mo st a ll th e ma te ria l is r em o ve d. T hi s is c au sed b y th e la r ge r cur va tu r e at t he be gi nn ing o f th e cu tti ng
ed ge g eo met ry a nd b y the f ac t th at th e ra ti o of th e to ol r adi us t o th e cut ti ng e dge r ad iu s is ap pr ox i ma te ly
10 0. T hi s r e su lts i n r el ati ve ly f lat t oo l ge ome tr y , as s ho wn i n Fi gur e 6 .
4.2. Experimental Results
T o validate the kinematic-geometrical model, grinding tests wer e conducted. For this purpose,
the spindle power of the dif fer ent grinding strategies was recor ded and compared with the courses of
the material r emoval rates fr om the simulation calculations, as shown in Figure 9 . Generally , for all test
series, the courses of the spindle power are highly similar to the courses of the simulation calculations.
However , small course deviations can be identified.
Inventions 2017 , 2 , 13 7 of 8
the highe s t achieved v a lue. Fo llow i ng the further angle po sition s, the curve is re gres sive and has
low values from angle po sition α = 16°. The course o f grinding st rategy III is similar but has a higher
p e ak t h an gr i n ding st r a t e g y II. There f or e, it do es not ap p e ar a d v a nt ageou s for p r ocess forc e s and
consequently for tool deflec tions. Hence, further c o nsideration s relate to grinding strategies I and II.
The course s o f the different grind i ng strategies I and II for g r ind i ng wat e rf al l r a d i i wit h a fo rm fact or
κ = 1. 5 di ffer only sl ight l y from t h e cou r ses of t h e eq u i va lent g r ind i ng st r a t e gi es for t h e i d ea l r a di i,
F i g u r e 8 . W i t h a n a n g l e p o s i t i o n o f α = 7 0 ° , a l m o s t a l l t h e m a t e r i a l i s r e m o v e d . T h i s i s c a u s e d b y t h e
lar g er curvat ure at the be ginnin g of th e cutting edg e geometry and by the f a ct tha t the ra ti o of the
t ool ra di us t o t h e cut t i ng edge ra di us is approx im at ely 1 0 0 . Th is r e su lt s in rel a t i vely f l at t ool
geometry, as shown in Figure 6.
4.2. Experime n t al Results
To vali dat e t h e kinem a t i c- geomet ric a l model, g r in ding tests were conducted. For thi s purpose,
the spindle power of the d i fferent gr ind i ng str a te gies was recorde d and comp ared with the course s
of the ma teria l removal rates f r om the si mula ti on calcula t i o ns, as shown i n Fi gure 9 . General l y , f o r
all test ser i es , the co urse s of the spind l e power ar e highly similar to the co urs e s o f the s i mulation
calc ul at ions . However, sm al l co urse dev i at ion s c a n be ident i fi ed.

Figure 9. Comparison of curve progressions of si m u lation d a ta and spindle power data .
For both cut t ing e d ge ge ometries, s l ight devi at io ns in t h e spindle power cou r ses c a n b e
observed i n compa r i s on to the si m u la ti on resul t s, F i g u re 9. The reason fo r this oc currence may be the
inertia of the tool spind l e’s continuous t u rn re gu lat i o n in combin at ion wit h t h e c o mparat ive l y short
process time of one tool path of approximately 8 s. Another i m porta n t fa ctor is the rela ti vely small
spindle lo ad and there f ore the less line ar beh a vi or between the ma teria l remov a l ra te a n d p r ocess
f o rces, as wel l a s the correct determina t i o n of the rel e va nt p a rt of the spi n dl e power course.
Neverthele ss, an aly s is of th e spind l e po wer dat a obta ined in the gr indin g tests c o nfirms the trend o f
the simu latio n resu lts. The validat i on of the simulat i o n model con f ir ms the benefi t of f o rm grindi ng
processes of c u tting edge microgeomet r ies for p l an n i ng str a tegies. The kinem a tic-geometric a l model
ena b les the prediction of ma ximum ma teria l re mova l ra tes a n d su bs eq u e ntl y i n di ca tes proces s
f o rces a n d potenti a l tool def l ecti ons. Furt hermore, the influence of t h e grin ding s t rategy on possib l e
da mages to the cutti ng edge ca n be a n al yzed.
5 . Summa ry
Within the presented work , a kin e matic - geometri c mo del was developed in o r der to an aly z e th e
complex contact condit ion s during grin ding c u tting e d g e mi c r og eome tr i e s . Ther e f or e, a si mu la ti on

Figure 9. Comparison of curve progressions of simulation data and spindle power data.
For both cutting edge geometries, slight deviations in the spindle power courses can be observed
in comparison to the simulation r esults, Figur e 9 . The r eason for this occurr ence may be the inertia of
the tool spindle’s continuous turn r egulation in combination with the comparatively short pr ocess time
of one tool path of approximately 8 s. Another important factor is the r elatively small spindle load
and ther efore the less linear behavior between the material r emoval rate and process for ces, as well
as the corr ect determination of the relevant part of the spindle power course. Nevertheless, analysis
of the spindle power data obtained in the grinding tests confirms the trend of the simulation r esults.
The validation of the simulation model confirms the benefit of form grinding processes of cutting edge
micr ogeometries for planning strategies. The kinematic-geometrical model enables the prediction
of maximum material r emoval rates and subsequently indicates pr ocess forces and potential tool
deflections. Furthermor e, the influence of the grinding strategy on possible damages to the cutting
edge can be analyzed.

Inventions 2017 , 2 , 13 8 of 8
5. Summary
W ithin the pr esented work, a kinematic-geometric model was developed in order to analyze the
complex contact conditions during grinding cutting edge micr ogeometries. Therefor e, a simulation
appr oach based on the intersection of geometric bodies was used. The grinding tool and workpiece
surface wer e defined as cylindrical point clouds and the cr oss section of the cutting edge micr ogeometry
was determined by numerical approximation. By moving the tool point cloud stepwise along the
coor dinates of the cutting edge micr ogeometry for each position, the interface between the workpiece
point cloud was calculated. Thus, the varying material removal rates could be determined and
compar ed with the spindle power data. The relatively even curve pr ogr essions show the correlation
between the material r emoval rate Q
w
and the proce ss forces. Therefor e, the geometric-kinematic
model can be used to analyze dif fer ent form grinding strategies for manufacturing various cutting edge
micr ogeometries with regar d to process for ces and consequently tool deflection. W ith the gather ed
information, the work results of form grinding cutting edge micr ogeometries, regar ding the form
accuracy and the chipping of the cutting edge, can be impr oved.
Acknowledgments:
The authors would like to thank the Deutsche Forschungsgemeinschaft (German Research
Foundation) for funding this resear ch within the project DFG UH 100/179-1.
Author Contributions: Both authors were equally involved in creating this paper .
Conflicts of Interest: The authors declare no conflict of interest.
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