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Modeling of Contact Forces for Brushing Tools
Eckart Uhlmann 1,2 and Anton Hoyer 1,*


Citation: Uhlmann, E.; Hoyer, A.
Modeling of Contact Forces for
Brushing Tools. Ceramics 2021,4,
397–407. https://doi.org/10.3390/
ceramics4030029
Academic Editors: Kevin Plucknett
and Gilbert Fantozzi
Received: 15 April 2021
Accepted: 1 July 2021
Published: 9 July 2021
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distributed under the terms and
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Attribution (CC BY) license (https://
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4.0/).
1Institute for Machine Tools and Factory Management, Technical University Berlin, Pascalstr. 8-9,
10587 Berlin, Germany; [email protected]
2Fraunhofer Institute for Production Systems and Design Technology, Pascalstr. 8-9, 10587 Berlin, Germany
*Correspondence: [email protected]; Tel.: +49-30-314-22781
Abstract:
Brushing with bonded abrasives is a flexible finishing process used for the deburring and
the rounding of workpiece edges as well as for the reduction of the surface roughness. Although in-
dustrially widespread, insufficient knowledge about the contact behavior of the abrasive filaments
mainly causes applications to be based on experiential values. Therefore, this article aims to increase
the applicability of physical process models by introducing a new prediction method, correlating the
contact forces of single abrasive filaments, obtained by means of a multi-body simulation, with the
experimentally determined process forces of full brushing tools during the surface finishing of ZrO
2
.
It was concluded that aggressive process parameters may not necessarily lead to maximum produc-
tivity due to increased tool wear, whereas less aggressive process parameters might yield equally
high contact forces and thus higher productivity.
Keywords: abrasive brushing; finishing; multi-body simulation; modeling; contact force; ZrO2
1. Introduction
Brushing with bonded abrasives is an industrial manufacturing process, which is
predominantly used for the deburring and rounding of metallic workpiece edges. Fur-
thermore, it has gained importance in the finishing of technical surfaces, mainly for the
reduction of the surface roughness [
1
–
3
]. The process is characterized by its flexible brush-
ing tools (Figure 1), usually consisting of a cast epoxy brush body, to which abrasive
filaments are attached (Figure 1a). These are composed of an extruded polymer matrix,
normally polyamide 6.12, and bonded abrasive grains, normally silicon carbide (SiC) or alu-
minum(III) oxide (Al
2
O
3
). However, especially during the finishing of ceramic workpieces,
hard abrasives, such as diamond or cubic boron nitride (cBN), are required [1,4].
The advantages of brushing with bonded abrasives are based on the high flexibility of
the abrasive filaments, which allow for the adaptation to complex workpiece geometries,
despite ordinarily shaped tools, and therefore the compensation of small geometric devi-
ations of tools, workpieces and machine systems, as well as tool trajectories. Additional
advantages are low process forces and temperatures, and also the potential utilization of
pre-existing machine systems designed for grinding and milling operations. A considerable
disadvantage of the process is the insufficient knowledge of the motion, the chipping, and
the wear behavior of the abrasive filaments, with the result that industrial processes are typ-
ically based on experiential values, making predictions for new processes difficult [1,5,6].
Regarding the tool specific parameters most relevant for achieving a low workpiece
surface roughness, small filament diameters d
f
, large filament lengths l
f
, and small grain
sizes d
g,
are needed, whereas contrary tool specifications are required for high material
removal rates [
3
,
7
,
8
]. A large brush body radius r
g
leads to a larger number of filaments
N
f
and to a more enhanced support between filaments than a small brush body radius r
g
,
and thereby to more efficient brushing processes. The essential process parameters are the
brushing velocity v
b
, the tangential feed rate v
ft
, and the penetration depth a
e
(Figure 1b).
Above all, the surface roughness of the workpieces can be successively reduced by multiple
Ceramics 2021,4, 397–407. https://doi.org/10.3390/ceramics4030029 https://www.mdpi.com/journal/ceramics
Ceramics 2021,4398
brushing cycles until a lower roughness limit is reached, which is primarily dependent on
the tool specification. Therefore, a low surface roughness can also be achieved with gentle
process parameters, specifically low brushing velocities v
b
, high tangential feed rates v
ft
,
and low penetration depths a
e
[
3
]. Furthermore, process heat can be dissipated by the use
of cooling lubricant to prevent the melting of the abrasive filaments, which would affect the
productivity adversely or even cause permanent tool damage. However, in some studies,
a reduction of the productivity during brushing with cooling lubricant is mentioned,
likely caused by a reduction of the effective Young modulus E
f
of the abrasive filaments
due to polyamide being prone to liquid absorption [
2
,
3
]. Within the scope of this article,
productivity is defined as the rate of change of the work result, meaning the surface
roughness or the material removal rate, while simultaneously considering the negative
influence of tool wear.
Ceramics 2021, 4 FOR PEER REVIEW 2 of 12
Above all, the surface roughness of the workpieces can be successively reduced by multi-
ple brushing cycles until a lower roughness limit is reached, which is primarily dependent
on the tool specification. Therefore, a low surface roughness can also be achieved with
gentle process parameters, specifically low brushing velocities v
b
, high tangential feed
rates v
ft
, and low penetration depths a
e
[3]. Furthermore, process heat can be dissipated
by the use of cooling lubricant to prevent the melting of the abrasive filaments, which
would affect the productivity adversely or even cause permanent tool damage. However,
in some studies, a reduction of the productivity during brushing with cooling lubricant is
mentioned, likely caused by a reduction of the effective Young modulus E
f
of the abrasive
filaments due to polyamide being prone to liquid absorption [2,3]. Within the scope of this
article, productivity is defined as the rate of change of the work result, meaning the sur-
face roughness or the material removal rate, while simultaneously considering the nega-
tive influence of tool wear.
Figure 1. Brushing with bonded abrasives; (a) round brush with diamond grains; (b) contact proportions during brushing,
schematically depicted on the basis of a single abrasive filament; (c) technological investigations with a single abrasive
filament during workpiece contact, capturing the filament tip deflection s
n
.
Various studies confirm a significant dependence of the productivity on the applied
contact forces, as high contact forces increase the penetration of the workpiece material by
the abrasive grains [1–3,5,6,9]. To gain an understanding of the process behavior of abra-
sive filaments, technological investigations with single filaments and full brushing tools
may be carried out, as well as numerical process simulations based on physical models.
In both cases, the knowledge transfer between single- and multi-filament models is chal-
lenging and a focus of contemporary research [9]. For example, contact force measure-
ments with single filaments at high brushing velocities v
b
are difficult (Figure 1c), due to
low force values along with a high predisposition of the experimental equipment towards
unwanted vibrations. Adversely, the use of numerical process models for the calculation
of a multitude of interacting filaments is still limited due to long computation times [9].
Hence, the aim of this article is the introduction of a new method to correlate the
numerically simulated contact forces of single abrasive filaments with the experimentally
determined process forces of full brushing tools, the acquisition of which is relatively un-
problematic. For this purpose, the contact impulse transmitted by the abrasive filaments
onto the workpiece is calculated in order to avoid the use of error-sensitive smoothing
algorithms and thus to increase the scope of application for numerically simulated brush-
ing parameters.
Figure 1.
Brushing with bonded abrasives; (
a
) round brush with diamond grains; (
b
) contact proportions during brushing,
schematically depicted on the basis of a single abrasive filament; (
c
) technological investigations with a single abrasive
filament during workpiece contact, capturing the filament tip deflection sn.
Various studies confirm a significant dependence of the productivity on the applied
contact forces, as high contact forces increase the penetration of the workpiece material by
the abrasive grains [
1
–
3
,
5
,
6
,
9
]. To gain an understanding of the process behavior of abrasive
filaments, technological investigations with single filaments and full brushing tools may be
carried out, as well as numerical process simulations based on physical models. In both
cases, the knowledge transfer between single-and multi-filament models is challenging
and a focus of contemporary research [
9
]. For example, contact force measurements with
single filaments at high brushing velocities v
b
are difficult (Figure 1c), due to low force
values along with a high predisposition of the experimental equipment towards unwanted
vibrations. Adversely, the use of numerical process models for the calculation of a multitude
of interacting filaments is still limited due to long computation times [9].
Hence, the aim of this article is the introduction of a new method to correlate the
numerically simulated contact forces of single abrasive filaments with the experimentally
determined process forces of full brushing tools, the acquisition of which is relatively
unproblematic. For this purpose, the contact impulse transmitted by the abrasive filaments
onto the workpiece is calculated in order to avoid the use of error-sensitive smoothing
algorithms and thus to increase the scope of application for numerically simulated brush-
ing parameters.
Ceramics 2021,4399
2. Materials and Methods
The chosen case of application was the surface finishing of zirconium dioxide, par-
tially stabilized with magnesium oxide (MgO-PSZ, or ZrO
2
for simplicity), its main do-
mains of application including dental and medical engineering as well as industrial fur-
nace linings [
10
,
11
]. The previously surface-ground workpieces feature dimensions of
200 ×200 ×200 mm3and an average roughness of Ra = 1.1 µm.
The brushing tools used were round brushes manufactured by C.Hilzinger-Thum
GmbH and Co.KG, Tuttlingen, Germany, with a brush body radius of r
g
= 140 mm, a tool
width of
bb= 20 mm,
a filament length of l
f
= 40 mm, a filament diameter of
df= 1.18 mm
and a filament density of
ρf
= 1.11 g/cm
3
. Due to the high hardness and the brittle
machining behavior of ZrO
2
, abrasive filaments composed of polyamide 6.12 and bonded
polycrystalline diamond with a grit size of 320 mesh and a mean grain size of
dg= 29.2 µm
were used [
12
]. Based on photographs of the brushing tools’ circumferences, an estimated
number of abrasive filaments per tool of N
f
= 10,500
±
600 was determined. For this,
a Hough transformation was applied to detect and count the circular filament tips of tool
segments, extrapolating their quantity for the entire brushing tool.
The technological investigations were carried out on a plane and profile grinding
machine of type Profimat MT 408 HTS, manufactured by Blohm Jung GmbH, Hamburg,
Germany, and process forces were measured with a quartz three-component dynamometer
of type 9257B, manufactured by Kistler Instrumente AG, Winterthur, Switzerland, to which
the workpieces were attached. Because the workpieces were planar and edge contacts were
not being investigated, only the process forces in the normal direction of the workpiece
surface were evaluated; their arithmetic mean value over the processing time t
p
being
defined as the tool contact normal force F
n,w
. Despite the low heat conductivity of ZrO
2
,
no cooling lubricant was used during brushing in order to decrease the number of possible
influences on the measurement of process forces.
To simulate the deformations and the contact forces of abrasive filaments, a multi-
body system based on the Lagrange formalism was implemented in MATLAB R2019b,
developed by The MathWorks, Inc., Natick, Massachusetts, USA. Using spherical coordi-
nates, the flexible filaments were subdivided into rigid segments, connected end-to-end by
rotational springs and dampers. After the appropriate reduction of the degrees of freedom,
the positions and the motions of the segments in three-dimensional space were distinctly
characterized by their minimal coordinates, namely their polar angles
ϕk
and azimuth
angles
θk
, as well as their respective angular velocities and accelerations. Incorporating the
Lagrange function L, the kinetic energy E
kin
, the potential energy E
pot
, the dissipation
energy D, and the conservative momentum M
k,ϕ
, a system of ordinary differential equa-
tions of second was formed, which could be solved numerically. Equation (1) shows the
differential equation used to calculate the polar angle
ϕk
of an arbitrary single segment
with index k, while the azimuth angle θkwas obtained analogously [9].
d
dt∂L
∂.
ϕK−∂L
∂ϕK
+∂D
∂.
ϕK
=MK,ϕwith L =Ekin −Epot and k =1, . . . , n. (1)
The results indicate that the calculated process forces do not vary significantly for
segment numbers of n
≥
25, so that n = 25 was chosen as a compromise between high
accuracy and low computation time. By the introduction of a rotating brush body and a
translationally moved workpiece in the form of boundary conditions, the contact between
abrasive filaments and arbitrary workpieces can be simulated in three-dimensional space.
To subsequently characterize the contact behavior, a nominal contact length l
d
was defined,
which can be deduced from the contact angle
Ωc
(Figure 2). Depending on the tool
specification and process parameters, abrasive filaments may exhibit dynamic oscillatory
behavior following the initial workpiece contact, which leads to the bouncing of the filament
tips on the workpiece surface. Therefore, the actual contact length l
c
was calculated as the
overall sum of the single contact lengths lc,i.
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Ceramics 2021,4400
Ceramics 2021, 4 FOR PEER REVIEW 4 of 12
Figure 2. Simulation of the deflection of a single abrasive filament during workpiece contact.
Analogously to the nominal contact length l
d
and the actual contact length l
c
, the
nominal contact time t
d
and the actual contact time t
c
were calculated, within which a fil-
ament tip was moved across the respective distance. Both the contact length l
c
and the
contact time t
c
can be calculated from the contact normal force F
n
, under the consideration
that time steps without a filament–workpiece contact yield a contact normal force of F
n
=
0 N (Figure 3). Due to the contact time t
c
being highly dependent on the brushing velocity
v
b
and the penetration depth a
e
, the contact time ratio ε
tc
was introduced on the grounds
of interpretability (Equation (2)). It corresponds to a dimensionless normalization of the
contact time t
c
by the nominal contact time t
d
and serves as a basis upon which to charac-
terize filament motions as either striking (ε
tc
→ 0) or sweeping (ε
tc
→ 1).
εtc = t
c
t
d
∈ [0,1]. (2)
It should also be noted that the contact time ratio ε
tc
equals the analogously calculated
contact length ratio ε
lc
, assuming a constant time step width Δt. Nonetheless, the contact
time ratio ε
tc
was used within the scope of this article due to the contact normal impulse
p
n
being based on the contact time t
c
and not the contact length l
c
(Equation (3)).
With regard to the dynamic oscillatory behavior of the abrasive filaments, the contact
time ratio ε
tc
is most likely a more robust measure than the previously established maxi-
mum filament tip deflection s
n,max
, meaning the maximum absolute value of the shortest
distance between deflected filament tip and undeformed filament during the fila-
ment-workpiece contact [13] (Figure 1c). Its main disadvantage becomes apparent while
studying the curve shape of the filament tip deflection s
n
over time t (Figure 3), which may
fluctuate considerably during the workpiece contact, where the deflection maximum is
located. However, the maximum filament tip deflection s
n,max
lacks usable information
about the workpiece contact itself, which would be crucial in order to understand the ef-
fects of highly dynamic filament behavior on the productivity.
Figure 2. Simulation of the deflection of a single abrasive filament during workpiece contact.
Analogously to the nominal contact length l
d
and the actual contact length l
c
, the nom-
inal contact time t
d
and the actual contact time t
c
were calculated, within which a filament
tip was moved across the respective distance. Both the contact length l
c
and the contact
time t
c
can be calculated from the contact normal force F
n
, under the consideration that
time steps without a filament–workpiece contact yield a contact normal force of
Fn=0N
(Figure 3).
Due to the contact time t
c
being highly dependent on the brushing velocity v
b
and the penetration depth a
e
, the contact time ratio
εtc
was introduced on the grounds of
interpretability (Equation (2)). It corresponds to a dimensionless normalization of the con-
tact time t
c
by the nominal contact time t
d
and serves as a basis upon which to characterize
filament motions as either striking (εtc →0) or sweeping (εtc →1).
εtc =tc
td
∈[0, 1]. (2)
It should also be noted that the contact time ratio
εtc
equals the analogously calculated
contact length ratio
εlc
, assuming a constant time step width
∆
t. Nonetheless, the contact
time ratio
εtc
was used within the scope of this article due to the contact normal impulse
pnbeing based on the contact time tcand not the contact length lc(Equation (3)).
With regard to the dynamic oscillatory behavior of the abrasive filaments, the con-
tact time ratio
εtc
is most likely a more robust measure than the previously established
maximum filament tip deflection s
n,max
, meaning the maximum absolute value of the
shortest distance between deflected filament tip and undeformed filament during the
filament-workpiece contact [
13
] (Figure 1c). Its main disadvantage becomes apparent while
studying the curve shape of the filament tip deflection s
n
over time t (Figure 3), which may
fluctuate considerably during the workpiece contact, where the deflection maximum is
located. However, the maximum filament tip deflection s
n,max
lacks usable information
about the workpiece contact itself, which would be crucial in order to understand the
effects of highly dynamic filament behavior on the productivity.
Ceramics 2021,4401
Ceramics 2021, 4 FOR PEER REVIEW 5 of 12
Figure 3. Contact normal force Fn and filament tip displacement sn over time t.
Further examination of the contact normal force Fn reveals that a meaningful maxi-
mum value is not easily deduced, because the curve shape is characterized by extreme
peaks and a multitude of contact-free regions (Figure 3), which can be attributed to the
dynamic oscillatory behavior of the abrasive filaments as well as to their complex defor-
mation [13,14]. Potential smoothing algorithms, such as median or Gaussian filters, would
thereby lead to significant deviations and consequently distort the curve shape. While the
contact normal force Fn shows consistent, steadily increasing curve shapes at low brushing
velocities vb [13], the described dynamic behavior necessitates novel methods of evalua-
tion, especially for high, industrially relevant brushing velocities vb.
Therefore, the contact normal force Fn was converted into the contact normal impulse
pn, (Equation (3)) [15]. Formally, the contact normal force Fn of a single abrasive filament
was integrated over the contact time tc. Because contact forces for both the process model
and after experimental data acquisition exist in the form of discrete values with distinct
indices i, the overall sum of all values Fn,i and subsequent multiplication with the time
step width Δt can be used for this purpose.
pn = Fn
tc
· dt = Δt · Fn,i
i. (3)
As for the inverse transform, dividing the contact normal impulse pn by the contact
time tc yields the equivalent contact normal force F
n (Equation (4)), which represents an
arithmetic averaging of the contact normal force Fn over the contact time tc.
F
n = pn
tc. (4)
To further project this method, which is only valid for single abrasive filaments, onto
full brushing tools, the contact normal impulse pn was multiplied with the estimated num-
ber of filaments Nf and the angular velocity ω to calculate the tool contact normal force
Fn,w (Equation (5)).
Fn,w = pn · Nf · ω
2π. (5)
process:
brushing with bonded
abrasives (simulation)
tool:
single filament
Abrafil n. G. (PA 6.12)
diamond, 320 mesh
rg= 140 mm
lf= 40 mm
df= 1.18 mm
ρf= 1.11 g/cm³
n = 25
workpiece:
plate, ground
MgO-PSZ (ZrO2)
200 ×200 ×20 mm³
process parameters:
vb= 10 m/s
vft = 0 mm/min
ae= 5 mm
contact normal force Fn
filament tip deflection sn
ae
vb
vft
360
time t
18
1.2
0.3
0
0.6
N
ms9
contact normal force F
n
24
−12
−24
0
mm
filament tip deflection s
n
s
n,max
tc = ∑tc,i
td
Figure 3. Contact normal force Fnand filament tip displacement snover time t.
Further examination of the contact normal force F
n
reveals that a meaningful max-
imum value is not easily deduced, because the curve shape is characterized by extreme
peaks and a multitude of contact-free regions (Figure 3), which can be attributed to the
dynamic oscillatory behavior of the abrasive filaments as well as to their complex de-
formation [
13
,
14
]. Potential smoothing algorithms, such as median or Gaussian filters,
would thereby lead to significant deviations and consequently distort the curve shape.
While the contact normal force F
n
shows consistent, steadily increasing curve shapes at low
brushing velocities v
b
[
13
], the described dynamic behavior necessitates novel methods of
evaluation, especially for high, industrially relevant brushing velocities vb.
Therefore, the contact normal force F
n
was converted into the contact normal impulse
p
n
, (Equation (3)) [
15
]. Formally, the contact normal force F
n
of a single abrasive filament
was integrated over the contact time t
c
. Because contact forces for both the process model
and after experimental data acquisition exist in the form of discrete values with distinct
indices i, the overall sum of all values F
n,i
and subsequent multiplication with the time step
width ∆t can be used for this purpose.
pn=Z
tc
Fn·dt =∆t·∑
i
Fn,i. (3)
As for the inverse transform, dividing the contact normal impulse p
n
by the contact
time t
c
yields the equivalent contact normal force
−
Fn
(Equation (4)), which represents an
arithmetic averaging of the contact normal force Fnover the contact time tc.
Fn=Pn
tc. (4)
To further project this method, which is only valid for single abrasive filaments,
onto full brushing tools, the contact normal impulse p
n
was multiplied with the estimated
number of filaments N
f
and the angular velocity
ω
to calculate the tool contact normal
force Fn,w (Equation (5)).
Fn,w=pn·Nf·ω
2π. (5)
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Ceramics 2021,4402
Physically, the tool contact normal force F
n,w
corresponds to the tool contact normal
impulse p
n,w
transmitted by the brushing tool onto the workpiece over the course of a
single second. It represents an extrapolation of a single abrasive filament while neglecting
filament interactions, but shows robust behavior compared to typical smoothing algorithms
if applied to abrasive filament contact forces.
3. Results
Utilizing the multi-body system based on the Lagrange formalism, a total of
120 contact simulations
were computed, each modeling the contact between a single abra-
sive filament and a planar workpiece of ZrO
2
, while considering brushing velocities of
vb≤30 m/s
and penetration depths of a
e≤
5 mm. The tangential feed rate v
ft
was not
varied, as it was not expected to have an influence due to the plain workpiece geometry,
no tangential or friction forces being investigated, and the tangential feed rate v
ft
being
several orders of magnitude smaller than the brushing velocity vb.
The maximum filament tip displacement s
n,max
, used to monitor dynamic oscillatory
filament behavior during previous research [
14
], increases approximately linearly with
the brushing velocity v
b
and degressively with the penetration depth a
e
(Figure 4a) but
otherwise provides insufficient information about the filament–workpiece contact due to
the absence of a time dependent component. The corresponding contact time t
c
shows a
highly regressive decrease with increased brushing velocity v
b
(Figure 4b), although this
effect is more prevalent for large penetration depths a
e
than for small penetration depths a
e
because of the increased contact angle
Ωc
(Figure 2). This implies that the brushing velocity
v
b
has a considerably greater impact on the contact time t
c,
as well as on subsequently
calculated parameters, than the penetration depth ae.
Ceramics 2021, 4 FOR PEER REVIEW 6 of 12
Physically, the tool contact normal force F
n,w
corresponds to the tool contact normal
impulse p
n,w
transmitted by the brushing tool onto the workpiece over the course of a
single second. It represents an extrapolation of a single abrasive filament while neglecting
filament interactions, but shows robust behavior compared to typical smoothing algo-
rithms if applied to abrasive filament contact forces.
3. Results
Utilizing the multi-body system based on the Lagrange formalism, a total of 120 con-
tact simulations were computed, each modeling the contact between a single abrasive fil-
ament and a planar workpiece of ZrO
2
, while considering brushing velocities of v
b
≤ 30
m/s and penetration depths of a
e
≤ 5 mm. The tangential feed rate v
ft
was not varied, as it
was not expected to have an influence due to the plain workpiece geometry, no tangential
or friction forces being investigated, and the tangential feed rate v
ft
being several orders
of magnitude smaller than the brushing velocity v
b
.
The maximum filament tip displacement s
n,max
, used to monitor dynamic oscillatory
filament behavior during previous research [14], increases approximately linearly with
the brushing velocity v
b
and degressively with the penetration depth a
e
(Figure 4a) but
otherwise provides insufficient information about the filament–workpiece contact due to
the absence of a time dependent component. The corresponding contact time t
c
shows a
highly regressive decrease with increased brushing velocity v
b
(Figure 4b), although this
effect is more prevalent for large penetration depths a
e
than for small penetration depths
a
e
because of the increased contact angle Ω
c
(Figure 2). This implies that the brushing ve-
locity v
b
has a considerably greater impact on the contact time t
c,
as well as on subse-
quently calculated parameters, than the penetration depth a
e
.
Figure 4. Filament–workpiece contact characteristics; (a) maximum filament tip displacement s
n,max
and (b) contact time t
c
for different brushing velocities v
b
and penetration depths a
e
.
Figure 4.
Filament–workpiece contact characteristics; (
a
) maximum filament tip displacement s
n,max
and (
b
) contact time t
c
for different brushing velocities vband penetration depths ae.
Ceramics 2021,4403
For a more suitable visualization, the contact time t
c
was divided by the nominal
contact time t
d
to calculate the contact time ratio
εtc
(Equation (2), Figure 5). Variation of
the process parameters shows that high brushing velocities v
b
lead to primarily striking
filament motions (
εtc →
0) because abrasive filaments, deflected by an initial workpiece
contact, are moved past the contact zone before they can swing back. In addition, an ap-
proximately proportional separating line between brushing velocity v
b
and penetration
depth a
e
can be observed, below which filament motions are primarily striking, and above
which the primarily sweeping region is characterized by a distinguishable tier.
Ceramics 2021, 4 FOR PEER REVIEW 7 of 12
For a more suitable visualization, the contact time t
c
was divided by the nominal con-
tact time t
d
to calculate the contact time ratio ε
tc
(Equation (2), Figure 5). Variation of the
process parameters shows that high brushing velocities v
b
lead to primarily striking fila-
ment motions (ε
tc
→ 0) because abrasive filaments, deflected by an initial workpiece con-
tact, are moved past the contact zone before they can swing back. In addition, an approx-
imately proportional separating line between brushing velocity v
b
and penetration depth
a
e
can be observed, below which filament motions are primarily striking, and above which
the primarily sweeping region is characterized by a distinguishable tier.
Figure 5. Contact time ratio ε
tc
for different penetration depths a
e
and brushing velocities v
b
.
In theory, dynamic oscillatory behavior and subsequent striking filament motions
lead to an increase of the process forces and thereby to an increase of the material removal
rate [16–19]. Although dynamic behavior can be both confirmed and controlled based on
technological investigations, a positive influence on productivity remains to be confirmed.
Nonetheless, the knowledge of the contact time ratio ε
tc
might prospectively be utilized to
predict the productivity of abrasive brushing processes. However, further technological
investigations need to be carried out to verify this.
Comparing the contact normal impulses p
n
of a single abrasive filament for varied
process parameters, a regressive dependence on the brushing velocity v
b
can be observed,
particularly for high penetration depths a
e
(Figure 6a). Similar to the contact time ratio ε
tc
,
although less distinct, a tier is formed along the proportional line between brushing ve-
locity v
b
and penetration depth a
e
, below which the contact normal impulse p
n
is low.
Except for the tier, the contact normal impulse p
n
qualitatively resembles the contact time
t
c
(Figure 4b) indicating that the contact normal force F
n
affects the contact normal impulse
p
n
considerably less than the contact time t
c
, Equation (3).
Figure 5. Contact time ratio εtc for different penetration depths aeand brushing velocities vb.
In theory, dynamic oscillatory behavior and subsequent striking filament motions
lead to an increase of the process forces and thereby to an increase of the material removal
rate [
16
–
19
]. Although dynamic behavior can be both confirmed and controlled based on
technological investigations, a positive influence on productivity remains to be confirmed.
Nonetheless, the knowledge of the contact time ratio
εtc
might prospectively be utilized to
predict the productivity of abrasive brushing processes. However, further technological
investigations need to be carried out to verify this.
Comparing the contact normal impulses p
n
of a single abrasive filament for varied
process parameters, a regressive dependence on the brushing velocity v
b
can be observed,
particularly for high penetration depths a
e
(Figure 6a). Similar to the contact time ratio
εtc
, although less distinct, a tier is formed along the proportional line between brushing
velocity v
b
and penetration depth a
e
, below which the contact normal impulse p
n
is low.
Except for the tier, the contact normal impulse p
n
qualitatively resembles the contact time
t
c
(Figure 4b) indicating that the contact normal force F
n
affects the contact normal impulse
pnconsiderably less than the contact time tc, Equation (3).
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Figure 6. Extrapolation of filament–workpiece contact forces; (a) contact normal impulse p
n
and (b) tool contact normal
force F
n,w
for different brushing velocities v
b
and penetration depths a
e
.
To explain the contradictory phenomenon where high brushing velocities v
b
lead to
low impulse transmissions and therefore to decreased productivity, the tool contact nor-
mal force F
n,w
is calculated with regard to angular velocity ω and the estimated number of
filaments N
f
(Equation (5)). Figure 6b illustrates that high brushing velocities v
b
amount
to a higher tool contact normal force F
n,w
than low brushing velocities v
b
, because the total
number of filament–workpiece contacts increases more rapidly with increasing angular
velocity ω than the impulse transmitted by a single filament decreases. Equally noticeable
is that the tool contact normal force F
n,w
only shows a linear increase with the penetration
depth a
e
for low brushing velocities v
b
. This gives the assumption that, contrary to previ-
ous research, high brushing velocities v
b,
together with large penetration depths a
e
, may
lead to unproductive brushing processes due to both parameters substantially contrib-
uting to tool wear [1,3,14].
Moreover, the tier observed in Figure 5 and Figure 6a results in a local maximum of
the tool contact normal force F
n,w
(Figure 6b) likely caused by a second contact area to-
wards the end of the filament–workpiece contact. This local maximum might be targeted
by means of appropriate process design, in order to maximize the impulse transmission
with only moderate brushing velocities v
b
. To what extent this phenomenon has an impact
on productivity remains to be determined by additional technological investigations.
Comparing the computed tool contact normal force F
n,w
with the experimentally de-
termined values, the variation of the brushing velocity v
b
shows that the fundamental be-
havior is reflected by the model, although all investigated brushing velocities v
b
lead to
deviations outside of the experimentally determined standard deviation (Figure 7a). Re-
Figure 6.
Extrapolation of filament–workpiece contact forces; (
a
) contact normal impulse p
n
and (
b
) tool contact normal
force Fn,w for different brushing velocities vband penetration depths ae.
To explain the contradictory phenomenon where high brushing velocities v
b
lead to
low impulse transmissions and therefore to decreased productivity, the tool contact normal
force F
n,w
is calculated with regard to angular velocity
ω
and the estimated number of
filaments N
f
(Equation (5)). Figure 6b illustrates that high brushing velocities v
b
amount to
a higher tool contact normal force F
n,w
than low brushing velocities v
b
, because the total
number of filament–workpiece contacts increases more rapidly with increasing angular
velocity
ω
than the impulse transmitted by a single filament decreases. Equally noticeable
is that the tool contact normal force F
n,w
only shows a linear increase with the penetration
depth a
e
for low brushing velocities v
b
. This gives the assumption that, contrary to previous
research, high brushing velocities v
b,
together with large penetration depths a
e
, may lead
to unproductive brushing processes due to both parameters substantially contributing to
tool wear [1,3,14].
Moreover, the tier observed in Figures 5and 6a results in a local maximum of the
tool contact normal force F
n,w
(Figure 6b) likely caused by a second contact area towards
the end of the filament–workpiece contact. This local maximum might be targeted by
means of appropriate process design, in order to maximize the impulse transmission with
only moderate brushing velocities v
b
. To what extent this phenomenon has an impact on
productivity remains to be determined by additional technological investigations.
Comparing the computed tool contact normal force F
n,w
with the experimentally
determined values, the variation of the brushing velocity v
b
shows that the fundamental
behavior is reflected by the model, although all investigated brushing velocities v
b
lead
to deviations outside of the experimentally determined standard deviation
(Figure 7a).
Regarding the penetration depth a
e
, the experimentally determined results show a de-
Ceramics 2021,4405
pendence that is not predicted by the model within the investigated parameter bound-
aries (Figure 7b).
Figure 7. Tool contact normal force Fn,w for different (a) brushing velocities vband (b) penetration depths ae.
In both cases, the most plausible cause of error may be the neglection of the filament
interactions, because the computation of the tool contact normal force F
n,w
is solely based
on the extrapolation of a single abrasive filament. For large penetration depths a
e
, it is
expected that abrasive filaments with either striking or sweeping motion behavior may be
pressed onto the workpiece surface by neighboring filaments, which would increase the
impulse transmission and thereby the process forces. Additional possible error causes could
be undetected manufacturing inaccuracies of the brushing tools as well as a slight waviness
of the abrasive filaments purposely induced by the manufacturer but not considered for the
model, for which perfectly cylindrical filaments are assumed. Furthermore, the filament–
workpiece contact is merely modeled for undeformed abrasive filaments, starting shortly
before the initial contact in order to minimize computation times, whereas subsequent tool
rotations and filament workpiece contacts at high brushing velocities v
b
might cause the
abrasive filaments to already be deflected.
4. Discussion
Within the scope of this article, a new method is introduced to correlate the numerically
simulated contact forces of single abrasive filaments with the experimentally determined
process forces of full brushing tools by interim calculation of the contact impulse. Based on
the presented work, the following conclusions can be drawn:
•
High brushing velocities v
b
may not compulsorily lead to maximum productivity,
but less aggressive process parameters might yield more productive results instead,
considering that the productivity is affected negatively by tool wear.
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Ceramics 2021,4406
•
Despite a variety of simplifications—which include the extrapolation of the total
number of filaments N
f
as well as the assumption of constant filament diameters d
f
and filament lengths l
f
,—measured and modeled tool contact normal forces F
n,w
are
in the same order of magnitude.
•
During the variation of the penetration depth a
e
in particular, discrepancies between
measured and modeled tool contact normal forces Fn,w arise.
•As a main cause of error, neglected filament interactions are suggested.
Due to the model inaccuracies, at this point in time, the model should only be applied
under the conditions of brushing tools without densely packed abrasive filaments and
small penetration depths a
e
. However, as industrial brushing processes usually require
small penetration depths of ae≤1 mm, the model is estimated to be partially applicable.
Additionally, the contact time ratio
εtc
is introduced in order to quantify the possibly
dynamic motion behavior of single abrasive filaments. This allows for a distinction between
primarily sweeping and primarily striking filament motions and might be used for future
research as an easily computed means to predict productivity while considering tool wear.
5. Outlook
Prospectively, technological investigations are planned to verify the influence of the
tool contact normal force F
n,w
and the contact time ratio
εtc
on the productivity, meaning on
the rates of change of the surface roughness and the material removal rate. Furthermore,
the interactions between filaments will be modeled similarly to the filament–workpiece
contact in order to explain the discrepancies between the current physical model and the
experimental results. For this purpose, the multi-body system will be compared with,
and possibly superseded by, commercially available software employing the discrete
element method, granting a compromise between modeling accuracy and computation
time. The finite element method will also be used to analyze the wear-related change of
the abrasive filament tip shape over time and its influence on the productivity of abrasive
brushing processes.
Further research should be carried out, investigating not only the process forces
exerted onto the surfaces of planar workpieces, but also workpieces with complex shapes
and especially workpiece edges, as edge deburring still remains the most important field
of application for abrasive brushing tools in industrial finishing processes, particularly for
metallic workpieces. The current implementation of the multi-body system permits the
investigation of both complex workpiece shapes and their edges by utilizing polynomial
splines of arbitrary degree, including them as boundary conditions while solving the
system of differential equations obtained from the Lagrange formalism. Of equal industrial
importance is the comparison of round brushing tools, as described within the scope of this
article, with brushing tools comprising filaments with axial orientation. Round brushing
tools were chosen for this research project because of their penetration depth a
e
being
independent from the brushing velocity v
b
, whereas other tool shapes may lead to a
reduction of the penetration depth a
e
and thus the tool contact normal force F
n,w,
with
increasing brushing velocity v
b
due to centrifugal forces deflecting the filaments outwards,
necessitating force controlled brushing processes as opposed to geometrically planned
tool paths.
Author Contributions:
Conceptualization, A.H., E.U.; methodology, E.U.; software, A.H.; validation,
A.H.; formal analysis, A.H.; investigation, A.H.; resources, A.H.; data curation, A.H.; writing—
original draft preparation, A.H.; writing—review and editing, E.U.; visualization, A.H.; supervision,
E.U.; project coordination, E.U.; funding acquisition, E.U. All authors have read and agreed to the
published version of the manuscript.
Funding:
This research was funded by Deutsche Forschungsgemeinschaft (DFG) within the scope
of the project “Analyse des Zerspan- und Verschleißverhaltens beim Bürstspanen mit abrasivem
Medium sprödharter Werkstoffe”, project number 392312434. The authors kindly thank the funder
for their support.
Ceramics 2021,4407
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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