Modeling of cutting forces in 1-D and 2-D ultrasonic vibration-assisted milling of Ti-6Al-4V

Abstract

To meet the modern demands for lightweight construction and energy efficiency, hard-to-machine materials such as ceramics, superalloys, and fiber-reinforced plastics are being used progressively. These materials can only be machined with great effort using conventional machining processes due to the high cutting forces, poor surface qualities, and the associated tool wear. Vibration-assisted machining has already proven to be an adequate solution in order to achieve extended tool lives, better surface qualities, and reduced cutting forces. This paper presents an analytical force model for longitudinal-torsional vibration-assisted milling (LT-VAM), which can predict cutting forces under intermittent and non-intermittent cutting conditions. Under intermittent cutting conditions, the relative contact ratio between the rake face and the sliding chip is utilized for modelling the shearing forces. Ploughing forces and shearing forces under non-intermittent cutting conditions are calculated by using an extended macroscopic friction reduction model, which can predict the reduced frictional forces under parallel and perpendicular vibration superimposition. The force model was implemented in MATLAB and can predict cutting forces without using any experimental vibration-assisted milling (VAM) data input.

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Vol.:(0123456789)
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https://doi.org/10.1007/s00170-021-08355-x
ORIGINAL ARTICLE
Modeling ofcutting forces in1‑D and2‑D ultrasonic vibration‑assisted
milling ofTi‑6Al‑4V
PhilippM.Rinck1· AlpcanGueray1· MichaelF.Zaeh1
Received: 1 May 2021 / Accepted: 5 November 2021
© The Author(s) 2021
Abstract
To meet the modern demands for lightweight construction and energy efficiency, hard-to-machine materials such as ceram-
ics, superalloys, and fiber-reinforced plastics are being used progressively. These materials can only be machined with great
effort using conventional machining processes due to the high cutting forces, poor surface qualities, and the associated tool
wear. Vibration-assisted machining has already proven to be an adequate solution in order to achieve extended tool lives,
better surface qualities, and reduced cutting forces. This paper presents an analytical force model for longitudinal-torsional
vibration-assisted milling (LT-VAM), which can predict cutting forces under intermittent and non-intermittent cutting condi-
tions. Under intermittent cutting conditions, the relative contact ratio between the rake face and the sliding chip is utilized for
modelling the shearing forces. Ploughing forces and shearing forces under non-intermittent cutting conditions are calculated
by using an extended macroscopic friction reduction model, which can predict the reduced frictional forces under parallel
and perpendicular vibration superimposition. The force model was implemented in MATLAB and can predict cutting forces
without using any experimental vibration-assisted milling (VAM) data input.
Keywords Cutting force· Vibration-assisted machining (VAM)· Milling· Ti 6 Al 4V· Friction reduction
1 Introduction
Ti-6Al-4V, also called grade 5 titanium, is a heat-treatable
α+β titanium alloy featuring biocompatibility, high strength
to weight ratio, and high corrosion resistance. As a result,
Ti-6Al-4V is used extensively in the aerospace industry, in
marine applications and medical implants. In the aerospace
industry, up to 95% of the raw part mass gets removed dur-
ing machining operations resulting in high resource con-
sumption and emissions during manufacturing [1].
In addition, the low thermal conductivity and high chemi-
cal reactivity with many cutting tool materials make titanium
and its alloys difficult to machine materials [2]. Machining
titanium is characterized by high tool-chip interface tem-
peratures due to its high strength and low thermal conductiv-
ity, which generates high temperatures during plastic defor-
mation. In addition, the low Young’s modulus of elasticity
causes deflection and rubbing problems during machining,
which further amplifies tool wear [3].
The term “vibration-assisted machining” refers to machin-
ing where an additional vibration is superimposed on the con-
ventional kinematics of the cutting process. Vibration-assisted
machining has been the subject of scientific investigation
since the 1950s and is currently used to overcome the tech-
nical limitations of conventional machining [4, 5]. Figure1
shows different types of vibration-assisted machining, which
can be distinguished according to tangential, feed and radial
vibration assistance (VA) depending on the resulting tooltip
trajectory. When applied tangentially to the cutting direction,
VA can cause the tool’s rake face to separate from the chip,
resulting in “intermittent cutting.” If the applied VA is in the
feed direction, chip thickness will vary during cutting, which
results in intermittent cutting at low feed rates [6, 7]. VA in
the radial direction can cause orthogonal cutting processes
to behave like oblique cutting [8] and vibrations in different
directions can be combined to create elliptical and 3-D paths.
* Philipp M. Rinck
philipp.rinck@iwb.tum.de
Alpcan Gueray
alpcan.gueray@tum.de
Michael F. Zaeh
michael.zaeh@iwb.tum.de
1 Institute forMachine Tools andIndustrial Management
(iwb), Technical University ofMunich, Boltzmanstr. 15,
85748Garching, Germany
/ Published online: 30 November 2021
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1 3
Rinck etal. [9] studied the effects of longitudinal (vibra-
tion normal to the cutting direction) and longitudinal-torsional
(vibration normal and in cutting direction) VAM under inter-
mittent and non-intermittent cutting conditions when milling
Ti-6Al-4V. Peripheral milling experiments have shown that
increasing vibration amplitudes result in lower cutting forces
and better surface finishes. During slotting, higher amplitudes
produce harder surfaces with increased compressive residual
stresses. The primary cutting edges of the tools used during
VAM show 20% less wear, but minor cutting-edge wear is
increased. Overall, LT-VAM performs better than longitudi-
nal VAM. Gao and Altintas [10] modeled the chatter stabil-
ity of synchronized elliptical VAM (vibrations normal to the
surface and in cutting direction). Vibrations normal to the
surface resulted in dynamic chip thickness and thus in forced
vibrations. At low cutting speeds, vibrations in cutting direc-
tion caused the periodic separation of tool and chip, result-
ing in intermittent contact and eliminating chatter vibrations.
At higher cutting speeds, intermittent contact no longer took
place and the effects of vibration assistance diminished. In
the case of Al 7050, the minimum stable depth of cut (DoC)
increased from 1.2 to 2.0 mm. Similarly, during half immer-
sion down milling of AISI 1045 steel, the minimum stable
DoC increased from 0.2 mm to 0.35 mm. Most of the research
done in the field of vibration assisted machining only concen-
trates on intermittent cutting conditions, where the influence
of the VA is more obvious. However, research conducted by
Suárez etal. [11] and Rinck etal. [9], presented the existence
of a phase where the tool-chip contact is not interrupted, but a
force reduction due to the vibration assistance is still evident.
Numerous methods exist for modeling cutting forces
during machining processes. According to Arrazola etal.
[12], these can be divided into artificial intelligence-based,
empirical, analytical, and numerical approaches. Analyti-
cal models describe the physical relations between the
input and output parameters, requiring no experimental
calibration and allowing simpler expansion of the models.
Merchant [13] modeled the cutting forces in 2-dimensional
orthogonal cutting, where the cutting force is a function
of the shear stress of the material (τs), the shear angle (Φ),
the rake angle (αr) and the friction angle (βa) (see Fig.5).
In addition, Merchant [14] proposed a model to predict the
shear angle (Φ), assuming the shearing of the material will
happen at the minimum cutting power. Based on his work,
numerous other analytical models were developed. These
models were able to calculate cutting forces, temperatures,
stresses, and strains based on material, workpiece and tool
properties [15–18].
In the case of 3-dimensional oblique cutting, the cutting
speed vector stands with the angle
𝛾
to the plane normal
to the cutting edge of the tool (plane x1z1 in Fig.6). The
inclined cutting edge introduces forces in all 3 cartesian
directions, which can be properly determined by knowing
the 5 oblique cutting angles, namely the normal (Φn) and
oblique (Φi) angle of the shear velocity (Vs), the normal
(θn) and oblique (θi) angle of the resulting shearing force
(FRes,s), and the chip flow angle (η) (see Fig.6). Shamoto
and Altıntas [19] developed a model to predict the 5 oblique
cutting parameters by using the tool geometry and the fric-
tion angle (βa), allowing the prediction of shearing forces in
oblique cutting operations.
The prediction of cutting forces by analytical or numeri-
cal models in vibration-assisted machining has been the
subject of several research projects [20]. Most analytical
models predict the cutting forces by multiplying the cut-
ting forces during conventional machining with the relative
contact ratio between tool and workpiece [21, 22]. In the
case of vibration-assisted micro-milling, Ding etal. [6] mod-
eled the cutting forces by calculating the relative contact
ratio and the chip thickness resulting from the previous tool
passes. However, this model only produces reliable results
for micro-milling, where low feed rates are used. Shamoto
etal. [8] developed a model which can predict the cutting
forces in a 3D elliptical cutting process using the thin shear
plane model. They assumed a constant shear direction and
friction angle when the tool and workpiece were in contact.
The shear angle calculation was applied either by using the
minimum energy principle of Merchant [14] or by using
the maximum shear stress principle of Krystoff [23]. Verma
etal. [24] developed a cutting force model for axial ultra-
sonic assisted milling where the effect of acoustic softening
of the workpiece material is considered with the help of a
modified Johnson-Cook model. Arefin etal. [25] developed
an analytical model that can predict the shear angle during
vibration-assisted orthogonal cutting. The proposed model
uses an energy-based approach that calculates the cutting
force during elastic deformation, plastic deformation, and
the elastic recovery steps in a vibration cycle to predict the
equivalent shear angle.
In addition, there are numerical models in the litera-
ture that represent the chip formation process in VAM
using the FEM [26]. The simulations are mostly limited
to the two-dimensional orthogonal cut, where a vibration
Fig. 1 Types of vibration-assisted machining [9]
1808 The International Journal of Advanced Manufacturing Technology (2022) 119:1807–1819
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1 3
is superimposed on the tool cutting edge. Patil etal. [27]
developed a numerical model for turning Ti 6Al 4V with
a vibration superposition in the cutting direction. For the
modeling, they use a combined Lagrangian-Euler approach
with a Johnson-Cook material model. The friction between
the cutting edge and the outgoing chip is assumed to be
constant. A simulative parameter study was conducted to
determine the influence of conventional process parameters
and vibration parameters on the cutting force components.
As the cutting speed increases from 10 to 30 m/min, a con-
tinuous cutting force reduction of 37 to 31 % is shown.
According to the model, an increasing frequency and an
increasing amplitude cause a higher cutting force reduction,
which was attributed by the authors to a lower engagement
time between the tool and the workpiece. A comparison
of the simulation results with experimental investigations
shows a good qualitative agreement of the results.
In the present work, a novel cutting force model for
VAM is introduced that can calculate cutting forces under
intermittent cutting and non-intermittent cutting condi-
tions. It combines a model for friction reduction under
the presence of ultrasonic vibrations with an analytical
cutting force model. In the following section, the kinemat-
ics of a longitudinal-torsional vibration-assisted helical
end mill will be modeled, the theory of macroscopic fric-
tion reduction will be explained, extended, and applied to
model the cutting forces under orthogonal and oblique cut-
ting conditions. Afterwards, the experimental settings and
the used ultrasonic actuator will be presented. The model
verification and the discussion of the results is explained
in Sect.4, followed by the conclusions in Sect.5.
2 Model development
In this chapter, a new force model is discussed which can
predict the cutting forces under longitudinal and longitudinal-
torsional vibration superposition. The first section is for
modeling the kinematics of a longitudinal-torsional vibration
superimposition to calculate the relative contact time between
the rake face of the cutting tool and the chip. Afterwards, the
macroscopic friction reduction model of Storck etal. [28]
will be extended for the case of simultaneous parallel and
perpendicular vibration assistance. In the final section, the
modeled relative contact ratio and the macroscopic friction
coefficient is used to extend a force model for VAM.
2.1 Relative contact ratio betweentool
andworkpiece duringLT‑VAM
In VAM, the effective movement of the cutting edge is
modified by the vibration superimposition. Due to the
longitudinal vibrations, each point of the milling tool
experiences an oscillation along the tool axis. In the case
of LT-VAM, an additional torsional vibration is superim-
posed again to act along the cutting direction. The result-
ing trajectory of the tooltip is given by
with
Al and At are the corresponding longitudinal and tor-
sional vibration amplitudes at the tooltip, r is the radius
of the tool, n is the spindle speed, fus is the ultrasonic
vibration frequency and
Δ𝜙
stands for a possible phase
shift difference between the longitudinal and the torsional
vibrations.
To interrupt the contact between the tool’s rake face and
the chip, the cutting edge must move away from the uncut
workpiece. By formulating the trajectory of the cutting edge,
the relative engagement during VAM can be calculated. The
position of a point on the tools end cutting edge, which moves
due to torsional vibration, is given by the following equations:
By differentiating Eq. (3) with respect to time, the result-
ing torsional vibration velocity vus,t and its maximum value
vt,max can be calculated as follows:
The vector of the torsional vibration is parallel to the
cutting speed vector (see Fig.2). In the case of a torsional
vibration, the cutting edge loses contact to the cutting zone
when the vibration velocity vus,t exceeds the cutting speed.
The vector of the longitudinal vibration, on the other hand,
(1)
�⃗
x
(t)=




rsin

2𝜋nt +At
rsin

𝜔us t+Δ𝜙)

+vft
rcos2𝜋nt +At
rsin𝜔ust+Δ𝜙
A
l
⋅sin

𝜔
us
t





(2)
𝜔us =2
⋅
𝜋
⋅
fus
(3)
x
(t)=A
t
⋅sin
(
𝜔
us
⋅t
)
(4)
vus,t
(t)=𝜔
us
⋅A
t
⋅cos
(
𝜔
us
⋅t
)
(5)
vt,max =𝜔us
⋅
At
,
,
Cutting
edge
,
,
Cutting
edge
Δ
,
Fig. 2 Relation between the helix angle and the relative contact ratio
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1 3
is perpendicular to the cutting speed vector. Because of the
helix angle λH of the milling tool, the longitudinal vibra-
tion can still cause a loss of contact between the tool of the
workpiece. The helix angle causes the cutting edge to move
away from the cutting zone by the amount Δxus,l in the event
of a longitudinal oscillation (see Fig.2).
Figure2 visualizes the trigonometric relationship
between the helix angle λH, the longitudinal vibration
amplitude Al, and the relative displacement between the
rake face and the chip due to the longitudinal vibration
Δxus,l. The relation can be formulated by the following
equation:
The corresponding longitudinal vibration velocity at the
cutting speed direction vus,l can be calculated as follows:
Thus, in case of a combined longitudinal and torsional
vibration superimposition, the combined vibration velocity
vus,lt can be obtained by combining Eqs. (4) and (7):
The maximum vibration velocity in cutting direction,
which results from a combined longitudinal-torsional VA
(vlt,max), can be determined according to Eq. (9) if the vibra-
tion amplitudes, frequency and phase shift are known:
If the value of vlt,max exceeds the cutting speed vc, separa-
tion between the chip and the rake face occurs during every
vibration cycle. To calculate the contact time tc during a
vibration period and the resulting relative engagement time,
the times of contact loss and re-entry between the tool and
the chip must be known. Figure3 shows the path covered
by the cutting edge in cutting direction with and without
(6)
Δ
x
us,l
(t)=A
l
⋅tan(𝜆
H
)⋅sin
(
𝜔
us
⋅t
)
(7)
vus
.
l
(t)=𝜔
us
⋅A
l
⋅tan
(
𝜆
H)
⋅cos
(
𝜔
us
⋅t
)
(8)
v
us,lt =𝜔us ⋅(At⋅cos
(
𝜔us ⋅t+Δ𝜙
)
+Al⋅tan
(
𝜆H
)
⋅cos
(
𝜔us ⋅t
)
)
(9)
vlt,max =max(vus,lt(t))
vibration superposition, illustrating the relationship between
the vibration superimposition and the times of tool exit and
re-entry.
The separation occurs when the cutting edge moves against
the cutting direction due to the vibration superposition. Since
the cutting edge exits at time tex, the relative speed of the
cutting edge is zero since the cutting speed and the vibration
speed cancel each other out. Hence, the exit time tex can be
calculated according to the following equation:
The re-entry to the cutting zone occurs at time ten, since
the displacement due to VA is equal to the distance covered
by the cutting motion with the speed vc. The simplified math-
ematical description is, therefore
where ∆xus.l stands for the displacement due to longitudinal
vibration and ∆xus,t for the displacement due to torsional
vibration. Once extended with the relevant input parameters,
Eq. (11) becomes
which can be numerically approximated to find the entering
time ta.
By knowing the exit (tex) and entry times (ten), the contact
time (tc) and the relative contact ratio (Tc) during VAM with
a helical end mill can be calculated by using the following
equation, where tP represents the duration of one oscillation
period:
When describing the kinematics and calculating the con-
tact ratio, it must be considered that elastic deformations
were neglected. In reality, the chip will first deform elasti-
cally and then plastically as the cutting edge enters the work-
piece. Similarly, the chip will initially spring back elastically
as the cutting edge exits, increasing the tool-workpiece con-
tact and resulting in a higher relative contact time between
tool and workpiece.
2.2 Macroscopic friction reduction atLT‑VAM
If the vibration velocity in cutting speed direction (see Eq.9)
is lower than the cutting speed, the rake face and the chip will
be in constant contact despite the vibration superimposition.
(10)
v
c−𝜔us ⋅(At⋅cos
(
𝜔us ⋅tex +Δ𝜙
)
+Al⋅tan
(
𝜆D
)
⋅cos
(
𝜔us ⋅tex
)
)=
0
(11)
Δ
xus,t
(
ten
)
−Δxus,t
(
tex
)
+Δxus,t
(
ten
)
−Δxus
,
t
(
tex
)
=vc
⋅
(
ten −tex
)
(12)
A
t⋅sin
(
𝜔us ⋅ten +Δ𝜙
)
−At⋅sin
(
𝜔us ⋅tex +Δ𝜙
)
+(Al⋅sin(𝜔us ⋅ten)−Al⋅sin(𝜔us ⋅tex
))
⋅tan
(
𝜆H
)
=vc⋅
(
ten −tex
)
(13)
T
c=
t
P
−(t
e
−t
a
)
t
P
=tc
f−1
us
Fig. 3 Position of the cutting edge in cutting direction [9]
1810 The International Journal of Advanced Manufacturing Technology (2022) 119:1807–1819
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1 3
However, due to the VA, the cutting speed and the chip flow
velocity will be modulated at high frequency, leading to
reduced mean friction between the cutting edge and the chip.
Storck etal. [28] modeled the macroscopic friction reduction
of a sliding body during parallel or perpendicular vibration
superposition (see Fig.4). The modeled reduction is due to
the continuous change of the sliding direction and can only
consider a parallel or a perpendicular vibration superposi-
tion. However, in case of LT-VAM, a combined parallel and
perpendicular superposition occur simultaneously at the
secondary (chip, rake face) and tertiary (machined surface,
flank face) deformation zones. Therefore, in this section, the
friction reduction model of Storck etal. [28] will be extended
to model the macroscopic friction reduction in case of a com-
bined parallel and perpendicular vibration superposition.
When a body slides over a surface with constant veloc-
ity vs with a superimposed vibration standing perpendicular
to the sliding direction, the resulting velocity vector con-
tinuously changes its value and direction due to the current
vibration velocity. Since the frictional force is independent
of the speed and is always directed against the movement
direction, any perpendicular vibration varies the direction
of the friction vector, but not the magnitude.
Figure4 visualizes the resulting friction reduction in
case of a parallel or perpendicular vibration superposi-
tion. For a perpendicular vibration (index ⊥), even low
vibration speeds result in a change of direction and thus a
friction reduction as seen in Fig.4. However, in case of a
parallel vibration (index ∥), the vibration velocity must be
greater than the sliding velocity to change the movement
direction. Therefore, as can be seen in Fig.4, a vibration
in the main movement direction results in a friction reduc-
tion if the parallel vibration velocity exceeds the sliding
velocity.
The perpendicular vibration velocity v⊥ is calculated by
where v⊥,max stands for the maximum perpendicular vibra-
tion velocity and can be calculated by using Eq.5. The
(14)
v
⟂(t)=v⟂
,max
⋅cos
(
𝜔
us
⋅t
)
velocity in the main sliding direction v∥ remains constant.
However, in case of a superimposed parallel vibration, the
velocity in the main sliding direction v∥ no longer has a con-
stant value. The resulting velocity in main sliding direction
can be calculated using the equation
where v∥,max stands for the maximum parallel vibration
velocity and Δϕ for a possible phase shift between the par-
allel and the perpendicular vibration. The absolute value
of the velocity vector can be calculated according to the
following equation:
By knowing the perpendicular and parallel components
of the current velocity vector, the direction of the result-
ing frictional force can be determined at any given time.
According to Coulomb’s law of friction, the magnitude of
the kinetic friction is only dependent on the normal force
FN and the kinetic friction coefficient µ. However, due to the
parallel and perpendicular vibrations, the direction of the
frictional vector changes continuously, causing a reduced
mean frictional force in main sliding direction FF,⊥,∥ over a
period T. The mean frictional force in the main movement
direction can be calculated as follows:
The quotient within the integral is the ratio between the
speed of the body in the main sliding direction and the over-
all speed of the body. By multiplying this quotient with the
friction coefficient and the normal force, the frictional force
acting on the main sliding direction can be found. After
determining the mean frictional force, the friction reduc-
tion coefficient μus,⊥,∥ can be calculated by the following
equation:
By using Eq. (18), it is now possible to calculate the mac-
roscopic friction reduction coefficient in the sliding direction
during a superimposed perpendicular and parallel vibration.
In case of longitudinal-torsional VAM, Eq. (18) can be used
to calculate the resulting friction reduction in the secondary
and tertiary deformation zones.
2.3 Modelling ofcutting forces inLT‑VAM
In the following subsections, a new force model will be
developed for vibration-assisted orthogonal and oblique
cutting, which can be used to determine the cutting forces
(15)
v∥
(t)=v
s
+v
∥,max
⋅cos
(
𝜔
us
⋅t+Δ𝜙
)
(16)
|
v(t)
|
=
√(
v∥(t)
)
2+
(
v⟂(t)
)2
(17)
F
F,⟂,∥=1
T
∫T
0
v
∥
(t)
|v(t)|
⋅𝜇⋅FN
dt
(18)
𝜇
us,⟂,∥=
F
F,⟂,∥
FF
=
F
F,⟂,∥
𝜇
⋅
FN
=1
T
∫
T
0
v
∥
(t)
|v(t)|dt
Fig. 4 Macroscopic friction reduction under parallel and perpendicu-
lar vibration superimposition [28]
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1 3
during longitudinal-torsional VAM. Both the orthogonal
and oblique models start with the distinction if the vibration
assistance results in intermittent cutting or non-intermittent
cutting. In case of intermittent cutting, the reduced shearing
forces will be modeled by using the relative contact ratio
between the tool and the chip (Tc). For non-intermittent cut-
ting, the reduced shearing forces will be modeled by utiliz-
ing the macroscopic friction reduction. For both cases, the
reduced ploughing forces will be modeled by using mac-
roscopic friction reduction, since the longitudinal-torsional
vibration assistance does not result in separation of the clear-
ance face and the machined workpiece during peripheral
milling.
2.3.1 Modeling ofcutting forces inorthogonal cutting
Due to simple, two-dimensional mechanics, the force model
is first explained using the orthogonal cut. The force com-
ponents in orthogonal cutting are calculated based on the
geometric relationships between the rake angle αr, the shear
angle Φs, and the friction angle βa. The friction angle is
determined by the friction coefficient acting between the
chip and the rake face of the tool, as described in Altintaş
and Lee [29]. In Fig.5, the vibration velocities resulting
from longitudinal-torsional vibration superposition are illus-
trated on the right-hand side. The torsional component vus,t
acts parallel to the cutting direction and the longitudinal
component vus,l acts perpendicular to it.
If the maximum combined vibration velocity in cutting
direction (see Eq.9) is lower than the cutting speed, there
will be no tool-workpiece separation due to the VA. How-
ever, the friction coefficient between the chip and the rake
face will be reduced on a macroscopic scale (see Sect.2.2).
The lower average friction in the chip flow direction will
allow chips to slide easier, reducing the shear angle and
forming thinner chips with a higher chip flow velocity when
compared to those in conventional cutting. The resulting
forces and angles due to the VA are visualized in Fig.5.
To model the macroscopic force reduction between the
chip and the rake face, the chip flow velocity must first be
determined. Assuming that the width of the deformed chip
does not change, the chip flow velocity in vibration assisted
machining vus,ch can be determined as a function of the cut-
ting speed vc, the maximum torsional vibration component
vus,t,max (see Eq.5) and the chip thickness ratio rus,c:
Using the geometric relationships shown in Fig.5, the chip
compression ratio under VA rus,c can be expressed as a func-
tion of the shear angle Φus,s and the rake angle of the tool αr:
The shear angle during orthogonal cutting can be deter-
mined by using the theory of Krystoff [23]. The theory
assumes that the shearing will happen in the plane where the
shear stress is maximal, which occurs when the angle between
the resultant force and the shear plane is 45°. Accordingly, the
shear angle under VA Φus,s is calculated as follows:
Using the friction reduction coefficient μ⊥,∥,ch at the rake
face and the conventional friction angle βa, the friction angle
under VA βus,a can be calculated:
The average friction reduction coefficient μ⊥,∥,ch between
the rake face and the chip is calculated by combining Eqs.18
and 19, since the chip flow is the main sliding direction and
longitudinal vibrations stand perpendicular to it. Thus,
the macroscopic friction reduction at the rake face result-
ing from a longitudinal-torsional VA can be calculated as
follows:
It must be considered that the chip flow velocity increases
due to the lower friction between the rake face and the chip.
However, an increase in chip flow velocity simultaneously
leads to a lower friction reduction. Therefore, Eqs. (19),
(20), (21), (22) and (23) must be calculated iteratively until
the chip thickness ratio converges.
By knowing the vibration-assisted shear and the friction
angles, vibration-assisted specific cutting pressures in tan-
gential (Kus,tc) and feed (Kus,fc) directions can be determined
for orthogonal cutting with
(19)
vus,ch
(t)=(v
c
+v
us,t,max
⋅cos
(
𝜔
us
⋅t+Δ𝜙
)
)⋅r
us,c
(20)
r
us,c=sin
(
Φus,s
)
cos
(
Φ
us,s
−𝛼
r)
(21)
Φus
,
s
=
𝜋
4
−
(
𝛽
us
,
a
−𝛼
r)
(22)
𝛽us,a
=tan
−1(
tan
(
𝛽
a)
⋅𝜇⟂
,∥,ch)
(23)
𝜇
⟂,∥,ch =1
T∫
T
0
v
us,ch
√(
v
us
,
sp)
2+
(
v
us
,
l
,
max
⋅cos
(
𝜔
us
⋅t
))
2
dt
(24)
K
us,tc =𝜏s
cos
(
𝛽us,a−𝛼r
)
sin
(
Φ
us,s)
⋅cos
(
Φ
us,s
+𝛽
us,a
−𝛼
r)
(25)
K
us,fc =𝜏s
sin
(
𝛽us,a−𝛼r
)
sin
(
Φ
us
,
s)
⋅cos
(
Φ
us
,
s
+𝛽
us
,
a
−𝛼
r)
Fig. 5 Cutting force diagram of conventional orthogonal cutting (left)
and vibration-assisted orthogonal cutting (right)
1812 The International Journal of Advanced Manufacturing Technology (2022) 119:1807–1819
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1 3
where τs is the shear yield stress of the respective material. To
calculate the resulting cutting forces, the reduced friction between
the flank face and the machined surface must be considered. The
macroscopic friction reduction coefficient μ⊥,∥,e at the clearance
face is calculated with the help of the cutting speed, which is
modulated by the torsional vibration component and the longitu-
dinal vibration assistance that stands perpendicular to it:
VA edge coefficients in tangential (Kus,te) and feed (Kus,fe)
directions are obtained by multiplying the friction reduction
coefficient with the conventional edge coefficients (Kte, Kfe),
while the conventional edge coefficients can be determined
according to Budak etal. [30]. Thus, the cutting forces under
VA-assisted orthogonal cutting can be calculated as follows:
where b stands for the cutting width and h stands for the
uncut chip thickness (see Fig.6). If the maximum combined
vibration velocity in cutting direction (see Eq.9) exceeds the
cutting speed, the chip and the rake face separate in every
vibration cycle. Since the tool and workpiece are not in con-
tact, shearing of the material does not occur and the shear-
ing forces diminish. When the rake face meets the chip, the
chip formation process resumes. Since the cutting force due
to shearing is mostly independent of the cutting speed, it is
assumed that the cutting forces during the contact phase are
equal to those in conventional cutting. Therefore, vibration-
assisted specific cutting coefficients are obtained by mul-
tiplying the contact ratio Tc with the conventional specific
cutting pressures. The flank of the tool is in constant contact
with the machined workpiece surface despite the intermittent
cutting conditions at the rake face. The vibration-assisted
edge coefficients can therefore be determined in an analogue
way as without interruption of the cut (see Eqs.26, 27 and
28). The resulting cutting force components during vibration
assisted orthogonal cutting under intermittent cutting condi-
tions are finally calculated as follows:
(26)
𝜇
⟂,∥,e=
1
T∫
T
0

vc+vus,t,max ⋅cos

𝜔us ⋅t+Δ𝜙
⋅
vc+vus,t,max ⋅cos𝜔us ⋅t+Δ𝜙2
+

vus,l,max ⋅cos

𝜔us ⋅t

2

−0.5
dt
(27)
F
us,t
=
Kus,tc ⋅b⋅h
+
Kte ⋅
𝜇
⟂,∥,e⋅
b
=K
us,tc
⋅b⋅h+K
us,te
⋅b
(28)
F
us,f
=K
us,fc ⋅
b
⋅
h+K
fe ⋅
𝜇
⟂,∥,e⋅
b
=K
us
,
fc
⋅b⋅h+K
us
,
fe
⋅b
(29)
F
us,t
=K
tc ⋅
b
⋅
h
⋅
T
c
+K
te ⋅
𝜇
⟂,∥,e⋅
b
=K
us,tc
⋅b⋅h+K
us,te
⋅b
(30)
F
us,f
=K
fc ⋅
b
⋅
h
⋅
T
c
+K
fe ⋅
𝜇
⟂,∥,e⋅
b
=K
us
,
fc
⋅b⋅h+K
us
,
fe
⋅b
It must be noted here that this model does not consider
elastic recovery of the chip during each vibration cycle. Due
to elastic recovery, the contact time between the tool and the
chip would be bigger in reality. Additionally, the model does
not account for tool wear and uses the simplification of a
uniform sliding contact between the rake face and the chip.
2.3.2 Modeling ofcutting forces inoblique cutting
The simple, two-dimensional model of orthogonal cutting
can be extended by inclining the cutting tool, causing force
components to exist in all 3 Cartesian coordinates. Due to
this fact, milling with an end mill with a helix angle greater
than zero (λH > 0) results in a three-dimensional shearing
force vector
FRes,s
. Figure 6 illustrates the oblique cutting
geometry with its forces and angles together with the respec-
tive longitudinal and torsional vibrations.
Like the case in orthogonal cutting, if the rake face and
chip do not separate during oblique cutting, reduced shear-
ing forces are modeled by using the macroscopic friction
reduction. For this purpose, the chip flow velocity needs
to be determined in the first step. The chip flow velocity in
longitudinal-torsional VAM is a function of inclination γ or
helix angle of the tool λH, the chip compression ratio rus,c
and the longitudinal and torsional vibration components as
in the following equation:
In contrast to orthogonal cutting, the direction of the
chip flow and the direction of the longitudinal oscilla-
tions are no longer perpendicular in oblique cutting. The
inclined cutting edge causes the chips to flow over the
rake face with a chip flow angle ηus, which is the angle
between the chip flow and the normal plane (see Fig.6).
To determine the macroscopic friction reduction at the
rake face, it is necessary to determine the longitudinal
(31)
v
us,ch =(vc+cos
(
𝜔ust+Δ𝜙
)
⋅vus,t,max
+cos
(
𝜔ust
)
⋅vus
,
l
,
max ⋅tan
(
𝜆D
)
)⋅rus
,c
Fig. 6 VA oblique cutting according to Altintas [31]
1813The International Journal of Advanced Manufacturing Technology (2022) 119:1807–1819
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1 3
vibration component that stands perpendicular to the chip
flow direction. For this purpose, three additional coordi-
nate systems are defined as shown in Fig.6 with the asso-
ciated angular relationships.
The 0th coordinate system is transformed into the 1st
coordinate system by rotating the 0th coordinate system
along its z-axis by γ. The 0th coordinate system is needed,
since the longitudinal vibrations only act along the y0 axis
and the torsional vibrations only act along the x0 axis.
The 1st coordinate system is then transformed to the 2nd
coordinate system by rotating the 1st coordinate system
along the y1 axis by αn. Finally, the 2nd coordinate system
is transformed into the 3rd coordinate system by rotating
the 2nd coordinate system along the x2 axis by η. In the 3rd
coordinate system, the rake face of the tool lies in the y3-z3
plane and the chips flow along the z3 axis. The longitudinal
vibration speed can therefore be expressed by the follow-
ing vector in the 3rd coordinate system basis (x3, y3, z3):
In the 3rd coordinate system, the
y3 component of the
longitudinal vibration velocity vector stands perpendicular
to the chip flow velocity and thus can be used to calculate
the macroscopic friction reduction between the rake face
and the chip (µ⊥,∥,r) under longitudinal-torsional VA:
To calculate specific cutting pressures, the vibration-
assisted normal friction angle βus,n must be determined
according to Eq. (22). By knowing the friction angle under
VA, the five oblique-cutting parameters (Φn, Φi, θi, θn,
η) in VAM can be determined according to Shamoto and
Altıntas [19]. The iterative calculation of the macroscopic
friction reduction µ⊥,∥,r and the oblique cutting parame-
ters are applied until the chip compression ratio under VA
(rus,c) converges to a fixed value. In each iteration step, the
chip flow angle η gets updated, which in turn results in a
different friction reduction at the rake face and a differ-
ent chip compression ratio under VA (rus,c), which can be
calculated by using the following equation:
Finally, by using the vibration-assisted oblique cutting
parameters, vibration-assisted specific cutting pressures in
tangential (Kus,tc), feed (Kus,fc) and radial (Kus,rc) directions
can be calculated by using the following equations which
are derived from Altintas [31]:
(32)
�����⃗
v
us,l=vus,l,max ⋅




−sin(𝛾)⋅cos

𝛼n

cos(𝛾)cos(𝜂)+sin(𝛾)sin𝛼nsin(𝜂)
cos(𝛾)sin(𝜂)−sin(𝛾)sin

𝛼
n
cos(𝜂)




(33)
𝜇
⟂,∥,r=
1
T∫
T
0
v
us,sp
√
(vus,sp)2+
(
vus,l,y3cos(𝜔ust)
)
2dt
(34)
Φ
us,n=tan−1rus,c
⋅cos
(
𝛼n
)
1−r
us,c
⋅sin
(
𝛼
n)
with
The macroscopic friction reduction coefficient μ⊥,∥,e
between the clearance face and the workpiece can be cal-
culated in an analogue way as for the orthogonal cut using
Eq. (26). Finally, the resulting cutting forces during
longitudinal-torsional vibration-assisted oblique cutting
can be calculated according to the following equations:
If the maximum combined vibration velocity in cutting
direction (see Eq.9) is higher than the cutting speed, the
rake face of the tool and the sliding chip will periodically
separate due to the VA. To predict the shearing forces,
the conventional specific cutting pressures are multiplied
by the contact ratio Tc. Since the clearance face of the
tool does not separate despite the VA, the force reduction
can be modeled by multiplying the macroscopic friction
reduction at the flank face (see Eq.26) with the respective
edge coefficients. Thus, the resulting cutting forces dur-
ing intermittent cutting can be calculated according to the
following equations:
Finally, by knowing the instantaneous force components
in tangential, feed and radial directions in oblique cutting,
(35)
K
us,tc =K1⋅
cos
(𝛽
us,n−
𝛼
n
)
+tan(
𝜂
)sin
(𝛽
us,n
)
tan(
𝛾)
√
K
2
(36)
K
us,fc =K1⋅
cos(𝛾)
−1
sin
(
𝛽us,n−𝛼n
)
√
K
2
(37)
K
us,rc =K1⋅
cos
(𝛽
us,n−
𝛼
n
)
tan(
𝛾
)+tan(
𝜂
)sin
(𝛽
us,n
)
√K2
(38)
K
1=
𝜏
s
sin
(
Φ
us,n)
(39)
K
2=cos2
(
Φ
us
,
n
+𝛽
us
,
n
−𝛼
n)
+tan2(𝜂)⋅sin
2(
𝛽
us
,
n)
(40)
F
t,us
=K
us,tc ⋅
b
⋅
h+K
te ⋅
𝜇
⟂,∥,e⋅
b
=K
tc,us
⋅b⋅h+K
te,us
⋅b
(41)
F
f,us
=K
us,fc ⋅
b
⋅
h+K
fe ⋅
𝜇
⟂,∥,e⋅
b
=K
fc
,
us
⋅b⋅h+K
fe
,
us
⋅b
(42)
F
r,us
=K
us,rc ⋅
b
⋅
h+K
re ⋅
𝜇
⟂,∥,e⋅
b
=K
rc,us
⋅b⋅h+K
re,us
⋅b
(43)
Fus
,
t=Ktc
⋅
Tc
⋅
b
⋅
h+Kte
⋅
𝜇
⟂,
∥
,
e
⋅
b
(44)
Fus,f=Kfc
⋅
Tc
⋅
b
⋅
h+Kfe
⋅
𝜇
⟂
,∥,e
⋅
b
(45)
Fus
,
r=Krc
⋅
Tc
⋅
b
⋅
h+Kre
⋅
𝜇
⟂,
∥
,
e
⋅
b
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1 3
forces in x-, y-, and z-directions in end milling can be cal-
culated according to Altintas [31].
For the sake of better understanding, a flowchart of the
oblique cutting model is presented in Fig.7.
3 Experimental setup
This chapter aims to explain the experimental setup and set-
tings that were used during the validation of the proposed
force model. For the VA peripheral milling experiments,
the longitudinal-torsional vibrations are generated by a
custom-made actuator, which has been previously used and
described in Rinck etal. [9]. As seen in Figs.8 and 9, the
actuator is positioned between the spindle and the tool. It
can be used to generate longitudinal or longitudinal-torsional
vibrations depending on the tool configuration. The actuator
was mounted on a Hermle UWF 900 3-Axis CNC milling
center.
Hufschmied cemented carbide milling tools with identi-
cal cutting-edge geometry were used for both the longi-
tudinal and longitudinal-torsional VAM experiments (see
Fig.10). However, the cutting-edge length and the total
length of the tools differ due to different vibration charac-
teristics. The tools used for longitudinal VAM milling had
a total length of 97mm and 10 mm length of the cutting
edges, whereas the tools for longitudinal-torsional VAM
had a total length of 36 mm and 20 mm length of the cut-
ting edges. The tools were connected to the actuator via
an M6 thread. The connection via thread ensures a good
transfer of the vibration amplitude to the tool with mini-
mal damping, but requires custom made tools. To ensure
Fig. 7 Flowchart of the oblique
cutting model
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1 3
sufficient concentricity of the tool, axial centering was per-
formed via a cylindrical surface. Longitudinal-torsional
VAM tools were further connected to a slotted sonotrode
to generate a torsional vibration output.
The workpiece was a 100 × 100 × 5 mm sized
Ti-6Al-4V block and was fixed on the Kistler 9257A
dynamometer.
Respective vibration amplitudes, frequencies and phase
shift between longitudinal torsional vibrations were meas-
ured with a Polytec-3-D-laser-Doppler-vibrometer. When
coupled with the actuator, the longitudinal-torsional mill
vibrates at a resonance frequency of 32.24 kHz and yields
a maximum longitudinal vibration amplitude of approxi-
mately 5.75 µm. Additionally, the longitudinal-torsional
mill has a torsional vibration output at the end cutting edge
that has 1.35 times the amplitude of longitudinal vibration.
Resulting longitudinal and torsional vibrations do not have
a phase shift (ΔΦ = 0).
The torsional vibration amplitude can thus be calculated
with the following equation:
Similarly, the longitudinal mill vibrates at the resonance
frequency of 31,645 kHz with a maximum amplitude of
approximately 5.75 µm.
For the validation of the force model, one set of con-
ventional milling (CM) experiment and three sets of lon-
gitudinal VAM (LVAM) and LTVAM experiments were
conducted. All experiments were performed with an axial
depth of cut of 5 mm (ap = 5 mm), radial depth of cut
of 0.5mm (ae = 0.5 mm), 0.035 mm feed per tooth (fz =
0.035 mm) and 80 m/min cutting speed (vc = 80 m/min).
During the conventional and the VAM experiments, all
parameters except for the vibration amplitude were kept
constant.
4 Results anddiscussion
To validate the vibration-assisted force model, VAM experi-
ments were carried out with longitudinal and longitudinal-
torsional vibration superposition at varying vibration ampli-
tudes. Each experiment was carried out three times and the
mean cutting force components in x-, y-, and z-directions
were determined. The resulting force components were then
compared with the modeled values.
To find the edge coefficients in CM, the conventional
specific cutting pressures in milling (Ktc, Krc, Kac) were
(46)
At=1.35
⋅
Al
Milling tool
for LTVAM
Screw
connector
Slotted
sonotrode
Tool holder
Stator coil
Rotor coil
Ultrasonic converter
with piezoelectric discs
Milling tool
for LVAM
Fig. 8 Actuator assembly with the longitudinal-torsional tool and lon-
gitudinal tool for VAM [9]
Fig. 9 Experimental setup [9]
Tool specifications
Material Cemented
carbide
Diameter 8mm
Helixangle 55 °
10 °
Rake angle 1°
Number of
teeth z 3
CoatingZrN
Clearance angle
Fig. 10 Specifications of the tools
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1 3
determined using Eqs. (35)–(37). Therefore, the shear
stress (τs) of Ti-6Al-4V and the friction angle (βa) were
taken from Budak etal. [30] and used as inputs for the
calculation of conventional and vibration-assisted specific
cutting pressures. For the validation, the conventional edge
coefficients (Kte, Kre, Kae) were calibrated directly from
the ploughing forces. The ploughing forces were found
by subtracting the shearing forces from the total cutting
force. By calibrating the edge coefficients directly from
the ploughing forces, accurate forces were modeled for
CM (see Fig.11).
After determining the conventional specific cutting
pressures and the edge coefficients, the analytical proce-
dure described in the previous chapter was used for cal-
culating the vibration-assisted specific cutting pressures
(Kus,tc, Kus,fc, Kus,rc) and edge coefficients (Kus,te, Kus,fe,
Kus,re). In Table1, the modeled specific cutting pressures
and the corresponding edge coefficients for conventional
and VAM are listed.
For a longitudinal vibration amplitude of 2.2 µm, the
combined vibration velocity in cutting direction (
vus,lt)
dur-
ing both longitudinal and longitudinal-torsional VAM was
less than the cutting speed (
vc
). Therefore, no separation
of the chip and the rake face took place. In the case of both
longitudinal and longitudinal-torsional VAM, vibration
amplitudes above 2.2 µm resulted in intermittent cutting.
As can be seen in Table1, the cutting coefficients
decrease with increasing vibration amplitudes due to the
vibration superposition. However, the specific cutting
pressure in axial direction (Kac) slightly increases under
non-intermittent cutting conditions. This noteworthy
remark is associated with macroscopic friction reduction.
The decreased friction angle (βa) results in an increased
chip flow angle (η), which in turn causes a higher specific
cutting coefficient in axial direction.
After modeling the specific cutting pressures and edge coef-
ficients, the cutting forces in longitudinal and longitudinal-
torsional VAM were calculated and compared with
the experimental data in Fig.11. The cutting force com-
ponents can be predicted with good accuracy within the
whole amplitude range. At an amplitude of 2.2 µm and
thus within the range of non-intermittent cutting, there is
an average error of 4% and 5% between the measured and
modeled forces for longitudinal and longitudinal-torsional
VAM, respectively. Thus, the increase of specific cutting
pressure under non-intermittent cutting conditions is com-
pensated by the decreased edge coefficient in axial direc-
tion. At a longitudinal vibration amplitude of 5.75 µm,
the average error between the experimental and modeled
forces is 5% for longitudinal VAM and 6% for longitudinal-
torsional VAM. All modeled cutting force components are
within the range of the standard deviation of the measured
values.
5 Conclusion andoutlook
This paper introduces a new analytical force model that
can predict the cutting forces during longitudinal and
longitudinal-torsional VAM. To achieve this, the relative
contact between the tool’s rake face and the chip was mod-
eled for a helical end mill, and an extended macroscopic
friction reduction model was proposed. The macroscopic
friction reduction model can calculate friction reduction
under a simultaneous parallel and perpendicular vibration
superimposition. The force model can predict the cutting
forces under intermittent and non-intermittent cutting con-
ditions. The predicted forces were in good agreement with
the experimental forces. In future work, the model can be
used to calculate the cutting forces for different workpiece
materials and different milling tool geometries.
Fig. 11 Comparison of modeled and experimental cutting forces
in longitudinal and longitudinal-torsional VAM (vc = 80 m/min,
ap=5mm, ae=0.5 mm, fz =0.035, fus =31,645kHz, Al=0–5.75
µm)
Table 1 Modeled specific cutting pressures and edge coefficients
Al
Ktc
Krc
Kac
Kte
Kre
Kae
𝜇m
N
mm
2
N
mm
2
N
mm
2
N
mm
N
mm
N
mm
CM 0 1965 497 1614 17.6 10 3.6
LVAM 2.2 1957 490 1616 17.1 9.7 3.5
LVAM 5.75 1365 345 1121 15.2 8.6 3.1
LTVAM 2.2 1913 454 1628 16.9 9.6 3.4
LTVAM 5.75 837 212 687 11.8 6.71 2.4
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1 3
Funding Open Access funding enabled and organized by Projekt
DEAL. This work was funded by the Deutsche Forschungsgemein-
schaft (DFG) within the research project “Machining of high perfor-
mance materials with ultrasonically modulated cutting speed” (Project
number 406283248).
Declarations
Ethics approval Not applicable.
Consent to participate Not applicable.
Consent for publication Not applicable.
Conflict of interest The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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