TECHNICAL REPORT

Effect of Machining System Dynamic Characteristic Balanceon Vibration Stability*
T. HASHIMOTO

In order to achieve high-precision cutting, vibration that occurs during machining must be suppressed. The
generated vibrations change due to the dynamic characteristic balance of the machining system. In this study, thephysical mechanisms of vibration phenomena were considered based on the zero-order approximation method forend milling.
Key Words: cutting, machining, vibration, dynamic characteristics, dynamics, anisotropy

1. Introduction
Cutting process using machine tools is widely usedin the production of metal components for automobiles,
aircraft, and other vehicles. A fundamental issue in
manufacturing is producing “good, fast, and cheap
products,” and much effort has also been directed atachieving high efficiency and high precision in the cutting
process. One problem that these efforts encounter is thevibration occurring during cutting (hereafter “machining
vibration”) known as “chatter.” This vibration is relative
vibration occurring between the cutting tool and theworkpiece that deteriorates the accuracy of the finishedsurface (Fig. 1) and causes abnormal tool wear, damage
to machine structures, and other problems, which lower
production efficiency1). Therefore, reducing machining
vibration is an important issue for cutting process.
Marks from vibration (chatter marks)

Fig. 1 Chattered surface* This paper was created based on Kyoto University Doctoral
Dissertation (Philosophy (Engineering)) No. 24145 (2022).

JTEKT ENGINEERING JOURNAL English Edition No. 1020E (2024)

A schematic of the mechanism causing machining
vibration is shown in Fig. 2. When cutting is started,
cutting force is generated according to the depth of cutof the tool. Subsequently, this force is applied to the tool,
workpiece, machine tool body, and other components ofthe mechanical system, relative vibration occurs between
the tool and the workpiece according to the frequency
response characteristics of the mechanical compliance(hereafter “dynamic characteristics”). Consequently, theamplitude of this vibration is superimposed on the depthof cut. The overall vibration system shown in Fig. 2 isreferred to as the “machining system” in this report.
Machining system
Mechanical system
Cutting Cutting force(dynamic characteristics)
process
Vibration (mainly displacement)

Fig. 2 Definition of machining system

Machining vibration is broadly classified as “forced
vibration” or “self-excited vibration” depending on
the mechanism of occurrence, and both are affected bydynamic characteristics. In recent years, studies havebeen conducted regarding the effects of anisotropy inmachining system dynamic characteristics on machining
vibration. For example, in end milling, it is known thatthe milling direction and the balance of machining system
dynamic characteristics change the likelihood of vibration
occurrence, meaning the stability boundary of self-excited
vibration2), 3). However, past studies have been limited toconfirming the phenomena by machining simulations andmachining tests and to mathematical considerations, andhave not offered a physical explanation of the mechanism.
Contrary to these studies, this report first explains the81−Effect of Machining System Dynamic Characteristic Balance on Vibration Stability−

mechanism of machining vibration occurrence and themethods for stability analysis of self-excited vibration.
Next, it presents the results of consideration of themechanism behind changes in the stability boundaryof end milling self-excited vibration resulting from the
dynamic characteristic balance based on zero-orderapproximation (ZOA)4), 5).

2. Vibration during Machining6)–8)
As described before, the machining vibration occurringin a machining system is classified as forced vibrationor self-excited vibration. When machining vibration
emerges as a problem, it becomes necessary to identify
the mechanism and causes of the vibration occurrence.
The following explains an overview of the causes andcountermeasures for forced vibration and self-excited
vibration.

2. 1 Forced Vibration Causes and Countermeasures

Forced vibration occurring in a machining system
is vibration caused by external change in the force ordisplacement and is classified as “force disturbance type”
or “displacement disturbance type”9).
A typical cause of force disturbance type forced
vibration is the cutting force which undergoesintermittent periodic variations during end milling andother machining. One measure to reduce this vibration
is avoiding resonance, that is to say, selecting machining
conditions so that the tooth passing frequency (productof the spindle rotation frequency and the number ofteeth) and integer multiples of this frequency do notmatch the natural frequency of the machining system.
Other measures include reducing the cutting force andimproving the dynamic stiffness of the tool system andthe workpiece system.
Causes of displacement disturbance type forced
vibration include vibration by the bearings, gears,
and motors that compose the machine tool, as well as
hydraulic pump pulsation and ground vibration. In thisvibration, changing the machining conditions is notalways effective, and it is necessary to enact measures
such as damping countermeasures at the vibration sourceor shutting off the vibration transmission path.

2. 2 Types and Causes of Self-Excited Vibration

The self-excited vibration occurring in a machining
system is unstable vibration that occurs due to theinteraction of the cutting and dynamic characteristics
even when a forced vibration cause is not present. The
main categories of this vibration are “regenerative type”
and “mode-coupling type”8).
Regenerative type self-excited vibration is also known
as “regenerative chatter,” and as shown in Fig. 3, the82

uncut portions caused by vibration during cutting produce
undulations in the finished surface. When the same
position is cut after a single rotation, this undulations
change the cutting force. This regeneration effect occurs
in many cutting methods.
On the other hand, mode-coupling type self-excited
vibration is produced by coupling between eigenmodes in
two directions which have close natural frequencies. In a
cutting method using a rotating tool such as end millingshown in Fig. 4, these mode-coupling effects become aproblem.
Workpiece(mass m)
c

x（t ）
n

x（t−T ）
Fr

Ft
ｋ

f

Cutting tool
* Assumed to be a rigid body.

X

Fig. 3 Plunge turning process with regenerative chatter(4)
Tool(end mill): Vibration (fluctuation of displacement): Fluctuation of cutting force
(3)

cx

(2)
(1)
((5))

kx
Coupled vibrationof (1) and (3)YcyX

WorkpiecekyMechanism of coupled vibration occurrence(1) Fluctuation of displacement x occurs,
the uncut chip thickness changes.
(2) The change in uncut chip thickness
causes fluctuation of cutting force Fy .
(3) The fluctuation of Fy causes fluctuationof displacement y , the uncut chip
thickness changes.
(4) The change in uncut chip thickness
causes fluctuation of cutting force Fx .
(5) The fluctuation of Fx causes fluctuationof displacement x ( (1)), and thereafter
the process of (1)−(4) will be repeated.

Fig. 4 Mode coupling in milling

2. 3 Stability Analysis of Regenerative ChatterTo address self-excited vibration, it is necessary
to understand the likelihood of self-excited vibration
occurring, in other words the “stability boundary.” First,
the analysis method for the regenerative chatter stability
boundary is explained in Fig. 3 above. It shows plunge
cutting (a form of turning) in which the tool-side is arigid body, and the workpiece-side is an elastic body.
m , c , and k in Fig. 3 are the mass of the workpiece,
damping coefficient, and stiffness respectively. When theworkpiece is rotated at rotation speed n , and machining
is performed by tool feed rate f , and X-direction vibration
x spontaneously occurs, not only does the uncut chip
thickness at that instant vary, but the uncut chip thickness
after one rotation (with time difference T ) also changes.
As a result, the normal force component F r of the cutting
force varies and the workpiece is vibrated again. Whenthe amplitude of vibration x (t ) at this time is larger than

JTEKT ENGINEERING JOURNAL English Edition No. 1020E (2024)

−Effect of Machining System Dynamic Characteristic Balance on Vibration Stability−

Regenerative
displacement
D h（ t ）−aK rDynamic cutting
force
D F（
x t）

Plunge cutting process

Present dynamic
displacementx（ t ）G（ i x ）Dynamic characteristics




e −Ts
Previous dynamic displacementDelayx（ t −T ）Fig. 5 Block diagram of plunge cutting system(= ω c/2π) at each spindle speed n . Figure 6 (a) is known
as the stability lobe diagram (SLD). The area above a lim is
the unstable area, and the area below it is the stable area.
a lim changes cyclically according to spindle speed n . For
example, when machining with a width of cut of 3 mm,
if the spindle speed is set to 4 000 min−1, regenerative
chatter occurs (point A in Fig. 6 (a)). Also, when the
cutting width is reduced to 2 mm in the same spindle
speed, machining becomes stable (point B). However, in
this case, the trade-off entails lower machining efficiency.
On the other hand, when leaving the cutting width at
3 mm and changing the spindle speed to 4 300 min−1 also
allows stable machining (point C). In this case, there is no
decline in machining efficiency.
Width of cut, mm

vibration x (t −T ) at the previous rotation, the vibration
amplitude continues to increase as machining progresses.
Regenerative chatter occurs in this manner, and themechanism is shown in the block diagram in Fig. 5.

a in Fig. 5 is the width of cut, and K r is the normal
specific cutting force (cutting force required for the unit
cutting cross section).
G (iω ) is the dynamic characteristic of the workpieceand is expressed in Formula ⑴ using m , c , and k
described above.
1−m x 2 ＋ic x＋k⑴

where i is the imaginary unit, and ω is the vibration
circular frequency. If the tool is more flexible than theworkpiece, G (iω ) is the dynamic characteristic of thetool. As the cutting force tangential force component
F t and the vibration in its direction mostly do not affectregenerative chatter in the plunge cutting shown in Fig. 3,
they have been omitted in Fig. 3 and Fig. 5.
Stability analysis of regenerative chatter means
identifying the stability of the vibration system in theFig. 5 block diagram. Although the details are omitted
here, according to the simplest formulation of thevibration system in Fig. 5, the cutting width which is theboundary condition for stability and instability (critical
width of cut) a lim is expressed by Formula ⑵.

a lim ＝ −1
2K r G （R i xc ）⑵

where G R (iω ) is the real part of dynamic characteristic G(iω ), and ω c is the circular frequency of the self-excited
vibration. This concludes the basic theory for stability
analysis of regenerative chatter.

2. 4 Countermeasures to Regenerative Chatter

Next, this section explains countermeasures toregenerative chatter. First, an example of the stability
analysis results is shown in Fig. 6. Parameters m , c , k
and K r are 3 kg, 300 N·s/m, 30 N/μm, and 450 MPa
respectively. Figure 6 (a) and (b) show the critical widthof cut a lim for regenerative chatter and chatter frequency f c

JTEKT ENGINEERING JOURNAL English Edition No. 1020E (2024)

Unstable

A

4
2
0
1 000

Stable
2 000

B
3 000

4 000

C
5 000

Spindle speed, min1(a) Critical width of cut a lim
Frequency, Hz

G（ i x ）＝

6

550

500
1 000

Natural frequency f n (503.3 Hz)
2 000

3 000

4 000

5 000

Spindle speed, min1(b) Chatter frequency f c

Fig. 6 Simulated stability in plunge cutting

Furthermore, according to Formula ⑵, when K r or
G R (iω c) is decreased, a lim can be averagely increased.
Therefore, lowering K r by increasing the rake angle of thetool or increasing workpiece stiffness k by adding workrests are also effective countermeasures to regenerative
chatter.
From Fig. 6 (b), chatter frequency f c is higher thannatural frequency f n, and varies periodically according
to spindle speed n . This is also a characteristic ofregenerative chatter.

2. 5 Stability Analysis of End Milling Self-Excited
Vibration and Its Countermeasures

End milling is one form of milling and one of the mostcommon cutting methods together with turning. A modelof this cutting is shown in Fig. 7. The end mill is long andthin, and due to its stiffness insufficiency the machining
vibration, particularly the self-excited vibration, can easily
become a problem. End milling is a cutting method where
both the regeneration effect and the mode-coupling effectact simultaneously. As the regeneration effect occurs
when the machined surface is cut by the next cutting edge83−Effect of Machining System Dynamic Characteristic Balance on Vibration Stability−

after it was previously cut by any one cutting edge, thetime cycle T regarding with the regeneration effect in end
milling is the tooth passing period. When the units for the
tooth passing period are seconds, and the units for spindle
speed n are min−1, and the number of teeth is expressedby N , the tooth passing period is calculated by 60/(nN ).
As the cutting force direction constantly changes during
end milling, it is necessary to model the vibration using
at least two degrees of freedom (DOF). Minute vibrationin the tool rotation axis direction (Z direction) can simply
be ignored, and the model is created with two orthogonalaxes in the tool radial direction. In this report, as shownin Fig. 7, a 2-DOF model is introduced with the feed
direction as the Y direction, and the tool radial direction
orthogonal to the feed direction as the X direction. In thismanner, the mode-coupling effect is also considered. In
Fig. 7, a is the axial depth of cut, R D is the radial depthof cut, f is the feed per tooth, k x and k y are the stiffnessof the end milling system in the X direction and the
Y direction, and c x and c y are similarly the damping
coefficients in the X direction and the Y direction.

X

h Workpiece

Tooth（ j −1 ）Tool (end mill)

Feed

Y

Tooth（ j ）

f

f
kx

Z

a
Rd

Fr
n
Tool
(end mill)

cx

piece

Work

Ft

Y

ky

hj

cy

X

Fig. 7 Schematics of cutting model for end milling

The mechanism which causes self-excited vibration
during end milling can be represented as the block
diagram in Fig. 8 by using “zero-order approximation.”
In Fig. 8, K t is the tangential specific cutting force, [A 0]is the DC component of the orientation matrix known as
the “cutting force coefficient matrix,” and [G (iω )] is the
dynamics matrix of the end milling system.
Dynamic cutting force Present dynamic displacementF（x（ t ）
D x（ t ）x t）
F（y（ t ）
D y（ t ） 1y t）
aK［［G（ i x ）］t A0 ］
2

The cutting force coefficient matrix is originally a timefunction and used to calculate the cutting force which
varies intermittently during end milling. However, it was
shown by Altintas and Budak4), 5) in the 1990s that whencalculating the stability boundary of self-excited vibration,
it is sufficient to consider only its DC component (thezero-order component), and this method is widely used
under names such as zero-order approximation (ZOA).
The DC component of the cutting force coefficient matrix[A 0] is expressed by Formula ⑶.
a 0xx［ A 0 ］＝ aa0xx ＝Nh［ −cos 2 h−2 k r h−k r sin 2 h ］hexst
4p

a0xy ＝Nh［ −sin 2 h ＋2 h ＋k r cos 2 h ］hexst
4p

a0yx ＝Nh［ −sin 2 h −2 h ＋k r cos 2 h ］hexst
4p

a0yy ＝Nh［ cos 2 h − 2 k rh＋ k r sin 2 h ］hexst
4p

Dynamic characteristics+–
Previous dynamic x（ t −T ）displacementy（ t −T ）

e

−Ts

Delay

Fig. 8 Block diagram of end milling system84⑶

where k r is the force ratio (ratio of normal force
component to tangential force component), h st is theentry angle, and hex is the exit angle. When the milling
direction is up-cut, hst is handled as 0°. When it is downcut, hex is handled as 180°. The stability boundary isaffected by the angle of hex − hst, which is called theimmersion angle.
As the off-diagonal components of the dynamic
characteristics are smaller than the diagonal components
in an ordinary mechanical structure, the off-diagonalcomponents of dynamic characteristics [G (iω )] areignored in this report, and [G (iω )] is defined in Formula⑷.
［ G（ i x ）］＝G xx（ i x ）00
G yy（ i x ）1

G xx（ i x ）＝

−mx x 2 ＋ic x x＋k x⑷

1−my x ＋ic y x＋k y

G yy（ i x ）＝Regenerative displacement

End milling process

a0xy
a0yy

0yx

2

Although the details are omitted, based on the above
conditions and assumptions, the stability boundary ofaxial depth of cut a lim is calculated from Formula ⑸.

a lim ＝ −

KR

Kt

（K ）

1＋

KI

2

⑸

R

where KR and KI are the real part and the imaginary partof K, which is the solution of the characteristic equation.
The characteristic equation when the off-diagonalcomponents of the dynamics matrix are ignored is shown

JTEKT ENGINEERING JOURNAL English Edition No. 1020E (2024)

−Effect of Machining System Dynamic Characteristic Balance on Vibration Stability−

in Formula ⑹ using the matrix components of [A 0] and[G (iω )].

a 0 K2 ＋ a 1 K ＋ 1 ＝ 0
a 0 ＝（ a0xx a0yy −a0xy a0yx ）G xx（ i x c ）G yy（ i x c ）⑹

a 1 ＝ a0xx G xx（ i x c ）＋ a0yy G yy（ i x c ）Axial depth of cut, mm

Next, the characteristics in the stability analysis resultsfor end milling using ZOA are explained below. In one
example, Fig. 9 shows the analysis results of dynamic
characteristic conditions with mx and my of 3 kg, cx and cyof 300 N·s/m, and kx and ky of 30 N/μm, and the cutting
model parameters from Table 1. The configuration in
Fig. 9 is approximately the same as Fig. 6. The differenceis that the vertical axis of the stability lobe diagram (SLD)in Fig. 9 (a) is the axial depth of cut. One difference in theanalysis results is that the chatter frequency f c shown in
Fig. 9 (b) is lower than the natural frequency f n dependingon the rotation speed. This characteristic appears whenthe mode-coupling effect in addition to the regeneration
effect is also acting simultaneously.
30
20

Unstable(Stable)

10
0
1 000

2 000

3 000

4 000

5 000

Spindle speed, min1

Frequency, Hz(a) Critical axial depth of cut a lim
550

500

Based on the above, the primary countermeasures
to self-excited vibration occurring during end milling
include the following.
(1) Selection of stable machining conditions based on thestability lobe diagram (SLD) (reducing axial depthof cut a , changing tooth passing period T (directly
changing spindle speed n ))(2) I mproving the dynamic characteristics of the tool
spindle system and the workpiece system (reducing
gain of dynamics matrix [G (iω )])(3) R educing the number of teeth and lowering theimmersion angle by reducing the radial depth of cut
and increasing the tool tip diameter (reducing gainof the DC component of the cutting force coefficientmatrix [A 0])(4) R educing specific cutting force K t(5) Using a suitably designed variable pitch/variable helix
end mill, and use of spindle speed variation (SSV)(reducing the regeneration effect by changing to nonconstant T )Also, possible means of reducing the mode-coupling
effect includes moving the natural frequencies of theeigenmodes for the two directions X and Y further apart,that is to say, adjusting the dynamic characteristic balanceof the end milling system. This will be explained in thenext section.

3. R elationship between Stability Boundaryand Dynamic Characteristic Balance for
End Milling Self-Excited Vibration

3. 1 Verification by Simulation and Testing2), 3)

Natural frequency f n (f n x f n y 503.3 Hz)

1 000

2 000

3 000

4 000

5 000

Spindle speed, min1(b) Chatter frequency f c

Fig. 9 Simulated milling stability

Table 1 Cutting model parameters
Specification

Conditions

Number of teeth

2

Milling direction

Up-cutEntry angle, exit angle

0，60°

Spindle speed
Tangential specificcutting force
Force ratio

1 000～5 000 min－1
1 500 MPa
0.3

JTEKT ENGINEERING JOURNAL English Edition No. 1020E (2024)

In the stability analysis of end milling self-excited
vibration shown in Fig. 9, the dynamic characteristics
are set as the same in the X direction and the Y direction.
Although these are ideal conditions, the dynamic
characteristics of an actual machining system are
frequently anisotropic. These are also the conditions
where the strongest mode-coupling effect is produced.
As described below, it is known that even when thenatural frequencies of the dynamic characteristics in two
directions are moved further apart, this does not always
improve the self-excited vibration stability boundary.
In comparison to the calculation conditions in Fig. 9(milling direction was set as up-cut), Fig. 10 shows
the stability analysis results of end milling under theconditions when only the X-direction stiffness kx or only
the Y-direction stiffness ky is changed to 35 N/μm, which
is an increase of 5 N/μm. The top parts of both Fig. 10(a) and (b) are stability lobe diagrams (SLDs), and thebottom parts show the chatter frequency f c. Also, f nx andf n y in Fig. 10 indicate the natural frequency in the X
direction and the Y direction. As shown by Fig. 10, thestability lobe diagram (SLD) differs largely depending85−Effect of Machining System Dynamic Characteristic Balance on Vibration Stability−

Frequency, Hz

Axial depthof cut, mm

on the relative sizes of f nx and f ny . The stability boundaryis higher when f nx < f ny as in Fig. 10 (b) than when f nx >f ny as in Fig. 10 (a). This trend was also confirmed by the
cutting test performed using the setup shown in Fig. 11.
On the other hand, the reverse trend is seen when the
milling direction is down-cut, and the stability boundary
improves when f nx > f ny . This improvement in the stability
boundary resulting from the dynamic characteristicbalance is also dependent on the force ratio and the radial
depth of cut. The effects are likely to occur when a sharp
cutting edge is used to cut a material such as aluminumalloy or carbon steel. Also, when the radial depth of cut is
set to approximately 1/2 of the tool diameter, the stability
boundary improvement effects become more conspicuous.
30
20

Unstable(Stable)100600Natural frequency in X direction f nx (543.6 Hz)

550
500

Natural frequency in Y direction f ny (503.3 Hz)
1 000
2 000
3 000
4 000
5 000
Spindle speed, min1

Axial depthof cut, mm(a) k x 35 N/m ( f nx ＞f ny )30
20

Unstable

10
StableFrequency, Hz

0Natural frequency in Y direction f ny (543.6 Hz)

600
550
500

Natural frequency in X direction f nx (503.3 Hz)
1 000
2 000
3 000
4 000
5 000
Spindle speed, min1(b) k y 35 N/m (f nx ＜f ny )Fig. 10 Simulated milling stability in anisotropic dynamics

End millf35.3 mmAccelerometer

Accelerometer
Workpiece

Workpiece

f

Jig

Jig
34.3 mm
W
V

35.3 mmU

Table

34.3 mm
W
V

U

Table

(a) Unstable condition ( f n x＞ f n y )(b) Stable condition ( f n x＜ f n y )Fig. 11 Experimental setup

86

End mill

3. 2 Mechanism of Stability Boundary
Improvement Resulting from Dynamic
Characteristic BalanceThis section considers the physical mechanism of
the phenomenon which causes the stability boundary ofend milling self-excited vibration change because of the
dynamic characteristic balance.
Using ZOA, in an end milling system where the offdiagonal components of the dynamics matrix are assumed
to be zero (Fig. 8), the cause which produces the modecoupling effect (interaction between the X direction andthe Y direction) is the off-diagonal components α0xy and
α0yx of the DC component of the cutting force coefficientmatrix [A 0 ] shown in Formula ⑶. As a result, thediagonal components α0xx and α0yy of [A 0] are considered
to primarily contribute to the regeneration effect. Basedon this concept, in the end milling system shown in Fig. 8,
stability analysis was performed under hypotheticalconditions assuming that the diagonal components or offdiagonal components of [A 0] are zero. The calculation
conditions are approximately the same as Fig. 9 (milling
direction was set as up-cut), and calculations were
performed for a total of three dynamic characteristic
conditions: the isotropic condition in Fig. 9 and the two
anisotropic conditions in Fig. 10. Furthermore, it is also
important that the stability boundary for self-excited
vibration in the end milling system demonstrates the trend
as below. In other words, when regenerative chatter isdominant, the stability boundary is primarily determinedby the Y-direction dynamic characteristic G yy for up-cut,
and primarily by the X-direction dynamic characteristic
G xx for down-cut. The reason is that in most cases therelative sizes of the absolute values of [A 0] diagonalcomponents α0xx and α0yy are |α0xx | < |α0yy | for up-cut and|α0xx | > |α0yy | for down-cut.
The analysis results are shown in Fig. 12. Figure 12 (a)shows the isotropic condition where kx and ky are both 30
N/μm, (b) shows the anisotropic condition where kx is 35
N/μm, and (c) shows the anisotropic condition where k y
is 35 N/μm. The top part of each figure is a stability lobe
diagram (SLD), and the bottom part shows the chatter
frequency f c. In addition to the analysis results with thecondition using only the diagonal components of [A 0](hereafter “diagonal components only” condition) andonly the off-diagonal components (hereafter “off-diagonalcomponents only” condition), the analysis results with allcomponents (same values as Fig. 9 and Fig. 10) are also
shown as the “original” condition.
First, we consider Fig. 12 (a). f c under the “diagonalcomponents only” condition is larger than the natural
frequencies (f nx , f ny ), and matches the characteristic ofregenerative chatter described in Section 2. 3. On theother hand, f c under the “off-diagonal components only”
condition is lower than the natural frequency depending

JTEKT ENGINEERING JOURNAL English Edition No. 1020E (2024)

−Effect of Machining System Dynamic Characteristic Balance on Vibration Stability−

100f nx, f ny

600
550
500
2 000

3 000

4 000

5 000

Axial depth of cut, mm

20Oﬀ-diagonal components only

30
20
10
0

Frequency, Hz

Axial depth of cut, mm

30

1 000

Diagonal components only

Frequency, Hz

Frequency, Hz

Axial depth of cut, mm

Original

f nx

600
550
500
1 000

f ny
2 000

3 000

4 000

5 000

30
20
10
0

f ny

600
550
500
1 000

f nx
2 000

3 000

4 000

5 000

Spindle speed, min1

Spindle speed, min1

Spindle speed, min1(a) k x  k y 30 N/m (f nx  f ny )(b) k x 35 N/m, k y 30 N/m (f nx ＞f ny )(c) k x 30 N/m, k y 35 N/m ( f nx ＜f ny )Fig. 12 Simulated milling stability in isotropic or anisotropic dynamics

on the rotation speed, and this trend is considered tobe the effect of the mode-coupling effect described in
Section 2. 4. That is to say, f c does not match under the“diagonal components only” condition and the “offdiagonal components only” condition, and f c under the“original” condition is approximately the intermediate
value between the “diagonal components only” condition
and the “off-diagonal components only” condition. The
reason that f c under the “original” condition is closer tothe f c under the “off-diagonal components only” conditionis assumed to be because the critical axial depth of cut
a lim under the “off-diagonal components only” conditionis smaller than the a lim under the “diagonal components
only” condition.
Next, we consider Fig. 12 (b) and (c). As describedabove, the self-excited vibration that occurs under thecondition considering only the diagonal componentsof [A 0] shows characteristics of regenerative chatter.
Furthermore, when the milling direction is up-cut, thestability boundary is bound to be dependent on the
Y-direction dynamic characteristic G yy . In contrast, inFig. 12 (b) and (c), f c under the “diagonal components
only” condition is larger than each of the Y-direction
natural frequencies, and there is no contradiction. On theother hand, under the “off-diagonal components only”
condition, which shows characteristics of mode-coupling
chatter, f c tends to be the intermediate value between
f nx and f ny . As a result, with the dynamic characteristicbalance in which f nx < f ny in Fig. 12 (c), the f c under the“diagonal components only” condition and the f c underthe “off-diagonal components only” condition takedifferent values. This can be interpreted as a state when
the regenerative chatter frequency and the mode-coupling
chatter frequency are separated. In addition, given thatthe f c calculated under the “original” condition either
approximately matches one of the two f c listed above oris an intermediate value between the two f c, and that theresult is the same as Fig. 12 (c), in the case of down-cut

JTEKT ENGINEERING JOURNAL English Edition No. 1020E (2024)with a dynamic characteristic balance where f nx > f ny , we
can interpret this to mean that when there is a dynamic
characteristic balance of f nx < f ny with up-cut, and f nx > f ny
with down-cut, one of either the regeneration effect or themode-coupling effect is dominant. This means that either
the two self-excited vibrations are not synchronized orelse the two self-excited vibrations each vibrates at anintermediate frequency separated from the frequency atwhich they are most unstable.
However, with the dynamic characteristic balance inwhich f nx > f ny in Fig. 12 (b), the f c under the “diagonalcomponents only” condition and the f c under the “offdiagonal components only” condition generally match at
most rotation speeds. The f c calculated under the “original”
condition also matches them. Although the details areomitted here, with the dynamic characteristic balance inwhich f nx < f ny with down-cut, the results are the same as
Fig. 12 (b). Therefore, this can be interpreted to mean thatwith a dynamic characteristic balance of f nx > f ny with upcut and a dynamic characteristic balance of f nx < f ny withdown-cut, the two self-excited vibrations (regenerative
type and mode-coupling type) occur in synchronization.
Based on the above, the physical mechanism which
improves the stability boundary during end milling when
the dynamic characteristic balance is f nx < f ny with upcut or f nx > f ny with down-cut can be explained mainly as
follows.
In the regenerative chatter and mode-coupling chatter
which occur in an end milling system, each chatter islikely to occur within a certain frequency range, and whenthese two frequencies are separated in a certain dynamic
characteristic balance, either the regeneration effect or
the mode-coupling effect becomes dominant. And thestability boundary is improved by the asynchronization of
the two self-excited vibrations.

87−Effect of Machining System Dynamic Characteristic Balance on Vibration Stability−

4. Conclusion
In this report, we first explained the mechanismof machining vibration occurrence and the methods
for stability analysis of self-excited vibration. Next,
we introduced the results of the consideration basedon zero-order approximation (ZOA) of the physicalmechanism behind changes in the stability boundary ofend milling self-excited vibration caused by the dynamic
characteristic balance. It is anticipated that the contents ofthis report will help to improve productivity and reduce
environmental impact in manufacturing workplaces.
Further development of the contents of this report will
be pursued to undertake the creation of innovativetechnologies for next-generation manufacturing.

References
1)

2)

3)

4)

5)

6)

7)

8)

9)

Y. Altintas, M. Weck: Chatter Stability of Metal Cutting
and Grinding, CIRP Annals, Vol. 53, No. 2 (2004) 619642.
T. Hashimoto, D. Kono, M. Furusawa, A. Matsubara:
Effect of Anisotropy of Dynamic Characteristics of
Cutting System on Vibration Stability, Journal of theJapan Society for Precision Engineering, Vol. 87 No. 2(2021) 238-244 (in Japanese).
T. Hashimoto, D. Kono, M. Furusawa, A. Matsubara:
Anisotropic Dynamic Characteristics of Machining
System Improving Vibration Stability in End Milling,
Journal of the Japan Society for Precision Engineering,Vol. 87 No. 11 (2021) 882-888 (in Japanese).
Y. Altintas, E. Budak: Analytical Prediction of Stability
Lobes in Milling, CIRP Annals, Vol. 44, No. 1 (1995)
357-362.
E. Budak, Y. Altintas: Analytical Prediction of Chatter
Stability in Milling - Part I: General Formulation, Journalof Dynamic Systems, Measurement, and Control, Vol.
120, No. 1 (1998) 22-30.
Y. Altintas: Manufacturing Automation: Metal Cutting
Mechanics, Machine Tool Vibrations, and CNC Design(Second Edition), Cambridge university press, (2012).
N. Suzuki: Chatter Vibration in Cutting, Part 1, Journalof the Japan Society for Precision Engineering, Vol. 76
No. 3 (2010) 280-284 (in Japanese).
E. Shamoto: Mechanism and Suppression of Chatter
Vibrations in Cutting, Denki-Seiko [Electric Furnace
Steel], Vol. 82 No. 2 (2011) 143-155 (in Japanese).
T. Hoshi: Phenomena in Chatter, Kogyo Chosakai (1977)(in Japanese).

T. HASHIMOTO**

88

Innovative Processing R&D Dept., Research &Development Division, Doctor of Engineering

JTEKT ENGINEERING JOURNAL English Edition No. 1020E (2024)