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IEEJ International Workshop on Sensing, Actuation, and Motion Control
Chatter Suppression in NC Machine Tools
by Cooperative Control of Spindle and Stage Motors
Satoshi Fukagawa∗a)
Shinji Ishii∗∗
Student Member,
Non-member,
Hiroshi Fujimoto∗
Yuki Terada∗∗
Senior Member
Non-member
The chatter vibration is well known as an undesirable phenomenon, so various methods for the chatter suppression
have been proposed. To vary the spindle speed in the cutting process is one of the methods for the chatter suppression.
However, the chatter vibration like as triangular velocity is not suppressed completely by the method of variable spindle
speed. This remained chatter is caused by that the spindle speed cannot follow the spindle speed reference. Moreover,
the triangular spindle speed variation need large torque and good response system of the spindle motor. This paper
proposes the method of chatter suppression by a cooperative control of the spindle and the stage motors to decrease
the spindle motor torque. The trajectory and the cooperative control are evaluated by simulations and experiments.
Furthermore, as a next work, the spindle speed and the stage speed trajectory to suppress the chatter vibration more
than only the triangular spindle speed variation is proposed.
Keywords: NC Machine tools, Endmilling, Self–excited chatter vibration, Cooperative control
1.
Introduction
1.1 Background
A machining is exceedingly important process to produce any product because it is adaptive
for various materials, high precise and much eﬀective. Until today, a considerable amount of study has been made on
the NC machine tools to attain higher accuracy and high efficiently in machining (1) (2) . Chatter vibration is one of the
causes to be worse the products quality. Chatter vibration
generalizes undesired vibration in cutting process. Thus, recently the chatter suppression has been focused on and many
papers has been published. Particularly, the tools and works
which are low stiﬀness and low damping induces chatter vibration. Chatter vibration is classified as self-excited chatter
vibration or forced chatter vibration.
Especially, self-excited chatter vibration is the most undesirable, being caused by the interaction between the tools
and works. If the chatter vibration becomes large enough,
the system becomes uncontrollable. The methods to suppress self-excited chatter is divided into four types (3) . Two of
these types analyze the conditions under self-excited chatter
does not occur, and cuts under these conditions (4) (5) . However, these approaches require parameters for the tools, and
the measurement of these parameters requires labor and specified skills.
Furthermore, due to tools being worn down and being deformed by heat, these parameters are constantly changing,
making accurate modeling of the system is diﬃcult. The third
Fig. 1. Schematic view of cutting process. The cutting
thickness is determined by the diﬀerence between previous and present tool position.
method is passive chatter suppression like using the variable
pitch cutting tools or the dumper (6) . However, this method
is not suﬃcient about robustness for variable parameter. The
fourth method is active self-excited chatter suppression (8) (7) .
For example, spindle motor velocity is varied in cutting process.
1.2 The Purpose of Study
In (7), the proposed
method to suppress chatter vibration is to use the higher amplitude of the spindle speed variation. In contrast, the author’s research group proposed a high frequency variation
speed control of the spindle to suppress self-excited chatter
vibration and demonstrated its eﬀectiveness. The method
proposed in (9), was shown to be eﬀective in suppressing
chatter by having the spindle speed follow a triangle wave
speed trajectory. However, in (9), there are two problems:
a) Correspondence to: [email protected]
∗
The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba, 227-8561 Japan
∗∗
DMG MORI SEIKI CO., LTD.
2-35-16 Meieki, Nakamura-ku, Nagoya City, Aichi 450-0002,
Japan
c 2015 The Institute of Electrical Engineers of Japan.
⃝
1
Chatter suppression in NC Machine by Cooperative Control (Satoshi Fukagawa et al.)
thickness to actual chip thickness is
h(s)
1
=
.· · · · · · · · · · · · · · · · · · (4)
−τs
h0 (s) 1 + (1 − e )K f aG(s)
Here, G(s) is the second order lag system shown in Fig. 2
and compliance transfer function describes how the machine
constructions vibrate. When the cutting situation is critical
state, the critical cutting width is defined as alim ,and chatter
frequency is defined ωc . Then, Eq. (5) is described as
Fig. 2. Block diagram of self-excited chatter vibration.
1 + (1 − e− jωc τ )K f alim (Re[G(ωc )] + jIm[G(ωc )]) = 0, · · (5)
where, Re[G(ωc )], Im[G(ωc )] are real part of G(ωc ) and
imaginary part of G(ωlim ). Thus, both sides of real part and
imaginary part are equal. Critical cutting width alim is compute as:
1
.· · · · · · · · · · · · · · · · · · · · · · · · · (6)
alim = 2K f Re[G(ωc )] ( 1 ) The triangle wave speed trajectory requires a discontinuous change in motor torque.
( 2 ) The chatter occurs in the point that the spindle speed
cannot follow the triangle wave speed trajectory
Therefore in addition to variable spindle speed control, a
cooperative variable feed speed control is implemented to
solve these problems above in this paper. The following
methods is proposed:
Equation. (6) shows that compliance of the machine structure
Re[G(ωc )] is in inverse proportion to critical cutting width
alim .
Proposed method
A velocity trajectory of the spindle and the stage are sinusoidal wave. This method archives to suppress the
chatter vibration and to decrease the required motor
torque.
2.
3.
Methods of Chatter Suppression
As shown in the previous section, the mechanism of the
self-excited chatter vibration is that the periodic change of
cutting thickness induces the periodic change of cutting
force. In a previous study, it was proposed that the variable
spindle speed suppressed the chatter vibration. Moreover,
in the authors research group, the method of high frequency
spindle speed variation was proposed and the eﬀect for chatter suppression was confirmed (9) . In this paper, this method
is regarded as a conventional method. And in this section,
the conventional method is mentioned and then the proposed
method is mentioned.
3.1 Chatter suppression by High Frequency Spindle
Speed Variation(conventional method)
In this subsection, the method of chatter suppression by high frequency
spindle speed variation (9) is introduced. To generalize amplitude and frequency of spindle speed variation, the parameters
ARV A and fRV F are defined as follows
The Mechanism of Self-Excited Chatter Vibration (9) (10)
Fig. 1 shows the schematic view of cutting process. Here,
the variation in relative position of a work and a tool induces
variation in cutting depth. The variation in cutting thickness determines the quality of cutting surface. The cutting
thickness is the diﬀerence between the previous tool position
and the present tool position, so the variation in previous cutting thickness aﬀects the present cutting thickness. This phenomenon is called “regeneration”. Because of the variation
in cutting thickness induced by regeneration, the cutting force
changes. So this variation in cutting force induces variation
in cutting thickness. This process consists of unstable closed
loop, therefore the self-excited chatter vibration occurs.
Fig. 2 shows the block diagram of self-excited chatter vibration (9) (10) . Here, a is the width of a tool. Nominal chip
thickness h0 is described as below:
NA
, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (7)
Nav
2π
=
. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (8)
Nav T
ARV A =
fRV F
2π
h0 = v s .· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (1)
ω
Here, T is the period of variation in spindle speed, Nav is the
average of spindle speed, and NA is the maximum value of
spindle speed. The variation in the spindle speed changes
the wave–number on cutting surface caused by vibration of
the tool in cutting of each teeth. Accordingly, the periodic
variation in cutting thickness is disturbed by change in the
spindle speed. Thus, chatter vibration is suppressed. Moreover, it is mentioned that the triangular velocity trajectory is
eﬀective for chatter suppression in (9) because to change the
wave–number on cutting surface constantly is important to
suppress chatter vibration.
3.2 Chatter Suppression by the Sinusoidal Velocity Trajectory Cooperative Control (proposed method)
The triangular velocity trajectory (9) of spindle needs discon-
Here, ω is the spindle speed. Displacement of tool y(t) is
excited by cutting force.
As Fig. 2 shows, h(s) is described as
h(s) = h0 (s) + y(s)e−τs − y(s),· · · · · · · · · · · · · · · · · · · · (2)
where, τ is a period of the spindle turning. Then, the thrust
cutting force Fn is proportional to the specific cutting force
K f and described as
Fn = aK f h(s).· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·(3)
This cutting force vibrates a work and cases displacement
y(s). Therefore, the transfer function from nominal chip
2
Chatter suppression in NC Machine by Cooperative Control (Satoshi Fukagawa et al.)
Fig. 3. The definition of nominal cutting thickness. C s and P s are the stage controller and plant. Cω and Pω are
the spindle controller and plant. The nominal cutting thickness is defined by feed speed v s and ω
Table 1.
Spindle
Parameter
ωav
A spindle
f spindle
v s av
A stage
Spindle, stage velocity
Feed
Table 2.
ĳ
Fig. 4. Sinusoidal velocity trajectory cooperative control (proposed method). The spindle and the stage velocity are sinusoidal. There is phase diﬀerence between the
spindle speed and the feed speed.
tinuous variation in motor torque, so that the bigger power
motor is required. The bigger motors need more expense.
Also it is diﬃcult to track the triangular velocity trajectory, so
synchronous motor was used as the spindle motor in (9). Although synchronous motors good in tracking control, usually
induction motors are used for most of machine tools. Therefore, this paper proposes cooperative control of spindle and
stage and these speed trajectory are sinusoidal. Fig. 4 shows
the conceptual diagram. The spindle and the feed speed is
sinusoidal and additionally, there is phase diﬀerence between
the spindle speed and the feed speed. This proposed method
does not requires discontinuous variation in motor torque of
spindle. Moreover, when the ratio of spindle speed variation
is small, the feed speed changes and then varies cutting thickness.
Model parameter for simulation.
Value
5 mm
300 Mpa
10 Ns2 /m
200 Ns/m
5 × 105 N/m
Here, ωav is the mean value of spindle speed, A spindle is the
amplitude coeﬃcient of the spindle speed variation defined
in Eq. (7), S spindle ( f spindle , t) is the shape function of spindle speed trajectory and f spindle is the frequency coeﬃcient
of spindle speed variation defined in Eq. (8). For example,
S spindle ( f spindle , t) in Eq. (8) is trigonometric function or triangle wave. In the same way, v s av is the average of feed speed,
A stage is the amplitude coeﬃcient of the feed speed variation,
S stage ( f spindle , t) is the shape function of stage speed trajectory
and f stage is the frequency coeﬃcient of stage speed variation.
Fig. 2 also is used for simulation, and Eq. (9) – (11) are used
to compute the valuable nominal cutting thickness and Eq.
(10) is used to compute τ. The τ is a period of the spindle
turning. Therefore, in Fig. 2, τ is changed approximately
depending on time.
4.1 Simulation by Sinusoidal Velocity Trajectory Cooperative Control (Proposed Method)
This proposed
method used sinusoidal velocity trajectory. The spindle and
the feed speed are defined as below:
Simulation of Chatter Suppression
To simulate the suppression of chatter vibration, cutting
thickness is defined as below and the block diagram is shown
Fig. 3. The nominal cutting thickness h0 (t) is
h0 (t) = v s (t)
Value
262 rad/s
0.3
0.7
2 mm/s
0.1
Parameter
Width of cut a
Specific cutting force K f
Dynamic mass M
Mechanical impedance B
Dynamic rigidity K
Time
4.
Reference parameter for simulation.
2π
,· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (9)
ω(t)
ω(t) = ωav + A spindle ωav sin ( f spindle ωav t), · · · · · · · · (12)
v s (t) = vav + A stage vav sin ( f stage ωav t − ϕ). · · · · · · · · · (13)
the spindle speed ω(t) is
ω(t) = ωav + A spindle ωav S spindle ( f spindle , t),· · · · · · · · · · · (10)
Here, ϕ is phase diﬀerence between the spindle speed and
the stage speed and the simulation parameter is Table. 1. The
optimal point of f stage , ϕ are decided by simulation of the each
condition f stage from 0 to 1 and ϕ from 0 to 2π. The f stage , ϕ
and the feed speed v s (t) is
v s (t) = v s av + A stage vav S stage ( f stage , t).· · · · · · · · · · · · · · · (11)
3
Chatter suppression in NC Machine by Cooperative Control (Satoshi Fukagawa et al.)
Fig. 8.
Table 3.
Fig. 5. Chatter vibration simulation dependence of f stage
and phase shift by sinusoidal trajectory when the spindle
RVF = 0.7. Chatter vibration tends to suppress around
phase shift = π/2 and RVF = 0.7
The reference parameters for experiment.
Parameter
ωav
A spindle
f spindle
v s av
A stage
f stage
−3
3
x 10
only spindle
spindle and stage
chatter h [mm]
2
1
0
Table 4.
0.02
0.04
0.06
time t [s]
0.08
0.1
0.12
Fig. 6. Chatter vibration simulation by sinusoidal trajectory (time–dependent). Some peaks of chatter vibration are suppressed.
4
3.5
x 10
only spindle
spindle and stage
3
Table 5.
chatter h [mm]
2.5
1.5
1
0.5
10
1
2
10
Frequency [Hz]
10
Value
6.8 × 10−3 kg · m2
7.8 × 10−3 Nm · s
0.47 NmA
8.5 × 10−3 kg · m2
1.4 Nm · s
0.715 Nm/A
Cutting condition.
Parameter
End mill flutes
End mill ϕ
Radial depth of cut
Axial depth of cut
Work piece
2
0
Value
90 rad/s
0.1
0.5
0.5 mm/s
0.2
0.5
The spindle and the stage parameters.
Parameter
J spindle
D spindle
Kt spindle
J stage
D stage
Kt stage
−1
−2
0
Experimental setup.
3
Fig. 7. Chatter vibration simulation by sinusoidal trajectory (FFT). The chatter vibration around 100 Hz is
suppressed.
Value
2
20 mm
5 mm
20 mm
C1100
Hz. It is known that the chatter vibration occurs around resonant frequency of the plant model (10) . The chatter vibration
frequency around 112 Hz is suppressed by proposed method.
5.
dependence of chatter vibration given by the simulation is
Fig. 5. This result explains that the chatter vibration tends to
be suppressed around ϕ = π/2 and f stage = 0.7. Moreover, in
this optimal parameters of ϕ = π/2 and f stage = 0.7, the time
dependent simulation and FFT results are shown in Fig 6, 7.
The reference parameters of this simulation are shown in Table.1. Table. 2 is Model parameter used in this simulation.
In Fig. 7, the vibration around 30 Hz are caused by the variation in the speed of spindle and the feed. Now, the resonant
frequency of the plant model used for simulation is about 112
Controlled System
Fig. 8 shows the experimental equipment. In this paper,
the spindle and the stage used velocity controller. The spindle and the stage models are described as transfer function
below
G(s) =
Kt
.· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (14)
Js + D
The parameters of the spindle and the stage are shown in Table. 4. Here, subscript “spindle” means the parameters of the
4
Chatter suppression in NC Machine by Cooperative Control (Satoshi Fukagawa et al.)
Reference
Response
Spindle, stage velocity
Spindle
Tstage
Feed
vs_av 㽢 Astage
Tspindle
Time
Fig. 11. Triangular velocity trajectory cooperative control (proposed method). The spindle and the stage velocity are triangular. When the spindle speed does not
change, the stage is moved.
Fig. 9. Experimental result FFT. The vibration from 580
Hz to 700 Hz is suppressed.
3
3
x 10
only spindle
spindle and stage
chatter h [mm]
2
1
0
1
2
0
0.02
0.04
0.06
time t [s]
0.08
0.1
0.12
Fig. 12. Chatter vibration by triangular trajectory
(time–dependent). Some peaks of chatter vibration are
suppressed.
4
3.5
Fig. 10. The experimental spindle speed of proposed
method in (9). The response cannot track the reference
at top of triangle spindle speed trajectory.
only spindle
spindle and stage
3
chatter h [mm]
2.5
spindle and “stage” means the parameters of the stage. The
PI controllers are used in both systems, and the pole assignment of the spindle is 50 Hz and the stage is 40 Hz. Then, the
stage system is semi closed loop control.
6.
x 10
2
1.5
1
0.5
Experimental Results
0
1
10
In this section, the proposed method is evaluated by experiments. The eﬃcacy of the proposed method is evaluated
by cutting experiment. The reference parameters are shown
in Table. 3 and the cutting conditions are shown in Table.
5. The eﬃcacy is evaluated by using the acceleration sensor. The the result of the cutting experiment is shown in Fig.
9. In Fig. 9, the waves shows FFT of acceleration. The
pink and broken line shows the acceleration of the spindle
with the spindle speed and the stage speed are constant. The
green and broken line shows the acceleration with the spindle
speed is sinusoidal trajectory and stage speed is constant. The
blue line shows the acceleration with proposed method. The
spindle and the stage have the phase diﬀerence of π/2. As
shown in Fig. 9, the chatter vibration appears from 580 Hz
to 700 Hz. Then, compared between the proposed method
and constant, the chatter vibration are much suppressed by
proposed method. Furthermore, compared between the proposed method and the conventional, the chatter vibration are
suppressed especially from 600 Hz to 700 Hz.
2
10
Frequency [Hz]
3
10
Fig. 13. Chatter vibration by triangular trajectory
(FFT). The chatter vibration around 100 Hz is suppressed.
7.
Another Trajectory as Next Works
In this section, another trajectory of the stage and the spindle is also proposed and simulated as next works.
7.1 Triangular Velocity Trajectory Cooperative Control (Next Proposed Method)
The chatter vibration is
not completely suppressed in (9). The reason is the spindle
speed cannot follow the triangle wave speed trajectory. The
remained chatter occurs in the point that the spindle speed
cannot follow the triangular spindle speed reference. The experimental wave of spindle speed is shown in Fig. 10. At the
top of triangular wave, the rate of the change in the spindle
speed is zero, so that the wave–number on cutting surface is
not varied in each cut and that induces periodic vibration in
5
Chatter suppression in NC Machine by Cooperative Control (Satoshi Fukagawa et al.)
Table 6.
Chatter suppression
Motor torque
Only spindle variation
Good
Normal
Characters of each method.
Sinusoidal cooperation (Proposed)
Good
Small
variation in spindle and stage velocity will optimized. Optimization of the spindle and feed speed variation is going to
make more chatter suppression.
cutting thickness. Thus, the chatter vibration occurs. Suppressing chatter vibration, it is essential to vary cutting thickness constantly.
Hence, At the point that the spindle speed cannot follow
the triangle wave speed trajectory, the feed speed is changed
to suppress chatter vibration more than only spindle speed
variation. This conceptual diagram is shown Fig. 11. Here,
a broken and red line shows the spindle speed and blue one
shows the feed speed. As shown Fig. 11, the feed speed
is changed when the rate of the variation in spindle speed is
zero. The aim of this method is to change cutting thickness
irregularly at all times.
7.2 Simulation by Triangular Velocity Trajectory Cooperative Control (Next Proposed Method)
The simulation model is Fig. 2. Reference parameters are shown in
Table.1 and Table. 2. Additionally, T spindle = 0.171 s, T stage =
0.0342 s. Fig. 12 shows the simulation result of time–
dependent chatter vibration by the triangular velocity trajectory. The red and broken line shows the chatter vibration h
with the variation in just spindle. The blue one is with the
variation in both the spindle and the stage. As shown in Fig.
13 which is the FFT graph of Fig. 12, the chatter vibrations
around 112 Hz witch is resonant frequency are more suppressed by the sinusoidal velocity trajectory cooperative control. Here, the vibrations around 30 Hz are caused by active
variation in the spindle and the feed speed.
8.
Triangle cooperation (Next proposed)
Very good
Normal
References
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Conclusion
In this paper, the cooperative control between the spindle
speed and stage speed for self–excited chatter suppression is
proposed. The proposed method using sinusoidal velocity
trajectory is evaluated by simulations and experiments. It is
eﬀective to suppress the chatter vibration and decrease the
discontinuous variation in motor torque. Moreover, it is possible that induction motors, which usually used for products
of machine tools can track the sinusoidal reference. This proposed method archives to save cost of machine tools.
The method of spindle and stage velocity trajectory using triangle trajectory is also evaluated as next works. This
method evaluated by simulation. The simulation shows effectiveness for chatter suppression more than the spindle velocity triangular variation and cooperative sinusoidal velocity
trajectory.
The characters of the proposed methods are shown in Table. 6. If to decrease motor torques is required, the sinusoidal
velocity trajectory cooperative variation is selected. If more
high precision products and decreasing chatter more than sinusoidal speed variation, the triangular cooperative variation
is chosen.
As the future works, triangular velocity trajectory cooperative control is going to be experiment. Moreover, it is necessary to optimize the spindle speed and the feed speed trajectory. The frequency response is going to analyzed and the
( 10 )
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