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Dynamical modeling of spindle with active magnetic bearing for milling
process
Etienne GOURC
1,a
, Sébastien SEGUY
1,b
, Gilles DESSEIN
2,c
1
Université de Toulouse; INSA; ICA (Institut Clément Ader); 135 avenue de Rangueil, F-31077
Toulouse cedex 4, France
2
Université de Toulouse; INPT-ENIT; LGP (Laboratoire Génie de Production), 47 avenue
d’Azereix, BP 1629, F-65016 Tarbes cedex, France
a
etienne.gourc@insa-toulouse.fr,
b
sebastien.seguy@insa-toulouse.fr,
c
gilles.dessein@enit.fr
Keywords: Dynamical; Milling; Spindle; Active Magnetic Bearing.
Abstract. A dynamical modeling of spindle with Active Magnetic Bearing (AMB) is presented. All
the required parameters are included in the model for stability analysis. The original map of stability
is generated by Time Domain Simulation. The major importance of forced vibrations is highlighted
for a spindle with AMB. Milling test are used to quickly evaluate the stability. Finally, the
simulation results are then validated by cutting tests on a 5 axis machining center with AMB.
Introduction
The productivity of machining process is still severely limited by machining vibrations, so called
chatter. The work of Tobias [1] have presented the surface regeneration as the major cause of
chatter.
One first method is to evaluate the asymptotic stability of the process, and it is possible to plot
the stability chart. This description introduced for turning has been extended by the work of Altintas
for milling [2]. Newly, improved methods have been developed with a more detailed stability
analysis [3-4]. For high-speed milling with highly interrupted cut and low helix angle, a new
unstable area has been detected, called period-doubling, or flip bifurcation [5].
Another approach is the Time Domain Simulation (TDS), based on numerical computation. In
this case, the equation of motion is numerically integrated. All the information is available like the
displacement, the tool position, or the cutting forces [6-8]. In some simple case, it’s even possible to
simulate the surface finish [8,9]. This technique is very powerful and it’s possible to model all the
nonlinear parameters of machining: ploughing or the fact that the tool leaves the cut under large
vibrations [6].
The development of spindles with active magnetic bearing became very fast during the last ten
years. A quite large bibliography is available, see for example [10-13]. The life span and their
robustness to force impact are major advantages of AMB, much larger than the conventional roller
bearing spindle [13]. In addition to these advantages, it is also very easy to use the included spindle
displacement sensors and feedback currents, for position and force measurement. For example,
Auchet et al. [14] developed an original method for indirect cutting force measurement by
analyzing the command voltage of AMB. Chen and Knospe developed approaches to maximize
damping, by using an additional AMB on the spindle, and in some case to it’s possible to control
chatter on dedicated simplified test bench [13, 15]. Kyung and Lee [16] have presented a study on
the stability of a spindle with AMB, but only for low spindle speeds corresponding to conventional
machining.
In this work, the dynamical modeling of a spindle with AMB is presented in the high-speed
domain (40000 rpm), in the area of the first Hopf lobe. The original AMB machining modeling is
presented, and the results are validated by experimental test. In section 2 the dynamical modeling is
presented. The stability results are studied in Section 3. Experimental tests are collected in Section
4. Finally, conclusions are gathered in Section 5.
Advanced Materials Research Vol. 423 (2012) pp 200-209
© (2012) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMR.423.200
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,
www.ttp.net. (ID: 195.83.11.66-21/12/11,20:35:30)
Dynamical modeling
Modeling of the spindle. The point M is located at the center of the rotor at the height z, this
displacement is defined by u(z, t) (Figure 1).
Figure 1. Modeling of the spindle.
The point M(0) is located at the end of the tool, the points M(A) and M(B) are at the height of
the A and B active magnetic bearings. The modal base is defined to represent the displacement
vibrations of the spindle:
(1)
φ
i
(z) is the i
th
modal shape, associated to the natural pulsation ω
i
and the modal displacement
coefficient q
i
(t), is obtained solving the motion equation:
(2)
m
i
, c
i
and k
i
are the modal mass, damping and stiffness of the i
th
mode. f
i
(t) is the projection of
the external forces F(t). This projection considers the fact that the displacement is different along
the height of the spindle:
)()0()( tFtf
ii
ϕ
=
(3)
The system of equations is defined by:
{ }
i
iiiiiii
tFtqktqctqm )()0()()()(
ϕ
=++
(4)
The analysis of the machining stability at high spindle speed requires taking into account the
spindle modal behavior. An analytical model of the spindle and measurements has shown that
gyroscopic effect on modal frequencies is not important. For example (Figure 2), the first modal
frequency (approximately 1 kHz) varies less 10% during spindle speed variation. We note a
doubling of this frequency due to the gyroscopic effect (Campbell diagram) and at 40000 rpm; two
close frequencies appear at 885 and 990 Hz. Thus this light variation can be disregarded and
gyroscopic effect won’t be taken into account in this study.
Also, the unbalance is not taken into account because a part of the control algorithm, not defined
here, is designed to center the rotation axis of the spindle at his inertia axis and thus avoid
unbalance force compensation.
∑
=
=
n
i
ii
tqztzu
1
)()(),(
ϕ
{
}
i
iiiiiii
tftqktqctqm )()()()( =++
Advanced Materials Research Vol. 423 201
Figure 2. Gyroscopic effect for the first modal frequency.
The rotor is free, and it is necessary to consider the rigid body mode of the rotor. In fact there is a
static component of the excitation force, which is balanced by the feedback control loop. As an
assumption, only the displacement at the bearing A is model, we consider that the B bearing is a
hinge. The mass of the rotor is m
r
and its length L. The rotor is model as a uniform bar. The θ angle
is considered very small (L
≫
x), the motion equation is:
(5)
The spindle dynamics is modeled in the xz plane, defined by its mass and the two first flexible
modes. The modal parameters were identified by hammer impact at the end of the tool, and the
mass rotor was measured.
Excitation by the cutting forces. The relationship between the chip thickness and the tool’s
vibration may create self-generated vibrations, which are different from forced or transitional
vibrations (Figure 3).
Figure 3. Dynamical model of the chatter milling.
)(
3
0
tFq
m
r
≈
202 Innovating Processes
A linear cutting law was used [5], the force is proportional to the chip thickness h(t):
(6)
g(t) is an step function that is equal to 1 when the tooth is cutting, and equal to 0 otherwise. K
1
is
a cutting coefficient defined by:
(7)
K
t
is the specific tangential cutting coefficient, A
p
is the axial depth of cut, α
x
is the x directional
milling force coefficient, is defined by:
(8)
k
r
is the reduced radial cutting coefficient, ϕ
st
and ϕ
ex
are the entry and exit cutting angle of the
tool, see [3]. The instantaneous chip thickness involved in Equation 6, is defined by:
(9)
Without vibrations, the chip thickness is equal to the feed per tooth f
z
. When vibration occurs,
the static chip thickness is modulated by the regenerative effect. The delay between two tooth pass
is a fundamental parameter for modeling self-excited vibrations, it is defined by:
(10)
With N the spindle speed, in rpm, and z the number of teeth of the tool. In real machining, the
tool vibrations still have finite amplitude. When the vibrations are too strong, the tool leaves the cut,
and no effort is applied during this time. In order to model this, a unit function r(h(t)) was
introduced. It can be described as follows:
• if h(t) < 0, i.e. chip thickness is zero, then r(h(t)) = 0, the cutting forces are nil, and no effort
is applied.
• if h(t) > 0, i.e. the tool is cutting, then r(h(t)) = 1, the cutting forces are calculated by a linear
cutting law (Equation 6).
Active Magnetic Bearing. The AMB generate a force, by the feedback intensity current i, in order
to maintain the rotor axis position x. The magnetic bearings are controlled in differential mode, i.e.
the current in the coils is the sum of the bias current i
0
, and the control current i (Figure 4). The bias
current i
0
>0 allows the double control force compared to no bias current, because the magnetic
forces on the spindle can only be attractive. The reactivity of the control is increased, but a heating
effect is possible only when i = 0.
Figure 4. Active Magnetic Bearings.
The nonlinear law is defined in the following equation:
(11)
i
0
is the bias current, x
0
is the nominal air gap, and k is the global magnetic permeability
calculated by:
[
]
))(()()()(
1
thrtgthKtF
c
=
xtp
KAK
α
2
1
1
=
[ ]
ex
st
rrx
kk
φ
φ
θθθα
)2sin(2)2cos(
2
1−−−=
)()()( txtxfth
z
−−+=
τ
Nz
60
=
τ
+
−
−
−
+
=)²(
)²(
)²(
)²(
),(
0
0
0
0
xx
ii
xx
ii
kxiF
amb
Advanced Materials Research Vol. 423 203
(12)
µ
0
is the vacuum magnetic permeability, also called magnetic constant, A
g
is the air gap area an n
is the number of turns. The knowledge of the control loop is an important aspect, because it
generates the forces applied to the spindle in reaction to the cutting forces. In this work, the
magnetic bearing manufacturer (MECOS) has given all the information about the control loop.
First, a semi-static control loop compensates the static forces. Secondly, there is a dynamic control
loop, similar to a proportional derivate regulator with band-cut filters. Third, there is the bias
current generation.
Time Domain Simulation with AMB
Numerical integration. The Time Domain Simulation of milling process requires to choice a
suitable resolution method, because there are many strong non-linearity: delay term, screen
function, cutting forces, maximum current limitations… The explicit schema of Runge-Kutta (2,3)
type [17] was used. In order to control the stability and the error the time step was adapted. This
algorithm is improved for strongly nonlinear problem. The modeling was developed in Matlab-
Simulink, and all the parameters are collected in Table 1. These parameters come for measures on
the spindle by various tests of magnetic bearings excitation.
Table 1 Parameters used for the simulation
Dynamical Values Cutting Values AMB Values
m
1
2.510 kg K
t
900 MPa i
0
3 A
c
1
305.56 Ns/m k
r
0.2 x
0
0.4 mm
k
1
9.3*10
7
N/m f
z
0.1 mm/tooth
ω
1
970 Hz R 6 mm
ϕ
1
0.01 A
e
4 mm
m
2
0.782 kg z 3
c
2
170.52 Ns/m α
x
-1.198
k
2
1.2*10
8
N/m
ω
2
1970 Hz
ϕ
2
0.08
m
r
21 kg
Results of simulation. The computation of the stability or instability by Time Domain Simulation
it’s not direct. The peak to peak criterion was used because is easy to compute and more reliable
[6]. The machining is considered as unstable if the peak to peak amplitude exceeds a given level.
The simulation chart obtained with a 0.1 mm criterion at tool’s end is presented Figure 5. The time
of computation is 6 hours on a desktop computer (Intel Core Duo 1.73 GHz – 1 Go RAM). It may
be noted that the dichotomy method may reduce the computing time [9], but full simulation allows
the absence of closed areas.
The stability map is presented on figure 5. It’s possible to divide the contribution of the stability
and the contribution of the forced vibrations. By removing the nonlinearity when the tool leave the
cut under large vibrations, i.e. assuming r(h(t)) = h(t), only the asymptotic stability is detected
(thick line). Great forced vibrations (dotted line) are present at 10000, 14000, 20000 and 40000
rpm. These areas are corresponding to simple forced vibrations but they affect the behavior of the
spindle by a safety mode (the spindle is stopped).
²
4
1
0
nAk
g
µ
=
204 Innovating Processes
Figure 5. Map of stability for a spindle with active magnetic bearing.
The evolution of peak to peak displacement function of the axial depth of cut is presented Figure
6. The (3) and (4) graphs show that after a linear part, an abrupt increase of the amplitudes is
present. This behavior corresponds to chatter at the critical depth of cut, linked to the asymptotic
instability. In graphs (2) and (3), the vibration amplitudes increase proportionally with the axial
depth of cut, like forced vibrations.
Figure 6. Displacement analysis.
A frequency analysis is used to determine the nature of these forced vibrations during
machining. Figure 7 show the frequency analysis of the displacement for 40000 rpm with an axial
depth of cut of 15 mm. Only the tooth passing frequency is present, with a frequency of 2000 Hz,
corresponding to the tooth passing frequency of a 3 tooth tool at 40000 rpm. Classically, for roller
bearing spindle, these cutting cases are associated with stable machining but with a high level of
vibrations. For milling with AMB, these forced vibrations are unsafe and the spindle is
automatically stopped by the control loop.
Advanced Materials Research Vol. 423 205
Figure 7. Frequency analysis for 40000 rpm.
In this paper, the concept of stability, widely used for roller bearing spindle, is developed by
including the forced vibrations. This new aspect would be explained by the superposition of the
asymptotic stability and the forced vibrations.
Experimental part
Cutting tests were conducted on a 5 axis high-speed milling center (Mikron UCP 600 Vario)
with a spindle with Active Magnetic Bearing (Ibag HF400M). The tool is a monolithic carbide end
mill, three teeth, 12 mm diameter, helix angle 30°, mounted on a HSK50E. Milling tests were
carried out on a solid block (80×80 mm) of aluminum 2017A. A schematic experimental diagram is
presented Figure 8. The part was down-milled with a feed per tooth of 0.1 mm, and a radial depth of
cut of 4 mm. The axial depth of cut is increased during the milling test from 0 mm to 12 mm. The
vibrations of the spindle were measured directly using the control signals of the magnetic bearings.
These sensors are included on the spindle but the bandwidth is limited to 2.5 kHz.
Figure 8. Experiment schematic diagram.
Vibrations analysis. Much information is available from the control signals, for example: the
position of the center of the rotor, the drive currents signals... In practice, it’s more reliable to use
the drive currents signals, because they naturally amplify the cutting force variations. In contrast,
the extracted information is indirectly related to the level of vibration.
The raw signal and the sampled signal once per tool revolution are presented in Figure 9 for N =
20000 rpm. For this case, it is difficult to identify the critical depth of cut only by the observation of
the measured signals. By sampling the signal at the tooth passing frequency, a loss of regularity of
the cut is observed at t = 0.5 s. Moreover, this machining with N = 20000 rpm is problematic,
206 Innovating Processes
because the vibrations are so strong and the spindle is automatically stopped (safety mode). Indirect
measurement of machining vibrations with the control signals of magnetic bearings is a reliable
means to control the milling stability.
Figure 9. Continuous time histories and 1/rev sampled signals for 20000 rpm.
Discussion simulations-experiments. The experimental results are collected on the map of stability
(Figure 10). A good correlation is observed between numerical simulations and milling tests. The
classical Hopf instability is found at 15000 rpm, 22000 rpm and 23000 rpm. Moreover, high forced
vibrations, predicted by the simulation, at 20000 rpm have been found by experiment. In this case
the spindle was automatically stopped by the control loop (safety). The test at 40000 rpm does not
show so much vibration than the 20000 rpm one, but the test at 37000 rpm shows high forced
vibrations much sooner than predicted. The discrepancies between the experiment and the modeling
are observed for some spindle speed. It is presently not clear what causes these discrepancies.
Possible causes could be: the too simple linear cutting law or the simplified AMB modeling.
Figure 10. Comparison between simulation and experiments.
Advanced Materials Research Vol. 423 207
Conclusions
In this paper, a dynamical modeling of spindle with Active Magnetic Bearing (AMB) is presented.
A complete but simple numerical model for machining with AMB is proposed. This model uses: the
AMB and the servo, the flexible mode of the rotor, the regenerative effect and the nonlinearity
when the tool leaves the cut under large vibrations. All this elements are naturally linked to the
electro-mechanical system and must absolutely be included in the model in order to obtain
acceptable results. For a spindle with AMB, it is indispensable to consider the strong forced
vibrations, because they can induce safety mode (stop of the spindle). The stability map is modified,
because new areas of large forced vibration appear for high spindle speed. Experimentally, the side
milling ramp test was used to easily find the stability limit. There is a good correlation between
simulation and experiment, which confirms the assumptions, made for the modeling. This original
model can be used, modified and improved easily. All the modeling equations are presented in the
paper, all the parameters are obtained by the machining properties and the resolution is made with
Matlab-Simulink.
Acknowledgments
This work has been supported by the French Ministry of Science. The authors would also like to
thank René LARSONNEUR (IBAG France) and Philippe LEDOUX (GF AgieCharmilles) for their
valuable suggestions.
References
[1] S. Tobias and W. Fishwick: Theory of regenerative machine tool chatter. Engineer 205 (1958),
pp. :199–203 238–239
[2] Y. Altintas and E. Budak: Analytical prediction of stability lobes in milling. Annals of the CIRP
44 (1995), pp. 357–362
[3] T. Insperger and G. Stépán: Uptaded semi-discretization method for periodic delay-differential
equations with discrete delay. International Journal for Numerical Methods in Engineering 61
(2004), pp. 117–141
[4] S.D. Merdol and Y. Altintas: Multi frequency solution of chatter stability for low immersion
milling. Transactions of the ASME, Journal of Manufacturing Science and Engineering 126
(2004), pp. 459–466
[5] B.P. Mann, T. Insperger, G. Stépán and P.V. Bayly: Stability of up-milling and down-milling,
part 2 : experimental verification. International Journal of Machine Tools and Manufacture 43
(2003), pp. 35–40
[6] S. Smith and J. Tlusty: Efficient simulation programs for chatter in milling. Annals of the CIRP
42 (1993), pp. 463–466
[7] L. Arnaud, O. Gonzalo, S. Seguy, H. Jauregi and G. Peigné: Simulation of low rigidity part
machining applied to thin walled structures. International Journal of Advanced Manufacturing
Technology 54 (2011), pp. 479–488
[8] M.L. Campomanes and Y. Altintas: An improved time domain simulation for dynamic milling
at small radial immersions. Transactions of the ASME, Journal of Manufacturing Science and
Engineering 125 (2003), pp. 416–422
[9] G. Peigné, H. Paris, D. Brissaud and A. Gouskov: Impact of the cutting dynamics of small
radial immersion milling operations on machined surface roughness. International Journal of
Machine Tools and Manufacture 44 (2004), pp. 1133–1142
208 Innovating Processes
[10] H. Bleuler, M. Cole, P. Keogh, R. Larsonneur, E. Malsen, R. Nordmann, Y. Okada, G.
Schweitzer and A. Traxler: Magnetic bearings, theory, design, and application to rotating
machinery. Springer (2009)
[11] A. Chiba, T. Fukao, O. Ichikawa, M. Oshima, M. Takemoto and D. Dorrell: Magnetic bearings
and bearingless drives. Elsevier (2005)
[12] N.C. Tsai and R.M Lee: Regulation of spindle position by magnetic actuator array.
International Journal of Advanced Manufacturing Technology 53 (2011), pp. 93–104
[13] C. Knospe: Active magnetic bearings for machining applications. Control Engineering Practice,
15 (2007), pp. 307–313
[14] S. Auchet, P. Chevrier, M. Lacour and P. Lipinski: A new method of cutting force measurement
based on command voltages of active electro-magnetic bearings. International Journal of
Machine Tools and Manufacture 44 (2004), pp. 1441–1449
[15] M. Chen and C. R. Knospe: Control approaches to the suppression of machining chatter using
active magnetic bearings. Control Systems Technology, IEEE Transactions 15 (2007), pp. 220–
232
[16] J.H. Kyung and C.W. Lee: Controller design for a magnetically suspended milling spindle
based on chatter stability analysis. JSME International Journal Series C-Mechanical Systems
Machine Elements and Manufacturing 46 (2003), pp. 416–422
[17] P. Bogacki and L.F. Shampine: A 3(2) pair of Runge-Kutta formulas. Applied Mathematics
Letters 2 (1989), pp. 321–325
Advanced Materials Research Vol. 423 209
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