2 Obtaining of Low Order Model via CAD, MKP and MBS Models
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Simulation Assessment of Suspension of Tool Vibrations during Machining
277
The dynamic compliance of machine tool or dynamic compliance only between
axis j and i could be expressed by simple way as
q ( s ) = α ( s ) Q ( s ) , or qi ( s ) = α i , j ( s ) Q j ( s )
(5)
The order of the obtained model is 428. Its eigenfrequencies were compared
with the measurements on the real machine - the accuracy was reached up to 10 %
in the excitation frequency range 0 - 280 Hz. The accuracy of static stiffness in
directions X and Y was up to 1 %.
3.3
Modification of LTI Model
It is suitable to transform the exported LTI model to the LTI with matrix A in a
modified canonic modal form (6) for the next reduction of the order.
A = diag ( A1 , A 2 , , A n )
All of matrices A k
(δ ± i
ω −δ
2
0
2
are 2 2 type and a couple of eigenvalues of LTI
) or (δ
k
δ
Ak =
− ω02 − δ 2
(6)
δ 2 − ω02
)
is projected as
k
ω02 − δ 2
δ
δ − δ 2 − ω 2
0
, or A k =
k
(7)
δ + δ 2 − ω02 k
The symbol δ or expression δ δ 2 − ω02 means damping factors and ω0 is
the natural frequency of the couple.
The application of such LTI form has two principal consequences: the original
model can be decomposed to a set of partial models α k ≡ ( A k , B k , Ck , Dk = 0 ) of
the second order, for which applies (8)
q ( s ) = αk ( s ) Q ( s )
k
= qk ( s )
(8)
k
see block model, Fig. 4a).
The second consequence results from the equivalent formulation of ( α k )i , j ( s )
by transfer function
(Q j ) s + (Q j )
1
0
s 2 + 2δ s + ω02
k
( α k )i , j ( s ) =
(9)
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Regarding the denominator of the transfer function it is obvious that LTI model
α k ( s ) describes the dynamic compliance of the (abstract) mass linked to the selected place of the machine tool by an ideal damper and ideal spring, see block
model, Fig. 4b).
b)
a)
Fig. 4 Block model of the system a) according to (8), b) considering (9)
The modeled dynamic compliance α ( s ) is according to (8) a sum of partial
dynamic compliances α k ( s ) , what can be utilized for the reduction of the LTI
model α ( s ) which respects the structure of the original model of the mechanical
system. It is possible to perform the reduction of α ( s ) by simple elimination of
partial models α k ( s ) with no significant contribution to the original model.
The order 428 of LTI model was reduced to 80 with relative accuracy
downgrade 7 % by this way.
4
Notes to the Minimization
The minimization according (2) (3) contains high number of local minima thus the
computation must be performed by methods which are not designed to fast detection of the closest local minimum. Although these methods can find the local extreme but the time cost might be unacceptable. Therefore it is worked with sets of
parameter values that correspond to the lowest achieved values of local extremes,
i.e. with locally optimal solutions. There are consequently eliminated sets which
would be difficult to implement as well as combinations of parameters with high
impact on the objective function.
Minimizations are done on the reduced LTI of the order 80. The impact of its
accuracy downgrade on the found parameters is eliminated by subsequent minimization on the full LTI of order 428. For that subsequent minimization, found parameter values are used as an initial estimation.
Simulation Assessment of Suspension of Tool Vibrations during Machining
5
279
Example of the Locally Optimal Solution
An example of the dynamic compliance of the machine tool with the integrated
mass A with locally optimal parameters summarizes Table 1. The dominant amplitude of the X axis is slightly decreased, of the Z axis is significantly decreased,
Fig. 5. a), the axis Y remains practically without changes. The exception is
represented by the dynamic compliance between axis X and Y (Fig. 5.b) where the
dominant amplitude was increased. The rest of compliance dominant amplitudes
show their decrease in tens of percent.
Table 1 Influence of the mass A with locally optimal parameters
Δi , j ( p opt ) [ % ]
X
Y
Z
X
-0.1
46.9
-38.5
Y
46.9
0.0
-49.8
Z
-38.5
-49.8
-72.0
a)
b)
Fig. 5 The integration of the mass A with the locally optimal parameters a) The amplitude
characteristics of the dynamic compliance of the machine in the Z axis (the best integration
influence) b) The amplitude characteristics of the dynamic compliance of the machine
between the X and Y axis (the worst integration influence)
The influence of the inaccuracy of the integrated mass A parameters on the machine dynamic compliance was evaluated through the worst behavior during the
random changes of the optimal values of the individual parameters inside the prescribed range.
There was observed significant improvement of the amplitude characteristic
during changes of all of parameters of the mass A in range ±2.5% (Table 2.),
e.g. Fig. 6.a). The exception was again represented by the dynamic compliance
between axis X and Y where the dominant amplitude got worse, Fig. 6.b).
280
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Table 2 The influence of changes of mass A parameters, range ±2.5%
Δi , j ( p ) [ % ]
X
Y
Z
X
-9.2
77.6
-39.3
Y
77.6
-13.7
-53.3
Z
-39.3
-53.3
-72.2
a)
b)
Fig. 6 Influence of the 2.5% random change of all of parameters of the mass A a) Amplitude characteristic of the dynamic compliance of the Z axis, b) Amplitude characteristic of
the dynamic compliance between axis X and Y
The changes of the purely mechanical parameters in the range of ±10% caused
in the worst case increasing of the dominant amplitude in the order of percent,
Table 3, Table 1 and Fig. 7.
Table 3 Influence of the changes in mechanical parameters of the mass A, range ±10%
Δi , j ( p ) [ % ]
X
Y
Z
X
3.9
49.9
-33.7
Y
49.9
-4.4
-53.1
Z
-33.7
-53.1
-73.7
a)
b)
Fig. 7 Influence of 10% random change in mechanical parameters of integrated mass A, the
worst case a) Amplitude characteristics of the Z axis, b) Amplitude characteristics of the
transfer from the X axis to the Y axis
Simulation Assessment of Suspension of Tool Vibrations during Machining
281
The all of demonstrated examples present almost complete suppress of the original dominant amplitudes in the frequency range 50-120 Hz. There is also observed new dominant amplitude at 9 Hz which is out of the typical work range of
the analyzed machine tool.
The solution is more robust to the changes of the mechanical parameters
compared to the changes of parameters which contain also geometric ones.
6
Conclusions
The proposed approach to the reduction makes possible to reduce the exported
state LTI model order of the model to approximately 20%. The maximal difference between the eigenfrequencies is then typically up to 10%.
The higher robustness of the dynamic compliance against the changes of the
purely mechanical parameters compared to the changes of geometrical parameters
seems to be general characteristic for all of locally optimal solutions. This fact is
positive because the practical achievement of the geometric parameters is easier
than of the mechanical ones.
The results can be used for strategic decisions concerning their utilization at the
engineering design phase. The implementation of the integrated mass A e.g. as
damper or absorber is then modeled in the same way and it is evaluated whether
the values of implementation parameters correspond with found locally optimal
parameters. The model is then integrated to the model of the whole machine tool
and the dynamical compliance is verified. The more detailed evaluation is consequently performed via co-simulations [10].
Acknowledgments. The present work has been supported by European Regional Development Fund in the framework of the research project NETME Centre under the Operational
Programme Research and Development for Innovation. Reg. Nr. CZ.1.05/2.1.00/01.0002,
id code: ED0002/01/01, project name: NETME Centre – New Technologies for Mechanical
Engineering.
References
[1] Neugebauer, R., Denkena, B., Wegener, K.: Mechatronic Systems for Machine Tools.
Annals of the CIRP 56(2), 657–686 (2007)
[2] Vetiška, J., Hadas, Z.: Using of Simulation Modelling for Developing of Active
Damping System. In: International Symposium on Power Electronics, Electrical
Drives, Automation and Motion, Sorrento, Italy, pp. 1199–1203 (2012)
[3] Banakh, L., Kempner, M.: Vibrations of mechanical systems with regular structure.
Springer (2010)
[4] Brezina, T., Vetiska, J., Hadas, Z.: Simulation Modelling of Machine Tools with
Flexible Parts as Mechatronic System. In: 6th Int. Conf. AIM, University of Defense,
Brno, Czech Republic, pp. 99–104 (2011)
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[5] Březina, T., Hadaš, Z., Vetiška, J.: Simulation Behavior of Machine Tool on the Base
of Structural Analysis in Multi-Body System. In: Proc. 15th International Conference
on Mechatronics, Mechatronika 2012, Praha, Czech Technical University in Prague,
pp. 347–350 (2012)
[6] Brezina, T., Hadas, Z., Vetiska, J.: Using of Co-simulation ADAMS-SIMULINK for
Development of Mechatronic Systems. In: Proc. 14th International Conference on Mechatronics, MECHATRONIKA 2011, Trenčín, TnUAD, Slovak Republic, pp. 59–63
(2011)
[7] Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press (2005)
[8] Craig, R.R., Bampton, M.C.: Coupling of substructures for dynamics analyses. AIAA
Journal 6(7), 1313–1319 (1968)
[9] Hadas, Z., Brezina, T., Andrs, O., Vetiska, J., Brezina, L.: Simulation Modeling of
Mechatronic System with Flexible Parts. In: 15th International Power Electronics and
Motion Control Conference and Exposition, EPE- PEMC 2012 ECCE Europe,
LS2e.1/1-7 (2012)
[10] Brezina, T., Vetiska, J., Hadas, Z., Brezina, L.: Simulation, Modelling and Control of
Mechatronic Systems with Flexible Parts. In: Jablonski, R., Brezina, T. (eds.) Proc.
9th Conf. Mechatronics 2011, pp. 569–578. Springer, Berlin (2011)
The Comparison of the Permanent Magnet
Position in Synchronous Machine
P. Svetlik
University of West Bohemia in Pilsen, Faculty of Electrical Engineering,
Univerzitni 26, 306 14, Pilsen, Czech Republic
psvetlik@kev.zcu.cz
Abstract. This paper deals with positioning of permanent magnets in synchronous
machine. The original electromagnetic design was calculated with permanent
magnets mounted on the rotor surface. Using 2D finite element method modeling,
the design results were reviewed and recalculated for different permanent magnet
position. The possibility of ferrites instead of neodymium permanent magnets was
also considered. However, the volume of these magnets must be several times
higher than original neodymium ones.
1
Introduction
The permanent magnets are used in many different electrical machines nowadays.
It is possible to find them in direct current, stepper, synchronous and many other
machines. The strongest permanent magnet material is a well known compound of
neodymium (Nd2Fe14B). The price of these magnets is higher in comparison with
ferrite magnets.
Because of this, many scientists are trying to reduce the volume of rare-earth
permanent magnets in general. One of the possible ways is lowering of rare-earth
permanent magnets volume by suitable positioning. The other way is to use other
types of magnets, ferrites for example. [2] [3]
The correct design of permanent magnets is a key matter in modern electrical
machines. The influence of the magnet position itself could be very important in
terms of the distribution of the magnetic flux density in the air gap. The focus of
this paper is a comparison of three rotor geometries with a different position of
permanent magnets and different types of permanent magnets. The fundamental
harmonic component of air gap’s magnetic flux density across all solutions is
analyzed and compared.
2
The Finite Element Analysis
For further simulations an existing design of synchronous machine with permanent neodymium magnets with parameters listed below was used. For results’
verification the fast Fourier analysis (FFA) was used [6].
T. Březina and R. Jabloński (eds.), Mechatronics 2013,
DOI: 10.1007/978-3-319-02294-9_36, © Springer International Publishing Switzerland 2014
283
284
P. Svetlik
Table 1 Simulated motor parameters
parameter name
unit
value
Power
Voltage
Nominal speed
Power factor
Air gap
Air gap mag. flux density
Length of the armature
Magnet width
Magnet thickness
[kW]
[V]
[rpm]
[-]
[mm]
[T]
[mm]
[mm]
[mm]
45
240
400
1,0
1.75
0.95
185
36
8
2.1
The Original Solution
The original design was calculated for permanent magnets attached to the surface
of the rotor. This mounting position is a very common solution used in full range
of synchronous machines. The original design was modified for usage of higher
type of permanent magnets, which affected obtained results [7] [10]. The fundamental harmonic component’s magnitude results, according to accomplished fast
Fourier analysis, near to 1.37 T (see fig. 2. and 3.) [6].
Table 2 Parameters of the neodymium magnet
magnet parameter
unit
value
Remanent mag. flux density
[T]
1.48
Coercive force
[kA/m]
891
Max. BH product
[kJ/m^3]
410
Fig. 1 The original solution with PM on the surface of the rotor
The Comparison of the Permanent Magnet Position in Synchronous Machine
285
2
B.n, Tesla
1
0
-1
-2
0
100
200
300
400
500
600
700
800
900
Length, mm
Fig. 2 The distribution of magnetic flux density in the air gap
Spectrum of Winding's Magnetic Flux Density
magnitude of harmonic component [T]
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
order of harmonic component [-]
Fig. 3 The FFA of the original solution
2.2
The Usage of Interior Mounted Permanent Magnets
Another possibility is usage of the interior permanent magnet position. Several
possibilities were tested with very promising results in comparison to original
design [7][10]. The volume of originally calculated magnets was kept unchanged
and magnets were put inside the rotor. Also the FFA [6] gives value of the
fundamental harmonic component of the air gap magnetic flux density higher than
1.4 T.
286
P. Svetlik
Fig. 4 The alternative solution with inner permanent magnets
2
B.n, Tesla
1
0
-1
-2
0
100
200
300
400
500
600
700
800
900
Length, mm
Fig. 5 The distribution of magnetic flux density in the air gap
Spectrum of Winding's Magnetic Flux Density
magnitude of harmonic component [T]
2
1.5
1
0.5
0
0
5
10
15
20
order of harmonic component [-]
Fig. 6 The FFA of the alternative solution
25
30
35
The Comparison of the Permanent Magnet Position in Synchronous Machine
287
This fact is caused by higher amount of power lines pervading the air gap and
also by better distribution of power lines. Because of this higher value of magnetic
flux density, it is possible to slightly reduce the volume of each magnet, which
means lower cost of the machine. The comparison of the air gap magnetic flux
density of these two solutions is shown in the fig.7.
Fig. 7 The comparison of the distribution of magnetic field in the air gap and nearby elements
2.3
The Implementation of Ferrite Permanent Magnets
The usage of ferrite PM has many benefits. They are cheap, highly available and
have higher work temperature in comparison to neodymium magnets. On the other
hand, the remanent magnetic flux density of ferrite magnet is very low hence the
volume of ferrite permanent magnets has to be several times higher than volume
of rare-earth permanent magnets. According to calculations, the volume of this
type of magnets would be bigger than the machine itself [5] [8]. For calculation the
properties of ferrite permanent magnet type Y30BH were used.
Because of very high required volume of ferrite permanent magnets, the maximum possible volume of magnets was used for analysis of this solution. In this
case the almost whole volume of the rotor is filled by permanent magnet material
and only thin parts of rotor are used as rotor poles.
Table 3 Parameters of the ferrite magnet
magnet parameter
unit
value
Remanent mag. flux density
[T]
0.38
Coercive force
[kA/m]
238
Max. BH product
[kJ/m^3]
30