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Enge1995-DiscreteElectromechanicalSystems.pdf
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
1
Modelling and Simulation of Discrete Electromechanical
Systems
Olaf Enge, Gerald Kielau, Peter Maier
Institute of Mechatronics at the Technical University Chemnitz,
Chemnitz, Germany
Abstract: Electromechanical systems can be regarded as physical structures characterized by interaction of electromagnetic elds with inertial bodies. Constitutive
equations describing the coupling of multibody dynamics with Kirchho's theory dene discrete electromechanical systems. The motion of an electromechanical system
will be understood as the motion of its representing point in its conguration space.
Based on the principle of virtual work the motion equations are Lagrange's equations
of second kind. The main goal is to show the automatic generation of these model
equations based on a unique approach using a dierential-geometric frame.
1
Introduction
Electromechanical systems (EMS) are physical structures characterized by interaction between
electromagnetic elds and inertial bodies [5]. The interaction can be expressed by constitutive
equations (force law) describing the coupling of Maxwell's theory and mechanics. Constitutive
equations describing the coupling between the dynamics of multibody systems (MBS) with a
nite degree of freedom and Kirchho's theory (as quasi stationary approximation of Maxwell's
theory) dene discrete EMS. A mathematical description oriented by classical analytical mechanics
and completed by some basic concepts and methods of graph theory to characterize topological
properties of electrical networks plays a fundamental role for a unique modelling and simulation of
discrete EMS. In view of such a unique approach, concepts, denitions and notations of analytical
multibody dynamics will be used [1]. That is justied because on the one hand analytical multibody
dynamics can be regarded as a possible special case of EMS and on the other hand its concepts
and denitions are developed very well to describe other physical structures in the same manner.
2
Electrical Systems
Using some basic concepts and methods of multibody dynamics [2], the electrical system dynamics
can be described in the same manner very practically. The application of Lagrange's approach
to electrical systems (ES) with lumped parameters is based on the concept of a multipole and
its representation by abstract 2-poles as well as Kirchho's theory and the principle of virtual
work. The internal dynamical behaviour of electrical construction elements (resistor, capacitor,
inductor etc.) can be represented by external measurements at the terminals using voltmeters and
ammeters. This leads to a model of a construction element by using a black box endowed with
a nite number of terminals. A black box having two terminals to measure voltage and current
according to gure 1 is called an electrical 2-pole (e.g. a coil, capacitor etc.). The terminals of
the voltmeter/ammeter have "+" and " " symbols. Positive (negative) indication means voltage
V (t) (<>) 0 and current I (t) (<>) 0. The conditions how to connect volt-/ammeters are dened by a
reference arrow, so that the initial point of which corresponds to "+" terminal and the nal point
2
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
corresponds to " " terminal. This way, a "coordinate system" (in the sense of mechanics) is
dened at each 2-pole. The 2-pole-element denes a branch of the ES. The measured quantities
V (t) and I (t) are called branch voltage and branch current. Their mathematical relationship in
general given by a nonlinear operator equation is called a branch relation (constitutive equation).
a
2 - POLE
+
V
+
A
-
b
2
1
ib
-
Figure 1:
Measurement instructions at 2-pole
A
ia
A
ic
c
V
Figure 2:
External measurements at 3-pole
ia = I 1 + I 2 ; ib = I 2 ; ic = I 1
Denition 1 An electrical n-pole is dened by a black box with n, 2 n < 1, terminals and the
following properties:
a) Voltages and currents can be measured between any two terminals at the same time;
b) Kirchho's laws must be satised related to the measured voltages and currents.
If the measurement results are independent on external inuences then the black box is called
proper n-pole else pseudo n-pole. The electrical properties of an n-pole can be characterized by
n voltages at any two
the set of all relations (determined by external measurements) between
2
n
terminals and the n currents through the terminals (gure 2). The 2 external measurements of
voltages at an n-pole dene a so-called complete measurement graph M . Its nodes represent the
terminals of the n-pole, and its branches represent the ctitious 2-poles, the voltages of which are
measured. Prescribing the voltages in an arbitrary frame G( M ), the other voltages in M are
uniquely determined due to Kirchho's voltage law. The currents measured in the branches of
the frame G( M ) determine uniquely the currents through the terminals of the n-pole. Hence, the
electrical properties of an n-pole can be characterized by (measured) relations between the n 1
voltages and the n 1 currents in G( M ).
Denition 2 A complete measurement graph M with exactly two nodes and one relation is called
an abstract 2-pole if it contains at least one of the both variables V , I of the measurement graph.
Abstract 2-poles are called connected if their graphs have common nodes. Following, an n-pole is
represented by n 1 abstract 2-poles which are connected tree-like. The abstract 2-poles are in
general pseudo 2-poles, because its branch relations can also contain variables V , I of other 2-poles
of the n-pole. Removing branches of isolators (I 0) from the representing tree of an n-pole and
identifying the nodes of short-circuit elements (V 0), one can get a representation of an n-pole
by a forest of not more then n 1 abstract 2-poles. This forest is called the pole-graph of the
n-pole.
2.1 Kirchho's Theory of Electrical Systems
Denition 3 An electrical system (ES) is a nite set of galvanically connected electrical multipoles
(nite multipole network).
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
3
After dening all pole-graphs, an ES is represented by a nite network of abstract 2-poles with
the network graph (containing B branches, N nodes, p components). Let G denote an arbitrary
(but xed) frame of and H (G) the coframe of G in . The corresponding fundamental cut and
the fundamental loop matrices are denoted by Q and A, respectively.
Then, Kirchho's laws become:
X
i2
X
i2
Ai j Vi = 0; j 2 H ("voltage law");
(2.1)
Qi j I i = 0; j 2 G ("current law"):
(2.2)
(2.1) describes the vanishing of the sum of all (oriented) voltages across each fundamental loop
j 2 H . (2.2) describes the vanishing of the sum of all (oriented) currents across each fundamental cut j 2 G. (Symbols of subgraphs are used simultaneously to denote corresponding index
sets.) Both linear equation systems have full rank (rank(A) = B N + p, rank(Q) = N p)
and are equivalent to Kirchho's voltage law or Kirchho's current law, respectively (bi j Vi = 0,
ai j I j = 0; bi j : mesh-branch-inzidence-matrix, ai j : node-branch-inzidence-matrix). Together with
the B branch relations of the abstract 2-poles (constitutive equations), (2.1) and (2.2) constitute
a system of 2B equations in order to determine the 2B functions Vi (t), I i (t), i 2 .
A and Q are related to each other by
Ai
j
=
Æi j ;
Qj i ;
i; j 2 H
i 2 G; j 2 H:
(2.3)
Hence, AT Q = 0.
2.2 Lagrange's Equations for ES in Charge Formulation
The Principle of Virtual Work
Resolving (2.1) and (2.2), three dierent sets of generalized coordinates can be introduced:
- Vi = Qi j Vj ; i 2 ; j 2 G, where Vj are the independent variables [3],
- I i = Ai j I j ; i 2 ; j 2 H , where I j are the independent variables [6],
- and a mixed one of that two forms [7].
In the following, the second form (the so-called "charge-approach") will be presented. (2.2) yields
I j = Qi j I i ; i 2 H; j 2 G;
and due to (2.3)
I j = Aj i ; 2 H; j 2 :
(2.4)
That means, each current I j , j 2 , can be represented by a linear combination of coframe currents
i , 2 H . (2.4) is called mesh-transformation (MT), it denes the kinematics of the ES.
Generalized coordinates of ES are introduced as follows:
Dening functions
qj (t) :=
Z t
0
I j ( ) d; j (t) :=
Z t
0
Vj ( ) d
(2.5)
4
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
as charge and ux of the abstract 2-pole j , j 2 , respectively, its branch relation has the form
_ j Vj = fj (q ; q_ ; q; t) or j = fj (q_ ; q; t) or qj = g0j (t);
(2.6)
where fj and g0j are given, dierentiable functions (notation: q := (q1 ; : : : ; qB ), := (1 ; : : : ; B )).
qj denotes the charge which has been moved in (0,t) caused by the current I j ; j has not necessarily
the meaning of "magnetic ux". An abstract 2-pole with the relation qj = g0j (t) is called current
generator, g0j (t) denotes its charge source. Hence, the equations (2.6) are the constitutive equations
of an ES (in the sense of mechanics).
Due to the mesh-transformation (2.4), a frame G must not contain current generators. Consequently, all current generators belong to the coframe H (G) and decomposite the index set
H (G) = H [ H0 ;
(2.7)
H : coframe branches not containing current generators,
H0 : coframe branches containing current generators,
so that (2.4) with (2.5) yields
qj = Aj q + q0j (t); q0j := Aj q Aj g0 (t); j 2 ; 2 H ; 2 H0 :
(2.8)
The set of branch charges fqj ; j 2 g is called a conguration of the ES. All charges q fullling
Kirchho's current law in integrated form ai j qj = 0 at time t determine the set
Lt := fqj j qj = Aj q + q0j (t); q 2 Rg
(2.9)
of all admissible congurations of the ES at time t.
In the following, latin indices are related to , greek indices to H , and the well-known summation
convention will be used. Denoting the power of H by m, (2.9) denes a 1-1 map of Lt to Rm .
Denition 4 The ES is said to be holonomic having the quasi degree of freedom m.
q = (q , 2 H ) is called its representing point, q 2 Rm ; Rm is called its conguration space,
the q , 2 H , are called (topologically generated) generalized coordinates of the ES.
The motion of the ES is described by C 2 -functions q = q(t). The state of the ES is given by (q_, q).
The ES is called scleronomic if Lt does not depend on t explicitly, otherwise it is called rheonomic
(H0 6= ;).
A virtual displacement of an ES is dened by a set of dierential increments of charges qj belonging
to a variation Æq , 2 H , at xed time t:
fÆqj j Æqj = Aj Æq ; j 2 ; Æq arbitraryg:
(Because Aj = Æj , j; 2 H , it is Æqj = 0 8j 2 H0 .)
analogously to mechanical systems [2].
(2.10)
Motion and state of an ES are dened
Axiom 1 (Principle of virtual work in view of the charge approach)
Let the constitutive equations be given as
_ i Vi = fi (q ; q_ ; q; t); i 2
n H0 :
Then
Æ0 A := Vi Æqi = 0
8Æqi virtual
denes the actual motion of an ES.
(2.11)
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
5
Assuming that the constitutive equations satisfy certain necessary and suÆcient conditions with
respect to the existence of a kinetic potential of rst order (Helmholtz-conditions), the axiom 1
yields the Lagrange equations of motion of the ES
(@_ )_
@ + @_ D = Q(S) ;
(2.12)
where (q;_ q; t) := Wm0 We V h is the Lagrangian (Wm0 - magnetic coenergy, We - electric energy,
V h - generalized potential) and D(q;_ q; t) := D(0) + D(1) denotes the dissipation function of the
ES. Q(S) := Ai Vi(S) jMT are those voltages in the fundamental circuit that cannot or should
not be represented by or D.
A Lagrange Model for ES
Let the constitutive equations of an ES be given by
Vi(L)
Vi(R)
Vi(C )
Vi0
=
=
=
=
_ i ; i (q_ ; t) = Lij (t)q_ j + i0 (t);
Rij (t)q_ j
(rheolinear resistors);
Cij (t)qj + Vi(0C ) ;
Vi0 (t)
(voltage generators)
with
Lij = Lji ; Cij = Cji ; @0 R[ij] 0;
(2.13)
where i0 (t), Vi0 (t), Vi(0C ) (initial voltage of a capacitor) are arbitrary and qi (t) := g0i (t), i 2 H0 .
Hence, the magnetic copotential reads
Wm0 (q;_ t) :=
1 i j
A A L q_ q_ + Ai [Lij q_0j (t) + i0 (t)]q_ ;
2 ij
q0j (t) := Aj g0 (t);
(2.14)
and the electric potential reads
1
We (q; t) := Ai Aj Cij q q + Ai [Cij q0j (t) + Vi0 (t) + Vi(0C )]q :
2
The dissipation function
1
D(1) (q;_ t) := Ai Aj R(ij) q_ q_ + Ai Rij q_0j (t)q_
2
(2.15)
(2.16)
and the generalized (gyroscopic) potential
1
1
V (q;_ q) := Ai Aj R[ij] q q_ r[ ] (q q_
2
2
q q_ )j<
can be assigned to the rheolinear resistors. Hence, the Lagrange model f; Dg of the ES is
(q;_ q; t) := Wm0 We V 12 Ai Aj [Lij (t)q_ q_ Cij (t)q q R[ij] q q_ ]+
+Ai f[Lij (t)q_0j (t) + i0 (t)]q_ [Cij (t)q0j (t) + Vi0 (t) + Vi(0C ) ]q g
D(q;_ t)
:=
1 i j
2 A A R(ij ) q_ q_
+ Ai Rij q_0j (t)q_ :
(2.17)
6
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
Both functions are quadratic in q_:
1
1
1
= g q_ q_ + g0 q_ + g00; D = r( ) q_ q_ + r q_
2
2
2
with
g (t)
g0 (q; t)
g00 (q; t)
r( ) (t)
:= l (t) Ai Aj Lij (t);
:= Ai [Lij (t)q_0j (t) + i0 (t) + 21 Aj R[ji] q ];
:= Ai fCij (t)[Aj q + 2q0j (t)] + 2[Vi0 (t) + Vi(0C ) ]gq ;
:= Ai Aj R(ij) (t);
r (t) := Ai Rij q_0j (t):
Then, the Lagrange's equations can be written in their explicit form
g q + q_ q_ + r( ) q_ + r = 0
(; 2 H ; ; 2 H [ f0g; q0 = t; q_0 = 1) with
%
%0
00
3
=
0 for ; % 2 H ;
1
1
0% = 2 Ai Aj % L_ ij + 4 Ai Aj % R[ij ] ;
1
:= @0 g0 2 @ g00 :
Discrete Electromechanical Systems
The Lagrange formalism is described for electromechanical systems (EMS) [4]. This will be done
based on both, analytical mechanics (i.e. its application in multibody dynamics) and the formalism
described for electrical systems in section 2. EMS can be regarded as physical structures characterized by interactions of electromagnetic elds with inertial bodies. The interaction is expressed by
constitutive equations describing the coupling between Maxwell's theory and mechanics. Constitutive equations, describing the coupling of multibody dynamics with Kirchho's theory (as quasi
stationary approximation of Maxwell's theory), dene discrete EMS. In the following, only such
systems will be regarded.
Denition 5 An EMS is a nite set of physical objects with mechanical and/or electrical multipoleproperties interacting among themselves by electrodynamical and/or electro-magneto-mechanical
coupling:
multibody dynamics [ Kirchho's theory.
3.1 Kinematics of Electromechanical Systems
The kinematics of an EMS is dened by the geometric constraints between the rigid bodies and
the topology of the electrical network represented by abstract 2-poles. The set fqi , xk s j i 2 ,
s = 1; : : : ; 6; k = 1; : : : ; K g is called a conguration of the EMS, where the qi denote the branch
charges ( = graph of the 2-pole-network) and the xk s denote the mechanical coordinates. All
branch charges qi and coordinates xk s , which full all constraints of the EMS at given time t (i.e.
geometric constraints and Kirchho's current law), determine the set
Lt :=
6; k = 1; :::; K ;
qi ; xk s jqi = Ai q + q0i (t); xk s = xk s (q ; t); i(q2 ; q;s) =2 1R;n:::;
; 2 H ; 2 J
(3.1)
of all admissible congurations of an EMS at time t. H denotes the subset of the coframe H (G)
of not containing current generators, J is an index set of mechanical coordinates q ; J \ H = ;.
7
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
Convention: In the following, ; ; ! 2 H , ; ; ; 2 J , a; b; c 2 H [ J , ; 2 H [ J [ f0g,
and q0 = t, q_0 = 1 have to be assumed. Lt is a 1-1 map to a cylindrical domain D Rn of
the conguration space, n := jH j + jJ j. n is called quasi degree of freedom of the EMS, and
q = (qa ) = (q ; q ) denotes its representing point. The mesh charges q and the mechanical
coordinates q are the generalized coordinates of the EMS. The motion of the EMS is given by
C 2 -functions q = q(t), and the state of the EMS is given by (q_, q). The EMS is called holonomic
if there are no nonintegrable kinematic constraints, otherwise it is called anholonomic. The EMS
is called scleronomic if Lt does not depend on t explicitly, otherwise it is called rheonomic.
3.2 Constitutive Equations
Let dk( ) be the applied forces acting on the bodies of the EMS, Vi denote the voltages of abstract
2-poles, and Q and v are the generalized forces and the mesh voltages, respectively. Hence,
Q := S @ xdk; v := Ai Vi jMT :
(3.2)
Assumtion:
The constitutive equations of an EMS
Q = Q(q_a ; qa ; t); v = v (qa ; q_a ; qa ; t)
(3.3)
are given by suÆcient smooth functions Q and v .
The simultaneous presence of q and q and their derivatives in these equations indicates the
electromechanical interaction. The description of Q and v by state functions (q;_ q; t) and
D(q;_ q; t) is necessary for a unique representation of the motion equations in Lagrange's notation:
Qa Æa @_a D
with
Q :=
a
(3.4)
va ; a 2 H Qa ; a 2 J:
(3.5)
Hence, the structure of the functions Qa and follows (not considering any physical background):
a) v = _ + u with = (q_ ; qa ; t), @_[ ] = 0 and u = u (q;_ q; t),
R
b) = + V with (q_ ; qa ; t) := dq_ , V (q;_ q; t) := !a (q; t)q_a + !0 ! q_ ,
u ; a = 2 H and Æ V [email protected] ! q_ .
c) Qa := @a + Æa V @_a D with Qa :=
a
[ a]
Q ; a = 2 J
The class representation
Q =
v =
_
Q(0)
+ Q(1)
+ Q(2)
;
(0)
(1)
u
u
u(2)
(3.6)
with
= @_ ;
u(0)
=
u(1)
=
u(2)
=
@ ;
Æ V;
[email protected] D;
Q(0)
=
Q(1)
=
Q(2)
=
@ ;
Æ V;
[email protected] D;
(Q(0)
a =
(Q(1)
a =
(Q(2)
a =
@a );
Æa V );
@_a D)
(3.7)
is a suÆcient condition in order to represent Q and v by state functions , D due to (3.4).
8
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
3.3 State Functions
The arbitraryly given state functions , V , D describe the classes K 0 , K 1 and K 2 according to
(3.7) . On the other hand, if the functions Qa are given by (3.3) and if there is a decomposition
(1)
(2)
Qa = Q(0)
a + Qa + Qa fullling the integrability conditions belonging to (3.7), the state functions
can be calculated:
the magnetomechanical copotential
:=
(q_Z
;q ;q )
dq_
(0) u(0)
dq + Q dq ;
(0;0;0)
(3.8)
the generalized electromechanical potential
V = ! (q; t)q_ !aq_a + !0 ;
(3.9)
!a , !0 dened by a PDE-system [4];
the dissipation function
D :=
(q_Z ;q_ )
(0;0)
u(2)
dq_
Q(2)
dq_ :
(3.10)
3.4 Kinetics of Electromechanical Systems
The kinetics of EMS is based on the principle of virtual work in Lagrange's notation.
Axiom 2 The actual motion of an EMS is characterized by the vanishing of the virtual work
Æ0 A := Vi Æqi + S Æx(dk xdm) = 0
8 Æqi ; Æx virtual
(3.11)
at any time t.
(S : Summation over all material points of the EMS, dk: applied forces arbitraryly distributed,
dm: mass element, x_ = dtd x(; q ; t): velocity, x = dtd x_ : acceleration of related to an inertial frame,
Vi : voltages of abstract 2-poles of the representing graph of the electrical network, qi : branch
charges of .)
Using the kinetic energy of the EMS
T (q_ ; q ; t) :=
1 2
S x_ dm
2
and the generalized forces Q and mesh voltages v
S Æxdk S @ xdkÆq QÆq ; Vi Æqi Vi Ai Æq v Æq ;
(3.11) yields
Æ0 A = v Æq + [ (@_ T )_ + @ T + Q ]Æq = 0
8 Æq ; Æq :
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
9
The motion equations of the EMS
v = 0; (@_ T )_
@ T = Q
(3.12)
are Kirchho's mesh equations and "mechanical" Lagrange's equations. With respect to (3.4) and
@ T 0, @_ T 0, they read in Lagrange's form
(@_a )_
@a + @_a D = 0; a 2 J [ H ;
:= T = T (q_ ; q ; t) + (q_ ; qa ; t) V (q;_ q; t):
(3.13)
(3.14)
If v , Q are given, , V und D can be calculated according to (3.8), (3.9) and (3.10). Parts of the
generalized forces v(S) , Q(S), which cannot or should not be represented by these state functions,
have to be taken into account on the right-hand side of (3.13):
(@_a )_
Q(aS)
@a + @_a D = Q(aS) ;
(
:=
v(S) ;
Q(S) ;
(3.15)
a = 2 H
a = 2 J:
and D together with Q(aS) determine the motion equations of the EMS. The formalism to get
these equations is independent of the EMS itself. f; D; Q(aS)g describes a mathematical model
of the EMS. Due to (3.15) a trivial model fT; 0; Qg always exists. f; Dg := f; D; 0g is called a
Lagrange model of the EMS.
3.5 The Structure of Lagrange's Equations of Motion
Starting from the Lagrange model f; Dg
= 21 g (q; t)q_ q_ ; D = 12 s (q; t) q_ q_
with
(@_a )_ = (ga q_ )_ = gab qb + @ ga q_ q_ ;
@a = 12 @a g q_ q_ ;
@_a D
= sa q_
and
a
1
:= (@ ga + @ ga
2
@a g );
the Lagrange's equations in explicit form read:
gab (q; t)qb + a (q; t)q_ q_ + sa q_ = 0:
Here is
gab := @_a @_b = gba ;
(3.16)
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
10
and with = T + V follows
(gab ) =
_ _
= Ai Aj Lij ;
g 0 ; g = @ @ = lP
u u + (1 s )(1 s )k ij k i k j ]:
0 g g = @_ @_ T = [m
k k i k i
k
Hence, gab is a direct sum of the two matrices g and g . Because of the quasi stationary
approximation of Maxwell's theory (@ 0), gab is independent of q . If g is a regular matrix,
gab is regular, too (g is regular because of the positive deniteness of the kinetic energy T ).
Assuming g is regular, gab (q; t) denes (in general) a time-dependent Riemannian metric in Rn .
The abc(q; t) are the time-dependent Christoel symbols of rst kind.
It is
1
1
= 2 @ g = 2 @ l ;
% 0;
! 0;
= 12 @ g! = 21 @ l! ;
0; ; ; ! 2 H ; ; ; % 2 J:
!
Using this, (3.16) yields
g q
[email protected] g q_ q_ +(2 b0 + sb )q_b + 00 +s0 = 0
1
%
! +(2 b0 + sb )q_b + 00 +s0 = 0:
g q + % q_ q_
2 @ g! q_ q_
(3.17)
This structure of Lagrange's equations follows from (3.16) with respect to a partitioning of the
generalized coordinates q = (q ; q ) and the special representation of the matrix gab as direct sum
of g and g due to the quasi stationary approximation of Maxwell's theory.
The Christoel symbols appearing in (3.17) can be calculated as:
% = (1 s )
0
=
0
00
0
=
=
=
0
=
00
=
K
X
k =%
"i qr [ mk Æij uk j k q uk r% + (1 s )(1 s% )#k ij k r k j k q% ];
1
@l
@ ! ;
2 0 [ ]
1
@ (
! );
2 0
@0 0 + @ !0 ;
0 ;
1 _ _
@ @ @ T @ @_ T ;
2 0 2 [ ] 1
@0 @_ T1 @ (T0 + 0 V0 ):
Some terms in (3.17) have a simple physical meaning. g q describes induced voltages as a
result of changing currents and @ g q_ q_ is the Coriolis voltage. The term 12 @ g! q_ q_! describes
generalized forces with electrical origin (Lorentz forces).
A special EMS (with inductivities, permanent magnets, resistors, current and voltage generators)
with the constitutive equations
i
V (R)
Vi0
= Lij q_ j + i0 ;
i; j 2 H;
Lij = Lij (q ); i0 = i0 (q );
= Rij q_ j ;
= Vi0 (t)
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Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
and the force laws
kk
= Kk (i) e(i) = Kr i Er i ;
Q = Æ ( V );
"mechanical" forces,
"electromagnetic" forces,
has Lagrange's equations of the following form:
Ai Aj [Lij q + @ Lij q_ q_ ] + Ai @ [Lij q_0j (t) + i0 ]q_ + Ai Vi0 (t)
= Ai Aj Rij q_ Ai Rij q_0j (t)
g q + % q_ q_% 21 Ai Aj @ Lij q_ q_ Ai @ [Lij q_0j (t) + i0 ]q_
=
4
K
P
k=
[Kk i uk i + Mk i k i ] + 12 @ Lij q_0i (t)q_0j (t) + @ i0 q_0i (t):
Examples
Generalized Electric Machine
The unique Lagrange approach to dynamic simulation of discrete EMS will be shown shortly by
the example of the generalized electric machine.
An idealized 2-pole 2-phase-machine is called generalized electric machine (gure 3). The memagnetic axis q
Rs
magnetic axis a
magnetic axis b
V60
R rV70
ϕ
V80
magnetic axis d
Rr
V50
Rs
rotor
stator
Figure 3: Generalized electric machine
chanical submodel consists of two bodies connected by a rotational joint. Both, rotor and stator
should have two lumped inductances, displaced by =2, which summarize all inductances of rotor
and stator respectively. The quasi stationary approximation of Maxwell's theory is used. The
magnetomechanical interaction is dened only by eld distribution in the air gap. The magnetic
material of rotor and stator should have a linear B-H-curve and no saturation. The structure
12
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
graph consists of four fundamental loops, each of them containing an inductance, a voltage generator and a resistor (gure 4). The relative angle between rotor and stator is the only mechanical
coordinate. The four charges in the fundamental loops of the structure graph are the generalized
electrical coordinates.
V50
V60
5
1
2
L,R
V70
6
L,R
V80
7
3
8
4
L,R
H := {1,2,3,4}
G := {5,6,7,8}
L,R
Figure 4: Graph of the generalized electric machine
The physical model yields the topology with the fundamental loop matrix
Ai = Æi 4 ; i 2 G; 2 H; H = H ; H0 = ;; jH j = 4:
The generalized coordinates are denoted by
q (qa ) (q ; q ) (qds ; qqs ; qar ; qbr ; ') (q1 ; q2 ; q3 ; q4 ; q5 )
(q : charge in fundamental loop ), and q_a are the generalized velocities of the electric machine.
The pseudo degree of freedom is ve.
The constitutive equations for a model of a smooth-air-gap machine read
i
Mel
Vi(R)
Vi(R)
Vi0
ML
=
=
=
=
=
=
Lij I j ;
1 L0 (')I i I j ;
2 ij
Rs I i ;
Rr I i ;
Vi0 (t);
ML(';
_ '; t)
0
a
0
c cos '
c sin '
B
(Lij ) = B
@
Lij = Lij (') = Lji ;
L0 (') = @L
@' ;
i = 1; 2; Rs = const:;
i = 3; 4; Rr = const:;
i = 5; 6; 7; 8;
= loading torque, Mr = k'_ (k > 0);
0
c cos '
b
dsin '
d sin ' e + f cos 2'
dcos '
f sin 2' e
c sin '
dcos '
f sin 2'
f cos 2'
1
C
C
A
with the simplifying assumtions
a = b = Ls ; c = d = M; e = Lr ; f = 0
(Lr , Ls - self-inductances of the rotor and the stator; M - mutual inductance).
Using this approach, the simulation tool alaska is able to generate automatically the complete
system of motion equations of the generalized electric machine. In this case, these are ve differential equations of second order. Hence, various investigations of the dynamics of dierent
rotational electric machines, like induction or synchronous drives, even DC-machines, can be done.
The unique Lagrange approach always guarantees the correct results with respect to generation of
motion equations.
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
13
Other Applications
Some other applications, which have been investigated using the simulation tool alaska, should
be shortly mentioned. In the case of the MAGLEV Transrapid, the main goal was to study some
aspects of controlling electromagnets which are unstable in the opened loop. The carrying and
tracking system is based on the attraction between the electromagnets in the vehicle and the
ferromagnetic material in the track. It works contactlessly. This MAGLEV-model consists of
ve rigid bodies, the cabin and four suspension bodies. The cabin and the suspension bodies
are coupled by springs. One magnet to carry the vehicle and one to track it are xed at each
suspension body. The drive is realized by stator windings in the track producing a travelling eld
(principle of linear synchronous motor). The parameters of the model (masses, stinesses etc.) are
comparable to the technical data of the Transrapid TR06/TR07. The electrical network of this
example consists of 28 branches (two branches for each magnet, six branches for drive windings on
each side of the track). The degree of freedom of the mechanical subsystem is 30, the quasi degree
of freedom of the complete system is 44. The control of the distance between the magnets and the
track is implemented by control of the current of each magnet separately. PD-, PID- and state
controls have been used in the entire model of the MAGLEV.
Another example is that of a planar motor driven directly. Some types of such drives with dierent
constructions and working principles are investigated at our institute. One of the planar DCmotors, e.g., consists of a stator with permanent magnets and a slide with inductivities. The
driving force is produced due to Lorentz's law. Another one is a hybrid stepper motor. Its stator
is a comb-like structured ferromagnetic plate being magnetically passive. The permanent magnets
and inductivities of the slide are arranged in such a way that the magnetic ux can be increased
and decreased properly. The reluctance principle is used to produce the force in every motion
direction.
In some special cases simplications usually done in the theory of electrical machines can cause a
mathematical problem. For instance, modelling of a 3-phase-machine with the same assumptions
like in section 4 yields a singular matrix of inductivities. That's why the metric of the whole EMS
is not regular and, if the simulation tool uses explicit integration codes only (like alaska), a new
model of the 3-phase-machine has to be designed. This can bo done using a linear transformation
with respect to generalized electrical coordinates.
The Lagrange approach allows to describe the "mechanical" and "electrical" subsystems of every
discrete EMS and the electro-magneto-mechanical interactions between them in a unique way.
Then, a coupling of dierent simulation tools can be avoided.
References
[1] Enge, O., Kielau, G., Maier, P.: Dynamiksimulation elektromechanischer Systeme. VDIFortschritt-Berichte, Reihe 20: Rechnerunterstutzte Verfahren, Heft 166. VDI-Verlag,
Dusseldorf, to appear.
[2] Maier, P.: Analytische Dynamik von Mehrkorpersystemen. ZAMM, 68(10):463{481, 1988.
[3] Maier, P., Steigenberger, J.: Zugang zur Theorie elektromechanischer Systeme mittels
klassischer Mechanik, Teil 1: Elektrische Systeme in Ladungsformulierung. Wissenschaftliche Zeitschrift TH Ilmenau, 20(6):105{123, 1974.
[4] Maier, P., Steigenberger, J.: Lagrange-Formalismus fur diskrete elektromechanische Systeme.
ZAMM, 59:717{730, 1979.
[5] Miu, D.K.: Mechatronics - Electromechanics and Contromechanics. Springer-Verlag, New
York-Berlin, 1993.
Third Conference on Mechatronics and Robotics, Paderborn, Germany, 1995
14
[6] Steigenberger, J., Maier, P.: Zugang zur Theorie elektromechanischer Systeme mittels klassischer Mechanik, Teil 2: Elektrische Systeme in Fluformulierung. Wissenschaftliche
Zeitschrift TH Ilmenau, 22(3):157{163, 1976.
[7] Steigenberger, J., Maier, P.: Zugang zur Theorie elektromechanischer Systeme mittels klassischer Mechanik, Teil 3: Elektrische Systeme in gemischter Formulierung. Wissenschaftliche Zeitschrift TH Ilmenau, 22(4):123{139, 1976.
Adresses of the authors
Dipl.-Ing. Olaf Enge
Dr. rer. nat. Gerald Kielau
Prof. Dr. sc. nat. Peter Maier
Institute of Mechatronics at the Technical University Chemnitz
Reichenhainer Srae 88
D-09126 Chemnitz, Germany
Tel.:
+49(0)371/531-4671
Fax:
+49(0)371/531-4669
e-mail: [email protected]
[email protected]
[email protected]
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