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Understanding Magnetic and Electrostatic Actuators Using Bond Graphs and a Mechanical Model
Understanding Magnetic and Electrostatic Actuators Using Bond Graphs and a Mechanical Model by
DEAN KARNOPP
Department of Mechanical CA 95616, U.S.A.
Engineering,
University
of California, Davis,
Magnetic actuators in relays, solenoids, and valve positioners and electrostatic loudspeakers and microphones store electrical and mechanical energy and are inherently nonlinear. They are harder to understand than transformer or gyrator transducers such as hydraulic pistons or voice coil drives which transduce power instantaneously and are essentially linear. Bond graph models show that magnetic and electrostatic transducers are mathematically analogous although electrically dual. An all-mechanical equivalent model shows the nonlinear behavior clearly and aids in the intuitive understanding of the problems associated with the use of such actuators.
ABSTRACT:
I. Zntroduction Two basic ways in which electrical energy can be converted to mechanical energy are represented by a movable core inductor and a movable plate capacitor. The magnetic transducer is widely used for relays, solenoid-operated valves and position servos, while the electrostatic actuator is mainly used for low force, high frequency applications as in loudspeakers. In both cases, energy conversion is accomplished through a storage process which is inherently nonlinear. This is in contrast to another class of transducer in which power is directly converted and the process is nearly linear. In a voice coil drive, force is directly proportional to electrical current and in an hydraulic piston, force is directly proportional to pressure. These transducers can be represented in a first approximation by ideal transformers or gyrators because energy is stored only because of parasitic rather than inherent effects. For the transducers to be discussed here, a direct computation of force generation on the basis of magnetic or electric fields is possible but inconvenient. Dynamic models of such actuators are normally derived based on overall energy considerations. For example see (1). For system dynamic studies, the “electrically linear” approximation often suffices and is a useful simplification, but the models cannot be both electrically and mechanically linear except in approximation for very small mechanical motion. Since the transducers can only produce attractive forces independent of the signs of the electrical variables, a mechanical return spring is usually necessary. Nonlinearity on the mechanical side makes it necessary to design the return spring
J" The Franklin 1nst1ture0016~003~~84$3.00+0.00
227
Dean Karnopp carefully. If the spring is too stiff, the performance of the actuator is reduced. If the spring is too weak, instability problems may arise in operation. The desire for a compact, inexpensive, lightweight actuator requires operation in a range for which nonlinearities are important. Often the resulting systems are sensitive to disturbing forces or Coulomb friction in some ranges. There are many ways to measure and characterize actuators, but not all are directly useful for dynamic studies. For example, the magnetic force as a function of current and position can be plotted based on static measurements, Dynamically, it is not really possible to drive the current on an inductive load. One can control the voltage on the actuator and use a current control loop, but with a finite saturation voltage, the controller cannot be perfect, and the current will be affected due to the back reaction from the actuator motion. A less conventional but dynamically correct model of a magnetic transducer would produce force as a function of position and flux or flux-linkage rather than current. Such models are conveniently incorporated in bond graphs. [See (2).] The only disadvantage to such models is that flux and flux-linkage are harder to measure than current. We shall show how the two versions of the model are related. The bond graph model automatically takes into account dynamic back-effects. We first show, using bond graph models, that the most basic models of magnetic and electrostatic actuators are closely analogous. To understand these actuators intuitively is not easy. An electrically equivalent circuit cannot represent the transduction process clearly except as a 4-terminal capacitance with certain nonlinear laws. On the other hand, an all-mechanical system can be sketched in which the nonlinear law appears in a visible way. This system may help to understand the functioning of such actuators.
ZZ. The Magnetic Actuator Figure l(a) shows a highly schematic diagram of a magnetic actuator. In this simplest possible model, the flux 4 follows the single magnetic circuit path through low reluctance iron and through the air gap with length x and area A. The mass m is restrained by a return spring and is acted upon by friction of coulombic character. The coil resistance is R and there are N turns. A variable voltage source drives the system. The bond graph of Fig. l(b) is constructed using the methods of (2). The gyrator relates the magneto-motive force M and flux rate 4 to the voltage e and current i, that is i = MjN,
(1)
4 = e/N. ’
(2)
The reluctance of the air gap is xl@, where ,uLois the permittivity of air assuming a uniform field in the gap. Neglecting the reluctance of the iron, the relation between flux and magnetomotive force is
Journal
228
of the Franklm Institute Pergamon Press Ltd.
Magnetic
and Electrostatic
Actuators
(a)
(b)
FIG. 1. The magnetic
actuator
: (a) schematic
diagram
and (b) bond graph.
This expression cannot hold for x + 0, since the reluctance of the iron remains even when the reluctance of the air gap vanishes. Also, when x is too large, most of the flux will bypass the air gap so we will assume Eq. (3) is valid only over a finite range. More complex models including leakage flux paths can be found in (2), but here we use only the simplest possible model to see the nature of the transduction process clearly. For constant x, the stored energy W can be evaluated as follows : (4) From energy C-field are
conservation
considerations,
the constitutive
laws of the 2-port,
(3)
(5) in which F, is the magnetic force corresponding to the assumed law of (3). The use of flux as a state variable produces the unexpected result, Eq. (5), that the force F,(x, 4) does not depend on x at all in this approximation. The expected xdependence does appear if we express F, as a function of current instead of flux. Combining (l), (3) and (4), we find
F,(x, i) =
p,,AN2i2 2x2
.
Measurements in which i is held constant therefore show a 1/x2 dependence over a useful range of x. This corresponds, however, to derivative causality on the magnetic Vol. 317, No. 4, pp. 227 -234, April Printed in Chest Bntam
1984
229
Dean Karnopp port of the C-field. The reaction combining (2), (3) and (1)
voltage
to an impressed
current
can be found by
N2p,Ai dx
N2pOA di e=_--_--X dt
x2
dt
in which one can recognize an inductance effect, L(x) di/dt as well as a velocity term. The integral causality forms, (3) and (4), are useful for dynamic simulation purposes, but the equivalent forms involving current equations (6) and (7) are useful experimentally.
ZZZ.The Electrostatic
Actuator
Figure 2(a) shows a schematic diagram of an electrostatic transducer shown as if it were a parallel plate capacitor with one movable plate. In practice, the mass and spring would usually represent the mass and stiffness of a flexible plate or membrane and the electrical circuit with current source and parallel resistor might well be replaced with a voltage source and series resistor. Figure 2 is arranged to correspond as much as possible to Fig. 1 to show the similarities of the two actuators. The bond graph of Fig. 2(b) is easily constructed corresponding to the sketch of Fig. 2(a). If we assumed that the capacitor can be described as an ideal parallel plate capacitor with capacitance E,,A/x where Q, is the permittivity of air, A is the plate area, and x the separation of the plates, we again have a linear electrical law e=
(Aiq > 0
where e is the voltage and q is the charge. As before, x cannot approach zero (because of arcing across the plates) nor can x be so large that the uniform electric field assumption behind (8) breaks down. (a)
FIG. 2. The electrostatic
actuator
: (a) schematic
diagram
and (b) bond graph. Journal
230
of the Franklin Pergamon
Institute Press Ltd.
Magnetic The stored energy, for x = constant,
is evaluated
and Electrostatic
Actuators
as follows :
The voltage and electric force are then
The mathematical analogy between (3), (4) and (5), and (8), (9) and (10) is clear. The C-field incorporates the physical analogy between magnetic and electric force generation. The bond graphs in Figs 1 and 2 are structurally equivalent. The gyrator in Fig. l(b) can be changed to a transformer by dualizing the electric circuit. The effort and flow roles of voltage and current are then interchanged and S, + S,, 1 + 0, R + R. The transformer can be replaced by a simple bond by scaling the efforts and flows by the parameter N. The result is that the bond graph of Fig. 2(b) can represent both actuator types. The state equations for the two systems in generalized variables are identical. Only the physical parameters differ between the two systems. Of course, more complex models of the two actuators would not necessarily be strictly analogous nor would the analogy remain if a different electric circuit drive would be used in Fig. 2. In these cases, strict analogy would hold only for the two C-fields.
IV. A Mechanically
Equivalent System
Bond graphs are a kind of equivalent circuit and they help to see how systems operate. Bond graphs often show analogies better than equations. For example, the correct equations for Fig. 1 in the form of (6) and (7) show no similarity to the equations for Fig. 2, (8) and (lo), although we see through the bond graphs that an analogy is possible. The strict mathematical analogy is seen when the integral causality equations (3) and (5) are compared with their counterparts (8) and (10). However, the essence of the transduction processes lies in the constitutive laws of the 2-port C-fields and is thus not immediately evident in the bond graph itself. An equivalent electric circuit for these systems is possible, but it is no better than a bond graph since it too must incorporate a 4-terminal (or 2-port) capacitive element with special constitutive laws. There is hope, however, that an all-mechanical system containing geometric nonlinear elements could represent the transducers in a visible way. Figure 3 shows a bond graph identical in structure to Fig. 2(b) but with only mechanical variables. We have not used the structure of Fig. l(b) since the inclusion of an all-mechanical gyrator (a high-speed gyroscope) would not improve the intuitive appeal of the model. The imposed velocity V corresponds to I in Fig. 2 or (after dualization) E in Fig. 1.
Dean Karnopp
FIG. 3. Bond graph for a mechanically equivalent system.
Similarly, F,, corresponds to e in Fig. 2 or M in Fig. 1 and y corresponds to 4 in Fig. 2 or$inFig. 1. The main problem is to find a proper mechanical C-field. Figure 4 shows a simple spring connected between x and y as a candidate C-field. In this case, the sign conventions are clearly visible and the C-field can be broken down into a l-port C and a O-junction. In the linear case, with 6 being deflection and k the spring constant, the energy W is W = ;k6’
= ;k(x+Y)‘.
(11)
Then the forces are F,=g=k(x+y), FY=z=
8Y
(12) k(x+Y).
(131
Obviously, since F, = F, always, even a nonlinear spring cannot produce C-field relations analogous to the transducer equations. The mechanical structure shown in Fig. 5 does, however, produce the required special forms for F, and F,. The slotted lever, which moves through small angles only, plays the role of the electrically linear transducer laws. For constant x, F, is proportional to y, but because of the quadratic cam, the level ratio varies with xl/‘. The deflection of the linear spring of constant k is zero when y is zero and has the form
F
Lo-
9
1
FX x
FIG. 4. A simple spring in C-field form. 232
Journal of the Franklm Institute Pergamon Press Ltd.
Magnetic and Electrostatic Actuators
FIG. 5. A mechanical
For x = constant,
system equivalent
to the two actuator
models.
the energy stored in the spring is
s Y
WX,Y)
=
F,d, = ; kh2 = ; ky2 ;.
(15)
0
Then
(16)
Fx=;y=
$ (
y2,
(17)
1
which corresponds to Eqs (3) and (8) or (5) and (10). Several unusual features of this type of system are evident. For example, for both positive and negative y, the lever pushes down on the spring shaft and the cam follower pushes down on the cam, tending to reduce x. This corresponds to the attractive forces generated by the actuators. For constant y, the force F, has nothing to do with x. This is necessary from energy considerations but is also clear from a direct free-body analysis. When x is changed, the spring carrier moves up or down and changes the spring force, but the slope of the cam under the follower changes also and the component F, remains constant. Vo, 317. No 4. pp 227 234. Apnl 19X4 Printed m Great Bntam
233
Dean Karnopp A horizontal
gives (16) upon angles)
force balance
of the spring shaft,
use of (14). The downforce
F from the slotted
lever is (for small
F=k6;.
(19)
On the cam, F, is related to F by the slope, (1/2)~-“~: F,
=
F;x-‘!~.
Equations (19) and (20) yield (17). In the real system, y is the result of applying the velocity V(t) to the damper just as the voltage E(t) is applied to the resistor in Fig. 1. Just as in (6) for constant current, one can see that for constant F, there is a strong x dependence in F,. Near x = 0, the cam slope is very steep. The relation between F, and F, is FXL2.
(21)
Thus, the mechanical system is strictly analogous to both the magnetic and the electrostatic transducers. If one can imagine how the mechanical system would react to velocity I’, then one can also imagine how the transducers would react. Naturally, the mechanical C-field could be combined with different other elements to model other actuator system models. V. Conclusions The simplest nonlinear models of magnetic and electrostatic transducers in bond graph form show a strict analogy. A mechanical model shows in a picturesque way the assumptions of electrical linearity and the inherent quadratic nonlinearity. It is relatively easy to imagine the difficulties of controlling this model by moving only one end of a damper connected to a lever. In particular, one can readily imagine the problems which a stick-slip friction element would cause when the transducer is used as a position servo element. This may be one of the cases in which a mechanically equivalent system can show complex dynamic relationship more clearly than an equivalent electric circuit. References (1) S. H. Crandall, D. C. Karnopp, E. F. Kurtz and D. C. Pridmore-Brown, “Dynamics of Mechanical and Electromechanical Systems”, McGraw-Hill, New York, Ch. 6, 1968. (2) D. Karnopp and R. C. Rosenberg, “System Dynamics : A Unified Approach”, John Wiley, New York, Sections 8.2 and 9.3, 1975.
234
Journalofthe
Franklin Institute Pergamon Press Ltd.
DEAN KARNOPP
Department of Mechanical CA 95616, U.S.A.
Engineering,
University
of California, Davis,
Magnetic actuators in relays, solenoids, and valve positioners and electrostatic loudspeakers and microphones store electrical and mechanical energy and are inherently nonlinear. They are harder to understand than transformer or gyrator transducers such as hydraulic pistons or voice coil drives which transduce power instantaneously and are essentially linear. Bond graph models show that magnetic and electrostatic transducers are mathematically analogous although electrically dual. An all-mechanical equivalent model shows the nonlinear behavior clearly and aids in the intuitive understanding of the problems associated with the use of such actuators.
ABSTRACT:
I. Zntroduction Two basic ways in which electrical energy can be converted to mechanical energy are represented by a movable core inductor and a movable plate capacitor. The magnetic transducer is widely used for relays, solenoid-operated valves and position servos, while the electrostatic actuator is mainly used for low force, high frequency applications as in loudspeakers. In both cases, energy conversion is accomplished through a storage process which is inherently nonlinear. This is in contrast to another class of transducer in which power is directly converted and the process is nearly linear. In a voice coil drive, force is directly proportional to electrical current and in an hydraulic piston, force is directly proportional to pressure. These transducers can be represented in a first approximation by ideal transformers or gyrators because energy is stored only because of parasitic rather than inherent effects. For the transducers to be discussed here, a direct computation of force generation on the basis of magnetic or electric fields is possible but inconvenient. Dynamic models of such actuators are normally derived based on overall energy considerations. For example see (1). For system dynamic studies, the “electrically linear” approximation often suffices and is a useful simplification, but the models cannot be both electrically and mechanically linear except in approximation for very small mechanical motion. Since the transducers can only produce attractive forces independent of the signs of the electrical variables, a mechanical return spring is usually necessary. Nonlinearity on the mechanical side makes it necessary to design the return spring
J" The Franklin 1nst1ture0016~003~~84$3.00+0.00
227
Dean Karnopp carefully. If the spring is too stiff, the performance of the actuator is reduced. If the spring is too weak, instability problems may arise in operation. The desire for a compact, inexpensive, lightweight actuator requires operation in a range for which nonlinearities are important. Often the resulting systems are sensitive to disturbing forces or Coulomb friction in some ranges. There are many ways to measure and characterize actuators, but not all are directly useful for dynamic studies. For example, the magnetic force as a function of current and position can be plotted based on static measurements, Dynamically, it is not really possible to drive the current on an inductive load. One can control the voltage on the actuator and use a current control loop, but with a finite saturation voltage, the controller cannot be perfect, and the current will be affected due to the back reaction from the actuator motion. A less conventional but dynamically correct model of a magnetic transducer would produce force as a function of position and flux or flux-linkage rather than current. Such models are conveniently incorporated in bond graphs. [See (2).] The only disadvantage to such models is that flux and flux-linkage are harder to measure than current. We shall show how the two versions of the model are related. The bond graph model automatically takes into account dynamic back-effects. We first show, using bond graph models, that the most basic models of magnetic and electrostatic actuators are closely analogous. To understand these actuators intuitively is not easy. An electrically equivalent circuit cannot represent the transduction process clearly except as a 4-terminal capacitance with certain nonlinear laws. On the other hand, an all-mechanical system can be sketched in which the nonlinear law appears in a visible way. This system may help to understand the functioning of such actuators.
ZZ. The Magnetic Actuator Figure l(a) shows a highly schematic diagram of a magnetic actuator. In this simplest possible model, the flux 4 follows the single magnetic circuit path through low reluctance iron and through the air gap with length x and area A. The mass m is restrained by a return spring and is acted upon by friction of coulombic character. The coil resistance is R and there are N turns. A variable voltage source drives the system. The bond graph of Fig. l(b) is constructed using the methods of (2). The gyrator relates the magneto-motive force M and flux rate 4 to the voltage e and current i, that is i = MjN,
(1)
4 = e/N. ’
(2)
The reluctance of the air gap is xl@, where ,uLois the permittivity of air assuming a uniform field in the gap. Neglecting the reluctance of the iron, the relation between flux and magnetomotive force is
Journal
228
of the Franklm Institute Pergamon Press Ltd.
Magnetic
and Electrostatic
Actuators
(a)
(b)
FIG. 1. The magnetic
actuator
: (a) schematic
diagram
and (b) bond graph.
This expression cannot hold for x + 0, since the reluctance of the iron remains even when the reluctance of the air gap vanishes. Also, when x is too large, most of the flux will bypass the air gap so we will assume Eq. (3) is valid only over a finite range. More complex models including leakage flux paths can be found in (2), but here we use only the simplest possible model to see the nature of the transduction process clearly. For constant x, the stored energy W can be evaluated as follows : (4) From energy C-field are
conservation
considerations,
the constitutive
laws of the 2-port,
(3)
(5) in which F, is the magnetic force corresponding to the assumed law of (3). The use of flux as a state variable produces the unexpected result, Eq. (5), that the force F,(x, 4) does not depend on x at all in this approximation. The expected xdependence does appear if we express F, as a function of current instead of flux. Combining (l), (3) and (4), we find
F,(x, i) =
p,,AN2i2 2x2
.
Measurements in which i is held constant therefore show a 1/x2 dependence over a useful range of x. This corresponds, however, to derivative causality on the magnetic Vol. 317, No. 4, pp. 227 -234, April Printed in Chest Bntam
1984
229
Dean Karnopp port of the C-field. The reaction combining (2), (3) and (1)
voltage
to an impressed
current
can be found by
N2p,Ai dx
N2pOA di e=_--_--X dt
x2
dt
in which one can recognize an inductance effect, L(x) di/dt as well as a velocity term. The integral causality forms, (3) and (4), are useful for dynamic simulation purposes, but the equivalent forms involving current equations (6) and (7) are useful experimentally.
ZZZ.The Electrostatic
Actuator
Figure 2(a) shows a schematic diagram of an electrostatic transducer shown as if it were a parallel plate capacitor with one movable plate. In practice, the mass and spring would usually represent the mass and stiffness of a flexible plate or membrane and the electrical circuit with current source and parallel resistor might well be replaced with a voltage source and series resistor. Figure 2 is arranged to correspond as much as possible to Fig. 1 to show the similarities of the two actuators. The bond graph of Fig. 2(b) is easily constructed corresponding to the sketch of Fig. 2(a). If we assumed that the capacitor can be described as an ideal parallel plate capacitor with capacitance E,,A/x where Q, is the permittivity of air, A is the plate area, and x the separation of the plates, we again have a linear electrical law e=
(Aiq > 0
where e is the voltage and q is the charge. As before, x cannot approach zero (because of arcing across the plates) nor can x be so large that the uniform electric field assumption behind (8) breaks down. (a)
FIG. 2. The electrostatic
actuator
: (a) schematic
diagram
and (b) bond graph. Journal
230
of the Franklin Pergamon
Institute Press Ltd.
Magnetic The stored energy, for x = constant,
is evaluated
and Electrostatic
Actuators
as follows :
The voltage and electric force are then
The mathematical analogy between (3), (4) and (5), and (8), (9) and (10) is clear. The C-field incorporates the physical analogy between magnetic and electric force generation. The bond graphs in Figs 1 and 2 are structurally equivalent. The gyrator in Fig. l(b) can be changed to a transformer by dualizing the electric circuit. The effort and flow roles of voltage and current are then interchanged and S, + S,, 1 + 0, R + R. The transformer can be replaced by a simple bond by scaling the efforts and flows by the parameter N. The result is that the bond graph of Fig. 2(b) can represent both actuator types. The state equations for the two systems in generalized variables are identical. Only the physical parameters differ between the two systems. Of course, more complex models of the two actuators would not necessarily be strictly analogous nor would the analogy remain if a different electric circuit drive would be used in Fig. 2. In these cases, strict analogy would hold only for the two C-fields.
IV. A Mechanically
Equivalent System
Bond graphs are a kind of equivalent circuit and they help to see how systems operate. Bond graphs often show analogies better than equations. For example, the correct equations for Fig. 1 in the form of (6) and (7) show no similarity to the equations for Fig. 2, (8) and (lo), although we see through the bond graphs that an analogy is possible. The strict mathematical analogy is seen when the integral causality equations (3) and (5) are compared with their counterparts (8) and (10). However, the essence of the transduction processes lies in the constitutive laws of the 2-port C-fields and is thus not immediately evident in the bond graph itself. An equivalent electric circuit for these systems is possible, but it is no better than a bond graph since it too must incorporate a 4-terminal (or 2-port) capacitive element with special constitutive laws. There is hope, however, that an all-mechanical system containing geometric nonlinear elements could represent the transducers in a visible way. Figure 3 shows a bond graph identical in structure to Fig. 2(b) but with only mechanical variables. We have not used the structure of Fig. l(b) since the inclusion of an all-mechanical gyrator (a high-speed gyroscope) would not improve the intuitive appeal of the model. The imposed velocity V corresponds to I in Fig. 2 or (after dualization) E in Fig. 1.
Dean Karnopp
FIG. 3. Bond graph for a mechanically equivalent system.
Similarly, F,, corresponds to e in Fig. 2 or M in Fig. 1 and y corresponds to 4 in Fig. 2 or$inFig. 1. The main problem is to find a proper mechanical C-field. Figure 4 shows a simple spring connected between x and y as a candidate C-field. In this case, the sign conventions are clearly visible and the C-field can be broken down into a l-port C and a O-junction. In the linear case, with 6 being deflection and k the spring constant, the energy W is W = ;k6’
= ;k(x+Y)‘.
(11)
Then the forces are F,=g=k(x+y), FY=z=
8Y
(12) k(x+Y).
(131
Obviously, since F, = F, always, even a nonlinear spring cannot produce C-field relations analogous to the transducer equations. The mechanical structure shown in Fig. 5 does, however, produce the required special forms for F, and F,. The slotted lever, which moves through small angles only, plays the role of the electrically linear transducer laws. For constant x, F, is proportional to y, but because of the quadratic cam, the level ratio varies with xl/‘. The deflection of the linear spring of constant k is zero when y is zero and has the form
F
Lo-
9
1
FX x
FIG. 4. A simple spring in C-field form. 232
Journal of the Franklm Institute Pergamon Press Ltd.
Magnetic and Electrostatic Actuators
FIG. 5. A mechanical
For x = constant,
system equivalent
to the two actuator
models.
the energy stored in the spring is
s Y
WX,Y)
=
F,d, = ; kh2 = ; ky2 ;.
(15)
0
Then
(16)
Fx=;y=
$ (
y2,
(17)
1
which corresponds to Eqs (3) and (8) or (5) and (10). Several unusual features of this type of system are evident. For example, for both positive and negative y, the lever pushes down on the spring shaft and the cam follower pushes down on the cam, tending to reduce x. This corresponds to the attractive forces generated by the actuators. For constant y, the force F, has nothing to do with x. This is necessary from energy considerations but is also clear from a direct free-body analysis. When x is changed, the spring carrier moves up or down and changes the spring force, but the slope of the cam under the follower changes also and the component F, remains constant. Vo, 317. No 4. pp 227 234. Apnl 19X4 Printed m Great Bntam
233
Dean Karnopp A horizontal
gives (16) upon angles)
force balance
of the spring shaft,
use of (14). The downforce
F from the slotted
lever is (for small
F=k6;.
(19)
On the cam, F, is related to F by the slope, (1/2)~-“~: F,
=
F;x-‘!~.
Equations (19) and (20) yield (17). In the real system, y is the result of applying the velocity V(t) to the damper just as the voltage E(t) is applied to the resistor in Fig. 1. Just as in (6) for constant current, one can see that for constant F, there is a strong x dependence in F,. Near x = 0, the cam slope is very steep. The relation between F, and F, is FXL2.
(21)
Thus, the mechanical system is strictly analogous to both the magnetic and the electrostatic transducers. If one can imagine how the mechanical system would react to velocity I’, then one can also imagine how the transducers would react. Naturally, the mechanical C-field could be combined with different other elements to model other actuator system models. V. Conclusions The simplest nonlinear models of magnetic and electrostatic transducers in bond graph form show a strict analogy. A mechanical model shows in a picturesque way the assumptions of electrical linearity and the inherent quadratic nonlinearity. It is relatively easy to imagine the difficulties of controlling this model by moving only one end of a damper connected to a lever. In particular, one can readily imagine the problems which a stick-slip friction element would cause when the transducer is used as a position servo element. This may be one of the cases in which a mechanically equivalent system can show complex dynamic relationship more clearly than an equivalent electric circuit. References (1) S. H. Crandall, D. C. Karnopp, E. F. Kurtz and D. C. Pridmore-Brown, “Dynamics of Mechanical and Electromechanical Systems”, McGraw-Hill, New York, Ch. 6, 1968. (2) D. Karnopp and R. C. Rosenberg, “System Dynamics : A Unified Approach”, John Wiley, New York, Sections 8.2 and 9.3, 1975.
234
Journalofthe
Franklin Institute Pergamon Press Ltd.