A MULTIPORT APPROACH TO MODELING ELECTROMECIIANICAL SYSTEMS Thesis for the Degree of M. S. MICHIGAN STATE UNIVERSITY JOHN FRANCIS MOYNES 1976 JHESIS ABSTRACT A MULTIPORT APPROACH TO MODELING ELECTROMECHANICAL SYSTEMS BY John F. Moynes In order to standardize the analysis, in particular computer analysis, of electromechanical systems, a multiport modeling approach on the electromechanical component level is used so that large systems may be assembled and studied directly. Electromechanical component models are developed and compiled in a catalog. In addition, their configurations into systems and analysis by state-space techniques is demonstrated. A MULTIPORT APPROACH TO MODELING ELECTROMECHANICAL SYSTEMS by John Francis Moynes A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1976 LIST OF FIGURES TABLE OF CONTENTS 0 OF ELECTROMECHANICAL COMPONENTS SYSTEMS CHAPTER I INTRODUCTION 1.1 Objectives . . 1.2 Organization . 1.3 Details . . . II A SAMPLE 2.0 M Transduction and Units 2.1 Solenoids . . 2.2 Sensors . . . 2.2.1 Resistive Sensors 2.2.2 Inductive Sensors . 2.2.3 Capacitive Sensors 2.2.4 Tachometers . . 2.3 Motors . . . 2.3 1 DC Motors . . III MODELING OF ELECTROMECHANICAL 3.1 Doorbell . . . 3.2 PA System . . 3.3 Feed—Forward Posit FIGURES . . . . . . . . CATALOG . . . . . . . . CONCLUSIONS . . . . . . . REFERENCES . . . . . . . APPENDIX . . . . . . . . ii ion Control Page UMP 16 18 23 35 58 59 61 62 Figure l. 2. \OQQO‘U‘IDW 10. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. LIST OF FIGURES Solenoid . . . . . . . . IC Component with Prefered Causality Complete Sensor Model . . . . . Schematic, LVDT . . . . . . . LVDT, Preliminary Model . . . . . Complete Augmented LVDT Model . . . Simplified LVDT . . . . . . Schematic for Variable Area Capacitive Complete Bond Graph of Capacitive Trans Simplified Capacitive Sensor . . . DC Tachometer Schematic . . . . . Bond Graph Development of Tachometer Word Bond Graph . . . . General Zeroth Order Motor Model Linear Zeroth Order Motor Model . Static Motor Model . . . . . Bearing Friction . . . . . . Piecewise Linear Static Model . . DC Motor Schematic . . . . Complete Permanent Magnet Motor Model Motor Models (a) (b) (c) . . . . Door Bell . . . . . . . Word Bond Graph . . . . System Model . . Fully Augmented System Model Enport Doorbell Model . . P. A. System . . . . Fully Augmented Bond Graph of Con O O O O O O O O O O O O O O O O O O O O Q. 0 O 0 O O O O O O denser Mi Linear Amplifer Model . . . . Loudspeaker Schematic . . . Loudspeaker Bond Graph . . . . Nonlinear P.A. System Model . . . . Bond Graph of Linearized Micr0phone . . Complete P. A. System Model . . . Feed Forward Postion Control Schematic . Device Level Bondgraph . . . Position Sensor . . . . Hybrid Device Level Model . . (a)(b)(c). . . . . iii ooooooooooonooooooooooooooooooooomoooooo Figure Page 44. Complete Model Without Augmentation . . 54 45. Complete Model, Fully Augmented . . . 55 46. Final System Model . . . . . . 56 47. O O O I O O O O I O O O 0 57 iv CHAPTER I INTRODUCTION A Multiport Approach to Electromechanical Components 1.1 Objectives In an effort to aid the design and development of electromechanical systems, it is the objective of this thesis to establish a model catalog of some basic electromechanical components to serve as an initiative for further development in this area of design. The future goal is a computer library of electromechanical components that can be apprOpriately selected and assembled to form desired electromechanical sys- tems for analysis and simulation. Further, it is the objective of this thesis to present the utility of the bondgraph approach to the problem of development, analysis, and simulation of electromcehanical system models. It should be noted that this thesis does not attempt to be comprehensive, but to demonstrate, by example, how models can be developed and systems assembled using bondgraph techniques. Given a component, a port denotes any power* flow into or out of the component. Therefore, a component has as ‘ * many external ports as it has external power exchanges. A * (or signal, if power is unimportant) 1 multiport component is one which has at least one external bond, where an external bond signifies the existence of a power flow. It is on this multiport level that modeling occurs. The multiport approach consists of modeling the energy storage, dissipation, transduction, and power ex- changes of the electromechanical component on a lumped- parameter basis. The versatility of the multiport approach is that its application is uniform in a variety of energy domains; therefore mixed energy domains are readily accomodated. Consequently, this approach lends itself to EM (electromech- anical) components. In EM components, there is an energy conversion which transforms electrical power to mechanical (typically rotational) power, as shown by the generalized 2-port EM component in Figure l-a, and its converse as shown in Figure l-b. Note that the difference between Figures l~a and l-b is in the directions of the arrows, which mark the direction of positive power flow. v and i (voltage and current) are the electrical power variables and T and m (torque and angular velocity) are the mechanical power var- iables. 1.2 Organization Using the bondgraph techniques as the principal tools of this construction, it is first necessary to model the essential parts prior to their assembly. Chapter II deals with the development of some basic EM component models which typify the field of EM devices. The primary areas’are sol- enoids, sensors, and motors. Chapter III treats the assembly of component models into EM systems and outlines some of the types of analysis that may be employed using the mulitport approach. In order to display the pattern of the multiport approach, three distinct types of systems are represented. 1.3 Details For those interested readers not familiar with bond- graphs and bondgraph computer analysis, informative references are "System Dynamics" by Karnopp and Rosenbergl, the ASME "Journal of Dynamic Systems, Measurement, and Control"2, and "A User's Guide to Enport—4" by R. C. Rosenberg.3 The major- ity of the EM device background used in this work was drawn from a text by E. S. Charkey, "Electromechanical System Components"4. It should be noted that for those interested in the multiport approach to other mixed energy domains, there is a thesis by M. R. Ray titled "The Multiport Approach to Modeling Fluid Power Systems".5 CHAPTER II A SAMPLE OF ELECTROMECHANICAL COMPONENTS 2.0 EM Transduction and Units In a mixed energy domain there is always the harsh reality of maintaining a consistent Set of units. Given the generalized electromechanical model of Figure 1, if this component is ideal, then the power in will equal the power out (V°i=T°w). This energy conversion typically occurs through a gyration effect, Figure 2-a, and care must be ex- hibited in the determination of the modulus (T) of the gyrator. The most direct solution to this possible problem is through the use of SI units, where both powers (volts x amps) and (newtonmeters x meters per second), are watts. This method allows the same modulus to be used in both transduction dir- ections (i.e., electrical-to-mechanical and mechanical—to- electrical). However, if British or other units are to be used, then an alternate course must be taken. Ideally, there could be one gyrator with two moduli, one for voltage-to- velocity and one for torque-to-current transduction; see Figure 2-b. Realistically from the viewpoint of Enport, a digital computer program for the analysis of bondgraphs, the notion of two moduli to a passive gyrator is unacceptable. On the other hand, through the use of signal activation the models in Figure 2-c, 2-d are acceptable and will yield consistent units so long as T1 and T2 are properly chosen. Then care must be taken to ensure that physical power is conserved. 2.1 Solenoids The first of the EM components to be introduced is the solenoid. The solenoid is essentially nonlinear in the sense that it lacks a dynamic linear representation for gross motion. However, it can be linearized effectively. The solenoid is basically a coil inductor with a moveable ferromagnetic core placed inside the coil along its axis, Figure 3. One port behaves as an electrical in- ductor while the other behaves as a mechanical spring. It is nonlinear because the position of the core has a non- linear influence on the strength of the field, and the field strength determines the force on the core, thereby affecting the position of the core. The describing equations of the solenoid will be developed in Section 3.1. Because of its dual energy stor- age nature, its ideal bondgraph equivalent element is the IC element, Figure 4. A more complete model deals with the electrical resistance of the coil, the inertia of the core, and the mechanical friction that the core will experience. Such a model is shown in Figure 5. 2.2 Sensors Resistive, Inductive, and Capacitive types are three main electromechanical sensor classifications. Models will be developed for these classes. In addition, a tachometer model will be developed. It is assumed that all models operate within their physical constraints regions. 2.2.1 Resistive Sensors In the resistive area there are strain gages and potentiometers. They both function by having a physical signal vary their resistance in an electrical circuit. The signal is the variable to be sensed, and the result- ing change in current (or voltage) is an indication of its variation. Such sensors frequently are used for the determination of linear and angular displacements and acceleration (i.e., motion prOportional to force). On first pass, a generalized resistive sensor is shown below: I b a where activated bond a has the signal to be measured. The power variables on bond b have the following form: ab = ¢r(fb' qa) where eb is the effort (voltage) on bond b, fb is the flow (current) on bond b, and qb is the displacement (angle or position) signal from bond a. This nonlinear resistive sensor may be employed as a potentiometer measuring angular displacement, as indicated below: C R I Is mtg—ales” "fi'f'IRK—z—lt—r—‘O —.——>ISF9utput 1s 6 voltage SE The source of flow from bond 1 represents angular velocity input. The capacitor (C3) serves as a symbolic integrator of the velocity input so that a displacement is sent to the R-Field. The remaining graph structure allows for a suitably-scaled voltage output. Although R is a nonlinear element, it typically has linear characteristics over a wide range of practical use. Where this linear relationship exists, the model can be considerably simplified to a linear model: C SFI-—-—-“1I—->-—\0——>—>ITF————>ISF \ where the TF modulus has the desired scaling. 2.2.2 Inductive Sensors Inductive sensors are generally used for determining linear displacement. They function on the principle of a moving ferromagnetic core in an electromagnetic field gen- erated by a coil inductor, the basic IC component. One widely used type of inductive sensor is the LVDT (Linear Variable Differential Transformer).4 The LVDT has three windings equally spaced on a cylindrical coil. The primary winding is the center coil, and the two adjacent coils are the secondary. A ferromagnetic core is placed axially inside the coils, creating a magnetic path among the three coils. When the primary winding is energized with an AC source, a voltage is induced in the secondary windings. Thus a position change by the core will result in a change in the output voltage. When the core is equidistant between the two secondary windings, the net output voltage is zero. A schematic of the device is shown in Figure 6. In the bondgraph modeling of the LVDT, the IC element is again introduced. Though typically a two— port device, in this application it acts as a three- port device. A preliminary model for the LVDT is shown in Figure 7.. This model neglects the resistance of the coils. A model with this taken into consideration is shown in Figure 8. Note that the source of flow on bond 1 is the dis- placement input, and the source of flow on bond 9 is the voltage output. Bond 1 may be activated since the back force should typically be negligible on the systems it is designed to sense. Although this is a nonlinear model with dynamics, two factors should be taken into consideration. The first is that the time response of this sensor must be significantly faster than that of the variable it will be measuring. Secondly, though the model is nonlinear, the input-output relationship over the typical usage range is linear. These two factors allow us to simplify the model considerably, such that the model in Figure 9 may be substituted. SF 1 is the displacement input variable, and the effort on bond 2 is treated as the voltage output. R is the desired scaling factor. 2.2.3 Capacitive Sensors Like the previous sensors, capacitive types predom- inantly are used for measuring linear and angular displace- ment and velocities. These sensors usually Operate in one of the two following ways; either a change in the area of the plates occurs while a fixed plate separation distance is maintained, or a change in the separation distance of two plates occurs while a constant area is maintained. A variable-area capacitor will be modeled as an example of a capacitive sensor. This sensor is useful in determining angular veloci- ties. There is a fixed plate and a moving plate, with the inertia of the moving plate assumed negligible. The moving plate is positioned to rotate past the fixed plate so that as it moves the common area between the two plates in- creases or decreases accordingly. This change in area affects the capacitance, C, so that C is a function of 6; see Figure 10,4 p. 38. A bondgraph of this sensor is shown in Figure 11.. As denoted by the complete, augmented graph, the variable area capacitor returns no significant 10 effort (i.e., torque) back on bond 1. However, the signal from bond 1 influences the capacitance on bond 2 in the following way: _ A(B)€ Cz— d where e = dielectric constant, A(6) = area (a function of 9), and d = distance between plates. Although the model is nonlinear, if the response of the model is much faster than the velocity to be measured, then over a certain operating region the input-output relationship is assumed linear, and the model can be sim— plified to the model in Figure 12. R is the appropriate scaling factor, and the source of effort (i.e., voltage) accounts for the bias. 2.2.4 Tachometers The last example of an electromechanical sensor is ono<1fthe most frequently used —- the tachometer. The tachometer produces a voltage output prOportional to an angular velocity input. A schematic is shown in Figure 13. The bondgraph of the tachometer is represented in Figure 14, where I1 is the rotary inertia of the input shaft and armature I is the inductance of the armature coil, R3 and 2 R4 represent the mechanical and electrical energy dissipa— tions respectively, and the gyrator models the interaction of the rotating armature winding in a permanent magnetic field. (Bonds 5 and 8 are activated, Figure 14-b, because 11 any back effects should be negligible to the system with appropriate matchup of the tachometer to the system). Further, the inductances and resistances can be neglected since the dynamics of the tachometer should be much faster than the over-all system response. (Obviously, if this is not the case, the model argument is not useful). The final model for the tachometer is merely a gyrator with two activated bonds, Figure l4-c. R is the scaling factor. Using the bond graph approach some rather complex sensors can be analyzed at a detailed level. However, they can be significantly simplified without losing their utility for restricted regions of Operation. 2.3 Motors The two major distinctions for motors fall under the headings AC powered or DC powered, but there are numerous types of motors under each heading. This section will deal with developing some basic types of DC motor models. The emphasis is on the DC motor rather then the AC motor, since the bondgraph deve10pment of the AC motor would tend to parallel that of the DC models in its apparent structure even thouth AC motors are considerably different in their magnetic field structure. Two good references for AC motor modeling are: The Unified Theory of Electromagnetic Machines, by M.G. Say10 and a doctoral thesis by L.L. Evans6 which has a bondgraph, multiport development of some AC motor models. 12 2.3.1 DC Motors The four basic types of DC motors will be discussed: the permanent magnet, shunt, series, and compound types. The basic motor model is a three-port device as shown in the word bondgraph of Figure 15. Note that if the power flows were changed in such a way that the rotational power was the input and the armature current was an output, then the model would be that of a generator instead of a motor. The approach in modeling these motors will be to begin with an ideal model (i.e., linear, lossless) and to develop that model into a static and then a dynamic model. These classifications -- ideal, static, and dynamic -- are called orders and are the zeroth, first, and second orders, respec- tively. Which electrical port is the input voltage depends on whether the motor is field-controlled or armature-controlled. In the development of the following models, all motors will be treated as armature-controlled. The zeroth order model, Figure 16, takes an input voltage and outputs an angular velocity. The third port determines the gyrator modulation coefficient. The third port is typically treated as signal influenced, due to low field back—interaction. If the field signal is constant, the gyrator is linear rather than modulated. The linear zeroth order model is shown in Figure 17. The next stage of model develOpment is the static model. On this level dissipation effects are introduced 13 into the model. On the electrical side of the motor there are the resistances of both the armature and field windings. However, the resistance of the field winding has no signi- ficance in the armature-controlled, linearized gyrator model. Mechanically the motor has friction between the rotor and its bearings and there are air friction losses7. Each of these dissipative effects can be modeled as a single “element as shown in Figure 18 which is the static motor model. So far nothing has been assumed regarding the linear— ity of the R-Field. (The model is good for both linear or nonlinear resistances. The electrical resistance is a linear relationship with a constant R term. The mechanical resistance is nonlinear as a result of bearing friction, which looks like coulomb friction, Figure 19. The bearing friction is a constant regardless of the motor speed except at w = 0 where it is indetermin- ate. However, the frictional air loss is taken as linear where the torque loss is proportional to the speed. In practice, the mechanical resistance is treated simply as a linear function oftflmaspeed. However, this technique can be improved upon by going to a piecewise linear model as shown in Figure 20. The effort source models the constant loss of the bearing friction as long as the direction of the angular velocity does not change or go to zero. Once there has been a sign change in speed, the model becomes inoperative until there is a corresponding sign change on 14 the effort source. The second order model (dynamic) is a rather complete physical model of the motor, which includes the components necessary to describe the major physical phenomenon associ- ated with the motor. The second model order includes the motor's inertia and the inductances of the windings of the rotor and stator. It is the wiring pattern of these windings that makes for the different types of motors. Model develOpment thus far has been in the nature of a generalized DC motor model with a linearized gyrator. Now, with the inclusion of the motor's inertia and the inductances, which are assumed linear, the specific types of DC motors will be discussed. Schematics of the four different types of motors are shown in Figure 21. The most common motor model is the permanent magnet motor, which has no field windings. Any DC motor model with a linearized gyrator will look like the permanent magnet motor model, Figure 22. The remaining models, Figure 23, are simply permuta- tions on the PM model, where all the models have modulated gyrators. Note that it is not necessary at this point to specify whether the resistances and inductances are linear or nonlinear. With these structural models available, it is only necessary to specify the type of motor and parame- ters desired in order that a motor can be included in an EM systems study. Further details on these models are 15 listed in the Catalog (Appendix A). Once all these component models have been developed, it is just a matter of assembling the components into sys- tems, inserting proper parameter values, and analyzing or simulating the system by standard bondgraph techniques. This is the subject of the next chapter. CHAPTER III MODELING OF ELECTROMECHANICAL SYSTEMS 3.1 Doorbell In this chapter, three EM systems will be modeled. The first will be a doorbell, Figure 24. The component or word bondgraph model for the doorbell is shown in Figure 25. Substituting the appropriate bondgraph models for the components yields a suitable system bondgraph, Figure 26. Since the load of the system is in series with the inertia and damping of the solenoid, they simply add respectively. Likewise, the spring can be included in the C field. For analysis, the model must be fully augmented, which includes assigning power flows and causality, Figure 27. The current version of Enport is linear, and there— fore it will not accept a nonlinear IC component. However, it is still possible to model the system structure on En- port, and even though it will not yield state equations, it will assign causality and power flows. This is possible by use of a C field and a gyrator, Figure 28, which yield the prOper port causalities. Having a complete model, the next task is to derive the equations of state, where the state vector is: [T x p]t. where T is the electrical flux linkage, x is core displacement 16 17 and p is the momentum of the mass. It is now necessary to know the port characteristics of the IC element. On the electrical side the current is where _ _ 2 L(x) = Lo + Ae B(X X0) A plot of L(x) versus x is shown in Figure 29-a, 8 where B [B>0] is the shape factor of the curve and x0 is the mag— netic center. Lo and A determine minimum and maximum ampli- tudes of the inductance. Since the IC element is energy-conservative, the mechanical force is determined as follows: integrating the power received over the time interval dt yields the net energy stored (W); differentiating this with respect to x will yield the mechanical force. The integral of the power is developed below: t A x AWA___ x W I (ei + fv)dt = I idl + J fdx = I L( ) + I fdx o o o o o voltage, 1 = current, f = force, and v = velocity. where e This may be integrated along any path,8 and the one chosen is shown in Figure 30, where for A = 0, f will be zero. Thus _ A2 W - 2L(x) and the force f is ... av. = flux). dx L(x) = dL(x) where L'(x) dx 18 A plot of L'(x) is shown in Figure 29-b. With the IC relations, it is now possible to complete the state equations. -_-RA A — LTET + E(t) i=9 m - _ 12L'(x) b p _ 2 L2(x) kx a? where R = 509, E(t) = 100 volts, k = 2.4 newtons/meter, b = .01 newton-sec/meter, m = .02 newton-secz/meter(kg) Parameters for these equations have been assumed, they are not from any specific model. This set of simultaneous nonlinear differential equations can then be solved by making use Of the digital computer. 3.2 PA System The following EM system model is of a public address system. It is a simplified version; however, it could easily be evolved to a more sophisticated level depending upon one's needs. The basic system is depicted in Figure 31. A condenser micrOphone Operates on the principle of a moving-plate capacitor, and though a moving-plate capaci- tor is nonlinear, it can be effectively linearized around an Operational equilibrium point. The schematic model Of the nonlinear microphone8 and the fully augmented bondgraph are shown in Figure 32. SE 1 19 is the sound input force and bond 9 is the output port to the amplifier. The mechanical compliance (C4) of the moying plate can be included in the C-Field; however, it will be kept separate for the sake Of clarity. Assuming that the amplifier Operates in a region of nonsaturation, the model for the amplifier is merely a transformer with the desired gain modulus and a suppressed back current, Figure 33; i.e., the micrOphone transmitts a signal tO the amp. In the schematic model, Figure 34,8 of the permanent magnet loudSpeaker, the gyrator models the ideal relation of the voice coil and cone assembly. The spring and dash- pot account for the cone's suspension, and the mass is that of the cone itself. The resistance and inductance are physical properties Of the voice coil. This linear model example does not take into account the higher modes Of vib— ration Of the cone, as well as the acoustical properties of the enclosure. Therefore, the model is limited to a low frequency range for good predictive accuracy. To fully account for the system, the driving load of the speaker should be included, i.e., the resistance and inertance of the air. These can be implicitly included in the speaker model in the R and I elements, respectively. However, for clarity the load will be placed outside the speaker model, as shown in Figure 35. These separate component models can now be linked together by their common power bonds, Figure 36. As 20 previously stated, the 2 port C-Field is the only nonlinear element, and it can be linearized arOund some prOperly chosen; Operating point. For a moving-plate capacitor, the capacitance is a function of x with the following relation: where Co is the calculated capacitance, and do is the dis- tance between the plates when x is equal to zero. The port relations for the 2 port C-Field are found using the same energy approach as in the IC element shown previously (a reference to the complete development is found in footnote 8, page 292). The mechanical force is found to be 2 F = _9__ Code and the electrical voltage is: do+X E = Codo Having these relations, the state equations for the micro- phone can be derived for the following state vector: [3] where p is momentum; q, electric charge: and x, linear displacement. The equations of state are: 21 . b 2 p = -kx - fig) - 2Cod + F(t) q _I% [:E(t) déogox :] } Equation 1 i=1, m where,IR = R + RL. The values of the parameters are: k = 176,400 newtons/meter m = .001one;::2;5ecz(kg) do: .000254 meters b = 5.9 newton-sec./meter Co: 10-6 farads R = 50. ohms E = 50. volts To effectively linearize the state equations Of the microphone, it is necessary to determine a stable equilib- rium point in terms Of x. The desired point will exist when the time derivatives are zero, and the eigenvalues indicate stability. Further it is desired that F(t) be set to zero, i.e., equilibrium pt. for no input. The re- sulting equations are: 2 =_:g_ . kx 2Codo Equation 2 _do+X ' . E(t)-—~Yififir—q Equatlon 3 where E(t) = Eo (constant). Now Equation 3 yields: = CodoEo do+X 22 By disposing of g from equation 2, the following relation for x exists: _ CoEo2 2do(l + X/do)2 kx = Substituting the parameter values and solving for x itera- tively yields a stable equilibrium point at R = 3.9 X 10-—5 meters, With a corresponding charge Of a = 5.9 X 10"5 coulombs. In order to make the micrOphone suitable for linear analysis the microphone will be linearized around the equilibrium point (2, a) where x(t) and q(t) have the form: x(t) 2 + xo(t) } Equation 4 q(t) = G + qo(t) where xo(t) and qo(t) are perturbations around the equili- brium point (2, a)8. Substituting Equation 4 into Equation 1, and neglecting the higher order terms Of qoand xo the following linearized equations are Obtained: A b F(t) -kxo - a}: - Egazwqo P ‘ 1 A A . qo=‘m- [((do + X)/Codo)qo + fixo] } Equat1on 5 o :E X0 m In matrix notation the linearized equations have the form: —~—« r12- 9"" _ ' — —— ~ P m Codo k P.T l ' _ _ (do + Q) _ q q 0 F(t), q° ‘ 0 IRCodo IRCodo ° + x 0 L5» ~1- ° 23 Equation 6. The linearized microphone then has the bondgraph shown in Figure 37. Note that the electrical source of effort has been eliminated and the capacitance of the Spring has been included in the C-Field. The C-Field is defined by the following matrix relation: 7 A m — ' _9L__ f = k COdO Xo Equation 7 L 9.91.1.3. e L Codo Coda _ Lq‘) The fact that the Off-diagional terms are equal implies that it is a conservative energy field.1 The final linear system bondgraph, Figure 38, is then a fifth order model, and while quite awkward to solve by hand, it is readily simulated by a digital computer. 3.3 Feed-forward Position Control The final example was taken from a paper by Kulkarni and Charyg. It is a feed-forward position control servo- mechanism. In their paper a solution was generated through several assumptions and by use of signal flow graphs. Using the same general model and assumptions, the bondgraph approach will be applied in an attempt to generate the same solution. In rather mixed notation a model Of the system is shown in Figure 39. The input shaft displacement is detected by a sensor which transmits an electrical signal which is then compared with the output shaft displacement. The comparison Of the 24 two results in an error signal. The error signal (in this study all electrical signals are voltages) is sent to the two sybsystems. Each subsystem consists of a sensor, amplifier, DC motor and apprOpriate gearing. In the system, there are two delays. Since these were ignored in the original paper, for the purpose of comparison they will be treated similarly here. However, it should be pointed out that the inclusion Of these delays would pose nO problem to the bondgraph approach. One error signal goes directly to an amplifier which Operates motor one, a DC, field-controlled motor. The position response Of motor one with respect to the load is then fed back, via another sensor, tO its amplifier, the comparator, and to a summer which is just ahead of the second subsystem. The original error signal and first subsystem response are appropriately summed with the resulting signal controlling subsystem two. Subsystem two has the same be- havior as subsystem one. Both subsystems one and two have the same parameters, though this need not be the case. In the model, it is assumed that there is no shaft compliance and that the two gear trains are ideal (i.e., lossless, mechanical transformers). Should it be necessary, shaft compliance and non-ideal gear trains may be readily and easily included in the complete model. Since the arma~ ture current of each motor is constant, the armature port effect is contained in the motor model, (i.e., the appro~ priate gyrator). The motor model is that of a DC shunt 25 motor. The complete device-level bondgraph Of the system with the appropriate power and signal flows is shown in Figure 40. The sensors are position sensors, where the angular velocity is integrated to angular position and then trans- formed tO a voltage signal, Figure 41.. The comparator is merely part of the bondgraph junction structure, and the amplifier is the linear model developed previously, assum— ing no saturation. TO complete the model, the load con- sists of a rotary inertia with damping. It should be noted that the load is positioned by the sum Of the velocities of the two subsystems as signified by bonds A and B at the zero junction in Figure 40. This model will now be modified into a hybrid model, Figure 42. This is not necessary from a bondgraph point Of View, but this was the manner of treatment in the orig- inal paper and for the purpose of comparison it will be followed.* In the hybrid model the inertia and damping of the load are scaled and included in the motor model Of each subsystem. The reason for this hybridization is straight forward. The true model contains three rotary inertias any two Of which may be treated as independent state varia- bles with the third being dependent. Rather than carry the complications of a dependent variable in the model, * The input signal and sensor are compounded and treated as a source of effort. 26 KulkarnianuiChary modeled the three inertias as two, where the load is taken into account by altering the inertia (J) and damping coefficient (b) Of each motor in subsystem one and two. Consequently, these two inertias both lead to independent state variables. For this model change to be correct, the following conditions must be true. When subsystem one acts upon the load there exists, during that time interval, no coupling between the load and subsystem two. Further, when sub- system two acts upon the load, in another time interval, there is no coupling between the load and subsystem one. In terms of the bondgraph approach (for ease and clarity the TFs and Rs have been dropped) the model in Figure 43-a was said to be dualized by the model in Figure 43-b, with appropriate changes to MlauuiM2. Although this may be a reasonable assumption, in actuality, it is not a correct dual model, as will be shown. The correct model of Figure 43-a, Figure 43-c, is an implicit I-Field which exhibits the cross coupling of bonds (1) and (2). The matrix relationship: V1 in 112 [P1] [V2] [in in] P2 p = momentum, v = velocity, is derived below via trans- mission matrix techniques. IE,: 27 These techniques are described in more detail later in the reference paper. For the model below, the following rela- tions hold (where e is force, f is velocity, and s is used in the context Of LaPlace transforms): I::I = I: “TI I::I A ea 1 0 e6 = 1 f1, I-TTg-S- '1— f5 — + IB e6 1 -mys e2 f5 0 l __ f2 4. Ic These matrix relations may be substituted to yield the relationship between [el f1] t and [e2 f2] t : 81 92 I X I X I f1 A B C f2 F_ms + m3 -(m7ms + m7m3 + m3m5)s—_ r_ez_1 m5 ms 1 -m7 4’ m5 ——- ——--—-—--—— f2 L sms m5 _ L .o This can then be placed in the form: Inf 7 mu + ms -m5 . p—EL‘ l m7m5 + msma + Ifl3m7 m7m5 + m5m3 + m3m7 S f -m5 ~(m5 + m3) __ _€_2_ L 2 Lm7m5 + msm3 + m3m7 m7m5 + m5m3 + m3m7 s 28 Since: = dE1 = dE2 e1 dt and e2 dt or in terms Of LaPlace: e1 s sp1 and e2 = sz Then: m T f2 _ P2:I Not only does the existence of the 112, 121 terms show the cross coupling, but the fact that they are equal proves that the I-Field is energy conservative, which we would ex- pect from physical reasoning. With the development Of suitable models for the multi- port components, it is merely a matter Of substituting them into the device level multiport model, Figure 42, to Obtain the complete model, Figure 44. Figure 45 is then the com- plete system model fully augmented. In the original study, the motor field inductances were considered negligible. Accordingly, those components may be deleted from Figure 45, leaving the final system model. Having obtained an analogous model to that of KulkarnianuiChary, there are now three bondgraph techniques that can be used to generate a solution for the transfer function between the input and the output. One technique consists Of generating the state equations, whereby the transfer function is Obtained through the application Of matrix techniques. Another technique is the transmission 29 matrices approach, and the last technique is the use of computer analysis which uses the computer program, Enport 4.2. In an effort to illustrate a manual approach, only the first two techniques will be applied. The state equations in matrix form are: F-p1_1 r--bs/J1 ’raokAkze/Re 0 0 1 _j?1— E, n/J1 0 0 0 x2 P3 0 'rzzkAkze/Re-bo/Ja ‘rzzkAk17/R3 p3 __i._J __ o o n/J3 o _ L_x.__ F—raokA/qu 0 + E(t) rzzKA/Re — O _ where plis the angular momentum of motor one, x2 is the geared shaft position Of motor one, and p3 and xu have the same respective designations, but for motor two. E(t) is the input variable, i.e., the desired shaft position. The output variable, 8 (the output shaft position), is the summation of the x2 plus x3. Using LaPlace techniques and substituting the numer- * ical parameters, Figure 46, the (SI - A) matrix is: ’s+-1 4o 0 07 -100 s o o 0 40 s + 1 4o __ o o -100 s_j The transfer function for 6(s)/E(s) is determined by * o 1" Where x = Ax + Eu 30 solving for X2 and X4 via Cramer's Rule: 9(8) = E(s) r3+1 1% o 07 r5+1 M) 40 0‘ -100 0 o o -100 s o o 0 40 s + 1 40 + o 40 40 40 L o o -100 s_‘ __ o' o o s_J [SI - AI .— Taking the determinant yields: 8(3) = 800052 + 8000s + 16x105 E(s) s“ + 253 + 800132 + 80005 + 16><106 which is the same relation Obtained in the original paper. Using the transmission matrix approach, a transfer function may be obtained without generating the state equations. A transmission matrix (M) has the form whereby the power variables (ei, fi) Of one bond of a two port device are related to the power variables of the other bond of the two port device, an example of which is shown below:1 e [:91] [“111 I“12] [:92] _11M__%;1 = f 1 2 f1 m21 mzz f2 The transmission matrix for subsystem one is developed * in the following stages. Subsystem One: I A I I l ‘32 31 * Bond 27A and the extra zero are included for ease of handling. 31 171/11 0 :| [€31] * _ o 0 £31 H32-31 M31~30 * 830 0 0 829 a? GY 29 = fao 1/r 0 £29 Mao-29 I K 1 629 1 R5 + 113 828 ____A 1 _____4 = 29 28 5 f29 0 1 £28 \ M29-2e R * 628 0 0 827 7?: TE -a>7—\ = £28 0 l/n f27 Mza-27 .M * C 827 0 0 I] 9273. I = 2 f27 CS 1 __ f27A ._?__x ____;> 7 0 27A M27-27A M32-27A = M32-31XM31—30XM30-29XM29-2axM2e-27XM27—27A * Zero due tO activation. 32 Hence: R5C2S(R5 + 115) M32-27A = nnr 0 This same manner yields: Rqus(R7 + 138) R6(R5 + I15) nHr 0 R3£R7 + I35) M15_19A = nfir 3hr 0 0 1 0 MIG—l7 = l/n O Man—26 = /n 0 0 0 0 The graph with transmission matrices is: SE—g—Ao———>1————>M 32-27A \ V V lL-*——-—3 0 —————$-M 1 —————J*SF au-zs 10 f = 0.0 In \ _____A \ l M15-19A 0 20 \ M 4.. Due to the nature of the activated bonds in M15_17 MIA—20 = M15—19A + MIG-17 RBCHS(R7 + I33) R8(R7 + I38) l + _ nnr 0 nHr O 33 From the bondgraph in Figure 47, the transfer function for elo/eg is obtained. All three transmission matrices are l x 2, since row two is null for each matrix due to activation. The flow variables are deleted, in manipulating the matrices * according to junction structure. es = 811 = 832 + 833 = M32e27A + MauM32e27A ea - (M32 + M34M32)(elo - e20) Equation 8 99 = 812 = elu + 813 = ”14820 + Mau(elo - 920) eg —- (Mlu’M3u7820 + Maue1o Equation 9 From Equation 9 comes the relationship: e9 - Mauelo = 820 Equation 10 Using Equation 10 for e20 in Equation 8 yields: e - M e 89 = (M32 + M34M32)(e10 - M1“ _3fi3:°) Simplifying: M32 + MauMaz l + e = I: M14 - M34 :] 9 MM +MM M32 + MauMaz + 3u( 32 _ 3“ 32) 810 Mlu Man Substituting the numerical values into the trans- mission matrices and the transmission matrices into the above equation yields the following function: 8000s2 + 8000s + 16x106 :;° = s“ + 2s3 + 800132 + 8000s + 16x105 ' which is the same as in the original paper. * M's have shortened form, MA-B becomes MA, 34 At this point two investigative paths are available. One would be to use the transfer function in a frequency response study, making use of available computer programs. The other would be to run some simulations on Enport to study its overall system behavior to some given inputs. FIGURES 1 w (a) (b) Figure l __;1__A GY ___l__s ___X__S GY T \ 1 .. w 1 w T TIIT2 v = Tm V = le 'I' = T1 T = T21 (a) (b) .___%L_:» GY ——-i%——A -——%}—4N GY L——i%—4> GY GY {/1/7T1 ///;%T{K\§k\\§ 1 T . V \I 0t——-\T \ 1Z1» m I \ / w GY ‘GY T2 T2 T. = T11 U) = TIV v = Tzw 1 = T2T (C) (d) Figure 2 35 36 I LIX) rIF .~_ -._+ ”J:‘I.““£-:1 .D, n.) {—X I y a ‘ ' e0 k. -eri:riq;l.w-- I Solenoid Figure 3 -—-§—-—IIC-—-§——I I X IC Component with Prefered Causality Figure 4 RIR] [b1 1 ——————— IC -—-———— l [L(x)] I[m] Complete Sensor Model Figure 5 I ,x I, .ww. ., _ + .3 L A g; e 'T B ‘- 'Out ‘ I 1 - 0......»- . I ’ I, ' .. __-..T\1 " Schematic, LVDT Figure 6 out «— + x-core position 37 R SE g 0 SFoutput SFinput If LVDT, Preliminary Model I Figure 7 I s5 R R R SE——-.;--—>Il—-2-—-“I t—T—tlt—T—AO—T—‘ISF Complete Augmented LVDT Model Figure 8 SFF——>l———\GY———2——AISF r Simplified LVDT Figure 9 38 I Fixed Plate C(e) = Age) :3 .I_ r2 where A(6) = E(eo + 6) moving + a t: \p _ plate e max I I capacitance ' I I I ias . I min max I! angular “a” rotation R 1.....-44/M L. 6 2;. :3 o c(e) — g m [‘ - w--m Schematic For-Variable Area Capacitive Transducer out O-—*E———)$ Figure 10 [ref. (4). p. 58] 39 SF - 2 SFI—-—>L——\C4——(—\;——{04—6——IlI4—“———SE I R Complete Bond Graph Of Capacitive Transducer Figure 11 1 2 SFiW GY l——————I 1 ————\I SF R Bias Simplified Capacitive Sensor Figure 12 ”‘pm ‘\. " ‘1 _ _ \ / " \ / \. __ \ .WI \ 1. £3". ” ' i (l) T (fairlv‘ if; 5 ea .x- " ',,r~- \' I 4’ 7“: ‘9 . (1" ) i,.;—" \ ’ I 5" \ . 1 ’ , .\ pm Tachometer Schematic Figure 13 1- ' - m .. I I 1'. f' If $13.... kilo-mm .U' 40 (a) I I ’ 4 1 2 35 1 6 GY 7 1 aa 1% 3 u / / R R (b) I-—-)—->GY-—>——>I R (c) Bond Graph Development of Tachometer Figure 14 I’f 1f ___%a—>.MOTOR ___%__S a Word Bond Graph Figure 15 41 “‘1. MGY ,L \— General Zeroth Order Motor Model Figure 16 ~ ‘iwI Linear Zeroth Order Motor Model Figure 17 SR“? Static Motor Model Figure 18 -w w+ -a Bearing Friction Figure 19 .\ / ______>.1 ______> GY-——————> 1._____Jx Piecewise Linear Static Model Figure 20 761-5 {it AI 11 {Ir—Mm .. I mId \‘I—‘f Armuhm' - -: Q PIJ, In It! Q:?7 Armature COMPOUISD MCI U H Shun! Iutld Mum: Held DC Motor Schematic Figure 21 43 I I P \ _.__A1_______1.GY______§1_____> V \ R R Complete Permanent Magnet Motor Model Figure 22 7H \ 7H SE -—————> 1 ——————5-MGY-——————3 1-——————3 \ \ R R Series Motor Model (a) I /R l - R I I MGY ———————> 1 ——-—> \1/ I, /\ I R SE ——————3 0 Compound Motor Model (b) Figure 23 44 1.1 V Compound Motor Model (c) Figure 23 - cOntinued Chime Solenoid If 4' (A Switch spring iLm 1:4,I/ A Striker Door Bell Figure 24 COMMAND ——————— SOLENOID ———————-l ———-———-LOAD VOLTAGE SPRING Word Bond Graph Figure 25 45 SE ——————— l ——————— IC —--————-1 -——————-I System Model Figure 26 FR] Th] 1 ————-\I IC ———AI 1 —————5I I [L(x)1 I [k] —————\I SE[E(t)I m] Fully Augmented System Model Figure 27 ————IGYI————C—————I IC Linear Representation I SE—All————->IGYI———Ac———>Il———>II Enport Doorbell Model Figure 28 46 Mayne-hr I'L'I'II'Y I I SOUND ———> (b) Figure 29 T (x,T) +¢+—> x Figure 30 CONDENSER MICROPHONE AMPLIFIER P. A. System Figure 31 SPEAKER l7, SOUN 47 R F(t b Am “i . é f”;j «T——E E (t) C(x) K 11* RL Mircophone Schematic C R SE Fully Augmented Bond Graph Of Condenser Microphone Figure 32 -——9-—AITF‘—————=d K Linear Amplifer Model Figure 33 48 b R L 3_[ :4qu '“5‘ ng m K Loudspeaker Schematic Figure 34 I """"""""""""""" H"— I Speaker llLoad I Rita] RIb] II i131 I I II I______;.1 -—————>IGY _____;> 1._____iL>1 I /\ II I I II N I IIL] CIK] Ilmll: I Loudspeaker Bond Graph Figure'35 49 em messes Homo: Emunmm .¢.m Hmmcflacoz If H m r A//L\\ _m0mDOm fllflTlllI N011WTI—IHIWBJJIO4IIILHTIIIU4II1HIIII— QZDOm U OMOA . memmmm maonmouowz 50 em seamen deco: Emummm .¢.m mumHmEOU Omoq memwmm . enormouowz em enemas Oconmouowz OONAHOOCHA mo sumac Odom afiE———4o mFL——— mlL———k+—— MOTOR ———l, LOAD GEARS I) A I SENSOR % 3 1 A 0 \ B SENSOR I. l—Al—AAMP—éMOTOR—AGEARS—él Device Level Bondgraph Figure 40 52 C I F——§——é 0-——+——AITF ——4>——4 n [Volts/Radian] k=l.0 Position Sensor Figure 41 I / x \A INPUT -—->0 —-—=~1 —)>*AMP —-=*MOTOR --—‘GEARS -)-=~SENSOR 1 : OUTPUT I ¢ “0..-.-.4‘- a. -¢——..-- -4“ 1 -->‘ 1 +-> AMP --‘-" MOTOR —-3 GEARS ->-> SENSOR Hybrid Device Level Model Figure 42 ‘-9.’--i 1 I-———-—\ IImZ] Dependent Causality (a) 53 —-———7IH . m m ' . . . Dual Hybr1d where m, and m3 accound for m1ss1ng m2 (b) 1 (1) ==|II:(2) 1 Figure 43 54 we mesmem coflumucmEmsd usonuez HOOOZ OumHmEOO _ H SO.IIIII H mm 55 me messes omucmfimsé waasm .Hmpoz wumadeou I7III N011 EIII\A K. mm _ Hz .3. 4|vI1 o m m /. // - I. OIIeeAIIlIHWIISIIIZAI HWIIEAIIem J] L d /U 0% m‘£——4—4-——74H / yr MI 56 we messes deco: sonmsm emcee hmTIIH . mm e WIMMI so 41%MI1fi TALMI. : . h 0 3 W M\ m.— Ao.oume 01 V IL mm .m o Parameter J1 = J2 = R5 = R7 = 57 Initialization for Figure 46 Inertia of the Motor and Load = 0.001 lbm-ft/ rad/sec2 Friction of the Motor and Load = 0.001 lbf-ft/ rad/sec n29-27 = n18-19 = Gear Ratio Reduction = 0.1 (i 10f27) R5 = R8 = Motor Field Resistance C2 = C. = Motor nan—26 = n32_31 = r30-29 = SE ———4> 0-——-4> 1 ————> M n16—17 400832) 12 ll 32 33 Sensor Sensitivity = n15-21 = D. C. Amplifier Gain r22-23 = Motor Field Constant 5.0 Ohms 32-27 3k'26 25 3*l ————A SF lh'ZO Figure 47 10 .e. f23= Velocity Integrating Constant = 1.0 1.0 volt/radian 400.0(i.e. 631 0.5 lbf-ft/amp f1o= 0.0 CATALOG The Catalog, which is included as Appendix A, contains a representative collection of common electromechanical system components. While the component models do not repre- sent all possible configurations, they are chosen to have considerable utility. The components are grouped by main function, (i.e., motors, sensors, etc.). ASide from general groupings, there is no other organization. As the catalog is expanded more formal organization may be required. Though consistancy Of representation is attempted, it is by no means a "hallmark", due to the nature Of some of the com— ponents. Where applicable, three models are presented for each component: ideal, static, and dynamic. 58 CONCLUSIONS Throughout this thesis, it has been shown how the Bondgraph Multiport Approach is well-suited to the modeling of electromechanical systems. In particular, this work has demonstrated the value of modeling electromechanical systems from a component level and indicated the ease with which the components can be assembled into larger systems, which may be organized at a simple or complex level accord- ing to the engineer's needs. In addition to the use of electromechanical compon- ents for electromechanical system models, the components can be used in conjunction with other multiport energy domains such as fluid power systems, for which an initial Component Catalog has been assembleds. With the develOpment of an Electromechanical Multi- port Component Catalog it is now feasible to think in terms of developing a computer library of electromechanical com- ponents. However, concerning future research and develop- ment in this area, two factors are Of primary concern in order to make this approach practically useful in industrial applications. The first is the development Of a large- scale electromechanical component catalog and the second is 59 60 that the Enport program must be able to handle a larger and more complex set of systems models than its current capabilities allow. (l) (2) (3) (4) (5) (6) (7) (8) (9) (10) REFERENCES Karnopp, D.C. and Rosenberg, R.C. System Dynamics; A unified Approach. New York: Wiley & Sons, 1975. ASME, Journal of Dynamic Systems, Measurement, and Control. Sept. 1972. Rosenberg, R.C. A Users GuidetxaEnport, New York: Wiley & Sons, 1974. Charkey, E.S. Electromechanical System Components, Wiley & Sons, New York,-l972. Ray, M. The Multiport Approach to ModelingFluid Power Systems, Michigan State University, Dept. of Mechanical Engineering, 1974, MS thesis. Evans, L.L. Simulation Techniques for the Study_of Nonlinear Magnetic Field Engineering, M.I.T. Dept. of Mechanical Engineering, 1968, PhD thesis. Szabados, B.; Sinha, N.K.; d: Cenzo, C.D. A Realistic Math Model for DC Motors, Control Engineering, Mar. 1972. Crandall, Darh0pp, Kurtz, and Pridmore-Brown, Dynamics of Mechanical & Electromechanical Systems. New York: McGraw-Hill, 1968. Kulkarni, S.K.; Chary, L.R., "An Objective Investigation Of a Feedforward Servomechanism" I.E.E.E. Transac- tions on Industrial Electronics and Control Instru- mentation, August 1973. Say, M.G., "Introduction to the Unified Theory Of Electromagnefic Machines New York: Pitman Publish- ing, 1971. 61 APPENDIX Catalog Contents Sensors Potentiometers LVDT VACT Tachometer DC Motors Series Shunt Permanent Magnet Compound Miscellaneous Microphone Solenoid Amplifier Loudspeaker 62 SENSORS Potentiometer ———%——¢'Potentiometer ——¥¥——3 Device Level Model IN . e _ 9 ”Q: +~R(O) = R( max ) nrjk*‘“£: max N I i I’"R2I Eout 6 Potentiometer Schematic SE I I. t—rdlt—T—‘O'é‘z—‘IRV—T—TF—s—Awd ’flSF input output Complete Potentiometer Model Parameter Identification - Complete Model C1 Velocity Integrator R23 Modulated Resistance, f3 = ¢(e3,6) R2 Load Resistance SE1 Circuit Voltage 63 64 SE(bias) 1 ' u F———A 1 Fék—J*O —€§—$ITF ——3—A|1 -—7—>lSF input 5 F——70 output * Simplified Potentiometer Model Parameter Identification - Simplified Model C1 Velocity Integrator TF23 Transformer (volts/radian) SE“ Bias Boltage _ RzEIN {E“ ’ Rle=07 + R2} LVDT (Linear Variable Differential Transformer) ——§——A L.V.D.T. ——¥——A~ Device Level Model Schematic of LVDT * Suitable within the following conditions: for R(e) = R(9max ‘ e) : oR2 emax 65 Complete LVDT Model Parameter Identification - Complete IC123 Inductance of Primary and Secondary Coils combined with the Capacitance effect from the Ferromagnetic Core 11 = ¢1(X1:A2,X) 12 = ¢2(11r12:X) Ru Primary Coil Resistance R5 Secondary Coil Resistance R5 Load Resistor SE9 Bias Voltage SPF—)I—AGY—z—VSF Simplified LVDT Model GY12 Gyrator (r - scaling factor, volts/rad/SOC) Variable Area Capacitive Transducer at; VOAOCOT. —LA 1 Device Level Model 66 (I) out L V V.A.C.T. Schematic SFoutput 6 SF t————4~C 4————10'4———4 ll¢———— SE input R Complete Capacitive Transducer Model Parameter Identification - Complete Model C12 Modulated Capacitance e1 = ¢(q,9) R3 Electrical Resistance SE“ Constant Voltage (EI) SPF—g—Gyé—lql—g—flSF 3 SE * Simplified Capacitive Transducer Model * EA where C(B) = a—(eo + 6) and EAR 3 d << 1 é(t) 67 Parameter Identification - Simplified Model GY12 Gyrator Scales e; to 6 SE3 Bias Voltage, EI Tachometer + TACH ..___\1_)__:. Device Level Model /'\\s F'""——" '.,"-:)J) T9 : ._4.. d I § \'./I A. Tachometer Schematic I 1 G.Y E2 . r Ideal Tachometer Model GY12 Gyrator (r-scale factor, volts/radian/sec) Static Tachometer Model Gle Electromagnetic Field Effect, Gyrator R3 Electrical Resistance Ru Mechanical Damping 68 Dynamic Tachometer Model GY12 Gyrator, Electromagnetic Field Effect R3 Electrical Resistance Ru Mechanical Damping Is Electrical, Armature Inductance Is Mechanical, Rotary Inertia DC MOTORS 4 hi J....__.o i‘ngLD AEQ‘IATURE ‘ {Jr «b PM FIELD! ,\‘ V~ $73 ARMATURE “* ’{IJ .J/fi L /\a/f§ L:__.___.-I.-.. <3”LOAD ‘"”“** gl/LBAD SERIES 'PERMANENT MAGNET _V__.__, A. J i 9 {SERIESP'SHUNT I ' FIELD FIELD ,LgfiRMa.PIELD Hi) A31 I --- 7’ fl ‘ il L—o fyfik LARM. ‘ L m _d;, (DAD , (i) IDAD SHUNT COMPOUND vf If vf If -—¥3>-M0T0R -—%—> -—¥§=+MGY t_%_l a a Word Bond Graph General Ideal Motor Model R R v I r3 r“ ——T—A{GY F—B—A 5 ‘ 1 ——T—=-GY ——3—A-1 Ideal Linear Motor Model Statlc Motor Model R SE R _7 3 . u /’ s /" 1 ____A __T_A.GY __?_A.1 __7_A. 6 Piecewise Linear Static Model 69 70 I I 15 /6 SE-—-7—\l-—1>GY 2\l____._.> / I R3 Ru Dynamic Series Motor Model R 3 ,1 Is \ / i ——‘T 11 I7 Ru Ia Dynamic Shunt Motor Model 71 R3 15 \ ‘fi \ / .\ I], 1\ 12 J Ig—————3 0 I3 \ K , $1.. SE —TF—3 0 -TT—3 1 ——T—A’MGY 2 ‘ l'—T;—3 \ \ R1. R5 Dynamic Compound Motor Model Parameter Identification Static Motor Model . GY12 Gyrator, Electromagnetic Field Effect R3 Armature Electrical Resistance Ru Mechanical Damping Piecewise Linear Static Model Gle Gyrator, Electromagnetic Field Effect R3 Armature Electrical Resistance Ru Air Friction Damping SE5 Bearing Friction where E5 = sign(velocity)*Iconstantl Dynamic Permanent Magnet Motor Model Gle Gyrator, Electromagnetic Field Effect R3 Armature Resistance Ru Mechanical Damping Is Armature Inductance Is Mechanical Rotary Inertia SE7 Command Voltage 72 Parameter Identification - Continued Dynamic Series Motor Model MGleg R3 Ru 159 Is SE7 Dynamic MGY129 SE13 Dynamic MGszlu R» R5 Is Modulated Gyrator, Nonlinear Electromagnetic Field Effect Armature and Field Resistance Mechanical Damping Armature and Field Inductance Mechanical Rotary Inertia Command Voltage Shunt Motor Model Modulated Gyrator, Nonlinear Electromagnetic Field Effect Field Resistance Mechanical Damping Field Inductance Armature Inductance Mechanical Rotary Inertia Command Voltage Compound Motor Model Modulated Gyrator, Nonlinear Electromagnetic Field Effect Shunt Field Resistance Series Field and Armature Resistance Mechanical Damping Shunt Field Inductance 73 Parameter Identification - Continued Ia Mechanical Rotary Inertia SE10 Command Voltage MISCELLANEOUS Microphone £142.); MICROPHONE ———‘T’——> X Device Level Model \‘ZVC‘C —_.. ..-.. Microphone Schematic I [1 lo 10 SE —5——>1——>c«c-—8—-1———\o 7 2/\3 6 C R SE Isa r?" ’- I9 H H Complete Nonlinear Microphone Model Parameter Identification — Complete Model C78 Moving Plate Capacitance F7 = @fiqa,x7),Fa = ¢2(q,x) I; Mass of Moving Plate C2 Mechanical Compliance of Moving Plate R3 Mechanical Damping Ru Electrical Circuit Resistance SE5 Bias Voltage R Load Resistor 74 75 R R N u 5 ..____.> __> 4__ ____3 SE 6 1 1 c 2 1 7 output 3 \ I Complete Linear Microphone Model Parameter Identification - Linearized Model C12 Linearized Capacitance and Compliance 13 Mass of Moving Plate Ru Mechanical Damping R5 Electrical Resistance Solenoid -——%——> SOLENOID € X Device Level Model 0.- -+x ' ‘ .__H_W L(x) . fl t n. I.-. t—u‘. _ i \+,,__ 13-, -. .. .- -. - _..._-,.... A. Solenoid Schematic —-—%—AJIC ——§—4fl L(x) x Ideal Solenoid Model 76 Parameter Identification - Ideal IC Electrical Inductance and Mechanical Capacitance of Moveable Core Inductor R '\ lo 5 ___7w Complete Solenoid Model Parameter Identification - Complete IC12 Same as above, F = ¢1(A,x), I = ¢2(A,x) I; Mass of Core Ru Electrical Circuit Resistance R5 Mechanical Damping Amplifier ———>——\‘1’ AMP ——-—\‘1’ K Device Level Model Ideal Linear Amplifier Model Parameter Identification - Ideal Linear TF12 Transformer, k-Gain 77 3 .__)__\ MFT __2___§ k(e3 or £3) 2 Ideal Nonlinear Amplifier Model Parameter Identification - Ideal Nonlinear MTF123 Modulated Transformer, k-Gain Determined by Signal of Bond 3 Loudspeaker —-JL—A SPEAKER ——§—4 1 x Device Level Model b R L 3—C 15:185.? " i‘.fE‘.:‘_:{é©= fi—mm-i Loudspeaker Schematic 1 GY 2 Ideal Speaker Model Parameter Identification — Ideal Model Gle Gyrator, Electormagnetic Field Effect Static Speaker Model 78 Parameter Identification — Static Model GY12 Gyrator, Electromagnetic Field Effect R3 Electrical Resistance Ru Mechanical Damping R R \ Dynamic Speaker Model Parameter Identification - Dynamic Model Gle Gyrator, Electromagnetic Field Effect 13 Electrical Inductance of Voice Coil In Mass of Speaker C5 Compliance of Speaker R5 Mechanical Damping of Speaker R7 Electrical Circuit Resistance MICHIGAN STQTE UNIV. LIBRRRIE 93ll 0ll llllll 6llll ll