BASIC BOND GRAPH ELEMENTS
3.1 BASIC 1-PORT ELEMENTS
3
FIGURE 3.1. The 1-port resistor: (a) bond graph symbol; (b) defining relation;
(c) representations in several physical domains.
to a vast network of power generation and distribution equipment, yet from the point of view of a system model, a relatively simple characterization of what is behind the wall outlet as a 1-port may suffice.
Here we first deal with the most primitive 1-ports. We consider, in order, 1-port elements that dissipate power, store energy, and supply power.
The 1-port resistor is an element in which the effort and flow variables at the single port are related by a static function. Figure 3.1 shows the bond graph symbol for the resistor, a typical graph of the constitutive relation betweeneand f, and sketches of resistors in several energy domains. The electrical resistor is anR-element if it can be characterized by a volt–current constitutive relationship such as
e=Ri.
Sinceeis an effort variable andi is a flow variable, this constitutive relationship exactly fits our definition of a linear 1-port resistor.
The mechanical dashpot is a 1-port resistor for the same reason as in the elec- trical resistor. If an ideal dashpot is characterized by a force–velocity relationship such as
F =bV ,
where b is the dashpot constant, then it is represented as a 1-port R element.
Since F is an effort and isV a flow, the constitutive relationship also fits our definition of a 1-port resistor.
The hydraulic example is a 1-port resistor because it is characterized by a pressure–volume flow rate relationship and pressure and volume flow rate are bond graph effort and flow variables. In most hydraulic cases the effort–flow relationship is more complicated than the first two examples. This is dealt with in great detail in Chapters 4 and 12; here we just specify that for turbulent flow through a restriction, the pressure–flow relationship is
P1−P2= 1
2A2ρQ|Q|
or
Q=A
2
ρ|P1−P2|sgn(P1−P2),
where A is the flow area. These relationships are effort–flow relationships and again fit our definition of a one-port resistor.
Usually, resistors dissipate energy; that is, power flows into the resistor but never comes out of it. From the point of view of a system connected to a resistor, over time, energy seems to disappear into a resistor. This must be true for simple electrical resistors, mechanical dampers or dashpots, porous plugs in fluid lines, and other analogous elements. Noting from Figure 3.1a that power flows into the port when the product ofeandf is positive according to the sign convention shown, we may deduce that power is always dissipated if the defining constitutive relation betweeneandf lies only in the first and third quadrants of thee–f plane as shown in Figure 3.1b, for then the productef is positive when bothe and f are positive or when both are negative. Because the resistance function cannot lie in the second or fourth quadrant, the law must pass through the origin.
When the relation betweeneandf for a 1-port resistor plots as a curved line as in Figure 3.1b, then the resistor is a nonlinear element. If the relation is a straight line, then it is alinear element. In the special case of a linear element, a coefficient, theresistance, or its inverse, theconductance, may be defined. These parameters are actually just the slopes of the straight line laws when plotted as e versusf orf versuse.
When a resistive element is assumed to be linear, it is conventional to indicate this on the bond graph by appending a colon (:) next to the −R and noting the physical symbol for the resistance parameter. This is done in Figure 3.1 for the electrical resistance and the mechanical dashpot. For the hydraulic resistor, no parameter is indicated since this is a nonlinear element and no single resistance parameter can be identified.
Table 3.1 shows the resistor constitutive laws in general form and in specific form for several energy domains. Note that for power-dissipating linear resistors, with the sign convention shown in Figure 3.1 and Table 3.1, the general resis- tance and conductance parameters,RandG, respectively, are positive, as are the corresponding parameters in the specific energy domains.
For simplicity, we establish the following arbitrary but useful rule: For passive (power dissipating) resistors, establish the power sign convention by means of
TABLE 3.1. The 1-Port Resistor,e
f R
General Linear SI Units for Linear
Relation Relation Resistance Parameter Generalized variables e=R(f ) e=Rf R=e/f
f =−R1(e) f =Ge=e/R
Mechanical translation F=R(V ) F=bV b=N-s/m V =−1R (F )
Mechanical rotation τ=R(ω) τ=cω c=N-m-s
ω=−1R (τ)
Hydraulic systems P=R(Q) P=RQ R=N-s/m5
Q=−R1(P)
Electrical systems e=R(i) e=Ri R=V/A=(ohm)
i=−1R (e) i=Ge
a half-arrow pointing toward the resistor. Then linear resistance parameters will be positive, and nonlinear relations will fall in the first and third quadrants of the e–f plane.
Since linear models are of great usefulness in certain fields (vibrations and electric circuits, for example), the linear versions of resistance relations in various energy domains are shown in Table 3.1 with the same notation employed in Chapter 2. The units of the linear resistance parameter are simply the units of effort divided by the units of flow. The units displayed in Table 3.1 are worth studying, since many of them may not be familiar. The only resistance unit dignified with its own name is the electrical ohm.
Next consider a 1-port device in which a static constitutive relation exists between an effort and adisplacement. Such a device stores and gives up energy without loss. In bond graph terminology, an element that relatese toq is called a 1-port capacitor orcompliance. In physical terms, a capacitor is an idealiza- tion of such devices as springs, torsion bars, electrical capacitors, gravity tanks, and hydraulic accumulators. The bond graph symbol, the defining constitutive relation, and some physical examples are shown in Figure 3.2.
As with the 1-port resistor, there are idealized linear compliance elements as well as nonlinear ones. In Figure 3.2b, a general nonlinear constitutive e, q relationship is shown. If the element can be assumed linear, then the e versusq curve will be a straight line and a compliance parameter can be defined such that e =q/C. Note that it is customary to define the linear compliance relationship using the inverse of the slope of the e versus q curve. The reason for this will become clear when more physical elements are presented in the next chapter.
For the linear case, it is customary to indicate the compliance parameter on the bond graph as shown in Figure 3.2.
FIGURE 3.2. The 1-port capacitor. (a) Bond graph symbol; (b) defining relation;
(c) representation in several physical domains.
The electrical capacitor, of capacitance C farads, is a compliance element because its idealized behavior is
e= q C, where q =
i dt is the charge on the capacitor. This fits perfectly with our definition of a linear 1-port capacitor. The spring of stiffness, k, is a 1-port capacitor because it is characterized by
F =kx,
TABLE 3.2. The 1-Port Capacitor, e
f= ˙qC
General Linear SI Units for Linear
Relation Relation Capacitance Parameter
Generalized q=C(e) q=Ce C=q/e
e=−C1(q) e=q/C 1/C=e/q
Mechanical X=C(F ) X=CF C=m/N
Translation F=−C1(X) F =kX k=N/m
Mechanical θ=C(τ ) θ =Cτ C=rad/N-m
Rotation τ=−C1(θ) τ =kθ k=N-m/rad
Hydraulic V =C(P ) V =CP C=m5/N
Systems P=−C1(V ) P =V/C
Electrical q=C(e) q=Ce C=A-s/V
Systems e=−C1(q) e=q/C =farad (F)
where x =
V dt is the relative displacement across the spring. This definition fits the general definition of a 1-port capacitor and the specific definition of a linear 1-port C-element as indicated in Table 3.2. In this case, as indicated on the bond graph, the compliance parameter is C =1/k.
A water storage tank is discussed in the next chapter, and it is ideally a linear compliance element. The torsional spring of stiffness, kτ, N-m/rad, is a linear 1-port compliance for the identical reason as in the linear spring. The compliance parameter for the torsional spring is 1/kτ, as indicated on the bond graph of Figure 3.2.
An air bladder is also discussed in the next chapter. There are circumstances in which it may be modeled as a linear C-element, at least for small pressure excur- sions. However, in general, compressing air is a nonlinear process. If the process was isentropic, then the behavior of the air bladder could be characterized by
P = P0V0γ
Vγ ,
where V = ∫Qdt, P0, V0 are initial pressure and volume in the bladder, and γ is the ratio of specific heats for air. This behavior is nonlinear, but still fits our general definition of a 1-port capacitor as indicated in Table 3.2. Thus, for the case here, the air bladder is a compliance element, but it is not a linear one, and no compliance parameter can be identified nor indicated on the bond graph.
Note that when a sign convention similar to that used for the resistor, namely, C, is used for theC-element, thenef represents power flowingtothe capacitor and
E(t)= t
0 e(t)f (t)dt+E0 (3.1)
represents the energy stored in the capacitor at any time t. The energy stored initially att =0 (if any) is calledE0.
Since from Eq. (2.2b) the displacementq is defined so thatfdt≡dq, and the constitutive relation of a C-element implies that e is a function ofq, e=e(q), then Eq. (3.1) can be rewritten as
E(q)=
q
q0e(q)dq+E0 (3.2)
where E0 is the energy stored whenq =q0. Usually, it is convenient to define the energy stored to be zero when the effort is zero. Then, ifq0 is that value of q at whiche =0, andE0=0, Eq. (3.2) may be written as
E(q)=
q
q0
e(q)dq. (3.2a)
The operation indicated in Eq. (3.2a) may be interpreted graphically as shown in Figure 3.3. As q varies, the area under the curve ofe versus q varies, and this area is equal to the stored energy,E. Theconservation of energy for —C is almost obvious. Ifq goes fromq0 toqˆ as in Figure 3.3a, then energy is stored;
ifq then ever returns toq0, the shaded area disappears and all the stored energy disappears. The power flow into the port, which resulted in the storage of energy, reverses and power flows out of the port. During the process, no energy is lost;
in other words, energy is conserved.
Table 3.2 summarizes the relationships characterizing capacitors. The units for linear capacitance parameters are given, and again it may be noted that only the electrical unit is given a name, the farad. For linear mechanical systems, it is common to use thespring constant,k, rather than the compliance,C≡1/k, which is analogous to the electrical capacitance, C, and the parameter C in generalized variables. In mixed electrical–mechanical systems, one must simply be careful to note whether a numerical parameter corresponds toC or the inverse ofC in a bond graph. Once again, the reader is urged to study the units shown, since some units will probably be unfamiliar.
A second energy-storing 1-port arises if the momentumpis related by a static constitutive law to the flowf. Such an element is called aninertiain bond graph terminology. The bond graph symbol for an inertia, the constitutive relation, and several physical examples are shown in Figure 3.4. The inertia is used to model inductance effects in electrical systems, and mass or inertia effects in mechanical or fluid systems.
The 1-port inertia is characterized by an f, p relationship as indicated in Figure 3.4b. If the relationship is linear, then it will plot as a straight line and the constitutive relationship will have the form f = p/I, where I is the inertia parameter andp = ∫edt. Note that, as with the linear compliance element, it is customary to define the inertance parameter as the inverse of the slope of the linear relationship of f versus p. Shown in Figure 3.4 is an electrical inductor
FIGURE 3.3. Area interpretation of stored energy for a 1-port capacitor: (a) nonlinear case; (b) linear case.
with inductanceL, a massm, a section of fluid-filled pipe with fluid inertia,If, and a rotating disk with moment of inertia, J. These are all examples of linear 1-port inertia elements. The inductor is ideally represented by the constitutive relationship i=λ/L, where λ= ∫edt; the mass is represented by V =p/m, wherep = ∫Fdt; and the rotating disk has the ideal behavior,ω=pτ/J, where pτ = ∫τdt. All these elements fit exactly our definition of a 1-port inertia as shown in Table 3.3. The fluid inertia is also a linear 1-port inertia, which is covered thoroughly in Chapter 4.
Using the sign convention I, the power flowing into the inertia is given by the expression in Eq. (3.1). In the present case, Eq. (2.2a) allows us to write
FIGURE 3.4. The 1-port inertia: (a) bond graph symbol; (b) defining relation; (c) rep- resentation in several physical domains.
edt≡dp, and iff =f (p), then Eq. (3.1) can be written thus:
E(p)=
p
p0
f (p)dp+E0. (3.3) If the energy is defined to vanish when f vanishes and if p0 corresponds to that point in the plot off versusp at whichf =0, then
E(p)= p
p0
f (p)dp. (3.3a)
The similarities between Eqs. (3.2) and (3.3) should be noted. Often the energy associated with a capacitor is calledpotential energy, whereas the energy associ- ated with an inertia is calledkinetic energy. These names are applied primarily to
FIGURE 3.5. Area interpretation of stored energy for 1-port inertia. (a) Nonlinear case;
(b) linear case.
mechanical systems. In electrical systems, the corresponding two forms of stored energy are sometimes calledelectric andmagnetic energy.
As in the case of the capacitor, if the constitutive relation of the inertia is plotted, then there is an area interpretation of the stored energy. This interpretation is shown in Figure 3.5, and, again, you should be able to demonstrate that any energy stored in an I-element can be recovered without loss.
Table 3.3 shows the constitutive relations for inertias and gives units for iner- tance parameters for the linear case. Since most engineering work is accomplished successfully using Newton’s law rather than the postulates of relativity, the rela- tion between velocity and momentum is linear and the mass or moment of inertia is the inertia parameter. Although it is common to think of mass as a ratio of force to acceleration from the equation
F =ma, a≡ ˙V , (3.4)
TABLE 3.3. The 1-Port Inertia,e= ˙p
f I
General Linear SI Units for Linear
Relation Relation Inertance Parameter
Generalized p=I(f ) p=If I=p/f
variables f =−11(p) f =p/I 1/I =f/p
Mechanical p=I(V ) p=mV m=N-s2/m=kg
translation V =−11(p) V =p/m
Mechanical pτ =I(ω) pτ =J ω J=N-m-s2=kg-m2
rotation ω=−11(pτ) tω=pτ/J
Hydraulic pp=I(Q) pp=IQ I=N-s2/m5
systems Q=−11(pp) Q=pp/I
Electrical λ=I(i) λ=Li L=V-s/A
systems i=−11(λ) i=λ/L =henrys (H)
the table gives the fundamental definition of a mass according to
p≡mV (3.5)
with
p˙ ≡F. (3.6)
Clearly, when Eq. (3.5) is differentiated with respect to time, and Eq. (3.6) is used, then Eq. (3.4) can be derived. If, on the other hand, Eq. (3.5) is replaced with a nonlinear relation,
p=I(V )= mV
(1−V2/c2)1/2, (3.7) wherem is the rest mass andc is the velocity of light, then Eqs. (3.5) and (3.6) hold for the special theory of relativity. See Reference [1, p. 19], for example.
Thus, there is some justification for the general constitutive relations given for mechanical systems, even though engineering is overwhelmingly concerned with the linear case. For electrical systems, however, the relation between the flux linkage variable (the time integral of the voltage) and the current in an inductor is nonlinear in typical cases. The use of the linear parameterLis then the result of a modeling decision. It is more satisfactory to generalizeλ=Litoλ=I(i) withλ˙ =ethan to try to generalizee=L di/dtto the nonlinear case.
As an aid in remembering the three 1-port relationships, the tetrahedron of state introduced in Figure 2.2 may be used. See Figure 3.6. We now know something about five of the six edges of the tetrahedron. The sixth edge, which stretches
FIGURE 3.6. The three 1-ports placed on the tetrahedron of state according to the vari- ables to which they relate.
between the vertices representing p and q, is hidden from view in Figure 3.6.
This is just as well, since no basic element will relate p andq.∗
Finally, two useful and rather simple 1-ports must be defined—theeffort source and theflow source: the 1-port sources are idealized versions of voltage supplies, pressure sources, vibration shakers, constant-flow systems, and the like. In each case, an effort or flow is either maintained reasonably constant, independent of the power supplied or absorbed by the source, or constrained to be some particular function of time. As an example of a constant-effort source, consider the gravity force on a mass. Near the surface of the earth, this force is essentially independent of the velocity of the mass. As an example of a time-varying source, the electrical wall outlet will serve. The wall outlet enforces a sinusoidal voltage across the power cord wires of most small appliances. Over a reasonable range of currents, the voltage is independent of fluctuations in the current. Of course, the voltage is actually affected by large currents, and a fuse will blow to protect the circuits if very large currents build up, but this simply means that the real outlet is not modeled exactly by an ideal source of effort. Real sources can be modeled as combinations of effort and flow sources combined with other bond graph elements.
Table 3.4 presents bond graph symbols and the constitutive relations for sources. In this table, physical names are given for the respective energy domains.
There are sources of velocity,SV, sources of force,SF, sources of pressure,SP, and so forth. In general, it is best to use the generalized name for the effort
∗One can, in fact, define an element corresponding to the hidden edge, the “memristor.” While interesting and occasionally useful, memristors can be represented in terms of other elements to be introduced later, so the memristor will not be considered to be a basic element. See G. F. Oster and D. M. Auslander, “The memristor: a new bond graph element,”Trans. ASME, J. Dynamic Systems, Measurement, and Control,94, Ser. G, no. 3, pp. 249–252 (Sept. 1972).
TABLE 3.4. The 1-Port Source Elements
Bond Graph Symbol Defining Relation
Generalized variables Se e(t) given,f(t) arbitrary
Sf f(t) given,e(t) arbitrary
Mechanical translation SF F(t) given,V(t) arbitrary
SV V(t) given,F(t) arbitrary
Mechanical rotation Sτ τ(t) given,ω(t) arbitrary
Sω ω(t) given,τ(t) arbitrary
Hydraulic systems SP P(t) given,Q(t) arbitrary
SQ Q(t) given,P(t) arbitrary
Electrical systems Se e(t) given,i(t) arbitrary
Si i(t) given,e(t) arbitrary
or flow source, Se or Sf, and denote next to the generalized name the specific energy domain being represented. This is done by example in most of the fol- lowing chapters. It should be further noted that the symbolsSE andSF for effort and flow source are often used in computer programs.
Typically, source elements are thought of as supplying power to a system. This accounts for the sign-convention half-arrow shown, which implies that whene(t) f(t) is positive, power flows from the source to whatever system is connected to the source. Later we will encounter modeling situations in which it is convenient to have the half-arrow pointing in the opposite direction to the directions shown in Table 3.4.
Since a source maintains one of the power variables constant or a specified function of time no matter how large the other variable may be, a source can supply an indefinitely large amount of power. This is, of course, not a realistic assumption, and real devices are not really ideal sources even though they may be modeled approximately by sources. As an example, consider the problem of predicting the current flowing from a 12-V automotive battery into a variable resistor connected to the battery. Figure 3.7 shows a circuit diagram, a bond graph, and a plot of voltage versus current. This is a static system operating at points at which the source characteristic intersects the resistor characteristics. For small currents (or high values of the resistanceR), the battery is almost a constant- voltage source. When the resistance is lowered toward zero, the predicted current approaches infinity. Actually, when the current gets large, the internal resistance in the battery reduces the voltage below the nominal 12 V. In fact, if the resistance approaches zero, as it will when a shorting bar is put across the battery terminals, the battery current will approach a finite value, labeled “short-circuit current” in Figure 3.7c. When more basic multiports have been defined, it will be possible to model the battery with an ideal source and a resistor in such a way that the actual characteristics in Figure 3.7 will be reproduced by the model. For now, we simply note that ideal sources are useful in modeling real devices but should