BASIC BOND GRAPH ELEMENTS
3.5 CAUSALITY AND BLOCK DIAGRAMS
permissible causality for a 0-junction is shown in the third row of Table 3.7.
Here the causal stroke on the end of bond 1 nearest the 0 indicates that e1 is an input to the junction and that all other bonds must have causal strokes at the end away from the 0. To interpret the diagram another way, the flows on bonds 2 and 3 are inputs to the 0-junction. These considerations are also expressed by the causal equations shown in Table 3.7. For a 3-port 0-junction, then, there are only three different permissible causalities in which each of the three bonds in succession plays the role assigned to bond 1 in the example shown in the table.
For an n-port 0-junction this description of the constraints on causality is still valid, and there are exactlyn different permissible causal assignments.
For a 1-junction the same considerations apply as for a 0-junction except that the roles of the efforts and flows are interchanged. Table 3.5 indicates that flows on all the bonds are equal and the efforts sum to zero. Thus, if the flow on any single bond is an input to the 1-junction, the flows on all other bonds are determined and must be considered outputs of the junction. Alternatively, when the efforts on all bonds except one are inputs to the 1-junction, the effort on the remaining bond is determined and must be an output of the junction. A typical permissible causality is shown in the fourth row of Table 3.7. In this example, bond 1 plays the special role of determining the common flow at the junction, and the remaining bonds supply effort inputs that suffice to determine the effort on bond 1. Clearly, there are three permissible causalities for a 3-port 1-junction, and there arenpermissible different causal assignments for ann-port 1-junction.
Although the causal considerations have been stated for all the basic multiports defined so far (summarized in Tables 3.6 and 3.7), it can hardly be clear as to what all the implications of causality are. The study of causality is very important, and bond graphs are uniquely suited to this study. However, only when some real system models have been assembled is it clear why causal information is so important. In the next chapter, system models are built up using the basic multiports just discussed. Using the rules of causality, it is then possible to predict many important features of these systems even before the constitutive laws for all the elements have been decided upon. For instance, it will be possible to predict the mathematical order of the system model before any equations are written and before a decision has been made about whether the model should be linear or nonlinear. In addition, causal considerations will prove invaluable in writing a set of state equations or in setting up computational program for system simulation.
When causal strokes are added to a bond graph, one may represent the infor- mation in the bond graph by a block diagram. For example, the block-diagram versions of the causal forms for the R,C, andI 1-ports shown in Table 3.6 are given in Figure 3.13. Similarly, block diagrams for 2-ports and 3-ports corre- sponding to entries in Tables 3.7 are shown in Figures 3.14 and 3.15. It should be possible for you to correlate the signal flow paths in the block diagrams with the equations in the tables and with the bond graph representation. Note that when one rigorously maintains the spatial arrangements with efforts above and to the left of bonds and flows below and to the right, the block diagrams have fixed patterns.
It may also be seen that block diagrams are more complex graphically than bond graphs because a single bond implies two signal flows on a block diagram.
A single causal mark indicates the signal flow directions on two (effort and flow) signals. Also, a single sign-convention half-arrow on a bond is replaced on the equivalent block diagram by a plus or minus sign at summation points.
FIGURE 3.13. Block diagrams for 1-ports.
FIGURE 3.14. Block diagrams for 2-ports.
FIGURE 3.15. Block diagrams for 3-ports.
Initially, block diagrams may be easier to understand than bond graphs because they contain redundant information. For systems with some complexity, however, block diagrams rapidly become so complicated that the conciseness of equiv- alent bond graphs is an advantage. For example, Figure 3.16 shows a block diagram equivalent to a bond graph model of the automotive drive train system of Figure 2.7. Note that the sign-convention half-arrows have yet to be put on
FIGURE 3.16. Interconnected drive train model. (a) Bond graph; (b) block diagram.
the bond graph, and the corresponding+and−signs do not appear in the block diagram near the circles representing signal summation.
After procedures for constructing bond graph models and adding causal strokes to them have been discussed in the following chapters, one option is to construct a block diagram from the bond graph. There are computer programs that accept block diagram models for the analysis and simulation of systems so there are some reasons for converting a bond graph model of a system to a block diagram.
In the following chapters we will focus more on deriving state equations for bond graph models but the ability to show a block diagram for a bond graph is often useful in explaining the assumptions behind a model to those unfamiliar with bond graph techniques. Also, particularly in the case of linearized system models, block diagrams are commonly used in the design of control systems.
It is also true that a detailed block diagram for a complex system model may
become so complex as to be virtually useless in understanding the assumptions that have been made.
REFERENCE
[1] S. H. Crandall, D. C. Karnopp, E. F. Kurtz, and D. C. Pridmore- Brown,Dynamics of Mechanical and Electromechanical Systems, New York:
McGraw-Hill, 1968.
PROBLEMS
3-1. A nonlinear dashpot has as its constitutive relation the “absquare law,”
F =AV|V|,
where F and V are the force and velocity across the dashpot and A is a constant. Plot this relation in a sketch, and indicate the bond graph sign convention implied ifA >0 and which causality the equation implies in the form given. Try to invert the constitutive law to yield the velocity as a function of the force.
3-2. A fluid of mass densityρ is pumped into an open-topped tank of areaA. If P is the pressure at the tank bottom andQ the volume flow rate, the tank is approximately a —C for slow changes in the volume of fluid stored. Is the —C a linear element in this case, and, if so, what is the capacitance?
Hint: It is useful to compute the height of fluid, h, as a function of the total volume of fluid as an intermediate step.
3-3. Reconsider Problem 3-2, but let the tank have sloping walls as shown.
What does this do to the constitutive law for the device?
3-4. Consider a uniform cantilever beam of length L, elastic modulus E, and area moment of inertiaI. If a forceF is applied at the tip of the beam, it will deflect. If the beam is supposed to be massless, decide what type of 1-port it is and compute its constitutive law.
3-5. Linear electrical inductors can be characterized by a law relating the volt- agee to the rate of change of current:
Ldi dt =e.
Convert this to a law relating the flow i to the momentum λ, which in this case is the time integral of e and is called the flux linkage. Plot a linear flow–momentum constitutive law, and show on the plot where the inductanceLappears. Now sketch a nonlinear flow–momentum law.
Convert the nonlinear law back to a relation between e and di/dt if this is possible.
3-6. A rigid pipe filled with incompressible fluid of mass densityρ has length Land cross-sectional areaA.
IfP1andP2are the pressures at the ends of the pipe and the volume flow rate isQ2, convince yourself that the bond graph shown correctly repre- sents the pipe in the absence of friction. Show that the correct constitutive law relating pressure momentum and volume flow is
pP2= t
P2dt =
ρL A
Q2
by writing Newton’s law for the slug of fluid in the pipe. (It should come as a surprise that small-area tubes have a lot of inertia whenP,Qvariables are used.)
3-7. An accumulator consists of a heavy piston in a cylinder. If the pressure is determined primarily by the weight of the piston, sketch the constitutive law for this 1-port device.
3-8. A flywheel is a uniform disk of radius R and thickness t and is made of a material of mass density ρ. Write the constitutive law for the 1- port representing the flywheel in its flow–momentum form. Evaluate the inertance parameter for a steel disk 1 in. in thickness and 10 in. in diameter.
3-9. Assume any needed dimensions for the hydraulic ram of Figure 3.8e, and write the constitutive law for this 2-port.
3-10. Repeat Problem 3-9 for the devices of Figure 3.8b andc.
3-11. In Figure 3.9c assume that the rotor has moment of inertia J and spins at a high angular rate . If the rotor is centered on an axle of length L, relate F1, V1,F2, V2, and thus demonstrate that the device is indeed approximately a —GY—.
3-12. Draw block diagrams for the following bond graphs, assuming all 1-ports are linear:
3-13. Draw a block diagram for the oscillator using the bond graph shown.
3-14. Consider an ideal rack and pinion with no friction losses:
If the pinion has radiusr and torque and speedτ and ω, and if the rack has velocity V and force F, what type of element would represent the device? Write the appropriate constitutive laws.
3-15. An electrodynamic loudspeaker is driven by a voice coil transducer described by Eq. (3.12). Show that if only the mass of the speaker cone is considered, then at the electrical terminals the device will act like a capacitor. The bond graph identity
e i
˙T
GY˙ F
V I = e
i C
is to be verified using the element constitutive equations directly.
3-16. An air spring is idealized as a piston in a cylinder with no leakage and no heat transfer through the cylinder walls. The process the air undergoes is assumed to be isentropic, such that
P Vγ =P0V0γ, whereV is the instantaneous volume,
V =V0−Apx,
V0 is the volume whenx =0, P0is atmospheric pressure, γ is the ratio of specific heats (≈1.4 for air), andP is the absolute cylinder pressure.
Derive the nonlinear constitutive relationship forF =F(x).
AP x F
Air
3-17. Three springs are used in a parallel configuration as shown. Sketch the constitutive behavior as seen from theF–v port. Show both compression and tension.
F
k
k x v
k δ1
δ2
3-18. Friction is always dissipative and is therefore a resistance in a bond graph model. Its constitutive behavior is sketched in an ideal sense.
(a) Discuss the only possible causality for this element.
(b) Discuss the problem of using this device near ν = 0. Propose a change to the constitutive behavior that will retain its fundamental character but avoid the problem near ν =0.
v
v v
R
Ff
Ff
Ff