MULTIPORT SYSTEMS AND BOND GRAPHS
2.4 INPUTS, OUTPUTS, AND SIGNALS
FIGURE 2.8. Experimental testing of a dc motor: (a) sketch of the test apparatus;
(b) block diagram showing input and output signal flow; (c) causal strokes added to multiport representation.
The meaning of causal strokes is summarized in Figure 2.9, in which both bond graphs and block diagrams are shown. Note that the half-arrow sign convention for power flow and the causal stroke are completely independent. Thus, usingA and B to stand for subsystems as in Figure 2.9, all the following combinations of sign convention and causal strokes are possible: A B, A B, A B, A B. The study of input– outputcausality, which is a uniquely useful feature of bond graphs, will be dealt with at length in succeeding chapters.
FIGURE 2.9. The meaning of causal strokes: (a) effort is output ofA, input toB; flow is output ofB, input toA; (b) effort is output ofB, input toA; flow is output ofA, input toB.
Finally, we come to the question of pure signal flow, or the transfer of informa- tion with negligible power flow, which we already encountered in the example of Figure 2.7. Multiports in principle all transmit finite power when interconnected.
This is correlated with the fact that both an effort and a flow variable exist when multiports are coupled. Thus, systems are interconnected by the matching of a pair of signals representing the power variables.
In many cases, however, systems are so designed that only one of the power variables is important, that is, so that a single signal is transmitted between two subsystems. For example, an electronic amplifier may be designed so that the voltage from a circuit influences the amplifier, but the current drawn by the amplifier has virtually no effect on the circuit. Essentially, the amplifier reacts to a voltage but extracts negligible power in doing so, as compared with the rest of the power levels in the circuit. No information can really be transmitted at exactly zero power, but, practically speaking, information can be transmitted at power levels that are negligible, as compared with other system power levels.
Every instrument is designed to extract information about some system variable without seriously disturbing the system to which the instrument is attached. An ideal ammeter indicates current but introduces no voltage drop, an ideal voltmeter reads a voltage while passing no current, an ideal pressure gage reads pressure with no flow, an ideal tachometer reads angular speed with no added torque, and the like. When an instrument reads an effort or flow variable, but with negligible power, there is a signal connection between subsystems without the back effect associated with finite power interaction.
The block diagrams of control engineering or the signal flow graphs that were developed first for electrical systems ideally show signal coupling. As Figure 2.8b shows, when multiports are considered, power interactions require a pair of bilat- erally oriented signals. The bond graph, in which each bond implies the existence of both an effort and a flow signal, is a more efficient way of describing mul- tiports than are block diagrams or signal flow graphs. Yet when the system is
dominated by signal interactions due to the presence of instruments, isolating amplifiers, and the like, then either an effort or a flow signal may be suppressed at many interconnection points. In such a case, a bond degenerates to a single signal and may be shown as anactive bond. The notation for an active bond is identical to that for a signal in a block diagram; for example,A−→e B indicates that the effort,e, is determined by subsystemAand is an input to subsystemB. Normally, this situation would be indicated by A e
f B, in which the flow, f, is determined byB and is an input toA. Wheneis shown as a signal (by means of the full arrow on the bond) or, in other words, an activated bond, the implication is that the flow,f, has a negligible effect onA.
When automatic control systems are added to physical systems, the control systems usually receive signals by means of nearly ideal instruments and affect the systems through nearly ideal amplifiers. The use of active bonds for such cases simplifies the analysis of the systems. Notice that in using bond graphs, one always assumes that multiports are coupled with both forward and back- ward effects unless a specific modeling decision has been made that a back effect is negligible and an active bond is specified. As we will see in Chapter 6, active bonds and block diagrams are often useful in studying control systems but for systems that involve real power interactions, the bond graph provides an ideal means for assuring the action–reaction effects on power bonds are properly represented.
In the following chapter we begin the detailed modeling of subsystems by con- sidering a basic idealized set of multiports that can be assembled to model the pertinent physical effects in a subsystem. At this detailed level, physical param- eters must be estimated and the rules of causality among ideal multiports must be discovered and obeyed in assembling the subsystem model from elemental multiports. As this process goes on, the notation and concepts briefly introduced in this chapter will become more familiar and useful.
PROBLEMS
2-1. Construct four tetrahedra of state similar to that shown in Figure 2.2 for the following four physical domains: mechanical translation, mechanical rotation, hydraulic systems, electrical systems. Replace e, f, p, and q with their physical counterpart variables, and list the dimensions of each variable.
2-2. For each multiport in Figure 2.1 construct a word bond graph similar to that shown in Figure 2.3. Construct several systems by bonding several multiports together.
2-3. Suppose a pump was tested by running it at various speeds and measuring the volume flow rate and torque for various pressures at the pump outlet.
Draw a schematic diagram, block diagram, and bond graph for the pump test analogous to those shown in Figure 2.8 for an electric motor test.
2-4. If the system of Figure 2.5 had the causality show how the signals flow by using a block diagram of the type used in Figure 2.8b for each multiport.
See also Figure 2.9.
2-5. Apply causal strokes in an arbitrary manner to each bond in the bond graph of Figure 2.6. Construct an equivalent block diagram for this system using one block for each multiport as in Figure 2.8b. Indicate the signal flow directions that correspond to your causal marks as in Figure 2.9.
2-6. Repeat Problem 2-5 for the system of Figure 2.7. (Note that active bonds act just like one-way signal flows in a block diagram, but that normal bonds each result in two signal flows fore andf.)
2-7. Consider the system of Problem 2-4. Identify the system input variables that come from the environment of the system, given the causal stroke pattern indicated. What variables are indicated as system outputs (and hence inputs to the environment)?
2-8. How long would a 100-W light bulb have to burn to use up the same energy that would be required to raise a 10-kg mass 30 m up in the earth’s gravity field?
2-9. Represent an electric drill as a multiport. Consider the switch position influence as occurring on an active bond. Apply causal strokes to your bond graph, assuming that the drill is plugged into a 100-V outlet and that the torque is determined by the material being drilled. Show a block diagram for the drill corresponding to your choice of causality at the ports.
2-10. If a positive-displacement hydraulic pump is 100% efficient (so that the mechanical power is always instantaneously equal to the hydraulic power) and if a torque of 5 N-m produces a pressure of 7.0 MPa, what is the relationship between volume flow and angular speed? (7.0 MPa=7.0× 106 N/m2.)
2-11. The slider-crank mechanism is the fundamental kinematic device in vir- tually all internal combustion engines. This device relates the rotational motion of the crankshaft to the reciprocating motion of the piston. In its most idealized representation, the slider-crank is massless, friction- less, and constructed from rigid components. Under these assumptions, the device is power conserving, in that τ ω=F ν, where τ is the torque
on the crankshaft, F is the force on the end of the connecting rod, ω is the angular velocity of the crank, andν is the velocity of the rod end. If we can derive howν andωare related, then we automatically know how F andτ are related.
l
θ, θ = ω
x F
v
τ α
(a) Slider-crank device
Slider-crank device (b) Word bond graph ωτ
F v .
As a word bond graph, the slider-crank will be represented as indicated in the figure. We will soon learn that this device is a modulated transformer.
Derive the relationship betweenν andω. Here is some help:
x=Rcosθ+lcosα lsinα=Rsinθ
Solve the second equation for sin α and then use cosα=
1−sin2α.
Substitute into the first equation. Then differentiate the result to relate
˙
x= −v to θ˙=ω. If you complete these steps, you will have derived ν=m(θ)ω, wherem(θ) is a function of the crank angle. Since this device is power conserving, we immediately know thatτ =m(θ)F.
Now try to derive the relationship between τ and F by using force and moment equilibrium conditions. You will find this far more difficult than deriving the velocity–angular velocity relationship.
2-12. The hydraulic system from Problem 1-9 has the word bond graph shown in the following figure. Causality has been assigned to the bonds. Identify the inputs and outputs for each element.
Tube Ps
Qa
Pc QI
Fm vm
Fs vs Fd vd Qs
Accumulator
Supply
pressure Cylinder Mass
Spring
Damper