3.7.  A GENERALIZED APPROACH FOR MODELING—THE PRINCIPLES OF CONSERVATION AND ANALOGY

Because of the large variety of control-system components that occur in practice, a generalized approach is useful for obtaining their mathematical model. Therefore, rather than pursue the presentation of further specific control-system components, this section will provide a generalized approach to deriving mathematical models.

There are several general principles that can be useful in serving as guides. The most important are the principle of conservation and the concept of analogous circuits.

A.  Principle of Conservation

The principle of conservation is a very important guideline for the derivation of a mathematical model. A statement of this concept is that

In terms of rates, the principle of conservation is stated as follows:

Exactly what is being conserved depends on the application. However, this principle is usually used to establish a balance or inventory of mass, energy, momentum, or charge.

The principle of conservation has been used several times throughout this chapter. For example, let us reconsider the hydraulic motor and pump power transmission system illustrated in Figure 3.21 and the electric hot-water heating system of Figure 3.27. In the case of the hydraulic motor and pump power transmission system, Eq. (3.123) was derived. It related the volume of oil flowing from the pump Q_P to its distribution in flow through the motor, Q_m(t), leakage around the motor, Q_l, and compressibility flow (accumulation), Q_c(t). Therefore, this equation was an application of the principle of conservation, where Eq. (3.153) was modified to the form

Figure 3.28  Block diagram for the system shown in Figure 3.27.

Similarly, in the electric hot-water heating system problem, Eq. (3.144) was derived. Let us rearrange it into the following form in order to demonstrate the principle of conservation:

This equation states that the heat flow supplied by the heating element, Q_h(t), plus the heat flow carried in by cold water entering the tank, Q_i(t), is distributed as heat flow into storage in the water in the tank, Q_c(t), heat flow lost by hot water leaving the tank Q_o(t), and heat flow through the insulation, Q_l(t). Therefore, the principle of conservation was applied, and the form of the principle applied was similar Eq. (3.155).

B.  Circuit Concept and Analogy

An alternative viewpoint to the principle of conservation is the concept of an analogous circuit. The basis for applying the principle of analogy is that two different physical systems can be described by the same mathematical model. This permits a generalization of ideas specific to a particular field in order that a broader understanding of a variety of apparently unrelated situations can be achieved.

The analogy concept can best be understood by focusing attention on some of the devices that have been covered in this chapter. For example, let us look at the electrical circuit of the armature-controlled dc servomotor illustrated in Figure 3.11a and the hydraulic motor and pump illustrated in Figure 3.21. Equations (3.61) and (3.132) described their respective mathematical models. The direct analogy between Eqs. (3.61) and (3.132) is self-evident. Furthermore, the development of these two models follows a very similar process. It is important to note how their respective relations in both cases enabled one to relate the two coupled sets of physical variables: electrical and mechanical for the armature-controlled dc servomotor, and hydraulic and mechanical for the hydraulic motor and pump transmission system.

This concept of analogy can be extended to relate electrical, mechanical (linear motion), mechanical (rotational), thermal, and hydraulic systems. For purposes of comparison. Table 3.4 illustrates a brief table of analogous quantities in different physical systems. It is important to note that there are two possible analogies between mechanical, and electrical systems. If torque (or force) is chosen to be analogous to current, then the mechanical circuit and electrical circuits look alike with inertia being analogous to capacitance and a spring being analogous to inductance. This is the system illustrated in Table 3.4. Another approach could be to choose torque (or force) to be analogous to voltage. Then the mechanical and electrical circuits will be analogous, with inertia being analogous to inductance and a spring being analogous to capacitance: it makes no difference which viewpoint is taken.

By using the method of analogs, complex mechanical (or hydraulic, etc.) systems can be drawn as equivalent circuit diagrams, for which Kirchhoff’s voltage and current laws can be utilized to obtain the mathematical model of the system. As an example of this approach, let us reconsider the complex mechanical system of Figure 3.5, using Kirchhoff’s voltage and current laws. By using the analogs of Table 3.4, the mechanical network of Figure 3.5 is redrawn as an equivalent electrical circuit in Figure 3.29. The node equations for Figure 3.29 are written by inspection as

where

Figure 3.29  Electrical analog of Figure 3.5.

Equations (3.157) and (3.158) are analogous to Eqs. (3.27) and (3.28), respectively. Substituting the analogous quantities into Eqs (3.157) and (3.158), and taking the inverse Laplace transforms we obtain the following set of equations:

Observe that Eqs. (3.159) and (3.160) are identical to Eqs. (3.27) and (3.28), respectively.

Electrical analogies have the advantage that they can be set up very easily in the laboratory. For example, a change in a particular parameter cna be accomplished very easily in the electric circuit to determine its overall effects and the electric circuit can be appropriately adjusted for the desired response. Afterwards, the parameters in the mechanical (or hydraulic, or thermal, etc.) system can be adjusted by an analogous amount to obtain the same desired response.