If it were supposed that a given type of instability might occur in the system as a result of a change in a particular variable Хг, and that a design modification in another parameter X2 could correct the situation, then the following method of assessing this instability could be used:

(a) Set up a mathematical simulation of the system to include the full dependence of both the system variable Хг and the parameter X2. (Xt may be the power-to-flow ratio and the parameter X2 may be a given feedback coefficient.)

(b) Disturb the model by a “kick” in a significant variable (say, pressure or flow) and observe the dynamic results. Figure 2.25 shows what the results might indicate.

(c) Calculate a damping factor from successive peaks of the transient.

damping factor A = (x2 — Xj)-1 In(уг/у2) (2.8)

where Уі and are defined in Fig. 2.26.

(d) Vary sensitive parameters for their effect on the damping factor; this sensitivity study would include both the system variable Хг and the design parameter X2. Figure 2.27 shows the result of such a sensitivity study. [The damping factor is a system characteristic and does not depend on the variable used for its calculation (mass flow shown in Fig. 2.25).]

(e) Invoke the design parameter needed to achieve the damping factor required. In Fig. 2.27, damping factors of 2 and above are acceptable while

those below about 1.5 show poor damping, neutral stability without damp­ing, and finally instability. Thus, if (say the power-to-flow ratio) can be 1 in this system, then X2 (the feedback coefficient) must be designed to be at least B.

This method can achieve very rapid results if an analog computation is used. Digital methods, which may be more comprehensive, can then be used to check the result.