A linear system satisfies the properties of superposition and homogeneity.

A system characterized by the relation is not linear, because the superposition

property is not satisfied. A system represented by the relation y = mx + b is not linear, because it does not satisfy the homogeneity property.

However, this second system may be considered linear about an operating point for small

changes and . When and , we have

Therefore, , which satisfies the necessary conditions.

The linearity of many mechanical and electrical elements can be assumed over a reasonably large range of the variables. This is not usually the case for thermal and fluid elements, which are more frequently nonlinear in character. Fortunately, however, one can often linearize nonlinear elements assuming small-signal conditions. This is the normal approach used to obtain a linear equivalent circuit for electronic circuits and transistors. Consider a general element with an excitation (through-) variable x(t) and a response (across-) variable y(t). Several examples of dynamic system variables are given in Table 3.1. The relationship of the two variables is written as

(3.6)

where indicates y(t) is a function of x(t). The normal operating point is designated

by . Because the curve (function) is continuous over the range of interest, a Taylor seriesexpansion about the operating point may be utilized. Then we have

(3.7)

The slope at the operating point,

is a good approximation to the curve over a small range of , the deviation from the operating point. Then, as a reasonable approximation, Equation (3.7) becomes

(3.8)

where m is the slope at the operating point. Finally, Equation (3.8) can be rewritten as the linear equation

(3.9)

Consider the case of a mass, M, sitting on a nonlinear spring, as shown in Figure 5(a).

The normal operating point is the equilibrium position that occurs when the spring force balances the gravitational force Mg, where g is the gravitational constant. Thus, we obtain

, as shown. For the nonlinear spring with , the equilibrium position is

. The linear model for small deviation is

Не знайшли потрібну інформацію? Скористайтесь пошуком google:

Генерація сторінки за: 0.005 сек.