Modern Control Engineering

128 Chapter 4 / Mathematical Modeling of Fluid Systems and Thermal Systems

x= 2 x 1

x= x 1

x= 0

x=–x 1

0

q

P

x=– 2 x 1

Figure 4–18

Characteristic curves

of the linearized

hydraulic

servomotor.

Equation (4–27) is a linearized mathematical model of the spool valve near the origin

Note that the region near the origin is most important in this

kind of system, because the system operation usually occurs near this point.

Figure 4–18 shows this linearized relationship among q, x,and The straight lines

shown are the characteristic curves of the linearized hydraulic servomotor. This family

of curves consists of equidistant parallel straight lines, parametrized by x.

In the present analysis we assume that the load reactive forces are small, so that the

leakage flow rate and oil compressibility can be ignored.

Referring to Figure 4–17(a), we see that the rate of flow of oil qtimesdtis equal to

the power-piston displacement dytimes the piston area Atimes the density of oil r.

Thus, we obtain

Notice that for a given flow rate qthe larger the piston area Ais, the lower will be the

velocitydydt. Hence, if the piston area Ais made smaller, the other variables re-

maining constant, the velocity dydtwill become higher. Also, an increased flow rate q

will cause an increased velocity of the power piston and will make the response time

shorter.

Equation (4–27) can now be written as

The force developed by the power piston is equal to the pressure difference times

the piston area Aor

=

A

K 2

aK 1 x-Ar

dy

dt

b

Force developed by the power piston=A ¢P

¢P

¢P=

1

K 2

aK 1 x-Ar

dy

dt

b

Ar dy=q dt

¢P.

(x–=0,¢p– =0, q–=0.)

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