Modern Control Engineering

Example Problems and Solutions 149

A–4–9. Derive the transfer function Z(s)/Y(s)of the hydraulic system shown in Figure 4–37. Assume that

the two dashpots in the system are identical ones except the piston shafts.

Solution.In deriving the equations for the system, we assume that force Fis applied at the right

end of the shaft causing displacement y. (All displacements y, w,andzare measured from re-

spective equilibrium positions when no force is applied at the right end of the shaft.) When force

Fis applied, pressure becomes higher than pressure Similarly,

For the force balance, we have the following equation:

(4–42)

Since

(4–43)

and

we have

Also, since

q 1 dt=A(dw-dz)r

we have

or

DefineA^2 Rr=B.(Bis the viscous-friction coefficient.) Then

(4–44)

Also, for the right-hand-side dashpot we have

Since

or

(4–45)

Substituting Equations (4–43) and (4–45) into Equation (4–42), we have

Taking the Laplace transform of this last equation, assuming zero initial condition, we obtain

k 2 Y(s)=Ak 2 +BsBW(s)+k 1 Z(s) (4–46)

k 2 y-k 2 w=k 1 z+Bw#

AAP 2 - Pœ 2 B=Bw#

w# =

q 2

Ar

=

AAP 2 - Pœ 2 B

A^2 Rr

q 2 =AP 2 - Pœ 2 BR, we obtain

q 2 dt=Ardw

w

• z

=

k 1

B

z

w# -z#=

k 1 z

A^2 Rr

q 1 =A(w# -z#)r

k 1 z=ARq 1

q 1 =

P 1 - Pœ 1

R

k 1 z=AAP 1 - P 1 œB

k 2 (y-w)=AAP 1 - Pœ 1 B+AAP 2 - Pœ 2 B

P 1 P 1 œ , or P 17 P 1 œ. P 27 P 2 œ.

R

F

R

k 1 k 2

P 1

q 1

Area=A

z

q 2

P (^19) w P 2 P (^29) w y
Figure 4–37
Hydraulic system.