Page 612 - The Mechatronics Handbook

0066_Frame_C20 Page 82 Wednesday, January 9, 2002 5:49 PM

The open loop transfer function is

2 2
⋅
⋅
V RET s n s A 1
------------------------------------ ---------------------------------------- --
G OL = ----------- = G C K OLV K TP 2 ⋅ ⋅ (20.53)
2
e s + 2zs n s + s n s + 2z A s A s + s A 2 s
2
Supposing G C to be constant, the static gain in open loop is
⋅
K 0 = G C K OLV K TP = G C K S ′K Q A C K TP (20.54)
⋅
------------------------------------
2
A C – K PQ g
The closed loop transfer function is

2 2
x
G CL = -------- = ----------------------------------------------------------------------------------------------------------------------------------------
K R G C K OLV s n s A
(
2
2
2
2
2
2
X set ( s + 2zs n s + s n ) s + 2z A s A s + s A )s + G C K OLV s n s A K TP
1
= ------------------------------------------------------------------------------ (20.55)
2
5
4
a 5 s + a 4 s + a 3 s + a 2 s + a 1 s + 1
3
where
z

2
1
a 5 = ------------------, a 4 = ------------------ ----- + ------ 
z A
2 2  
s n s A K 0 s n s A
s n s A K 0

z A


2
1
z
1
1
1
------------ +
a 3 = ----- ----- + 4z A z ------ , a 2 = ----- ----- + ------ , a 1 = -----



2
s A
K 0 s n 2 s n s A K 0 s n s A K 0
The transfer function between output and disturbance, said dynamic compliance, is
2
2
(
x K OLF ts 1–( ) s + 2zs n s + s n )s A 2
G FCL = ---- = – ---------------------------------------------------------------------------------------------------------------------------------------- (20.56)
2
2
( s + 2zs n s + s n ) s + 2z A s A s + s A )s + 2 2
2
(
2
F e G C K OLV s n s A K TP
Static compliance is
x K OLF K PQ 1
---- = --------------------------- = ------------------------------------ = ----------------------------------- (20.57)
G C K OLV K TP G C K S ′K Q A C K TP G C K S ′K P A C K TP
F e s=0
having put
Pa
∂
K PQ = K Q K P = --------- pressure gain ------
------,
P L
K P ∂ A V A , P m 2
V0 L0
©2002 CRC Press LLC