Page 101 - Fluid Power Engineering

Hydraulic Transmission Lines 75

Considering the effect of oil compressibility in the last portion,
the following relations can be deduced:

dP
Q − Q = C L (3.16)
2 L
dt
P = P
2 L (3.17)
Applying Laplace transform to these equations, then, after rear-
rangement, the following equation is obtained:
⎡ Ps ()⎤ ⎡ 1 0⎤⎡ Ps ()⎤ ⎡ Ps ()⎤
L
L
2
⎢ Qs () ⎥ = ⎢ Cs 1 ⎥⎢ Qs () ⎥ ⎥ = C ⎢ Qs () ⎥ (3.18)
⎣ 2 ⎦ ⎣ ⎦⎣ L ⎦ ⎣ L ⎦
2
where C = Whole line capacitance = πDL 4/ B, m /Pa
3
C = Capacitance matrix
The transfer matrix relating the line parameters P , Q , P , and Q can be
o o L L
deduced by eliminating the assumed internal variables, P , P , Q , and Q .
1 2 1 2
⎡ Ps()⎤ ⎡ Ps ()⎤ ⎡ Ps ()⎤ ⎡ Ps ()⎤
⎢ o ⎥ = R ⎢ 1 ⎥ = RI ⎢ 2 ⎥ = RIC ⎢ L ⎥
⎣ Qs() ⎦ ⎣ Qs () ⎦ ⎣ Qs () ⎦ ⎣ Qs () ⎦
o
2
1
L
or
⎡ Ps ()⎤ ⎡ ICs + RCs + 1 Is R P (s)⎡ s ⎤
+ ⎤
2
L
⎢ Qs () ⎥ = ⎢ Cs 1 ⎥⎢ Qs () ⎥ (3.19)
o
⎣ o ⎦ ⎣ ⎦⎣ L ⎦
This equation defines the relation between the pressures and flow
rates at both of the line extremities in the transient conditions, assum-
ing a single oil lump.
Example 3.1 Find the transfer function relating the pressures and flow rates
at the two extremities of a closed end line.

For a closed end line, Q = 0.
L
P = ( ICs + RCs + )1 P
2
o L
Q = CsP
o L
or
P 1 P 1
L = and L =
2
P o ICs + RCs + 1 Q o Cs