Page 106 - Fluid Power Engineering

80 Cha pte r T h ree

The Two-Lump Model
Assuming more than one lump, the hydraulic resistance, inertia, and
capacitance of each lump in an n-lump model are given by the fol-
lowing expressions:

Lump resistance R = 128μ( Ln)/ (3B.5)
n π D 4
4ρ( Ln)/
Lump inertia I = (3B.6)
n π D 2
π DL n ⎛ 1( / ) 5 D ⎞
2
Lump capacitance C = ⎜ + ⎟ (3B.7)
n
4 ⎝ B 4 Eh⎠
The following are the equations describing the two-lump model:

P − P = R Q o (3B.8)
11
o
2
dQ
P − P = I o (3B.9)
11 1 L 2
dt
dP
Q − Q = C 1 L (3B.10)
o 1 L 2 dt
P − P = R Q 1 L (3B.11)
1
21
L
2
dQ
P − P = I 1 L (3B.12)
21 L 2 dt
dP
Q − Q = C 2 L (3B.13)
L
1
L
dt
The transfer matrix of the two-lump model can be deduced as
follows:
2
⎡ Ps ()⎤ ⎡ Ps()⎤ ⎡ ICs + R Cs + 1 I s R ⎤ ⎡Ps() ⎤
+
2
⎢ o ⎥ = [R I C ] ⎢ L ⎥ = ⎢ ⎢ 2 2 2 2 2 2 ⎥ ⎢ L ⎥
2
2
2 2
L
L
o
2
⎣ Qs () ⎦ ⎣ Qs() ⎦ ⎣ Cs 1 ⎦ ⎣ Qs() ⎦
⎡a a ⎤⎡Ps () ⎤
= ⎢ 11 12 ⎥⎢ L ⎥ (3B.14)
a
⎣ 21 a 22 ⎦⎣ Qs () ⎦
L
FIGURE 3B.2 The two-lump model.