Page 108 - Fluid Power Engineering

82 Cha pte r T h ree

and

⎡ 1 R ⎤ ⎡ 1 Is⎤ ⎡ 1 0⎤
R = ⎢ 4 ⎥ , I = ⎢ 4 ⎥ and C = ⎢ ⎥ (3B.24)
4 4 4
⎣ 0 1 ⎦ ⎣ 0 1 ⎦ ⎣ Cs 1 ⎦
4
3
48
4
3
2
2
47
3 3
c = I Cs + 4 I R Cs + ( 6 I R C 4 + 7 I C )s 6
11 4 4 4 4 4 4 4 4 4 4
+ ( 21IR C 3 + 4I R C 4 )s 5
2
3
4 4 4 4 4 4
+ ( 15IC 2 + 21IIR C 3 + R C ) s 4
2
2
4
4
4 4 4 4 4 4 4
+ 7 ( RC 3 + 30 I R C ) s + (115RC 2 + 10I C s 2
2
2
3
3
)
4 4 4 4 4 4 4 4 4

+ 10RC s + 1 (3B.25)
4 4
Higher-Order Models
Higher-order models of transmission lines can be deduced by assuming
a greater number of fluid lumps. The mathematical models describing
these models can be deduced systematically, as shown earlier.
The dynamic behavior of hydraulic transmission lines can be studied
by calculating the transient response or the frequency response. These cal-
culations can be carried out by using the detailed mathematical models or
by using the transfer functions, deduced from the transfer matrix.
Case Study
The lumped parameter model is used to calculate the transient response
of a hydraulic transmission line. The studied line had the following
parameters:
Pipe length L = 18 m
Pipe diameter D = 0.01 m
Dynamic viscosity of oil μ = 0.1215 Ns/m 2
Oil density ρ = 868 kg/m 3
9
Bulk modulus of oil B = 1.96 × 10 N/m 2
e 128μ
×
9
Line resistance R = L = 8.912 10 Pa s/m 3
π D 4

ρ L 4 ρ L
×
8
Line inertia I = = = 1.9893 10 kg/m 4
A π D 2
2
Line capacitance C = V = π DL ⎛ 1 + 5 D ⎞ ⎟
⎜
B 4 ⎝ B 4 Eh⎠
e
×
5
= 7.213 10 − 13 m / N

The line is connected to a constant pressure source and return line
by the directional control valves (DCV) (a) and (b), respectively (see
Fig. 3B.3). The experiment was conducted using the following sequence: