Page 103 - Fluid Power Engineering

Hydraulic Transmission Lines 77

3.7 Nomenclature
3
C = Whole line capacitance, m /Pa
C = Capacitance matrix
d, D = Tube inner diameter, m
I = Whole line inertia, kg/m 4
I = Inertia matrix
3
Q = Maximum flow rate, m /s
max
R = Whole line resistance, Ns/m 5
R = Resistance matrix
v = Mean fluid velocity, m/s
ΔP = Pressure losses, Pa
μ = Dynamic viscosity, Ns/m 2
ξ = Local loss coefficient
ρ = Fluid density, kg/m 3
2
ν = Kinematic viscosity, m /s

Appendix 3A The Laplace Transform
When a differential equation expressed in terms of time, t, is operated
on by a Laplace integral, a new equation results, which is expressed
in terms of a complex term (s). The Laplace transform translates the
time-dependent function from the time domain to the frequency (or
Laplace) domain. The transformed equation is in pure algebraic form
and may be manipulated algebraically.

The Direct Laplace Transform
The direct Laplace transform is given by the following expression:
∞
∫
Xs() = L [( )] = xt e dt
−
st
xt
()
(3A.1)
0
The Inverse Laplace Transform
The inverse Laplace transform is an integral operator that enables a
transform from the Laplace domain to the time domain.
1 σ + iR
xt() = Lim ∫ 1 X s e ds (3A.2)
st
(
)
2π i R→∞ σ 1 − iR
Actually, all functions in the time domain have a direct Laplace
transform, but some of the functions in the Laplace domain have no
inverse Laplace transform.
Properties of the Laplace Transform
The following are the basic properties of the Laplace transform:
1. L [ ( )ft ± f 2 ( )] = F 1 () F 2 () (3A.3)
s ±
t
s
1
2. L [( )] t = aF ( ) s (3A.4)
af