Page 48 - Fluid Power Engineering

Hydraulic Oils and Theor etical Backgr ound 25

force, and the friction force. The spool motion is described by the
following equation:

2
F = m dx + f dx + kx (2.28)
dt 2 v dt

where F = Driving force, N
k = Spring stiffness, N/m
m = Mass of moving parts, kg
x = Spool displacement, m

Assuming zero initial conditions, and applying Laplace transfor-
mation to the equation of motion, the following transfer function can
be deduced:
Xs() 1 K
Gs() = = = (2.29)
+
2
Fs() ms + f s k s 2 2ζ
v + s + 1
ω 2 ω
n n
where ζ= Damping coefficient, proportional to the oil viscosity;
πμ DL
ζ =
2 ckm
k = Spring stiffness, N/m
K = Gain, K = 1/k, m/N
ω = Natural frequency, ω = km, rad/s
/
n n
2.2.2 Oil Density

Definition
The density is the mass per unit volume: ρ= m/V. The hydraulic oils
are of low compressibility and volumetric thermal expansion.
Therefore, under ordinary operating conditions, the oil density is
practically constant. The density of mineral hydraulic oils ranges
3
from 850 to 900 kg/m . The oil density affects both the transient and
steady state operations of the hydraulic systems. The hydraulic
losses in throttling elements and transmission lines are dominated
mainly by the inertia and friction losses. The effect of oil inertia on
these elements is discussed in this chapter.

Effect of Density on Hydraulic System Operation

Orifice Flow Orifices, short-tube or sharp-edged, are a basic means of
control in fluid power systems. This section aims at deriving equa-
tions for the flow rate of fluid through orifices and evaluating the ef-
fect of fluid viscosity and inertia. In most cases, the orifice flow occurs
at high Reynolds numbers. Such flow is referred to as turbulent flow,
but this term does not have quite the same meaning as the pipe flow.