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3
frames, Ziegler (1998), p.497. Assuming that the
liquid motion is characterized by the ideal fluid
flow along a representative streamline, the stan-
dard form of the non-stationary Bernoulli equation
has to be extended to account for the relative
fluid motion with respect to a moving reference
system
A
, see e.g. Hochrainer (2005). For the
symmetric arrangement the liquid stroke is
u u u
u
H
u
H
p
− =
p
2
n p
+
O
,
2
1
0
(2)
a
a
V
A
H
=
0
a
H
where
V
0
,
p
0
,
n
denote the gas volume and the
equilibrium gas pressure as well as the poly-
tropic index, respectively. The invention of the
gas spring also protects the TLCGD from exces-
sive vibrations because extreme stroke amplitudes
cause a nonlinear stiffening effect, thereby limit-
ing the maximum liquid displacement. For ex-
tremely slow vibrations the gas spring will oper-
ate under isothermal conditions (n=1), at high
frequencies an adiabatic change will occur and
the polytropic index becomes
n
=1.4. For all
other operating conditions
n
is in the range of
1
= =
1
2
and the equation of motion is given
by
2
1
(
)
−
+
∫
L u
2
gu
sin
β
= −
p
−
p
a
⋅
e
'
ds
'
,
eff
2
1
A
t
ρ
1
H B
A
A
L
=
2
+
H
eff
B
(1)
kg m
denote the constant
of gravity, and the density, e.g. of water, respec-
where
g
and
ρ
=
1000
3
≤
n
.
. To prevent the rigid piping from
buckling due to low pressure during the gas ex-
pansion phase, the equilibrium gas pressure should
be chosen above the atmospheric pressure. Since
all discussed advantageous effects of the gas spring
have been proved experimentally under labora-
tory conditions by Khalid (2010) it is one of the
most promising developments of recent research
activities.
To include energy losses by the experimen-
tally observed turbulent fluid flow an averaged
nonlinear pressure loss
δ
L
u
is added to the
equation of motion. The head loss factor
δ
L
ac-
counts for the losses in the elbows and the hy-
draulic roughness of the pipe walls. If necessary,
damping can be reduced by a smooth and con-
tinuous change in the cross sectional area from
A
B
to
A
H
, however at the price of an additional
nonlinearity, or increased by the insertion of ad-
ditional hydraulic orifices into the liquid path. In
the design stage and for the sake of tuning this
nonlinear damping term is equivalently linearized
by substituting viscous damping where the linear
viscous damping coefficient
ζ
1 4
2
∫
tively. The scalar integral expression
a
⋅
e
'
ds
'
A
t
1
accounts for the moving reference frame, with
e
'
t
and
a
A
denoting the relative streamline's tan-
gential direction and the absolute acceleration of
the moving reference point
A
. If the pipe endings
are kept open (classical TLCD) the pressure dif-
ference
p p
2
−
vanishes approximately, if the
pipes are sealed (the novel proposal of the TLC-
GD) the liquid stroke
u
will compress and expand
the enclosed gas mass thereby building up a gas-
spring. Under the assumption that the TLCGD
operating range is limited to low frequencies, the
inertia of the gas is small, and a quasi-static ap-
proach applies to approximate the pressure dif-
ference
p p
2
−
. With the weakly nonlinear
polytropic material law for ideal gases the pres-
sure difference can be sufficiently well approxi-
mated in the range
u H
a
≤
0 3
by just consid-
ering the linearized portion in the series
/
.
=
4
U
δ
3
π
A
max
L
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