Fluid transmission lines and fittings were found to significantly affect the dynamic and steady state operation of the engineering system in which they are incorporated. Therefore, a better understanding of the operation of a system can be obtained by including the performance of the transmission lines and fittings within the system model. The most accurate model of a transmission line or fitting is obtained by using a distributed parameter model. However, a distributed parameter model tends to be very complex. This complexity can be avoided without significant loss of model accuracy by using a lumped parameter model when the length of the fluid path through the transmission line or fitting is short. This article develops a lumped parameter model for short fluid transmission lines and fittings, describes the conditions under which the model is valid, and presents the model parameters associated with the servo hydraulic system of the Deep Space Network (DSN) 70-meter antennas

Bartos, R. D.

English

NASA Technical Reports Server

TDA Progress Report 42-109
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.... _#ii
I,q;
May 15, 1992
N92-29378
Dynamic Modeling of Fluid Transmission Lines
of the DSN 70-Meter Antennas by Using
a Lumped Parameter Model
R. D. Bartos
Ground Antennas and Facilities EngineeringSection
Fluid transmission lines and fittings have been found to significantly affect the
dynamic and steady state operation of the engineering system in which they are
incorporated. Therefore, a better understanding of the operation of a system can be
obtained by including the performance of the transmission lines and fittings within
the system model. The most accurate model of a transmission line or fitting is
obtained by using a distributed parameter model. However, a distributed parameter
model tends to be very complex. This complexity can be avoided without significant
loss of model accuracy by using a lumped parameter model when the length of the
fluid path through the transmission line or fitting is short. This article develops a
lumped parameter model for short fluid transmission lines and fittings, describes
the conditions under which the model is valid, and presents the model parameters
associated with the servo hydraulic system of the DSN 70-meter antennas.
I. Introduction
All fluid transmission lines and fittings are physically
distributed systems because the resistance to fluid flow re-
sulting from the fluid viscosity, inertial, and compressibil-
ity effects are spread out over the length of the component.
Modeling fluid lines and fittings with distributed param-
eters generally results in a very complex model. Under
certain conditions in which the line or fitting is consid-
ered to be short, a lumped parameter model may be used
so that the mathematical equations describing the physi-
cal behavior are greatly simplified. A lumped parameter
model approximates the behavior of the actual line or fit-
ting without considering the distribution of the parameter
over the length of the component. This article develops
a theoretical model for fluid transmission lines and fit-
tings from a lumped parameter point of view by using
the methodology recommended by Meritt [1] and Wat-
ton [2]. A description of the conditions under which the
lumped parameter model is valid is also included in the ar-
ticle. The model is subsequently used to mathematically
describe the behavior of the control lines found within the
servo hydraulic system of the DSN 70-meter antennas.
II. Theoretical Model Development
The lumped parameter model developed in this section
provides a means of approximating the behavior of short
transmission lines and fittings. The model is based on the
equation of state for a fluid:
162
https://ntrs.nasa.gov/search.jsp?R=19920020135 2020-03-24T07:16:49+00:00Z
dn
dP = fl-'- (1)
Po
where P is the fluid pressure, fl is the effective bulk mod-
ulus, p is the density, and P0 is the average density. The
effective bulk modulus is given by the relationship
1 1 l
+ (2)
where )31 is the bulk modulus of the fluid and tic is the
effective bulk modulus of the container, fl! describes how
the fluid density changes with respect to pressure, while
fl¢ describes how the fluid container expands with respect
to pressure. The fluid bulk modulus is determined exper-
imentally for a liquid and is given by
fll = 7P0R00 (3)
for an ideal gas, where 3' is the specific heat ratio, R is
the ideal gas constant, and 60 is the average gas tempera-
ture. When the container is a circular conduit, the effective
bulk modulus of the container, tic, is given by the equation
[2,3]
1 2 [(l+v)(a+_)2+(1-v)a 2]
_° - Z_ (2a + 6 (4)
where a is the internal radius of the conduit, E is the
modulus of elasticity, _ is the conduit wall thickness, and v
is Poisson's ratio. Note that the effective bulk modulus will
change for different types of fluid containers. When several
containers are combined together, as shown in Fig. 1, the
system can be represented as a single container by using
the equation
V0 ½ ½ ½ ¼
7=T1+_+5+...+5 (5)
where V0 is the total volume, fl is the effective bulk mod-
ulus, and fli and Vii are the effective bulk modulus and
volume of the ith container, respectively.
The development of a dynamic model for a short line or
fitting using the lumped parameter representation involves
expressing Eq. (1) as
OP fl Op
_ × (6)
8t Po Ot
The continuity relationship for a control volume is given
by
0. _ .0 [ aVo]t v0 ¢71
where V0 is the total volume, Qin is the flow into the con-
trol volume, Qou_ is flow out of the control volume, and
aVo/Ot is the boundary deformation of the control volume.
If the volumetric expansion of the fluid container with re-
spect to pressure is expressed in terms of tic, then aVo/at is
zero. Substituting Eq. (7) into Eq. (6) and setting cgVo/cgt
to zero yields
cgt - Vo Qi. - __, Qo_. (8)
The pressure drop across a fitting or pipe where the
lumped parameter model is valid can be ignored when the
pressure drop across the fluid line is negligible with respect
to the system operating pressures. This article considers
only fluid lines and fittings with negligible pressure drops.
A detailed presentation on modeling pressure drops across
fluid conduits can be found in [3].
The conditions under which the lumped parameter
model is valid are formulated by considering the speed,
co, at which pressure waves traverse the fluid component
given by the equation
_o= V_ ° (o)
Hence, the time it takes a pressure wave to travel down
and back along a line or fitting of length L is
2L
r = -- (10)
CO
Therefore, a lumped parameter model is only valid when
the relationship
2L 1
-- < (11)
co 7
holds where f is the highest frequency of system oscillation
that exists [2].
163
III. Model Parameters
The parameters of the model of the hydraulic control
lines were determined for each control line of the 70-meter
antenna servo hydraulic system based on the component
layout of the 70-meter antenna servo hydraulic system
shown in Figs. 2 and 3. The model parameters are pre-
sented in Table 1 and were computed by using Eqs. (2),
(4), (5), and (8), along with the physical properties shown
in Table 2. The maximum pressure drop between the ser-
vovalve and a hydraulic motor due to fluid viscosity is
expected to be less than 69,000 Pa (10 PSI) under both
tracking and slewing conditions. Hence, the pressure drop
due to fluid viscosity is not modeled since it is not con-
sidered to be significant. Fluid accelerations in the lines
K = 5.216 x 101°N/m 5 (124.0 lbf/in, s)
[Elevation control no. 2] (16)
The summations of the inlet and outlet flow rates into a
control line found in Eq. (12) must be specified as bound-
ary conditions by using the servovalve and motor models.
The validity of the model presented in Eq. (12) is de-
termined by using Eqs. (9) and (11). The largest distance
between the servovalve and hydraulic motor is approxi-
mately 20.83 m (820 in.). From Eq. (9), and Tables 1 and
2, the speed of sound is determined to be approximately
1230 m/see (48,500 in./sec). Commands are currently sent
also cause a pressure drop across the lines. However, the
pressure drop across a line due to fluid acceleration is be- comes
lieved to have a negligible effect on system performance
since the antenna accelerations are both small in mag-
nitude and short in duration under tracking conditions.
Therefore, the fluid pressure in each of the control lines is
described by
o--?= K Q_. - Qo,,, (12)
to the servovalve at a rate of 20 hertz, hence, Eq. (ll) be-
2L 1
-- = 0.034 sec < "7 = 0.05 sec (17)
co !
which shows that a pressure wave is able to travel from the
servovalve to the hydraulic motors and back to the servo-
valve between the time that two consecutive commands
are sent to the servovaive. Therefore, the proposed model
is valid.
where
K = 4.756 x 101°N/m s (113.0 lbf/in. 5)
[Azimuth control no. 1]
K--- 4.655 X 101°N/m s (110.6 lbf/in. 5)
[Azimuth control no. 2]
K = 5.220 x 10'°N/m 5 (124.1 Ibf/in. s)
[Elevation control no. 1]
(13)
(14)
(15)
IV. Conclusion
This article presented a method of approximating the
performance of short fluid transmission lines and fittings
by using a lumped parameter model and computed the pa-
rameters required to model tile control lines of the DSN
70-meter antenna servo hydraulic system. It has been
found to be more advantageous to use a lumped parame-
ter model than the distributed parameter model because
of the mathematical simplicity of the lumped parame-
ter model. This simplicity allows the lumped parameter
model to be implemented easily within computer simula-
tion packages.
Refe re nces
[1] H. E. Merritt, Hydraulic Control Systems, New York: John Wiley and Sons, 1967.
[2] J. Watton, Fluid Power Systems Modeling, Simulation, Analog, and Microcom-
puter Control, New York: Prentice Hall, 1989.
[3] F. M. White, Fluid Mechanics, 2nd ed., New York: McGraw-Hill, 1986.
164
Table1.TransmissionlineparametersforthecontrollinesoftheDSN70-meterantennas.
Total Effective bulk
Transmission line
volume modulus
Azimuth control no. 1 0.02710 m 3 1.2888 x 109 Pa
(1653.6 in. 3) (1.8693 X l0 s psi)
Azimuth control no. 2 0.02770 m 3 1.2895 X l09 Pa
(1690.4 in. 3') (1.8703 X 10 _ psi)
Elevation control no. 1 0.02455 m 3 1.2815 × 109 Pa
(1498.1 in. 3) (1.8587 X 1135 psi)
Elevation control no. 2 0.02457 m 3 1.2816 x 109 Pa
(1499.6 in. s) (1.8588 x 10 z psi)
Table 2. Physical properties.
Fluid bulk modulus
Fluid density
Fluid kinematic viscosity
Inside diameter of 1-1n. pipe
Outside diameter of 1-in. pipe
Inside diameter of 1.5-in. pipe
Outside diameter of 1.5-in. pipe
Inside diameter of 2-in. pipe
Outside diameter of 2-in. pipe
Modulus of elasticity of stainless steel pipe
Poisson's ratio for stainless steel
Volume of 1.5-in. tee
Volume of 2-in. tee
Volume of 1-in. hose, 66-in. long
Bulk modulus of hose
1.379 x I09 Pa (2 x 105 psi)
850 kg/m 3 (7.95 x 10 -s lbf-sec_/in. 4)
27.7 m2/sec (1090 infl/sec)
2.070 cm (0.815 in.)
3.340 cm (1.315 in.)
3.399 cm (1.338 in.)
4.826 cm (1.900 in.)
4.285 cm (1.687 in.)
6.032 cm (2.375 in.)
2.0684 X 10 xl Pa (30 x 10 s psi)
0.25
2.147 × 10-4m3(13.1 in. 3)
4.015 x 10-4m3(24.5 in. 3)
8.495 × 10-4m3(51.8 in. 3)
4.32 X 109 Pa (6.26 xlO s psi)
165
_2
½
(33
V 3
Fig. 1. Combination of fluid containers.
166
PASSIVE ACTIVE I I SHORTINGSERVOVALVE SERVOVALVE VALVE
MANIFOLD I MANIFOLD iCONTROL NO, 1 CONTROL NO. 2
2-in. PIPE 12-in. PIPE
111 in. LONG I 125 n, LONGI
t
/
L 2-in. TEE =1 1 I =
I
I126 in. LONG131 in, LONG i 124 in, LONG I
l t I
r_ 1,5-in. TEE _ _ 1.5-in. TEE _ r_ 1.5-in, TEE _ _ 1.5-in. TEE
J,_,op,_,_,-,nP,P_,-,nP,P_I'-'oP":'_II'-'r'P'P_,-,nP,P_,-,nP,P_
'0''n'ON_lI 130 in. LONG132 in. LONG564 in. LONG1&52 in, LONGI141 n. LONG 138 in. LONG
; t t l 1 t t
I,-,o,-,osEI,-,r,Hos',I ,-'nHOS':I II'n"Os':l[I'°HOS I I''r'"OS':lI''n"OSe66 in. LONG 66 in. LONG 66 in. LONG 66 in. LONG 66 in. LONG 66 in. LONG 66 in. LONG
_E I._.Psc.M VAI_E VA
I VALVE I_-Psc-_I VA VE VA PSC VA _G_._J I-_
I _OTO_O.,I'-_s_---I_o_o,NO.=I MOTOR NO. 3 I_I"Psc"'_i--oo---_ _-o_-_
Fig. 2. DSN 70-meter antenna aervo hydraulic system azimuth component layout.
1,5-in. PIPE
129 n. LONG
I
564 in. LONGI
I 1-in. HOSE I66 in. LONG
Psc't VAIVEIPAc,V_EI
l
1,
MOTOR NO, 4
167
1
t 1.5-in. PIPE115 n. LONG
96 in. LONG
I 1-in. HOSE66 in. LONG
1.5-in. TEE
1 1
PASSIVE
SERVOVALVE
MANIFOLD ICONTROL NO. 1
2-in. PIP_E
253 in. LONG I
2-in. TEE
i nPP ]l137 in. LONG
1-in, HOSE ] I66 in. LONG
1
1.5-in. PIPE
143 in. LONG
I
_ 1.g-in. TEE
1-in. PIPE J96 in. LONG
I
I
1-in. HOSE I
I66 in. LONG
I REUEF/ / RE
VALVE _I-Psc. _ VA
L_
I,
MOTOR NO, 1
[ _,o_,P_t
_'n_ON_II
1
I,,o.o_1I66 in. LONG
EF @V_ Psc
1
1.5-in. PIPE
109 in. LONG
I
_ 1.5-in. TEE
1-in. PIPE I86 in. LONG
I
1-in. HOSE I66 in. LONG
1-in. PIPE
145 in. LONG
t t
MOTOR NO. 2
I 1-in, HOSE66 in. LONG
.IP_c_EF/ I--_vA VE I"Psc'l vA
_J IPAc
1
I 1
I MOTOR NO. 3
1 1
I SHORTING
VALVE
MANIFOLD
CONTROL NO. 2
I
2-in. PIPE
255 n. LONG
I
2-in, TEE :
1-in. PIPE
86 in. LONG
J
1.5-in, PIPE148 n, LONG
I
1.5-in. TEE
I
I 1-in. HOSE I66 in. LONG
I ,,oP,P_I145 in. LONG I
I
I
1-in, HOSE I
66 in. LONG I
I REI,EF/ / RE
I c" AGE/ L_
rEFI
NO. 4 I
Fig. 3. DSN 70-meter antenna servo hydraulic system elevation component layout.
168
aco
E
f
i
K
L
P
PB
PD
PR
Ps
Psc
Q.
Internal radius of conduit
Average speed of sound in a fluid
Modulus of elasticity
Highest system frequency, in hertz
Index, current
Constant associated with a particular fluid line
Length of fluid transmission line or fitting
Pressure
Brake pressure
Drain pressure
Return pressure
Supply pressure
Supercharge pressure
Volumetric flow rate into case drain
Volumetric flow rate into a control volume
Appendix
Glossary
Qou$
R
v_
vo
Zj
"7
6
Oo
11
P
Po
7"
Volumetric flow rate out of a control volume
Constant in ideal gas law
Volume of the ith container
Total volume of a control volume
Effective bulk modulus
Effective bulk modulus of the container
Bulk modulus of the fluid
Bulk modulus of the ith container
Specific heat ratio
Conduit wall thickness
Average gas temperature
Poisson's ratio
Fluid density
Average fluid density
Time delay
169


Dynamic modeling of fluid transmission lines of the DSN 70-meter antennas by using a lumped parameter model

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