ОДНОШАРОВИЙ АЛГОРИТМ МАРШОВОГО РОЗРАХУНКУ НАДЗВУКОВОГО ОБТІКАННЯ РАКЕТ З РОЗБИТТЯМ РОЗРАХУНКОВОЇ ОБЛАСТІ НА ДЕКІЛЬКА ПІДОБЛАСТЕЙ
DOI: https://doi.org/10.15407/itm2026.01.023 Hydraulic pipeline systems in liquid-propellant rocket engines (LPREs) are numerous and diverse. In staged-combustion LPREs, use is made of reconfigurable hydraulic systems, in which the propellant component flow directions change during an engine start-u...
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| author | CHERNIAVSKYI, O. S. DOLGOPOLOV, S. I. SHEVCHENKO, S. A. |
| author_facet | CHERNIAVSKYI, O. S. DOLGOPOLOV, S. I. SHEVCHENKO, S. A. |
| author_institution_txt_mv | [
{
"author": "O. S. CHERNIAVSKYI",
"institution": "https:\/\/orcid.org\/0009-0000-6401-7512 Oles Honchar Dnipro National University 72 Nauky Ave., Dnipro 49045, Ukraine; e-mail: o.s.cherniavskyi@gmail.com"
},
{
"author": "S. I. DOLGOPOLOV",
"institution": "https:\/\/orcid.org\/0000-0002-0591-4106 Institute of Technical Mechanics of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine 15 Leshko-Popel St., Dnipro 49005, Ukraine; e-mail: dolmrut@gmail.com"
},
{
"author": "S. A. SHEVCHENKO",
"institution": "https:\/\/orcid.org\/0000-0002-5495-7479 Institute of Transport Systems and Technologies of the National Academy of Sciences of Ukraine 5 Pysarzhevskoho St., Dnipro 49005, Ukraine; e-mail: sergiishevch@gmail.com"
}
] |
| author_sort | CHERNIAVSKYI, O. S. |
| baseUrl_str | https://journal-itm.dp.ua/ojs/index.php/ITM_j1/oai |
| collection | OJS |
| datestamp_date | 2026-07-13T20:30:24Z |
| description | DOI: https://doi.org/10.15407/itm2026.01.023
Hydraulic pipeline systems in liquid-propellant rocket engines (LPREs) are numerous and diverse. In staged-combustion LPREs, use is made of reconfigurable hydraulic systems, in which the propellant component flow directions change during an engine start-up. A reliable engine start-up requires a smooth transition from the start-up propellant supply to the main propellant supply. The objective of this study is to develop an approach to the mathematical simulation of dynamic processes in a branched reconfigurable hydraulic feed system. A methodological framework for modeling dynamic processes in such systems is proposed. It involves determining the frequency responses of several configurations of a hydraulic system realized at different stages of its operation as a distributed-parameter system. The next step is the construction of a lumped-parameter mathematical model with fluid motion and continuity equations in lumped parameters. Lumped compliances are typically introduced in the hydraulic network model at pipeline junctions. Their number and values are selected so that the frequency responses of the distributed- and lumped-parameter models for each hydraulic system configuration may be in agreement within a prescribed accuracy. To demonstrate the proposed approach, a test reconfigurable hydraulic system is considered. Two configurations of the hydraulic system are set off: from the start-up tank to the gas generator and from the pump outlet to the gas generator. For both configurations, frequency responses of the corresponding distributed-parameter systems are determined. A lumped-parameter mathematical model of dynamic processes in the hydraulic system under analysis is developed. The values of the lumped compliances at the network nodes are identified such that the frequency responses of the distributed- and lumped-parameter models are in satisfactory agreement. It is shown that the values of the lumped compliances remain practically unchanged for different hydraulic system configurations, boundary conditions, or propellant mixture ratios in the gas generator.
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14. Pérez-Roca S., Marzat J., Piet-Lahanier H., Langlois N., Galeotta M., Farago F., Le Gonidec S. Model-based robust transient control of reusable liquid-propellant rocket engines. IEEE Transactions on Aerospace and Electronic Systems. 2021. V. 57. Iss.1. Pp. 129-144.https://doi.org/10.1109/TAES.2020.3010668
15. Degtyarev A. V., Shulga V. A., Zhivotov A. I., Dibrivny A. V. The development of lox-kerosene liquid rocket engines family for perspective launch vehicles of Yuzhnoye SDO based on proven technologies. Aerospace Technic and Technology. 2013. No. 1. Pp. 44-50. (In Russian).
16. Prokopchuk А. А., Shulga V. А. The advanced liquid rocket engine line of SDO "Yuzhnoye" for the creation of new families of launch vehicles. Kosm Nauka Technol. 2015. V. 21, No. 5. Pp. 28-35. (In Russian).https://doi.org/10.15407/knit2015.05.028
17. Prokopchuk O. O., Shulga V. A. New and advanced liquid rocket engines of the Yuzhnoye SDO. Kosm. Teh. Raket. Vooruž. 2024. No 1 (121). Pp. 9-18. ( in Ukrainian).https://doi.org/10.33136/stma2024.01.009
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19. Fox J. A. Hydraulic Analysis of Unsteady Flow in Pipe. London: Red Globe Press, 1977. 216 pp.https://doi.org/10.1007/978-1-349-02790-3
20. Pylypenko O., Dolgopolov S., Nikolayev O., Khoriak N., Kvasha Yu., Bashliy I. Determination of the thrust spread in the Cyclone-4M first stage multi-engine propulsion system during its start. Science and Innovation. 2022. V. 18. No. 6. Pp. 97-112.
21. Dolgopolov S., Cherniavskyi O., Shevchenko S. Mathematical modeling of dynamic processes in the branched reconfigurable fuel feed system of a liquid‑propellant rocket engine. CEAS Space Journal. 2025. 14 pp. https://doi.org/10.1007/s12567-025-00691-y
22. Shevyakov A. A., Kalnin V. M., Naumenkova M. V., Dyatlov V. G. Theory of Automatic Control of Rocket Engines. Moscow: Mashinostroyeniye, 1978. 288 pp. (in Russian).
23. Belyaev E. N., Chervakov V. V. Mathematical Modeling of LPRE. Moscow: MAI-PRINT Publ., 2009. 280 рp. (In Russian).
24. Yan Z., Peng X., Cheng Y., Wu J. Modeling and simulation of system dynamics for spacecraft propulsion system. Applied Mechanics and Materials. 2012. Vs. 229-231. Pp. 2112-2116.https://doi.org/10.4028/www.scientific.net/AMM.229-231.2112
25. Altshul A. D. Hydraulic Resistances. Moscow: Nedra, 1970. 216 pp. (In Russian). |
| first_indexed | 2026-04-04T01:00:15Z |
| format | Article |
| fulltext |
23
UDK 621.454.2 https://doi.org/10.15407/itm2026.01.023
O. S. CHERNIAVSKYI1, https://orcid.org/0009-0000-6401-7512
S. I. DOLGOPOLOV2, https://orcid.org/0000-0002-0591-4106
S. A. SHEVCHENKO3, https://orcid.org/0000-0002-5495-7479
FEATURES OF THE MATHEMATICAL SIMULATION OF DYNAMIC
PROCESSES IN THE RECONFIGURABLE PROPELLANT FEED HYDRAULIC
SYSTEM OF LIQUID-PROPELLANT ROCKET ENGINES
1 Oles Honchar Dnipro National University, 72 Nauky Avenue, Dnipro, 49045, Ukraine;
e-mail: o.s.cherniavskyi@gmail.com
2Institute of Technical Mechanics of the National Academy of Sciences of Ukraine and State Space Agency of
Ukraine, 15 Leshko-Popelya Str., Dnipro, 49005, Ukraine; e-mail: dolmrut@gmail.com
3 Institute of Transport Systems and Technologies of the National Academy of Sciences of Ukraine, 5 Pisarzhev-
sky Str., Dnipro, 49005, Ukraine; e-mail: sergiishevch@gmail.com
Гідравлічні трубопровідні системи в рідинних ракетних двигунах (РРД) є численними та різномані-
тними. У РРД з допалюванням генераторного газу відомі гідравлічні системи змінної структури, у яких
під час запуску двигуна змінюється напрямок потоків компонентів палива. Для надійного запуску РРД
необхідно забезпечити плавний перехід від живлення пусковим пальним до живлення основним пальним.
Метою цієї роботи є розробка підходу до математичного моделювання динамічних процесів у розгалуже-
ній гідравлічній системі живлення зі змінною структурою. Розроблено методичний підхід до математич-
ного моделювання динамічних процесів у розгалуженій гідравлічній системі зі змінною структурою. Він
передбачає визначення частотних характеристик кількох конфігурацій гідравлічної системи, які реалізу-
ються на різних етапах її роботи, як системи з розподіленими параметрами. Надалі складається математи-
чна модель із зосередженими параметрами, що містить рівняння руху та нерозривності рідини у зосере-
джених параметрах. Зосереджені податливості розміщуються у розрахунковій гідравлічній схемі зазвичай
у місцях розгалуження трубопроводів. Їхню кількість і значення задають таким чином, щоб частотні хара-
ктеристики систем з розподіленими та зосередженими параметрами кожної конфігурації гідравлічної
системи змінної структури узгоджувалися з заданою точністю. Для демонстрації запропонованого підходу
розглянуто тестову гідравлічну систему змінної структури. Виділено дві конфігурації гідравлічної систе-
ми: від пускового бачка до газогенератора та від виходу з насоса до газогенератора, для яких визначено
частотні характеристики для систем з розподіленими параметрами. Побудовано математичну модель ди-
намічних процесів у аналізованій гідравлічній системі із зосередженими параметрами. Визначено значен-
ня зосереджених податливостей у вузлах, що дозволяють задовільно узгодити частотні характеристики з
розподіленими та зосередженими параметрами. Встановлено, що значення зосереджених податливостей
практично не змінюються за різних конфігурацій гідравлічної системи, граничних умов та співвідношень
компонентів палива в газогенераторі.
Ключові слова: рідинний ракетний двигун, система живлення змінної структури, розгалужений гі-
дравлічний тракт, математичне моделювання, імпедансний метод, частотна характеристика.
Hydraulic pipeline systems in liquid-propellant rocket engines (LPREs) are numerous and diverse. In
staged-combustion LPREs, use is made of reconfigurable hydraulic systems, in which the propellant component
flow directions change during an engine start-up. A reliable engine start-up requires a smooth transition from the
start-up propellant supply to the main propellant supply. The objective of this study is to develop an approach to
the mathematical simulation of dynamic processes in a branched reconfigurable hydraulic feed system. A meth-
odological framework for modeling dynamic processes in such systems is proposed. It involves determining the
frequency responses of several configurations of a hydraulic system realized at different stages of its operation as
a distributed-parameter system. The next step is the construction of a lumped-parameter mathematical model with
fluid motion and continuity equations in lumped parameters. Lumped compliances are typically introduced in the
hydraulic network model at pipeline junctions. Their number and values are selected so that the frequency re-
sponses of the distributed- and lumped-parameter models for each hydraulic system configuration may be in
agreement within a prescribed accuracy. To demonstrate the proposed approach, a test reconfigurable hydraulic
system is considered. Two configurations of the hydraulic system are set off: from the start-up tank to the gas
generator and from the pump outlet to the gas generator. For both configurations, frequency responses of the
corresponding distributed-parameter systems are determined. A lumped-parameter mathematical model of dynam-
ic processes in the hydraulic system under analysis is developed. The values of the lumped compliances at the
network nodes are identified such that the frequency responses of the distributed- and lumped-parameter models
are in satisfactory agreement. It is shown that the values of the lumped compliances remain practically unchanged for
different hydraulic system configurations, boundary conditions, or propellant mixture ratios in the gas generator.
Keywords: liquid-propellant rocket engine, reconfigurable feed system, branched hydraulic circuit, math-
ematical simulation, impedance method, frequency response.
© O. S. Cherniavskyi, S. I. Dolgopolov, S. A. Shevchenko, 2026
The article is an open access article distributed underthe terms and conditions of the Creative
Commons Attributions ( CC BY) license (https/creativecommons.org/licenses/by/4.0/)
ISSN 1561-9184 (Print) ISSN 2616-6380 (Online) Технічна механіка. 2026. № 1.
https://doi.org/10.15407/itm2026.01.023
mailto:o.s.cherniavskyi@gmail.com
mailto:dolmrut@gmail.com
mailto:sergiishevch@gmail.com
24
Introduction. Pipeline systems are widely used in industry, transportation,
construction, and municipal utilities [1]. Particularly stringent reliability require-
ments are imposed when transporting liquids and gases in the chemical and petro-
leum industries, nuclear power engineering [2], and shipbuilding [3]. The most
severe operating conditions, associated with structural vibrations of pipelines, oc-
cur in aviation and, especially, rocket technology [4, 5]. In liquid-propellant rocket
engines (LPREs), over a relatively short time interval, the pipeline systems may
first be exposed to cryogenic propellant components during filling and subsequent-
ly to high temperatures and intense heat fluxes from the regions where propellant
combustion occurs, namely the gas generators and combustion chambers.
Hydraulic and pneumatic pipeline systems in LPREs are numerous and di-
verse. Mathematical modeling of dynamic processes in these systems is subject to
the same stringent requirements as the modeling of individual LPRE units and as-
semblies. At present, mathematical simulation of LPRE operating processes shows
a tendency to focus on individual engine components [6, 7] and to incorporate an
extensive range of phenomena and effects observed at different stages of LPRE
operation [8–10]. This enables effective in-service diagnostics of rocket engines
[11], investigation of LPRE dynamic characteristics [12], decision-making on
LPRE reuse [13], and development of LPRE control loops [14].
A well-known LPRE configuration with gas-generator afterburning and non-
hypergolic propellant components employs a starting fuel for ignition [15 – 17].
This starting fuel is contained in a dedicated ampoule; after the ampoule is opened,
the fuel is supplied to the ignition cavities of the gas generator and the combustion
chamber, where it ignites spontaneously upon contact with the oxidizer. After
LPRE start and transition to the main fuel, the hydraulic path that includes the
starting-fuel ampoule is effectively shut off due to its high hydraulic resistance. In
steady-state operation, the engine is supplied with the main fuel through the prima-
ry hydraulic line.
The described ignition scheme and the transition from starting fuel to the main
fuel are based on the use of a reconfigurable propellant feed hydraulic system. For
reliable LPRE start-up, it is necessary to ensure a smooth transition from starting-
fuel supply to main-fuel supply. This transition changes the propellant mixture
ratio in the gas generator and can significantly affect the stability of turbopump
operating parameters during start-up.
The objective of this study is to develop an approach for mathematical model-
ing of dynamic processes in a branched reconfigurable propellant feed hydraulic
system during its operation.
1. Mathematical Modeling of Dynamic Processes in Branched Pipelines.
Pipeline systems in LPREs are often long and highly branched. The flow in these
systems is predominantly turbulent, and wave processes may be significant. For
most of these hydraulic systems, except for special cases, the characteristic flow
velocity is much lower than the speed of sound; therefore, the convective terms in
the unsteady momentum and continuity equations can be neglected [18]. On this
basis, mathematical modeling of dynamic processes in LPRE pipeline systems
should be formulated by solving the partial differential equations governing un-
steady flow and continuity [18]:
25
,
z
,
+ + =
+ =
p m k
m
A t A
m A p
z tc 2
1
0
0
(1)
where p and – are the fluid pressure and mass flow rate; t – time; z – axial co-
ordinate along the pipeline; A m – the pipeline cross-sectional area; k – the
equivalent linear friction coefficient per unit pipeline length
=
p A
k
l m
2
; p –
the hydraulic pressure loss; l – the length of the pipeline section; m –the steady-
state mass flow rate; c – the speed of sound in the fluid flowing through a pipeline
with elastic walls.
To solve system (1), the method of characteristics is often used [5, 19]. A less
common approach is the impedance method [18], despite its successful application
to a wide range of problems related to LPRE hydraulic system dynamics. In partic-
ular, the impedance method has recently been used to model the start-up of various
propulsion systems [20], to simulate transient processes in a hydraulic system con-
taining lumped gas cavities and undissolved gas in the liquid [20], and to analyze
coupled longitudinal oscillations of the pipeline structure and the liquid.
For branched pipeline systems, the impedance method is more appropriate be-
cause it enables compact, universal models to be developed both for the initial
LPRE start-up period during filling of the hydraulic lines with propellant compo-
nents and for simulating dynamic processes during engine run-up to steady-state
regime and under steady-state conditions. When the impedance method is applied,
the solution of system (1) for a pipeline segment between nodes m and n can be
represented as a passive two-port (four-terminal) network:
,
,
= +
= +
n m m m m n m
n nm m nn m
p b p b m
m b p b m
where mp and np – are the pressure deviations at the inlet and outlet of the
two-port network from their steady-state values; nm and mm – are the mass-
flow-rate deviations at the inlet and outlet from their steady-state values of the
two-port network; mmb , mnb , nmb and nnb – are the elements of the transfer
matrix of the pipeline segment, which for a distributed-parameter system are de-
fined as follows:
( ),= m mb ch l ( ),= − m n Bb Z sh l
( )
,
= −nm
B
sh l
b
Z
( ) ,= nnb ch l
where – complex wave propagation “constant” per unit pipeline
length = Z Y ; BZ – characteristic (wave) impedance of the pipeline,
=B
Z
Z
Y
; Z – hydraulic series impedance per unit pipeline length,
( ) = +Z j k
A
1
; Y – shunt admittance (conductance) per unit pipeline length,
=
A j
Y
c 2
; j – imaginary unit; – angular frequency.
26
Using the passive two-port network representation, the input impedance
( )mZ j and the gain (transfer) factor ( )W j of the considered pipeline segment
can be determined when the output impedance is known ( )nZ j .
( ) ( )
( )
( ) ( )
−
= =
−
m m n nn n
m
m nm n m m
p j b b Z j
Z j
m j b Z j b
,
( )
( )
( ) ( )
= = +
n
m m m n
m
p j
W j b b
p j Z j1
1
.
The dynamic gain of the entire hydraulic system composed of pipeline seg-
ments con-nected in series is determined as the product of the gain factors of the
individual segments. At branching points where three or more segments are con-
nected, the frequency characteristics ( )Z j and are obtained from the conditions
of mass-flow balance for the incoming and outgoing flows, and from equality of
pressures in all connected branches [18].
With boundary conditions specified at the ends of the pipeline system, the im-
pedance-method approach presented enables the solution of system (1) for a pipe-
line network of virtually any complexity. The solution yields the natural (resonant)
frequencies of fluid oscillations and the results of a stability analysis for the dis-
tributed-parameter pipeline system. In contrast to the method of characteristics, the
impedance method also provides insight into the frequency range of the dynamic
processes involved.
For a reconfigurable hydraulic system, the impedance-based approach pre-
sented requires refinement. It is proposed to consider several configurations of
such a system that occur at different stages of its operation. The transition from
one configuration to another during start-up simulation is accomplished either by
the operation of the check valves or, in a forced manner, by programmatically clos-
ing some valves and opening others [21].
2. Lumped-Parameter Mathematical Model. To simulate transient process-
es in the hydraulic system during LPRE start-up, a lumped-parameter model of the
hydraulic system dynamics is commonly used [4, 20]. Its construction relies on
solutions of the differential equation system (1) obtained for the hydraulic line
treated as a distributed-parameter system. In general, the proposed approach to
mathematical modeling of dynamic processes in a branched reconfigurable propel-
lant feed hydraulic system comprises several stages.
At the first stage, the frequency characteristics of several configurations of the
reconfigurable hydraulic system that occur at different stages of its operation are
determined by treating the system as a distributed-parameter system. Next, a
lumped-parameter mathematical model is formulated, including the lumped-
parameter forms of the fluid momentum and continuity equations.
( ) ,
,
= + +
= − +
n m m m m
n m m m
p p R j J m
m j C p m
where mR , mJ – are the coefficients of hydraulic (resistive) and inertial re-
sistance of the pipeline segment; mC – is the lumped compliance of the pipeline
segment [22–24].
27
The coefficients mJ are determined from the geometric characteristics of the
hydraulic lines, whereas the coefficients mR are obtained from engine balance
calculations or from the corresponding pressure-drop relations for hydraulic losses
[25]. Lumped compliances are typically placed in the computational hydraulic
scheme at pipeline branching nodes or at locations associated with long pipeline
sections. The values of the compliance coefficients mC are selected so that, for
each configuration of the reconfigurable hydraulic system, the frequency charac-
teristics of the distributed-parameter and lumped-parameter models match within a
prescribed accuracy over a specified frequency range.
The lumped-parameter mathematical model is then used directly to simulate
transient processes in the reconfigurable hydraulic system.
Compared with the method of characteristics [19], determining transient pro-
cesses using the approach proposed in this work includes one additional intermedi-
ate stage, namely, the evaluation and matching of the frequency characteristics of
the hydraulic system represented by distributed-parameter and lumped-parameter
models. However, this stage is not redundant, because it enables a compact math-
ematical model to be constructed that describes the dynamic processes over a
guaranteed frequency range, which is required for transient process simulation in a
reconfigurable LPRE propellant feed system.
3. Example of a Reconfigurable Propellant Feed Hydraulic System. To
demonstrate the proposed approach, we consider the simplest reconfigurable hy-
draulic system, namely the fuel feed system of the gas generator of a test restarta-
ble staged-combustion LPRE. This hydraulic system includes (see Fig. 1) three
hydraulic lines: from the starting tank ( Bp ) to the branching node ( p3 ), from the
pump outlet ( pp ) to the branching node, and from the branching node to the gas
generator ( ggp ). The first and second lines contain check valves (labeled valve 1
and valve 2 in the scheme). The third line includes a solenoid valve, labeled valve 3,
which is intended to isolate the fuel cavity from oxidizer vapors and purge gases.
oggm – oxidizer flow rate to the gas generator;
fggm , fpm , fbm , fm 23 , fm 12 – fuel flow rates in the corre-
sponding sections; p1 , p2 , p3 – pressures at the nodes;
ggp – pressure in the gas generator; Bp – pressure in the starting
tank; pp – pressure at the fuel pump outlet
Fig. 1 – Computational scheme of the feed system of the test LPRE
28
The reconfigurable hy-
draulic system operates as
follows. Upon a command
from the control system,
valve 3 is opened. At this
initial time, the pump shaft
is not yet spinning up, and
the pressure at the fuel pump
outlet pp is much lower
than the pressure in the start-
ing tank Bp . Therefore, the
fuel flows from the starting
tank through the branching
node ( p3 ) to the gas genera-
tor. Check valve 2 prevents
fuel from entering the pump
line. After ignition of the
propellant components in the
gas generator and sufficient
spin-up of the pump shaft,
the pressure at the fuel pump
outlet pp becomes higher
than the starting-tank pres-
sure Bp . Check valve 1 clo-
ses, the fuel flow from the
starting tank stops, and the
gas generator is supplied with
fuel from the pump line.
Thus, during operation of
the reconfigurable hydraulic
system, the configuration of
the hydraulic path supplying fuel to the gas generator changes. Initially, fuel flows
from the starting tank ( Bp ) through the branching node ( p3 ) to the gas generator
( ggp ). Subsequently, fuel is supplied from the pump ( pp ) through the branching
node ( p3 ) to the gas generator ( ggp ). Therefore, mathe-matical modeling of dy-
namic processes in the reconfigurable hydra-ulic system under consideration re-
quires developing dynamic models for each of these two branches.
4. Results of Mathematical Modeling. The frequency characteristics were
determined for two selected branches of the reconfigurable hydraulic system: from
the starting tank to the gas generator ( )
gg
B
p
j
p
and from the fuel pump outlet to
the gas generator ( )
gg
P
p
j
p
. In evaluating these frequency characteristics, each
check valve was assumed to be either fully open or fully closed. In the fully open
state, the valve provides minimal hydraulic resistance; in the fully closed state, the
0.0
0.5
1.0
1.5
0 200 400
1
2
3
f, Hz
( )
j
p
p
P
gg
mod
a
-6
-4
-2
0
0 200 400
1
3
2
f , Hz
( ),arg
j
p
p
P
gg
rad
b
1 – distributed-parameter model; 2 – lumped-parameter
model; 3 – lumped-parameter model with =jC 0
Fig. 2 – Magnitude (a) and phase (b) of the gain factor
( )
gg
P
p
j
p
29
hydraulic line containing the check valve is blocked. The boundary condition at
the gas generator cross-section (impedance ( )
gg
fgg
p
j
m
) was specified using the
equations for determining the gas-generator pressure under choked discharge of
the combustion products.
gg gg
ogg fgg sgg
gg gg gg
= + −
V dp
m m m
R T dt
,
gg
gggg gg
sgg gg
gggg gg
+
−
=
+
A p
m
R T
1
12
1
. (2)
Assuming ogg =m const and gg gg =R T const , we obtain, where ggV – is the
volume of the gas generator combustion zone; ggR – is the specific gas constant of
the combustion products in the gas generator; ggT – is the temperature of the
gg
gggg gg
gg gg
gg gg gg gggg gg
+
−
+ =
+
fgg
V A
j p m
R T R T
1
12
1
combustion prod-
ucts in the gas generator;
gg – is the specific heat
ratio of the combustion
products in the gas genera-
tor; ggA – is the critical
(throat) area of the gas
generator.
The frequency de-
pendences of the gain fac-
tors ( )
gg
B
p
j
p
and
( )
gg
P
p
j
p
for the dis-
tributed-parameter mo-dels
are shown in Figs. 2 and 3.
They were obtai-ned under
boundary conditions
corres-ponding to the nom-
inal propellant mixture ra-
tio in the gas generator.
The frequency range over
which gain factors
( )
gg
B
p
j
p
and ( )
gg
P
p
j
p
are
presented was selected to
include at least one reso-
nant frequency.
0.0
0.5
1.0
1.5
0 200 400
1
2
3
f, Hz
( )
j
p
p
B
gg
mod
а
-8
-6
-4
-2
0
0 200 400
1
2
3
f , Hz
( ),arg
j
p
p
B
gg rad
б
1 – distributed-parameter model; 2 – lumped-parameter
model; 3 – lumped-parameter model with =jC 0
Fig. 3 – Magnitude (a) and phase (b) of the gain factor
( )
gg
B
p
j
p
30
For the feedline from the pump, the resonant frequency is 459 Hz, whereas for
the feedline from the starting tank, two resonant frequencies fall within the same
range – 272 Hz and 541 Hz. This range was selected with a margin, because the
working frequency range typically considered in mathematical modeling of low-
frequency dynamic processes in LPREs does not exceed 100 Hz [4]. It should be
noted that the “gas generator-starting tank” line exhibits a larger number of reso-
nant frequencies, which is attributed to its greater length.
When constructing the lumped-parameter dynamic model of the hydraulic
system, the large length of the “gas generator–starting tank” line was taken into
account. For this purpose, this line was divided into three sections and therefore
includes two intermediate nodes (nodes p1 and p2 in Fig. 1). Another node is lo-
cated at the branching point ( p3 ). Lumped compliances С1 , С 2 andС 3 . are
placed at these nodes. In the pipelines between the nodes, the liquid is treated as
incompressible and its dynamics are described by the momentum equations. The
hydraulic resistances introduced by the check valves are determined by their effec-
tive flow areas and depend primarily on the pressure drop between the valve inlet
and outlet.
According to the proposed approach, the lumped-parameter dynamic model of
the reconfigurable fuel hydraulic system can be written as follows:
fb
fb B 1 fb fb= − −
dm
J p p m
dt
2 , (3)
fb f12= −
dp
C m m
dt
1
1 , (4)
f12
f12 1 2 f12 f12= − −
dm
J p p m
dt
2
, (5)
f12 f23= −
dp
C m m
dt
2
2 , (6)
( )f
f23 3 f23 V1 f23= − − +
dm
J p p m
dt
223
2 , (7)
( )fp
fp p 3 fp V2 fp= − − +
dm
J p p m
dt
2
, (8)
f23 fp fgg= + −
dp
C m m m
dt
3
3 , (9)
( )fgg
fgg gg fgg V3 fgg= − − +
dm
J p p m
dt
2
3 , (10)
where – is the fuel density; fb , f12 , f23 , fp , fgg – are hydraulic resistance
coefficients determined from hydraulic calculations; fbJ , f12J , f23J , fpJ , fggJ –
are inertial resistance coefficients obtained from the geometric characteristics of
the pipelines; V1 , V2 , V3 – are the hydraulic resistance coefficients of the
check valves, determined by their effective flow areas V1A , V2A , V3A :
V1
V1
=
A
1
2
, V2
V2
=
A
1
2
, V3
V3
=
A
1
2
. (11)
Thus, the dynamic behavior of the analyzed lumped-parameter reconfigurable
hydraulic system is described by a system consisting of the ordinary differential
31
equations (3)–(10), the expressions for the hydraulic resistance coefficients of the
check valves (11), and the gas-generator boundary conditions (2).
The values of the lumped compliancesС1 , С 2 and С 3 are selected to achieve
the required accuracy in matching the frequency characteristics of the hydraulic
system obtained from the distributed-parameter and lumped-parameter models.
Figures 2 and 3 show the gain factors of the lumped-parameter hydraulic system
( )
gg
B
p
j
p
and ( )
gg
P
p
j
p
, obtained with the same compliance values С1 , С 2
and С 3 , used for both gain factors. An analysis of Figs. 2 and 3 indicates that the
gain factors predicted by the distributed-parameter and lumped-parameter models
are in satisfactory agreement over the considered frequency range. In addition, it
was found that the gain factors ( )
gg
B
p
j
p
and ( )
gg
P
p
j
p
obtained under different
boundary conditions and different propellant mixture ratios in the gas generator,
can also be described satisfactorily by the lumped-parameter system using the
same compliance values С1 , С 2 and С 3 . This makes it possible to use constant
compliance values , and С 3 throughout the entire engine start-up process.
Introducing compliances С1 and С 2 into the lumped-parameter model made
it possible to reproduce two resonances in the frequency response ( )
gg
B
p
j
p
. The
introduced compliance values were set equal to the compliance at the pipeline
branching node.
= =
C
C
C
2
2
3
1.04, = =
C
C
C
1
1
3
0.65.
Figures 2 and 3 also present the gain factors of the lumped-parameter hydrau-
lic system ( )
gg
B
p
j
p
and ( )
gg
P
p
j
p
obtained with zero values of the lumped
compliances С1 , and . These results indicate that, for the lumped-parameter
model describes the dynamic processes satisfactorily within the frequency range
up to 70 Hz. This model is simpler and can be used for mathematical modeling of
low-frequency dynamic processes; however, in this case the system equations take
a slightly different form. The absence of compliances implies equality of the flow
rates at the node. Therefore, to determine the nodal pressure, instead of the conti-
nuity equation it is necessary to use a set of momentum equations in which the
time derivatives of the flow rates are set to zero.
Conclusions.
1. A methodological approach to the mathematical modeling of dynamic pro-
cesses in a branched reconfigurable hydraulic system has been developed. First,
the frequency characteristics of several configurations of the reconfigurable hy-
draulic system occurring at different stages of its operation are determined using a
distributed-parameter model. Next, a lumped-parameter mathematical model is
formulated, incorporating the corresponding fluid momentum and continuity equa-
tions. The inertial resistance coefficients are determined from the geometric char-
acteristics of the hydraulic lines, whereas the coefficients of hydraulic resistance
32
are obtained from engine balance calculations or from the corresponding pressure-
drop relations for hydraulic losses. Lumped compliances are placed in the compu-
tational hydraulic scheme, typically at pipeline branching nodes or at locations
associated with long pipeline sections. Their number and values are selected so
that, for each configuration of the reconfigurable hydraulic system, the frequency
characteristics of the distributed-parameter and lumped-parameter models match
within a prescribed accuracy over a specified frequency range.
2. To demonstrate the proposed approach, a reconfigurable hydraulic system
was considered, namely the fuel feed system of the gas generator of a test restarta-
ble LPRE employing a gas-generator afterburning (staged-combustion) cycle. Two
configurations of the hydraulic system were identified: from the starting tank to
the gas generator, and from the fuel pump outlet to the gas generator. For each
configuration, the frequency characteristics were determined using distributed-
parameter models. One resonant frequency of 459 Hz was identified in the pump
line, whereas two resonant frequencies, 272 Hz and 541 Hz, were found in the
longer line from the starting tank. A lumped-parameter model of the dynamic pro-
cesses in the analyzed hydraulic system was then developed. It includes one node
at the pipeline branching point and two nodes along the long pipeline between the
starting tank and the gas generator. The values of the lumped compliances at the
nodes were determined to achieve satisfactory agreement between the frequency
characteristics of the distributed-parameter and lumped-parameter models. It was
found that the lumped compliance values remain virtually unchanged under differ-
ent system configurations, boundary conditions, and propellant mixture ratios in
the gas generator.
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Received on February 18, 2026;
approved on March 26, 2026;
published on March 31, 2026
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|
| id | oai:ojs2.journal-itm.dp.ua:article-169 |
| institution | Technical Mechanics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-14T01:01:02Z |
| publishDate | 2026 |
| publisher | текст 3 |
| record_format | ojs |
| resource_txt_mv | journal-itmdpua/c0/4d3810668bbdedd13e4620024d7eabc0.pdf |
| spelling | oai:ojs2.journal-itm.dp.ua:article-1692026-07-13T20:30:24Z FEATURES OF THE MATHEMATICAL SIMULATION OF DYNAMIC PROCESSES IN THE RECONFIGURABLE PROPELLANT FEED HYDRAULIC SYSTEM OF LIQUID-PROPELLANT ROCKET ENGINES ОДНОШАРОВИЙ АЛГОРИТМ МАРШОВОГО РОЗРАХУНКУ НАДЗВУКОВОГО ОБТІКАННЯ РАКЕТ З РОЗБИТТЯМ РОЗРАХУНКОВОЇ ОБЛАСТІ НА ДЕКІЛЬКА ПІДОБЛАСТЕЙ CHERNIAVSKYI, O. S. DOLGOPOLOV, S. I. SHEVCHENKO, S. A. liquid-propellant rocket engine, reconfigurable feed system, branched hydraulic circuit, mathematical simulation, impedance method, frequency response. рідинний ракетний двигун, система живлення змінної структури, розгалужений гідравлічний тракт, математичне моделювання, імпедансний метод, частотна характеристика. DOI: https://doi.org/10.15407/itm2026.01.023 Hydraulic pipeline systems in liquid-propellant rocket engines (LPREs) are numerous and diverse. In staged-combustion LPREs, use is made of reconfigurable hydraulic systems, in which the propellant component flow directions change during an engine start-up. A reliable engine start-up requires a smooth transition from the start-up propellant supply to the main propellant supply. The objective of this study is to develop an approach to the mathematical simulation of dynamic processes in a branched reconfigurable hydraulic feed system. A methodological framework for modeling dynamic processes in such systems is proposed. It involves determining the frequency responses of several configurations of a hydraulic system realized at different stages of its operation as a distributed-parameter system. The next step is the construction of a lumped-parameter mathematical model with fluid motion and continuity equations in lumped parameters. Lumped compliances are typically introduced in the hydraulic network model at pipeline junctions. Their number and values are selected so that the frequency responses of the distributed- and lumped-parameter models for each hydraulic system configuration may be in agreement within a prescribed accuracy. To demonstrate the proposed approach, a test reconfigurable hydraulic system is considered. Two configurations of the hydraulic system are set off: from the start-up tank to the gas generator and from the pump outlet to the gas generator. For both configurations, frequency responses of the corresponding distributed-parameter systems are determined. A lumped-parameter mathematical model of dynamic processes in the hydraulic system under analysis is developed. The values of the lumped compliances at the network nodes are identified such that the frequency responses of the distributed- and lumped-parameter models are in satisfactory agreement. It is shown that the values of the lumped compliances remain practically unchanged for different hydraulic system configurations, boundary conditions, or propellant mixture ratios in the gas generator. REFERENCES 1. Thorley A. R. D. Fluid Transients in Pipeline Systems. London: City University, 2004. 304 pp. 2. Su H., Sheng L., Zhao Sh., Lu Ch., Zhu R., Chen Y., Fu Q. Water hammer characteristics and component fatigue analysis of the essential service water system in nuclear power plants. Processes. 2023. V. 11. Iss. 12. 3305.https://doi.org/10.3390/pr11123305 3. Quan L., Gao J., Guo C., Fu C. Analysis of water hammer and pipeline vibration characteristics of submarine local hydraulic system. J. Mar. Sci. Eng. 2023. V.11, Iss.10. 1885.https://doi.org/10.3390/jmse11101885 4. Pylypenko O. V., Dolgopolov S. I., Nikolayev O. D., Khoriak N. V. Mathematical modeling of the transient processes in propulsion system of the upper stage of the Cyclone-4M launch vehicle. Science and Innovation 2024. V. 20. No.1. Pp. 49-67. 5. Das D., Padmanabhan P. Study of pressure surge during priming phase of start transient in an initially unprimed pump-fed liquid rocket engine. Propulsion and Power Research. 2022. V. 11. No. 3. Pp. 353-375.https://doi.org/10.1016/j.jppr.2022.07.003 6. Das D., Waghmare Sh., Padmanabhan P., Kumaresan V., Sudhakar D. P. Control of opening duration in a pneumatically operated valve with two-fluid combination and quadratic damping. Journal of Engineering and Applied Science. 2023. V. 70. 41.https://doi.org/10.1186/s44147-023-00207-7 7. Cherniavskyi O. S., Dolgopolov S. I., Shevchenko S. A. Modeling of a check valve operation in the reconfigurable hydraulic feed system of a liquid rocket engine. Teh. Meh. 2025. No. 4. Pp. 19-30.https://doi.org/10.15407/itm2025.04.019 8. Dolgopolov S. I., Nikolayev O. D., Khoriak N. V. Dynamic interaction between clustered liquid propellant rocket engines under their asynchronous start-ups. Propuls. Power Res. 2021. V. 10. Iss. 4. Pp. 347-359.https://doi.org/10.1016/j.jppr.2021.12.001 9. Li H., Guo Y., Xu K., Yan X. Simulation of characteristics of staged combustion cycle rocket engine and control valve based on AMESim/Simulink. Journal of Aerospace Power. 2024. V. 39. Iss. 1. 8 pp. https://doi.org/10.13224/j.cnki.jasp.20210401 (In Chinese). 10. Su Q., Wang J., Yan M., Sun Z., Huang W., Zha B. Dynamic characteristics of LOX/Kerosene variable thrust liquid rocket engine test system based on general modular simulation method. International Journal of Aerospace Engineering. 2022. 14 pp. https://doi.org/10.1155/2022/2171471https://doi.org/10.1155/2022/2171471 11. Huang P., Yu H., Wang T. A study using optimized LSSVR for real-time fault detection of liquid rocket engine. Processes. 2022. V.10. Iss. 8. 1643.https://doi.org/10.3390/pr10081643 12. Dong L., Nie W., Li G. Modeling, simulation and experimental study of squeeze engine system. Journal of Physics: Conference Series. 2022. V. 2288. 012003.https://doi.org/10.1088/1742-6596/2228/1/012003 13. Naderi M., Karimi H., Guozhu L. Modeling the effect of reusability on the performance of an existing LPRE. Acta Astronautica. 2021. V. 181. Iss. 4. Pp. 201-216.https://doi.org/10.1016/j.actaastro.2020.12.001 14. Pérez-Roca S., Marzat J., Piet-Lahanier H., Langlois N., Galeotta M., Farago F., Le Gonidec S. Model-based robust transient control of reusable liquid-propellant rocket engines. IEEE Transactions on Aerospace and Electronic Systems. 2021. V. 57. Iss.1. Pp. 129-144.https://doi.org/10.1109/TAES.2020.3010668 15. Degtyarev A. V., Shulga V. A., Zhivotov A. I., Dibrivny A. V. The development of lox-kerosene liquid rocket engines family for perspective launch vehicles of Yuzhnoye SDO based on proven technologies. Aerospace Technic and Technology. 2013. No. 1. Pp. 44-50. (In Russian). 16. Prokopchuk А. А., Shulga V. А. The advanced liquid rocket engine line of SDO "Yuzhnoye" for the creation of new families of launch vehicles. Kosm Nauka Technol. 2015. V. 21, No. 5. Pp. 28-35. (In Russian).https://doi.org/10.15407/knit2015.05.028 17. Prokopchuk O. O., Shulga V. A. New and advanced liquid rocket engines of the Yuzhnoye SDO. Kosm. Teh. Raket. Vooruž. 2024. No 1 (121). Pp. 9-18. ( in Ukrainian).https://doi.org/10.33136/stma2024.01.009 18. Pylypenko V. V., Zadontsev V. A., Natanzon M. S. Cavitation Oscillations and Dynamics of Hydraulic Systems. Moscow: Mashinostroyeniye, 1977. 352 p. (In Russian). 19. Fox J. A. Hydraulic Analysis of Unsteady Flow in Pipe. London: Red Globe Press, 1977. 216 pp.https://doi.org/10.1007/978-1-349-02790-3 20. Pylypenko O., Dolgopolov S., Nikolayev O., Khoriak N., Kvasha Yu., Bashliy I. Determination of the thrust spread in the Cyclone-4M first stage multi-engine propulsion system during its start. Science and Innovation. 2022. V. 18. No. 6. Pp. 97-112. 21. Dolgopolov S., Cherniavskyi O., Shevchenko S. Mathematical modeling of dynamic processes in the branched reconfigurable fuel feed system of a liquid‑propellant rocket engine. CEAS Space Journal. 2025. 14 pp. https://doi.org/10.1007/s12567-025-00691-y 22. Shevyakov A. A., Kalnin V. M., Naumenkova M. V., Dyatlov V. G. Theory of Automatic Control of Rocket Engines. Moscow: Mashinostroyeniye, 1978. 288 pp. (in Russian). 23. Belyaev E. N., Chervakov V. V. Mathematical Modeling of LPRE. Moscow: MAI-PRINT Publ., 2009. 280 рp. (In Russian). 24. Yan Z., Peng X., Cheng Y., Wu J. Modeling and simulation of system dynamics for spacecraft propulsion system. Applied Mechanics and Materials. 2012. Vs. 229-231. Pp. 2112-2116.https://doi.org/10.4028/www.scientific.net/AMM.229-231.2112 25. Altshul A. D. Hydraulic Resistances. Moscow: Nedra, 1970. 216 pp. (In Russian). DOI: https://doi.org/10.15407/itm2026.01.023 Гідравлічні трубопровідні системи в рідинних ракетних двигунах (РРД) є численними та різноманітними. У РРД з допалюванням генераторного газу відомі гідравлічні системи змінної структури, у яких під час запуску двигуна змінюється напрямок потоків компонентів палива. Для надійного запуску РРД необхідно забезпечити плавний перехід від живлення пусковим пальним до живлення основним пальним. Метою цієї роботи є розробка підходу до математичного моделювання динамічних процесів у розгалуженій гідравлічній системі живлення зі змінною структурою. Розроблено методичний підхід до математичного моделювання динамічних процесів у розгалуженій гідравлічній системі зі змінною структурою. Він передбачає визначення частотних характеристик кількох конфігурацій гідравлічної системи, які реалізуються на різних етапах її роботи, як системи з розподіленими параметрами. Надалі складається математична модель із зосередженими параметрами, що містить рівняння руху та нерозривності рідини у зосереджених параметрах. Зосереджені податливості розміщуються у розрахунковій гідравлічній схемі зазвичай у місцях розгалуження трубопроводів. Їхню кількість і значення задають таким чином, щоб частотні характеристики систем з розподіленими та зосередженими параметрами кожної конфігурації гідравлічної системи змінної структури узгоджувалися з заданою точністю. Для демонстрації запропонованого підходу розглянуто тестову гідравлічну систему змінної структури. Виділено дві конфігурації гідравлічної системи: від пускового бачка до газогенератора та від виходу з насоса до газогенератора, для яких визначено частотні характеристики для систем з розподіленими параметрами. Побудовано математичну модель динамічних процесів у аналізованій гідравлічній системі із зосередженими параметрами. Визначено значення зосереджених податливостей у вузлах, що дозволяють задовільно узгодити частотні характеристики з розподіленими та зосередженими параметрами. Встановлено, що значення зосереджених податливостей практично не змінюються за різних конфігурацій гідравлічної системи, граничних умов та співвідношень компонентів палива в газогенераторі. ПОСИЛАННЯ 1. Thorley A. R. D. Fluid Transients in Pipeline Systems. London: City University. 2004. 304 p. 2. Su H., Sheng L., Zhao Sh., Lu Ch., Zhu R., Chen Y., Fu Q. Water Hammer Characteristics and Component Fatigue Analysis of the Essential Service Water System in Nuclear Power Plants. Processes 11. 2023. 14 p. https://doi.org/10.3390/pr11123305 3. Quan L., Gao J., Guo C., Fu C. Analysis of water hammer and pipeline vibration characteristics of submarine local hydraulic system. J. Mar. Sci. Eng. 2023. 11. 1885. https://doi.org/10.3390/jmse11101885 4. Pylypenko O. V., Dolgopolov S. I., Nikolayev O. D., Khoriak N. V. Mathematical modeling of the transient processes in propulsion system of the upper stage of the Cyclone-4M launch vehicle. Science and Innovation. 2024. 20(1). P. 49–67. 5. Das D., Padmanabhan P. Study of pressure surge during priming phase of start transient in an initially unprimed pump-fed liquid rocket engine. Propulsion and Power Research. 2022. 11(3). P. 353–375. https://doi.org/10.1016/j.jppr.2022.07.003 6. Das D., Waghmare Sh., Padmanabhan P., Kumaresan V., Sudhakar D. P. Control of opening duration in a pneumatically operated valve with two-fuid combination and quadratic damping. Journal of Engineering and Applied Science. 2023. Р. 20. https://doi.org/10.1186/s44147-023-00207-7 7. Cherniavskyi O. S., Dolgopolov S. I., Shevchenko S. A. Modeling of a check valve operation in the reconfigurable hydraulic feed system of a liquid rocket engine. Tech. Mech. 2025. № 4. P. 19–30. https://doi.org/10.15407/itm2025.04.019 8. Dolgopolov S. I., Nikolayev O. D., Khoriak N. V. Dynamic interaction between clustered liquid propellant rocket engines under their asynchronous start-ups. Propuls. Power Res. 2021. 10(4), P. 347–359. https://doi.org/10.1016/j.jppr.2021.12.001 9. Li H., Guo Y., Xu K., Yan X. Simulation of characteristics of staged combustion cycle rocket engine and control valve based on AMESim/Simulink. Journal of Aerospace Power. 2024. 39(1), 8 р. https://doi.org/10.13224/j.cnki.jasp.20210401 (in Chinese) 10. Su Q., Wang J., Yan M., Sun Z., Huang W., Zha B. Dynamic characteristics of LOX/Kerosene variable thrust liquid rocket engine test system based on general modular simulation method International. Journal of Aerospace Engineering. 2022. 14 p. https://doi.org/10.1155/2022/2171471 11. Huang P., Yu H., Wang T. A Study Using Optimized LSSVR for Real-Time Fault Detection of Liquid Rocket Engine. 2022. Processes 10. P. 16. https://doi.org/10.3390/pr10081643 12. Dong L., Nie W., Li G. Modeling, simulation and experimental study of squeeze engine system. Journal of Physics: Conference Series. 2022. P. 8. https://doi:10.1088/1742-6596/2228/1/012003 13. Naderi M., Karimi H., Guozhu L. Modeling the effect of reusability on the performance of an existing LPRE. Acta Astronautica. 2021. 181(4), P. 201–216. https://doi.org/10.1016/j.actaastro.2020.12.001 14. Pérez-Roca S., Marzat J., Piet-Lahanier H., Langlois N., Galeotta M., Farago F., Le Gonidec S. Model-based robust transient control of reusable liquid-propellant rocket engines. IEEE Transactions on Aerospace and Electronic Systems. 2021. 57(1), P. 129–144. https://doi.org/10.1109/TAES.2020.3010668 15. Degtyarev A. V., Shulga V. A., Zhivotov A. I., Dibrivny A. V. The development of lox-kerosene liquid rocket engines family for perspective launch vehicles of Yuzhnoye SDO based on proven technologies. Aerospace Technic and Technology. 2013. No 1. P. 44–50. (In Russian). 16. Prokopchuk А. А., Shulga V. А. The advanced liquid rocket engine line of SDO “Yuzhnoye” for the creation of new families of launch vehicles. Space Science and Technology. 2015. 21(5), P. 28–35. (In Russian). https://doi.org/10.15407/knit2015.05.028 17. Prokopchuk O. O., Shulga V. A. New and advanced liquid rocket engines of the Yuzhnoye SDO. Kosm. teh. Raket. vooruž. 2024. No 1 (121). P. 9–18. ( in Ukrainian) https://doi.org/10.33136/stma2024.01.009 18. Pylypenko V. V., Zadontsev V. A., Natanzon M. S. Cavitation Oscillations and Dynamics of Hydraulic Systems. Mechanical Engineering. Moscow: Mashinostroenie. 1977. 352 p. (In Russian). 19. Fox J. A. Hydraulic Analysis of Unsteady Flow in Pipe. London: Red Globe Press. 1977. 216 p. https://doi.org/10.1007/978-1-349-02790-3 20. Pylypenko O., Dolgopolov S., Nikolayev O., Khoriak N., Kvasha Yu., Bashliy I. Determination of the thrust spread in the Cyclone-4M first stage multi-engine propulsion system during its start. Science and Innovation. 2022. Vol. 18, № 6. P. 97–112. 21. Dolgopolov S., Cherniavskyi O., Shevchenko S. Mathematical modeling of dynamic processes in the branched reconfigurable fuel feed system of a liquid‑propellant rocket engine. CEAS Space Journal. 2025. P. 14. https://doi.org/10.1007/s12567-025-00691-y 22. Shevyakov A. A., Kalnin V. M., Naumenkova M. V., Dyatlov V. G. Theory of automatic control of rocket engines. Moscow: Mashinostroenie. 1978. 288 p. (in Russian). 23. Belyaev E. N., Chervakov V. V. Mathematical Modeling of LPRE. Moscow: MAI-PRINT Publ. 2009. 280 р. (in Russian). 24. Yan Z., Peng X., Cheng Y., Wu J. Modeling and simulation of system dynamics for spacecraft propulsion system. Applied Mechanics and Materials. 2012. Vols 229-231. Pp. 2112–2116. https://doi.org/10.4028/www.scientific.net/AMM.229-231.2112 25. Altshul, A. D. Hydraulic Resistances. M: Nedra, 1970. 216 p. (in Russian). текст 3 2026-03-31 Article Article application/pdf https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/169 Technical Mechanics; No. 1 (2026): Technical Mechanics; 23-33 Институт технической механики Национальной академии наук Украины и Государственного космического агентства Украины; № 1 (2026): Technical Mechanics; 23-33 ТЕХНІЧНА МЕХАНІКА; № 1 (2026): ТЕХНІЧНА МЕХАНІКА; 23-33 en https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/169/71 Copyright (c) 2026 Technical Mechanics |
| spellingShingle | рідинний ракетний двигун система живлення змінної структури розгалужений гідравлічний тракт математичне моделювання імпедансний метод частотна характеристика. CHERNIAVSKYI, O. S. DOLGOPOLOV, S. I. SHEVCHENKO, S. A. ОДНОШАРОВИЙ АЛГОРИТМ МАРШОВОГО РОЗРАХУНКУ НАДЗВУКОВОГО ОБТІКАННЯ РАКЕТ З РОЗБИТТЯМ РОЗРАХУНКОВОЇ ОБЛАСТІ НА ДЕКІЛЬКА ПІДОБЛАСТЕЙ |
| title | ОДНОШАРОВИЙ АЛГОРИТМ МАРШОВОГО РОЗРАХУНКУ НАДЗВУКОВОГО ОБТІКАННЯ РАКЕТ З РОЗБИТТЯМ РОЗРАХУНКОВОЇ ОБЛАСТІ НА ДЕКІЛЬКА ПІДОБЛАСТЕЙ |
| title_alt | FEATURES OF THE MATHEMATICAL SIMULATION OF DYNAMIC PROCESSES IN THE RECONFIGURABLE PROPELLANT FEED HYDRAULIC SYSTEM OF LIQUID-PROPELLANT ROCKET ENGINES |
| title_full | ОДНОШАРОВИЙ АЛГОРИТМ МАРШОВОГО РОЗРАХУНКУ НАДЗВУКОВОГО ОБТІКАННЯ РАКЕТ З РОЗБИТТЯМ РОЗРАХУНКОВОЇ ОБЛАСТІ НА ДЕКІЛЬКА ПІДОБЛАСТЕЙ |
| title_fullStr | ОДНОШАРОВИЙ АЛГОРИТМ МАРШОВОГО РОЗРАХУНКУ НАДЗВУКОВОГО ОБТІКАННЯ РАКЕТ З РОЗБИТТЯМ РОЗРАХУНКОВОЇ ОБЛАСТІ НА ДЕКІЛЬКА ПІДОБЛАСТЕЙ |
| title_full_unstemmed | ОДНОШАРОВИЙ АЛГОРИТМ МАРШОВОГО РОЗРАХУНКУ НАДЗВУКОВОГО ОБТІКАННЯ РАКЕТ З РОЗБИТТЯМ РОЗРАХУНКОВОЇ ОБЛАСТІ НА ДЕКІЛЬКА ПІДОБЛАСТЕЙ |
| title_short | ОДНОШАРОВИЙ АЛГОРИТМ МАРШОВОГО РОЗРАХУНКУ НАДЗВУКОВОГО ОБТІКАННЯ РАКЕТ З РОЗБИТТЯМ РОЗРАХУНКОВОЇ ОБЛАСТІ НА ДЕКІЛЬКА ПІДОБЛАСТЕЙ |
| title_sort | одношаровий алгоритм маршового розрахунку надзвукового обтікання ракет з розбиттям розрахункової області на декілька підобластей |
| topic | рідинний ракетний двигун система живлення змінної структури розгалужений гідравлічний тракт математичне моделювання імпедансний метод частотна характеристика. |
| topic_facet | liquid-propellant rocket engine reconfigurable feed system branched hydraulic circuit mathematical simulation impedance method frequency response. рідинний ракетний двигун система живлення змінної структури розгалужений гідравлічний тракт математичне моделювання імпедансний метод частотна характеристика. |
| url | https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/169 |
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