Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model
A mathematical model describing the nonlinear dynamical interaction of a launch vehicle (LV) and its marching liquid-propellant engine in the active phase of the LV flight is developed on the basis of the finiteelement discretization of the "propulsion – LV structure" self-oscillating syst...
Збережено в:
| Дата: | 2018 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут технічної механіки НАН України і НКА України
2018
|
| Назва видання: | Технічна механіка |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/173894 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model / O.D. Nikolayev, I.D. Bashliy, N.V. Khoryak // Технічна механіка. — 2018. — № 2. — С. 17-29. — Бібліогр.: 18 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-173894 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1738942020-12-27T01:26:02Z Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model Nikolayev, O.D. Bashliy, I.D. Khoryak, N.V. A mathematical model describing the nonlinear dynamical interaction of a launch vehicle (LV) and its marching liquid-propellant engine in the active phase of the LV flight is developed on the basis of the finiteelement discretization of the "propulsion – LV structure" self-oscillating system using three-dimensional and onedimensional finite elements. An approach to the computation of the liquid-propellant launch vehicle selfoscillation parameters self-oscillations of the liquid launch vehicle under POGO instability is developed. На основе конечно-элементной дискретизации автоколебательной системы «Жидкостная ракетная двигательная установка (ЖРДУ) – корпус ракеты-носителя (РН)» с использованием трехмерных и одномерных конечных элементов разработана математическая модель, описывающая нелинейное динамическое взаимодействие корпуса двухступенчатой РН (как сложной оболочечной конструкции с жидкостью) и ее маршевой ЖРДУ на активном участке полета РН, и развит подход к определению параметров автоколебаний жидкостной РН при ее продольной неустойчивости (POGO). На основі скінченно-елементної дискретизації автоколивальної системи «Рідинна ракетна двигунна установка (РРДУ) – корпус ракети-носія (РН)» із використанням тривимірних і одновимірних кінцевих елементів розроблено математичну модель, що описує нелінійну динамічну взаємодію корпусу двоступеневої РН (як складної оболонкової конструкції, що містить рідину) і її маршової РРДУ на активній частині польоту РН, і розвинено підхід до визначення параметрів автоколивань рідинної РН при її поздовжньої нестійкості (POGO). 2018 Article Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model / O.D. Nikolayev, I.D. Bashliy, N.V. Khoryak // Технічна механіка. — 2018. — № 2. — С. 17-29. — Бібліогр.: 18 назв. — англ. 1561-9184 http://dspace.nbuv.gov.ua/handle/123456789/173894 629.784:621.64:532.542 en Технічна механіка Інститут технічної механіки НАН України і НКА України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A mathematical model describing the nonlinear dynamical interaction of a launch vehicle (LV) and its marching liquid-propellant engine in the active phase of the LV flight is developed on the basis of the finiteelement discretization of the "propulsion – LV structure" self-oscillating system using three-dimensional and onedimensional finite elements. An approach to the computation of the liquid-propellant launch vehicle selfoscillation parameters self-oscillations of the liquid launch vehicle under POGO instability is developed. |
| format |
Article |
| author |
Nikolayev, O.D. Bashliy, I.D. Khoryak, N.V. |
| spellingShingle |
Nikolayev, O.D. Bashliy, I.D. Khoryak, N.V. Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model Технічна механіка |
| author_facet |
Nikolayev, O.D. Bashliy, I.D. Khoryak, N.V. |
| author_sort |
Nikolayev, O.D. |
| title |
Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model |
| title_short |
Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model |
| title_full |
Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model |
| title_fullStr |
Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model |
| title_full_unstemmed |
Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model |
| title_sort |
computation of the pogo self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model |
| publisher |
Інститут технічної механіки НАН України і НКА України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/173894 |
| citation_txt |
Computation of the POGO self-oscillation parameters in the dynamic "propulsion – rocket structure" system by using a 3d structural model / O.D. Nikolayev, I.D. Bashliy, N.V. Khoryak // Технічна механіка. — 2018. — № 2. — С. 17-29. — Бібліогр.: 18 назв. — англ. |
| series |
Технічна механіка |
| work_keys_str_mv |
AT nikolayevod computationofthepogoselfoscillationparametersinthedynamicpropulsionrocketstructuresystembyusinga3dstructuralmodel AT bashliyid computationofthepogoselfoscillationparametersinthedynamicpropulsionrocketstructuresystembyusinga3dstructuralmodel AT khoryaknv computationofthepogoselfoscillationparametersinthedynamicpropulsionrocketstructuresystembyusinga3dstructuralmodel |
| first_indexed |
2025-07-15T10:42:21Z |
| last_indexed |
2025-07-15T10:42:21Z |
| _version_ |
1837709281008287744 |
| fulltext |
17
UDC 629.784:621.64:532.542
O. D. NIKOLAYEV1, I. D. BASHLIY2, N. V. KHORYAK3
COMPUTATION OF THE POGO SELF-OSCILLATION PARAMETERS IN THE
DYNAMIC "PROPULSION – ROCKET STRUCTURE" SYSTEM BY USING A 3D
STRUCTURAL MODEL
Institute of Technical Mechanics
of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine,
15 Leshko-Popel St., 49005, Dnipro, Ukraine;
e-mail: 1odnikolayev@gmail.com; 2ibloha@i.ua; 3khoryak@i.ua
На основі скінченно-елементної дискретизації автоколивальної системи «Рідинна ракетна двигунна
установка (РРДУ) – корпус ракети-носія (РН)» із використанням тривимірних і одновимірних кінцевих
елементів розроблено математичну модель, що описує нелінійну динамічну взаємодію корпусу двоступе-
невої РН (як складної оболонкової конструкції, що містить рідину) і її маршової РРДУ на активній частині
польоту РН, і розвинено підхід до визначення параметрів автоколивань рідинної РН при її поздовжньої
нестійкості (POGO).
У запропонованому підході корпус рідинної ракети розглядається як складна багатозв'язкова диси-
пативна система «конструкція РН – рідке паливо в баках» і схематизується тривимірними скінченними
елементами, що дозволяє досліджувати просторові коливання корпусу РН і рідкого палива в його баках.
Моделювання низькочастотної динаміки насосів РРДУ виконується на основі розробленої в Інституті
технічної механіки Національної академії наук України і Державного космічного агентства України
(ІТМ НАНУ і ДКАУ) теорії кавітаційних автоколивань в насосних системах. При моделюванні низькочас-
тотних процесів в динамічній системі «РРДУ – корпус РН» враховуються нелінійності, які є найбільш
суттєвими при чисельному рішенні нелінійної задачі про POGO-коливання рідинної ракети – нелінійна
залежність об'єму і постійної часу кавітаційних каверн від режимних параметрів насосів та нелінійна
залежність декрементів коливань корпусу РН від амплітуд його коливань.
Проведено чисельне моделювання POGO-автоколивань двоступеневої РН загальною масою 165 т із
масою рідини в баках окислювача і пального першого ступеня 130 т. Для розрахункового випадку резона-
нсної взаємодії корпусу РН із РРДУ визначено параметри граничного циклу динамічної системи «РРДУ –
корпус РН». Показано, що при автоколиваннях переміщення елементів конструкції корпусу ракети, тисків
та витрат пального в елементах рідинного ракетного двигуна відбуваються з частотою 15,9 Гц, близькою
до частоти II -го тону власних поздовжніх коливань корпусу.
Розроблене науково-методичне забезпечення може бути використано для теоретичного визначення
параметрів автоколивань перспективних рідинних РН (в тому числі ракет, які мають складну просторову
конфігурацію корпусу) по відношенню до пружних поздовжніх і поперечних коливань конструкції, а та-
кож для оцінки динамічного навантаження конструкцій РН.
На основе конечно-элементной дискретизации автоколебательной системы «Жидкостная ракетная
двигательная установка (ЖРДУ) – корпус ракеты-носителя (РН)» с использованием трехмерных и одно-
мерных конечных элементов разработана математическая модель, описывающая нелинейное динамиче-
ское взаимодействие корпуса двухступенчатой РН (как сложной оболочечной конструкции с жидкостью)
и ее маршевой ЖРДУ на активном участке полета РН, и развит подход к определению параметров автоко-
лебаний жидкостной РН при ее продольной неустойчивости (POGO).
В предлагаемом подходе корпус жидкостной ракеты рассматривается как сложная многосвязная
диссипативная система «конструкция РН – жидкое топливо в баках» и схематизируется трехмерными
конечными элементами, что позволяет исследовать пространственные колебания корпуса РН и жидкого
топлива в его баках. Моделирование низкочастотной динамики насосов ЖРДУ выполняется на основе
разработанной в Институте технической механики Национальной академии наук Украины и Государст-
венного космического агентства Украины (ИТМ НАНУ и ГКАУ) теории кавитационных автоколебаний в
насосных системах. При моделировании низкочастотных процессов в динамической системе «ЖРДУ –
корпус РН» учитываются нелинейности, наиболее существенные при численном решении нелинейной
задачи о POGO-колебаниях жидкостной ракеты – нелинейная зависимость объема и постоянной времени
кавитационных каверн от режимных параметров насосов и нелинейная зависимость декрементов колеба-
ний корпуса РН от амплитуд его колебаний.
Проведено численное моделирование POGO-автоколебаний двухступенчатой РН общей массой 165 т
с массой жидкости в баках окислителя и горючего первой ступени 130 т. Для расчетного случая резонансно-
го взаимодействия корпуса РН и ЖРДУ определены параметры предельного цикла динамической системы
«ЖРДУ – корпус РН». Показано, что при автоколебаниях перемещения элементов конструкции корпуса
ракеты, давлений и расходов в элементах жидкостного ракетного двигателя происходят с частотой
15,9 Гц., близкой к частоте II -го тона собственных продольных колебаний корпуса.
Разработанное научно-методическое обеспечение может быть использовано для теоретического оп-
ределения параметров автоколебаний перспективных жидкостных РН (в том числе ракет, имеющих слож-
ную пространственную конфигурацию корпуса) по отношению к упругим продольным и поперечным
O. D. Nikolayev, I. D. Bashliy, N. V. Khoryak, 2018
Техн. механіка. – 2018. – № 2.
18
колебаниям конструкции, а также для оценки динамического нагружения конструкций РН.
A mathematical model describing the nonlinear dynamical interaction of a launch vehicle (LV) and its
marching liquid-propellant engine in the active phase of the LV flight is developed on the basis of the finite-
element discretization of the "propulsion – LV structure" self-oscillating system using three-dimensional and one-
dimensional finite elements. An approach to the computation of the liquid-propellant launch vehicle self-
oscillation parameters self-oscillations of the liquid launch vehicle under POGO instability is developed.
In the proposed approach, the rocket structure is considered as a complex multiply connected dissipative
system "LV structure – liquid propellant in tanks" and is schematized by three-dimensional finite elements, which
allows investigating the spatial vibrations of the LV structure and the liquid propellant in the tanks. Modeling of
the low-frequency dynamics of the rocket engine pumps is performed on the basis of the theory of cavitation self-
oscillations in pumping systems developed at the Institute of Technical Mechanics of the National Academy of
Sciences of Ukraine and the State Space Agency of Ukraine (ITM of NASU and SSAU). The most significant
nonlinearities in the numerical solution of the non-linear problem of liquid-propellant rocket POGO oscillations,
namely, the nonlinearity of the dependence of the cavitation volume and the cavitation time constant on the pump
operational parameters and the nonlinearity of the dependence of the LV structure oscillation decrements on the
LV structure vibration amplitudes, were taken into account in the model of the system low-frequency dynamics.
Numerical modeling of POGO self-oscillations of a two-staged LV with a total mass of 165 tons and with a
mass of 130 tons of the propellant in the first stage tanks is carried out. For the computation case of the resonant
interaction of the LV structure and the liquid-propellant rocket engine (LPRE), the limiting cycle parameters of
the dynamic "LPRE – LV structure" system are determined. It is shown that in the case of LV POGO self-
oscillations the structural elements vibrate and the pressures and the flow rates in the liquid-propellant rocket
engine oscillate at a frequency of 15.9 Hz, which is close to the natural frequency of the second mode of the struc-
tural longitudinal oscillations.
The scientific software developed may be used in the theoretical determination of the POGO self-oscillation
parameters of prospective liquid-propellant rockets (including rockets whose structure has a complex spatial con-
figuration) with respect to elastic longitudinal and transverse oscillations of the LV structure and in assessing
dynamic loads on LV structures.
Keywords: POGO instability, liquid-propellant rocket, self-oscillations, lon-
gitudinal structural vibrations, mathematical modeling, pump cavitation, low-
frequency propulsion system dynamics, three-dimensional finite elements.
1. Introduction. Longitudinal oscillations of the launch vehicle structure
(POGO-oscillations according to the NASA terminology) with frequencies from 3
Hz to 50 Hz – 100 Hz, arising due the loss of liquid rocket longitudinal stability
during the flight, can lead to disruption of the integrity of the launch vehicle (LV),
to operation failure of the LV control system and to other emergency situations
[1]. Hence, ensuring the longitudinal stability of liquid-propellant rockets is an
urgent task, requiring its solution in the design stage of new LV and in moderniza-
tion stage of old-designed LV [1 – 6].
At present, the problem of POGO oscillations as a problem of theoretical pre-
diction of the amplitude limiting level of longitudinal self-excited vibration of liq-
uid rockets in the leading research aerospace institutes is not solved. For example,
the American Aerospace Corporation [7], which for many years carries out the
most responsible research on POGO oscillations of liquid rockets, consider the
solution of this problem by mathematical modeling methods is impossible because
of the poor knowledge of nonlinear dynamic processes in systems and assemblies
of liquid rocket propulsion system (mainly, in cavitating pumps).
The problem of determining the amplitudes of the POGO oscillations at the
LV longitudinal instability on the base of a nonlinear mathematical model of the
"liquid propellant propulsion – rocket structure" dynamic system was first consid-
ered by Dr. M. Natanzon. He substantiated the "mechanism" of the longitudinal
self-oscillations of the rocket structure and showed the determining role of the
nonlinearities in the characteristics of LRE pumps for the development of longitu-
dinal oscillations of liquid rockets [1]. However, his POGO nonlinear analysis of a
liquid rocket was based on qualitative methods and estimates.
19
The nonlinear theory of longitudinal oscillations of liquid LVs was created
and further developed in works of the ITM NASU and SSAU under of Academi-
cian V. Pilipenko [in particular, 8 – 11]. Mathematical models of low-frequency
dynamics of cavitation pumps of liquid rocket engines have been developed, which
allow to carry out not only qualitative but also quantitative analysis of the effect of
cavitation phenomena in LRE pumps on the dynamic characteristics of a propul-
sion system [8], and also carry out theoretical forecasts of longitudinal stability
and determine the parameters of longitudinal oscillations of liquid rockets in the
LV flight. It was shown in [9] that the nonlinearities of the characteristics of LRE
cavitating inducer-centrifugal pumps are a factor limiting the growth of the ampli-
tudes of longitudinal oscillations of liquid rockets. An approach to the numerical
solution of the problem of determining the parameters of the limiting cycle of lon-
gitudinal oscillations of a liquid rocket using the method of harmonic linearization
was proposed in [10].
In the papers [11–12], the longitudinal components of the LV load factor os-
cillations were determined on the basis of nonlinear mathematical models of the
low-frequency dynamics of the system "liquid propellant propulsion – rocket struc-
ture". However, the LV structure longitudinal oscillations were described in these
models on the basis of the generally accepted simplification [1], i.e. by equations
in generalized normal coordinates, taking into account several (mostly three) lower
vibration modes, and the structural shapes of free longitudinal oscillations of the
investigated LV and liquid propellants in LV tanks were calculated without ac-
counting for damping and interaction of propellant with an LV elastic structures.
In work [13] the developed mathematical model of low-frequency dynamics of the
"liquid propellant propulsion – rocket structure" system is proposed, in which lon-
gitudinal oscillations of the LV structure are described as oscillations of a
branched chain of oscillators with damping effects. However, the model presented
was linear. It took into account only the lowest modes of natural longitudinal oscil-
lations of liquid propellant in the tanks. Thus, perspective directions of the further
development of the nonlinear mathematical model of low-frequency dynamics of
the "liquid propellant propulsion – rocket structure" system used to the study of
LV longitudinal stability and dynamic loads are the allowance for the interaction
of various shapes of fluid oscillations in LV tanks with oscillations of LV elastic
structure and dissipative losses at these oscillations.
2. Statement of the problem. The aim of this work is the mathematical mod-
eling of LV oscillations at which to describe the spatial oscillatory motion of LV
structure holds its three-dimensional finite element schematization, and the one-
dimensional finite elements schematization are used to describe the low-frequency
dynamics of liquid propellant rocket engines (LRE).
In accordance with the proposed approach, the development of a mathematical
model of POGO self-oscillations of a liquid LV is based on the following posi-
tions:
– parameters of self–excited oscillations of a liquid LV are determined on the
basis of a nonlinear mathematical model of the low-frequency dynamics of the
closed "LRE – LV structure" system with "frozen" coefficients;
– modeling of the spatial vibrational motion of the LV structure with liquid
propellant in tanks is carried out on the basis of the three-dimensional finite ele-
ment model of the system of liquid filled shell structures using modern computer
means of finite element analysis;
20
– spatial fluctuations of liquid propellants in the tanks are modeled taking into
account the motion of the free surface of the liquid and the interaction of the liquid
with the tank structure [15];
– when describing the LRE low-frequency dynamics, aggregates, hydraulic
and gas paths are considered as one-dimensional finite elements [2];
– allowance for the energy dissipation of liquid propellant oscillations and
elastic vibrations of the LV structure is carried out on the basis of the viscous fric-
tion model with the use of experimental values of the vibration decrements of the
LV structure and the nonlinear dependences of the decrements of its oscillations
on the vibration amplitudes;
– cavitation phenomena in LRE pumps are taken into account on the basis of
the hydrodynamic model of cavitation oscillations in pumping systems [8];
– the main nonlinearities are taken into account in model of the low–
frequency processes in the «LRE – LV structure» dynamic system: the nonlinear
dependences of the volume of the cavitation caverns and the cavitation time con-
stant on the operational parameters of the pumps [8], the dependence of the dec-
rements of the LV structure oscillations on the vibration amplitudes [17].
The proposed approach allows to investigate the POGO oscillations of liquid
LVs and determine the parameters of limiting cycle taking into account the com-
plex character of the interaction of the spatial oscillations of liquid propellant in
tanks, the elastic vibrations of the LV structure with nonlinear dissipative forces in
the spatial deformations of the LV structure and low-frequency processes in the
LPE.
3. Mathematical modeling of spatial vibrations of the liquid LV elastic
structure by the use of the finite element method. The liquid LV structure is a
complex hydromechanical system "the LV structure – liquid propellant in tanks".
As a rule, it has a complex configuration, is performed with variable wall thickness
of load-carrying structures of propellant tanks and includes various load-bearing
elements.
Using the capabilities of modern CAE–systems [14, 15], based on the finite
element method, allows in most cases to take into account the design features of
the LV structure during mathematical modeling of its dynamic interaction with the
LRE. In accordance with the proposed approach, the problem of numerical model-
ing and analysis of the spatial vibrations of a liquid rocket structure with frequen-
cies up to 100 Hz is solved in the following formulation:
– the LV structure is considered as a complex system of filled liquid shell
structures;
– the LV structure spatial vibrations are modeled as vibrations of the "LV
structure – liquid propellant in tanks" dissipative system using finite element
methods and modern computer finite element analysis [13].
A mathematical model describing the LV structure spatial vibrations taking in-
to account energy dissipation is represented by the matrix equation
0)(uK)(uD)(uM ttt ssssss , (1)
where sM , sK , sD are mass matrix, stiffness matrix and, accordingly, damping
factors matrix of the system "LV structure – liquid propellant in tanks", having a
size of ss nn 66 ; sn is number of nodes of the finite element model of the "LV
structure – liquid propellant in tanks" system; su is vector of system node move-
21
ments: Sn
iiss 1]u[u , ][u (x,y)
si
(x,z)
si
(y,z)
si
z
si
y
si
x
sisi u,u,u,u,u,u ; x , y , z are axis of
the Cartesian coordinate system, whose center is located on the axis of symmetry
of the LV structure at any point ),,( AAA zyxA , convenient for analyzing the dis-
turbed motion (the axis z coincides with the LV longitudinal axis and is the axis of
symmetry of the LV structure); (z)
si
(y)
si
(x)
si u,u,u are projections of the displacement
of the n-th node on the axes x , y , z ; (x,y)
si
(x,z)
si
(y,z)
si u,u,u are angles of rotation of
the n-th node around the axes x , y , z ; t is time.
Here and in the paper, the first derivative of a function with respect to time is
denoted by a variable with a point, and the second derivative by a variable with
two points: dttdt ss /)(u)(u ; 22 /)(u)(u dttdt ss .
The values of the damping coefficients for describing the vibrational motions
of the LV structure and the liquid propellants in equation (1) were selected in ac-
cordance with [16].
In modeling the spatial oscillations of liquid propellant in tanks, the following
generally accepted assumptions were made regarding the properties and nature of
liquid motion [1, 14]:
– liquid propellant is a homogeneous ideal compressible liquid;
– the motion of the fluid is irrotational;
– the forces of surface tension of liquid propellants are small, and their influ-
ence on the parameters of fluid and structure vibrations can be neglected.
The feature of modeling of spatial oscillations of the liquid propellants in LV
tanks was account the interaction of the liquid propellants with the tank elastic
structure and interaction of longitudinal and transverse oscillations of the liquid in
the tanks.
For modeling the spatial vibrations of the LV elastic structure, it was assumed
that the condition of joint deformation of the liquid and the tank structure is ful-
filled for the LV tanks (i. e. the condition of "sticking" of liquid on the tank wall).
This means that for nodes of the finite-element model of the system "LV structure
– liquid propellant in tanks" corresponding to the interface between the media
"wetted surface of the tank shell – liquid in the tank", displacement of finite ele-
ments "3D-liquid" [14] and shell elements of the tank structure are equal to each
other in the directions of the three axes of the local coordinate system. Further-
more, it was assumed that the condition of equality of nodal displacements in the
direction of the normal to the tank shell (the condition of ideal sliding of the liquid
along the wall of the tank) is fulfilled at the nodes of the finite element model sys-
tem "LV structure – liquid propellant in the tanks", corresponding media interface
of "wetted surface of the tank shell – the free surface of the liquid in the tank" for
elements «3D-liquid" and shell elements of the tank structure.
The pressure of liquid propellant in the nodes of the finite elements "3D liq-
uid" used in the schematization of liquid filling in the LV tank was determined in
the developed model by the equation
,uBGp fT (2)
where Tp is vector by fn6 length, the components of which are the pressure val-
ues in the finite elements "3D liquid"; fn is total number of finite element nodes
22
"3D liquid" in tanks; G , B are matrixes with size ff nn 66 : G is bulk elasticity
matrix, B is transition matrix, connecting deformations and node movements of
"3D-liquid" elements; fF n
i
(x,v)
fi
(x,z)
fi
(y,z)
fi
z
fi
y
fi
x
fi
n
ifif u,u,u,u,u,u 11 ][]u[u is vector
of nodal displacements of the "3D fluid" elements, which is part of the vector of
nodal displacements of the system (1): sf uu .
4. Development of a mathematical model of the dynamic interaction of
the first stage LV rocket engine and the LV structure. At flight conditions with
POGO phenomena the elastic LV structure and liquid propulsion forms a self-
oscillating "LRE – LV structure" dynamic system with positive feedback on the
thrust of the rocket engine [1], [2]. In accordance with [1], the elastic longitudinal
vibrations of the LV structure, caused by the longitudinal dynamic part of the en-
gine thrust. This vibrations cause pressure fluctuations on the bottom of the tank,
which lead to propellant pressure oscillations at the entrance to the engine. The
latter cause oscillations in the LRE combustion chamber pressure and engine thrust
oscillations that excite the LV structure vibrations. At unfavorable amplitude and
phase relationships between the LV structure oscillations and the thrust of the liq-
uid rocket engine (such relationships are realized in the resonant interaction of the
LV structure with the LPRE), these oscillations increase and the "LRE – LV struc-
ture" dynamic system loses stability. In case of stability loss of a linear dynamical
system, the amplitudes of oscillations of this system increase without limit. The
influence of nonlinear factors in the dynamic "LRE – LV structure" system with a
stable limit cycle consists mainly of the fact that: as the LV oscillations amplitudes
increase to a certain level, the resonant frequency of propellant oscillations in the
LRE feed line decreases and moves away from the frequency of the longitudinal
oscillations of the LV structure [1]. Accordingly, a "detuning" of the resonance
relations occurs in the dynamic system, which leads to a decrease in the intensity
of the feedback, and consequently, to a decrease in the oscillating energy influx.
When the influx of energy into the "LRE – LV structure" system is completely
compensated for by dissipative losses, the system sets to the self-oscillation mode.
The nonlinear mathematical model of the low-frequency dynamics of the sus-
tainer LRE, considered as an autonomous dynamic system "feed lines – LRE", can
be represented in general form by the equation
0),u,u,u(F teeee , (3)
where ],,[u 1 eneee uu is vector of variables of the system under consideration;
т
111 )],,,[),u,u,u(F t,u,,uu,u,u,u(Ft
eee eneeneeneeieeee is column vector con-
sisting of nonlinear functions eiF ( eni ,,1 ), which describe the low-frequency
processes in the LRE (superscript « т » denotes the vector transposition operation).
A feature of the mathematical model of the low-frequency dynamics of the
propulsion system used in modeling its dynamic interaction with the LV structure
was the description of the cavitation phenomena in the LRE pumps based on the
nonlinear hydrodynamic model of cavitation oscillations in systems with high-
speed centrifugal pumps with inducer [8]. The mathematical model of the dynam-
ics of cavitating pumps includes the equation of the dynamics of cavitation cav-
erns, the equation of fluid continuity in the flowing part of the pump, and the equa-
tion for determining the fluid pressure at the pump outlet:
23
dt
dVTB
W
GVkpp K
K
CP
KCP 1
2
1
1
*
1 2
),(
, (4)
12 GG
dt
dVg K , (5)
dt
dGJVpppp HKHH
2
12 )~(~ , (6)
where 1p , 1G are pressure and flow rate of liquid at the pump inlet; CPp is the
pump breakdown pressure; KV – volume of cavitation cavity; ),( 1
* GVk K is cavi-
tation number dependence *k from the volume of cavitation caverns and the flow
rate of fluid at the pump inlet;
2
2
1CPW
is speed head at the pump inlet; KTB ,1 are
cavitation elasticity and cavitation time constant; is fluid density; g is free fall
acceleration; 22 , Gp are the pressure and flow rate of liquid at the pump outlet;
Hp is pump head; )~(~
KH Vp is cavitation function of pump; KV~ is relative cavita-
tion cavity volume; HJ is coefficient of inertial resistance of fluid in the flowing
part of the pump.
The main nonlinear factors in this model are the dependences of the number of
cavitation *k and the cavitation time constant KT (or variable 11 BTE K ) in the
flowing part of the pump on the pressure 1p and flow rate 1G of the propellant at
the pump inlet.
The force effect of the LV structure on the marching LRE in the nonlinear
model of their dynamic interaction was taken into account in the equations of the
unsteady isothermal motion of a real compressible fluid on all sections of LRE the
hydraulic tract.
In the case, when the modeling is carried out taking into account only the lon-
gitudinal components of the oscillations of the parameters of the "LPRE – LV
structure" closed dynamic system, the motion of the liquid in the k-th section of
the hydraulic tract of the LPRE is described by the following equations [2]:
)/(2
1 gunhgGaGIpp k
sjCTkkkkkkk , (7)
1 kkkk GGpC , (8)
where kk Gp , are pressure and weight flow rate of fluid in the k -th element of
the LRE hydraulic path; kh is the projection length r of the k -th element on the
longitudinal axis of the LV; kC , ,ka kI – the concentrated compliance, the co-
efficient of hydraulic and inertial resistance for k -th section, respectively;
k
sju is longitudinal acceleration of j -th the LV structural finite element, to which
is rigidly fastened the k -th element of the LRE hydraulic tract.
The propellant pressure Bp at the tank outlet into the feed lines during LV
flight time this structural elastic oscillations was determined as follows:
24
CTB
din
TBHBTiHBB nHPpppp , (9)
where HBp the supercharge pressure of the tank; Tip is pressure in the finite
«3D-fluid» element, corresponding to the lower pole of the propellant tank; is
propellant density; BH is liquid propellant height at tank; CTn is the LV quasi-
static overload; din
TBp is dynamic component of propellant pressure at tank bottom,
due to nodal movements fu of finite «3D-fluid» element at the LV structure elas-
tic vibrations.
It follows from (3), (7) – (9), that low-frequency processes in the marching
LRE under the action of perturbations due to elastic longitudinal oscillations of the
LV structure are described by a system of equations
),u(F̂),u,u,u(F tt sseeee , (10)
where Т
61 )],ˆ[),u(F̂ t,u,u(Ft
snssisss is vector of perturbations acting in the LV
flight from the LV structure side on the LRE.
The mathematical model of the LV structure elastic oscillations arising from
the action of perturbing forces on it from the side of the operating LRE can be
written in the following form:
),u(F̂)(uK)(uD)(uM tttt eessssss , (11)
where ),u(F̂ tee is the vector of disturbing forces acting to the LV structure from
the side of the working liquid rocket engine.
The basic perturbing force in equation (11) is applied to the elements of the
LV structure in the place where the propulsion system is attached to rocket struc-
ture. This force is due to the deviation of the LRE thrust force R from its qua-
sistatic value CR :
C
din
CCC PPRRR / , (12)
where СP , din
CP are the steady-state value of the pressure and the dynamic pres-
sure in the combustion chamber.
In addition, the perturbing forces act on the LV structure in the elements of
the LRE hydraulic tract. The dynamic components kF of these forces for the k -th
element of the hydraulic tract are determined by the expression
kk
kz
sikfk
k
kkkkk AAg
G
umL
g
G
ApApF 11
1
2
2
11
, (13)
where kk Gp , are pressure and flow rate in the k -th element of the hydraulic
path; kA , 1kA , kL are the the area of the inlet and outlet cross-section and, re-
spectively, the length of the k -th element of the hydraulic tract; kfm is mass of
propellant in the k -th element of the hydraulic tract; i is number of finite element
in the LV structural model, with which is in the force interaction with the k -th
25
element of the hydraulic tract; z
siu is the displacement projection of the i -th
element of the LV structure on the longitudinal axis of the rocket.
A nonlinear mathematical model of the low-frequency dynamics of a closed
self-oscillating system "LRE – LV structure", determined the dynamic interaction
of the liquid rocket propulsion and the LV structure in the active part of flight, is
described by the system of equations (10) and (11). In the equation of forced elas-
tic vibrations of the rocket structure (8), taking into account the main non-linear
factor – the non-linear dependence of the amplitude of the structure oscillations on
the decrements – is carried out by means of a corresponding change in the values
of the matrix elements of the damping coefficients sD using the data given
in [17].
3. An example of the implementation of the proposed approach to the
computation of the LV self-oscillation parameters in case of longitudinal in-
stability the liquid launch vehicle. Numerical modeling of the dynamic processes
caused by the interaction of the LV structure and its first stage LRE was carried
out with reference to the hypothetic two-staged LV with a spacecraft. The LV first
stage is equipped with a liquid propulsion system. The length of the considered LV
is 38 m; diameter is 3.9 m; the total mass of the launch vehicle is 165 tons; the
mass of the liquid propellant in the oxidant and the fuel tanks of the LV first stage
is about 130 tons; The mass of the spacecraft is 3 tons, the total mass of the shell
structures of LV is 10 tons.
The finite element model
of the LV structure with liquid
propellant in the tanks included
1440 "elastic shell" elements,
496 "3D liquid" elements,
576 "3D mass" elements,
96 elements of "concentrated
mass" and the calculated grid
obtained in the CAE system
consisted of Ns = 2795 nodes.
This model was supplemented
by a model of the LRE low-
frequency dynamics, in which
the hydraulic and gas paths,
LREP pumps were represented
by one-dimensional finite ele-
ments (Fig. 1). The thrust force
of the LRE was relied on the
"3D mass" elements imitating
the vibrational motion of the
propulsion engine.
The Table 1 presents the
calculated values of the natural
frequencies jf of the LV struc-
ture free elastic vibrations in the
frequency range up to 50 Hz
and the values of the corre-Fig. 1
26
sponding effective (generalized) masses z
jm in the longitudinal direc-
tion 10,,1j . From these results it follows that the modes of the LV housing
with natural frequencies 1f = 12.4 Hz, 2f = 16 Hz and 6f = 39.3 Hz have the larg-
est generalized masses in the longitudinal direction.
The frequencies 1f , 2f , 6f are in the range of the oscillation frequencies of
the liquid propellant in the oxidizer and fuel feed lines, therefore, in this case the
dynamic interaction of the LRE and the LV structure can lead to the development
of longitudinal (POGO) LV self-induced oscillations with amplitudes that are un-
acceptable for the integrity of the LV and working capacity its systems
Analysis of the LV POGO stability of the LV performed on the basis of the lin-
earized model of the dynamic "LRE – LV structure" system for the relative operating
time of the first stage LRE max/Tt = 0.5 showed that the system is unstable at the fre-
quency close to 15.9 Hz. The loss of longitudinal stability of the liquid rocket under
investigation resulted from the resonance interaction of the longitudinal oscillations
of the LV structure and fluid oscillations in the oxidant feed line (with the natural
structure oscillation frequency of 16 Hz, and the frequency of the fluid in the oxidant
feed line of 16.1 Hz).
To determine the parameters of the limit cycle of longitudinal self-oscillations
of the LV, the mathematical model of the dynamic
"LRE – LV structure" system (7) – (8) was used,
in which the matrix equation (7) included nonlin-
ear equations of low-frequency dynamics of cavi-
tating pumps. To describe the cavitation phenom-
ena in pumps, the nonlinear hydrodynamic model
of V. Pilipenko was used [8]. Parameters of the
limiting cycle of POGO oscillations were deter-
mined by numerical integration of the nonlinear
system of ordinary differential equations (7) – (8)
by the Runge–Kutta method.
As a result of the system numerical integra-
tion, it is established that the considered "LRE –
LV structure" dynamic system goes to the limit
cycle with a frequency of self-oscillations of 15.8
Hz, close to the natural frequency of the second mode of the structural longitudinal
oscillations. The parameters of auto-oscillations of this system are determined: the
dynamic components of the pressure in the LV tanks, the movement and vibration
acceleration of the elements of the LV structure, as well as the flow rates and pres-
sures in the hydraulic elements of the liquid rocket engine.
Below are the calculated dependencies of a number of parameters of the "LRE
– LV structure" dynamic system versus time at the self-excited oscillation operat-
ing mode (in the resonance interaction of the liquid propulsion system and the LV
structure). Comparing the calculated dependences of the oxidant pressure P1 at the
entrance to the main rocket engine (Fig. 2) and the dynamic component of the
pressure Pkc in the combustion chamber (Fig. 3) versus time for auto-oscillations of
the LV, it is easy to see that the pressure oscillations at the entrance to the propul-
sion system and the pressure in the combustion chamber occur with the same fre-
quency of 15.8 Hz – the frequency of POGO self-oscillations.
Table 1
j jf , Hz z
jm , kg
1 12,4 42117
2 16,0 45633
3 25,2 548
4 33,2 1254
5 37,6 498
6 39,3 32461
7 41,0 1103
8 41,4 209
9 41,8 7408
10 46,3 135
27
The oscillations of the above calculated parameters have a non-harmonic (im-
pulse) character, since nonlinear dependences of the volume KV of cavitation cav-
erns in the flowing part of the cavitating inducer (Figure 4) and the catitation time
constant KT (or the parameter 1Е given in Fig. 5) from the cavitation number k and
the flow rate parameter q ( 321 qqq ) were used in the mathematical model (7) -
(8) of the dynamic system. Note that the cavitation number and flow rate are deter-
mined by the pressure and flow rate of the liquid at the pump inlet [8, 9].
–
As can be seen from Fig. 2, 3, the non-harmonic nature of the liquid oscilla-
tions is particularly pronounced in the oscillations of the oxidant pressure at the
engine inlet. It should be emphasized that this type of pressure oscillations at the
inlet to the cavitating pump corresponds to the flight data of liquid rockets in
which the POGO phenomenon was recorded at [18].
1E , МPа·s/m3
0 0.1 0.2 0.3 0.4
1
2
q1
q2
q3
k
Fig. 5
k
2
4
6
0.1 0.20
q1 q2 q3
33,10 mVk
Fig. 4
0
5
10
15
20
8 8.2 8.4 8.6 8.8 9
Fig. 2
t, s
p1, bar
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
8 8.2 8.4 8.6 8.8 9
Fig. 3
t, s
pkc, bar
28
As a result of the system numerical study, the POGO amplitudes of various
LV structure elements, including the spacecraft (in the cross section of its center of
mass), are determined at the self-excited oscillation operating mode. It is shown
that after the "LRE – LV structure" dynamic system has entered the limit cycle
mode, the values of vibration amplitudes of the spacecraft reach 0.13 g (Fig. 6).
Conclusions. An approach to the mathematical modeling of POGO self-
oscillations of liquid rockets by use of three-dimensional finite-element discretiza-
tion for the LV structure with liquid filling of its propellant tanks, the propulsion
systems by use of by one-dimensional finite elements discretization and with help
of nonlinear hydrodynamic model of cavitation phenomena in LRE pumps is pro-
posed.
Applied to the liquid two-staged rocket, a nonlinear mathematical model of
the low-frequency dynamics of the "LRE – LV structure" system is developed,
which describes the interaction of spatial oscillations of liquid propellant in tanks,
the LV elastic structural vibrations and low-frequency processes in the LRE on the
active part of the LV flight. It has been determine that the considered "LRE – LV
structure" dynamic system goes to the limit cycle with the self-oscillations fre-
quency of 15.8 Hz, close to the natural frequency of the LV structural second
modes of longitudinal oscillations. The parameters of self-oscillations of this sys-
tem are determined: the dynamic components of the pressure in the LV tanks, the
movement and vibration acceleration of the elements of the LV structure, as well
as the flow rates and pressures in the hydraulic elements of the liquid rocket en-
gine.
The proposed approach makes it possible to investigate the POGO oscillations
of liquid rockets (including rockets having a complex spatial configuration of pro-
pellant tanks and an asymmetric arrangement of stages) and to determine the pa-
rameters of the limit cycle of the rocket POGO self-oscillations. The use of this
approach allows one to take into account the interaction of the spatial oscillations
of liquid propellant in tanks, the elastic vibrations of the LV structure and low-
frequency processes in the LRE, and also the nonlinear dissipative forces in the
elastic deformations of structure.
The developed approach can be used to estimate the LV oscillation amplitudes
and the dynamic loads of the rocket structures of liquid launch vehicles under de-
sign, which, according to the results of the POGO linear analysis, are unstable with
respect to longitudinal oscillations at flight time phase.
t, s
0.2
-0.2
0
nz, g
Fig. 6
-0.10
-0.05
0.00
0.05
0.10
8 8.2 8.4 8.6 8.8 9
29
1. Natanson M. S. Longitudinal self-oscillation of liquid rocket. М : Mechanical engineering, 1977. 208 p.
2. Oppenheim B. W., Rubin S. Advanced Pogo Stability Analysis for Liquid Rockets. Journal of Spacecraft and
Rockets. 1993. Vol. 30. No. 3. P. 360 – 383.
3. Qingwei Wang, Shujun Tan, Zhigang Wu, Yunfei Yang, Ziwen Yu. Improved modelling method of Pogo
analysis and simulation. Acta Astronautica1. 07(2015). P. 262–273.
URL: http://dx.doi.org/10.1016/j.actaastro.2014.11.034.
4. Swanson L. A., Giel T. V. Design Analysis of the Ares I POGO Accumulator. AIAA 2009-4950. 45th
AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit (2–5 August 2009, Denver, Colorado).
URL: http://dx.doi.org/10.2514/6.2009-4950.
5. Zhihua Zhao, Gexue Ren, Ziwen Yu, Bo Tang, Qingsong Zhang. Parameter Study on Pogo Stability of Liquid
Rockets. Journal of Spacecraft and Rockets. 2011. Vol. 48. No. 3. P. 537 – 541. (doi: 10.2514/1.51877).
6. Junbeom Kim, Sang Joon Shin, Jongho Park, and Youdan Kim. Structural Modeling Reflected Nonlinearity for
Longitudinal Dynamic Instability (POGO) Analysis of Liquid Propellant Launch Vehicles in Preliminary
Design Phase. AIAA SPACE 2015 Conference and Exposition, AIAA SPACE Forum, (AIAA 2015-4594).
URL: http://dx.doi.org/10.2514/6.2015-4594
7. Dotson K. W., Phuong Than. Procedure for Mission-Specific Pogo Stability Analyses and Risk Assessments.
Spacecraft and Launch Vehicle Dynamic Environments Workshop Proceeding. (22 June 2005, El Segundo,
CA). The Aerospace Corporation, 19 p.
8. Pilipenko V. V. Cavitational self-oscillation. Kiev: Naukova Dumka, 1989. 316 p. (in Russian).
9. Pilipenko V. V., Zadonzev A. P., Grigoriev A. P., Belezkiy A. S. Estimation of amplitudes of liquid launch
vehicles longitudinal oscillations // Mechanics in Aviation and Astronautics. М., 1995. P. 27–34 (in Russian).
10. Belezkiy A. S. Estimation of amplitudes of liquid launch vehicles longitudinal oscillations by method of
harmonic linearization. Technical mechanics. 1993. Issue.2. P.58–63. (in Russian).
11. Pilipenko V. V., Dovgotko N. I., Dolgopolov S. I., Nikolayev O. D., Serenko V. A., Khoriak N. V. Theoretical
determination of amplitudes of liquid launch vehicle longitudinal oscillations. Space science and technology.
1999. Vol. 5. № 1. P. 90–96. (in Russian).
12. Pilipenko V. V., Dovgotko N. I., Pilipenko O. V., Nikolayev O. D., Pirog V. A., Dolgopolov S. I., Hodorenko
V. F., Khoriak N.V., Bashliy I. D. Theoretical prediction of spacecraft longitudinal vibrations of Cyclone-4
liquid launch vehicle. Technical mechanics. 2011. № 4. P. 30–36. (in Russian).
13. Khoriak N. V., Nikolayev O. D. Mathematical modeling of interaction of longitudinal oscillations of liquid
launch vehicle structure and dynamic processes in the propulsion system. Technical mechanics. 2010. № 3. P.
27–37. (in Russian).
14. Chigarev A. V., Kravchuk A. S., Smaluk A. F. ANSYS for engineers. Handbook. М : Mechanical engineering,
2004. 512 p. (in Russian).
15. Bashliy I. D., Nikolayev O. D. Mathematical modeling of spatial oscillations of shell structures with liquid
using modern computer-aided design and analysis tools. Technical mechanics. 2013. № 2 P. 12–22.
16. Nikolayev O. D., Bashliy I. D. Mathematical modeling of spatial oscillations of liquid in a cylindrical tank
with tank structure longitudinal vibrations. Technical mechanics. 2012. № 2. P. 14-22. (in Russian).
17. Nikolayev O. D., Khoriak N. V., Serenko V. A., Klimenko D. V., Hodorenko V. F., Bashliy I. D. Mathematical
modeling of longitudinal oscillations of liquid launch vehicle structure taking into account of dissipative
forces. Technical mechanics. 2016. № 2. P. 16–31. (in Russian).
18. Jarvinen, W & Kennoy, J & A. Kiefling, L & Odum, R & G. Papadopoulos, J & S. Ryan, R. (1970). A study
of Saturn AS-502 coupling longitudinal structural vibration and lateral bending response during boost. Journal
of Spacecraft and Rockets. 7. 10.2514/3.29884.
Получено 15.05.2018,
в окончательном варианте 29.05.2018
|