Some ways of hydrodynamic fin application for underwater supercavitating vehicles
The paper deals with considering the two non-traditional ways of application of hydrodynamic fins for a high-speed underwater supercavitating vehicles. The techniques for active roll stabilization and course control of the moving supercavitating vehicle are developed that use the regulation of the r...
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Інститут гідромеханіки НАН України
2018
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| Цитувати: | Some ways of hydrodynamic fin application for underwater supercavitating vehicles / V.N. Semenenko, O.I. Naumova // Гідродинаміка і акустика. — 2018. — Т. 1, № 3. — С. 355-371. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1742982021-01-12T01:26:37Z Some ways of hydrodynamic fin application for underwater supercavitating vehicles Semenenko, V.N. Naumova, O.I. The paper deals with considering the two non-traditional ways of application of hydrodynamic fins for a high-speed underwater supercavitating vehicles. The techniques for active roll stabilization and course control of the moving supercavitating vehicle are developed that use the regulation of the roll angle by means of both the special roll fin, and the automatic error-closing control system. The examples of a computer simulation of the course of maneuvering of the supercavitating vehicle controlled with the vertical hydrodynamic fins having zero roll angle stabilization are given along with the cases of the roll angle regulation. A method for determining the equilibrium motion parameters (balancing) of the supercavitating vehicle is developed for the case when a pair of identical horizontal cavity-piercing fins is used for complete or partial compensation of the vehicle's weight. The examples of a computer simulation of the motion of the supercavitating vehicle with horizontal fins in both the planing avoidance mode, and the combined mode are presented. It is shown that the steady-state longitudinal motion of the balanced supercavitating vehicle in the planing avoidance mode is stable "in the small", in contrast to its motion with planing in the cavity. It is found that, the horizontal fins in the combined motion mode can play a damping role suppressing the supercavitating vehicle motion instability "in the small", however, after a long time interval, the motion loses the global stability. The computer simulation suggests that the activation of the automatic depth stabilization makes the supercavitating vehicle motion stable in general in all the examined cases. Also it is revealed that course maneuverability of the supercavitating vehicle controlled with the vertical fins is maximal when starting the balancing in planing avoidance mode, but it deteriorates dramatically when starting the balancing in the combined mode. Рассмотрены два нетрадиционных способа применения гидродинамических рулей при движении высокоскоростных подводных суперкавитирующих аппаратов. Разработан метод активной стабилизации движения суперкавитирующего аппарата по крену и метод управления его движением по курсу путем регулирования угла крена с помощью специального руля крена и автоматической системы управления с обратной связью. Приведены примеры компьютерного моделирования маневрирования суперкавитирующего аппарата по курсу при управлении с помощью вертикальных гидродинамических рулей со стабилизацией нулевого угла крена и путем регулирования угла крена. Разработан метод нахождения равновесных значений параметров движения (балансировки) суперкавитирующего аппарата в случае, когда для полной или частичной компенсации его веса используется пара одинаковых горизонтальных рулей, проникающих из каверны в воду. Приведены примеры компьютерного моделирования движения суперкавитирующего аппарата с горизонтальными рулями в режиме без глиссирования и в смешанном режиме. Показано, что установившееся продольное движение сбалансированного суперкавитирующего аппарата без глиссирования, в отличие от его движения в режиме глиссирования в каверне, является устойчивым "в малом". Установлено, что в смешанном режиме движения горизонтальные рули могут играть демпфирующую роль, подавляя неустойчивость движения аппарата "в малом", однако на большом интервале времени движение теряет устойчивость в целом. Моделирование показало, что применение автоматической стабилизации движения по глубине во всех рассмотренных случаях делает его устойчивым в целом. Показано также, что курсовая маневренность такого аппарата при управлении с помощью вертикальных рулей максимальна при начальной балансировке в режиме без глиссирования, но резко ухудшается при начальной балансировке в смешанном режиме. Розглянуто два нетрадиційні способи застосування гідродинамічних рулів при русі високошвидкісних підводних суперкавітуючих апаратів. Розроблено метод активної стабілізації руху суперкавітуючого апарату за креном і метод керування його рухом за курсом шляхом регулювання кута крену за допомогою спеціального руля крену й автоматичної системи керування зі зворотним зв'язком. Наведено приклади комп'ютерного моделювання маневрування суперкавітуючого апарату за курсом при керуванні за допомогою вертикальних гідродинамічних рулів зі стабілізацією нульового кута крену і шляхом регулювання кута крену. Розроблено метод знаходження рівноважних значень параметрів руху (балансування) суперкавітуючого апарату у випадку, коли для повної чи часткової компенсації ваги апарату використовується пара однакових горизонтальних рулів, які проникають з каверни у воду. Наведено приклади комп'ютерного моделювання руху суперкавітуючого апарату з горизонтальними рулями в режимі без глісування і в змішаному режимі. Показано, що усталений поздовжний рух збалансованого суперкавітуючого апарату без глісування, на відміну від його руху в режимі глісування в каверні, є стійким "в малому". Встановлено, що в змішаному режимі руху горизонтальні рулі можуть грати демпфуючу роль, пригнічуючи нестійкість руху апарату "в малому", однак на великому інтервалі часу рух втрачає стійкість в цілому. Моделювання показало, що застосування автоматичної стабілізації руху по глибині у всіх розглянутих випадках робить його стійким в цілому. Показано також, що курсова маневровість такого апарату при керуванні за допомогою вертикальних рулів максимальна при початковому балансуванні в режимі без глісування, але різко погіршується при початковому балансуванні в змішаному режимі. 2018 Article Some ways of hydrodynamic fin application for underwater supercavitating vehicles / V.N. Semenenko, O.I. Naumova // Гідродинаміка і акустика. — 2018. — Т. 1, № 3. — С. 355-371. — Бібліогр.: 18 назв. — англ. 2616-6135 DOI: doi.org/10.15407/jha2018.03.355 http://dspace.nbuv.gov.ua/handle/123456789/174298 532.528 en Гідродинаміка і акустика Інститут гідромеханіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The paper deals with considering the two non-traditional ways of application of hydrodynamic fins for a high-speed underwater supercavitating vehicles. The techniques for active roll stabilization and course control of the moving supercavitating vehicle are developed that use the regulation of the roll angle by means of both the special roll fin, and the automatic error-closing control system. The examples of a computer simulation of the course of maneuvering of the supercavitating vehicle controlled with the vertical hydrodynamic fins having zero roll angle stabilization are given along with the cases of the roll angle regulation. A method for determining the equilibrium motion parameters (balancing) of the supercavitating vehicle is developed for the case when a pair of identical horizontal cavity-piercing fins is used for complete or partial compensation of the vehicle's weight. The examples of a computer simulation of the motion of the supercavitating vehicle with horizontal fins in both the planing avoidance mode, and the combined mode are presented. It is shown that the steady-state longitudinal motion of the balanced supercavitating vehicle in the planing avoidance mode is stable "in the small", in contrast to its motion with planing in the cavity. It is found that, the horizontal fins in the combined motion mode can play a damping role suppressing the supercavitating vehicle motion instability "in the small", however, after a long time interval, the motion loses the global stability. The computer simulation suggests that the activation of the automatic depth stabilization makes the supercavitating vehicle motion stable in general in all the examined cases. Also it is revealed that course maneuverability of the supercavitating vehicle controlled with the vertical fins is maximal when starting the balancing in planing avoidance mode, but it deteriorates dramatically when starting the balancing in the combined mode. |
| format |
Article |
| author |
Semenenko, V.N. Naumova, O.I. |
| spellingShingle |
Semenenko, V.N. Naumova, O.I. Some ways of hydrodynamic fin application for underwater supercavitating vehicles Гідродинаміка і акустика |
| author_facet |
Semenenko, V.N. Naumova, O.I. |
| author_sort |
Semenenko, V.N. |
| title |
Some ways of hydrodynamic fin application for underwater supercavitating vehicles |
| title_short |
Some ways of hydrodynamic fin application for underwater supercavitating vehicles |
| title_full |
Some ways of hydrodynamic fin application for underwater supercavitating vehicles |
| title_fullStr |
Some ways of hydrodynamic fin application for underwater supercavitating vehicles |
| title_full_unstemmed |
Some ways of hydrodynamic fin application for underwater supercavitating vehicles |
| title_sort |
some ways of hydrodynamic fin application for underwater supercavitating vehicles |
| publisher |
Інститут гідромеханіки НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/174298 |
| citation_txt |
Some ways of hydrodynamic fin application for underwater supercavitating vehicles / V.N. Semenenko, O.I. Naumova // Гідродинаміка і акустика. — 2018. — Т. 1, № 3. — С. 355-371. — Бібліогр.: 18 назв. — англ. |
| series |
Гідродинаміка і акустика |
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AT semenenkovn somewaysofhydrodynamicfinapplicationforunderwatersupercavitatingvehicles AT naumovaoi somewaysofhydrodynamicfinapplicationforunderwatersupercavitatingvehicles |
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2025-07-15T11:14:18Z |
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2025-07-15T11:14:18Z |
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1837711290953367552 |
| fulltext |
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 355355–371371.
UDC 532.528
SOME WAYS OF HYDRODYNAMIC FIN APPLICATION
FOR UNDERWATER SUPERCAVITATING VEHICLES
V. N. Semenenko†, O. I. Naumova
Institute of Hydromechanics of NAS of Ukraine
Zhelyabov Str., 8/4, 03057, Kyiv, Ukraine
†E-mail: vnsvns60@gmail.com
Received 25.09.2017
The paper deals with considering the two non-traditional ways of application of hy-
drodynamic fins for high-speed underwater supercavitating vehicles. The techniques
for active roll stabilization and course control of the moving supercavitating vehicle
are developed that use the regulation of the roll angle by means of both the special
roll fin, and the automatic error-closing control system. The examples of a computer
simulation of the course of maneuvering of the supercavitating vehicle controlled with
the vertical hydrodynamic fins having zero roll angle stabilization are given along with
the cases of the roll angle regulation. A method for determining the equilibrium motion
parameters (balancing) of the supercavitating vehicle is developed for the case when a
pair of identical horizontal cavity-piercing fins is used for complete or partial compen-
sation of the vehicle’s weight. The examples of a computer simulation of the motion
of the supercavitating vehicle with horizontal fins in both the planing avoidance mode,
and the combined mode are presented. It is shown that the steady-state longitudinal
motion of the balanced supercavitating vehicle in the planing avoidance mode is stable
“in the small”, in contrast to its motion with planing in the cavity. It is found that, the
horizontal fins in the combined motion mode can play a damping role suppressing the
supercavitating vehicle motion instability “in the small”, however, after a long time
interval, the motion loses the global stability. The computer simulation suggests that
the activation of the automatic depth stabilization makes the supercavitating vehicle
motion stable in general in all the examined cases. Also it is revealed that course ma-
neuverability of the supercavitating vehicle controlled with the vertical fins is maximal
when starting the balancing in planing avoidance mode, but it deteriorates dramati-
cally when starting the balancing in the combined mode.
KEY WORDS: supercavitating vehicle, control, maneuvering, fins, roll, computer sim-
ulation
1. INTRODUCTION
A natural way to attain a very high speed in water is to form the motion in the supercav-
itation regime, when a cavity filled by water vapour or gas is formed around a body with the
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Fig. 1. Two modes of a SC-vehicle motion:
a — planing in a cavity mode, b — planing avoidance mode
help of a special nose cavitator [11]. In this case the water drag to the motion is dramatically
decreased due to minimization of both the friction drag and the base drag.
The regime of motion of a supercavitating vehicle (briefly a SC-vehicle) is traditional,
when the vehicle is planing by its small tail part along the lower cavity wall (the planing
in a cavity mode, see Fig. 1Fig. 1a). In this case the vehicle weight 𝐹𝑔 is compensated by the
hydrodynamic lift on the wetted body part 𝐹𝑠𝑦, and an arising moment relatively to the
vehicle mass center is compensated by the moment of the lift 𝐹𝑛𝑦 created by the cavitator
inclination.
The SC-vehicle dynamics is very complex due to the nonstationary behavior of the cavity
and the complex discontinuous interaction of the body with the cavity walls. In particular,
the steady-state longitudinal motion of the SC-vehicle in the planing in a cavity mode is
unstable on depth, therefore there is a necessity of active depth stabilization of the SC-
vehicle motion. In practice, the depth stabilization of the SC-vehicle motion is realized by the
automatic error-closing regulation of the cavitator inclination angle 𝛿𝑧 (𝛿-stabilization) [22].
The second problem is choosing an optimal strategy of control of the SC-vehicle motion,
that ensures specified parameters of its course maneuverability [33,44].
In our works [55–88], the comparative analysis of the following three methods of the SC-
vehicle motion control was given:
1) inclination of the cavitator with two degrees of freedom (𝛿-control);
2) deflection of the propulsor thrust vector with two degree of freedom (𝜂-control);
3) using vertical and horizontal hydrodynamic cavity-piercing fins (𝑓 -control).
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 355355–371371.
In this case we supposed that the SC-vehicle moves in the traditional planing in a cavity
mode (see Fig. 1Fig. 1a). The obtained conclusions are briefly listed in Section 33.
The paper [99] considers an alternative scheme of the SC-vehicle motion avoiding the
planing of the vehicle tail part along the cavity wall, which results in instability of the SC-
vehicle motion (the planing avoidance mode). In this case a pair of horizontal cavity-piercing
fins located in the vehicle tail part are used to compensate the vehicle weight 𝐹𝑔 (see Fig. 1Fig. 1b).
The work [1010] is based on a similar idea, in this case the weight compensation and stability
of the SC-vehicle longitudinal motion is ensured by means of an ring-type tail wing. The
work [1111] considers a possibility of maintenance of the planing avoidance mode by regulating
the gas supply into a ventilated cavity.
Use of hydrodynamic fins as operation controls of the SC-vehicle motion, which serve
directly to create transversal control forces (as in [44, 88]), may be considered as traditional.
This article considers two non-traditional ways of application of hydrodynamic fins for SC-
vehicles.
A method of active roll stabilization of the SC-vehicle and a method of course control
of the SC-vehicle motion by regulating the roll angle 𝜃 (𝜃-control, which is analogous to
the “bank-to-turn” control) are investigated in Sections 44 and 55. Examples of computer
simulation of course maneuvering of the SC-vehicles at both the 𝑓 -control with the zero roll
angle 𝜃-stabilization, and the 𝜃-control are given.
A method for determination of the balanced values of the motion parameters (balancing)
of the SC-vehicle with horizontal fins in the planing avoidance mode (see Fig. 1Fig. 1b) and in the
combined mode, i. e., if both the horizontal fins and planing of the vehicle body in a cavity
are presented, is considered in Section 66. Examples of computer simulation of motion of the
SC-vehicle balanced in both the planing avoidance mode and the combined mode are given
in Section 77.
2. CALCULATION METHOD AND DESIGN MODEL
For computer simulation of dynamics of the self-propelled guided underwater SC-vehicles
with ventilated supercavities we use the approximation mathematical model of 3D motion of
supercavitating bodies based on the principle of independence of the unsteady cavity section
expansion by G. V. Logvinovich [11]. It includes the following equations and relations:
∙ – equations of 3D-dynamics of a solid body with six degrees of freedom in the body
coordinates;
∙ equations for calculation of an unsteady cavity shape and its location in the flow
coordinates with taking into account the cavity distortions caused by both the cavitator
inclination and the gravity effect [1212,1313];
∙ equations for pressure in a ventilated cavity 𝑝𝑐(𝑡), which is determined by a difference
between a rate of gas supply into the cavity and a rate of gas loss from the cavity [1313,1414];
∙ relations connecting the acting hydrodynamic forces and moments in the body coordi-
nates with the current body and cavity parameters, where the force components when
the vehicle is planing in a cavity are calculated by E. V. Paryshev’s formulae [77, 1515];
to calculate forces on the fins and dimensions of cavities forming past the fins, we use
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 355355–371371.
Fig. 2. Sketch of the SC-vehicle design model
the known relations of the linear theory of the supercavitating hydrofoils with a small
aspect ratio [88].
All the calculations were performed for the SC-vehicle design model, which is shown
in Fig. 2Fig. 2. Its main parameters are the following: the length 𝐿 = 6.0 m, the cylindrical
part diameter 𝐷𝑏 = 340 mm, the cavitator diameter 𝐷𝑛 = 70 mm, the mass 𝑚 = 600 kg,
the mass center position 𝑥𝑐 = 3.0 m, the moments of inertia relatively to the model axes
𝐼𝑥 = 8.0 kg·m2, 𝐼𝑦 = 𝐼𝑧 = 900 kg·m2. The vehicle has the aft tube with the length 1.0 m
and diameter 200 mm, where a nozzle of a rocket or hydrojet engine is located in practice.
The starting parameters of the model motion in the cruise phase are the following: veloc-
ity 𝑉 = 120 m/s, depth 𝐻 = 10 m. In this case the cavitation number is 𝜎 = 0.02, the cavity
length is 𝐿𝑐 = 6.565 m, the volumetric gas supply rate into the cavity is �̇�𝑖𝑛 = 72.125 l/s.
The balanced values of the parameters in the model without fins steady-state motion in the
planing in a cavity mode are the following: the cavitator inclination angle 𝛿𝑧 = −5.774∘, the
pitch angle 𝜓 = 0.371∘, the propulsor thrust 𝐹𝑝𝑟 = 23.219 kN.
The model may have a pair of identical vertical fins 1, pair of identical horizontal fins 2,
and the bottom roll fin 3. All the fins have a rectilinear shape in plan and a wedge shape
in cross section (see Fig. 3Fig. 3). Here, 𝑐𝑓 is the fin chord; ℎ𝑓 is the fin span; 𝛽𝑓 is the wedge
semiangle in the fin section; 𝛿𝑓 is the fin deflection angle; 𝛼𝑓 is the fin effective angle of
attack. Axes of both the flow coordinates 𝑂𝑥0𝑦0𝑧0 and the body coordinates 𝑂𝑥1𝑦1𝑧1 are
shown in Fig. 3Fig. 3 for the vertical fin case. Real position and dimensions of the fins are given
in Table 1Table 1.
The vertical fins 1 are intended for the SC-vehicle course maneuvering by deflection on
the angle 𝛿𝑓𝑣 (see Fig. 3Fig. 3). The horizontal fins 2 are intended for balancing the SC-vehicle by
deflection on the angle 𝛿𝑓ℎ. The roll fin 3 is intended for regulating the vehicle roll angle 𝜃
by deflection on the angle 𝛿𝑓𝑏.
3. FEATURES OF CONTROL OF SC-VEHICLE MOTION
A course maneuverability of an underwater vehicle is its ability to move with a specified
radius of a turning circle 𝑟 in the horizontal plane. A simple analysis shows that the high-
speed SC-vehicles can turn with a minimal radius being (2 . . . 3) orders larger than in the
continuous flow, because surfaces of the vehicle contact with water are small [55]. As was
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 355355–371371.
a b
Fig. 3. Scheme of the SC-vehicle fin:
a — side view, b — cross section
Table 1. Position and dimensions of the design model fins
Fins 𝑥𝑓 , m 𝑐𝑓 , mm ℎ𝑓 , mm 𝛽𝑓 ,
∘
horizontal 4.80 100.0 130.0 3.0
vertical 2.30 100.0 130.0 3.0
roll fin 4.94 50.0 80.0 3.0
shown in work [1616], possibility of the SC-vehicle course maneuvering is additionally limited
owing to the cavity deformation and, as a consequence, the vehicle wetting. A simple method
of estimation of the minimal radius of a turning circle for the SC-vehicle with a disk cavitator
is given in [1616].
The analysis has shown that the most effective operating control for the SC-vehicle motion
is the cavitator (𝛿-control). However, the 𝛿-control is often insufficient to attain admissible
according to [1616] radii of a turning circle, because the cavitator inclination angle is limited
by a some maximal value 𝛿𝑚𝑎𝑥. In addition, the cavitator is usually used as an operating
control of an automatic system of the motion depth stabilization [22,66]. Therefore, in practice
one has to use the hydrodynamic cavity-piercing fins (𝑓 -control) for the SC-vehicle course
maneuvering.
The SC-vehicle maneuvering with the help of 𝑓 -control is associated with a significant
increase in the motion drag. In the case of high-speed SC-vehicles, the cavity-piercing fins
are flown in the supercavitation regime. In this case the efficiency of such an operating
control is decreased in comparison with the case of the continuous flow. In addition, the
stepwise changing the transversal forces accompanied by a hysteresis effect may occur when
varying the fin effective angle of attack 𝛼𝑓 (see [1717]).
Since the transversal forces on fins of each pair are usually different owing to the cavity
asymmetry, the total axial moment arises:
𝑀𝑓𝑥 = 𝐹𝑣2𝑦𝑣2 − 𝐹𝑣1𝑦𝑣1 + 𝐹ℎ2𝑧ℎ − 𝐹ℎ1𝑧ℎ1, (1)
where 𝑦𝑣1, 𝑦𝑣2, 𝑧ℎ1, 𝑧ℎ2 are distances from the points of the transversal force application
(𝐹𝑣1, 𝐹𝑣2 on the vertical fins, 𝐹ℎ1, 𝐹ℎ2 on the horizontal fins, respectively) up to the vehicle
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 355355–371371.
longitudinal axis. Here, index 1 is related to both the lower vertical fin and the left horizontal
fin, index 2 is related to both the upper vertical fin and the right horizontal fin. The moment
𝑀𝑓𝑥 leads to increasing the vehicle roll angle 𝜃 and, hence, to unfavorable changing the
control force direction. Indeed, the horizontal projections of the forces acting on the vertical
fins are decreased with increasing the roll angle. Hence, the 𝑓 -controlled SC-vehicle motion
must be stabilized not only on depth, but on the roll as well.
It was shown in work [88] that the frontal location of the vertical fins (the “duck” scheme)
is the most effective for the SC-vehicle course maneuvering. When displacing the vertical
fins to the aft, their efficiency is decreased approximately by a linear law. The traditional
for surface ships and submarines aft location of fins is unacceptable for the SC-vehicle course
maneuvering.
The method of 𝜂-control by deflection of the propulsor thrust vector is commonly used
in aviation and rocketry. But as applied to the SC-vehicle motion 𝜂-control is equivalent to
the 𝑓 -control with the aft fins from a hydrodynamic point of view, i. e., it is unacceptable.
4. ROLL STABILIZATION OF SC-VEHICLE MOTION
As was said above, the roll stabilization must be ensured when the 𝑓 -controlled SC-vehicle
course maneuvering. The SC-vehicle roll stabilization may be realized by both the passive
methods and the active methods. The passive hydrostatic stabilization of the zero roll angle
is realized if the vehicle has the metacentric height (as for surface ships and submarines).
The metacentric height ℎ𝑀 is a distance between a center of the vehicle cross section and of
the vehicle mass center. If the vehicle mass center is lower than the cross section center on
the value ℎ𝑀 , then the restoring moment relatively to the vehicle longitudinal axis occurs
with the vehicle roll on the angle 𝜃:
𝑀𝑔𝑥 = −ℎ𝑀𝑚𝑔 cos𝜓 sin 𝜃, (2)
where 𝜓 is the model pitch angle. As a result, the vehicle will make low-frequency oscillation
on roll about the value 𝜃 = 0 during motion. In other words, the vehicle is statically stable
relatively perturbations of the zero roll angle when ℎ𝑀 > 0.
In practice, as applied to the SC-vehicles, the efficiency of the passive hydrostatic roll
stabilization is limited because the really achievable values of ℎ𝑀 are small. A method of the
passive hydrodynamic roll stabilization with the help of a 𝑉 -shaped system of cavity-piercing
fins is known as well [1818]. The fin parameters are selected in such a way that the fins create
a total restoring axial moment with the vehicle roll.
We propose a method of active hydrodynamic stabilization of the specified roll angle of
the SC-vehicle with the help of a special roll fin. The roll fin 3 is located in the vehicle
bottom tail part in the zone of the vehicle planing along the lower cavity wall (see Fig. 2Fig. 2).
Dimensions of the roll fin may be essentially smaller than dimensions of the vertical fin 1,
because it serves not to create the transversal force, as at the 𝑓 -control, but only to create
an axial moment. The law for the automatic system of error-closing regulating the angle of
the roll fin deflection 𝛿𝑓𝑏 for maintenance of the roll angle 𝜃 = 𝜃2 is set in the form of the
“plain autopilot” [22]:
𝛿𝑓𝑏(𝑡) = −𝑢1[𝜃(𝑡− 𝑡1) − 𝜃2] − 𝑢2𝜔𝑥(𝑡− 𝑡1), (3)
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a b
Fig. 4. Operation of the automatic system of the zero roll angle stabilization:
a — dependence 𝛿𝑓𝑏(𝑡), b — dependence 𝜃(𝑡)
where 𝑢1 ≥ 0, 𝑢2 ≥ 0 are the feedback coefficients (transfer ratios); 𝑡1 is the lag time of the
actuator operation; 𝜔𝑥 is the angular velocity relatively to the vehicle longitudinal axis. In
this case the design restriction for varying the roll fin deflection angle has to been introduced:
|𝛿𝑓𝑏(𝑡)| ≤ 𝛿𝑓𝑏1, 𝑡 > 𝑡𝑏𝑒𝑔, (4)
where 𝑡 = 𝑡𝑏𝑒𝑔 is time of beginning of the 𝜃-stabilization system operation.
Fig. 4Fig. 4 shows an example of operation of the automatic system of stabilizing the zero roll
angle at computer simulation of the 𝑓 -controlled SC-vehicle course maneuvering. Graphs
of varying the roll fin deflection angle 𝛿𝑓𝑏 (Fig. 4Fig. 4a) and corresponding attained magnitudes
of the roll angles 𝜃 (Fig. 4Fig. 4b) on the starting interval 0 < 𝑡 < 5.0 s are given in Fig. 4Fig. 4.
The calculations were performed for the following control parameters: 𝛿𝑓𝑣 = 5.0∘, 𝑢1 = 0.1,
𝑢2 = 0.3, 𝛿𝑓𝑏1 = 10.0∘.
As can be seen, both the functions 𝛿𝑓𝑏(𝑡) and 𝜃(𝑡) take the form of steady quasiperiodic
oscillation with the fundamental frequency 6.80 Hz on completion of the transient process. In
this eample, the following maneuver parameters were attained: �̇� = 7.645∘/s, 𝑟 = 0.888 km.
Here �̇� is the speed of varying the path angle 𝜒 = 𝜙− 𝛽; 𝑟 is the radius of a turning circle:
�̇� = 𝑉
𝑑𝜒
𝑑𝑠
, 𝑟 =
𝑉
�̇�
, (5)
where 𝜙 is the yaw angle; 𝛽 is the sliding angle; 𝑠 is the angular position of the vehicle mass
center along the path.
We note for comparison, that when the 𝜃-stabilization of the zero roll angle was absent in
this example, the motion of the model lasted for only 0.3 s, after that the model was wetted.
In this case the model roll angle was 𝜃 = 140∘ at the time 𝑡 = 0.3 s.
5. COURSE 𝜃-CONTROL OF SC-VEHICLE MOTION
The method of 𝜃-control of the SC-vehicle motion on course is based on that the effective
angle of the cavitator inclination in the horizontal plane 𝛿𝑦 is changed with varying the
SC-vehicle roll angle 𝜃. Hence, projections of vector of the force acting onto the cavitator
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are changed too. Indeed, the vehicle roll on the angle 𝜃 with preliminary inclination of
the cavitator in the diametral plane 𝑂𝑥1𝑦1 on the angle 𝛿′𝑧 is equivalent to the cavitator
inclination in the two transversely-spaced planes on the angles 𝛿𝑧 and 𝛿𝑦 when 𝜃 = 0. The
angles 𝛿′𝑧, 𝜃, 𝛿𝑧, and 𝛿𝑦 are linked by relations:
tg 𝛿𝑧 =
tg 𝛿′𝑧
cos 𝜃
, tg 𝛿𝑦 =
tg 𝛿′𝑧
sin 𝜃
. (6)
The cavitator inclination on the angle 𝛿𝑦 creates the transversal component of the force
𝐹𝑛𝑧 [55]. Thus, the cavitator is the real active operating control at the 𝜃-control of the SC-
vehicle motion. Therefore, the 𝜃-control is as effective as the 𝛿-control. The required vehicle
roll angle 𝜃2 ̸= 0 is attained with the help of the special roll fin 3 (see Fig. 2Fig. 2). To solve a
problem of setting and further maintenancing of the required roll angle 𝜃2, we proposed to
use the following algorithm which is analogous to the “reverse of rudder” algorithm when
turning surface ships.
1. It is supposed that the roll fin is absent on time interval 0 < 𝑡 < 𝑡𝑏𝑒𝑔 and the model
moves without roll: 𝜃 = 0, 𝜔𝑥 = 0.
2. The roll fin is deflected on the specified angle 𝛿𝑓𝑏 at the time 𝑡 = 𝑡𝑏𝑒𝑔. After that both
the model roll angle 𝜃 and the angular velocity 𝜔𝑥 begin to increase.
3. The angle of the roll fin deflection is reversed at the time when 𝜃 = 𝜃1/2, where 𝜃1 is
the some test roll angle. As a result, the angular velocity 𝜔𝑥 begins to decrease, but
the model roll angle 𝜃 continues to increase.
4. At the time when 𝜔𝑥 = 0 (in this case the roll angle possesses the value 𝜃 = 𝜃2) the
roll fin angle deflection is set to zero, and then the automatic system of the roll angle
𝜃2 stabilization (3)(3) is activated.
We note that presence of the metacentric height ℎ𝑀 > 0 plays a negative role when using
the 𝜃-control of the SC-vehicle motion. Indeed, when the required roll angle 𝜃2 is stabilized
by the automatic control system (3)(3), it must permanently overcome the axial moment (2)(2)
tending to set the roll angle 𝜃 = 0.
Fig. 5Fig. 5 shows an example of calculating the process of setting and automatic 𝜃-stabilization
of the specified roll angle 𝜃2 = 15.0∘ for the SC-vehicle without vertical fins. The calculation
was performed for the following parameters: 𝜃1 = 17.6∘, 𝑡𝑏𝑒𝑔 = 1.0 s, 𝑢1 = 0.1, 𝑢2 = 0.3,
𝛿𝑓𝑏1 = 10.0∘. A graph of the dependence 𝛿𝑓𝑏(𝑡) is given in Fig. 5Fig. 5a, a graph of corresponding
variation of the model roll angle 𝜃(𝑡) is given in Fig. 5Fig. 5b.
Table 2Table 2 gives results of calculation of the average speed of the model course turning �̇�
and the radius of a turning circle 𝑟 for a number of the roll angles 𝜃. The calculation for each
column of Table 2Table 2 were performed for fixed magnitude of the roll angle 𝜃 on the distance
1.0 km when 𝑥𝑐 = 2.5 m and ℎ𝑀 = 0.
Fig. 6Fig. 6 shows the calculated path configurations of the model mass center (projections
onto the horizontal plane) at the 𝜃-control for a number of the regulated roll angles 𝜃2 on
the distance 4.0 km. Comparison with [55] shows that the 𝜃-control of the SC-vehicle motion
for its course maneuvering is quite as effective as the 𝛿-control.
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a b
Fig. 5. Operation of the automatic system of setting and stabilizing the specified roll angle:
a — dependence 𝛿𝑓𝑏(𝑡), b — dependence 𝜃(𝑡)
Table 2. Dependence of speed of the SC-model turning and radius
of a turning circle on the roll angle
𝜃,∘ 0 2.0 5.0 10.0 20.0 30.0 40.0
�̇�,∘/s 0 −0.417 −0.743 −0.849 −1.715 −2.711 −3.942
𝑟, km ∞ 37.5 16.5 8.09 4.01 2.54 1.74
Fig. 6. Path configurations of the 𝜃-controlled SC-model:
1 — 𝜃2 = 6.61∘, 2 — 𝜃2 = 20.85∘, 3 — 𝜃2 = 44.24∘
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6. SC-VEHICLE BALANCING WITH FINS
Let us consider the SC-vehicle motion mode, when a pair of horizontal cavity-piercing
fins is used for full or partial compensation of the vehicle weight 𝐹𝑔 (see Fig. 1Fig. 1b).
The SC-vehicle steady-state longitudinal motion is considered as balanced if a sum of
all the forces acting, and a sum of their moments relatively to the model mass center are
equal to zero. To determine balanced values of the parameters, we use the numerical method
described in work [1414]. In this case, the cavitator force 𝐹𝑛, the horizontal fin force 𝐹𝑓 , the
planing force 𝐹𝑠, the propulsor thrust 𝐹𝑝𝑟, and the body weight 𝐹𝑔 act onto the vehicle. A
set of equations of the SC-vehicle balance in projections onto the body coordinates has the
form:
𝐹𝑛𝑥 + 𝐹𝑠𝑥 + 𝐹𝑓𝑥 + 𝐹𝑔 sin𝜓 − 𝐹𝑝𝑟 cos 𝜂𝑧 = 0,
𝐹𝑛𝑦 + 𝐹𝑠𝑦 + 𝐹𝑓𝑦 + 𝐹𝑔 cos𝜓 − 𝐹𝑝𝑟 sin 𝜂𝑧 = 0,
𝑀𝑛𝑧 +𝑀𝑠𝑧 +𝑀𝑓𝑧 + 𝐹𝑝𝑟 sin 𝜂𝑧(𝐿− 𝑥𝑐) = 0,
(7)
where 𝜂𝑧 is the thrust vector deflection; 𝑀𝑛𝑧 = −𝐹𝑛𝑦𝑥𝑐; 𝑀𝑠𝑧 = 𝐹𝑠𝑦(𝑥𝑠−𝑥𝑐); 𝑀𝑓𝑧 = 𝐹𝑓𝑦(𝑥𝑓 −
𝑥𝑐); 𝑥𝑠, 𝑥𝑓 are the points of application of the forces 𝐹𝑠 and 𝐹𝑓 , respectively. Eliminating
𝐹𝑝𝑟 from equations (7)(7), we obtain a set of two equations with four unknowns:
𝐹1(𝛿𝑧, 𝛿𝑓ℎ, 𝜂𝑧, 𝜓) = 0, 𝐹2(𝛿𝑧, 𝛿𝑓ℎ, 𝜂𝑧, 𝜓) = 0. (8)
Fixing any two of the four angles 𝛿𝑧, 𝛿𝑓ℎ, 𝜂𝑧, 𝜓, one can determine from it the balanced
values of two other angles with the help of the Newton’s iteration algorithm [1414].
Fig. 7Fig. 7 shows character of influence of both the mass center position �̄�𝑐 = 𝑥𝑐/𝐿 and the
horizontal fin position �̄�𝑓ℎ = 𝑥𝑓ℎ/𝐿 onto the balanced parameters of the model with the
specified mass 𝑚. Graphs of dependence of the balanced values 𝛿𝑧 and 𝛿𝑓ℎ on �̄�𝑐 are given
in Fig. 7Fig. 7a for fixed 𝜂𝑧 = 0, 𝜓 = 0, �̄�𝑓ℎ = 0.8. Graphs of dependence of the balanced values
𝛿𝑧 and 𝛿𝑓ℎ on �̄�𝑓ℎ are given in Fig. 7Fig. 7b for fixed 𝜂𝑧 = 0, 𝜓 = 0, �̄�𝑐 = 0.5.
We note that the problem of balancing the SC-vehicle with horizontal fins may not have
a solution for the specified vehicle mass 𝑚 and admissible ranges of varying the parameters
a b
Fig. 7. Balanced angles 𝛿𝑧 and 𝛿𝑓ℎ when 𝜂𝑧 = 0, 𝜓 = 0:
a — dependence on �̄�𝑐, b — dependence on �̄�𝑓ℎ
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𝑥𝑐, 𝑥𝑓ℎ, 𝛿𝑧, 𝛿𝑓ℎ, 𝜂𝑧, 𝜓. Also, it may have an ambiguous solution caused by nonmonotonic
dependence 𝑐𝑓𝑦(𝛿𝑓ℎ) for SC-fins having a wedge-shaped cross section [1717]. So, in the case
of Fig. 7Fig. 7b the SC-model may not be balanced when �̄�𝑓ℎ < 0.204, and it has two balanced
values 𝛿𝑓ℎ for each 0.572 < �̄�𝑓ℎ < 0.789.
7. DYNAMICS OF SC-VEHICLE BALANCED WITH FINS
Computer simulation allows a difference in dynamic behavior of the SC-vehicle to be
shown, which was balanced in the planing in a cavity mode (see Fig. 1Fig. 1a), in the planing
avoidance mode (see Fig. 1Fig. 1b), and in the combined mode.
As is known, the longitudinal motion of the SC-vehicle without fins in the planing in
a cavity mode is unstable “in the small”. In this case the stability loss occurs in a “soft”
oscillatory manner, and increasing the model angular oscillation amplitude is limited by
interaction between the model and the cavity walls. In this case to prevent increasing the
vehicle mass center deflection from the horizontal path 𝑦 one should apply the automatic
𝛿-stabilization of motion on depth, which makes the motion global stable [22,66].
On the contrary, computer simulation showed that the motion of the SC-model with
the horizontal fins in the planing avoidance mode is stable “in the small”. In this case an
allowable range of deflections of the pitch angle 𝜓 from its balanced value exists when the
solution fast tends to the balanced one.
In the combined mode, i. e., if both the horizontal fins and the model planing in a cavity
are presented, computer simulation shows the complex SC-model dynamics which essentially
depends on the starting values of the parameters. In this case, it turned out that the balanced
pitch angle 𝜓 and, hence, the model tail edge immersion in water ℎ have a basic importance.
As calculation showed, the horizontal fins may play a damping role for relatively great value
of ℎ, suppressing the motion instability “in the small”.
Fig. 8Fig. 8 gives a comparison of graphs of dependence 𝜓(𝑠) in the combined mode of the SC-
model motion for two starting values 𝜓0 = 0.4 and 𝜓0 = 0.5. The graphs have the following
nomenclature: 𝑠 = 𝑠/𝐿 is the distance along the path; 𝑦 = 𝑦/𝐿 is the deflection of the model
mass center from the horizontal path; magnitudes of the angle 𝜓 are plotted in degrees. It
a b
Fig. 8. Dynamics of the SC-model balanced in the combined mode:
a — 𝜓0 = 0.4, b — 𝜓0 = 0.5
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Table 3. Effect of starting balancing on maneuverability of the 𝑓 -controlled SC-model
No. 1 2 3 4 5 6 7 8
𝜓0,
∘ 0.371 0 0.1 0.2 0.3 0.4 0.5 0.6
ℎ0, mm 1.07 3.76 12.98 23.23
𝛿𝑓ℎ,
∘ 1.130 0.930 0.730 0.530 −4.123 −9.750 −13.072
�̇�, ∘/𝑠 −4.942 −10.693 −8.468 −5.307 −5.076 −3.085 −3.034 −2.764
𝑟, km 1.383 0.642 0.810 1.291 1.348 2.208 2.254 2.433
was accepted in the calculation that �̄�𝑐 = 0.5, �̄�𝑓ℎ = 0.8.
In case Fig. 8Fig. 8a the model was balanced at the following parameters: 𝜓 = 0.4, 𝜂𝑧 = 0,
𝛿𝑧 = −5.948∘, 𝛿𝑓ℎ = −4.138∘, 𝐹𝑝𝑟 = 23.896 KN. As can be seen, in this case the dynamic
behavior of the SC-model is quite similar to the case of motion in the planing in a cavity
mode without fins [66]. After loss of stability “in the small”, the model performs angular
oscillation with the frequency 7.8 Hz. It reaches the free water surface after 5.1 s.
In case Fig. 8Fig. 8b the model was balanced at the following parameters: 𝜓 = 0.5, 𝜂𝑧 = 0,
𝛿𝑧 = −5.642∘, 𝛿𝑓ℎ = −9.756∘, 𝐹𝑝𝑟 = 25.656 KN. As can be seen, in this case high-frequency
angular oscillation of the model does not arise. However, after a long time interval the motion
loses the global stability. In both the cases, using the depth 𝛿-stabilization of motion [66]
makes it stable in global.
A question about the effect of starting balancing the 𝑓 -controlled SC-vehicle on its course
maneuverability is of interest. Table 3Table 3 gives results of calculation of the SC-model motion
after distance 1 km when 𝛿𝑓𝑣 = −5∘ for a number of the starting balanced pitch angles 𝜓0.
The fin parameters are given in Table 1Table 1. When calculation both the roll 𝜃-stabilization of
motion (3)(3) and the depth 𝛿-stabilization of motion [66] were activated.
In Table 3Table 3, the balanced values of both the model aft edge immersion in water ℎ0 and the
horizontal fin deflection angle 𝛿𝑓ℎ, average values of the model turning speed �̇�, and average
values of the radius of a turning circle 𝑟 are given. The column 1 of the table corresponds
to the case of the planing in a cavity mode without horizontal fins. The columns 2 to 5
correspond to the case of starting balancing in the planing avoidance mode. The columns 6
to 8 correspond to the case of starting balancing in the combined mode.
As can be seen, the SC-vehicle course maneuverability is maximal at the starting balanc-
ing in the planing avoidance mode, but it dramatically deteriorates at the starting balancing
in the combined mode.
8. CONCLUSIONS
A method of active roll stabilization of the SC-vehicle motion with the help of both
the bottom roll fin and the automatic error-closing control system has been devised. It
was shown that the stabilization of the zero roll angle is necessary when the 𝑓 -controlled
SC-vehicle course maneuvering. The same method can be used for the course 𝜃-control
of the SC-vehicle motion by setting the specified roll angle 𝜃2 and the further automatic
𝜃-stabilization of this roll angle.
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Since the active operating control at the 𝜃-control of the SC-vehicle motion is the cavita-
tor, the 𝜃-control has the same advantages and limitations as the 𝛿-control [1616]. As a result
of the comparative analysis of the advantages and disadvantages of each of the four methods
of course controlling the SC-vehicle motion, it can be concluded that the proposed 𝜃-control
method is optimal within its efficiency.
A method of determination of the balanced motion parameters (balancing) of the SC-
vehicle in the case, when a pair of identical horizontal cavity-piercing fins is used for full or
partial compensation of the vehicle weight, has been devised.
It was shown that the longitudinal motion of the balanced with the horizontal fins SC-
vehicle in the planing avoidance mode is stable “in the small”. In the combined mode, i. e.,
if both the horizontal fins and the planing of the model body in a cavity are presented,
the computer simulation shows the complex SC-vehicle dynamic behaviour which essentially
depends on the starting value of the pitch angle 𝜓. For relatively big values of 𝜓 the horizontal
fins may play a damping role, suppressing the SC-vehicle motion instability “in the small”.
However, after a long time interval motion loses the global stability.
In all the examined cases, activation of the automatic depth 𝛿-stabilization [66] makes the
SC-vehicle motion stable in global. It was shown that the 𝑓 -controlled SC-vehicle course
maneuverability is maximal at the starting balancing in the planing avoidance mode, but it
dramatically deteriorates at the starting balancing in the combined mode.
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[14] Semenenko V. N., Naumova O. I. Dynamics of a partially cavitating underwater vehi-
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[15] Paryshev E. V. On unsteady planning of a body over liquid curvilinear surface //
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[16] Савченко Ю. Н., Семененко В. Н., Савченко Г. Ю. Особенности маневрирова-
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18(90), № 1. — С. 79–82.
[17] Savchenko Y. N., Semenenko V. N. Special features of supercavitating flow around
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[18] Savchenko Y. N. Control of supercavitation flow and stability of supercavitating motion
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[3] I. N. Kirschner, D. C. Kring, A. W. Stokes, N. E. Fine, and J. J. S. Uhlman, “Control
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 355355–371371.
[16] Y. N. Savchenko, V. N. Semenenko, and G. Y. Savchenko, “Features of manoeuvring
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pp. 660–672, 2001.
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tion of bodies,” in Supercavitating Flows, pp. 14(1–29), RTO/NATO, 2002.
В. М. Семененко, О. I. Наумова
Деякi способи застосування гiдродинамiчного руля
для суперкавiтуючих пiдводних апаратiв
Розглянуто два нетрадицiйнi способи застосування гiдродинамiчних рулiв при русi
високошвидкiсних пiдводних суперкавiтуючих апаратiв. Розроблено метод актив-
ної стабiлiзацiї руху суперкавiтуючого апарату за креном i метод керування йо-
го рухом за курсом шляхом регулювання кута крену за допомогою спецiального
руля крену й автоматичної системи керування зi зворотним зв’язком. Наведено
приклади комп’ютерного моделювання маневрування суперкавiтуючого апарату
за курсом при керуваннi за допомогою вертикальних гiдродинамiчних рулiв зi ста-
бiлiзацiєю нульового кута крену i шляхом регулювання кута крену. Розроблено
метод знаходження рiвноважних значень параметрiв руху (балансування) супер-
кавiтуючого апарату у випадку, коли для повної чи часткової компенсацiї його ваги
використовується пара однакових горизонтальних рулiв, якi проникають з кавер-
ни у воду. Наведено приклади комп’ютерного моделювання руху суперкавiтуючого
апарату з горизонтальними рулями в режимi без глiсування i в змiшаному режимi.
Показано, що усталений поздовжнiй рух збалансованого суперкавiтуючого апара-
ту без глiсування, на вiдмiну вiд його руху в режимi глiсування в кавернi, є стiйким
«в малому». Встановлено, що в змiшаному режимi руху горизонтальнi рулi можуть
грати демпфуючу роль, пригнiчуючи нестiйкiсть руху апарату «в малому», однак
на великому iнтервалi часу рух втрачає стiйкiсть в цiлому. Моделювання показа-
ло, що застосування автоматичної стабiлiзацiї руху по глибинi у всiх розглянутих
випадках робить його стiйким в цiлому. Показано також, що курсова маневренiсть
такого апарату при керуваннi за допомогою вертикальних рулiв максимальна при
початковому балансуваннi в режимi без глiсування, але рiзко погiршується при
початковому балансуваннi в змiшаному режимi.
КЛЮЧОВI СЛОВА: суперкавiтуючий апарат, керування, маневрування, рулi, крен,
комп’ютерне моделювання
В. Н. Семененко, О. И. Наумова
Некоторые способы применения гидродинамического руля
для суперкавитирующих подводных аппаратов
Рассмотрены два нетрадиционных способа применения гидродинамических рулей
при движении высокоскоростных подводных суперкавитирующих аппаратов. Раз-
работан метод активной стабилизации движения суперкавитирующего аппарата
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 355355–371371.
по крену и метод управления его движением по курсу путем регулирования угла
крена с помощью специального руля крена и автоматической системы управле-
ния с обратной связью. Приведены примеры компьютерного моделирования ма-
неврирования суперкавитирующего аппарата по курсу при управлении с помощью
вертикальных гидродинамических рулей со стабилизацией нулевого угла крена
и путем регулирования угла крена. Разработан метод нахождения равновесных
значений параметров движения (балансировки) суперкавитирующего аппарата в
случае, когда для полной или частичной компенсации его веса используется пара
одинаковых горизонтальных рулей, проникающих из каверны в воду. Приведены
примеры компьютерного моделирования движения суперкавитирующего аппарата
с горизонтальными рулями в режиме без глиссирования и в смешанном режиме.
Показано, что установившееся продольное движение сбалансированного суперка-
витирующего аппарата без глиссирования, в отличие от его движения в режи-
ме глиссирования в каверне, является устойчивым «в малом». Установлено, что в
смешанном режиме движения горизонтальные рули могут играть демпфирующую
роль, подавляя неустойчивость движения аппарата «в малом», однако на большом
интервале времени движение теряет устойчивость в целом. Моделирование пока-
зало, что применение автоматической стабилизации движения по глубине во всех
рассмотренных случаях делает его устойчивым в целом. Показано также, что кур-
совая маневренность такого аппарата при управлении с помощью вертикальных
рулей максимальна при начальной балансировке в режиме без глиссирования, но
резко ухудшается при начальной балансировке в смешанном режиме.
КЛЮЧЕВЫЕ СЛОВА: суперкавитирующий аппарат, управление, маневрирова-
ние, рули, крен, компьютерное моделирование
371
INTRODUCTION
CALCULATION METHOD AND DESIGN MODEL
FEATURES OF CONTROL OF SC-VEHICLE MOTION
ROLL STABILIZATION OF SC-VEHICLE MOTION
COURSE -CONTROL OF SC-VEHICLE MOTION
SC-VEHICLE BALANCING WITH FINS
DYNAMICS OF SC-VEHICLE BALANCED WITH FINS
CONCLUSIONS
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