CC BY
20
UDC 532.582.5
DOI: 10.18698/0236-3941-2020-1-56-69
LATERAL MOTION OF TOWED UNDERWATER VEHICLE WITHIN THE PROBLEM OF CONTINENTAL SHELF MONITORING
V.T. Grumondz1 R.V. Pilgunov1,2 M.V. Vinogradov2 N.V. Maykova1
v.grumondz@gmail.com prvl 983s04@mail.ru vinmaxc@gmail.com mayvik@yandex.ru
1 Moscow Aviation Institute (National Research University),
Moscow, Russian Federation
2JSCGNPP Region, Moscow, Russian Federation
Abstract
Lateral motion dynamics was studied of a robotic towed underwater system designed to monitor the continental shelf and consisting of a towed vehicle and a tow wireline. In regard to underwater vehicles of the type in question, it is quite correct to represent spatial motion in the form of a super-position consisting of two flat motions, i.e., longitudinal motion in the vertical plane and lateral motion in the horizontal plane. Dynamics of the towed system longitudinal motion within the monitoring problem was considered in a previously published work by the authors. The present work is its natural continuation and development traditionally accepted in the problems of the underwater vehicles spatial motion mechanics. Diagram of the towed vehicle operation and its hydrodynamic characteristics are presented; besides, mathematical model of a wireline and also a model of the wireline-towed vehicle system lateral motion were constructed. Probable steady system motions were analyzed, issues of balancing, as well as those of the towed vehicle dynamic stability when moving at a constant depth were considered. Results of numerical calculations were provided. The results obtained were considered in conjunction with the results of the authors' above mentioned work related to the towed vehicle longitudinal motion and make it possible to select such system parameters that provide the specified character of spatial movements in the process of monitoring the continental shelf taking into consideration the need to perform turns in the horizontal plane at changing directions and to ensure vertical maneuvers when avoiding underwater obstacles
Keywords
Towed underwater vehicle, continental shelf monitoring, lateral motion dynamics, horizontal circulation, motion stability
Received 11.01.2019 Accepted 01.08.2019 © Author(s), 2020
Introduction. Problems of analyzing the dynamics of a mechanical system consisting of tow-wireline-towed vehicle motion were considered in works [1-12] and in a number of other papers, which brief review is presented in work [6]. Basically, these works are devoted to studying a system steady longitudinal motion. Statement of the problem raised in this work and devoted to studying the towed vehicle lateral, as well as statement of problem in analyzing its longitudinal motion considered in work [6] during the continental shelf monitoring process, put forward a number of new stringent requirements to a towed system dynamics in comparison with works [1-5] and [7-12]. These requirements include very accurate angular stabilization of the towed vehicle differential longitudinal axis with respect to the given direction, ensuring a wide range of depths in the established horizontal motion and corresponding towered vehicle balancing state under conditions of hydrostatic force excess over gravity, requirement to the wireline length, which should be not less than some limiting length to guarantee an acceptably low level of interference from the towing ship on the towed vehicle acoustic observation system. Finally, there is a requirement to ensure that the towed vehicle could track the sea bottom relief and bypass the underwater obstacles. Paper [6] is devoted to studying these issues in regard to the case of wireline-towed vehicle system longitudinal motion. The present work is a natural continuation and development of paper [6]. It addresses the issues of towed vehicle lateral motion and ensuring the towed vehicle circulation motion in the horizontal plane, which makes it possible to close the system operation scheme. The problem under consideration considers that the wireline ensures balance in the vertical plane, and control in the horizontal plane is carried out by deflecting vertical rudders of the towed vehicle.
Problem statement. As in paper [6], a tow-wireline-towed vehicle mechanical system is considered designed to being used on the continental shelf in order to solve a wide range of problems associated with the sea bottom, such, for example, as searching for various potentially hazardous objects, inspection of underwater engineering structures, studying the underwater relief, creating the depth map, searching for sunken objects and many others. A somewhat more detailed review of possible tasks is provided in papers [1-6].
As in paper [6], it is assumed that one end (running end) of the flexible wireline is fastened to the towed vehicle (TV) with an attachment point, the other (root) end is wound on the winch of the ship winch system installed on board the towing ship. The scheme of tow-wireline-towed vehicle system operation is presented in Fig. 1.
Specified scanning zone
TV scanning zone Carrier motion trajectory (S-turn)
Scanning zones overlap area
Fig. 1. Tow-wireline-towed vehicle system scheme of operation
The following notations is used here: bar and Zflr are width and length of the specified scanning zone; rtum is the tow trajectory turning radius; btack is the tow tack width, i.e., the distance between rectilinear sections of its trajectory; bscan is the width of the towed vehicle scanning zone.
When scanning the observation area, the tow moves for a while in a straight line, then turns around and moves in the opposite direction, etc. (see Fig. 1). Moreover, in the problem statement under consideration, the towed vehicle before turning could be moved to the tow using a winch. In other words, scanning of the water basin is carried out by tacks (S-turns). In this case, scanning zones of the towed vehicle should be overlapped with a certain margin for reliable coverage of the entire examined area. The indicated action scheme, in addition to procedures specified in paper [6], includes a turn in horizontal plane carried out after passing through each rectilinear section.
Relative arrangement of the tow-wireline-towed-towed vehicle system elements during a steady-state circulation, as well as the necessary symbolic notations at the wireline balance are shown in Fig. 2.
As in paper [6], it is supposed that the body of a towed vehicle is a feathered body of revolution, on which wings for deepening and steering controls are installed. Geometrical parameters of the towed vehicle, i.e., wingspan, limiting body and empennage linear dimensions and empennage arrangement on the body, are subject to design limitations caused by ensuring the reliable operation of the acoustic system monitoring the underwater space,
Fig. 2. Scheme of the tow-wireline-towed vehicle system elements arrangement and wireline balance parameters at steady towing in projection on the tow symmetry plane (a) and on the horizontal plane (b)
as well as by arrangement of the towed vehicle on board the tow, when the towed vehicle is not used for its intended purpose, and the tow itself is operating as the towed vehicle carrier. It is assumed that the towed vehicle motion in the turning section occurs after the towed vehicle is pulled to the tow for a short distance and is in established circulation in the horizontal plane. The water surface is considered flat and undisturbed; no sea currents are present. Wireline (flexible connection) is assumed to be weighty, absolutely flexible, inextensible, having some fixed strength limit determined by the T* ultimate tensile force. The cx0 longitudinal and the cx90 transverse hydro-dynamic wireline characteristics, as well as the pT wireline material density are considered to be known.
It is required to build algorithms and corresponding relations that make it possible (jointly using the results indicated in paper [6]) to select geometric parameters of the towed vehicle, balancing angle values of the installed wings and controls deflection in such a way that, under conditions of significantly positive towed vehicle buoyancy, p = A - G > 0, steady horizontal motion of the towed vehicle and of the entire towed cable system is ensured in the horizontal plane turning sections, when changing motion direction to the opposite in operating modes that satisfy the restrictions in и and L parameters.
Model of the towed vehicle hydrodynamic characteristics in spatial motion. Hydrodynamic scheme of the towed vehicle is accepted as it is in paper [6]. In regard to the towed vehicle hydrodynamic characteristics in spatial motion, the following expressions should be accepted:
V
Сха(аь ßb 5z) = c* (аь ßfc) + 4(aki ßfc) 5
5s = o
cya(o-k, ß/bStf^c^abßfcJL =0+
rh
Cza(uk, ßb Sv) = cz(ocb Pifc)|8 =0+c|(otit,ßfc)5y;
mx (ak, ßfc, 5E) = mx (ak, ßfc)|g = Q + mbxEbE-,
my (ab ßb5v) = my (ak, ß^)| =o + m8y(ak,$k)bv; rnz(ak,$k,8H) = mz(ak,$k)\5 =Q+ml(ak,
where ak and (3^ are the towered vehicle attack and slip angles; 8h and 8y are deflection angles of its horizontal and vertical rudders. Dependences of the spatial velocity hydrodynamic coefficients include: the cxa drag force; the cya lift force; the cZfl lateral force; the mx roll moment; the my yaw moment; the mz different moment of the towed vehicle from a^ and angles obtained experimentally, and are presented (at zero rudder deflection angles) in Fig. 3.
Rotational derivatives and damping coefficients, as in paper [6], are calculated using the method by K.K. Fedyaevsky [13-15].
Wireline model at the tow steady circulation motion. Wireline motion (equilibrium) equations. The general problem of spatial equilibrium in the system under consideration could be divided into two: the problem of the wireline equilibrium and the problem of the towed vehicle equilibrium. Let us consider the first of them.
Since the wireline motion is steady, equations of its motion turn into equilibrium equations during circulation for an elementary wireline section (wireline equilibrium differential equations) in the cylindrical coordinate system [1]:
dT pdT
R2 cos2 a - p sin a cos cp;
= cxgo-a>c R2 sin2 oc -—cos a cos ф -
cos a sin cp
R
(1)
dcp _
dl Silx
d% _ cos a dy
dl R ' dl
Silx 4 ^
dR .
= sin a cos ф; — = sin a sin ф. dl
Ш2 - 10 •■h
mz -5' ■h)
0°,
10° Рё)
™z 20° ßj
vH--.ee)
U^'O-MU)
<=za(20°.ße)
- ___
15 tN 5 •>4 V у \
Fig. 3. Dependences of the towed vehicle cx,cy,cz,mx,my, mz spatial hydrodynamic coefficients on the Pjt slip angle at the a^ - -10°, -5°, 0,10°, 20° angle of attack values (indicated first in parentheses along the ordinate axis) for zero values of the 8h and 8y rudder deflection angles
System (1) could be considered in conjunction with the wireline spatial equilibrium equations system in the following form [1]:
dT . da 1 / . 9 \
— = -rn cosz a - p sin a cos cp; — = — U90 sinz a —p cos a cos a> ;
— = —-—(ssin3 a cos a + psincp); (2) dl T sina v '
dx dv dz
— = cos a; — = sin a cos cp; — = sin a sin cp, dl dl dl
■»x(20-°.ßä)
which in the case of wireline equilibrium in the vertical plane, i.e., for <p = 0, takes the following form:
dT 0 . da 1 ( . -, \
— = -ro cosz a - p sin a; — = — I rgo sinz a-pcosal; dl dl T
dx dy
— = cos a; — = sin a. dl dl
In system (2), e is the empirical coefficient characterizing the lateral flow around the wireline, and especially in case of a smooth wireline s = 0 [3].
The following notations are used in the wireline system equations (l)-(3), see Fig. 2: T is the local tension force; a is the local towing angle; cp is the local angle of lateral inclination; % is the local steering angle of the flow incident on the wireline during circulation; R is the distance from the current wireline point to the tow axis of circulation.
The jço, ro, p parameters (respectively, the wireline resistance force per unit length at 90° and 0 angles of attack and the wireline mass per unit length in water) are equal to:
pV2 pV2 ndT2
rgo =cX9o—^—dT; r0=cx0—^—clT; p = {pT-p) g■
Signs of the terms in equations (l)-(3) imply a countdown from the root end (connected to the tow) of the wireline.
By attaching boundary conditions (see below) to the system (1) for each of the T, a, cp, %, y, R wireline equilibrium parameters, a form of its equilibrium
could be obtained during circulation for the specified L wireline length, Vc linear and rac angular circulation velocities, as well as for the 5y, 8h rudder deflection angles.
Towered vehicle spatial motion equations. Let us write down dynamic equations of the towed vehicle spatial motion in the form of two vector equations [14]:
^ + ¿5 xQ = FE; (4)
at
— + (bxK + ÛxQ = ME, (5)
dt
where Q is the momentum resultant vector; K is the towed vehicle momentum resultant moment relative to the beginning of the 0\xyz bound
coordinate system; dQjdt and dK.jdt are local derivatives of the Q and K
vectors in the Oxyz bound coordinate system; U is the pole velocity vector of the bound coordinate system relative to the Earth coordinate system; co is the angular velocity vector of bound axes relative to the Earth axes; Fe and Me are the resultant vector and the resultant moment of external forces applied to the towed vehicle.
The Q and K vectors are determined by the following expressions:
Q = m(U + (t>xrcy, (6)
K = J(b + m(rcxU), (7)
where rc is the radius vector of the towed vehicle center of mass in the bound axes, rc = ixc+jyc+kzc; f is the towed vehicle inertia tensor relative to the axes passing through its center of mass and parallel to the bound axes.
System of the towed vehicle motion should be closed in numerical integration by the known kinematic relations [14], which are not presented here for their obviousness, and should be written down in projections on the axis of the bound coordinate system.
Towed vehicle steady circulation. Let us accept the following assumptions for the towed vehicle steady horizontal circulation [13-15]:
(qx=(oz = 0, (Oy « G)c = const, y = 0.
Moreover, from the last equality and taking into account the conditions of motion at a constant depth:
tgScosafc -cosysina^ -siny tgP = 0
it follows that at =
Then, equilibrium equations of the towed vehicle in circulation would take the following forms:
Xt+Cxa(Vc, ajt, pfc, 8y, 8H)^Sw-(G-A)sinS-- (m + X33 ) (OyVz - (X35 - mxc) aj = 0;
P Vc
Cya (at, 8ff) — Sm+Yt -(G-A)cosS = 0; (8)
Pvc
Cza(Vc, Pfc, 8y, cOy)^—Sm+Zt+(m + Xn)cOyVx=0; PVC
my (Vc, pfc, 8y, d)y ) —-— SmLk ~(mxc -X35)ayVx = 0;
PVr.
- [(Усу - Ук)А ~(Ус ~ Ук)с~\ sinö = °>
where the attached masses and the attached static moments could be calculated on the basis of paper [16].
Moments in the last two equations of system (8) are registered relative to the K point (see Fig. 1), i.e., in such a way as not to be dependent on the Xt, Yt, Zt towing force values taken in the projections on the towed vehicle bound axes.
Taking into consideration the last remark, system (8) is being divided: the last two equations form a closed system for the given Vc, 5y, and coc, and could be solved independently of the first three, which provides the ajt> Pa: desired values.
The first three equations (8), taking into account dependencies for the projections of the towed vehicle generalized velocities on the bound axes
in their decision will give the Xt,Yt, Zt towing forces values equal to:
where 7^, 7^, T^ is the towing force essence in projections on the axis of the Ofyr\t, coordinate system connected to the tow. After solving system (9) in regard to the 7^, T^, T^ values, it is possible to calculate the 7^, ock> 9k values of the T, a, cp wireline equilibrium parameters at its K running end, which are required for solving system (1):
However, as was indicated, the system (8), which provides values of the Xt,Yt, Zt forces in its solution, and thereby the 7^, 7^, Tq values, should be solved not only when the 8я rudders deviation angles are set, but also at the Vc towed vehicle circulation velocity equal to Vc = соIn turn, the Vc and Rc parameters themselves are, as a rule, not known in advance (except for
vx = Vc cos a*; cos ßfc and Vz = Vq sin ßjt,
Xt = T,V cos ajt cos ßjt + Tq sin - %; cos sin ß^; Yt = -T^ sin a*; cos ßfc + Тц cos ak + X\ sin ak sin ßjt; Zt = 7^ sin ßfc + T; cos ßfc,
(9)
the case when the Rc required radius of the towed vehicle circulation is specified), and only the Rtow and Vjow = ©c Rtow corresponding parameters in regard to the tow are defined. In the case when the Rtnw tow circulation radius is specified, the problem of finding the wireline equilibrium form turns out to be a boundary problem and requires an iterative procedure for jointly solving the towed vehicle equilibrium (8) and the wireline (1) equation system at horizontal circulation of the mechanical system. Moreover, as boundary conditions for the x and y parameters of system (1), the following could be set:
X\K =0 and y\K = 0.
Fig. 4 presents comparative graphs illustrating some results of calculating the wireline equilibrium parameters for both options in setting the system (1) boundary conditions with respect to the R parameter. In this case, the lower index A in the notation of parameters in the graphs corresponds to the Rtow setting and index B — to the i?c setting. Let us also note that along abscissa in all the graphs the I curvilinear coordinate values are marked, i.e., the wireline current length is counted from its K running end. The a, cp, % wireline angular parameters are given in degrees.
Fig. 4. Alterations in the wireline a (a), (p (b), %(c), R(d) equilibrium parameters during the tow-wireline-towed vehicle system steady circulation characterized by the
following values: the L = 180 wireline dimensionless length, the Rc = Rtv =170 initial dimensionless radii, the (Oc = 0,7 °/s angular circulation velocity, Vc = 6 m/s linear circulation velocity and the cxa = 2,5; cya = 5; cza = -1 hydrodynamic
coefficient values
Lateral stability of TV steady motions. When studying stability of steady motion, where:
Vo = |V| = const, (£>y = GOyo = G)c = const, p = pfc0 = const, l|/ = l|/0,
the system of first approximation in regard to equations of the lateral perturbed controlled motion in the horizontal plane at zero roll is written down in the following form [13]:
(bupt+bi2)AV + bi3ptA\\r + (bi5pt+bi6)A$k (hipt + hi )AV+(b23pt + ¿>24) +{b26pt+ hn) Ap* = b28A8v; (b31pt+b32)AV+(h3pt +b34) Acoy +(b36pt +b37) Apfc = b38A8v,
where blk =-^—-, pt=— is the time differentiation operator.
cos (flfco) dt
Under the assumption that AV" = 0, the system of lateral perturbed motion equations takes the following form:
(b'23pt+ b'24 )Aasy+(b26pt+ b27) Apfc = 0;
(b'33pt + b'M ) Ag)y +(b36pt + b37) Apfc = 0.
In regard to the controlled lateral motion under perfect control, and when the control law could be written down as:
A5y = fciAv|/ + k2ptA\\i,
the following equations of the perturbed motion first approximation are obtained:
[b23pf + (¿24 -k2b22 )pt -k\b2% ] Av|/ + (b26pt + b27 ) Apfc = 0; [b33pf + (¿>34 - k2b32 ) pt - fci&38 ] Av|/ + (b36pt + b37 ) Apfc = 0,
where
b22=-(m + Xu)®yO COSOCfco COSPfco--pS V0 (<l pfco + c3 P^0 + cf 8B)-czv ^^(oyo-, b23 = - (A,26 + mxc) cos afco;
rav P SL
b24 = cos afco
- (m + Я,ц ) cos afco cos pfc0 - c'
у
z
V0;
¿27 =
&26 =(m + A,ii)Vo cosßfco;
' . pSV2" (m + Xn)(oy0 cosafco sin ßfc0---—
(cUko+ЩоС'з);
Ь32 =(т + Я,ц)соуо cosafco cosßfc0-- (mj ßfco +W3 ß|0 +тъу ôyB)pSLV0 ~ЩУ ^^ю^о;
¿34 =
¿33 =(ly +^55) cosajto; (X26 + m xc ) Vq cos a fco cos ßfc0 - m^
pSL
cosafco;
¿36 = -(^-26 +mxc)V0 cosßfco;
pSL
¿37 = -(^26 +mxc)(oyoVo cosafco sinPfc0--— V^2 (mÇ + 3$2k0);
b38=msy ^Vq2.
Introducing new notations:
fl = &23> b = {b2±-k2b22), c = kib2&, d = b26p + b27, =¿33, h=(bM-k2b32), C\ = &1&38, di=b36p + b37,
we obtain
[op^ + fcpi - c] A\|/ + i/APfc = 0; [flip2 + bipt - Ci ] A\|/ + rfiAPfc = 0. It follows that stability conditions have the following form: Dfc>0, fc = Ô3; D1D2-D0D3> 0,
where
D3 = &23&36 -&33^26î £>2 = ¿23 ¿37 + (¿24-k2 ¿28 ) ¿36 -¿33 ¿27 -(¿»34 ¿38 ) ¿265
A = (¿24 — k2 ¿28 ) ¿37 -fcl ¿28 ¿36 "(¿34 ~h ¿38 ) ¿27 "¿¿I ¿38 ¿36! D0 = -hb26h6 + hh&b27.
Conclusions. A mathematical model was constructed in regard to steady horizontal circulation motion of an underwater towed vehicle within the problem of monitoring the continental shelf, as well as the algorithm for such motion determination. Possibility of realizing these movements was demonstrated taking into consideration fulfillment of the balancing conditions. The problem was solved in constructing the lateral dynamic stability conditions for the considered horizontal steady motion of a towed system designed to monitor the near-bottom shelf area taking into account both strict restrictions on the range of depths of the towed vehicle motion and on the trim angles, and significant positive buoyancy of the towed vehicle. Together with the results of paper [6], the constructed algorithms make it possible to solve the problems of spatial towing of a towed vehicle of the type under consideration.
Translated by K. Zykova
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Grumondz V.T. — Dr. Sc. (Phys.-Math.), Professor, Department of Aircraft Flight Dynamics and Control, Moscow Aviation Institute (National Research University) (Volokolamskoe shosse 4A, Moscow, 125993 Russian Federation).
Pilgunov R.V. — Assistant, Department of Aerodynamic System Design, Moscow Aviation Institute (National Research University) (Volokolamskoe shosse 4A, Moscow, 125993 Russian Federation); Research Assistant, JSC GNPP Region (Kashirskoe shosse 13A, Moscow, 115230 Russian Federation).
Vinogradov M.V. — Computing Engineer of the 2nd rank, JSC GNPP Region (Kashirskoe shosse 13A, Moscow, 115230 Russian Federation).
Maykova N.V. — Assist. Professor, Department of Aerodynamic System Design, Moscow Aviation Institute (National Research University) (Volokolamskoe shosse 4A, Moscow, 125993 Russian Federation).
Please cite this article as:
Grumondz V.T., Pilgunov RV., Vinogradov M.V., et al. Lateral motion of towed underwater vehicle within the problem of continental shelf monitoring. Herald of the Bauman Moscow State Technical University, Series Mechanical Engineering, 2020, no. 1, pp. 56-69. DOI: https://doi.org/10.18698/0236-3941-2020-l-56-69
