Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/ecf/PHDK32.pdf Äàòà èçìåíåíèÿ: Wed Feb 3 16:14:55 2016 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:56:06 2016 Êîäèðîâêà:

ISSN 1028 3358, Doklady Physics, 2016, Vol. 61, No. 1, pp. 32­36. © Pleiades Publishing, Ltd., 2016. Original Russian Text © E.V. Vetchanin, A.A. Kilin, 2016, published in Doklady Akademii Nauk, 2016, Vol. 466, No. 3, pp. 293­297.

MECHANICS

Free and Controlled Motion of a Body with a Moving Internal Mass through a Fluid in the Presence of Circulation around the Body
E. V. Vetchanina, b and A. A. Kilinc
Presented by Academician V.V. Kozlov April 20, 2015 Received April 27, 2015

Abstract--The free and controlled motion of an arbitrary two dimensional body with a moving internal mass and constant circulation around the body in an ideal fluid is studied. Bifurcation analysis of the free motion is performed (under the condition of a fixed internal mass). It is shown that the body can be moved to a given point by varying the position of the internal mass. Some problems related to the presence of a nonzero drift of the body with a fixed internal mass are noted. DOI: 10.1134/S1028335816010110

The problem about the motion of a rigid body in a fluid is classified as one of classical and long studied problems of hydrodynamics. The first results in this area were obtained by G. Kirchhoff [1], H. Lamb [2], S.A. Chaplygin [3], and V.A. Steklov [4] using ideal fluid theory. A recent qualitative analysis of the motion of a rigid body in an ideal fluid can be found in [5­7]. We also note that, from a practical point of view, of great interest is the problem about the controlled motion of a rigid body in a fluid and, in particular, the motion without varying the body's shape due to the displacement of internal elements. This problem was considered in a series of recent works [8­13]. For example, the theoretical possibility of such a motion was demonstrated in [8]. Later, in [9, 10, 13], a num ber of examples of the body control by moving point masses inside the body were presented. In [11], the motion control by means of an internal flywheel and Flettner rotor allowing one to vary the circulation around the body was considered. In this work, we consider the controlled motion of a hydrodynamically asymmetric body in an ideal fluid in the presence of circulation around the body. In this process, the control is implemented by displacing a

point mass inside the body. We demonstrate the com plete controllability of this system based on the Rashevsky­Chow theorem and generalize some results from [6] about the free motion of a rigid body to the case of its hydrodynamical asymmetry. EQUATIONS OF MOTION AND FIRST INTEGRALS Let us consider the two dimensional problem of motion of a rigid body with mass M in an infinite vol ume of an ideal incompressible fluid; inside the body, a point mass m moves along a certain curve = ((t), (t)) (Fig. 1). We assume that the fluid is at rest at infinity and circulation around the body is equal to a constant (by Lagrange's theorem) . To describe the motion, we introduce a fixed coordinate system Oxy and a moving coordinate system O1 rigidly attached to the body so that the point O1 coincides with the cen ter of mass of the body. The position of the body in absolute space is determined by the radius vector r = (x, y) of the point O1 and the angle of rotation of the system O1 about the point O1. Let v = (v1, v2) denote the absolute velocity of the point O1 referred to the moving axes, and , the angular velocity of the body. Then the following kinematic relations hold: · q = Q w,
T

Kalashnikov Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, Udmurtia, 426069 Russia b Udmurt State University, Universitetskaya ul. 1, Izhevsk, Udmurtia, 426034 Russia c Steklov Mathematical Institute, ul. Gubkina 8, 119991 Moscow, Russia e mail: eugene186@mail.ru 32

a

cos sin 0 Q = ­ sin cos 0 , 0 1 0

(1)

where q = (x, y, )T is the vector of generalized coor dinates and w = (v1, v2, )T is the vector of quasi


FREE AND CONTROLLED MOTION OF A BODY WITH A MOVING INTERNAL MASS

33

velocities. According to [6], the equations of motion of the system under consideration can be represented in the form of PoincarÈ equations on the group E(2) L L d L L = + cos + sin , x y dt v 1 v2 L L L d L = ­ ­ sin + cos , v1 x y dt v 2 d L L L L = v2 ­ v1 + dt v1 v 2 with Lagrangian L = 1 ( Aw, w ) + ( , w ) + ( u, w ) , 2 where a 0 ­ 1 A = 0 a , 2 ­ b ­ ( x sin ­ y cos ) 2 = ­ ( x cos + y sin ) 2 ­ ( x sin ­ y cos ) ­ ( x cos + y sin u= · a1 = M + 1 + m , m b= = , m · · , · ­ M + 2 + m , m
2

y .. (, )






m O1

(2)
O

x

Fig. 1. Two dimensional body with an internal mass.

(3)

p x = ( a 1 v 1 ­ + u 1 ) cos ­ ( a 2 v 2 + + u 2 ) sin + y + sin ­ cos , p y = ( a 1 v 1 ­ + u 1 ) sin + ( a 2 v 2 + + u 2 ) cos ­ x ­ cos ­ sin , K = ­ v 1 + v 2 + b + u 3 + xp
2 2 ­ yp x + ( x + y ) . 2 y

(4)

, )

FREE MOTION Let us consider the free motion of the system under consideration; i.e., we assume that (t) = const and (t) = const. In this case, as is well known [3], the equations for velocities w are separated from the com plete system (1), (2) and, in the chosen notation, take the form d ( a 1 v 1 ­ ) = ( a 2 v 2 + ) ­ v 2 ­ , dt da ( 2 v 2 + ) = ­ ( a 1 v 1 ­ ) + v 1 + , dt (5) d ( ­ v1 + v2 + b ) dt = v 2 ( a 1 v 1 ­ ) ­ v 1 ( a 2 v 2 + ) + v 1 ­ v 2 . Equations (5) admit the energy integral T=
2 2 2 1 ( a 1 v 1 + a 2 v 2 + b ) ­ v 1 + v 2 = h (6) 2

a2 =
2

I + 6 + m ( + ) , m = , m = . m

Here, 1 and 2 are the coefficients of added masses (without loss of generality, we set 1 > 2), 6 is the coefficient of the added moment of inertia, I is the central moment of inertia of the body, is the fluid density, and and are coefficients associated with the asymmetry of the body; they are determined by integrals depending on the body shape [3]. Equations (1) and (2) form a closed system of equations describing the controlled motion of the body in a fluid. The system of equations (1), (2) admits first inte grals of motion [6]
DOKLADY PHYSICS Vol. 61 No. 1 2016

and one more additional integral, quadratic in veloci ties: F = ­ v1 + v2 + b
2 2 + 1 ( ( a 1 v 1 ­ ­ ) + ( a 2 v 2 + ­ ) ) = f . 2

(7)

The integral (7) relates to the integrals (4) by F = K + px + py . Thus, reduced system (5) is integrable [3]. 2
2 2


34 h 10 8 6 4 2 0 ­10 c ­5 0 fA fB
1

VETCHANIN, KILIN h2 h1 c c2 b

a

f1* 5

f2* fC 10 f

Fig. 2. Bifurcation diagram of the free system.

Bifurcation Analysis Let us first consider the problem of the existence of stationary solutions of the system (5). Using straight forward substitution into (5), it is easy to demonstrate that the following proposition holds. Proposition 1. The system (5) admits a unique one parameter family of fixed points 0 + 0 v1 = , a1 0 ­
2

tionary solutions (8) of the reduced system (5) and can be obtained in the form of curves given parametrically by substituting (8) into integrals (6) and (7). An exam ple of the bifurcation diagram for the parameters a1 = 1, a2 = 0.5, b = 2, = 1, = 0.1, and = 0.2 is given in Fig. 2. In Fig. 2, the curve a corresponds to the parameter values 0 (­, /a1), the curve b h1 to the values 0 (/a1, /a2); and the curve c h2 to the values 0 (/a2, +). The solid lines correspond to stable fixed points; the dashed lines, to unstable points. The domain of admissible values of integrals h and f is shown in gray. As is seen from Fig. 2, three types of phase portraits of the system in the plane (v1, v2) are possible depend ing on the value of the integral f. Figure 3 shows the types of these phase portraits corresponding to values of the integral F = fA, fB, and fC presented in Fig. 2. Critical values of the integral f separating different types of phase portraits can be found, e.g., as coordi nates of the cusp c1 and c2 of the branches of the bifur cation diagram; at these points, the following equali ties are satisfied: dF = dT = 0 . d 0 d 0 Absolute Motion Let us now consider the issue of the motion of the system under consideration in absolute space. For this purpose, we represent the first two equalities (4) in the following form: a 1 v 1 ­ = ( p x ­ y ) cos + ( p y + x ) sin + , a 2 v 2 + = ­ ( p x ­ y ) sin + ( p y + x ) cos + . (9)

0 ­ 0 v2 = , a2 0 ­

2

(8)

= 0 , where 0 is the family parameter varying in the inter vals (, /a1), (/a1, /a2), and (/a2, +). By an appropriate choice of a coordinate system, we can take and in Eqs. (5) and solutions (8) to be zero. Therefore, in what follows we consider the case = = 0. Let us construct the bifurcation diagram of the system (5) in the plane of first integrals (f, h). The branches of the bifurcation diagram correspond to sta
v2 2 1 0 ­1 a () 4 3 2 1 0 ­1 ­2 ­3 ­4 ­2 ­1 0 1 2 ­2 ­1
a

Eliminating the parameter from the system (9), we obtain the relation between the coordinates x, y and
(b) 6 4 2
h1 b c a b

(c)
h2

0 ­2 ­4 ­6

h1

0

1

2

3

­4

­2

0

2

4 v1

Fig. 3. Phase portraits corresponding to different values of the integral F: (a) fA < f * , (b) f * < fB < f * , and (c) fC > f * . Points 1 1 2 2 a and b correspond to the minimum of energy T; point c, to the maximum; and points h1 and h2, to the saddle points. DOKLADY PHYSICS Vol. 61 No. 1 2016


FREE AND CONTROLLED MOTION OF A BODY WITH A MOVING INTERNAL MASS

35

velocities v1, v2, on the common level set of the integrals px, py: ( a 1 v 1 ­ ­ ) + ( a 2 v 2 + ­ ) = ( px ­ y ) + ( py + x ) .
2 2 2 2

(10)

Equation (10) in the coordinates x, y is an equation ­p of a circumference with a center at the point y , px and variable radius
2 2 1/2 1 ( ( a 1 v 1 ­ ­ ) + ( a 2 v 2 + ­ ) ) . (11) At a fixed level of kinetic energy T, the velocities v1, v2, and are bounded functions of time; therefore, the radius R is also bounded. Thus, the following proposition holds. Proposition 2. On a fixed level set of the integrals (4), the arbitrary motion of the system (5) occurs in an ­p p annular domain with the center at the point y , x .

where x = px ­ y and y = py + x. Substituting expression (14) into the kinematic relationships (1), we obtain the equations of motion for the body on a fixed level set of integrals (4) in the standard form lin ear in the controls: · z = S ( V0 ( z ) + V1 ( z ) u1 + V2 ( z ) u2 ) , (15) T ­1 S = diag ( Q A , E 2 ) , V0 ( z ) = + x cos + y sin , ­ x sin + y cos , 2 K + px + py ­ x ­ y , 0, 0 , 2 V 1 ( z ) = ( ­ 1, 0, , 1, 0 ) , V 2 ( z ) = ( 0, ­ 1, ­ , 0, 1 ) . Here, z = (x, y, , , )T is a vector in the extended phase space and En is the unit matrix of dimensions · · n â n; the velocities u1 = and u2 = of the moving mass (and not its coordinates) are used as controls. In (15), the vector field SV0 corresponds to the free motion of the body and is called the drift (according to control theory), and the vector fields SV1 and SV2 cor respond to the control actions. In the proof of controllability, we rely on a general ization of the Rashevsky­Chow theorem for a system with drift [14]. In this theorem, in addition to the completeness of the linear span of vector fields and their commutators, it is required that an everywhere dense set of Poisson stable points exists for the free motion in the system's phase space. First, let us consider the issue of the completeness of the vector fields. The matrix S is nondegenerate at all points of the phase space; therefore, according to [15], it is sufficient to demonstrate the completeness of the linear span of the vector fields (16) and their com mutators. Consider six vector fields V 0, V 1, V 2, V V
02 01 T T 2 2 2 2 T

R=

(16)

Note 1. The maximum and minimum radii of the ring in which the body moves can be found as con strained extrema of radius R (11) on a fixed level set of the integrals T and F by maximizing and minimizing, respectively, of the function = R + 1 ( T ­ h ) + 2 ( F ­ f ) , (12) where 1 and 2 are the undetermined Lagrange mul tipliers and h and f are given level sets of the integrals T and F, respectively. Note 2. Note that stationary regimes (8) in absolute space are associated with the motion in a circle with the radius R=
2 2 + 0 + ­ 0 a 1 0 ­ a 2 0 ­ 1/2

.

(13)

COMPLETE CONTROLLABILITY OF MOTION Let us consider the issue of complete controllability of motion on a fixed level set of first integrals (4). For this purpose, we express velocities w from Eqs. (4) in terms of the coordinates q and controls and : · x cos + y sin + ­ · ­1 ­ x sin + y cos + ­ w=A , (14) 2 2 2 2 2 K + p x + p y ­ x ­ y ­ ( ­ ) ·· 2
DOKLADY PHYSICS Vol. 61 No. 1 2016

= [ V 0, V 1 ] , V
12

= [ V 0, V 2 ] ,

= [ V 1, V 2 ] ,

(17)

where [·, ·] is the Lie bracket. By straightforward cal culations, it is easy to show that the linear span of vec tor fields (17) is complete everywhere in the phase space except on the surface of codimension 2 spec ified in explicit form by the equations ­ (y ­ x) = 0, ( cos ­ sin ) ( + x ) + ( sin + cos ) y + x + (y + x ) = 0.
2 2 2

(18)


36

VETCHANIN, KILIN

It is evident that such degeneration of the manifold on which the rank of the linear span decreases does not lead to the appearance of any obstacles for controlla bility. Let us now consider the issue of the Poisson stabil ity of the drift. As was shown by Chaplygin in [3], the system (5) in the case of a fixed internal mass is inte grable. Furthermore, as was shown in the previous sec tion, the motion of the system is bounded on a fixed level set of the integrals (4). It follows that there exists an everywhere dense set of Poisson stable points for the free motion (drift). Thus, in view of the fact that the system (15) describes a flow in the extended space , the following theorem is valid. Theorem 1. An arbitrary body moving in a fluid (in the presence of circulation around it) with a given initial velocity can be moved from any initial position to any end position using the appropriate bounded motion of the internal mass. Note that the body does not stop after switching the control off in the end position but continues its free motion in the annular domain. Thus, in spite of the fact that we can move the body to a given point, we cannot keep it there without applying additional efforts. As an example, let us consider the issue of the motion of the body from one point of the space (x0, y0) to another point (x1, y1) on a fixed level set of the inte grals px, py, and K and require that the velocities v1, v2, and be equal to zero at the initial and end points. In · addition, we suppose that at the initial instant (0) = · (0) = 0. From the condition of preserving the inte grals of motion, we determine the velocity of the inter nal mass at the end point: · = 1 ( x cos + y sin + ) , m 1 · = ( ­ x sin + y cos + ) . m

changing the values of the integrals (4) so that the end position is a fixed point of the total system. It is evident that a change in the level set of integrals is possible only in other models, e.g., due to forces of viscosity or using additional mechanisms, such as the Flettner rotor [11]. ACKNOWLEDGMENTS We are grateful to A.V. Borisov for discussions of the results. The work of E.V. Vetchanin (Introductory part and section "Free motion") was supported by the Russian Science Foundation under grant 14 19 01303 and performed at the Kalashnikov Izhevsk State Technical University. The work of A.A. Kilin (sections "Equations of motion" and "Complete controllability of motion") was supported by the Russian Science Foundation under grant 14 50 00005 and performed at the Steklov Mathematical Institute of the Russian Academy of Sciences. REFERENCES
1. G. Kirchhoff and K. Hensel, Vorlesungen Ýber mathema tische Physik. Mechanik (BG Teubner, Leipzig, 1874). 2. H. Lamb, Hydrodynamics (Dover, New York, 1945). 3. S. A. Chaplygin, Tr. Tsentr. Aerogidrodinam. Inst., No. 19, 300 (1926). 4. V. A. Steklov, Soobshch. Khar'k. Matem. Obshch. 2, 209 (1891). 5. A. V. Borisov, V. V. Kozlov, and I. S. Mamaev, Regul. Chaotic Dyn. 12, 531 (2007). 6. A. V. Borisov and I. S. Mamaev, Chaos 16, 013118 (2006). 7. J. Vankerschaver, E. Kanso, and J. E. Marsden, Regul. Chaotic Dyn. 15, 606 (2010). 8. V. V. Kozlov and S. M. Ramodanov, J. Appl. Math. Mech. (Engl. Transl.) 65, 579 (2001). 9. V. V. Kozlov and D. A. Onishchenko, J. Appl. Math. Mech. (Engl. Transl.) 67, 553 (2003). 10. A. A. Kilin, S. M. Ramodanov, and V. A. Tenenev, Non linear Dyn. Mob. Robot. 2, 115 (2014). 11. S. M. Ramodanov, V. A. Tenenev, and D. V. Treschev, Regul. Chaotic Dyn. 17, 547 (2012). 12. S. Childress, S. E. Spagnolie, and T. Tokieda, J. Fluid Mech. 669, 527 (2011). 13. E. V. Vetchanin, I. S. Mamaev, and V. A. Tenenev, Regul. Chaotic Dyn. 18, 100 (2013). 14. B. Bonnard, C. R. Acad. Sci. Paris, SÈr. 1 292, 535 (1981). 15. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, Regul. Chaotic Dyn. 17, 258 (2012).

(19)

Since it is required to stop the body at the end point (x1, y1), the right hand sides of Eqs. (19) are constant; therefore, in order for the body to stay in place, the internal mass must move with a constant velocity. Since the motion of the internal mass is bounded by the body, a stop at a given point of the space for an arbitrarily long time is impossible. Thus, for practical control of the system considered, the complete controllability according to Theorem 1 is insufficient. It is also necessary to solve the problem of stabilization of the body position at the end point. This can be implemented by regular maneuvers near the end point, which will require additional energy, or by

Translated by A. Nikol'skii
DOKLADY PHYSICS Vol. 61 No. 1 2016