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The self-induced rotational and oscillatory motions of an aerodynamic pendulum
Journal of Applied Mathematics and Mechanics 77 (2013) 360–368
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Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The self-induced rotational and oscillatory motions of an aerodynamic pendulum夽 B.Ya. Lokshin, V.A. Samsonov ∗ Moscow, Russia
a b s t r a c t The behaviour of an oscillatory mechanical system with alternating dissipation is considered taking an aerodynamic pendulum as an example. The phase portraits are investigated, their rearrangements are studied and the critical values of a parameter are determined. The equilibrium positions of the pendulum and the self induced rotational and oscillatory states of the motion are determined and their stability is investigated. © 2013 Elsevier Ltd. All rights reserved.
For oscillatory mechanical systems, it is usually possible to give a fairly accurate description of a static force action from static experiments, for example. The problem of describing a dynamic action that depends on the velocity of the motion is completely different. Both experimentally and theoretically, the determination of the dissipative or antidissipative nature of the action of a force on a moving object gives rise to considerable difficulties. Nevertheless, the study of the behaviour “in the large” of systems with variable dissipation is necessary since it has important applications (see, for example, Refs 1–3). A mathematical model of a velocity-dependent force action is usually of a heuristic nature and contains parameters. In studying the behaviour of a system as a function of the parameter values, the investigation of the existence or absence of closed phase trajectories and of both stable phase trajectories (attractors) as well as unstable phase trajectories (repellors) is of special importance since the self-induced rotational and oscillatory motions of the initial system correspond to such closed trajectories. Object of the investigation. The behaviour of a specific oscillating system with one degree of freedom is considered, that is, an aerodynamic pendulum.4 Such a pendulum is a thin flat plate, rigidly fastened along the support, while the other end of the support is attached to a fixed cylindrical hinge (the rocking axis of the pendulum). The whole construction is located in a steady air flow, the velocity of which is equal to V. It is assumed that the characteristic size of the plate in the transverse cross section (the width of the plate) in the direction “along the support” is much greater than its size in the direction “transverse to the support”. The cross section of this aerodynamic pendulum in the rocking plane is shown in Fig. 1. The plate AB of width 2b is rigidly fastened to its geometric centre G on the support OG and can rotate together with the support as a whole about an axis perpendicular to the plane of the sketch and passing through the point O. We will neglect the interaction of the support with the flow and friction in the rocking axis and only take account of the aerodynamic action that is exerted by the flow of the medium on the plate. It is well known in applied aerodynamics that, for such bodies, the aerodynamic action reduces to a drag force D and a cross-wind force L. The point C of intersection of the line of action of the resultant force R = D + L with the plate is called the centre of pressure. We shall take, as usual, the magnitudes D and L of the aerodynamic action in the following form
where Cd (␣) and Cl (␣) are aerodynamic coefficients and S is the area of the plate. We shall assume that, at each instant, the force R due to the action of the medium is completely determined by the vector Vc of the velocity of the centre of pressure with respect to the medium, that is, by the instantaneous values of the magnitude Vc of this velocity and the angle ␣ (the effective angle of attack) between the vector Vc and the support. The vector U, the absolute velocity of motion of the ˙ where r is the length of the support, ϑ is the angle of rotation of the centre of pressure, is equal to U = V + Vc and U = U = (r − (␣)), support, (␣) is the displacement of the centre of pressure C, measured from the geometric centre G, where the function (␣) is assumed to be positive if the centre of pressure is shifted towards the axis of rotation and negative otherwise.
夽 Prikl. Mat. Mekh., Vol. 77, No. 4, pp. 501–513, 2013. ∗ Corresponding author. E-mail address: [email protected] (V.A. Samsonov). 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.11.004
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Fig. 1.
Equations of motion of the pendulum. The behaviour of the pendulum can be described by the equations4,5 (1) (2) The following dimensionless parameters and variables are introduced here
(3) where J is the moment of inertia of the whole structure with respect to the point O, a derivative with respect to the time is denoted by a dot and Cn (␣) = Cl (␣)cos␣ + Cd (␣)sin␣ is the normal force coefficient. In accordance with the quasistatic flow model, the relations q(␣) and e(␣) are determined from static experiments in wind tunnels and are assumed to be known. For example, graphs of these relations for plane rectangular plates of different elongation have been presented in Ref. 6. Hence, model (1), (2) of the pendulum motion contains two functions of the angle of attack, q(␣) and e(␣), that are only determined by the shape of the pendulum and two structural parameters: the dimensionless length of the support l and the dimensionless quantity a which is inversely proportional to the moment of inertia of the structure. Note that the model of the aerodynamic moment,4,5 introduced on the right-hand side of Eqs (1), implicitly contains the dependence on the angular velocity . In applied aerodynamics, this relation is usually introduced in terms of the so-called rotational derivatives and, has a linear form. It can be hoped that the model used will have a wider range of application. We shall assume that the functions q(␣) and e(␣) are specified, continuous and triply differentiable in the interval [-,] and possess the following properties (a derivative with respect to the angle ␣ is denoted by a prime here): a) the function q(␣) is an odd function with respect to ␣ = 0, an even function with respect to ␣ = /2, and q(␣) > 0 when
b) the function e(␣) is an even function with respect to ␣ = 0, an odd function with respect to ␣ = /2, and e(␣) > 0, e (␣) < 0 when
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We further mention a special feature of Eqs (1) and (2). Under the assumptions made with regard to the symmetry of the flow over the pendulum, the system, in addition to the solution (ϑ, , ␣, u), also has a solution (-ϑ, -, -␣, u). Henceforth, it is therefore sufficient to consider just the half strip Q = {(, ) : ∈ [− , ], ≥ 0} as the involute of a part of the phase cylinder. Properties of the kinematic equations. Solving kinematic equations (2) for ␣ and u and substituting relations ␣ = ␣(, ) and u = u(, ) obtained into Eq. (1) could have been an approach to the study the behaviour of the pendulum for different values of the parameters and the initial data. However, when e = / 0, Eqs (2) can have a non-unique solution. We will consider this question in greater detail and show that, in the interval (4) exactly three solutions of these equations exist. Actually, it follows from the second equation of (2) when ϑ = - /2 that: 1) cos ␣ = 0 or 2) u = 0. In the first case, the solution has the form (5) 1 in ␣. For ∈ (1/(l + em ), and exists for all values of . In the second case (u = 0), from the first equation of (2) we have the equation e(␣) = l − ˜ where ␣() ˜ is the unique positive solution of the equation obtained. Note that 1/(l − em )), it has two symmetric solutions ±␣(),
These solutions can be represented in the form (6) So, the interval (4) exists in the phase plane of the variables (ϑ, ) such that, at each point of this interval, there are exactly three solutions (5), (6) of the kinematic equations (2). Outside this interval on the line ϑ = /2, Eqs (2) only have one solution (5). We now consider the behaviour of the solutions for small deviations ϑ and from the interval (4). To be specific, we choose ∈ (1/l, 1/(l − em )). Solutions (5) and (6) then have the form
The increments ␣k , uk (k = 1, 2, 3) correspond to the deviations ϑ and . In the neighbourhood of the first solution, from Eqs (2) we have
It is clear that a solution ␣, u of these equations exists regardless of the sign of the increments ϑ and . Consequently, the first solution ␣1 , u1 is determined in the phase plane to the left and to the right of the interval (4) considered with respect to ϑ. In the neighbourhood of the second (and third) solution, we have
Since ␣ ˜ ∈ (0, /2) and the increment u for this solution can only be positive, the solution ␣, u of these equations only exists for positive values of ϑ. Hence, the second and third solutions ␣2 , u2 , ␣3 , u3 are only determined in a certain domain H1 ⊂ Q adjacent to the interval (4) on the right: these solutions do not extend to the left of it (a cut-off of the surface ␣ = ␣(, ) of its own kind). A similar situation also holds for the interval
The first solution to the left and to the right of interval (4), while the second and third solutions only exist in a certain domain H2 ⊂ Q, adjacent to the interval on the left. Hence, there are two domains, H1 and H2 , adjacent to interval (4) where the function ␣(ϑ, ) is triple-valued (for convenience, we shall assume that ␣(− /2, 1/1) = ± /2. These domains are projections of the two convolutions of the surface ␣(ϑ, ) with one common point = − /2, = 1/l onto the (ϑ,) plane. The boundary of the domain H1 consists of three pieces: of the part of interval (4) (the segment = − /2, ∈ (1/l, 1/(l − em ))) in which ␣ has three values: ␣1 , ␣2 , ␣3 and two envelopes, the equations of which are given in the following implicit form
(7)
where < /2 and the parameter ␣ ∈ (0, /2). In one of the envelopes the solutions ␣1 and ␣2 are identical, and in the other envelope
␣1 and ␣3 are identical. All three solutions ␣1 =␣2 = ␣3 = ␣ * are identical at the point * , * of intersection of the envelopes (the cuspidal point). The value of ␣* is uniquely determined from the equation (8) Substituting the value of ␣* into Eqs (7), we also find * , *. The boundary of the domain H2 is similarly specified but, in Eqs (7), it is necessary to take
At the cuspidal point on this boundary we have ␣1 = ␣2 = ␣3 = − ␣∗ , which follows from Eq. (8).
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Fig. 2.
We will now consider a special case as an example. Suppose
Then, from Eq. 8, we obtain cos2 ␣ * =2/3 such that, at the cuspidal points, we have
From Eqs (7), we find the values of the coordinates of the cuspidal points:
For illustration, the geometrical locations of the points that are solutions of the kinematic equations (2) when ␣ = const are depicted in the phase plane when ≥ 0 (Fig. 2) and an enlarged fragment of this figure is shown on the right. In the general case, when l > em , the shape of the domains H1 and H2 is similar to that shown, although their dimensions change somewhat. When l < em , as l becomes smaller the “middle” point = − /2, = 1/l of the domain of non-uniqueness is raised upwards and, when l → 0, it departs to infinity. The evolution of the domains H1 and H2 in the case of the example considered is shown in Fig. 3 for different values of l. The boundary of a domain when l = em is distinguished by the bold line. Since, in the real motion of a pendulum, the character of the flow over it and the force action on the part of the medium are more likely to change continuously than abruptly, we shall henceforth assume that, in crossing a domain of non-uniqueness in the value of the angle ␣, the choice has to be made of retaining as far as possible the continuity of its change during the motion. The point depicted when > 0 in the domain Q shifts from the left to the right such that, on entering the domains H1 and H2 and during the motion within them, a continuous branch of ␣(ϑ, ) certainly exists and a jump from one branch another can only occur on leaving these domains. ¯ u¯ = 1 hold at all the equilibrium positions of the Equilibrium positions. It follows from Eqs (1) and (2) that the equalities ω ¯ = 0, ˛ ¯ = ϑ, ¯ () ≡ . pendulum () ≡ , ¯ Three stationary points: M1 (0, 0), M2 (, 0), M3 (− , 0), to which the values ␣1 = 0, ␣2 = , ␣3 = − correspond, in the domain Q in the phase plane of the variables (ϑ,) certainly exist. The position of the pendulum “along the flow” corresponds to the first stationary point and the reversed position of the pendulum (“against the flow”) corresponds to the points M2 and M3 . When l > em , there are no other equilibrium positions of the pendulum considered. ¯ 0) and M5 (−, ¯ 0) exist, where ¯ = ˛r is the unique positive root of the equation When l < em , a further pair of stationary points: M4 (, e(␣) = l. Two symmetrical deflected positions of the pendulum correspond to these points, at which the centre of pressure (the point C) coincides with the point of fastening O. In particular, when l = 0, there are two symmetrical positions of the pendulum when the plate is orthogonal to the free stream. We shall henceforth denote any of the stationary values of ϑ and ␣ corresponding to one of the indicated ¯ and ␣ ¯ =␣ points by ¯ ( ¯ when = 0). The situation may be different in the case of pendulums with another configuration (a plate that is not along a support, is not of rectangular shape, is not flat, etc). ¯ , 0), we will denote the small deviations from the Analysis of the stability of the equilibrium positions. For each of the points Mk ( k ¯ , corresponding solution , ¯ ␣, ¯ u¯ by x, y, z and so that
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Fig. 3.
The point M1 (0,0). Taking account of the fact that z = x + (l − em )y, = 0, the equations for the small deviations can be represented in the following form
Henceforth, q0 qp (0), e0 = e (0), . . ... Since q0 > 0, the point M1 (0,0) is stable when l > em and, when l > em , it is unstable (a saddle point). When l = em , the characteristic equation has a double zero root and it is impossible to make a judgement regarding stability using the first approximation. In this case, the equations in the deviations can be represented, up to terms of the third order, in the form (9) These equations have a first integral
and V(x, y) ≥ 0 by virtue of the inequalities e0 < 0 and q0 > 0. Consequently, in such a case the point M1 (0,0) is the centre, to terms of the third order of infinitesimals. We shall only confirm this for the case when l = em since the inclusion of higher-order derivatives of the experimentally determined functions q(␣) and e(␣) makes no sense. The point M2 (,0) (or M3 (-,0)). Taking account of the equation
we conclude that the second equation of system (9) has the form
It is clear that the points M2 (,0) and M3 (-,0) are saddle points. The point M4 (␣r ,0) (or M5 (-,0)). It follows from the kinematic equations that = 0, z = x, and the second equation in system (9) therefore reduces to the form
Since e (˛r ) < 0, no conclusion regarding stability can be drawn on the basis of the first approximation equations. Taking account of terms of the second order of infinitesimals and expressing z to terms of the second order: z = x(1 − ye (␣r )cos␣r ), we write the equations in the deviations using a suitable change of variables in the following form
(10)
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where a and b are constant numbers. The quantity b, which is proportional to the coefficient of xy, is certainly positive, and the quantity a, which is proportional to the coefficient on x2 , can have any sign. In system (10), we now change to a system of polar coordinates by putting
and then, for sufficiently small values of , we have
where the notation
has been introduced. ˜ 3 () makes a “mean” contribution to the change in the quantity . Since () is a decreasing function for It is seen that only the term R sufficiently small then, when ab > 0, the trivial solution of system (10) is unstable, and, if ab < 0, it is asymptotically (but not exponentially) stable. Returning to the initial variables and notation, we obtain that, when l ∈ (0, em ), the points M4 (˛r , 0) and M5 (− ˛r , 0) are stable (unstable) foci if the inequality (11) is satisfied (not satisfied). We emphasize that, for these points, the linear part of the “position” force has the nature of a “restoring” force and their stability or instability is determined by the sign of the coefficient of the non-linear part of the position force (by the sign of the coefficient a in system (10)). The results of the qualitative analysis of the stability of the points M4 (␣r , 0) and M5 (− ␣r , 0) are completely confirmed by the results of a stability study using a well-known rigorous method.7 ¯ − ␣r = /2) corresponds, is of special interest when analysing The case when l = 0, to which a position of the plate across the flow ( the stability of the points M4 (␣r , 0) and M5 (− ␣r , 0). In this case, terms of the third order of infinitesimals have to be taken into account. As a result of the corresponding calculations, from Eqs (1) and (2) we have
and again the equations obtained can be reduced using the change of variables to the form (the coefficient a˜ has any sign and b˜ > 0)
The trivial solution of this system is asymptotically stable, which follows from Lyapunovs theorem. Actually, the function
˜ 2 z 2 by virtue of that is positive definite in a certain neighbourhood of the origin of coordinates, has a negative definite derivative V˙ = −bz 1 2 the system of equations constructed. Hence, when l = 0, the points M4 (/2.0) and M5 (− /2.0) are asymptotically stable. So, when l > em , the position “along the flow” is asymptotically stable and, when l = em , it is stable but not asymptotically, while the position “against the flow” is unstable. There are no other equilibrium positions. When 0 < l < em , the “along the flow” and “against the flow” positions are unstable and the points M1 (0,0), M2 (, 0) and M3 (− , 0) are saddle points. A pair of symmetric deflected (stable or unstable) equilibrium positions exists. Since system (1), (2) is dissipative as a whole then, in the case of values of l for which there are no equilibrium states, the creation of a stable cycle would be expected which encompasses one or several stationary points and, possibly, one or several pairs of stable and unstable cycles (by a cycle, we means any closed phase trajectory that both encompasses the phase cylinder and also lies in the strip < . When l = 0, the “along the flow” and “against the flow” positions also remain unstable but the “across the flow” position of the pendulum is asymptotically stable (the points M4 (/2, 0) and M5 (− /2, 0) are stable foci). Cyclic motions of the pendulum. The qualitative analysis carried out gives a specific representation of the behaviour of the pendulum close to the equilibrium positions and of the corresponding phase trajectories in the neighbourhood of the stationary points, that is, when |ω| 1. It is known that, when |ω| 1, the action of the medium is of a dissipative nature, that is, it ensures a tendency towards a reduction in the energy. However, an antidissipative character3,5 of the action of the medium may manifest itself in certain bounded domains of phase space. In the problem considered, the determination of the integral character of the action of the medium for “mean” values of is complicated by the lack of uniqueness of the solution of the kinematic equations. This forces us to turn to numerical modelling as a method of studying the behaviour of the pendulum for different values of the structural parameter, that is, the length of the support. A plate with an elongation 8 was chosen for the calculations. The value of the dimensionless parameter a was taken to be equal to 0.001 and the dimensionless length l of the support was varied over a range from 0 to 3.
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Fig. 4.
As a result of the numerical experiment, it was established that the form of the motions of the pendulum “in the large” depends considerably on the values of l; several critical values of l were distinguished and, when these are passed through, there is a fundamental reconstruction of the phase portrait of the system. The relations between the maximum values of the angular velocity with respect to ϑ, max , and the characteristic phase trajectories are shown in Fig. 4 as a function of the length of the support. The critical values of the length of the support are denoted by lk (k = 1, 2, . . ., 6) and, here, l1 = em , S+ and S− are the values of max on the separatrix entering the saddle point M2 (, 0) and emerging from the saddle point M3 (− , 0) closest to these points, C+ are the values of max on a stable cycle that does not encompass the phase cylinder, and R+ and R− are the values of max for stable (the + sign) and unstable (the - sign) cycles encompassing the phase cylinder (the curves R− are shown by dashed lines). We will now briefly illustrate the results presented using phase portraits (PP). Since the values of the “amplitudes” of the phase trajectories are shown in Fig. 4, in the PP (Fig. 5) we shall restrict ourselves to just a qualitative depiction of these trajectories, distinguishing the most characteristic of them: the separatrices and cycles by means of bold lines. When l > l1 = em according to Fig. 4, the separatrix S− emerging from the saddle point M3 (− , 0) is located below the separatrix S+ entering the saddle point M2 (, 0). All the remaining phase trajectories contract into the origin of the coordinates, the point M1 (0,0), the unique attractor to which the stable “along the flow” equilibrium position of the pendulum corresponds. When the length of the support is reduced (when l > em ), both the separatrices S− and S+ approach close to the ϑ axis. The first critical value of the length of the support is l = l1 = em . In this case, the point M1 (0,0) is the centre up to terms of the third order of infinitesimals. The form of the PP is unchanged in other respects and is not illustrated in Fig. 5. As soon as the length of the support becomes less than the first critical value, the character of the PP changes qualitatively. The point M1 (0,0) ceases to be stable (it becomes a saddle point) and a pair of unstable foci M4 (␣r , 0) and M5 (− ␣r , 0) is generated. Hence, all these three points are unstable, and the saddle points M2 (, 0) and M3 (− , 0) are also unstable. By virtue of the dissipativeness of the system as a whole, the separatrices emerging from the point M1 (0,0) cannot depart to infinity, as a consequence of which closed trajectories necessarily arise. At first, there is one closed trajectory C+ as shown in Fig. 5 (l = 0.55) which encompasses all three points M1 (0,0), M4 (␣r , 0) and M5 (− ␣r , 0). It is as though this cycle inherits the properties of the unique attractor that existed when l > em . This unusual bifurcation (a Hopf type bifurcation) is caused by the existence of dynamic symmetry in the system and the characteristic non-linearity of the position force. The trajectory C+ is a unique attractor to which all phase trajectories (apart from S+ ) contract both from within a cycle and from the outside it. When there is a further reduction in the value of l, the cycle C+ expands, the separatrices S− and S+ descend and, at a certain new critical value of the length of the support l = l2 < l1 (the calculations showed that l2 ∼ = 0.4245), the two separatrices coincide, forming a loop of separatrices. The cycle C+ settles on this very loop, having expanded up to the limit (from = - to = ). All the more sweeping oscillations of the pendulum that does not execute one complete revolution correspond to unwinding phase trajectories and, with each successive oscillation, the time interval between the successive stops of the pendulum increases. A rotation of the pendulum in one and the same direction corresponds to the phase trajectories located outside this loop. and the time of each revolution increases monotonically to infinity. When l < l2 , the separatrices S− and S+ (S− is now higher than S+ ) are formed from the loop and a closed stable trajectory R+ (Fig. 5, l = 0.3) encompassing the phase cylinder (a second trajectory R+ is also formed here in the domain < 0). A stable rotational state of motion of the pendulum corresponds to the trajectory R+ . As the length of the support is reduced in the interval 0 < l < l2 , the separatrices continue to “descend” and the trajectories R+ separate, such that the angular velocity of the pendulum in the rotational state increases. A new periodic trajectory arises in the case of the next critical value l = l3 (l3 ∼ = 0.267 < l2 ) in the domain of fairly high values of and it is metastable: all the phase trajectories located above it contract to it, and the trajectories located below it are repelled from it and descend to the stable trajectory R+ .
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367
Fig. 5.
When the length of the support is further reduced, this metastable trajectory splits into a pair of periodic trajectories: an unstable trajectory R− and a stable trajectory R+ located above it, which gradually recede from one another as the value of l decreases. The trajectory 1 1 R− separates the domains of attraction of the stable trajectories R+ and R+ . In this case, the domain of attraction of the trajectory R+ remains 1 1 quite spacious. From any initial fixed position apart from the equilibrium positions, the pendulum “independently” reaches a self induced means, in particular, that the pendulum rotational state corresponding to the trajectory R+ . The existence of the unstable trajectory R− 1 cannot “independently” reach a state of self induced rotation corresponding to the trajectory R+ and a preliminary rotation is necessary to 1 reach it. With a subsequent decrease in the length of the support, a further metastable cycle is generated at a certain value l = l4 (l4 ∼ = 0.24 < l3 = 0.267) for the specific plate considered) which, on continuing to reduce l, decomposes into a pair of periodic trajectories: a stable trajectory R+ and, above it, an unstable trajectory R− , but both of these are located below the trajectory R+ . The corresponding 2 2 phase portrait is qualitatively represented in Fig. 5 (l = 0.2) in which the states R− and R− are represented by the bold dashed line. The 2 1 regular subdivision of the phase space into domains of attraction of the stable rotational states is clear. The pendulum now independently but not R+ as was the case earlier. just reaches the self-induced rotational state R+ 2 As l decreases, the trajectory R+ descends approaching the separatrices S− and S+ , and, at the critical value l = l5 ∼ = 0.00085, the trajectory 2 + − + R2 coincides with the separatrices S and S and, at the same time, they all pass through the point M1 (0,0) to coincide now with the separatrices of the point M1 (0,0) and form two loops. Within the loops, the points M4 (␣r , 0) and M5 (− ␣r , 0) are unstable foci. + When there is a further reduction in the value of the parameter l, the orbitally stable cycles C+ 1 and C2 , encompassing the points M4 (␣r , 0) and M5 (− ␣r , 0) respectively, are removed from these loops, as illustrated by the phase portrait in Fig. 5 (l = 6 × 10−4 ). The points M4 (␣r , 0) + and M5 (− ␣r , 0) still remain as unstable foci and the trajectories emerging from them are wound onto the cycles C+ 1 and C2 from within. All − the remaining trajectories located below R2 are wound onto these cycles from the outside. Oscillatory motions of the pendulum that change (after a finite number of complete revolutions) into one of the two stable self-induced oscillatory states correspond to these trajectories. When l is reduced, the above-mentioned cycles contract and, finally when l = l6 ∼ = 0.00045, each of them collapses to the stable point M4 (␣r , 0) or M5 (− ␣r , 0). For all l from the interval l ∈ (0, l6 ), these points are asymptotically stable foci (although they are not exponentially stable, so that the attenuation is very slow). The decaying oscillations of the pendulum about the equilibrium position close to the “across the flow” position now correspond to trajectories located below the trajectory R− . 2 In conclusion, we shall dwell on the case when l = 0. The problem of the behaviour of such a pendulum (a wind vane) has already been studied by Zhukovskii8 and the following was noted by him: “If such a wind vane is placed against the wind, it becomes perpendicular to
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the direction of the wind and has good stability in this position. But, if we impart a rotational motion to the wind vane in one direction or another, it will continue to rotate in the specified direction” and, in the state of self-induced rotation, “experiment gives approximately ∼ = 0.5”. It is curious that, even within the second-order dynamical system considered here, based on a quasistatic model of aerodynamic action on a plate, these properties have received not only qualitative but also quantitative confirmation ( ∼ = 0.45, see Fig. 4). Moreover, the estimate of the initial angular velocity ( ∼ = 0.39) that has to be surmounted in order to reach the domain of attraction of the self-induced rotational state R+ is clear from Fig. 4. The high velocity self-induced rotational states R+ and R− have not been discussed previously, possibly due to the difficulty of their 1 1 practical realization and on account of features of the specific profile used in the calculations. Acknowledgement This research was supported by the Russian Foundation for Basic Research (11-08-00444, 12- 01-00364). References 1. 2. 3. 4. 5. 6. 7. 8.
Dosayev MZ, Kobrin AI, Lokshin BYa, Samsonov VA, Selyutskii YuD. Constructive Theory of Small-Scale Wind Power Generators. Moscow: Izd MGU; 2007. Barbashin YeA, Tabuyeva VA. Dynamical Systems with a Cylindrical Phase Space. Moscow: Nauka; 1969. Shamolin MV. Methods of Analysing Dynamical Systems with Variable Dissipation in Dynamics of Solids. Moscow: “Ekzamen”; 2007. Lokshin BYa, Samsonov VA. Computational-analytical investigation of the behaviour of an aerodynamic pendulum. Vestn MGU Ser 1 Mathematics Mechanics 1996;6:50–2. Lokshin BYa, Samsonov VA. Problem of the Motion of a Body in a Resisting Medium. Qualitative Analysis. Moscow: Izd Mosk Univ; 2012. Tabachnikov VG. Stationary characteristics of wings at low velocities over the whole range of angles of attack. Tr TsAGI 1974;(Issue 1621):79–83. Malkin IG. Theory of Stability of Motion. Oak Ridge TN: US Atom Energy Commission; 1952. Zhukovskii NYe. On the fall in air of light elongated bodies rotating about their longitudinal axis. In: Collected paper. Moscow:Gostekhizdat; 1949;4;41–68.
Translated by E.L.S.
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Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The self-induced rotational and oscillatory motions of an aerodynamic pendulum夽 B.Ya. Lokshin, V.A. Samsonov ∗ Moscow, Russia
a b s t r a c t The behaviour of an oscillatory mechanical system with alternating dissipation is considered taking an aerodynamic pendulum as an example. The phase portraits are investigated, their rearrangements are studied and the critical values of a parameter are determined. The equilibrium positions of the pendulum and the self induced rotational and oscillatory states of the motion are determined and their stability is investigated. © 2013 Elsevier Ltd. All rights reserved.
For oscillatory mechanical systems, it is usually possible to give a fairly accurate description of a static force action from static experiments, for example. The problem of describing a dynamic action that depends on the velocity of the motion is completely different. Both experimentally and theoretically, the determination of the dissipative or antidissipative nature of the action of a force on a moving object gives rise to considerable difficulties. Nevertheless, the study of the behaviour “in the large” of systems with variable dissipation is necessary since it has important applications (see, for example, Refs 1–3). A mathematical model of a velocity-dependent force action is usually of a heuristic nature and contains parameters. In studying the behaviour of a system as a function of the parameter values, the investigation of the existence or absence of closed phase trajectories and of both stable phase trajectories (attractors) as well as unstable phase trajectories (repellors) is of special importance since the self-induced rotational and oscillatory motions of the initial system correspond to such closed trajectories. Object of the investigation. The behaviour of a specific oscillating system with one degree of freedom is considered, that is, an aerodynamic pendulum.4 Such a pendulum is a thin flat plate, rigidly fastened along the support, while the other end of the support is attached to a fixed cylindrical hinge (the rocking axis of the pendulum). The whole construction is located in a steady air flow, the velocity of which is equal to V. It is assumed that the characteristic size of the plate in the transverse cross section (the width of the plate) in the direction “along the support” is much greater than its size in the direction “transverse to the support”. The cross section of this aerodynamic pendulum in the rocking plane is shown in Fig. 1. The plate AB of width 2b is rigidly fastened to its geometric centre G on the support OG and can rotate together with the support as a whole about an axis perpendicular to the plane of the sketch and passing through the point O. We will neglect the interaction of the support with the flow and friction in the rocking axis and only take account of the aerodynamic action that is exerted by the flow of the medium on the plate. It is well known in applied aerodynamics that, for such bodies, the aerodynamic action reduces to a drag force D and a cross-wind force L. The point C of intersection of the line of action of the resultant force R = D + L with the plate is called the centre of pressure. We shall take, as usual, the magnitudes D and L of the aerodynamic action in the following form
where Cd (␣) and Cl (␣) are aerodynamic coefficients and S is the area of the plate. We shall assume that, at each instant, the force R due to the action of the medium is completely determined by the vector Vc of the velocity of the centre of pressure with respect to the medium, that is, by the instantaneous values of the magnitude Vc of this velocity and the angle ␣ (the effective angle of attack) between the vector Vc and the support. The vector U, the absolute velocity of motion of the ˙ where r is the length of the support, ϑ is the angle of rotation of the centre of pressure, is equal to U = V + Vc and U = U = (r − (␣)), support, (␣) is the displacement of the centre of pressure C, measured from the geometric centre G, where the function (␣) is assumed to be positive if the centre of pressure is shifted towards the axis of rotation and negative otherwise.
夽 Prikl. Mat. Mekh., Vol. 77, No. 4, pp. 501–513, 2013. ∗ Corresponding author. E-mail address: [email protected] (V.A. Samsonov). 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.11.004
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Fig. 1.
Equations of motion of the pendulum. The behaviour of the pendulum can be described by the equations4,5 (1) (2) The following dimensionless parameters and variables are introduced here
(3) where J is the moment of inertia of the whole structure with respect to the point O, a derivative with respect to the time is denoted by a dot and Cn (␣) = Cl (␣)cos␣ + Cd (␣)sin␣ is the normal force coefficient. In accordance with the quasistatic flow model, the relations q(␣) and e(␣) are determined from static experiments in wind tunnels and are assumed to be known. For example, graphs of these relations for plane rectangular plates of different elongation have been presented in Ref. 6. Hence, model (1), (2) of the pendulum motion contains two functions of the angle of attack, q(␣) and e(␣), that are only determined by the shape of the pendulum and two structural parameters: the dimensionless length of the support l and the dimensionless quantity a which is inversely proportional to the moment of inertia of the structure. Note that the model of the aerodynamic moment,4,5 introduced on the right-hand side of Eqs (1), implicitly contains the dependence on the angular velocity . In applied aerodynamics, this relation is usually introduced in terms of the so-called rotational derivatives and, has a linear form. It can be hoped that the model used will have a wider range of application. We shall assume that the functions q(␣) and e(␣) are specified, continuous and triply differentiable in the interval [-,] and possess the following properties (a derivative with respect to the angle ␣ is denoted by a prime here): a) the function q(␣) is an odd function with respect to ␣ = 0, an even function with respect to ␣ = /2, and q(␣) > 0 when
b) the function e(␣) is an even function with respect to ␣ = 0, an odd function with respect to ␣ = /2, and e(␣) > 0, e (␣) < 0 when
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We further mention a special feature of Eqs (1) and (2). Under the assumptions made with regard to the symmetry of the flow over the pendulum, the system, in addition to the solution (ϑ, , ␣, u), also has a solution (-ϑ, -, -␣, u). Henceforth, it is therefore sufficient to consider just the half strip Q = {(, ) : ∈ [− , ], ≥ 0} as the involute of a part of the phase cylinder. Properties of the kinematic equations. Solving kinematic equations (2) for ␣ and u and substituting relations ␣ = ␣(, ) and u = u(, ) obtained into Eq. (1) could have been an approach to the study the behaviour of the pendulum for different values of the parameters and the initial data. However, when e = / 0, Eqs (2) can have a non-unique solution. We will consider this question in greater detail and show that, in the interval (4) exactly three solutions of these equations exist. Actually, it follows from the second equation of (2) when ϑ = - /2 that: 1) cos ␣ = 0 or 2) u = 0. In the first case, the solution has the form (5) 1 in ␣. For ∈ (1/(l + em ), and exists for all values of . In the second case (u = 0), from the first equation of (2) we have the equation e(␣) = l − ˜ where ␣() ˜ is the unique positive solution of the equation obtained. Note that 1/(l − em )), it has two symmetric solutions ±␣(),
These solutions can be represented in the form (6) So, the interval (4) exists in the phase plane of the variables (ϑ, ) such that, at each point of this interval, there are exactly three solutions (5), (6) of the kinematic equations (2). Outside this interval on the line ϑ = /2, Eqs (2) only have one solution (5). We now consider the behaviour of the solutions for small deviations ϑ and from the interval (4). To be specific, we choose ∈ (1/l, 1/(l − em )). Solutions (5) and (6) then have the form
The increments ␣k , uk (k = 1, 2, 3) correspond to the deviations ϑ and . In the neighbourhood of the first solution, from Eqs (2) we have
It is clear that a solution ␣, u of these equations exists regardless of the sign of the increments ϑ and . Consequently, the first solution ␣1 , u1 is determined in the phase plane to the left and to the right of the interval (4) considered with respect to ϑ. In the neighbourhood of the second (and third) solution, we have
Since ␣ ˜ ∈ (0, /2) and the increment u for this solution can only be positive, the solution ␣, u of these equations only exists for positive values of ϑ. Hence, the second and third solutions ␣2 , u2 , ␣3 , u3 are only determined in a certain domain H1 ⊂ Q adjacent to the interval (4) on the right: these solutions do not extend to the left of it (a cut-off of the surface ␣ = ␣(, ) of its own kind). A similar situation also holds for the interval
The first solution to the left and to the right of interval (4), while the second and third solutions only exist in a certain domain H2 ⊂ Q, adjacent to the interval on the left. Hence, there are two domains, H1 and H2 , adjacent to interval (4) where the function ␣(ϑ, ) is triple-valued (for convenience, we shall assume that ␣(− /2, 1/1) = ± /2. These domains are projections of the two convolutions of the surface ␣(ϑ, ) with one common point = − /2, = 1/l onto the (ϑ,) plane. The boundary of the domain H1 consists of three pieces: of the part of interval (4) (the segment = − /2, ∈ (1/l, 1/(l − em ))) in which ␣ has three values: ␣1 , ␣2 , ␣3 and two envelopes, the equations of which are given in the following implicit form
(7)
where < /2 and the parameter ␣ ∈ (0, /2). In one of the envelopes the solutions ␣1 and ␣2 are identical, and in the other envelope
␣1 and ␣3 are identical. All three solutions ␣1 =␣2 = ␣3 = ␣ * are identical at the point * , * of intersection of the envelopes (the cuspidal point). The value of ␣* is uniquely determined from the equation (8) Substituting the value of ␣* into Eqs (7), we also find * , *. The boundary of the domain H2 is similarly specified but, in Eqs (7), it is necessary to take
At the cuspidal point on this boundary we have ␣1 = ␣2 = ␣3 = − ␣∗ , which follows from Eq. (8).
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Fig. 2.
We will now consider a special case as an example. Suppose
Then, from Eq. 8, we obtain cos2 ␣ * =2/3 such that, at the cuspidal points, we have
From Eqs (7), we find the values of the coordinates of the cuspidal points:
For illustration, the geometrical locations of the points that are solutions of the kinematic equations (2) when ␣ = const are depicted in the phase plane when ≥ 0 (Fig. 2) and an enlarged fragment of this figure is shown on the right. In the general case, when l > em , the shape of the domains H1 and H2 is similar to that shown, although their dimensions change somewhat. When l < em , as l becomes smaller the “middle” point = − /2, = 1/l of the domain of non-uniqueness is raised upwards and, when l → 0, it departs to infinity. The evolution of the domains H1 and H2 in the case of the example considered is shown in Fig. 3 for different values of l. The boundary of a domain when l = em is distinguished by the bold line. Since, in the real motion of a pendulum, the character of the flow over it and the force action on the part of the medium are more likely to change continuously than abruptly, we shall henceforth assume that, in crossing a domain of non-uniqueness in the value of the angle ␣, the choice has to be made of retaining as far as possible the continuity of its change during the motion. The point depicted when > 0 in the domain Q shifts from the left to the right such that, on entering the domains H1 and H2 and during the motion within them, a continuous branch of ␣(ϑ, ) certainly exists and a jump from one branch another can only occur on leaving these domains. ¯ u¯ = 1 hold at all the equilibrium positions of the Equilibrium positions. It follows from Eqs (1) and (2) that the equalities ω ¯ = 0, ˛ ¯ = ϑ, ¯ () ≡ . pendulum () ≡ , ¯ Three stationary points: M1 (0, 0), M2 (, 0), M3 (− , 0), to which the values ␣1 = 0, ␣2 = , ␣3 = − correspond, in the domain Q in the phase plane of the variables (ϑ,) certainly exist. The position of the pendulum “along the flow” corresponds to the first stationary point and the reversed position of the pendulum (“against the flow”) corresponds to the points M2 and M3 . When l > em , there are no other equilibrium positions of the pendulum considered. ¯ 0) and M5 (−, ¯ 0) exist, where ¯ = ˛r is the unique positive root of the equation When l < em , a further pair of stationary points: M4 (, e(␣) = l. Two symmetrical deflected positions of the pendulum correspond to these points, at which the centre of pressure (the point C) coincides with the point of fastening O. In particular, when l = 0, there are two symmetrical positions of the pendulum when the plate is orthogonal to the free stream. We shall henceforth denote any of the stationary values of ϑ and ␣ corresponding to one of the indicated ¯ and ␣ ¯ =␣ points by ¯ ( ¯ when = 0). The situation may be different in the case of pendulums with another configuration (a plate that is not along a support, is not of rectangular shape, is not flat, etc). ¯ , 0), we will denote the small deviations from the Analysis of the stability of the equilibrium positions. For each of the points Mk ( k ¯ , corresponding solution , ¯ ␣, ¯ u¯ by x, y, z and so that
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Fig. 3.
The point M1 (0,0). Taking account of the fact that z = x + (l − em )y, = 0, the equations for the small deviations can be represented in the following form
Henceforth, q0 qp (0), e0 = e (0), . . ... Since q0 > 0, the point M1 (0,0) is stable when l > em and, when l > em , it is unstable (a saddle point). When l = em , the characteristic equation has a double zero root and it is impossible to make a judgement regarding stability using the first approximation. In this case, the equations in the deviations can be represented, up to terms of the third order, in the form (9) These equations have a first integral
and V(x, y) ≥ 0 by virtue of the inequalities e0 < 0 and q0 > 0. Consequently, in such a case the point M1 (0,0) is the centre, to terms of the third order of infinitesimals. We shall only confirm this for the case when l = em since the inclusion of higher-order derivatives of the experimentally determined functions q(␣) and e(␣) makes no sense. The point M2 (,0) (or M3 (-,0)). Taking account of the equation
we conclude that the second equation of system (9) has the form
It is clear that the points M2 (,0) and M3 (-,0) are saddle points. The point M4 (␣r ,0) (or M5 (-,0)). It follows from the kinematic equations that = 0, z = x, and the second equation in system (9) therefore reduces to the form
Since e (˛r ) < 0, no conclusion regarding stability can be drawn on the basis of the first approximation equations. Taking account of terms of the second order of infinitesimals and expressing z to terms of the second order: z = x(1 − ye (␣r )cos␣r ), we write the equations in the deviations using a suitable change of variables in the following form
(10)
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where a and b are constant numbers. The quantity b, which is proportional to the coefficient of xy, is certainly positive, and the quantity a, which is proportional to the coefficient on x2 , can have any sign. In system (10), we now change to a system of polar coordinates by putting
and then, for sufficiently small values of , we have
where the notation
has been introduced. ˜ 3 () makes a “mean” contribution to the change in the quantity . Since () is a decreasing function for It is seen that only the term R sufficiently small then, when ab > 0, the trivial solution of system (10) is unstable, and, if ab < 0, it is asymptotically (but not exponentially) stable. Returning to the initial variables and notation, we obtain that, when l ∈ (0, em ), the points M4 (˛r , 0) and M5 (− ˛r , 0) are stable (unstable) foci if the inequality (11) is satisfied (not satisfied). We emphasize that, for these points, the linear part of the “position” force has the nature of a “restoring” force and their stability or instability is determined by the sign of the coefficient of the non-linear part of the position force (by the sign of the coefficient a in system (10)). The results of the qualitative analysis of the stability of the points M4 (␣r , 0) and M5 (− ␣r , 0) are completely confirmed by the results of a stability study using a well-known rigorous method.7 ¯ − ␣r = /2) corresponds, is of special interest when analysing The case when l = 0, to which a position of the plate across the flow ( the stability of the points M4 (␣r , 0) and M5 (− ␣r , 0). In this case, terms of the third order of infinitesimals have to be taken into account. As a result of the corresponding calculations, from Eqs (1) and (2) we have
and again the equations obtained can be reduced using the change of variables to the form (the coefficient a˜ has any sign and b˜ > 0)
The trivial solution of this system is asymptotically stable, which follows from Lyapunovs theorem. Actually, the function
˜ 2 z 2 by virtue of that is positive definite in a certain neighbourhood of the origin of coordinates, has a negative definite derivative V˙ = −bz 1 2 the system of equations constructed. Hence, when l = 0, the points M4 (/2.0) and M5 (− /2.0) are asymptotically stable. So, when l > em , the position “along the flow” is asymptotically stable and, when l = em , it is stable but not asymptotically, while the position “against the flow” is unstable. There are no other equilibrium positions. When 0 < l < em , the “along the flow” and “against the flow” positions are unstable and the points M1 (0,0), M2 (, 0) and M3 (− , 0) are saddle points. A pair of symmetric deflected (stable or unstable) equilibrium positions exists. Since system (1), (2) is dissipative as a whole then, in the case of values of l for which there are no equilibrium states, the creation of a stable cycle would be expected which encompasses one or several stationary points and, possibly, one or several pairs of stable and unstable cycles (by a cycle, we means any closed phase trajectory that both encompasses the phase cylinder and also lies in the strip < . When l = 0, the “along the flow” and “against the flow” positions also remain unstable but the “across the flow” position of the pendulum is asymptotically stable (the points M4 (/2, 0) and M5 (− /2, 0) are stable foci). Cyclic motions of the pendulum. The qualitative analysis carried out gives a specific representation of the behaviour of the pendulum close to the equilibrium positions and of the corresponding phase trajectories in the neighbourhood of the stationary points, that is, when |ω| 1. It is known that, when |ω| 1, the action of the medium is of a dissipative nature, that is, it ensures a tendency towards a reduction in the energy. However, an antidissipative character3,5 of the action of the medium may manifest itself in certain bounded domains of phase space. In the problem considered, the determination of the integral character of the action of the medium for “mean” values of is complicated by the lack of uniqueness of the solution of the kinematic equations. This forces us to turn to numerical modelling as a method of studying the behaviour of the pendulum for different values of the structural parameter, that is, the length of the support. A plate with an elongation 8 was chosen for the calculations. The value of the dimensionless parameter a was taken to be equal to 0.001 and the dimensionless length l of the support was varied over a range from 0 to 3.
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Fig. 4.
As a result of the numerical experiment, it was established that the form of the motions of the pendulum “in the large” depends considerably on the values of l; several critical values of l were distinguished and, when these are passed through, there is a fundamental reconstruction of the phase portrait of the system. The relations between the maximum values of the angular velocity with respect to ϑ, max , and the characteristic phase trajectories are shown in Fig. 4 as a function of the length of the support. The critical values of the length of the support are denoted by lk (k = 1, 2, . . ., 6) and, here, l1 = em , S+ and S− are the values of max on the separatrix entering the saddle point M2 (, 0) and emerging from the saddle point M3 (− , 0) closest to these points, C+ are the values of max on a stable cycle that does not encompass the phase cylinder, and R+ and R− are the values of max for stable (the + sign) and unstable (the - sign) cycles encompassing the phase cylinder (the curves R− are shown by dashed lines). We will now briefly illustrate the results presented using phase portraits (PP). Since the values of the “amplitudes” of the phase trajectories are shown in Fig. 4, in the PP (Fig. 5) we shall restrict ourselves to just a qualitative depiction of these trajectories, distinguishing the most characteristic of them: the separatrices and cycles by means of bold lines. When l > l1 = em according to Fig. 4, the separatrix S− emerging from the saddle point M3 (− , 0) is located below the separatrix S+ entering the saddle point M2 (, 0). All the remaining phase trajectories contract into the origin of the coordinates, the point M1 (0,0), the unique attractor to which the stable “along the flow” equilibrium position of the pendulum corresponds. When the length of the support is reduced (when l > em ), both the separatrices S− and S+ approach close to the ϑ axis. The first critical value of the length of the support is l = l1 = em . In this case, the point M1 (0,0) is the centre up to terms of the third order of infinitesimals. The form of the PP is unchanged in other respects and is not illustrated in Fig. 5. As soon as the length of the support becomes less than the first critical value, the character of the PP changes qualitatively. The point M1 (0,0) ceases to be stable (it becomes a saddle point) and a pair of unstable foci M4 (␣r , 0) and M5 (− ␣r , 0) is generated. Hence, all these three points are unstable, and the saddle points M2 (, 0) and M3 (− , 0) are also unstable. By virtue of the dissipativeness of the system as a whole, the separatrices emerging from the point M1 (0,0) cannot depart to infinity, as a consequence of which closed trajectories necessarily arise. At first, there is one closed trajectory C+ as shown in Fig. 5 (l = 0.55) which encompasses all three points M1 (0,0), M4 (␣r , 0) and M5 (− ␣r , 0). It is as though this cycle inherits the properties of the unique attractor that existed when l > em . This unusual bifurcation (a Hopf type bifurcation) is caused by the existence of dynamic symmetry in the system and the characteristic non-linearity of the position force. The trajectory C+ is a unique attractor to which all phase trajectories (apart from S+ ) contract both from within a cycle and from the outside it. When there is a further reduction in the value of l, the cycle C+ expands, the separatrices S− and S+ descend and, at a certain new critical value of the length of the support l = l2 < l1 (the calculations showed that l2 ∼ = 0.4245), the two separatrices coincide, forming a loop of separatrices. The cycle C+ settles on this very loop, having expanded up to the limit (from = - to = ). All the more sweeping oscillations of the pendulum that does not execute one complete revolution correspond to unwinding phase trajectories and, with each successive oscillation, the time interval between the successive stops of the pendulum increases. A rotation of the pendulum in one and the same direction corresponds to the phase trajectories located outside this loop. and the time of each revolution increases monotonically to infinity. When l < l2 , the separatrices S− and S+ (S− is now higher than S+ ) are formed from the loop and a closed stable trajectory R+ (Fig. 5, l = 0.3) encompassing the phase cylinder (a second trajectory R+ is also formed here in the domain < 0). A stable rotational state of motion of the pendulum corresponds to the trajectory R+ . As the length of the support is reduced in the interval 0 < l < l2 , the separatrices continue to “descend” and the trajectories R+ separate, such that the angular velocity of the pendulum in the rotational state increases. A new periodic trajectory arises in the case of the next critical value l = l3 (l3 ∼ = 0.267 < l2 ) in the domain of fairly high values of and it is metastable: all the phase trajectories located above it contract to it, and the trajectories located below it are repelled from it and descend to the stable trajectory R+ .
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Fig. 5.
When the length of the support is further reduced, this metastable trajectory splits into a pair of periodic trajectories: an unstable trajectory R− and a stable trajectory R+ located above it, which gradually recede from one another as the value of l decreases. The trajectory 1 1 R− separates the domains of attraction of the stable trajectories R+ and R+ . In this case, the domain of attraction of the trajectory R+ remains 1 1 quite spacious. From any initial fixed position apart from the equilibrium positions, the pendulum “independently” reaches a self induced means, in particular, that the pendulum rotational state corresponding to the trajectory R+ . The existence of the unstable trajectory R− 1 cannot “independently” reach a state of self induced rotation corresponding to the trajectory R+ and a preliminary rotation is necessary to 1 reach it. With a subsequent decrease in the length of the support, a further metastable cycle is generated at a certain value l = l4 (l4 ∼ = 0.24 < l3 = 0.267) for the specific plate considered) which, on continuing to reduce l, decomposes into a pair of periodic trajectories: a stable trajectory R+ and, above it, an unstable trajectory R− , but both of these are located below the trajectory R+ . The corresponding 2 2 phase portrait is qualitatively represented in Fig. 5 (l = 0.2) in which the states R− and R− are represented by the bold dashed line. The 2 1 regular subdivision of the phase space into domains of attraction of the stable rotational states is clear. The pendulum now independently but not R+ as was the case earlier. just reaches the self-induced rotational state R+ 2 As l decreases, the trajectory R+ descends approaching the separatrices S− and S+ , and, at the critical value l = l5 ∼ = 0.00085, the trajectory 2 + − + R2 coincides with the separatrices S and S and, at the same time, they all pass through the point M1 (0,0) to coincide now with the separatrices of the point M1 (0,0) and form two loops. Within the loops, the points M4 (␣r , 0) and M5 (− ␣r , 0) are unstable foci. + When there is a further reduction in the value of the parameter l, the orbitally stable cycles C+ 1 and C2 , encompassing the points M4 (␣r , 0) and M5 (− ␣r , 0) respectively, are removed from these loops, as illustrated by the phase portrait in Fig. 5 (l = 6 × 10−4 ). The points M4 (␣r , 0) + and M5 (− ␣r , 0) still remain as unstable foci and the trajectories emerging from them are wound onto the cycles C+ 1 and C2 from within. All − the remaining trajectories located below R2 are wound onto these cycles from the outside. Oscillatory motions of the pendulum that change (after a finite number of complete revolutions) into one of the two stable self-induced oscillatory states correspond to these trajectories. When l is reduced, the above-mentioned cycles contract and, finally when l = l6 ∼ = 0.00045, each of them collapses to the stable point M4 (␣r , 0) or M5 (− ␣r , 0). For all l from the interval l ∈ (0, l6 ), these points are asymptotically stable foci (although they are not exponentially stable, so that the attenuation is very slow). The decaying oscillations of the pendulum about the equilibrium position close to the “across the flow” position now correspond to trajectories located below the trajectory R− . 2 In conclusion, we shall dwell on the case when l = 0. The problem of the behaviour of such a pendulum (a wind vane) has already been studied by Zhukovskii8 and the following was noted by him: “If such a wind vane is placed against the wind, it becomes perpendicular to
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the direction of the wind and has good stability in this position. But, if we impart a rotational motion to the wind vane in one direction or another, it will continue to rotate in the specified direction” and, in the state of self-induced rotation, “experiment gives approximately ∼ = 0.5”. It is curious that, even within the second-order dynamical system considered here, based on a quasistatic model of aerodynamic action on a plate, these properties have received not only qualitative but also quantitative confirmation ( ∼ = 0.45, see Fig. 4). Moreover, the estimate of the initial angular velocity ( ∼ = 0.39) that has to be surmounted in order to reach the domain of attraction of the self-induced rotational state R+ is clear from Fig. 4. The high velocity self-induced rotational states R+ and R− have not been discussed previously, possibly due to the difficulty of their 1 1 practical realization and on account of features of the specific profile used in the calculations. Acknowledgement This research was supported by the Russian Foundation for Basic Research (11-08-00444, 12- 01-00364). References 1. 2. 3. 4. 5. 6. 7. 8.
Dosayev MZ, Kobrin AI, Lokshin BYa, Samsonov VA, Selyutskii YuD. Constructive Theory of Small-Scale Wind Power Generators. Moscow: Izd MGU; 2007. Barbashin YeA, Tabuyeva VA. Dynamical Systems with a Cylindrical Phase Space. Moscow: Nauka; 1969. Shamolin MV. Methods of Analysing Dynamical Systems with Variable Dissipation in Dynamics of Solids. Moscow: “Ekzamen”; 2007. Lokshin BYa, Samsonov VA. Computational-analytical investigation of the behaviour of an aerodynamic pendulum. Vestn MGU Ser 1 Mathematics Mechanics 1996;6:50–2. Lokshin BYa, Samsonov VA. Problem of the Motion of a Body in a Resisting Medium. Qualitative Analysis. Moscow: Izd Mosk Univ; 2012. Tabachnikov VG. Stationary characteristics of wings at low velocities over the whole range of angles of attack. Tr TsAGI 1974;(Issue 1621):79–83. Malkin IG. Theory of Stability of Motion. Oak Ridge TN: US Atom Energy Commission; 1952. Zhukovskii NYe. On the fall in air of light elongated bodies rotating about their longitudinal axis. In: Collected paper. Moscow:Gostekhizdat; 1949;4;41–68.
Translated by E.L.S.