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ISSN 1560-3547, Regular and Chaotic Dynamics, 2014, Vol. 19, No. 1, pp. 116­139. c Pleiades Publishing, Ltd., 2014.

The Dynamics of a Rigid Body with a Sharp Edge in Contact with an Inclined Surface in the Presence of Dry Friction
Ivan S. Mamaev1* and
1

Tatiana B. Ivanova2**

Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia 2 Faculty of Physics and Energetics, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
Received March 28, 2013; accepted December 31, 2013

Abstract--In this paper we consider the dynamics of a rigid bo dy with a sharp edge in contact with a rough plane. The bo dy can move so that its contact point is fixed or slips or loses contact with the support. In this paper, the dynamics of the system is considered within three mechanical mo dels which describe different regimes of motion. The boundaries of the domain of definition of each mo del are given, the possibility of transitions from one regime to another and their consistency with different co efficients of friction on the horizontal and inclined surfaces are discussed. MSC2010 numbers: 70F40, 70E18 DOI: 10.1134/S1560354714010080 Keywords: ro d, Painlev´ paradox, dry friction, loss of contact, frictional impact e

Contents
1 2 INTRODUCTION MAT 2.1 2.2 2.3 2.4 HEMATICAL MODELS DESCRIBING VARIOUS TYPES OF MOTION An Inverted Pendulum A Sliding Rod A Free Rod Problems of Consistent Change of Motion Regimes 117 118 119 120 121 121 123 123 124 124 127 127 132 132 136

3

THE BOUNDARIES OF THE REGION OF POSSIBLE MOTIONS IN DIFFERENT MODELS, THEIR INTERSECTION AND REGIONS OF PARADOXES 3.1 The Region of Possible Loss of Contact 3.2 RPM of the Sliding Ro d 3.3 RPM of the Inverted Pendulum QUALITATIVE ANALYSIS OF THE DYNAMICS, CHANGE OF REGIMES AND PARADOXES 4.1 A Co efficient of Friction that is Smaller than a Critical Value ( ) 4.2 An Example of the Tra jectory with a Change of Motion Regimes for < 4.3 A Co efficient of Friction that is Larger than a Critical Value for a Horizontal Plane ( = 0, > ) 4.4 A Co efficient of Sliding Friction that is Larger than a Critical Value for an Inclined Plane ( > 0, > )
*

4

**

E-mail: mamaev@rcd.ru E-mail: tbesp@rcd.ru

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THE DYNAMICS OF A RIGID BODY WITH A SHARP EDGE 5 CONCLUSION ACKNOWLEDGMENTS REFERENCES 137 138 138

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1. INTRODUCTION In this paper we consider the planar motion of a body with a sharp edge in contact with a rough plane, more precisely, of a homogeneous rod with a single point of contact with the supporting plane, but the results obtained can be generalized to the case of more complex bodies and contacting surfaces. In the future, various mathematical models providing a tool for explaining the resulting inconsistencies (paradoxes) in the system under consideration can be extended to include more complex systems (for example, Euler's disk) and to explain dynamical effects, in particular, losses of contact observed during their motions [6, 10]. Since the plane realizes a unilateral constraint and the body can move in such a manner that its end which is in contact with the plane is either fixed or slips or loses contact with the support, we consider the dynamics of the system within the framework of three mechanical models which describe various regimes of motion. We also assume that the Coulomb law of dry friction is valid at the point of contact with the plane. But, as is well known, in systems in which Coulomb friction along with absolute rigidity of the body and the surface is assumed, various paradoxical situations may arise where the solution either does not exist or is nonunique under the same initial conditions. Such paradoxes, called the Painlev´ paradoxes, with respect to the planar motion of a body with a single point of contact were e apparently first explored by E. A. Bolotov [1], who continued a discussion started by P. Painlev´ [14], e L. Lecornu [11], M. de Sparre [2], F. Pfeiffer [15] and others. In the course of that discussion, in order to solve the paradoxes obtained in the Painlev´ ­ Klein system, the authors introduced additional e hypotheses on the presence of elastic deformations at the point of contact and instantaneous stop (jamming) when the boundary of a paradoxical region is reached. An analogous approach for a system with many degrees of freedom is developed by Le Suan An in [16]. We note that there are many well-known mechanical systems with Coulomb friction in which the paradoxes of nonuniqueness or nonexistence of a solution occur (for example, a braking system [23], a ladder resting on a horizontal floor and a vertical wall [25], the Painlev´ ­ Appel system [14] and e others). Among recent studies on paradoxes in systems with Coulomb friction we mention the papers by N. A. Fufaev [3], Yu. I. Neimark [12] and V. A. Samsonov [22], which investigate various types of motions of the systems using the analysis of phase tra jectories depending on the values of system parameters (the clearness and efficacy of applying phase plane analysis was demonstrated by the authors of [13, 22] using a braking system as in [23]). In contrast to the above-mentioned (physical) approaches to solving the paradoxes there also exist other approaches, which are more formal and do often not elucidate the mechanical meaning of the result obtained, but require the use of rather cumbersome mathematical tools. In Stewart [24] the paradox of nonexistence of a solution is attributed to the presence of "impulsive" forces arising from a collision with the surface. The author of [24] uses the "principle" of maximum of dissipation, the theory of differential variational inequalities, complementarity conditions and other nontrivial mathematical methods. However, in the problem of the fall of a homogeneous rod the developed theory is applied by Stewart only to the nonexistence paradox, although this system is known to have regions of initial conditions for which the solution is nonunique (see, e.g., [9]). Moreover, for the "solution of the paradox" obtained the mechanical meaning is unclear and is not elucidated by the author. Some of the results presented in this paper for a homogeneous rod were previously obtained in [4, 21]. In particular, the authors of [4] present the critical value of the coefficient of friction at which the paradoxes of nonexistence and nonuniqueness of the planar motion of the rod arise, and carry out a very detailed analysis of the behavior of the system near the singular point on the boundary of the region of loss of contact for coefficients of friction larger than a critical value. Rozenblat [21] explores the conditions which must be satisfied by the initial data and parameters of the body in the motion without loss of contact, i.e., he actually shows only the
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boundary on which the normal acceleration of the point of contact becomes positive. Nevertheless, the analyses presented in the above-mentioned papers are definitely not sufficient to gain insight into the dynamics of the entire system. The possibility of loss of contact between a point of the homogeneous rod with the plane is also discussed in the papers by A. P. Ivanov [5, 6, 8, 9] as applied to the motion of systems with unilateral constraints in more general settings. In particular, in [8], using a geometrical approach based on the introduction of some auxiliary parameter space divided into non-intersecting regions according to the number of possible types of motion, necessary and sufficient conditions have been obtained for one-valued solvability of the problem of motion of a body on a surface with friction. In the case of slipping and rest the possibility of simultaneous existence of three solutions (loss of contact, rest and slipping of the contact point) is shown. The authors of [9] examine in detail the motion of a rod on an absolutely rough support, define the regions of inconsistency of the sign of normal reaction and normal acceleration (paradoxes) and show that the elimination of motions of the point of contact with slipping from consideration leads to the onset of the paradox of nonexistence of a solution, which is not observed if we take into account the boundedness of the real coefficient of friction. This paper is a supplement to and a generalization of the previously obtained results for the planar motion of a body with a single point of contact on a rough plane. To prove most assertions, we use the method of phase portrait construction, supplemented by the construction of boundaries of the regions of possible motions of the relevant models. This method is illustrative and allows one, on the one hand, to avoid cumbersome analytical calculations, but, on the other hand, to introduce a classification of various solutions depending on the system parameters. 2. MATHEMATICAL MODELS DESCRIBING VARIOUS TYPES OF MOTION Consider the planar motion of a body with a single point of contact, C , with a rough plane. We introduce the following notation (Fig. 1): is the angle of inclination of the plane to the horizon, (x, y ) are the coordinates of the center of mass G of the body relative to the fixed coordinate system (the axis Ox is directed along the plane), is the angle between the straight line CG and the plane, m is the mass of the body, I is the mass moment of inertia, Fig. 1. A body with a single point of contact with an l is the distance from the center of mass to the contact point of inclined plane. the body. We shall assume that the center of mass of the rod is acted upon by an external force whose normal and tangential components are equal to Pn and Pt , respectively (if there are no other forces but gravity mg , then Pn = mg cos , Pt = mg sin ). We denote the normal and tangential components of the reaction forces acting on the rod at the contact point from the plane by Rn and Rt , respectively. As noted above, it is impossible to describe the motion of the system by means of one mechanical model, since, depending on the initial conditions, the point of contact is either fixed or slips or loses contact with the support. As a result, it is necessary to use at least three mechanical models describing various regimes of motion: the model of an inverted pendulum, that of a sliding rod and that of a free rod. Where necessary, we shall denote the indices of the variables for the description of each of the systems by p (pendulum ), s (sliding ) and f (free ), respectively. In the analysis of the system we shall assume that ­ the contact with the plane is a point contact, ­ the Coulomb law of dry friction is valid at the point of contact. We shall make here no other additional hypotheses associated with the problems of realization of such a contact and with possible restrictions on the coefficient of friction.
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For convenience, we shall write all equations of motion in dimensionless form. To do this, we choose l/g to be the unit of time and l the unit of distance, i.e., we make a change of coordinates and time in the equations: x x, l I , ml2 y y, l Pt , mg g dt dt. l Rn , mg Rt . mg (2.1)

Moreover, we define the following dimensionless quantities: = = N= R = (2.2)

In what follows, the system dynamics will be illustrated with the case where the body is a homogeneous rod and I = ml2 /3, = 1/3, [0, ]. 2.1. An Inverted Pendulum This model corresponds to the case where the contact point C neither slips nor loses contact with the surface (Fig. 2a), i.e., in this case two constraints are imposed on the rod: xc const, yc 0, (2.3)

where (xc ,yc ) are the coordinates of the point of contact which are related to the coordinates of the center of mass (x, y ) by xc = x - cos , yc = y - sin . (2.4)

Fig. 2. Schematic of the mechanical dynamics models with the acting forces (the sign of identical equality means that a corresponding constraint is imposed, while the sign of equality in the case (c) corresponds to a special choice of initial conditions).

Hence, the reaction from the plane at the point C has a normal component N and a tangential component R . The equations of motion of the system are x = R + , ¨ y = N - 1, ¨ = R sin - N cos . ¨ (2.5)

The area of applicability of this model is defined by the conditions of continuous contact and absence of slipping (i.e., the reaction force lies inside the cone of friction): N > 0, where is the coe Thus, for those model, the region manifold (with an N > |R | ,

fficient of friction. regimes of motion of the system which are described by the inverted pendulum of possible motions (hereinafter referred to as the RPM) is a three-dimensional edge) defined by , N = Np (, )
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M3 = {(xc ,, ) | xc = const, 0 p

0, N -|R | = fr (, )

0}.

(2.6)

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Remark 1. The parameterized by change of motion at various values

necessity to consider in this case the family of two-dimensional manifolds (, ) the coordinate of the point of contact xc stems from the fact that under a possible regimes (inverted pendulum and sliding rod) the point of contact comes to rest of xc .

Differentiating the constraint equation (2.3) using (2.4), we find the components of the reaction force from the plane: +sin2 - sin cos - 2 sin , 1+ sin cos - ( +cos2 ) R = - 2 cos . 1+ N = Np (, ) = Substituting these expressions into (2.5), we obtain: ­ the dynamics of the inverted pendulum in the phase space M3 is governed by p =- ¨ 1 (cos + sin ), 1+

(2.7)

­ the trajectories of the system are the level lines of the energy integral 1+ 2 +sin - cos = const. E= 2 2.2. A Sliding Rod In this model the point of contact C moves without loss of contact on a plane with velocity V = xc (Fig. 2b), which corresponds to a single constraint: yc 0. (2.8) Hence, the reaction force at the contact point has only the normal component N . There is also a friction force acting at the point of contact T = -N , = sgnV. The equations of motion of the system are x = - N , ¨ y = N - 1, ¨ = -N sin - N cos . ¨ (2.10) The region of applicability of this model is defined by the inequality N > 0, xc = 0. Thus, in the sliding rod model the RPM consists of two four-dimensional manifolds M4(+) = {(, , xc , xc ) | 0 s M4(-) s = {(, , xc , xc ) | 0 , N = Ns (, , xc , xc ) , N = Ns (, , xc , xc ) 0, xc > 0}, 0, xc < 0}, (2.11) (2.9)

which are adjacent to each other along the manifold {xc = 0}. Differentiating (2.4) using (2.8) and (2.10), we obtain the condition of conservation of constraint in the form yc = -B (, )+A()N = 0, ¨ (2.12) 1 B (, ) = 1 - 2 sin , A() = +cos2 + cos sin . The sign B (, ) in this case has been chosen by analogy with the notation that is often used in the literature (see, e.g., [4, 7, 19]). From (2.12) we find the reaction force and the relevant friction force: N = Ns (, ) = B (, ) , A() T = -N = - B (, ) . A() (2.13)

Substituting (2.13) into (2.10), we obtain:
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­ the dynamics of the sliding rod in the phase space M4 is governed by s (2 sin - 1)( sin +cos ) , +cos2 + cos sin ( sin +cos ) 2 ( +1) - sin - + , V= 2 + cos sin +cos = ¨ ­ the total energy +1 2 V 2 + - V sin +sin - (xc +cos ) 2 2 decreases due to the friction force: E= dE = -|V |Ns . dt Since velocity rod we s and = (2.15)

(2.14)

the first of Eqs. (2.14) explicitly depends only on = sgnV = ±1 and not on the value of V , in the analysis of the region of possible motions and phase tra jectories of the sliding hall pro ject them onto the plane (, ). There will be two of such pro jections: when = 1 -1. 2.3. A Free Rod

The equations of motion of the system at the instant of loss of contact, when no reaction forces act on the body from the support (Fig. 2c), have the simplest form x = , ¨ y = -1, ¨ = 0. ¨ (2.16)

Differentiating (2.4) using (2.16), we find that the normal component of the acceleration of the point of contact at the instant of loss of contact depends only on , and can be represented as yc = -B (, ) = -1+ 2 sin . ¨ (2.17)

We note that B (, ) is the same function defined above for the sliding rod (2.12). The area of applicability is determined by the fact that this model is used only for the analysis of the loss of contact. Therefore, among all tra jectories we need to consider only those whose initial conditions lie on the submanifold which corresponds to the contact of the point C with the fixed plane and is defined by M4 : yc = y - sin = 0, 0 yc = y - cos = 0. (2.18)

Thus, in the six-dimensional phase space of the free system we are interested only in phase tra jectories that start on the submanifold M4 in its small neighborhood and specify some five0 dimensional manifold M5 = {(, , xc , xc ,yc , yc ) | 0 f , yc |
t=0

= 0, yc |

t=0

= 0, 0

yc

1 , 0

yc

2 },

(2.19)

where 1 and 2 are arbitrarily small positive constants. 2.4. Problems of Consistent Change of Motion Regimes In this paper we consider the motion of the body only before the instant of fal l onto the supporting plane or before the loss of contact. In the case where the models under consideration completely describe the dynamics of the system, its general phase tra jectory (depending on the initial conditions) may consist of portions of the tra jectories of the inverted pendulum and the sliding rod which must match consistently. At the instant of loss of contact the tra jectory must terminate on the boundary of the domain of definition of the free system. This implies that the boundaries of the domain of definition of all the three models must consistently match, too.
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However, as is well known, in systems with Coulomb friction a consistent description of motion by means of such simple models in the entire region of possible parameters is, as a rule, impossible, since paradoxical situations occur. The classical work of Painlev´ [14] gives examples of systems with bilateral constraints which e exhibit paradoxes of nonuniqueness of solutions within the framework of one model due to the ambiguity of definition of the reaction force with large coefficients of friction. Systems with unilateral constraints within the framework of one model can have no nonuniqueness paradoxes, since the sign of vertical reaction is uniquely defined by the conditions of the model under consideration. Nevertheless, systems with unilateral constraints may exhibit nonuniqueness paradoxes due to a possible overlap of RPMs of various mathematical models. Moreover, in the phase space there may exist regions in which none of the models is defined, which corresponds to the paradox of nonexistence of a solution. These paradoxes may arbitrarily be called algebraic. We note that a partial description of the areas of algebraic paradoxes for the homogeneous rod is given, for example, in works [4, 9]. Moreover, such systems may display dynamical paradoxes due to the fact that the directions on the tra jectories of various models can be unmatched on the common boundaries of the RPMs (which corresponds to nonexistence of a solution). At some parameter values, regions may arise where at zero velocity of the contact point (V = 0) the motions of the inverted pendulum are correctly defined, but at small deviations (|V | > 0) all tra jectories in the sliding rod model tend to leave the neighborhood of the manifold {V = 0}, i.e., from the mathematical point of view, instability arises. Thus, one of the problems associated with the dynamics of this system is the analysis of consistency and the degree of smoothness during a transition from the tra jectory described by one mathematical model to a tra jectory described by another model, i.e., under a change of the regime of motion. We show the three models schematically as sectors on a circular diagram (Fig. 3). These sectors are divided by segments corresponding to portions of the boundaries of the domain of definition of these systems Bij along which they may come into contact with each other (the indices i, j denote the corresponding model). We note that the dimension of the boundaries is different, although they look the same in the schematic. For example, the boundary Bps is two-dimensional and is p defined by the equality fr (, ) = 0 (see (2.6)), while the dimension of the boundary Bsf is equal to three, and it is given by the equation Ns (, , xc , xc ) = 0 in the fourdimensional space (see (2.11)). The boundary Bpf is also two-dimensional and is given by the relation Np (, ) = 0. On the boundaries of the loss of contact Bpf and Bsf we consider only the transition in one direction, namely, that of the loss of contact (the fall back onto the plane requires the use of an additional impact model). On the other hand, on the boundary Bps , transitions in both directions are in principle possible. The reaction may go beyond the cone of Fig. 3. Schematic of possible transifriction, and in this case the point of contact begins to slip. tions between motion regimes which Moreover, during the motion of the point of contact (xc =0) are described by the mathematical models used. the tra jectory can reach the boundary (xc = 0) under the action of the friction force, and, depending on the sign p of fr (, ), the point of contact stops (i.e., it passes to the inverted pendulum regime) or continues to slip, but xc changes sign. This diagram is simplified: in reality some boundaries may be absent (for example, if one of the regions completely lies inside another without intersecting the third one). We point out some problems which are analyzed below. ­ There is a fairly well-known hypothesis relating to the dynamics of such systems, which states that there is no boundary between the inverted pendulum system and the free system: the RPM of the inverted pendulum is located "inside" the RPM of the sliding rod, i.e., the point of contact always slips before detachment. Below we show that if the coefficient of friction is smal ler than a
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critical value, this hypothesis is true, and the measure of the trajectories of the inverted pendulum which reach the boundary Bpf is zero. ­ As pointed out above, within the framework of the models, a transition from the motion with slipping to the inverted pendulum regime is in principle possible, i.e., the system can reach the constraint xc = const. We show that in this case a noncontradictory (nonparadoxical) passage to the constraint is real ly possible. ­ We also show that if the coefficient of friction is larger than a critical value, paradoxical situations of various types may arise. An important feature of all the three models is that both their tra jectories and the boundaries of the RPM are pro jected onto the plane (, ) without singularities (folds, self-intersections etc.). This makes it possible to use a more illustrative graphical method for the analysis, which in many cases allows lengthy calculations to be replaced with a series of figures. 3. THE BOUNDARIES OF THE REGION OF POSSIBLE MOTIONS IN DIFFERENT MODELS, THEIR INTERSECTION AND REGIONS OF PARADOXES The boundaries of regions of possible motions (RPM) are actually the boundaries of the corresponding phase spaces which are defined above for each of the models. In this section we will consider the RPM of each of the models in more detail and define the regions of their overlap (regions of algebraic paradoxes), if any. In the regions where the RPMs of various models match or overlap we will additionally examine the phase tra jectories to identify and explore possible dynamical paradoxes in this system. 3.1. The Region of Possible Loss of Contact According to (2.17), during the loss of contact the acceleration of the point of contact is ¨ yc = -B (, ). It is obvious that if B (, ) < 0 (that is, yc > 0, the hatched region in Fig. 4), ¨ the body loses contact with the plane and begins free motion. Otherwise (B (, ) > 0) the body seeks to deform the supporting surface, that is, it reaches a constraint, and has to be described by one of the other two models. Thus, the boundary of the loss of contact does not depend on the parameters and and is defined by the equation B (, ) = 0. (3.1)

Fig. 4. The region in which yc > 0 (hatched area). The solid line is the boundary line of the RPM of the free ¨ rod (2.17).

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3.2. RPM of the Sliding Rod According to (2.11) and (2.13), the region of possible motions in M4 does not depend on the s parameter and is defined by the equation B (, ) > 0, A() that is, the quantities A() and B (, ) have to be of the same sign. From the relationships (2.12) and (3.2) it follows that there exists a critical value = = 2 (1 + ), such that -- if , the function A() is positive definite for all [0, ], therefore, the region Ns > 0 is determined by the inequality 2 > 1/ sin , (3.3) that is, it does not depend on and is bounded by the curves (3.1) (see Fig. 5a); -- if > , there is an interval in which the function A() is negative. In this case the condition Ns > 0 0 is equivalent to the following conditions (see Figs. 5b and 5c): ­ B (, ) > 0 if = 1, (1 ,2 ) and if = -1, (3 ,4 ), ­ B (, ) < 0 if = 1, (1 ,2 ) and if = -1, (3 ,4 ), where i , i = 1,... , 4, are the roots of the function A() at the corresponding value . (3.2)

Fig. 5. The pro jection of RPM of a sliding rod onto the plane (, ) (dotted filling) at = 0, = ±1 and various values of . If = i (the roots of A()), the denominator of the reaction force Ns (2.13) vanishes.

3.3. RPM of the Inverted Pendulum According to (2.6), the RPM of the inverted pendulum in M3 is defined by two conditions. p 1. The reaction force is Np > 0, that is, according to (2.7), 2 < +sin2 - cos sin . sin (1 + ) (3.4)

Note that the curve Np = 0 corresponding to the boundary of the region of the loss of contact does not depend on .
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2. The overall reaction is within the cone of friction, that is, the following condition is satisfied fr (, ) = Np -|R | or, according to (2.7), (cos - sin ) (sin (1 (cos + sin ) (sin (cos + sin ) 2 < (1 (cos - sin ) 2 > + + - cos ) - ( + ) ) - cos )+ ( - ) ) if R > 0, (3.5) if R < 0. 0,

Typical RPMs which are defined by relations (3.4) and (3.5) for different values of and = 0 are shown in Fig. 6, and those for > 0 are shown in Fig. 7.

Fig. 6. RPM of the inverted pendulum for different values of and = 0 (dark grey). The boundary of the loss of contact (3.4) is shown as a dashed line. Light grey denotes the region in which Np > 0.

According to relations (3.5), for = 0 there is a critical value = such that ­ f or
(1) (1) (1)

=

1 2 (1 + )

,

(see Fig. 6a) the RPM is not simply connected (there are two holes), (see Fig. 6c) the RPM is a simply connected domain.

­ f or >

For = 0 the symmetry of the boundaries of RPM relative to = /2 is broken (Fig. 7). In this (1) case (1) is split into two critical values ± : (1 + 42 +4 2(1 + 2 ) (1 + (1) + = 42 +4 moreover, there is another critical value
(1) -

=

2(1 + 2 )



) - (1 + 2)2 , - 2 )+ (1 + 2)2 ; - 2

= These critical values are such that

(2)

= .

­ for < (2) (see Fig. 7a) the RPM is disconnected: it consists of three regions one of which is not simply connected,
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Fig. 7. RPM of the inverted pendulum for = 0.25 (- = 0.41, + = 1.25, (2) = 0.25) and different values of (dark grey). If no other external forces but gravity are applied to the center of mass, this value of corresponds to the angle of inclination of the plane = /12. The boundary of the loss of contact (3.4) is shown as a dashed line. Light grey denotes the region in which Np > 0.

(1)

(1)

­ for (2) < < two holes), ­ f or ­ f or
(1) - (1) +

(1) - (1) +

(see Fig. 7c) the RPM is connected, but not simply connected (there are (see Fig. 7d) the RPM has only one hole,

<<

< (see Fig. 7f ) the RPM is a simply connected domain.

The curves which are determined by the equalities fr = 0 and Np = 0 from (3.4) and (3.5) intersect each other only at two points (see Figs. 6 and 7), whose value can be determined analytically: , 0 = ± 1+ 2 . (3.6) 0 = /2+ arcsin 2 1+ Moreover, from the inequalities defining the RPM of the inverted pendulum, | R | < Np , 0 < Np ,

it follows that the interior of the cone of friction defined by the inequality fr > 0 (3.5) lies within the region determined by the inequality Np > 0.
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Therefore, for analysis of the motion of the inverted pendulum it is enough to consider the boundaries determined by the inequality fr > 0 (3.5). After a separate discussion of the boundaries of RPM of each system, we will consider the possibility of transitions from one regime of motion to another for different values of and and discuss their consistency. As noted above, an important feature of the models considered is the possibility of correct projection of al l phase trajectories and boundaries of RPMs on the plane (, ). In this case most propositions can be proved not by analytical calculations but by the method of phase portrait construction, supplemented by the construction of the boundaries of the RPMs of the corresponding models. Therefore, in what follows we usually omit relevant calculations in the proofs. 4. QUALITATIVE ANALYSIS OF THE DYNAMICS, CHANGE OF REGIMES AND PARADOXES In this section, without making a complete qualitative analysis, we describe only several dynamical effects observed in the system depending on the values of the parameters and . 4.1. A Coefficient of Friction that is Smaller than a Critical Value ( )

In Fig. 8 we have combined the RPMs of various models for analysis of possible situations and represented inside the corresponding RPMs the phase tra jectories of the inverted pendulum (a, b, c) and pro jections of the phase tra jectories of the sliding rod onto the plane (, ) for = -1 (d, e, f, the pro jection with = 1 will be antisymmetric relative to = /2) and shown the directions of motion on them for various coefficients of friction and = 0. Analogous graphs for > 0 are shown in Fig. 9. This makes it possible to illustrate the peculiarities of transitions under the change of motion regimes and to analyze the consistency of directions on the phase tra jectories on the boundaries of the RPM. We note that in the case of a horizontal plane ( = 0) the RPMs of all the three models and the phase tra jectories of the inverted pendulum are symmetric about = /2. On the inclined plane ( > 0) this symmetry is broken. As seen from the figures, the RPMs of all three models consistently match only along the boundary lines (except for the obvious overlap of the RPM of the inverted pendulum with the RPM of the sliding rod on the plane xc = 0). This suggests that for the system under consideration has no algebraic (related to the overlap of the RPMs of various models) and dynamical (related to the consistent match of phase tra jectories on the boundaries of the RPMs) paradoxes, i.e., the mechanical models used are sufficient to describe the system, and all transitions between them are consistent. We prove this by separately considering all possible transitions between the models under consideration. 1. Inverted p endulum ­ sliding ro d. We recall that the tra jectories in the inverted pendulum model are located on the plane xc = 0, whereas for the tra jectories of the sliding rod xc = 0. In order to ascertain the possibility of changing the regimes under consideration, we analyze the behavior of the tra jectories of the sliding rod in a small neighborhood of the plane xc = 0. We show that the inverted pendulum regime inside the cone of friction (i.e., under the condition fr > 0, the corresponding regions are denoted in dark grey in Figs. 8 and 9) is stable with respect to small deviations from the constraint xc = 0: all tra jectories of the sliding rod in the corresponding small neighborhood are directed to the plane xc = 0 . To show this, we define the direction of acceleration of the point of contact xc V in the ¨ 4 near x = 0. In the region where the inverted p endulum is a stable solution, the c phase space Ms acceleration of the point of contact near xc = 0 must be directed to the plane, i.e., the equality < 0, while = -1 requires that V > 0. The direction of acceleration of the = 1 requires that V point of contact in this case is shown schematically in Fig. 10a. But if in a neighborhood of some region on the plane xc = 0 the acceleration V is directed to the plane only from one side (for example, as shown in Figs. 10b and 10c), the point of contact
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Fig. 8. Phase tra jectories of the inverted pendulum (a), (b), (c) and pro of the sliding rod onto the plane (, ) for = -1 (d), (e), (f ), = 0 and inverted pendulum is denoted in dark grey, that of the sliding rod is shown of the free system is shaded. The point of intersection of the separatrices is

jections of the phase tra jectories various . The RPM of the as a dotted fill pattern, and that 0 = arctg(1/).

with the corresponding values of , will not reach the constraint xc = const, and after the stop it will start moving in the opposite direction. It should be noted that the boundaries of the RPM of the inverted pendulum fr = 0 and the boundaries of sign definiteness of V coincide, since the boundary fr = 0 is defined by the condition ¨ N = |R | with xc 0 (the boundary of the cone of friction, see (3.5)), and the boundary of the change in sign of V is defined by the analogous condition xc = 0 with F = -N (the value of ¨ horizontal reaction on the boundary of the cone of friction). Figure 11 shows regions in which the acceleration of the contact point near xc = 0 with = ±1 and = 0 is directed to the plane (, ). It can be seen that in the region fr > 0 the direction of acceleration of the contact point is everywhere directed to the plane. Thus, when in the region fr > 0, xc = 0, the inverted pendulum is a "stable" solution. From Fig. 11 we can also determine in what direction the sliding starts when the boundary of the RPM of the inverted pendulum is reached. For example, sliding to the left starts during the motion through the point P in Fig. 11 ( = 0.47, = 0.39) when the boundary of the cone of friction fr = 0 is crossed, since in the region which the phase tra jectory reaches the acceleration V is directed to the plane xc = 0 with = 1, and when = -1, it is directed from the plane, which corresponds to the case illustrated in Fig. 10b. We also note that the regions of uncertainty of the sliding direction (when simultaneously the acceleration of the point of contact can be directed from the plane in any direction) are absent for (Fig. 10d). It is seen in Figs. 8 and 9 that when the boundary fr = 0 is crossed, the direction of motion along the phase tra jectories does not change. Thus, the following proposition holds Prop osition 1. When , a transition between the systems of the inverted pendulum and the sliding rod occurs consistently.
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Fig. 9. Phase tra jectories of the inverted pendulum (a), (b), (c) and pro jections of the phase tra jectories of the sliding rod onto the plane (, ) for = -1 (d), (e), (f ), = 0.25 and various . The RPM of the inverted pendulum is denoted in dark grey, that of the sliding rod is shown as a dotted fill pattern and that of the free system is shaded. The point of intersection of the separatrices is 0 = arctg(1/).

Fig. 10. Various directions of acceleration of the contact point: (a) the inverted pendulum is a stable solution, (b) the solution for xc = 0 is a rod sliding to the right, (c) the solution for xc = 0 is a rod sliding to the left, (d) the direction of sliding is not defined.

This proposition can also be proved analytically by checking the direction on the phase tra jectories on the boundary fr = 0, which was actually done in the definition of regions of "stability" of the inverted pendulum. We also note that the point of intersection of the separatrices 0 = arctg(1/) always lies outside the RPM of the sliding rod on the plane xc = 0, hence, the phase tra jectory never reaches this point. An example of such a motion with the change from the regime of the inverted pendulum to that of the sliding rod will be given below (Section 4.2). 2. Sliding ro d ­ free system. The pro jections of the RPM of the sliding rod and the regions of possible loss of contact are adjacent to the free system only along the curve B (, ) = 0. Thus, ­ for B (, ) < 0 (shaded area in Fig. 8, see also Fig. 4) the reaction force is Ns < 0, and for a small deviation of the point of contact from the plane, according to (2.17), its acceleration is yc > 0, which corresponds to a unique solution -- the loss of contact, ¨
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Fig. 11. Regions in which the acceleration of the contact point near xc = 0 with = ±1 and = 0 is directed to the plane (, ) (denoted in grey): (a) V < 0 for = 1 and (b) V > 0 for = -1.

­ for B (, ) > 0 (the region shown as a dotted fill pattern in Fig. 8) the reaction force is Ns > 0, which corresponds to sliding without loss of contact or to immobility of the point of contact (for xc =0, fr > 0, dark grey area in Fig. 8a). When the deviation of the point of contact from the plane is small, the acceleration of the free system is yc < 0, which corresponds to a stable conservation ¨ of contact with the plane. We now check the consistency of the direction of motion of the point of contact on the boundary between the sliding rod and the free system. Since on the boundary itself yc = 0, to ascertain the directions of motion of the point of contact ¨ ... ¨ in the free system, it is necessary to calculate the quantity y c for yc = 0. By virtue of (2.17) we obtain cos , > 0, ... sin3/2 3 y c = cos = - cos , < 0. sin3/2 ... y Thus, for the upper branch of the boundary (3.1) ... (i.e., > 0) we obtain ... c > 0 if < /2 ... y c < 0 if > /2, for the lower branch we obtain y c < 0 if < /2 and y c > 0 if > /2. and ... In order to show that there exist no tra jectories reaching the boundary (3.1) at y c < 0, we calculate the tangent vector ± to the tra jectory of the system (2.14) at the point = /2, = ±1. We find ± = (±1, 0), hence, on the segment under consideration all tra jectories of the sliding rod are directed opposite to the boundary (Fig. 12). Therefore, the following proposition holds.

Fig. 12. Pro jections of the RPM and the phase tra jectories of the sliding rod onto the plane (, ) for = 0, = 0.5, = -1. The region in which there could be tra jectories with an inconsistent direction of motion during the loss of contact is denoted in grey.

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Prop of the plane of the

osition 2. When , the region fil led with trajectories reaching "inconsistent portions" boundary with the free system is equal to zero, that is, the sliding rod loses contact with the when reaching the boundary (3.1) consistently, and during the loss of contact the acceleration contact point yc is everywhere zero. ¨

3. Inverted p endulum ­ free system. When = 0, the boundary of the transition between the inverted pendulum and the region of the loss of contact of the free system on the plane xc = 0 consists of only two points = /2, = ±1 (see Figs. 8 and 13a), which are actually unreachable.

Fig. 13. The RPM of the inverted pendulum intersect each other only at the points = / to the functions B (, ) = 0 and fr = 0 (right is given of the parameter values ( = 1, = 3) and fr = 0 appear.

and the boundary of the free system: (a) when = 0, they 2, = ±1, (b) when = 0.5 and = = 4/3, the tangents branch) at the point of intersection coincide, (c) an example at which two points of intersection of the curves B (, ) = 0

As the value of increases, the point of intersection shifts to the region > /2 (Fig. 13b). As the value of increases, the region corresponding to the cone of friction fr > 0 expands, and, starting from some values of and , there are two points of intersection of the curves B (, ) = 0 and fr = 0 (Fig. 13c), i.e., a region appears which the family of phase curves of the inverted pendulum can reach at yc > 0. ¨ In this nonuniqueness region, at least two solutions -- the inverted pendulum and the free system -- are simultaneously defined and stable. Analysis of the relative positions of the tangents to the functions fr = 0 and B (, ) = 0 yields a region on the plane of parameters (, ) which give rise to the above-mentioned nonuniqueness (Fig. 14). As evident from the figure, the values are outside the region of nonuniqueness for any values of . Hence we can deduce the following Prop osition 3. When , the point of contact always slips before the loss of contact.

Fig. 14. Grey denotes a region of values of the parameters and at which the boundary of the free system B (, ) = 0 passes inside the RPM of the inverted pendulum.

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4.2. An Example of the Tra jectory with a Change of Motion Regimes for <



We give an example illustrating the motion of the system for < , = 0 with a change of the motion regime and the sliding direction. Figure 15 presents graphs showing the dependence of (t), (t) and V (t) obtained under the initial conditions inside the RPM of the inverted pendulum: (0) = 0.4, (0) = 0.5, V (0) = 0, = 0, = 0. 5 . (4.1)

Fig. 15. Time dependences of the angle , the angular velocity and the velocity of the point of contact V for a homogeneous rod with the initial conditions (4.1).

The curves obtained correspond to the following real motion: at the initial instant of time the center of mass begins to rise ((0) > 0), the point of contact remains fixed until time t = 0.16 (at the point C1 ). Then the point of contact begins to slip to the left (V < 0), the rod continues to deviate anticlockwise. The angular velocity gradually decreases to zero, and at t = 0.73, on reaching the maximum angle of deviation max = 0.58, the rod changes the direction of rotation (now clockwise). After starting to move to the left, the point of contact stops under the action of the friction force at t = 1.84 (at the point C2 ) and begins to slide to the right until it falls onto the plane at t = 2.04. At t < 0.16 the phase tra jectory V (, ) lies in the plane (, ) until it crosses the boundary fr = 0 (at the point C1 in Fig. 16). Further, after reaching the region of stable motion on the left (see Figs. 10b and 11), the tra jectory passes across the region V < 0 and crosses again the plane (, ) at t = 1.84, = 0.16, = -0.75 (point C2 in Fig. 16), i.e., outside the RPM of the inverted pendulum (see also Fig. 8b). This intersection corresponds to the change in the direction of motion of the contact point. Until the motion of the contact point starts, the energy is conserved and then decreases until the fall onto the plane (Fig. 17). 4.3. A Coefficient of Friction that is Larger than a Critical Value for a Horizontal Plane ( = 0, > ) Consider in more detail the system with > . In this case, the boundaries of the RPMs of various models and phase curves look different and new qualitative effects occur. Remark 2. A critical value of the coefficient of friction for the homogeneous rod = 4/3 is practically unreachable in a natural experiment. Nevertheless, by means of added masses one can significantly decrease the critical value of for the entire system (see, e.g., [19]). As in the previous section, Fig. 18 combines the RPM of the inverted pendulum and pro jections of the RPM of the sliding rod and the free system on the plane (, ). Inside the corresponding RPMs, the figure represents the phase tra jectories of the inverted pendulum for xc = 0 (Figs. 18a and 18c) and pro jections of the phase tra jectories of the sliding rod onto the plane (, ) for = 1 (Fig. 18b) and = -1 (d) for = 0, = 3. As evident from Fig. 18, in contrast to the case , the plane (, ) contains not only overlapping regions of the RPM (the region of nonuniqueness is
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Fig. 16. The phase tra jectory V (, ) (thick solid line) obtained under the initial conditions (4.1). The RPM of the inverted pendulum is denoted in grey. At the point C1 the phase curve crosses the boundary fr = 0 and at the point C2 the plane (, ). The direction of sliding of the contact point changes in this case: the tra jectory has a break.

Fig. 17. Graph showing the dependence of the system's energy with the initial conditions (4.1). At t < 0.16 the energy is conserved and then, due to the action of the friction force, it decreases (2.15).

Fig. 18. The RPM and phase tra jectories of the inverted pendulum (a, c) and pro jections of the RPM and phase tra jectories of the sliding rod onto the plane (, ) for = ±1 (b, d), = 0 and = 3. The RPM of the inverted pendulum is denoted in dark grey, that of the sliding rod is shown as a dotted fill pattern and that of the free system is shaded.

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the shaded area with a dotted fill pattern in Fig. 18), but also regions in which none of the models is defined (regions of nonexistence of a solution). We note that the nonexistence of a solution implies that the mechanical models used are insufficient to describe the dynamics of the system (with the initial conditions from this region). Let us consider this region in more detail. As shown above, for > there is a region (1 ,2 ) with = 1 and a region (3 ,4 ) ¨ with = -1, where i , i = 1,... , 4, are the roots of the function A() (2.12), in which for yc < 0 the normal reaction Ns (2.13) takes negative values (unshaded region in Fig. 18). This means that for xc = 0 the sliding rod model cannot be used in the above-mentioned region (Ns < 0) (Figs. 18b and 18d), and a loss of contact must occur due to the unilateral constraint. On the other hand, the acceleration of the contact point is yc < 0 and is directed to the plane, i.e., in the absence of ¨ reaction the contacting body tends to deform the support. Remark 3. Paralleling the conclusions drawn in [13, 16, 22], we can assume that when the system reaches the above-mentioned nonexistence region, the so-called "frictional impact" (tangential impact) [1] occurs, at which (in the case of one degree of freedom) dynamical jamming, i.e., an instantaneous stop occurs, resulting in a possible loss of contact of the rod with the plane (bounce). A more detailed study of this phenomenon requires the use of additional models of elasticity theory, in particular, allowance for local deformation at the contact point, but this exceeds the scope of this paper. The plane xc = 0 (in the inverted pendulum model) also contains a region of nonexistence of a solution -- in the above-mentioned interval (1 ,i+1 ), i = 1, 3, outside the RPM, denoted in grey in Figs. 18a and 18c. We now turn to a more detailed analysis of the system in the nonuniqueness region and to a separate consideration of transitions between the models when the motion regimes change. 1. Sliding ro d ­ free system. When > , regions of nonuniqueness of a solution arise, i.e., regions in which the sliding rod and the free system are defined simultaneously (shaded region with a dotted fill pattern in Fig. 18). Inside this region A() < 0, B (, ) < 0, i.e., the conditions imposed on the definition of the RPM ¨ of the sliding rod (Ns > 0) and the free system (yc > 0) are satisfied simultaneously. In the phase space M4 , two tra jectories emanate from any point with the initial conditions s corresponding to this nonuniqueness region. One of these tra jectories corresponds to the loss of contact (the coordinate yc and the velocity yc become larger than zero), and the other stays in this space (yc = yc = 0) and corresponds to the sliding rod, and any deviation from the plane of contact leads to the loss of contact. Furthermore, the tra jectory of the sliding rod may intersect the surface B (, ) = yc = 0 (this portion of the boundary is shown as a heavy line in Figs. 18b ¨ and 18d). After that the tra jectory reaches the nonexistence region, which was discussed above. Remark 4. In order to answer the question of which of these solutions can be realized in this case in practice, it is necessary to carry out natural experiments. We recall that depending on the realization method [15], both solutions can be realized in an analogous situation in the system with a bilateral constraint (where nonuniqueness arises). As in the case of nonexistence of a solution, a theoretical analysis of such a system requires the use of more complex models incorporating deformations of contacting surfaces of the supporting plane or the rod (see also [7, 15, 17, 18]). Outside the above-mentioned nonuniqueness region, the RPM of the sliding rod consistently matches the RPM of the free system on the surface B (, ) = yc = 0 (the reasoning is analogous ¨ ). to the case 2. Inverted p endulum ­ free system. As in the case , for = 0 the boundary of the transition between the inverted pendulum and the free system on the plane xc = 0 consists only of two points = /2, = ±1 (see Figs. 8, 13a and 14), i.e., the measure of the tra jectories reaching the boundary is zero.
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3. Inverted p endulum ­ sliding ro d. A special feature of this case (as compared with the case of small ) is that on the plane xc = 0 the RPM of the inverted pendulum is adjacent not only to the RPM of the sliding rod but also to the region of nonexistence of a solution. Moreover, there is a set of tra jectories (of nonzero measure) of the inverted pendulum which reach these boundaries (such boundaries are shown as heavy lines in Figs. 18a and 18c). Thus, as the system evolves, it reaches regions of the nonexistence paradox and, as noted above, has to be described using more complex dynamical models. For this case we now consider the stability of motions of the inverted pendulum with respect to the slipping of the point of contact (i.e., with respect to small perturbations of velocity xc in a neighborhood of the plane xc = 0). Figure 19 shows regions in which the acceleration V = xc near xc = 0 is directed to the plane ¨ (, ) with = ±1. It is seen in the figures that on the interval (0,3 ) (4 ,1 ) (2 , ) (where there is no intersection with the nonexistence region for = ±1), as in the case , the inverted pendulum is a "stable" solution: if during the motion the system has reached the constraint xc = const inside the above-mentioned region, then it remains sub ject to this constraint until the intersection with one of the lines = i , i = 1,... , 4.

Fig. 19. Regions in which at = 1 (a) and = -1 (b), = 0, = 3 the acceleration V = xc near xc = 0 is ¨ directed to the plane (, ), denoted in grey.

In the remaining zones, the RPM of the inverted pendulum is adjacent to both the region of existence and the region of nonexistence of a solution in the sliding rod model (as seen from Fig. 18, they change places when the sign of changes). Under a small deviation of xc in a direction where the motion of the sliding rod is possible, the system tends to return to the constraint xc = 0, under the deviation in a different direction we reach a region in which the motion cannot be described within the models considered (the region of "frictional impact" due to which the system can also return to the constraint xc = 0). In the absence of small deviations from the constraint xc = 0 towards the nonexistence region, if the system has reached the constraint xc = const inside the RPM of the inverted pendulum, it remains sub ject to this constraint until the intersection with the boundary fr = 0, after which a consistent transition to sliding to the left or to the right is possible, depending on what boundary of the cone of friction the phase tra jectory has reached: ­ for the boundary fr = 0 shown as a heavy line in Fig. 18a, sliding to the left begins (since the acceleration V is directed from the plane (, ) with = -1, i.e., towards V < 0, where the sliding rod is defined, Ns > 0), ­ for the boundary fr = 0 shown as a heavy line in Fig. 18c, sliding to the right begins (since the acceleration V is directed from the plane (, ) for = 1, i.e., in the direction of V > 0, where the sliding rod is also defined, Ns > 0).
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4.4. A Coefficient of Sliding Friction that is Larger than a Critical Value for an Inclined Plane ( > 0, > ) Since > , all peculiarities associated with the change in sign of the function A(), namely, the appearance of the nonexistence region (unshaded area in Fig. 20) and intersection of the RPM of the sliding rod and the free system (shaded area with a dotted fill pattern in Fig. 20), described in detail in the previous section, are also present in this case. We will not consider them here again.

Fig. 20. The RPM and phase tra jectories of the inverted pendulum (a), (c) and pro jections of the RPM and the phase tra jectories of the sliding rod onto the plane (, ) for = ±1 (b), = 1 and = 3. The RPM of the inverted pendulum is denoted in dark grey, that of the sliding rod is shown as a dotted fill pattern and that of the free system is shaded.

In this case, as opposed to the previous ones, a situation is possible where the boundary of the free system B (, ) = 0 passes inside the RPM of the inverted pendulum. This region of parameters is denoted in grey in Fig. 14. Let us choose specific values from this region ( = 1 and = 3) and analyze the arising dynamical effects. To do this, we combine the RPM and phase tra jectories of the inverted pendulum (Figs. 20a and 20c) and the pro jections of the RPM and phase tra jectories of the sliding rod onto the plane (, ) for = ±1 (Figs. 20b and 20d). In contrast to the cases considered above, with this choice of parameters regions appear on the plane (, ) in which several conditions are satisfied simultaneously (an example of such a region is given on the enlarged fragment in Figs. 20a and 20c): 1) on the plane xc = 0 the motion of the inverted pendulum is possible, 2) on the plane xc = 0 andeverywhereinits neighborhood (xc = 0) the loss of contact is possible,
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3) in a neighborhood of the plane xc = 0 for = 1 there exists a solution for the sliding rod, while for = -1 there exists no solution (Ns < 0), moreover, as shown in Fig. 21, in the case = 1 on the tra jectories in the sliding rod model V > 0, i. e., they tend to leave the small neighborhood of xc = 0, 4) in regions of uniqueness of solutions (both in the inverted pendulum model and for the sliding rod) there exists a set of tra jectories of nonzero measure which reach the boundary of this region.

Fig. 21. Regions in which at = 1 (a) and at = -1 (b), = 1, = 3 the acceleration V = xc near xc = 0 ¨ is directed to the plane (, ) (denoted in grey).

Thus, in this case (inclined plane) regions of paradoxes may arise not only under a special choice of initial conditions, but also in the process of motion, when the system correctly described by one of the models reaches a region in which nonuniqueness of the solution is possible. 5. CONCLUSION In this paper we have used methods of qualitative dynamical systems theory to analyze the problem of planar motion of a rod on a rough inclined plane, which is described within the model of an absolutely rigid body and the Coulomb law of dry friction. Let us briefly summarize the main results. 1. It is shown that if the coefficient of friction is smaller than a critical value ( ), the inverted pendulum and sliding rod models correctly describe the motion of the system until the loss of contact. Before detachment the point of contact always slips (the measure of tra jectories of the inverted pendulum that reach the region of the loss of contact is zero). 2. For > and the horizontal plane = 0, regions of paradoxes of both nonuniqueness and nonexistence of solutions appear, and the tra jectories from correct regions can reach the regions of paradoxical behavior. 3. Larger coefficients of friction ( > ) and the inclined plane > 0 additionally give rise to a region of the loss of contact, which is reached by solutions both in the sliding rod and inverted pendulum models. A hitherto unresolved problem is the experimental verification of the motion pattern of a rigid body with a sharp edge for large coefficients of friction ( > ) and the ascertainment of real dynamical effects to which the nonuniqueness and nonexistence paradoxes lead (by analogy with the experiment of Prandtl for the Painlev´ ­ Klein system, as described in [15]). However, we note e articles where the possibility of specifying initial conditions under which the phase tra jectories can reach the paradoxical region is shown for various modifications of the system under consideration (for example, the inverted pendulum on a slider (IPOS), the compass biped model and the rimless
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wheel model, which are described by Y. Or [19, 20]), and where the phenomenon of tangential impact is demonstrated in the region corresponding to nonexistence of a solution [26]. Other unresolved problems include qualitative analysis of the dynamics of a top with a sharp edge and a disk which move with slipping on a plane in the presence of dry friction. In [9] it is pointed out that losses of contact with the support may occur in such systems. ACKNOWLEDGMENTS The authors thank A. V. Borisov, A. P. Ivanov, D. V. Treschev and A. A. Kilin for comments and useful discussions. This research was supported by the analytical departmental target program "Development of Scientific Potential of Higher Schools" for 2012­2014, No 1.1248.2011 "Nonholonomic Dynamical Systems and Control Problems" and by the Grant of the President of the Russian Federation for Support of Leading Scientific Schools NSh-2964.2014.1. The work of T. B. Ivanova was supported by the Grant of the President of the Russian Federation for Support of Young Candidates of Science MK-2171.2014.1. REFERENCES
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