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Bifurcations of the relative equilibria of a heavy bead on a rotating hoop with dry friction
Journal of Applied Mathematics and Mechanics 78 (2014) 460–467
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Bifurcations of the relative equilibria of a heavy bead on a rotating hoop with dry friction夽 A.A. Burov ∗ , I.A. Yakushev Moscow, Russia
a r t i c l e
i n f o
Article history: Received 23 January 2014
a b s t r a c t The sliding of a heavy bead, threaded on a thin circular hoop, rotating with a constant angular velocity around a vertical axis, situated in its plane and, in the general case, not passing through its vertical diameter, is considered. It is assumed that dry friction acts between the bead and the hoop. A set of unisolated positions of relative equilibrium of the bead on the hoop is obtained, and their dependence on the problem parameters is investigated. The results are presented in the form of bifurcation diagrams. The stability properties of the unisolated relative equilibria obtained are discussed. © 2015 Elsevier Ltd. All rights reserved.
Investigations of the existence and stability of unisolated equilibria in systems with friction apparently date back to the papers by Krementulo.1,2 Shortly after, a general theory of the stability of equilibria in systems with dry friction was developed.3 Methods of investigating the stability of such equilibria, based on the general theory of systems with discontinuous right-hand sides have recently been proposed.4–6 Bifurcations of equilibria in systems with friction have been investigated,7,8 and also the bifurcations of the phase portraits of such systems.9 The dependence of families of unisolated equilibria on the parameters have been investigated10,11 in two-dimensional and three-dimensional cases. An example of bifurcational sets of the “symmetrical pitchfork” type was investigated in Ref. 12 in the problem of the motion of a heavy bead on a circle, rotating around its vertical diameter. As is well known, when there is no friction this problem gives the classical example of a “pitchfork” type bifurcation (see, for example, Ref. 13). In this paper, using the above example, we investigate “asymmetrical pitchfork” type bifurcation sets, and also their splitting with the formation of “horseshoe” and “saddle-node” bifurcation sets. Phase portraits, not encountered previously,12 corresponding to different values of the bifurcation parameters, are constructed. The problem considered is the simplest version of the problem of the motion of mechanical systems with mobile masses, widely employed for self-balancing in systems with rotating components (see, for example, Ref. 14). Different aspects of the dynamics of such systems have been actively investigated.15–17 It should be noted that, in the dynamics of systems with dry friction, the obvious simplicity of the formulation of the problem is sometimes not only combined with some complexity of its solution, but also enables one to clarify the mechanical nature of the observed phenomena, such as, for example, static friction (see the recent paper by Kozlov,18 in which an idea arising from investigations by Tomlinson19 and Prandtl20 is developed).
1. Statement of the problem and the equations of motion in redundant coordinates Consider the motion of a heavy point mass – a bead P of mass m, threaded on a circular hoop of radius with centre at the point O, rotating with constant angular velocity around the vertical axis, lying in its plane, a distance a ≥ 0 from the vertical diameter. We will assume that a dry friction force with friction coefficient acts between the bead and the hoop. The motion of the bead can be described using Lagrange’s equations of the first kind in a mobile system of coordinates connected with the hoop. Suppose Oxyz is this system of coordinates with origin at the centre of the hoop, the y axis of which is directed vertically upwards,
夽 Prikl. Mat. Mekh., Vol. 78, No. 5, pp. 645–655, 2014. ∗ Corresponding author. E-mail address: [email protected] (A.A. Burov). http://dx.doi.org/10.1016/j.jappmathmech.2015.03.004 0021-8928/© 2015 Elsevier Ltd. All rights reserved.
A.A. Burov, I.A. Yakushev / Journal of Applied Mathematics and Mechanics 78 (2014) 460–467
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Fig. 1.
the horizontal x axis is in the plane of the hoop and is directed from its axis of rotation, while the horizontal z axis is perpendicular to the plane of the hoop and supplements the x and y axes to a right triple (Fig. 1). In the mobile system of coordinates the axis of rotation is specified by the straight line x = −a, z = 0 for an arbitrary value of y, the position of the circle P is specified by the coordinates (x, y, z), while the constraints, restricting its motion, are defined by the relations
(1.1) ˙ is the velocity of the bead in the mobile system of coordinates and r = (r , r )1/2 . Since its transfer velocity ˙ y, ˙ z) Suppose r = (x, e = (z, 0, − (x + a)), the kinetic energy of the system, free from constraints, and also the potential energy, have the form
where g is the gravitational acceleration and Lagrange’s equations are written as follows:
(1.2) (1.3) Equations (1.2) can be represented as
where a is the acceleration of the bead in the mobile system of coordinates, FC and Fc are the Coriolis and centrifugal forces, FN is the gravitational force, N is the normal reaction of the hoop, and F is the friction force. Using the unit vectors of the tangent, inner normal and binormal to the hoop at the point P:
we obtain
where, in the case of sliding (r = / 0)
(1.4)
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It is more convenient to introduce dimensionless variables and parameters, by making the following replacements
(1.5) Retaining the dot above a symbol as the notation for a derivative with respect to the new time, taking expressions (1.5) into account, the equations of the constraints (1.1) and the Lagrange function (1.3) can be written as
(1.6) To determine the Lagrange multipliers 1 and 2 we calculate the first and second derivatives with respect to time from the identities which specify the constraints. They have the form (1.7) (1.8) Since the friction force vector touches the hoop at the point P, we have
and substituting into identities (1.7) and (1.8) the expressions for the second derivatives from Eqs (1.2), we can represent 1 and 2 , and also the equations of motion, in the form
(1.9) By the Amonton – Coulomb law, the value of the friction force is constrained by the relation (1.10) Substituting expressions (1.9) for 1 and 2 into inequality (1.10) we obtain the condition
which, in the case of slipping, reduces to an equality, where, by relation (1.4),
since the direction of the friction force is opposite to the slipping direction. In the case when the bead is in equilibrium with respect to the hoop, the following inequality is satisfied (1.11) Remark. In Ref. 12, when calculating the normal reaction, only its component lying in the plane of the hoop, was taken into account. We are grateful to A. P. Ivanov for pointing out this restriction. 2. The existence of relative equilibria and their properties The relative equilibria equations. If the bead is in equilibrium with respect to the hoop, its relative velocity, and also the Coriolis force, are equal to zero. Then the friction force acting along the tangent to the hoop compensates the sum of the tangential components of the gravitational force and the centrifugal force, i.e., (2.1) We will introduce the following notation
Using the first of equations (1.6), Eq. (2.1) reduces to the form (2.2)
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Together with the first equality of (1.6) and inequality (1.11), this forms a system for determining the relative equilibria, which, after substituting expression (2.2) for F, takes the form (2.3) In order to follow the boundaries of the region ∈ R2 (x, y), specified by the first of relations (2.3), it is convenient to introduce the parameters
The boundary ∂ of the region is then given by the relation
and consists of two hyperbolae, represented in the form (2.4) Henceforth we will devote the main attention to the most interesting case when a < 1. In general, the set of relative equilibria depends on the angular velocity of rotation of the hoop. The relative equilibria, for which the bead is situated on the axis of rotation, are an exception. These solutions exist if the axis of rotation intersects the hoop and inequality (1.11) is satisfied. For these
The last inequality, by virtue of the assumptions made above, can be represented in the form
In the general case, expressing y from equality (2.4) and substituting the result into the constraint equation, we can obtain a fourth-degree equation in x with fairly lengthy coefficients, having the form
(2.5) When a = a , Eq. (2.5) allows of the solution x = − a. When
Eq. (2.5) has a double root x = 0. In this case system (2.3) has the solutions
and also the solutions
which only exist when the following conditions are satisfied
Equation (2.5) has the root x = 1 when 2 (± l + a) − = 0, taken with the minus sign, and the root x = −1 for the same condition taken with the plus sign. In these relative equilibria the bead is situated at points of the horizontal diameter, the greatest and least distances from the axis of rotation, respectively. Remark. The functions L and R introduced above differ from the familiar function sign(x) in the fact that they are supplemented to zero, and are convenient not only for solving problems of the mechanics of systems with dry friction, but also problems of elementary mathematics.21 Parametric analysis of the relative equilibrium positions. The sets of relative equilibria, defined by system (2.3), and also the phase portraits of the system being investigated depend on the two parameters (a, ). A cumbersome analysis of this relationship enables us to distinguish four fundamental types of bifurcational diagrams, defined by the values of the parameter a. The case a = 0 has been investigated in some detail.12 We will consider the case when 0 < a < a , the critical case a = a and, finally, the case a < a < 1; when a = / 0 these cases are decomposed by the values of the parameter into subcases with a common number 19. It is best to use the angular variable to consider these.
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Fig. 2.
3. The equations of motion and bifurcation diagrams using the angular variable The configuration manifolds of the mechanical system considered with one degree of freedom is a circle. It is natural to introduce into it the angular coordinate , read-off counterclockwise from the downward radius. Then
Introducing the notation
when = / 0 we can write the equations of motion in the form (3.1) When ˙ = 0 the system remains in equilibrium when the following condition is satisfied (3.2) while in the opposite case the motion continues according to the law
The boundaries of the set of relative equilibria, defined by inequality (3.2), are given by the relations
In the (, ) plane when a = 0 the set of relative equilibria fills a “pitchfork”, symmetrical about the = 0 axis (the left upper part of Fig. 2). When 0 < a < a the set of relative equilibria also fills a “pitchfork”, which, however, is now asymmetrical (the left lower part of
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Fig. 3.
Fig. 2). The case a = a is a special, critical one: the crosspiece, connecting the two “prongs” of the pitchfork with the third “prong” and the “handle” narrows down to a single point (the right upper part of Fig. 2). Finally, when a < a < 1, separation of the pair of “prongs” from the “third” prong and the “handle” connected to them has already occurred (the right lower part of Fig. 2). 4. Phase portraits Analytical properties. The structure of the vector field, defined by Eqs (3.1) in the (, ) ˙ plane, depends considerably on the signs of the angular velocity ˙ and of the function Q(, ), ˙ where the phase plane is split into four regions, in which the vector field when ˙ > 0 and Q > 0(Q < 0) is directed “upwards to the right” (“downwards to the right”), when ˙ < 0 and Q > 0(Q < 0) it is directed “upwards to the left” (“downwards to the left”), at points of the curve Q = 0 it is directed along the horizontal: to the right in the upper half-plane and to the left in the lower half-plane, at points of the horizontal axis, different from equilibria, it is directed along the vertical: upwards if Q > 0 and downwards if Q < 0. In addition to the sets of “elliptic type” (“E-type”) and “hyperbolic type” (“H-type”) unisolated equilibria, discussed previously in Ref. 12 (see Ref. 12, Fig. 4a and b respectively), sets of “elliptic-hyperbolic type” (“EH-type”) unisolated equilibria are observed in the system, in the neighbourhood of which the vector field is represented in Fig. 3 (the lines Q = 0 are shown as the dot-dash curve). The arrows, which, for brevity, are only shown on some curves, define the direction of the phase flow. For the set of EH-type equilibria all the inner points are at traction points. In the neighbourhood of the left (right) end point, the behaviour of the trajectory recalls the behaviour in the neighbourhood of an elliptic (hyperbolic) singular point. Thus, in the neighbourhood of the right-hand point there is a pair of separatrices, entering at this point, and one (unlike the usual saddle point) emerging separatrice. 5. Phase portraits We can construct phase portraits in the (, ) ˙ plane for each pair of values of the parameters (a, ). The number of cases that arise is fairly large, and in a number of cases the observed patterns of the dynamics are similar. We will confine ourselves to citing certain typical patterns. Thus, in Fig. 4a we present the case when one family of E-type equilibria and one family of H-type equilibria coexist in the phase portrait; small values of correspond to this case. In Fig. 4b we show the case when one family of H-type equilibria, one family of EH-type equilibria and one family of E-type equilibria (moderate values of ) coexist in the phase portrait. In Fig. 4c we show the case when two families of H-type equilibria and two families of E-type equilibria (large values of ) coexist in the phase portrait. Finally, in Fig. 4d we show the phase portrait for the critical value of the parameter when a = 0. Note that phase portraits similar to those shown in Fig. 4b and c, but differing from them in being symmetrical, were constructed earlier (see Ref. 12, Fig. 5a and b). The phase portrait shown in Fig. 4c is of a type that does not occur in the case a = 0 considered earlier.12 The phase portrait presented in Fig. 4d, was omitted in Ref. 12.
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Fig. 4.
Fig. 5.
6. A possible interpretation of the existence and stability of unisolated equilibria in the framework of the Tomlinson – Prandtl theory The presence of sets of unisolated equilibrium positions and the properties of their stability can be fairly clearly interpreted in the Tomlinson – Prandtl model of dry friction19,20 (see also Ref. 18). We will assume that a bead of unit mass is threaded onto a cosinusoid of small amplitude A situated in the vertical Oxy plane having a certain microroughness. The potential energy of the bead, due to the gravitational force, is given in the form
The gravitational acceleration is assumed to be equal to unity. The parameter ␣ defines the frequency of the roughnesses. We will assume that the bead is connected to the vertical axis by a spring so that the elastic interaction potential has the form
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Finally, we will assume that the Oxy plane rotates uniformly around the y axis with constant angular velocity . Then, we have the following expression for the potential of the centrifugal force
and the changed potential, defining the relative equilibrium positions of the bead, is written as
The relative equilibrium positions are found from the equation (6.1) all the roots of which are situated in the section [− x* , x* ], x* = A/|a|>0 and alternate, giving the function Ua minimum and maximum values alternately. Here, since
the relative equilibrium x = 0 is stable when a + A␣ > 0 and unstable when a + A␣ < 0. Suppose a > 0 (a < 0), i.e. the elastic (centrifugal) force predominates over the centrifugal (elastic) force. Then, for fairly large values of x, the function U (x) increases (decreases) monotonically, and xh , the greatest root of Eq. (6.1), can give the smooth function U(x) only a locally minimum (maximum) value. The same holds for x – the smallest root of Eq. (6.1). Hence, the roots of Eq. (6.1) alternately give the function U(x) locally minimum and locally maximum values, beginning with the locally minimum (maximum) and ending with the locally minimum (maximum) values. This means that a sequence of alternating stable and unstable relative equilibria occurs, beginning and ending with a stable (unstable) equilibrium. When a→ ∞ these equilibria accumulate in the section [− x* , x* ] and become a set of unisolated equilibria of the system with E-type (H-type) Coulomb friction. A graph of U(x) is shown in Fig. 5a and b for a > 0 and a < 0 respectively. A similar approximation of the set of unisolated EH-type equilibria with isolated equilibria can be made using forces that depend nonlinearly on the position of the bead. The potential U(x) of these forces can have the form shown in Fig. 5c. Acknowledgements This research was supported by the Russian Foundation for Basic Research (12-01-00536-a, 12-08-00637-a) and by the Federal Special Purpose Programme (14.740.11.0995). References 1. Krementulo VV. Investigation of the stability of a gyroscope taking into account dry friction on the axis of the inner cardan ring (gimbal). J Appl Math Mech 1959;23(5):1382–6. 2. Krementulo VV. Stability of a gyroscope having a vertical axis of the outer ring, with dry friction in the gimbal axes taken into account. J Appl Math Mech 1960;24(3):843–9. 3. Pozharitskii GK. On the stability of equilibrium states for systems with dry friction. J Appl Math Mech 1962;26(1):3–16. 4. Van de Wouw N, Leine RI. Stability of stationary sets in nonlinear systems with set-valued friction. In: Proc. 45th IEEE Conf. Decision and Control and European Control Conf. (CDC2006). 2006. p. 3765–70. 5. Leine RL, van de Wouw N. Stability properties of equilibrium sets of nonlinear mechanical systems with dry friction and impact. Nonlinear Dynamics 2008;51(4):551–83. 6. Leine RL, van de Wouw N. Stability and convergence of mechanical systems with unilateral constraints. Lecture Notes in Applied and Computational Mechanics, 36. Berlin: Springer; 2008, 236. 7. Leine RL, van Campen DH. Bifurcation phenomena in non-smooth dynamical systems. Europ J Mechanics A Solids 2006;25:595–616. 8. Leine RL. Bifurcations of equilibria in non-smooth continuous systems. Physica D 2006;223:121–37. 9. Teuifel A, Steindl A, Troger H. The classification of non-smooth bifurcations for friction oscillator. In: Problems of Analytical Mechanics and Stability Theory. Collection of Scientific Papers in the Memory of Acad. V. V. Rumyantsev. Moscow: IPU Ros Akad Nauk; 2009. p. 161–75. 10. Ivanov AP. Bifurcations in systems with friction: fundamental models and methods. Nelineinaya Dinamika 2009;5(4):479–98. 11. Ivanov AP. Principles of the Theory of Systems with Friction. Izhevsk: RKhD; 2011. 12. Burov AA. On bifurcations of relative equilibria of a heavy bead sliding with dry friction on a rotating circle. Acta mechanica 2010;212(3–4):349–54. 13. Rubanovskii VN, Samsonov VA. Stability of Steady Motions in Examples and Problems. Moscow: Nauka; 1988. 14. Blekhman II. Synchronization in Nature and Technology. Moscow: Nauka; 1981. 15. Pankova NV, Rubanovskii VN. The stability and bifurcations of the steady rotations of a free rigid body and of a point mass elastically connected to it. Izv Akad Nauk SSSR MTT 1976;4:14–8. 16. Gorbenko AN. On the stability of self-balancing of a rotor with the help of balls. Strength of Materials 2003;35(3):305–12. 17. van de Wouw N, van den Heuvel MN, van Rooij JA, Nijmeijer H. Performance of an automatic ball and balancer with dry friction. Int J Bifurcation and Chaos 2005;15:65–82. 18. Kozlov VV. On the dry-friction mechanism. Dokl Physics 2011;56(4):256–7. 19. Tomlinson GA. A molecular theory of friction. The London, Edinburgh, and Dublin Philos. Mag and Journal of Sciences 1929;7:905–39. 20. Prandtl L. Ein Gedankenmodell zur kinetischen Theorie der festen Köper. ZAMM 1928;8(2):85–106. 21. Burov AA. Small sigma and problems with absolute values. Kvant 2012;1:36–8.
Translated by R. C. G.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Bifurcations of the relative equilibria of a heavy bead on a rotating hoop with dry friction夽 A.A. Burov ∗ , I.A. Yakushev Moscow, Russia
a r t i c l e
i n f o
Article history: Received 23 January 2014
a b s t r a c t The sliding of a heavy bead, threaded on a thin circular hoop, rotating with a constant angular velocity around a vertical axis, situated in its plane and, in the general case, not passing through its vertical diameter, is considered. It is assumed that dry friction acts between the bead and the hoop. A set of unisolated positions of relative equilibrium of the bead on the hoop is obtained, and their dependence on the problem parameters is investigated. The results are presented in the form of bifurcation diagrams. The stability properties of the unisolated relative equilibria obtained are discussed. © 2015 Elsevier Ltd. All rights reserved.
Investigations of the existence and stability of unisolated equilibria in systems with friction apparently date back to the papers by Krementulo.1,2 Shortly after, a general theory of the stability of equilibria in systems with dry friction was developed.3 Methods of investigating the stability of such equilibria, based on the general theory of systems with discontinuous right-hand sides have recently been proposed.4–6 Bifurcations of equilibria in systems with friction have been investigated,7,8 and also the bifurcations of the phase portraits of such systems.9 The dependence of families of unisolated equilibria on the parameters have been investigated10,11 in two-dimensional and three-dimensional cases. An example of bifurcational sets of the “symmetrical pitchfork” type was investigated in Ref. 12 in the problem of the motion of a heavy bead on a circle, rotating around its vertical diameter. As is well known, when there is no friction this problem gives the classical example of a “pitchfork” type bifurcation (see, for example, Ref. 13). In this paper, using the above example, we investigate “asymmetrical pitchfork” type bifurcation sets, and also their splitting with the formation of “horseshoe” and “saddle-node” bifurcation sets. Phase portraits, not encountered previously,12 corresponding to different values of the bifurcation parameters, are constructed. The problem considered is the simplest version of the problem of the motion of mechanical systems with mobile masses, widely employed for self-balancing in systems with rotating components (see, for example, Ref. 14). Different aspects of the dynamics of such systems have been actively investigated.15–17 It should be noted that, in the dynamics of systems with dry friction, the obvious simplicity of the formulation of the problem is sometimes not only combined with some complexity of its solution, but also enables one to clarify the mechanical nature of the observed phenomena, such as, for example, static friction (see the recent paper by Kozlov,18 in which an idea arising from investigations by Tomlinson19 and Prandtl20 is developed).
1. Statement of the problem and the equations of motion in redundant coordinates Consider the motion of a heavy point mass – a bead P of mass m, threaded on a circular hoop of radius with centre at the point O, rotating with constant angular velocity around the vertical axis, lying in its plane, a distance a ≥ 0 from the vertical diameter. We will assume that a dry friction force with friction coefficient acts between the bead and the hoop. The motion of the bead can be described using Lagrange’s equations of the first kind in a mobile system of coordinates connected with the hoop. Suppose Oxyz is this system of coordinates with origin at the centre of the hoop, the y axis of which is directed vertically upwards,
夽 Prikl. Mat. Mekh., Vol. 78, No. 5, pp. 645–655, 2014. ∗ Corresponding author. E-mail address: [email protected] (A.A. Burov). http://dx.doi.org/10.1016/j.jappmathmech.2015.03.004 0021-8928/© 2015 Elsevier Ltd. All rights reserved.
A.A. Burov, I.A. Yakushev / Journal of Applied Mathematics and Mechanics 78 (2014) 460–467
461
Fig. 1.
the horizontal x axis is in the plane of the hoop and is directed from its axis of rotation, while the horizontal z axis is perpendicular to the plane of the hoop and supplements the x and y axes to a right triple (Fig. 1). In the mobile system of coordinates the axis of rotation is specified by the straight line x = −a, z = 0 for an arbitrary value of y, the position of the circle P is specified by the coordinates (x, y, z), while the constraints, restricting its motion, are defined by the relations
(1.1) ˙ is the velocity of the bead in the mobile system of coordinates and r = (r , r )1/2 . Since its transfer velocity ˙ y, ˙ z) Suppose r = (x, e = (z, 0, − (x + a)), the kinetic energy of the system, free from constraints, and also the potential energy, have the form
where g is the gravitational acceleration and Lagrange’s equations are written as follows:
(1.2) (1.3) Equations (1.2) can be represented as
where a is the acceleration of the bead in the mobile system of coordinates, FC and Fc are the Coriolis and centrifugal forces, FN is the gravitational force, N is the normal reaction of the hoop, and F is the friction force. Using the unit vectors of the tangent, inner normal and binormal to the hoop at the point P:
we obtain
where, in the case of sliding (r = / 0)
(1.4)
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It is more convenient to introduce dimensionless variables and parameters, by making the following replacements
(1.5) Retaining the dot above a symbol as the notation for a derivative with respect to the new time, taking expressions (1.5) into account, the equations of the constraints (1.1) and the Lagrange function (1.3) can be written as
(1.6) To determine the Lagrange multipliers 1 and 2 we calculate the first and second derivatives with respect to time from the identities which specify the constraints. They have the form (1.7) (1.8) Since the friction force vector touches the hoop at the point P, we have
and substituting into identities (1.7) and (1.8) the expressions for the second derivatives from Eqs (1.2), we can represent 1 and 2 , and also the equations of motion, in the form
(1.9) By the Amonton – Coulomb law, the value of the friction force is constrained by the relation (1.10) Substituting expressions (1.9) for 1 and 2 into inequality (1.10) we obtain the condition
which, in the case of slipping, reduces to an equality, where, by relation (1.4),
since the direction of the friction force is opposite to the slipping direction. In the case when the bead is in equilibrium with respect to the hoop, the following inequality is satisfied (1.11) Remark. In Ref. 12, when calculating the normal reaction, only its component lying in the plane of the hoop, was taken into account. We are grateful to A. P. Ivanov for pointing out this restriction. 2. The existence of relative equilibria and their properties The relative equilibria equations. If the bead is in equilibrium with respect to the hoop, its relative velocity, and also the Coriolis force, are equal to zero. Then the friction force acting along the tangent to the hoop compensates the sum of the tangential components of the gravitational force and the centrifugal force, i.e., (2.1) We will introduce the following notation
Using the first of equations (1.6), Eq. (2.1) reduces to the form (2.2)
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Together with the first equality of (1.6) and inequality (1.11), this forms a system for determining the relative equilibria, which, after substituting expression (2.2) for F, takes the form (2.3) In order to follow the boundaries of the region ∈ R2 (x, y), specified by the first of relations (2.3), it is convenient to introduce the parameters
The boundary ∂ of the region is then given by the relation
and consists of two hyperbolae, represented in the form (2.4) Henceforth we will devote the main attention to the most interesting case when a < 1. In general, the set of relative equilibria depends on the angular velocity of rotation of the hoop. The relative equilibria, for which the bead is situated on the axis of rotation, are an exception. These solutions exist if the axis of rotation intersects the hoop and inequality (1.11) is satisfied. For these
The last inequality, by virtue of the assumptions made above, can be represented in the form
In the general case, expressing y from equality (2.4) and substituting the result into the constraint equation, we can obtain a fourth-degree equation in x with fairly lengthy coefficients, having the form
(2.5) When a = a , Eq. (2.5) allows of the solution x = − a. When
Eq. (2.5) has a double root x = 0. In this case system (2.3) has the solutions
and also the solutions
which only exist when the following conditions are satisfied
Equation (2.5) has the root x = 1 when 2 (± l + a) − = 0, taken with the minus sign, and the root x = −1 for the same condition taken with the plus sign. In these relative equilibria the bead is situated at points of the horizontal diameter, the greatest and least distances from the axis of rotation, respectively. Remark. The functions L and R introduced above differ from the familiar function sign(x) in the fact that they are supplemented to zero, and are convenient not only for solving problems of the mechanics of systems with dry friction, but also problems of elementary mathematics.21 Parametric analysis of the relative equilibrium positions. The sets of relative equilibria, defined by system (2.3), and also the phase portraits of the system being investigated depend on the two parameters (a, ). A cumbersome analysis of this relationship enables us to distinguish four fundamental types of bifurcational diagrams, defined by the values of the parameter a. The case a = 0 has been investigated in some detail.12 We will consider the case when 0 < a < a , the critical case a = a and, finally, the case a < a < 1; when a = / 0 these cases are decomposed by the values of the parameter into subcases with a common number 19. It is best to use the angular variable to consider these.
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Fig. 2.
3. The equations of motion and bifurcation diagrams using the angular variable The configuration manifolds of the mechanical system considered with one degree of freedom is a circle. It is natural to introduce into it the angular coordinate , read-off counterclockwise from the downward radius. Then
Introducing the notation
when = / 0 we can write the equations of motion in the form (3.1) When ˙ = 0 the system remains in equilibrium when the following condition is satisfied (3.2) while in the opposite case the motion continues according to the law
The boundaries of the set of relative equilibria, defined by inequality (3.2), are given by the relations
In the (, ) plane when a = 0 the set of relative equilibria fills a “pitchfork”, symmetrical about the = 0 axis (the left upper part of Fig. 2). When 0 < a < a the set of relative equilibria also fills a “pitchfork”, which, however, is now asymmetrical (the left lower part of
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Fig. 3.
Fig. 2). The case a = a is a special, critical one: the crosspiece, connecting the two “prongs” of the pitchfork with the third “prong” and the “handle” narrows down to a single point (the right upper part of Fig. 2). Finally, when a < a < 1, separation of the pair of “prongs” from the “third” prong and the “handle” connected to them has already occurred (the right lower part of Fig. 2). 4. Phase portraits Analytical properties. The structure of the vector field, defined by Eqs (3.1) in the (, ) ˙ plane, depends considerably on the signs of the angular velocity ˙ and of the function Q(, ), ˙ where the phase plane is split into four regions, in which the vector field when ˙ > 0 and Q > 0(Q < 0) is directed “upwards to the right” (“downwards to the right”), when ˙ < 0 and Q > 0(Q < 0) it is directed “upwards to the left” (“downwards to the left”), at points of the curve Q = 0 it is directed along the horizontal: to the right in the upper half-plane and to the left in the lower half-plane, at points of the horizontal axis, different from equilibria, it is directed along the vertical: upwards if Q > 0 and downwards if Q < 0. In addition to the sets of “elliptic type” (“E-type”) and “hyperbolic type” (“H-type”) unisolated equilibria, discussed previously in Ref. 12 (see Ref. 12, Fig. 4a and b respectively), sets of “elliptic-hyperbolic type” (“EH-type”) unisolated equilibria are observed in the system, in the neighbourhood of which the vector field is represented in Fig. 3 (the lines Q = 0 are shown as the dot-dash curve). The arrows, which, for brevity, are only shown on some curves, define the direction of the phase flow. For the set of EH-type equilibria all the inner points are at traction points. In the neighbourhood of the left (right) end point, the behaviour of the trajectory recalls the behaviour in the neighbourhood of an elliptic (hyperbolic) singular point. Thus, in the neighbourhood of the right-hand point there is a pair of separatrices, entering at this point, and one (unlike the usual saddle point) emerging separatrice. 5. Phase portraits We can construct phase portraits in the (, ) ˙ plane for each pair of values of the parameters (a, ). The number of cases that arise is fairly large, and in a number of cases the observed patterns of the dynamics are similar. We will confine ourselves to citing certain typical patterns. Thus, in Fig. 4a we present the case when one family of E-type equilibria and one family of H-type equilibria coexist in the phase portrait; small values of correspond to this case. In Fig. 4b we show the case when one family of H-type equilibria, one family of EH-type equilibria and one family of E-type equilibria (moderate values of ) coexist in the phase portrait. In Fig. 4c we show the case when two families of H-type equilibria and two families of E-type equilibria (large values of ) coexist in the phase portrait. Finally, in Fig. 4d we show the phase portrait for the critical value of the parameter when a = 0. Note that phase portraits similar to those shown in Fig. 4b and c, but differing from them in being symmetrical, were constructed earlier (see Ref. 12, Fig. 5a and b). The phase portrait shown in Fig. 4c is of a type that does not occur in the case a = 0 considered earlier.12 The phase portrait presented in Fig. 4d, was omitted in Ref. 12.
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Fig. 4.
Fig. 5.
6. A possible interpretation of the existence and stability of unisolated equilibria in the framework of the Tomlinson – Prandtl theory The presence of sets of unisolated equilibrium positions and the properties of their stability can be fairly clearly interpreted in the Tomlinson – Prandtl model of dry friction19,20 (see also Ref. 18). We will assume that a bead of unit mass is threaded onto a cosinusoid of small amplitude A situated in the vertical Oxy plane having a certain microroughness. The potential energy of the bead, due to the gravitational force, is given in the form
The gravitational acceleration is assumed to be equal to unity. The parameter ␣ defines the frequency of the roughnesses. We will assume that the bead is connected to the vertical axis by a spring so that the elastic interaction potential has the form
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Finally, we will assume that the Oxy plane rotates uniformly around the y axis with constant angular velocity . Then, we have the following expression for the potential of the centrifugal force
and the changed potential, defining the relative equilibrium positions of the bead, is written as
The relative equilibrium positions are found from the equation (6.1) all the roots of which are situated in the section [− x* , x* ], x* = A/|a|>0 and alternate, giving the function Ua minimum and maximum values alternately. Here, since
the relative equilibrium x = 0 is stable when a + A␣ > 0 and unstable when a + A␣ < 0. Suppose a > 0 (a < 0), i.e. the elastic (centrifugal) force predominates over the centrifugal (elastic) force. Then, for fairly large values of x, the function U (x) increases (decreases) monotonically, and xh , the greatest root of Eq. (6.1), can give the smooth function U(x) only a locally minimum (maximum) value. The same holds for x – the smallest root of Eq. (6.1). Hence, the roots of Eq. (6.1) alternately give the function U(x) locally minimum and locally maximum values, beginning with the locally minimum (maximum) and ending with the locally minimum (maximum) values. This means that a sequence of alternating stable and unstable relative equilibria occurs, beginning and ending with a stable (unstable) equilibrium. When a→ ∞ these equilibria accumulate in the section [− x* , x* ] and become a set of unisolated equilibria of the system with E-type (H-type) Coulomb friction. A graph of U(x) is shown in Fig. 5a and b for a > 0 and a < 0 respectively. A similar approximation of the set of unisolated EH-type equilibria with isolated equilibria can be made using forces that depend nonlinearly on the position of the bead. The potential U(x) of these forces can have the form shown in Fig. 5c. Acknowledgements This research was supported by the Russian Foundation for Basic Research (12-01-00536-a, 12-08-00637-a) and by the Federal Special Purpose Programme (14.740.11.0995). References 1. Krementulo VV. Investigation of the stability of a gyroscope taking into account dry friction on the axis of the inner cardan ring (gimbal). J Appl Math Mech 1959;23(5):1382–6. 2. Krementulo VV. Stability of a gyroscope having a vertical axis of the outer ring, with dry friction in the gimbal axes taken into account. J Appl Math Mech 1960;24(3):843–9. 3. Pozharitskii GK. On the stability of equilibrium states for systems with dry friction. J Appl Math Mech 1962;26(1):3–16. 4. Van de Wouw N, Leine RI. Stability of stationary sets in nonlinear systems with set-valued friction. In: Proc. 45th IEEE Conf. Decision and Control and European Control Conf. (CDC2006). 2006. p. 3765–70. 5. Leine RL, van de Wouw N. Stability properties of equilibrium sets of nonlinear mechanical systems with dry friction and impact. Nonlinear Dynamics 2008;51(4):551–83. 6. Leine RL, van de Wouw N. Stability and convergence of mechanical systems with unilateral constraints. Lecture Notes in Applied and Computational Mechanics, 36. Berlin: Springer; 2008, 236. 7. Leine RL, van Campen DH. Bifurcation phenomena in non-smooth dynamical systems. Europ J Mechanics A Solids 2006;25:595–616. 8. Leine RL. Bifurcations of equilibria in non-smooth continuous systems. Physica D 2006;223:121–37. 9. Teuifel A, Steindl A, Troger H. The classification of non-smooth bifurcations for friction oscillator. In: Problems of Analytical Mechanics and Stability Theory. Collection of Scientific Papers in the Memory of Acad. V. V. Rumyantsev. Moscow: IPU Ros Akad Nauk; 2009. p. 161–75. 10. Ivanov AP. Bifurcations in systems with friction: fundamental models and methods. Nelineinaya Dinamika 2009;5(4):479–98. 11. Ivanov AP. Principles of the Theory of Systems with Friction. Izhevsk: RKhD; 2011. 12. Burov AA. On bifurcations of relative equilibria of a heavy bead sliding with dry friction on a rotating circle. Acta mechanica 2010;212(3–4):349–54. 13. Rubanovskii VN, Samsonov VA. Stability of Steady Motions in Examples and Problems. Moscow: Nauka; 1988. 14. Blekhman II. Synchronization in Nature and Technology. Moscow: Nauka; 1981. 15. Pankova NV, Rubanovskii VN. The stability and bifurcations of the steady rotations of a free rigid body and of a point mass elastically connected to it. Izv Akad Nauk SSSR MTT 1976;4:14–8. 16. Gorbenko AN. On the stability of self-balancing of a rotor with the help of balls. Strength of Materials 2003;35(3):305–12. 17. van de Wouw N, van den Heuvel MN, van Rooij JA, Nijmeijer H. Performance of an automatic ball and balancer with dry friction. Int J Bifurcation and Chaos 2005;15:65–82. 18. Kozlov VV. On the dry-friction mechanism. Dokl Physics 2011;56(4):256–7. 19. Tomlinson GA. A molecular theory of friction. The London, Edinburgh, and Dublin Philos. Mag and Journal of Sciences 1929;7:905–39. 20. Prandtl L. Ein Gedankenmodell zur kinetischen Theorie der festen Köper. ZAMM 1928;8(2):85–106. 21. Burov AA. Small sigma and problems with absolute values. Kvant 2012;1:36–8.
Translated by R. C. G.