geometric representation of detachment conditions in systems with unilateral constraint

ISSN 1560-3547, Regular and Chaotic Dynamics, 2008, Vol. 13, No. 5, pp. 435–442. c© Pleiades Publishing, Ltd., 2008.

NONHOLONOMIC MECHANICS

Geometric Representation of Detachment Conditions inSystems with Unilateral Constraint

A.P. Ivanov*

A.N.Kosygin Moscow State Textile UniversityMalaja Kaluzhskaja ul. 1, 19991 Moscow, Russia

Received August 14, 2008; accepted September 8, 2008

Abstract—Mechanical systems with unilateral constraints that can be represented in thecontact mode on the phase plane are considered. On the phase plane we construct domainsthat satisfy the following conditions 1) a detachment from the constraint is impossible; 2) thesign of the constraint reaction corresponds to its unilateral character. These conditions areequivalent for an ideal constraint [1, 2], but they can differ in the presence of friction [3].Trajectories without detachments belong to intersections of these domains. A circular discmoving on a horizontal support with viscous friction and a disc with the sharp edge moving onan icy surface [4, 5] are considered as examples.Usually for the control of contact conservation one uses only the second condition from above,which can lead to invalid qualitative conclusions.

MSC2000 numbers: 70E18, 70E50, 70G70

DOI: 10.1134/S1560354708050067

Key words: unilateral constraint, detachment conditions

1. FORMULATION OF THE PROBLEM

Consider a mechanical system with two degrees of freedom and a geometrical unilateralconstraint

Λ(q) � 0, q ∈ R2. (1.1)

The system has two main modes of motion: continuous contact in some interval of time implies theequality Λ = 0, while the strong inequality Λ > 0 corresponds to the weakened constraint (in thiscase the constraint does not affect the motion). In the contact mode one can decrease the systemorder using the relation

Λ(q) = 0, Λ(q) = 0. (1.2)

This leads to a reduced system. One can consider its phase space as a plane (θ, θ), where thegeneralized coordinate θ is chosen in the coordinate space for convenience reasons.

We suppose that external forces do not depend on time; in this case the phase portrait ofthe initial system gives the full description of the dynamics in the contact mode. Contact modedomains are indispensable additions to this portrait. For constructing such domains one should usethe following two rules.

1) For points (θ, θ) ∈ Ω the value Λ(q), calculated using (1.2) for the zero constraint reaction,is negative. In other words, the constraint attenuation in the domain Ω is impossible.

2) A contact mode domain Ω0 is defined by the condition that the normal constraint reactiondoes not vanish (i.e., the sign of this reaction is constant and corresponds to the sign in (1.1)).

*E-mail: [email protected]

435

436 IVANOV

For an ideal constraint these conditions are equivalent, i.e., Ω = Ω0 [1, 2]. In this case thedynamics of a satellites bundle is an example for the application of the phase plane method [6].

In the general case in presence of friction (including the absolute roughness) it holds Ω �= Ω0 [3].Here the phase space is separated into the following four parts.

1) In the domain Ω ∩ Ω0 the contact is continuous;

2) in the domain Ω ∩ Ω0 (the bar over a symbol of a set denotes the complement of this set)the constraint is weekened;

3) the paradox situation of the non-uniqueness of motion, when it is possible both thedetachment and the continuous contact, corresponds to the set Ω ∩ Ω0;

4) another paradox situation, when both these types of motion are impossible, corresponds tothe set Ω ∩ Ω0.

Example. A heavy circular disc of the radius a is moving in a fixed vertical plane. This system issubject to the ideal bilateral constraint: the center of mass moves along a fixed vertical line and alsoto the nonideal unilateral constraint in the form of a horizontal plane with viscous friction (Fig. 1a).Consider the inertial reference frame OXY such that its origin lies on the plane of support and itsordinate axis contains the mass center G. As Lagrange coordinates we chose the ordinate y of thecenter of mass and the angle θ between the vector CG (C is the geometrical center of the disc) andthe absciss axis. The constraint (1.1) can be written in the form

Λ = y − h(θ) � 0, h(θ) = a + b sin θ, b = |GC|. (1.3)

The friction is given by the formula

F = −μv, v = θh(θ), (1.4)

where v is the sliding velocity in a contact point and μ is the coefficient of viscous friction.The kinetic and potential energies of the moving disc, detached from the support, have the form

T =12my2 +

12Cθ2, Π = mgy. (1.5)

In order to calculate these energies in the contact mode one should express y, y via θ, θ usingrelations (1.2), (1.3) and substitute these expressions into (1.5). Here m is mass of the body, g isthe acceleration due to gravity and C is the moment of inertia of the body with respect to the GZaxis that is orthogonal to the plane of the motion.

One can find the equation of motion of the body in the contact mode using the work-kineticenergy theorem

dT = −dΠ + Fvdt. (1.6)

Substituting expressions (1.2)–(1.5) into (1.6) one gets the relation

2(C + mh′2)θ + 2mh′h′′θ2 = −mgh′ − μh2θ, (1.7)

where a prime means the derivation with respect to θ.To construct the domains Ω and Ω0 we calculate the second derivative of the function (1.3)

Λ = y − h′′θθ2 − h′θ, y = m1N − g, (1.8)

where N is the normal reaction of the support. Using (1.7), one can express (1.8) in the form

Λ =1m

N +μh2h′θ − g(2C + mh′2) − 2Ch′′θ2

2(C + mh′2). (1.9)

REGULAR AND CHAOTIC DYNAMICS Vol. 13 No. 5 2008

GEOMETRIC REPRESENTATION OF DETACHMENT CONDITIONS 437

Fig. 1.

Since for an unilateral constraint it holds N � 0 the domains Ω and Ω0 can be defined by thefollowing conditions

Ω : − g(2C + mh′2) − 2Ch′′θ2 < 0, (1.10)

Ω0 : μh2h′θ − g(2C + mh′2) − 2Ch′′θ2 < 0 (1.11)

(formula (1.11) differs from (1.10) by presence of the friction term).Comparing these inequalities, one can note that in the domain determined by the formula

θ cos θ > 0, (1.12)

condition (1.11) implies (1.10), i. e., Ω0 ⊂ Ω. For the opposite sign in inequality (1.12) it holds theopposite inclusion Ω ⊂ Ω0. On Fig. 1b the domains Ω and Ω0 are marked by the horizontal andinclined hatching correspondingly. The difference between these sets is bigger for a bigger coefficientof friction. Here is also the phase portrait of the system on the cylinder evolvent (θ, θ) ∈ (−π, π)×R.The equilibrium positions θ = π/2 and θ = −π/2 are correspondingly the saddle point and thestable focus. The location of domains Ω and Ω0 with respect to each other shows that the continuousphase curve can penetrate into the domain Ω ∩ Ω0, but the domain Ω ∩ Ω0 can be reached onlyprovided the initial conditions belong to the domain.

Remark. The partition of the phase space using domains Ω and Ω0 can be made also for phasespaces of greater dimension, but for the phase plane it is more complete and obvious.

The similar method can be used for systems with more degrees of freedom if its motion in contactmode allows the representation on the phase plane. This situation can appear if an unilateralconstraint imposes additional restrictions on the motion. The problem of a thin disc motion on


438 IVANOV

ice will be considered below. In this problem the friction eliminates the motion in the directionorthogonal to the nodal line.

2. DISC ON A SMOOTH SUPPORT

At first we find the domain Ω in the problem of the motion of a heavy thin homogeneous discalong a perfectly smooth horizontal plane. Consider the following coordinate systems. The first oneOXY Z is inertial with its origin on the support plane and the straight up axis OZ. An origin ofthe second system GX ′Y ′Z ′ coincides with the mass center of the body and its axis GZ ′ is thebody’s line of symmetry. Consider coordinates (x, y, z) of the point G and Euler angles ϑ,ψ, φ asgeneralized coordinates.

The kinetic and potential energies of the body and the unilateral constraint are expressed bythe following formulae

T =12m(x2 + y2 + z2) +

12A

(θ2 + ψ2 sin2 θ

)+

12C

(ψ cos θ + φ

)2,

Π = mgz, Λ = z − a sin θ � 0,(2.1)

where m is the disc mass; A and C are correspondingly the equatorial and axial moments of inertia.The equations of motion in the detachment mode (Λ > 0)

d

dt

(∂T

∂q

)− ∂T

∂q= −∂Π

∂q, q = (x, y, z, θ, φ, ψ) (2.2)

admit the following first integrals, corresponding to cyclic variables

∂T

∂x= px,

∂T

∂y= py,

∂T

∂φ= pφ,

∂T

∂ψ= pψ (2.3)

and also the energy integral

h = T + Π = const. (2.4)

In the contact mode (Λ = 0) the normal reaction N acts onto the disc. The system (2.2) canbe adapted to this action if one includes appropriate terms into the right hand sides of equationsin variables z and θ; in this situations equations (2.3), (2.4) remain valid. This allows decreasingthe order of the system (2.2) using the Routh method. Finally one gets the following conservativesystem with one degree of freedom

d

dt

(∂T

∂θ

)− ∂T

∂θ= −Π′(θ),

T =12(A + ma2 cos2 θ)θ2, Π = mga sin θ +

(pψ − pφ cos θ)2

2A sin2 θ.

(2.5)

Let us find the second derivative of the function Λ in (2.1), which determines presence of thecontact

Λ = U + V N =1m

N − g + aθ2 sin θ − aθ cos θ. (2.6)

To calculate the value of U one should express θ from (2.5) and substitute it into (2.6). Due tointegrals (2.3) an analysis of equation (2.2) with respect to the variable θ leads to the same result.

The contact conservation condition Λ = 0, N � 0 is expressed by the inequality

U = −g + aθ2 sin θ + aττ ′ cos θ � 0,

τ =pψ − pφ cos θ

A sin θ, τ ′ =

pφ − pψ cos θ

A sin2 θ,

(2.7)



which is equivalent to the relation

θ2 � Φ(θ),

Φ(θ) =g

a sin θ− cos θ

A2 sin4 θ(pφ − pψ cos θ) (pψ − pφ cos θ) .

(2.8)

Inequality (2.8) determines the domain Ω = Ω0 of possible disc motion on the phase plane inthe contact mode. The shape of this domain depends on the number and positions of roots of theequation Φ(θ) = 0, equivalent to the equality

κ sin3 θ = cos θ (α − cos θ) (1 − α cos θ) ,

κ =gA2

ap2M

, α =pm

pM,

(2.9)

where pM is the constant pφ or pψ with the biggest absolute value and pm is another constant.Thus, |α| � 1.

Fig. 2.

The graphic solution of equation (2.9) is represented on Fig. 2; here fl(θ) and fr(θ) areexpressions from the left and the right hand side of equation (2.9) correspondingly. There aretwo parameters in the problem, which determine the relative position of these curves.

1) In the case α = 0, corresponding to the relation pφpψ = 0, in (2.9) it holds fr(θ) < 0 thatimplies Φ(θ) > 0 for all θ ∈ (0, π). The domain Ω is represented on Fig. 3a.

2) If α = 1, then pψ = pφ �= 0 and after simple transformations equation (2.9) takes the form

2κ = μ(1 − μ)(1 + μ), μ = tanθ

2. (2.10)

For κ > κ0 =√

3/9 equation (2.10) has no positive roots, it holds Φ(θ) > 0 for all θ ∈ (0, π) andthe domain Ω is qualitatively similar to the domain on Fig. 3a. For κ < κ0 equation (2.10) hastwo positive roots and in the interval between them it holds Φ(θ) < 0, i.e., U > 0 for all θ. Thedomain Ω has two components (see Fig. 3b).

3) For values α ∈ (0, 1) the domain Ω has the shape as on Figs. 3a and 3b, depending on thevalue of κ.

4) The case α < 0 is reduced to the previous one by the substitution θ → π − θ. Now one canconstruct the domain Ω by the symmetric reflection of previous diagrams with respect to theline θ = π/2.

To study the disc dynamics one should construct on the phase plane the domain Ω and levellines of the reduced energy, defined by

T + Π = const. (2.11)


440 IVANOV

Fig. 3.

The contact mode of motion corresponds to trajectories that are completely in Ω. Key points ofthe phase portrait are stationary points of the function Π, which can be found from equation

mga cos θ + Aττ ′ = 0. (2.12)

Depending on values of first integrals, equation (2.12) has one or three roots; the correspondingphase portraits have a unique elliptic fixed point (Fig. 4a) or a hyperbolic fixed point and twoelliptic fixed points (Fig. 4b) [7, 8]. Equilibrium positions of system (2.5) (i. e., stationary motionsof the disc) are always in Ω. Besides, when the domain Ω is disconnected (Fig. 3b), one of theequilibrium position is in the left component of Ω and two other equilibrium positions (if theyexist) are in the right component of Ω.

Fig. 4.

This analysis is valid for the case α > 0; one can obtain results for the opposite case usingsymmetry considerations from above.

3. A DISC WITH THE SHARP EDGE ON AN ICY SURFACE

In this case the friction is anisotropic: the disc can freely rotate about its axis and the velocityvector of a contact point is parallel to a horizontal disc diameter [4]. Also this system has the energy



integral (2.4) and the angular momentum integrals (the last two relations in (2.3)); besides thereis the following integral

vx +a

A

(pψ ln tan

θ

2− pφ ln sin θ

)= c0, (3.1)

where vx is the projection of velocity of the disc center onto the horizontal diameter. Now, usingthe above kinematics constraint, one can decrease the order of the system; its trajectories on thephase plane (θ, θ) lies on the curves

(A + ma2

)θ2 = 2h − 2mga sin θ − mv2

x − Aτ2 − 1C

p2φ (3.2)

that are level lines of the energy integral.Inequality (2.7) includes phase variables as well as values of integrals pφ and pψ. Therefore the

domain Ω, defined by (2.7), has the same shape as in case of a smooth plane (Figs. 3a and 3b).To construct the domain Ω0 one should calculate the second time derivative of the function Λ

defined in (2.1). Differentiating (3.2) and using the equality v′x = −aτ one gets(A + ma2

)θ = −mga cos θ + mavxτ − Aττ ′. (3.3)

Substituting this expression into (2.6) we find

Λ = U0 + V0N, V0 = m−1,

U0 = −g + aθ2 sin θ +a cos θ

A + ma2

(mga cos θ − mavxτ + Aττ ′) .

(3.4)

The equality Λ = 0 is possible for some N � 0 provided the following inequality holds:

aθ2 sin θ � g − a cos θ

A + ma2

(mga cos θ − mavxτ + Aττ ′) . (3.5)

Contrary to condition (2.7), inequality (3.5), defining the domain Ω0, contains also an arbitraryconstant c0 due to (3.1). The phase portrait of system (3.2) is changing while c0 is varying. Thedomain Ω0 is also changing but it has invariant vertical asymptotes as θ → +0 and θ → π − 0.For θ = π/2 boundary points of this domain are unchanged too. Clearly, the right hand side ofinequality (3.5) is linear in c0. Therefore for any values of pφ and pψ not equal to zero simultaneouslyand for any point on the phase plane with θ �= 0, π/2, π, it is possible to find values of c0 such thatthe domain Ω0 contains (or does not contain) this point. Depending on values of integrals thedomain Ω0 has the form shown on Figs. 3a, 3b or 3c.

Note that, similarly to the case of smooth support, the system is symmetric with respect tothe substitution pψ → −pψ, θ → π − θ, i.e., the equation of motion and domains Ω and Ω0 areunchanged (this substitution converts the orientation of the OZ axis). Therefore one can restrictoneself with an analysis of the case pφ � 0, pψ � 0. Consider some subcases.

1. For pφ = pψ = 0 one has τ = 0 and the right hand side in (3.5) does not depend on c0. Thecomparison of inequalities (2.8) and (3.5) shows that Ω0 ⊂ Ω. In equation (3.2) it holds vx = const,therefore the system is dynamically equivalent to the inverted pendulum. Its phase portrait is shownon Fig. 5a.

2. For pφ = pψ �= 0 system (3.2) can have two or none equilibrium positions depending on avalue of c0. Typical phase portraits are shown on Figs. 5b and 5c. Note that equilibrium positionsare necessarily in Ω0, but not necessarily in Ω. In this situation there is a non-uniqueness of themotion, i.e., both equilibrium (corresponding to a stationary motion of the body) and detachmentof the disc from the support are possible.

3. For pφ �= pψ all phase trajectories of the system (3.2) are bounded; the corresponding modelphase portraits are shown on Fig. 4. Likewise the previous case all equilibrium positions arenecessarily in Ω0, but not necessarily in Ω.


442 IVANOV

Fig. 5.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation of Basic Research (grant 08-01-00718).The author is grateful to A.V. Borisov and to participants of the Seminar on Nonlinear Dynamicsfor idea of this paper and useful discussion.

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geometric representation of detachment conditions in systems with unilateral constraint

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