On the regularization of impact without collision: the Painlevé paradox and compliance
OOn the regularization of impact without collision: thePainlev ´e paradox and compliance S. J. Hogan and K. Uldall Kristiansen ∗ August 28, 2018
Abstract
We consider the problem of a rigid body, subject to a unilateral constraint, inthe presence of Coulomb friction. We regularize the problem by assuming compliance(with both stiffness and damping) at the point of contact, for a general class of normalreaction forces. Using a rigorous mathematical approach, we recover impact withoutcollision (IWC) in both the inconsistent and indeterminate Painlev´e paradoxes, in thelatter case giving an exact formula for conditions that separate IWC and lift-off. Wesolve the problem for arbitrary values of the compliance damping and give explicitasymptotic expressions in the limiting cases of small and large damping, all for a largeclass of rigid bodies.
Keywords—
Painlev´e paradox, impact without collision, compliance, regularization
In mechanics, in problems with unilateral constraints in the presence of friction, the rigidbody assumption can result in the governing equations having multiple solutions (the inde-terminate case) or no solutions (the inconsistent case). The classical example of Painlev´e[29, 30, 31], consisting of a slender rod slipping along a rough surface (see Fig. 1), is thesimplest and most studied example of these phenomena, now known collectively as Painlev´eparadoxes [2, 4, 5, 32, 34]. Such paradoxes can occur at physically realistic parameter valuesin many important engineering systems [22, 23, 25, 27, 28, 36, 38].When a system has no consistent solution, it can not remain in that state. Lecornu [21]proposed a jump in vertical velocity to escape an inconsistent, horizontal velocity, state. Thisjump has been called impact without collision (IWC) [12], tangential impact [13] or dynamicjamming [28]. Experimental evidence of IWC is given in [38]. ∗ S. J. Hogan: Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UnitedKingdom. K. Uldall Kristiansen: Department of Applied Mathematics and Computer Science, TechnicalUniversity of Denmark, 2800 Kgs. Lyngby, DK. We prefer to avoid describing this phase of the motion as sliding because we will be using ideas frompiecewise smooth systems [11], where sliding has exactly the opposite meaning. a r X i v : . [ m a t h . D S ] M a r IWC occurs instantaneously. So it must be incorporated into the rigid body formulation[6, 15] by considering the equations of motion in terms of the normal impulse, rather thantime. However, this process has been controversial [3, 35], because it can sometimes lead toan apparent energy gain in the presence of friction.G´enot and Brogliato [12] considered the dynamics around a critical point, correspondingto zero vertical acceleration of the end of the rod. They proved that, when starting in aconsistent state, the rod must stop slipping before reaching the critical point. In particular,paradoxical situations cannot be reached after a period of slipping.One way to address the Painlev´e paradox is to regularize the rigid body formalism.Physically this often corresponds to assuming some sort of compliance at the contact point A , typically thought of as a spring, with stiffness (and sometimes damping) that tend to therigid body model in a suitable limit. Mathematically, very little rigorous work has been doneon how IWC and Painlev´e paradoxes can be regularized. Dupont and Yamajako [7] treatedthe problem as a slow fast system, as we will do. They explored the fast time scale dynamics,which is unstable for the Painlev´e paradoxes. Song et al. [33] established conditions underwhich these dynamics can be stabilized. Le Suan An [1] considered a system with bilateralconstraints and showed qualitatively the presence of a regularized IWC as a jump in verticalvelocity from a compliance model with diverging stiffness. Zhao et al. [37] considered theexample in Fig. 1 and regularized the equations by assuming a compliance that consisted ofan undamped spring. They estimated, as a function of the stiffness, the orders of magnitudeof the time taken in each phase of the (regularized) IWC. Another type of regularizationwas considered by Neimark and Smirnova [26] who assumed that the normal and tangentialreactions took (different) finite times to adjust.In this paper, we present the first rigorous analysis of the regularized rigid body formalism,in the presence of compliance with both stiffness and damping. We recover impact withoutcollision (IWC) in both the inconsistent and indeterminate cases and, in the latter case, wepresent a formula for conditions that separate IWC and lift-off. We solve the problem forarbitrary values of the compliance damping and give explicit asymptotic expressions in thelimiting cases of small and large damping. Our results apply directly to a general class ofrigid bodies. Our approach is similar to that used in [16, 17] to understand the forwardproblem in piecewise smooth (PWS) systems in the presence of a two-fold.The paper is organized as follows. In Section 2, we introduce the problem, outline someof the main results known to date and include compliance. In Section 3, we give a summaryof our main results, Theorem 1 and Theorem 2, before presenting their derivation in Sections4 and 5. We discuss our results in Section 6 and outline our conclusions in Section 7. Consider a rigid rod AB , slipping on a rough horizontal surface, as depicted in Fig. 1.The rod has mass m , length 2 l , the moment of inertia of the rod about its center of mass S is given by I and its center of mass coincides with its center of gravity. The point S hascoordinates ( X, Y ) relative to an inertial frame of reference ( x, y ) fixed in the rough surface.The rod makes an angle θ with respect to the horizontal, with θ increasing in a clockwisedirection. At A , the rod experiences a contact force ( F T , F N ), which opposes the motion.Figure 1: The classical Painlev´e problem.The dynamics of the rod is then governed by the following equations m ¨ X = − F T , (1) m ¨ Y = − mg − F N ,I ¨ θ = − l (cos θF N − sin θF T ) . where g is the acceleration due to gravity.The coordinates ( X, Y ) and ( x, y ) are related geometrically as follows x = X + l cos θ, (2) y = Y − l sin θ. We now adopt the scalings (
X, Y ) = l ( ˜ X, ˜ Y ) , ( x, y ) = l (˜ x, ˜ y ) , ( F T , F N ) = mg ( ˜ F T , ˜ F N ) , t = ω ˜ t, α = ml I where ω = gl . For a uniform rod, I = ml , and so α = 3 in this case.Then for general α , (1) and (2) can be combined to become, on dropping the tildes,¨ x = − ˙ θ cos θ + α sin θ cos θF N − (1 + α sin θ ) F T , (3)¨ y = − θ sin θ + (1 + α cos θ ) F N − α sin θ cos θF T , ¨ θ = − α (cos θF N − sin θF T ) . To proceed, we need to determine the relationship between F N and F T . We assume Coulombfriction between the rod and the surface. Hence, when ˙ x (cid:54) = 0, we set F T = µ sign( ˙ x ) F N , (4)where µ is the coefficient of friction. By substituting (4) into (3), we obtain two sets ofgoverning equations for the motion, depending on the sign of ˙ x , as follows:˙ x = v, (5)˙ v = a ( θ, φ ) + q ± ( θ ) F N , ˙ y = w, ˙ w = b ( θ, φ ) + p ± ( θ ) F N , ˙ θ = φ, ˙ φ = c ± ( θ ) F N , where the variables v, w, φ denote velocities in the x, y, θ directions respectively and a ( θ, φ ) = − φ cos θ, (6) b ( θ, φ ) = − φ sin θ,q ± ( θ ) = α sin θ cos θ ∓ µ (1 + α sin θ ) ,p ± ( θ ) = 1 + α cos θ ∓ µα sin θ cos θ,c ± ( θ ) = − α (cos θ ∓ µ sin θ )for the configuration in Fig. 1. The suffices q ± , p ± , c ± correspond to ˙ x = v ≷ x = v = 0 and a ( θ, φ ) + q + ( θ ) F N < < a ( θ, φ ) + q − ( θ ) F N , (7)where ˙ v in (5) ± for v ≷ v = 0, by using the Filippov vector-field [11]. Simplecomputations give: Proposition 1
The Filippov vector-field, within the subset of the switching manifold ˙ x = v = 0 where (7) holds, is given by ˙ y = w, (8)˙ w = b ( θ, φ ) + S w ( θ ) F N , ˙ θ = φ, ˙ φ = S φ ( θ ) F N , where S w ( θ ) = q − ( θ ) q − ( θ ) − q + ( θ ) p + ( θ ) − q + ( θ ) q − ( θ ) − q + ( θ ) p − ( θ ) = 1 + α α sin θ , (9) S φ ( θ ) = q − ( θ ) q − ( θ ) − q + ( θ ) c + ( θ ) − q + ( θ ) q − ( θ ) − q + ( θ ) c − ( θ ) = − α cos θ α sin θ . (cid:50) Remark 1
Our results hold for mechanical systems with different q ± , p ± and c ± in (6) andeven dependency on several angles θ ∈ T d , e.g. the two-link mechanism of Zhao et al. [38].Note that both S w and S φ in (9) are independent of µ , even for general c ± , q ± and p ± . (cid:50) In order to solve (5) and (8), we need to determine F N . The constraint-based methodleads to the Painlev´e paradox. The compliance-based method is the subject of this paper. In order that the constraint y = 0 be maintained, ¨ y (= ˙ w ) and F N form a complementaritypair given by ˙ w ≥ , F N ≥ , F N · ˙ w = 0 . (10)Note that F N ≥ F N and y satisfy the complementarity conditions0 ≤ F N ⊥ y ≥ . (11)In other words, at most one of F N and y can be positive.For the system shown in Fig. 1, the Painlev´e paradox occurs when v > θ ∈ (0 , π ),provided p + ( θ ) <
0, as follows: From the fourth equation in (5), we can see that b is thefree acceleration of the end of the rod. Therefore if b >
0, lift-off is always possible when y = 0 , w = 0. But if b <
0, in equilibrium we would expect a forcing term F N to maintainthe rod on y = 0. From ˙ w = 0 we obtain F N = − bp + (12)since v >
0. If p + >
0, which is always true for θ ∈ ( π , π ), then F N ≥
0, in line with(11). But if p + <
0, which can happen if θ ∈ (0 , π ), then F N < F N isin an inconsistent (or non-existent ) mode. On the other hand, if b > p + < F N > y = 0 and hence F N is in an indeterminate (or non-unique ) mode. It is straightforward to show that p + ( θ ) < µ > µ P ( α ) ≡ α √ α. (13)Then the Painlev´e paradox can occur for θ ∈ ( θ , θ ) where θ ( µ, α ) = arctan 12 (cid:16) µα − (cid:112) µ α − α ) (cid:17) , (14) θ ( µ, α ) = arctan 12 (cid:16) µα + (cid:112) µ α − α ) (cid:17) . For a uniform rod with α = 3, we have µ P (3) = . For α = 3 and µ = 1 . in the ( θ, φ )-plane, as in Fig. 2. Along θ = θ , θ , we have p + ( θ ) = 0.These lines intersect the curve b ( θ, φ ) = 0 at four points: φ ± , = ± (cid:112) csc θ , . G´enot andBrogliato [12] showed that the point P : ( θ, φ ) = ( θ , √ csc θ ) is the most important andanalyzed the local dynamics around it. The rigid body equations (1) are unable to resolvethe dynamics in the third and fourth quadrants. So we regularize these equations usingcompliance. Compare with Figure 2 of G´enot and Brogliato [12], where the authors plot the unscaled angular velocity ωφ vs. θ , for the case g = 9 . − , l = 1 m. Figure 2: The ( θ, φ )-plane for the classical Painlev´e problem of Fig. 1, for α = 3 and µ = 1 . P has coordinates ( θ , √ csc θ ), where θ is given in (14). In the first quadrantcentered on P , we have b > , p + <
0, so the dynamics is indeterminate (non-unique). Inthe second quadrant, b > , p + > b < , p + > p + = 0 unless also b = 0. In the fourthquadrant, b < , p + < y = 0 is satisfied, there exists no positive value of F N , contradicting (11). We assume that there is compliance at the point A between the rod and the surface, whenthey are in contact (see Fig. 1). Following [7, 24], we assume that there are small excursionsinto y <
0. Then we require that the nonnegative normal force F N ( y, w ) is a PWS functionof ( y, w ): F N ( y, w ) = [ f ( y, w )] ≡ (cid:26) y > { f ( y, w ) , } for y ≤ , (15)where the operation [ · ] is defined by the last equality and f ( y, w ) is assumed to be a smoothfunction of ( y, w ) satisfying ∂ y f < , ∂ w f < . The quantities ∂ y f (0 ,
0) and ∂ w f (0 ,
0) repre-sent a (scaled) spring constant and damping coefficient, respectively. We are interested inthe case when the compliance is very large, so we introduce a small parameter (cid:15) as follows: ∂ y f (0 ,
0) = (cid:15) − , ∂ w f (0 ,
0) = (cid:15) − δ. (16)This choice of scaling [7, 24] ensures that the critical damping coefficient ( δ crit = 2 in theclassical Painlev´e problem) is independent of (cid:15) . Our analysis can handle any f of the form f ( y, w ) = (cid:15) − h ( (cid:15) − y, w ) with h (ˆ y, w ) = − ˆ y − δw + O ((ˆ y + w ) ) . (17)But, to obtain our quantitative results, we truncate (17) and consider the linear function h (ˆ y, w ) = − ˆ y − δw, (18)so that F N ( y, w ) = (cid:15) − (cid:2) − (cid:15) − y − δw (cid:3) . (19)In what follows, the first equation in (5) will play no role, so we drop it from now on.Then we combine the remaining five equations in (5) with (15) and (16) to give the followingset of governing equations that we will use in the sequel˙ y = w, (20)˙ w = b ( θ, φ ) + p ± ( θ ) (cid:15) − [ − (cid:15) − y − δw ] , ˙ θ = φ, ˙ φ = c ± ( θ ) (cid:15) − [ − (cid:15) − y − δw ] , ˙ v = a ( θ, φ ) + q ± ( θ ) (cid:15) − [ − (cid:15) − y − δw ] , For (cid:15) > F N in these expressions with the square bracket (cid:15) − [ − (cid:15) − y − δw ] (see also Lemma 5 below). We now present the main results of our paper, Theorem 1 and Theorem 2. Theorem 1 showsthat, if the rod starts in the fourth quadrant of Fig. 2, it undergoes a (regularized) IWCfor a time of O ( (cid:15) ln (cid:15) − ). The same theorem also gives expressions for the resulting verticalvelocity of the rod in terms of the compliance damping and initial horizontal velocity andorientation of the rod. Theorem 1
Consider an initial condition ( y, w, θ, φ, v ) = (0 , O ( (cid:15) ) , θ , φ , v ) , v > , (21) within the region of inconsistency (non-existence) where p + ( θ ) < , b ( θ , φ ) < , (22) and q + ( θ ) < , q − ( θ ) > , a (cid:54) = 0 . Then the forward flow of (21) under (20) returns to { ( y, w, θ, φ, v ) | y = 0 } after a time O ( (cid:15) ln (cid:15) − ) with w = e ( δ, θ ) v + o (1) , (23) θ = θ + o (1) ,φ = φ + (cid:26) − c + ( θ ) q + ( θ ) + S φ ( θ ) S w ( θ ) (cid:18) e ( δ, θ ) + p + ( θ ) q + ( θ ) (cid:19)(cid:27) v + o (1) ,v = o (1) , as (cid:15) → . During this time y = O ( (cid:15) ) , w = O (1) so that F N = O ( (cid:15) − ) . The function e ( δ, θ ) , given in (61) below, is smooth and monotonic in δ and has the following asymptoticexpansions: e ( δ, θ ) = p − ( θ ) − p + ( θ ) q − ( θ ) p + ( θ ) − q + ( θ ) p − ( θ ) δ − (cid:0) O ( δ − ln δ − ) (cid:1) for δ (cid:29) , (24) e ( δ, θ ) = (cid:115) p + ( θ )( p − ( θ ) − p + ( θ )) q + ( θ )( q − ( θ ) − q + ( θ )) (cid:32) − (cid:112) S w ( θ )2 (cid:32) π − arctan (cid:32)(cid:115) − S w ( θ ) p + ( θ ) (cid:33)(cid:33) δ + O ( δ ) (cid:33) for δ (cid:28) . (25) (cid:50) Theorem 2 is similar to Theorem 1, but now the rod starts in the first quadrant of Fig. 2.This theorem also gives an exact formula for initial conditions that separate (regularized)IWC and lift off.
Theorem 2
Consider an initial condition ( y, w, θ, φ, v ) = (0 , (cid:15)w , θ , φ , v ) , w < w ∗ ≡ − λ − ( θ ) b ( θ , φ ) p + ( θ ) < , (26) with λ − defined in (33) below, within the region of indeterminacy (non-uniqueness) where p + ( θ ) < , b ( θ , φ ) > , (27) and q + ( θ ) < , q − ( θ ) > , a (cid:54) = 0 . Then the conclusions of Theorem 1, including expressions(23), (24) and (25), still hold true as (cid:15) → . For w > w ∗ lift-off occurs directly after atime O ( (cid:15) ) with w = O ( (cid:15) ) . During this period y = O ( (cid:15) ) , so F N = O (1) . (cid:50) Remark 2
These two theorems have not appeared before in the literature. In the rigidbody limit ( (cid:15) → absence of damping [37]. Wegive exact and asymptotic expressions for key quantities as well as providing a geometricinterpretion of our results, for a large class of rigid bodies, in the presence of a large class ofnormal forces, as well as giving a precise estimate for the time of (regularized) IWC, all inthe presence of both stiffness and damping. Note that we are not attempting to describe allthe dynamics around P . There is a canard connecting the third quadrant with the first andthe analysis is exceedingly complicated [18] due to fast oscillatory terms. Instead, we follow[37] and consider that the rod dynamics starts in a configuration with p + ( θ ) < (cid:50) The proof of Theorem 1 is divided into three phases, illustrated in Fig. 3. These phases area generalisation of the phases of IWC in its rigid body formulation [38]. • Slipping compression (section 4.2): During this phase y , w and v all decrease. Thedynamics follow an unstable manifold γ u of a set of critical points C , given in (31)below, as (cid:15) →
0. Along γ u the normal force F N = O ( (cid:15) − ) and v will therefore quicklydecrease to 0. Mathematically this part is complicated by the fact that the initialcondition (21) belongs to the critical set C as (cid:15) → • Sticking (section 4.3): Since F N = O ( (cid:15) − ) and q + q − < v ≡ y = ˙ w > F N = 0 as (cid:15) → • Lift-off (section 4.4): In the final phase F N = 0, lift off occurs and the system eventuallyreturns to y = 0. Before we consider the first phase of IWC, we apply the scaling y = (cid:15) ˆ y, (28)also used in [7, 24], which brings the two terms in (19) to the same order. Now letˆ F N (ˆ y, w ) ≡ (cid:15)F N ( (cid:15) ˆ y, w ) = [ − ˆ y − δw ] . (29)Equations (20) then read: ˆ y (cid:48) = w, (30) w (cid:48) = (cid:15)b ( θ, φ ) + p ± ( θ ) ˆ F N (ˆ y, w ) ,θ (cid:48) = (cid:15)φ,φ (cid:48) = c ± ( θ ) ˆ F N (ˆ y, w ) ,v (cid:48) = (cid:15)a ( θ, φ ) + q ± ( θ ) ˆ F N (ˆ y, w ) , with respect to the fast time τ = (cid:15) − t where () (cid:48) = ddτ . This is a slow-fast system in non-standard form [24]. Only θ is truly slow whereas (ˆ y, w, φ, v ) are all fast. But the set ofcritical points C = { (ˆ y, w, θ, φ, v ) | ˆ y = 0 , w = 0 } , (31)for (cid:15) = 0 is just three dimensional. System (30) is PWS [16, 17]. We now show that (30) + contains stable and unstable manifolds γ s,u when the equivalent rigid body equations exhibita Painlev´e paradox, when p + ( θ ) <
0. The saddle structure of C within the fourth quadranthas been recognized before [1, 7, 33].0 Proposition 2
Consider the system (30) + . Then for p + ( θ ) < there exist smooth stableand unstable sets γ s,u ( θ , φ , v ) , respectively, of (ˆ y, w, θ, φ, v ) = (0 , , θ , φ , v ) ∈ C con-tained within ˆ F N ≥ given by γ s,u ( θ , φ , v ) = (cid:26) (ˆ y, w, θ, φ, v ) | w = λ ∓ ˆ y, θ = θ , φ = φ − c + ( θ ) λ − ∓ [1 + δλ ∓ ]ˆ y,v = v + q + ( θ ) p + ( θ ) λ ∓ ( θ )ˆ y, ˆ y ≤ (cid:27) , (32) with λ ∓ ( θ ) ≶ given in (33) below. (cid:50) Proof
Consider the smooth system, (30) ˆ F N = − ˆ y − δw , obtained from (30) by setting ˆ F N = − ˆ y − δw . The linearization of (30) ˆ F N = − ˆ y − δw about a point in C then only has two non-zeroeigenvalues: λ ± ( θ ) = − δp + ( θ )2 ± (cid:112) δ p + ( θ ) − p + ( θ ) , (33)satisfying λ ± = − p + ( θ ) (1 + δλ ± ) . (34)For p + ( θ ) < λ − < < λ + . The eigenvectors associated with λ ± are v ± = (cid:16) , λ ± , , c + p + ( θ ) λ ± , q + p + ( θ ) λ ± (cid:17) T . Therefore the smooth system (30) ˆ F N = − ˆ y − δw has a (stable, un-stable) manifold γ s,u tangent to v ∓ at (ˆ y, w, θ, φ, v ) = (0 , , θ , φ , v ). But then for ˆ y ≤ F N (ˆ y, λ ± ˆ y ) = − (1 + δλ ± )ˆ y = λ ± p + ( θ ) ˆ y ≥
0, by (34). Hence the restrictions of γ s,u in(32) to ˆ y < C for the PWS system (30) ˆ F N =[ − ˆ y − δw ] . (cid:4) Remark 3
For the smooth system (30) ˆ F N = − ˆ y − δw , the critical manifold C perturbs by Fenichel’stheory [8, 9, 10] to a smooth slow manifold C (cid:15) , being C ∞ O ( (cid:15) )-close to C . A simple cal-culation shows that C (cid:15) : ˆ y = (cid:15) b ( θ,φ ) p + ( θ ) (1 + O ( (cid:15) )) , w = O ( (cid:15) ). Since b ( θ, φ ) < C (cid:15) ⊂ { ˆ y > } for (cid:15) sufficiently small. Therefore the manifold C (cid:15) is invariant for the smoothsystem (30) ˆ F N = − ˆ y − δw only. It is an artifact for the PWS system (30) ˆ F N =[ − ˆ y − δw ] since thesquare bracket vanishes for ˆ y >
0, by (15). (cid:50)
Remark 4
Our arguments are geometrical and rely on hyperbolic methods of dynamicalsystems theory only. Therefore the results remain unchanged qualitatively if we replace thepiecewise linear ˆ F N in (29) with the nonlinear version ˆ F N (ˆ y, w ) = [ h (ˆ y, w )], where h (ˆ y, w ) = − ˆ y − δw + O ((ˆ y + w ) as in (17), having (18) as its linearization about ˆ y = w = 0. We wouldobtain again a saddle-type critical set C with nonlinear (stable, unstable) manifolds γ s,u . (cid:50) Following the initial scaling (28) of this section, we now consider the three phases of IWC.1 (a) (b)
Figure 3: The limit (cid:15) → w, ˆ y, v )-variables and (b) a projection ontothe ( w, ˆ y )-plane. The slipping compression phase, shown in red, where ˆ y , w and v > γ u (32) of a critical set C , givenin (31). The sticking phase (in blue) is described by Filippov [11]. Finally the lift-off phase(in green) occurs and we return to ˆ y = 0. In both figures the grey region is where ˆ F N > Now we describe the first phase of the regularized IWC: slipping compression , which endswhen v = 0. We define the following section (or switching manifold ), Π shown in Fig. 3(a):Π = { (ˆ y, w, θ, φ, v ) | v = 0 } . (35)Proposition 3 describes the intersection of the forward flow of initial conditions (21) with Π;in other words, the values of ˆ y, w, θ, φ at the end of the slipping compression phase. Proposition 3
The forward flow of the initial conditions (21) under (30) intersects Π in γ u ∩ Π + o (1) ≡ (cid:26) (ˆ y, w, θ, φ, | ˆ y = − p + ( θ ) q + ( θ ) λ + ( θ ) v + o (1) , w = − p + ( θ ) q + ( θ ) v + o (1) θ = θ + o (1) , φ = φ − c + ( θ ) q + ( θ ) v + o (1) , (cid:27) , (36) as (cid:15) → . (cid:50) Remark 5
The o (1)-term in (36) is O ( (cid:15) c ) for any c ∈ (0 ,
1) (see also Lemma 3 below). (cid:50)
We prove Proposition 3 using Fenichel’s normal form theory [14]. But since (30) ˆ F N =[ − ˆ y − δw ] ispiecewise smooth, care must be taken. There are at least two ways to proceed. One way isto consider the smooth system (30) ˆ F N = − ˆ y − δw , then rectify C (cid:15) by straightening out its stableand unstable manifolds. Then (30) ˆ F N = − ˆ y − δw will be a standard slow-fast system to which2Fenichel’s normal form theory applies. Subsequently one would then have to ensure that con-clusions based on the smooth (30) ˆ F N = − ˆ y − δw also extend to the PWS system (30) ˆ F N =[ − ˆ y − δw ] .One way to do this is to consider the following scaling κ : ˆ y = r ˆ y , w = r w , (cid:15) = r , (37)zooming in on C at ˆ y = 0 , w = 0. In terms of the original variables, y = (cid:15) ˆ y , w = (cid:15)w . Thescalings (ˆ y, w ) and (ˆ y , w ) have both appeared in the literature [5, 7, 24].In this paper we follow another approach (basically reversing the process described above)which works more directly with the PWS system. Therefore in Section 4.2.2 we study thescaling (37) first. We will show that the (ˆ y , w )-system contains important geometry of thePWS system (significant, for example, for the separation of initial conditions in Theorem 2).Then in Section 4.2.3 we connect the “small” (ˆ y = O ( (cid:15) ) , w = O ( (cid:15) )) described by (37) with the“large” (ˆ y = O (1) , w = O (1)) in (30) by considering coordinates described by the followingtransformation: κ : ˆ y = − r , w = r w , (cid:15) = r (cid:15) . (38)For y < κ between κ and κ : κ : r = − r y , w = − w y − , (cid:15) = − y − . (39)The coordinates in κ (38) appear as a directional chart obtained by setting ¯ y = − r, ˆ y, ¯ w, ¯ (cid:15) ) (cid:55)→ (ˆ y, w, (cid:15) ) given by ˆ y = r ¯ˆ y, w = r ¯ w, (cid:15) = r ¯ (cid:15), r ≥ , (ˆ y, ¯ w, ¯ (cid:15) ) ∈ S = { (ˆ y, ¯ w, ¯ (cid:15) ) | ˆ y + ¯ w + ¯ (cid:15) = 1 } . (40)The blowup is chosen so that the zoom in (37) coincides with the scaling chart obtained bysetting ¯ (cid:15) = 1. The blowup transformation blows up C to ¯ C : r = 0 , (ˆ y, ¯ w, ¯ (cid:15) ) ∈ S a space( θ, φ, v ) ∈ R of spheres. The main advantage of our approach is that in chart κ we can focus on C ∩ { ˆ y − w > − δ − , ˆ y < } (or simply w < δ − in (38)) of C , the grey area in Fig. 3, where r − ˆ F N (ˆ y, w ) = (cid:2) − ˆ y − δw (cid:3) = − ˆ y (cid:16) δ ˆ y − w (cid:17) > , (41)and the system will be smooth. This enables us to apply Fenichel’s normal form theory [14]there. All the necessary patching for the PWS system is done independently in the scalingchart κ . Also chart κ enables a matching between the two scalings that have appeared inthe literature: κ ∩ { ˆ y < , r = 0 } , visible within r = 0 of (38), and system (30) (cid:15) =0 , visiblewithin (cid:15) = 0 of (38). More accurately, the chart ¯ y = − r = r (cid:113) (cid:15) + w , ˆ y = − / (cid:113) (cid:15) + w , ¯ w = w / (cid:113) (cid:15) + w , ¯ (cid:15) = (cid:15) / (cid:113) (cid:15) + w , with ˆ y + w + (cid:15) = 1 . See [20] for further details on directional and scaling charts. Note that (40) is not a blowup transformation in the sense of Krupa and Szmolyan [19], where geometricblowup is applied in conjunction with desingularization to study loss of hyperbolicity in slow-fast systems.We will not desingularize the vector-field here. κ Let ˆ F N, (ˆ y , w ) = (cid:15) − ˆ F N ( (cid:15) ˆ y , (cid:15)w ) = [ − ˆ y − δw ] . Then applying chart κ in (37) to thenon-standard slow-fast system (30) gives the following equations:ˆ y (cid:48) = w , (42) w (cid:48) = b ( θ, φ ) + p + ( θ ) ˆ F N, (ˆ y , w ) ,θ (cid:48) = (cid:15)φ,φ (cid:48) = (cid:15)c + ( θ ) ˆ F N, (ˆ y , w ) ,v (cid:48) = (cid:15) (cid:16) a ( θ, φ ) + q + ( θ ) ˆ F N, (ˆ y , w ) (cid:17) , Equation (42) is a slow-fast system in standard form: (ˆ y , w ) are fast variables whereas( θ, φ, v ) are slow variables. By assumption (22) of Theorem 1, b < p + < F N, (ˆ y , w ) ≥
0, we have w (cid:48) < (cid:15) =0 . The critical set C of the smooth system (42) ˆ F N, = − ˆ y − δw , given by C = { (ˆ y , w , θ, φ, v ) | ˆ y = b ( θ, φ ) p + ( θ ) , w = 0 } , (43)lies within ˆ y >
0. So C is an invariant of (42) ˆ F N, = − ˆ y − δw but an artifact of the PWSsystem (42) ˆ F N, =[ − ˆ y − δw ] , as shown in Fig. 4(a) (recall also Remark 3). (a) (b) Figure 4: (a) Phase portrait (42) (cid:15) =0 for b < C of (42) ˆ F N, = − ˆ y − δw , given by (43), is an artifact of the PWS system (42) ˆ F N, =[ − ˆ y − δw ] .(b) Phase portrait (42) (cid:15) =0 for b > C is a saddle-typecritical manifold for the PWS system, γ u is given by (44), γ s by (64) and w ∗ by (65). Thegrey region is now where ˆ F N, >
0. Orbit segments outside this region are parabolas turningdownwards and upwards in (a) and (b), respectively. Dashed lines indicate backward orbits,from initial conditions on the w -axis. Similar figures from numerical computations appearin [5].4The unstable manifold γ u of C in the smooth system (42) ˆ F N, = − ˆ y − δw is given by γ u ( θ , φ , v ) = (cid:26) (ˆ y , w , θ , φ , v ) | ˆ y = b ( θ , φ ) p + ( θ ) + s, w = λ + ( θ ) s, s ≤ − b ( θ , φ ) p + ( θ ) (cid:27) (44)and its restriction to the subset ˆ y ≤ w ≤ F N, ≥
0, is locally invariant for thePWS system (42) ˆ F N, =[ − ˆ y − δw ] .In chart κ , initial conditions (21) now become:(ˆ y , w , θ, φ, v ) = (0 , O (1) , θ , φ , v ) . (45)In Lemma 1, we determine the values of the variables during the slipping compression phase,starting from initial conditions (45), as seen in chart κ , and show that the system remainsclose to γ u ( θ , φ , v ). Lemma 1
Consider Λ = { (ˆ y , w , θ, φ, v ) | ˆ y = − ν − } with ν > small. Then the forwardflow of (45) under (42) intersects Λ in z ( (cid:15) ) ≡ ( − ν − , w c ( ν ) + O ( (cid:15) ) , θ + O ( (cid:15) ) , φ + O ( (cid:15) ) , v + O ( (cid:15) )) . (46) where w c ( ν ) = − λ + ν − (1 + o (1)) , ν → . (cid:50) Proof
Consider the layer problem (42) (cid:15) =0 . Then θ ( τ ) = θ , φ ( τ ) = φ , v ( τ ) = v . Since b <
0, initial conditions (45) with w > y = 0 with w <
0, see Fig. 4(a).Therefore we consider w (0) ≤ ˆ F N, = − ˆ y − δw , (cid:15) =0 to findˆ y ( τ ) = b ( θ , φ ) p + ( θ ) + k + e λ + τ + k − e λ − τ , (47) w ( τ ) = k + λ + e λ + τ + k − λ − e λ − τ . Here k + = λ − b/p + − w (0) λ + − λ − < , k − = w (0) − λ + b/p + λ + − λ − < λ − < < λ + the solution remains within ˆ F N, > τ > γ u ( θ , φ , v ) as τ → ∞ . Therefore there exists a time τ c = τ c ( ν ) > is reached. Then at τ = τ c ,we have z (0) = ( − ν − , w c , θ , φ , v ) , where w c ( ν ) ≡ w ( τ c ) = k + λ + e λ + τ c + k − λ − e λ − τ c = − λ + ν − (1 + o (1)) <
0. To obtain z ( (cid:15) ) we apply regular perturbation theory and the implicitfunction theorem using transversality to Λ for (cid:15) = 0. (cid:4) For (cid:15) > φ, v ) will vary by O (1)-amount as ˆ y , w → −∞ . But thevariables ( φ, v ) are fast in (30) and slow in (42). To describe this transition we change tochart κ .5 κ Writing the non-standard slow-fast PWS system (30) ˆ F N =[ − ˆ y − δw ] in chart κ , given by (38),gives the following smooth (as anticipated by (41)) system (cid:15) (cid:48) = (cid:15) w , (48) w (cid:48) = (cid:15) b ( θ, φ ) + p + ( θ ) (1 − δw ) + w ,θ (cid:48) = (cid:15)φ,φ (cid:48) = c + ( θ ) r (1 − δw ) ,v (cid:48) = (cid:15)a ( θ, φ ) + q + ( θ ) r (1 − δw ) ,r (cid:48) = − r w , on the box U = { ( (cid:15) , w , θ, φ, v, r ) | (cid:15) ∈ [0 , ν ] , w ∈ [ − λ + − ρ, − λ + + ρ ] , r ∈ [0 , ν ] } , for ρ > w < δ − ) and ν as above. Notice that z ( (cid:15) ) from (46) inchart κ becomes z ( (cid:15) ) ≡ κ ( z ( (cid:15) )) : r = (cid:15)ν − , w = w c ν + O ( (cid:15) ) , (cid:15) = ν, (49)using (39). Clearly z ( (cid:15) ) ∈ κ (Λ ) ⊂ U , κ (Λ ) being the face of the box U with (cid:15) = ν .In this section we will for simplicity write subsets such as { ( (cid:15) , w , θ, φ, v, r ) ∈ U | · · · } by { U | · · · } . Lemma 2
The set M = { U | r = 0 , (cid:15) = 0 , w = − λ + } is a set of critical points of(48). Linearization around M gives only three non-zero eigenvalues − λ + < , λ − − λ + < , λ + > , and so M is of saddle-type. The stable manifold is W s ( M ) = { U | r = 0 } whilethe unstable manifold is W u ( M ) = { U | (cid:15) = 0 , w = − λ + } . In particular, the D unstablemanifold γ u ( θ , φ , v ) ⊂ W u ( M ) of the base point ( (cid:15) , w , θ, φ, v, r ) = (0 , , θ , φ , v , ∈ M is given by γ u ( θ , φ , v ) = (cid:26) U | w = − λ + ( θ ) , θ = θ , φ = φ − c + ( θ ) p + ( θ ) λ + ( θ ) r , (50) v = v − q + ( θ ) p + ( θ ) λ + ( θ ) r , r ≥ , (cid:15) = 0 (cid:27) . (cid:50) Proof
The first two statements follow from straightforward calculation. For γ u ( θ , φ , v ),we restrict to the invariant set: (cid:15) = 0, w = − λ + and solve the resulting reduced system. (cid:4) Remark 6
Notice that the set γ u ( θ , φ , v ) is just γ u ( θ , φ , v ) in (32) written in chart κ for (cid:15) = 0. (cid:50) Notice that z (0) ⊂ W s ( M ). The forward flow of z (0) is described for τ ≥ τ c by writingsolution (47) to the layer problem (42) (cid:15) =0 in chart κ using κ , to get (cid:15) ( τ ) = − (cid:18) b ( θ , φ ) p + ( θ ) + k + e λ + τ + k − e λ − τ (cid:19) − = − k − e − λ + τ (1 + O ( e − λ + τ + e ( λ − − λ + ) τ )) , τ → ∞ (51) w ( τ ) = − (cid:0) k + λ + e λ + τ + k − λ − e λ − τ (cid:1) (cid:18) b ( θ , φ ) p + ( θ ) + k + e λ + τ + k − e λ − τ (cid:19) − = − λ + (1 + O ( e − λ + τ + e ( λ − − λ + ) τ )) , τ → ∞ . z ( (cid:15) ) ⊂ { (cid:15) = ν } up until r = ν , with ν sufficientlysmall, by applying Fenichel’s normal form theory. Lemma 3
Let c ∈ (0 , and set Λ = { U | r = ν } . Then for ν and ρ sufficiently small, theforward flow of z ( (cid:15) ) in (49) intersects Λ in (cid:26) Λ | w = − λ + + O ( (cid:15) c ) , θ = θ + O ( (cid:15) ln (cid:15) − ) , φ = φ − c + ( θ ) p + ( θ ) λ + ( θ ) ν + O ( (cid:15) c ) ,v = v − q + ( θ ) p + ( θ ) λ + ( θ ) ν + O ( (cid:15) c ) (cid:27) . (52) (cid:50) as (cid:15) → . Proof
By Fenichel’s normal form theory we can make the slow variables independent ofthe fast variables ( (cid:15) , w , r ): Lemma 4
For ν and ρ sufficiently small, then within U there exists a smooth transforma-tion ( (cid:15) , w , φ, v, r ) (cid:55)→ ( ˜ φ, ˜ v ) satisfying ˜ φ = φ + c + ( θ ) p + ( θ ) λ + ( θ ) r + O ( r ( w + λ + )) , (53)˜ v = v + q + ( θ ) p + ( θ ) λ + ( θ ) r + O ( r ( w + λ + ) + (cid:15) ) , which transforms (48) into (cid:15) (cid:48) = (cid:15) w , (54) w (cid:48) = (cid:15) b ( θ, ˜ φ ) + p + ( θ ) (1 − δw ) + w + O ( (cid:15) ) ,θ (cid:48) = (cid:15) ˜ φ, ˜ φ (cid:48) = 0 , ˜ v (cid:48) = 0 ,r (cid:48) = − r w . (cid:50) Proof
Replace r by νr in (48) and consider ν small. Then (cid:15) = r = 0, w = − λ + is asaddle-type slow manifold for ν small. The result then follows from Fenichel’s normal formtheory [14]. Using φ = ˜ φ + O ( r ) together with r (cid:15) = (cid:15) in the w -equation then gives thedesired result.To prove Lemma 3 we then integrate the normal form (54) with initial conditions z ( (cid:15) ) from(49) from (a reset) time τ = 0 up to τ = T , defined implicitly by r ( T ) = ν . Clearly θ ( T ) = θ + O ( (cid:15)T ), ˜ φ ( T ) = ˜ φ , ˜ v ( T ) = ˜ v . Then, from (51), Gronwall’s inequality and thefact that 1 − λ − λ − >
1, we find T = λ − ln (cid:15) − (1 + o (1)) (55) (cid:15) ( T ) = (cid:15)ν − w ( T ) = − λ + (1 + O ( e − λ + T + e ( λ − − λ + ) T + (cid:15) )) = − λ + + O ( (cid:15) c (1 − λ − λ − ) + (cid:15) c ) = − λ + + O ( (cid:15) c ) , (56)7for c ∈ (0 , θ = θ ( T ), φ = φ ( T ) and v = v ( T ) in (52)from (53) in terms of the original variables. (cid:4) To complete the proof of Proposition 3 we then return to (30) using (38) and integrateinitial conditions (52) within { ˆ y = − r = − ν } , up to Π : v = 0 given in (35), using regularperturbation theory and the implicit function theorem. This gives (36) which completes theproof of Proposition 3. After the slipping compression phase of the previous section, the rod then sticks on thesliding manifold Π given in (35), with (ˆ y, w, θ, φ ) given by (36). This is a corollary of thefollowing lemma:
Lemma 5
Suppose a (cid:54) = 0 , q + < , q − > . Consider the (negative) function F ( θ, φ ) = (cid:40) a ( θ,φ ) q + ( θ ) if a > , a ( θ,φ ) q − ( θ ) if a < . Then there exists a set of visible folds at: Γ (cid:15) ≡ { (ˆ y, w, θ, φ, v ) ∈ Π | ˆ y + δw = (cid:15) F ( θ, φ ) } , (57) of the Filippov system (30), dividing the switching manifold Π : v = 0 into (stable) sticking: Π s ≡ { (ˆ y, w, θ, φ, v ) ∈ Π | ˆ y + δw < (cid:15) F ( θ, φ ) } , and crossing upwards (downwards) for a > ( a < ): Π c ≡ { (ˆ y, w, θ, φ, v ) ∈ Π | ˆ y + δw > (cid:15) F ( θ, φ ) } . (cid:50) Proof
Simple computations, following [11]; see also Proposition 1. (cid:4)
The forward motion of (36) within Π s ⊂ Π for (cid:15) (cid:28) y (cid:48) = w, (58) w (cid:48) = (cid:15)b ( θ, φ ) + S w ( θ ) [ − ˆ y − δw ] ,θ (cid:48) = (cid:15)φ,φ (cid:48) = S φ ( θ ) [ − ˆ y − δw ] , here written in terms of ˆ y and the fast time τ , until sticking ends at the visible fold Γ (cid:15) . Notethis always occurs for 0 < (cid:15) (cid:28) y (cid:48)(cid:48) = w (cid:48) >
0, for [ − ˆ y − δw ] > (cid:15) = 0. From (58), θ = θ , a constant, andˆ y (cid:48) = w, (59) w (cid:48) = S w ( θ ) [ − ˆ y − δw ] ,φ (cid:48) = S φ ( θ ) [ − ˆ y − δw ] . (cid:15) = 0 as initial conditions, given by(ˆ y (0) , w (0) , φ (0)) = (cid:18) − p + ( θ ) q + ( θ ) λ + ( θ ) v , − p + ( θ ) q + ( θ ) v , φ − c + ( θ ) q + ( θ ) v (cid:19) , (60)up until the section Γ : ˆ y + δw = 0 shown in Fig. 3(a), where sticking ceases for (cid:15) = 0, byLemma 5 and (57) (cid:15) =0 . We then obtain a function e ( δ, θ ) > v (21)with the values of (ˆ y, w, φ ) on Γ , at the end of the sticking phase. Proposition 4
There exists a smooth function e ( δ, θ ) > and a time τ s > such that: . (ˆ y ( τ s ) , w ( τ s ) , φ ( τ s )) ∈ Γ with ˆ y ( τ s ) = − δe ( δ, θ ) v ,w ( τ s ) = e ( δ, θ ) v ,φ ( τ s ) = φ + (cid:26) − c + ( θ ) q + ( θ ) + S φ ( θ ) S w ( θ ) (cid:18) e ( δ, θ ) + p + ( θ ) q + ( θ ) (cid:19)(cid:27) v , (cid:50) where (ˆ y ( τ ) , w ( τ ) , φ ( τ )) is the solution of (59) with initial conditions (60). The function e ( δ, θ ) is monotonic in δ : ∂ δ e ( δ, θ ) < , and satisfies (24) and (25) for δ (cid:29) and δ (cid:28) ,respectively. Proof
The existence of τ s is obvious. Linearity in v follows from (60) and the linearityof (59). Since ˙ˆ y = w , we have e >
0. The φ -equation follows since φ (cid:48) = S φ ( θ ) S w ( θ ) w (cid:48) . Themonotonicity of e as function δ is the consequence of simple arguments in the ( w, ˆ y )-planeusing (59) and the fact that w (0) in (60) is independent of δ while ˆ y (0) = ˆ y ( δ ) decreases(since λ + is an increasing function of δ ). To obtain the asymptotics we first solve (59) with δ (cid:54) = √ S w ( θ ) . Simple calculations show that e ( δ, θ ) = ξ + ξ − ( λ + − ξ − ) p + q + λ + e ξ + τ s , (61)suppressing the dependency on θ on the right hand side, where ξ ± = − δS w ± (cid:112) δ S w − S w ,and τ s is the least positive solution of e ( ξ + − ξ − ) τ s = ξ − ( λ + − ξ + ) ξ ( λ + − ξ − ) . For δ (cid:29) ξ ± are real and negative. Hence τ s = ξ + − ξ − ln (cid:16) ξ − ( λ + − ξ + ) ξ ( λ + − ξ − ) (cid:17) . Now using ξ + = − S w δ (1 + O ( δ − ) , ξ − = S w ξ − = − δ − (1 + O ( δ − ), and λ + = − p + δ (1 + O ( δ − )) , we obtain ξ + τ s = O ( δ − ln δ − ) , and hence e ( δ, θ ) = − S w − p + q + S w δ − (cid:0) O ( δ − ln δ − ) (cid:1) , (62)as δ → ∞ . For δ (cid:28) ξ ± are complex conjugated with negative real part. This gives τ s = iξ + − ξ − ) ( φ − πn ) , φ = arg (( λ + − ξ + ) ξ − ) > , n = (cid:98) φ/π (cid:99) . Using the asymptotics of ξ ± λ + we obtain τ s = π − arctan (cid:16)(cid:113) − Swp + (cid:17) √ S w − δ (1 + O ( δ )) , and then e ( δ, θ ) = − (cid:112) p + ( p + − S w ) q + (cid:32) − √ S w (cid:32) π − arctan (cid:32)(cid:115) − S w p + (cid:33)(cid:33) δ + O ( δ ) (cid:33) , (63) (cid:4) as δ → + . Simple algebraic manipulations of (62) and (63) using (9) give the expressionsin (24) and (25). Remark 7
The critical value δ = δ crit ( θ ) ≡ √ S w ( θ ) gives a double root of the characteristicequation. Note that δ crit ( π ) = 2 for the classical Painlev´e problem, as expected (see section2.2). (cid:50) For 0 < (cid:15) (cid:28) (cid:15) . We therefore perturb from (cid:15) = 0 asfollows:
Proposition 5
The forward flow of (36) under the Filippov vector-field (58) intersects theset of visible folds Γ (cid:15) o (1) -close to the intersection of (36) (cid:15) =0 with Γ described in Proposi-tion 4. (cid:50) Proof
Since the (cid:15) = 0 system is transverse to Γ we can apply regular perturbation theoryand the implicit function theorem to perturb τ s continuously to τ s + o (1). The result thenfollows. (cid:4) Beyond Γ (cid:15) we have ˆ F N ≡ (cid:15) = 0 we have ˆ y (cid:48) = w and w (cid:48) = θ (cid:48) = φ (cid:48) = v (cid:48) = 0. By Proposition 5 and regular perturbation theory, we obtain the desired result inTheorem 1. In terms of the original (slow) time t , it follows that the time of IWC is of order O ( (cid:15) ln (cid:15) − ) (recall (55)). As (cid:15) →
0, IWC occurs instantaneously, as desired.
Here, by assumption (27), we have b >
0. Now C ⊂ { ˆ y < } , where C is given by (43);see also Fig. 4(b). The stable manifold of C ∩ { θ = θ , φ = φ , v = v } is: γ s ( θ , φ , v ) = (cid:26) (ˆ y , w , θ , φ , v ) | ˆ y = b ( θ , φ ) p + ( θ ) + s, w = λ − ( θ ) s, s ≤ − b ( θ , φ ) p + ( θ ) , (cid:27) (64)with λ − defined in (33). γ s intersects the w -axis in γ s ∩ { ˆ y = 0 } : w = w ∗ ≡ − λ − ( θ ) b ( θ , φ ) p + ( θ ) < , (65)0and divides the negative w -axis into (i) initial conditions that lift off directly ( w > w ∗ ,blue in Fig. 4(b)) and (ii) initial conditions that undergo IWC before returning to ˆ y = 0( w < w ∗ , green in Fig. 4(b)). (A canard phenomenon occurs around w = w ∗ for0 < (cid:15) (cid:28) w < w ∗ the remainder of the proof of Theorem 2 on IWC in the indeterminantecase then follows the proof of Theorem 1 above. The quantity e ( δ, θ ) relates the initial horizontal velocity v of the rod to the resultingvertical velocity at the end of IWC. It is like a “horizontal coefficient of restitution”. Theleading order expression of e ( δ, θ ) in (24) for δ (cid:29) µ , in general. Usingthe expressions for q ± and p ± in (6), together with (62), we find for large δ that e ( δ, θ ) = α α ) sin(2 θ ) δ − (cid:0) O ( δ − ln δ − ) (cid:1) , θ ∈ ( θ , θ ) . (66)The limit δ → ∞ is not uniform in θ ∈ ( θ , θ ).The expression for δ (cid:28) does depend upon µ , in general. Using(6) and (63), for δ = 0, we have: e (0 , θ ) = (cid:115) (1 + α cos θ − µα sin θ cos θ )( α sin θ cos θ − µ (1 + α sin θ )) α sin θ cos θ (1 + α sin θ ) (67)We plot e (0 , θ ) in Fig. 5(a) for α = 3 and µ = 1 .
4. Fig. 5(b) shows the graph of e ( δ,
1) and e ( δ, .
2) along with the approximations (dashed lines) in (24) and (25).In the inconsistent case, described by Theorem 1, the initial conditions (21) are verysimilar to those assumed by [37]. As in their case, these conditions would be impossible toreach in an experiment without using some form of controller . Nevertheless, it should bepossible to set up the initial conditions in (21) to approach the rigid surface from above, asit appears to have been done in [38] for the two-link manipulator system.The indeterminate case described by Theorem 2 is characterised by an extreme expo-nential splitting in phase space, due to the stable manifold of C in the κ -system (43).For example, the blue orbit in Fig. 4(b) lifts off directly with w = O ( (cid:15) ). But on theother side of the stable manifold, the green orbit undergoes IWC and then lifts off with w = O (1). The initial conditions in Theorem 2 correspond to orbits that are almost grazing( ˙ y = w = O ( (cid:15) ) , ¨ y = ˙ w = b >
0) the compliant surface at y = 0. In Fig. 6 we illustratethis further by computing the full Filippov system (5) (cid:15) =10 − for two rods (green and blue)initially distant by an amount of 10 − above the compliant surface ( y ≈ .
1, see also t = 0in Fig. 6(a)). Fig. 6(a) shows the configuration of the rods at different times t = 0, t = 0 . To see this, fix any ˆ y <
0. Then by applying the approach in section 4.2 backwards in time, it followsthat the backward flow of (21) for b < κ -dynamics) intersectsthe section { ˆ y = ˆ y } at a distance which is o (1)-close to γ s ∩{ ˆ y = ˆ y } as (cid:15) →
0. Here γ s is the stable manifoldof C for (cid:15) = 0. But cf. (30) (cid:15) =0 , the horizontal velocity v (and hence the energy) increases unboundedlyalong γ s in backwards time. This increase occurs on the fast time scale τ . (a) (b) Figure 5: (a) Graph of e (0 , θ ) from (67), where θ , are given by (14). (b) Graph of e ( δ, θ )for θ = 1 and θ = 1 .
2, where the dashed lines correspond to the approximations obtainedfrom (24) and (25). For both figures, α = 3 and µ = 1 . t = 0 . t = 1. Up until t = 0 .
5, the two rods are indistinguishable. At t = 0 .
5, grazing( ˙ y = w ≈ − ) with the compliant surface y = 0 occurs where θ ≈ . φ ≈ . v ≈ .
00 (so b ≈ . p + ≈ − . τ , and therefore subsequently lifts off from y = 0 with w = O (1). Incomparison the blue rod lifts off with w ≈ − . At t = 1 the two rods are clearly separated.Fig. 6(b) shows the projection of the numerical solution in Fig. 6(a) onto the ( w, ˆ y )-plane(compare Fig. 3(b)). The blue orbit lifts off directly. The green orbit, being on the other sideof the stable manifold of C , follows the unstable manifold (red) until sticking occurs. Thenwhen ˆ F N = 0 at ˆ y + δ ˆ w = 0 (dashed line), lift off occurs almost vertically in the ( w, ˆ y )-plane.Fig. 6(c) and Fig. 6(d) show the vertical velocity w and horizontal velocity v , respectively,for both orbits over the same time interval as Fig. 6(b); note the sharp transition for thegreen orbit around t = 0 .
5, as it undergoes IWC. In Fig. 6(c), we include two dashed lines w = ev and w = − p + q + v , corresponding to our analytical results (23) and (36), which alsohold for the indeterminate case (from Theorem 2), in excellent agreement with the numericalresults. We have considered the problem of a rigid body, subject to a unilateral constraint, in thepresence of Coulomb friction. Our approach was to regularize the problem by assuming acompliance with stiffness and damping at the point of contact. This leads to a slow-fastsystem, where the small parameter (cid:15) is the inverse of the square root of the stiffness.Like other authors, we found that the fast time scale dynamics is unstable. Dupont andYamajako [7] established conditions in which these dynamics can be stabilized. In contrast,2 (a) (b)(c) (d)
Figure 6: (a) Dynamics of the Painlev´e rod described by the Filippov system (5) for µ = α = 3, δ = 1 and (cid:15) = 10 − in the indeterminate case. The green and blue rods are separatedat t = 0 by a distance of 10 − . At around t = 0 .
5, impact with the compliant surface occurs.The green rod experiences IWC whereas the blue rod lifts off directly. (b) Projection ontothe ( w, ˆ y )-plane. (c) and (d) w and v as functions of time near t = 0 . κ to numerically compute stability boundaries [7, 24]or phase plane diagrams [5].The main achievement of this paper is to rigorously derive these, and other, results thathave eluded others in simpler settings. For example, the work of Zhao et al. [37] assumes nodamping in the compliance and uses formal methods to provide estimates of the times spentin the three phases of IWC. They suggest that their analysis can “ · · · roughly explain whythe Painlev´e paradox can result in [IWC].”. In contrast, we assumed that the compliance has3 both stiffness and damping, analysed the problem rigorously, derived exact and asymptoticexpressions for many important quantities in the problem and showed exactly how and whythe Painlev´e paradox can result in IWC. There are no existing results comparable to (23),(24) and (25) for any value of δ .Our results are presented for arbitrary values of the compliance damping and we are ableto give explicit asymptotic expressions in the limiting cases of small and large damping, allfor a large class of rigid bodies, including the case of the classical Painlev´e example in Fig. 1.Given a general class of rigid body and a general class of normal reaction, we have beenable to derive an explicit connection between the initial horizontal velocity of the body andits lift-off vertical velocity, for arbitrary values of the compliance damping, as a function ofthe initial orientation of the body. References [1] Le Suan An. The Painlev´e paradoxes and the law of motion of mechanical systems withCoulomb friction.
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