DYNAMICS OF PSEUDO-RIGID BALL IMPACT ON RIGID ...2. Pseudo-rigid Ball Impact 2.1 Problem description...

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To appear in the International Journal of Non-linear Mechanics DYNAMICS OF PSEUDO-RIGID BALL IMPACT ON RIGID FOUNDATION Eva KANSO and Panayiotis PAPADOPOULOS * Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740, USA Table of Contents 1. Introduction 2. Pseudo-rigid Ball Impact 2.1 Problem description 2.2 Equations of motion 3. Impact Map 4. Impact Dynamics 4.1 Numerical results 4.2 Symplecticity and quasi-periodicity 5. Closure References Abstract This paper concerns the dynamics induced by the ideally elastic normal impact of a linearly elastic pseudo-rigid sphere on a rigid, stationary foundation. An impact map is derived and studied by numerical and analytical means. Periodic, quasi-periodic, and chaotic response is observed consistently with the symplectic nature of the map. Keywords: Dynamics, pseudo-rigid, impact, chaos, Lyapunov exponents, symplectic map, KAM theory. * Corresponding author 1

Transcript of DYNAMICS OF PSEUDO-RIGID BALL IMPACT ON RIGID ...2. Pseudo-rigid Ball Impact 2.1 Problem description...

  • To appear in the

    International Journal of Non-linear Mechanics

    DYNAMICS OF PSEUDO-RIGID BALL

    IMPACT ON RIGID FOUNDATION

    Eva KANSO and Panayiotis PAPADOPOULOS∗

    Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740, USA

    Table of Contents

    1. Introduction

    2. Pseudo-rigid Ball Impact

    2.1 Problem description2.2 Equations of motion

    3. Impact Map

    4. Impact Dynamics

    4.1 Numerical results4.2 Symplecticity and quasi-periodicity

    5. Closure

    References

    Abstract

    This paper concerns the dynamics induced by the ideally elastic normal impact of alinearly elastic pseudo-rigid sphere on a rigid, stationary foundation. An impact mapis derived and studied by numerical and analytical means. Periodic, quasi-periodic,and chaotic response is observed consistently with the symplectic nature of the map.

    Keywords: Dynamics, pseudo-rigid, impact, chaos, Lyapunov exponents, symplectic map,KAM theory.

    ∗Corresponding author

    1

  • Pseudo-rigid sphere impact

    1 Introduction

    The problem of a rigid ball bouncing between a fixed wall and an oscillating wall was first

    proposed by Fermi to study cosmic ray acceleration [1]. The Fermi model and a closely

    related one of a ball bouncing on a single oscillating wall under the influence of gravity arise

    in a number of physical applications, see Tufillaro et al. [2, Chapter 1]. A simplified version

    of the latter model in which the wall imparts momentum to the ball while maintaining a

    fixed position leads to the so-called standard map. This two-dimensional, area-preserving

    map has been extensively analyzed as a prototypical system exhibiting both regular and

    chaotic response, see, e.g., Chirikov [3], Greene [4], Lichtenberg and Lieberman [5], and

    Guckenheimer and Holmes [6, Section 2.4].

    All rigid bouncing ball problems neglect the strain energy of the ball, although this

    energy may constitute an appreciable portion of the total energy and can significantly

    affect the dynamics of the system. In order to account for the strain energy, one needs to

    introduce a set of deformation measures, as done recently in a study of pseudo-rigid ball

    impact by Solberg and Papadopoulos [7]. A pseudo-rigid body is a continuum capable of

    undergoing only spatially homogeneous deformation. The origins of pseudo-rigid mechanics

    are traced to the work of Slawianowski [8,9]. A general theory for such constrained continua

    has been formulated by Cohen [10] and Muncaster [11,12], and is comprehensively discussed

    in a monograph by Cohen and Muncaster [13].

    The present work explores in depth the dynamics of ideally elastic impact between an

    isotropic, linearly elastic pseudo-rigid ball and a rigid stationary foundation. To this end,

    an exact impact map is derived from the governing equations of motion. A numerical study

    of this map at various energy levels reveals that an energy surface typically consists of pe-

    riodic, quasi-periodic and chaotic trajectories. Chaos occurs at a given energy level due to

    the transversal intersections of the stable and unstable manifolds of hyperbolic points. The

    branches of these manifolds generate a sea of connected chaos surrounding islands charac-

    terized by dominantly regular behavior. The chaotic nature of the sea is evidenced by the

    calculation of Lyapunov characteristic exponents. The observed response is analogous to

    that predicted by the KAM theory1 originally developed for nearly integrable Hamiltonian

    systems and later extended to associated symplectic maps. The impact map derived for

    the pseudo-rigid ball problem is shown to be symplectic on a given energy surface. Yet,

    1After Kolmogorov [14], Arnold [15], and Moser [16].

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  • E. Kanso and P. Papadopoulos

    the system is not strictly Hamiltonian, owing to the non-smoothness of the response at

    impact.

    The organization of the paper is as follows: The impact problem is specified and its

    equations are formulated in Section 2. A derivation of the impact map is presented in

    Section 3, followed by a discussion of the impact dynamics in Section 4. A summary of the

    findings is given in Section 5.

    2 Pseudo-rigid Ball Impact

    2.1 Problem Description

    The dynamical system analyzed here arises from a model originally proposed and studied

    by Solberg and Papadopoulos [7]. In this model, a pseudo-rigid body impacts normally on

    a stationary rigid foundation under the influence of a uniform gravitational field g. The

    body is taken to be initially spherical with outer radius R, mass density ρ0, volume V0,

    and total mass m. The material is assumed homogeneous, isotropic, and linearly elastic

    with Young’s modulus E and Poisson’s ratio ν.

    Solberg and Papadopoulos [7] showed that at impact the body is subject to a normal

    indeterminate force acting at the material point of contact with the rigid foundation. To

    eliminate this indeterminacy, they assumed that the impact is ideally elastic, in the sense

    that it conserves the total energy of the pseudo-rigid body. Between any two consecutive

    impacts, the motion of the center of mass is that of a point mass in a constant gravita-

    tional field, while the body undergoes free vibration relative to its undeformed spherical

    configuration. Due to the nature of the loading, only three of the six modes of vibration

    are excited by impact. Of these, the two deviatoric modes have identical eigenvalues due

    to material and geometric symmetries. In an attempt to further simplify the analysis, it is

    now assumed that Poisson’s ratio is equal to zero. As a result, a single vibration mode is

    excited by impact, namely the one in the direction of the impact force. This reduces the

    system to two-degrees of freedom, one associated with the rectilinear motion of the mass

    center and another with the excited mode of vibration.

    For concreteness, the reference configuration is defined to be that of the undeformed

    sphere in contact with the rigid foundation, as shown in Figure 1(a). The infinitesimal

    strain due to the deformation of the ball relative to its reference configuration is denoted

    by u and is geometrically interpreted in Figure 1(b). In addition, the position of the mass

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  • Pseudo-rigid sphere impact

    center x is measured in the direction of gravity from a point O placed at the center of the

    sphere in the reference configuration, see Figure 1(c).

    2.2 Equations of Motion

    A Lagrangian approach is used to formulate the equations of motion for the two-degree of

    freedom pseudo-rigid sphere. The resulting equations agree with those derived in [7] using

    the momentum balances for the pseudo-rigid body.

    Apart from impact, the two degrees of freedom x and u are uncoupled and a Lagrangian

    L = T − V (1)

    is proposed, where T is the total kinetic energy of the system,

    T =1

    2mẋ2 +

    1

    2E0u̇

    2 , (2)

    and V is the total potential energy,

    V = −mgx +1

    2EV0u

    2 . (3)

    In equation (2), E0 is given by

    E0 =

    V0

    ρ0r2dv , (4)

    where r is the radial coordinate to a material point of the sphere in the reference configu-

    ration2. Starting from (4), it can be readily verified that

    E0 = kmR2 , (5)

    where k is a dimensionless parameter satisfying 0 ≤ k ≤ 1. The extreme values k = 0

    and k = 1 correspond, respectively, to the mass being concentrated at the center r = 0

    and the exterior boundary r = R.

    The motion of the sphere is subject to two constraint conditions. The first precludes

    self-penetration and can be expressed as

    φ1(u) = u > −1 . (6)

    This constraint will be enforced at the outset by the appropriate choice of initial conditions,

    see Section 3. The second constraint, written in the form

    φ2(x, u) = x + Ru ≤ 0 , (7)

    2It can be shown that for spherically symmetric bodies, the component matrix of the referential Euler

    tensor is diagonal with entries equal to E0.

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  • E. Kanso and P. Papadopoulos

    ensures that the sphere does not penetrate the rigid foundation. Impact events occur at

    t = ti, i = 1, 2, . . ., when condition (7) is satisfied as an equality. Also, continuity of x

    and u in time necessitates that the pre-impact velocity of the material point of contact be

    positive, namely that

    ẋ(t−i ) + Ru̇(t−

    i ) > 0 . (8)

    Employing the Lagrange multiplier method, it is concluded that the forces needed to

    impose the constraint condition (7) as a strict equality at t = ti must be of the form

    Fx = − p∂φ2

    ∂x= −p δ(t − ti) ,

    Fu = − p∂φ2

    ∂u= −R p δ(t − ti) ,

    (9)

    where p is to be determined. Notice that the negative sign is used in (9) only for con-

    venience. Invoking Lagrange’s equations and taking into account (1)-(3) and (9) leads

    to

    mẍ − mg = − p δ(t − ti) ,

    E0ü + EV0u = − Rp δ(t − ti) .(10)

    Integrating equations (10) in the time interval (t−i , t+

    i ) yields the jump conditions

    mẋ+i = mẋ−

    i − pi ,

    E0u̇+

    i = E0u̇−

    i − Rpi ,(11)

    where (·)+i = (·)(t+

    i ) and (·)−

    i = (·)(t−

    i ), while pi =∫ t+

    i

    t−i

    pδ(t − ti) dt. The Lagrange

    multiplier pi can be determined by further assuming that the total energy E = T + V of

    the system is conserved during impact. Since the potential energy V exhibits no jumps,

    energy conservation during impact reduces to

    1

    2m(ẋ+i )

    2 +1

    2E0 (u̇

    +

    i )2 =

    1

    2m(ẋ−i )

    2 +1

    2E0 (u̇

    i )2 . (12)

    Substituting (11) into (12) yields the non-trivial solution

    pi =2k

    1 + km(ẋ−i + Ru̇

    i ) . (13)

    As already noted in [7], the multiplier pi is independent of Young’s modulus. By virtue

    of (13), the post-impact velocities ẋ+i and u̇+

    i can be obtained from (11) as functions of

    the pre-impact velocities ẋ−i and u̇−

    i in the form

    ẋ+i =1 − k

    1 + kẋ−i −

    2k

    1 + kRu̇−i ,

    u̇+i = −2

    1 + k

    1

    Rẋ−i −

    1 − k

    1 + ku̇−i .

    (14)

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  • Pseudo-rigid sphere impact

    Clearly, the impact events couple the response of x and u by redistributing kinetic energy

    between the rectilinear motion of the mass center and the free vibration of the sphere.

    The preceding system can be non-dimensionalized by introducing the set of variables

    t = ωt , x =x

    R, u = u , (15)

    where ω is the natural frequency of the sphere given by

    ω =

    EV0

    E0. (16)

    The dimensionless Lagrangian

    L = T − V (17)

    can be obtained from equations (1)-(3) by invoking (15). Here, T and V are the dimen-

    sionless kinetic and potential energies of the system expressed respectively as3

    T =1

    2ẋ

    2 +1

    2ku̇2 (18)

    and

    V = −gx +1

    2ku2 , (19)

    where

    g =g

    ω2 R. (20)

    The inequality constraint (7) is now expressed as

    x(t) + u(t) ≤ 0 , (21)

    while condition (8) on the pre-impact velocity of the contact point reduces to

    ẋ−

    i + u̇−

    i > 0 . (22)

    Dimensionless counterparts of the equations of motion (10) are similarly deduced in the

    form

    ẍ− g = − p δ(t − ti) ,

    ü + u = −1

    kp δ(t − ti) ,

    (23)

    where

    p =p

    mω2R. (24)

    3Note that all time derivatives of dimensionless variables are taken relative to dimensionless time�.

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  • E. Kanso and P. Papadopoulos

    As in the dimensional case, the impulse pi at t = ti ensures conservation of the total energy

    E = T+V in (t−i , t+

    i ). Also, in correspondence to (14), the dimensionless equations relating

    post- to pre-impact velocities are written as

    ẋ+

    i =1 − k

    1 + kẋ−

    i −2k

    1 + ku̇−i ,

    u̇+i = −2

    1 + kẋ−

    i −1 − k

    1 + ku̇−i .

    (25)

    3 Impact Map

    The solution of (23) in the time interval (ti, ti+1) between two successive impacts is

    x(t) = xi + ẋ+

    i (t − ti) +1

    2g (t − ti)

    2 ,

    u(t) = ui cos(t − ti) + u̇+

    i sin(t − ti) .

    (26)

    Therefore, it suffices to study how a point in the phase space at time t−i maps to the

    corresponding point at t−i+14. To this end, let τi+1 = ti+1 − ti denote the time to next

    impact. With the aid of (25)1, one can readily conclude from (26) that

    τi+1 =1

    g

    [

    ẋ−

    i+1 −1 − k

    1 + kẋ−

    i +2k

    1 + ku̇−i

    ]

    . (27)

    Appealing to condition (21) in equality form, it is evident that points of the phase space

    at t−i , i = 1, 2, . . ., can be thought of as belonging to a three-dimensional Euclidean space

    with coordinates (x, ẋ−, u̇−) 5. Taking into account (25) and (27), equations (26) lead to

    g (x−i+1 − xi) =1

    2(ẋ−i+1)

    2 −1

    2

    (1 − k

    1 + kẋ−

    i +2k

    1 + ku̇−i

    )2

    ,

    x−

    i+1 = xi cos[1

    g

    (

    ẋ−

    i+1 −1 − k

    1 + kẋ−

    i +2k

    1 + ku̇−i

    )

    ]

    +( 2

    1 + kẋ−

    i +1 − k

    1 + ku̇−i

    )

    sin[1

    g

    (

    ẋ−

    i+1 −1 − k

    1 + kẋ−

    i +2k

    1 + ku̇−i

    )

    ]

    ,

    u̇−i+1 = xi sin[1

    g

    (

    ẋ−

    i+1 −1 − k

    1 + kẋ−

    i +2k

    1 + ku̇−i

    )

    ]

    −( 2

    1 + kẋ−

    i +1 − k

    1 + ku̇−i

    )

    cos[1

    g

    (

    ẋ−

    i+1 −1 − k

    1 + kẋ−

    i +2k

    1 + ku̇−i

    )

    ]

    .

    (28)

    The map (28) is smooth and invertible with a smooth inverse, hence is a diffeomorphism.

    4Equivalently, one may choose to study the mapping from time� +i

    to� +i+1.

    5An alternative choice would be the set ( � , ẋ−, ˙� −), since it is clear from (21) that � = −x at impact.

    7

  • Pseudo-rigid sphere impact

    Since the total system energy is conserved throughout the motion, any trajectory of

    the impact map must lie on the two-dimensional ellipsoidal energy surface

    E(xi, ẋ−

    i , u̇−

    i ) =1

    2(ẋ−i )

    2 +1

    2k(u̇−i )

    2 − gxi +1

    2kx2i = constant . (29)

    In fact, the trajectories are restricted to only the semi-ellipsoid defined by the pre-impact

    condition (22). In addition, the constraint (6), in conjunction with the equality form of

    (21), implies that xi < 1, which places a restriction on the potential energy V at impact,

    expressed as

    V(xi) =1

    2kx2i − gxi <

    1

    2k − g . (30)

    The supremum of V is attained when the kinetic energy T tends to zero, in which case

    V → E. Thus, the restriction (30) can be equivalently stated as

    E <1

    2k − g . (31)

    4 Impact Dynamics

    4.1 Numerical results

    Trajectories of the impact map are numerically generated for given initial conditions

    (x0, ẋ−

    0, u̇−

    0). The initial conditions are chosen to satisfy (22), as well as (31). An impact

    point (xi+1, ẋ−

    i+1, u̇−

    i+1) of a given trajectory is calculated from (xi, ẋ−

    i , u̇−

    i ) by substituting

    (25) and (26) in (21) and solving it as an equality for time to next impact τi+1. The

    smallest positive root of this equation is computed using a variant of the bisection method.

    This guarantees that (22) is satisfied at impact, therefore the inequality constraint (21) is

    enforced throughout the motion of the sphere. Subsequently, τi+1 is substituted back in

    (26) to yield the impact point at ti+1.

    The phase space of the map is depicted in Figure 2 for E = 0.01, k = 0.2 and g = 0.04.

    For this choice of parameters, the map possesses one elliptic fixed point that corresponds

    to the periodic motion shown in Figure 3. A dense set of quasi-periodic solutions is formed

    around the elliptic fixed point and is contained inside the major island in Figure 2. Further,

    two sets of period-six points are found to exist outside this major island. The first set

    consists of elliptic points surrounded by the minor islands of dominantly regular behavior,

    while the second set consists of hyperbolic points located in the chaotic region between

    successive elliptic points. The elliptic period-six motion is illustrated in Figure 4. The

    8

  • E. Kanso and P. Papadopoulos

    period-six points are computed by composing the impact map six times with itself and

    solving the resulting non-linear algebraic system by a standard iterative method. The

    stability type of these points is assessed by examining the eigenvalues of the Jacobian matrix

    associated with the so-constructed map. An approximation to this matrix is obtained by

    substituting all partial derivatives by first-order differences.

    The stable and unstable manifolds of the heteroclinic hyperbolic period-six points in-

    tersect transversally, as illustrated in Figure 5. Thus, they prevent the quasi-periodic

    trajectories from further expanding around the elliptic fixed point of the map. Indeed, the

    major island is contained by two intersecting stable and unstable branches whose oscilla-

    tions remain small until they reach the vicinity of the hyperbolic points. A close-up to

    one of the hyperbolic points in Figure 6 reveals a detail of these intersecting stable and

    unstable branches. The other two intersecting branches undergo enormous stretching and

    oscillatory bending, as evident by the segments traced in Figure 5. These intersections

    indicate the existence of a hyperbolic invariant set of points on which the impact map

    (28) is topologically equivalent to a horseshoe map as guaranteed by the Smale-Birkhoff

    theorem 6. However, a formal proof of the existence of such a set is currently unavailable.

    As shown in [7], the sphere may experience a sequence of densely clustered impacts before

    it globally separates from the foundation, only to come again to contact due to gravity. In

    the chaotic region of Figure 2, the initial conditions for such successive clusters of impact

    appear to be completely uncorrelated.

    The globally connected character of the chaotic region in Figure 2 is asserted by a cal-

    culation of Lyapunov characteristic exponents of the map following the numerical method

    proposed by Benettin et al. [18]. Recall that the Lyapunov exponents measure the mean

    exponential rate of divergence of nearby trajectories, hence vanish identically for initial

    conditions resulting in regular trajectories. On the other hand, a region of connected chaos

    is characterized by a positive maximal Lyapunov exponent that is independent of the choice

    of chaotic trajectory. These exponents are defined by

    λj = limi→∞

    ln ‖JiJi−1 . . . J1vj‖ , (32)

    where vj , j = 1, 2, 3, is a unit vector in R3 with respect to the Euclidean norm ‖ · ‖ and Ji

    6For a formal statement and a proof of the theorem see [6, Theorem 5.3.5].

    9

  • Pseudo-rigid sphere impact

    is the Jacobian of the map (28), expressed as

    Ji =

    ∂xi+1∂xi

    ∂xi+1

    ∂ẋ−i

    ∂xi+1

    ∂ ˙� −

    i

    ∂ẋ−i+1

    ∂xi

    ∂ẋ−i+1

    ∂ẋ−i

    ∂ẋ−i+1

    ∂ ˙� −

    i

    ∂ ˙� −i+1

    ∂xi

    ∂ ˙� −i+1

    ∂ẋ−i

    ∂ ˙� −i+1

    ∂ ˙� −

    i

    . (33)

    The components of Ji are readily obtained by solving a system of nine linear algebraic equa-

    tions resulting from differentiating (28) with respect to xi, ẋ−

    i , and u̇−

    i , as in [19]. Figure 7

    shows the values of the Lyapunov exponents for three distinct trajectories that explore

    the chaotic sea in Figure 2. The three maximal values converge to approximately 0.235

    as the number of iterations increases. The two non-vanishing exponents are equal and

    opposite, as in the case of symplectic maps, see [20, Theorem 8]. Vanishing of the third

    exponent is due to the use of a three-dimensional form in describing a system that lives on

    a two-dimensional energy manifold.

    The presence of regular behavior in the system depends crucially on the energy E.

    Figure 8 depicts several trajectories for E = 0.001, k = 0.2 and g = 0.04 with dominantly

    regular behavior around the elliptic periodic points of the map. The regular behavior

    observed here, as well as in Figure 2, does not appear to depend continuously on initial

    conditions. Further, the regular trajectories appear to be progressively destroyed as the

    total energy increases.

    4.2 Symplecticity and quasi-periodicity

    The objective of this section is to establish the symplectic property of the impact map on

    any given two-dimensional manifold of constant energy.

    It is well-known that the flow in phase space resulting from Lagrange’s equations (23)

    away from impact is symplectic, see, e.g., [21, Section 3.5]. Indeed, given the one-form

    Θ � =∂L

    ∂ẋdx +

    ∂L

    ∂u̇du , (34)

    associated with the Lagrangian L(x, u, ẋ, u̇) on the phase space, the flow can be shown to

    conserve the two-form

    Ω � = −dΘ � . (35)

    Taking into account the specific form of the Lagrangian functional in (17-19), it is imme-

    diately seen that

    Θ � = ẋdx+ ku̇du , (36)

    10

  • E. Kanso and P. Papadopoulos

    and

    Ω � = dx ∧ dẋ + kdu ∧ du̇ , (37)

    where ∧ denotes the standard exterior product between forms.

    To deduce the symplectic property of the impact map, first note that the restriction

    Θ � ,i of the one-form Θ � to the subspace of impact points defined by the equality condition

    in (21), is given as

    Θ � ,i = (ẋi − ku̇i)dxi . (38)

    Interestingly, Θ � ,i is well-defined at impact, namely Θ+� ,i = Θ

    � ,i, as can be verified by

    appealing to (25). Likewise, the two-form Ω � , when restricted on the subspace of impact

    points, takes the form

    Ω � ,i = dxi ∧ (dẋi − kdu̇i) . (39)

    The impact map is now symplectic, in the sense that it conserves Ω � ,i on the energy

    surface defined by (29) in the subspace of impact points. This can be easily argued, since

    equation (25) guarantees that

    Ω+� ,i = Ω−

    � ,i , (40)

    while the Lagrangian structure of the system between two successive impacts implies that

    Ω−� ,i+1 = Ω+� ,i . (41)

    Hence, the condition

    Ω−� ,i+1 = Ω−

    � ,i (42)

    holds true and the symplectic property of the map on the two-dimensional energy manifold

    is established. This result is consistent with the findings in the recent work by Fetecau

    et al. [17], where Lagrangian flows subject to elastic impacts are shown to conserve a

    symplectic structure in a suitably extended phase space.

    The regular behavior exhibited by the impact map in the neighborhood of periodic

    points is remarkably similar to that of symplectic twist maps, such as the standard map [5].

    The presence of quasi-periodic trajectories in symplectic twist maps is well-understood

    within the context of the KAM theory [22]. Although the impact map cannot be readily

    categorized as a twist map, it is conjectured here that its symplectic property is responsible

    for the observed regular behavior.

    11

  • Pseudo-rigid sphere impact

    5 Closure

    The elastic impact of an pseudo-rigid sphere onto a rigid foundation gives rise to a piecewise

    differentiable Lagrangian system in which the impact loading induces a mixture of regular

    and chaotic response depending on the material parameters. The resulting impact map

    possesses a symplectic structure on any constant energy manifold despite the discontinuities

    in the momenta at impact. A rigorous justification for the existence of regular trajectories

    in the vicinity of the periodic points of the map along the lines of the KAM theory remains

    an outstanding question.

    Acknowledgments

    The authors would like to thank Professor Andrew Szeri of the University of California,

    Berkeley for useful discussions on various aspects of this research. Also, the authors are

    grateful to Professor Jerry Marsden of the California Institute of Technology for supplying

    them with preprints of his recent work on collision dynamics.

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    [13] H. Cohen and R.G. Muncaster. The Theory of Pseudo-rigid Bodies. Springer-Verlag,New York, 1988.

    [14] A.N. Kolmogorov. Preservation of conditionally periodic movements with small changein the Hamiltonian function. Dokl. Akad. Nauk. SSSR (Reprinted in MacKay andMeiss(1987)), 98:527–530, 1954.

    [15] V.I. Arnold. Proof of A. N. Kolmogorov’s theorem on the preservation of quasi-periodicmotions under small perturbations of the Hamiltonian. Usp. Mat. Nauk. SSSR, 18,no. 5:13–40, 1963.

    [16] J. Moser. Stable and Random Motions in Dynamical Systems. Princeton UniversityPress, Princeton, 1973.

    [17] R.C. Fetecau, J.E. Marsden, M. Ortiz, and M. West. Nonsmooth Lagrangian mechan-ics and variational collision integrators. (to appear), 2002.

    [18] G. Benettin, L. Galgani, A. Giorgilli, and J. Strelcyn. Lyapunov characteristic ex-ponents for smooth dynamical systems and for Hamiltonian systems; A method forcomputing all of them. Part 2: Numerical Application. Mecc., pages 21–31, 1980.

    [19] C.R. Oliveira and P.S. Goncalves. Bifurcations and chaos for the quasiperiodic bounc-ing ball. Phys. Rev. E, 56:4868–71, 1997.

    [20] G. Benettin, L. Galgani, A. Giorgilli, and J. Strelcyn. Lyapunov characteristic ex-ponents for smooth dynamical systems and for Hamiltonian systems; A method forcomputing all of them. Part 1: Theory. Mecc., pages 9–20, 1980.

    [21] R. Abraham and J.E. Marsden. Foundations of Mechanics. Addison-Wesley, Reading,2nd edition, 1978.

    [22] V.I. Arnold and A. Avez. Problèmes Ergodiques de la Mécanique Classique. Gauthier-Villars, Paris, 1967.

    13

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    PSfrag replacements

    (a) (b) (c)

    R OO

    u

    Ru

    x

    Figure 1: Pseudo-rigid ball impact: (a) reference configuration; (b) motion between twoconsecutive impacts; (c) typical impact event.

    14

  • −0.20

    0.20.4

    −0.4

    −0.2

    0

    0.2

    0.4

    −0.1

    0.1

    0.3

    PSfrag replacements

    x

    ẋ−

    u̇−

    Figure 2: Regular and chaotic trajectories of the map for E = 0.01, k = 0.2, and g = 0.04corresponding to 8 initial conditions and approximately 45,000 points.

    15

  • 0 2 4 6 8 10 12 14 16−0.4

    −0.3

    −0.2

    −0.1

    0

    0.1

    0.2

    PSfrag replacements

    −x(t)

    u(t)

    t

    −x, u

    Figure 3: Period-one solution corresponding to the elliptic fixed point for E = 0.01, k = 0.2,and g = 0.04.

    16

  • 0 10 20 30 40 50 60−0.5

    −0.4

    −0.3

    −0.2

    −0.1

    0

    0.1

    0.2

    PSfrag replacements

    −x(t)

    u(t)

    t

    −x, u

    Figure 4: Solution corresponding to the elliptic period-six points for E = 0.01, k = 0.2,and g = 0.04.

    17

  • −0.20

    0.20.4

    −0.4

    −0.2

    0

    0.2

    0.4

    −0.1

    0.1

    0.3

    PSfrag replacements

    x

    ẋ−

    u̇−

    Figure 5: Selected sections of the stable and unstable manifolds of the period-six hyperbolicpoints at E = 0.01, k = 0.2, and g = 0.04. For illustrative purposes, the sections aresuperposed on a subset of the regular and chaotic trajectories shown in Figure 2.

    18

  • 0.21 0.21160.0925

    0.0932

    PSfrag replacements

    x

    Figure 6: Detailed plot of the stable and unstable branches near a hyperbolic point forE = 0.01, k = 0.2, and g = 0.04, illustrated in green and blue, respectively.

    19

  • 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    x 105

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    −0.2

    −0.1

    0

    0.1

    0.2

    0.3

    PSfrag replacements

    Number of iterations

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    Figure 7: Lyapunov exponents corresponding to three distinct trajectories in the chaoticsea of Figure 2 for E = 0.01, k = 0.2, and g = 0.04.

    20

  • 0

    0.2

    0.4 −0.05

    0.05

    0.05

    0.15

    PSfrag replacements

    ẋ−

    x

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    Figure 8: Regular and chaotic trajectories of the map for E = 0.001, k = 0.2, and g = 0.04corresponding to 6 initial conditions and approximately 35,000 points.

    21