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Repeated Binary Direct Collinear Impacts Under Incremental Contact Laws With Permanent Indentation: A Hybrid Systems Formulation

Mihails Milehins, Dan B. Marghitu

arXiv:2609.06138v3physics.class-phcs.SYeess.SY

TL;DR

When contact force vanishes while indentation remains, rigid-body impact models need separate rules for contact switching and the interface’s residual state. This paper formulates repeated collinear impacts as a hybrid system with those rules at the interface, establishes analytical properties, and reports numerical collision sequences with evolving contact and flight durations.

  • Problem

    Some incremental contact laws reach zero force at nonzero indentation, leaving deformation at separation that body equations alone do not represent.

    Source: §I · p. 1, B. Contact Interface · p. 8

  • Method

    The paper models the contact interface as a hybrid system carrying the contact law and state, coupled to the bodies through relative velocity and contact force.

    Source: §I · p. 2, B. Contact Interface · p. 10

  • Results

    The formulation establishes passivity and completeness under stated conditions, examines solution branching, and simulations show contact duration growing while flight duration diminishes.

    Source: §VII · pp. 23–24

  • Takeaways & Limitations

    The simulations exhibit repeated-impact behavior that an instantaneous-impact treatment cannot express, including changing contact and flight durations.

    Source: §VII · pp. 23–24

    The modeled bodies are convex and axisymmetric, move along one axis, meet at a single point, and have small indentation; simulation parameters are illustrative and not fitted to measurements.

    Source: §VII · p. 24

Abstract

from arXiv · show

Incremental contact laws specify the normal contact force through a differential equation carrying an internal state, driven by the indentation and its rate. In some, the force is extinguished at a nonzero indentation, whether by plastic deformation or by an elastic aftereffect, so that a residual deformation remains at the separation. Such laws sit uneasily within rigid body dynamics, which admits no deformation. The tension is tolerable when the indentation is small relative to the bodies, so that it may be carried constitutively rather than geometrically. Even then, the contact law alone does not determine the interaction of the bodies. Because force and indentation no longer vanish together, conditions for the commencement and termination of contact must be supplied separately. So must the fate of the deformation and internal state at separation, neither of which the equations of motion contain. This article formulates the repeated direct collinear impact of two convex bodies under external forces as a hybrid dynamical system. The contact interface is modeled as a massless element carrying the contact law and its state, coupled to the bodies through relative velocity and an interaction force dictated by the contact state. Consequently, all switching and resets are confined to the interface, leaving the geometry and the equations of motion of the bodies unaltered. The principal analytical properties of the resulting formulations are established, among them passivity and completeness; the branching of solutions at the onset and termination of contact is also examined. The framework is demonstrated through numerical simulations.

I. INTRODUCTION

The introduction motivates a hybrid-system formulation for repeated impacts when contact force vanishes before residual indentation does. The paper makes contact onset, separation, and interface-state treatment explicit, and places the model within a standard hybrid-system framework.

I. INTRODUCTION

  • Residual indentation at zero contact force can arise from yielding or delayed viscoelastic recovery, so it does not by itself identify plasticity.The key modeling distinction is whether indentation vanishes when force does, not which dissipation mechanism dominates.Source: §I · p. 1
  • Repeated contacts make the treatment of residual indentation and interface state consequential, while prior practices vary and are often left unspecified.The desired formalism places contact onset, termination, and state treatment on the same footing as the force law.Source: §I · p. 2

II. CONTRIBUTIONS AND OUTLINE

  • The model formulates repeated direct collinear impacts under arbitrary applied loads as a hybrid system, with switching and resets confined to the contact interface.The bodies’ equations of motion remain free of switching and resets, and their geometry is unaltered.Source: §I · p. 2, §II · p. 3
  • The analysis establishes passivity and conditional completeness, and examines solution branching at contact onset and termination.Completeness is established for continuous bounded applied forces when the contact law’s storage function confines its internal state.Source: §II · p. 3

A. Basics

  • Hybrid-system evolution tracks ordinary time during flows and increments a discrete counter at each transition.Inputs may be hybrid signals, though the external loads considered later are prescribed as functions of ordinary time.Source: A. Basics · pp. 3–4
  • The formulation uses hybrid systems, whose flow and jump sets govern continuous evolution and abrupt transitions, respectively.The framework represents a system through flow and jump sets, corresponding maps, and an output map.Source: A. Basics · pp. 3–4

B. Integration

The section defines integration over a hybrid time domain by accumulating contributions across its successive flow intervals. For inputs induced by a continuous ordinary-time signal, this hybrid integral reduces to the ordinary integral over elapsed time.

B. Integration

  • The integral over a hybrid time domain sums the integrals over its successive flow intervals.The domain is generated by an ordered sequence of interval endpoints.Source: B. Integration · p. 4
  • For a hybrid signal induced by a continuous ordinary-time input, its integral over a truncated hybrid domain is defined across the intervals it contains.The induced signal restricts the same continuous input to each flow interval.Source: B. Integration · p. 4
  • The interval contributions combine into the ordinary-time integral because the consecutive flow intervals cover the elapsed-time interval.This identifies the hybrid integral of an induced signal with integration over ordinary time.Source: B. Integration · p. 4

C. Dissipativity

The section defines dissipativity through a storage function and nonnegative dissipation during flows and jumps. It derives an energy balance and applies it to passive systems and storage growth.

C. Dissipativity

  • Dissipativity requires a nonnegative storage function whose flow and jump inequalities bound its change by a supply rate.The system is passive when its supply rate is the inner product of input and output.Source: C. Dissipativity · p. 5
  • The section defines supply and accumulated flow and jump dissipated energies over a hybrid trajectory.The energies are accumulated over continuous intervals and discrete transitions, respectively.Source: C. Dissipativity · p. 5
  • For a dissipative hybrid system, the change in storage plus flow and jump dissipation equals the supplied energy.The flow and jump dissipation terms are nonnegative, yielding an upper bound on storage change by supplied energy.Source: C. Dissipativity · p. 5
  • Under the stated storage-growth condition, storage remains bounded along a solution up to any attained hybrid time.The argument uses the dissipativity bound and the growth condition on the storage function.Source: C. Dissipativity · pp. 5–6

D. Stability

The section states a completeness criterion for maximal solutions and introduces the one-dimensional binary collision model. The collision setup represents the bodies by their relative displacement and center-of-mass motion under contact and external forces.

D. Stability

  • A maximal solution is complete when it starts in a forward-invariant set and remains bounded on every finite ordinary-time interval, under the stated hybrid basic conditions.The criterion also requires that the set of states reached by the solution up to each finite time be bounded.Source: D. Stability · p. 6

A. Binary Direct Collinear Collisions of Rigid Bodies

  • The collision model assumes compact, convex, axisymmetric bodies translating along their common axis, with contact force acting along that axis.External forces are continuous and bounded, and act along the axis without exerting moments.Source: A. Binary Direct Collinear Collisions… · pp. 6–7
  • Relative displacement is the signed gap: positive when bodies are apart, zero at touch, and negative when they overlap.The model’s contact detection can therefore use a single scalar quantity under these geometric constraints.Source: A. Binary Direct Collinear Collisions… · pp. 6–7
  • The model separates relative motion from center-of-mass motion, which evolves under the sum of the external forces.The relative-motion and center-of-mass descriptions together recover the individual bodies’ motion.Source: A. Binary Direct Collinear Collisions… · p. 7

B. Contact Interface

The contact interface is modeled as a hybrid subsystem that carries the incremental law’s indentation and internal state, receives relative velocity, and returns force to the bodies. Separate contact conditions and assumptions govern when the interface resets, allowing residual deformation to be represented without changing rigid-body geometry.

B. Contact Interface

  • The model assumes indentation is small relative to the bodies and that a later contact engages an unindented region or follows complete recovery.These assumptions let each contact be treated as a fresh loading and the interface be reset without recording deformation in body geometry.Source: B. Contact Interface · p. 8
  • Contact begins when the bodies meet while approaching and ends when force reaches zero as they recede, even if they still overlap.The termination criterion is force extinction, not the vanishing of the gap.Source: B. Contact Interface · p. 8
  • The interface keeps its own indentation, law state, and contact-regime flag because indentation can differ from relative displacement after separation with overlap.At termination it is restored to its undeformed configuration; a new contact begins with the interface undeformed.Source: B. Contact Interface · pp. 8, 10
  • The law is specified by continuous maps governing indentation, internal-state evolution, and force, with the interface initially undeformed and its internal state at rest.The force map must vanish at zero indentation and state for every approach rate, ensuring force continuity at contact onset; passivity is desirable but not required.Source: B. Contact Interface · p. 9
  • The interface exchanges relative velocity and force with the bodies, confining contact switching and state resets to the interface rather than altering the bodies’ equations.Its force is the negative of the force exerted on it by the bodies.Source: B. Contact Interface · pp. 8, 10
  • In the Bouc-Wen-Maxwell law, a dashpot is in series with elastic and hysteretic branches, and internal deformation continues at fixed indentation.The branch deflections sum to indentation, distinguishing this arrangement from the preceding law.Source: B. Contact Interface · p. 10

C. The Interconnected System

The interconnected system closes the body and interface models by passing relative velocity to the interface and returning its negative force to the bodies. The paper specifies the resulting system directly, while noting that a formal subsystem interconnection construction is not developed.

C. The Interconnected System

  • The body subsystem supplies relative velocity to the interface, whose returned force with opposite sign becomes the contact force acting on the bodies.The coupling is unambiguous because the contact force affects relative acceleration, not the velocity supplied to the interface.Source: C. The Interconnected System · p. 11
  • The system Hf is specified directly rather than derived through a formal hybrid-system interconnection of its subsystems.The formal apparatus for that construction is not developed, and the paper proves properties of the systems directly.Source: C. The Interconnected System · p. 11
  • The full system takes external forces on both bodies as inputs and their center-of-mass velocities as outputs.Its state includes body motion and the interface state; its dimensions depend on the contact law’s internal-state dimension.Source: C. The Interconnected System · p. 11
  • For an infinite-mass body, center-of-mass components can be deleted and the remaining relative-motion system Hr used.The center of mass then coincides with that of the heavier body.Source: C. The Interconnected System · p. 11

V. PROPERTIES

The formulated systems satisfy the stated hybrid basic conditions, and the paper establishes passivity results when the underlying contact law is passive. These results cover the contact interface and the reduced and full body systems.

A. Background

  • Hc, Hf, and Hr satisfy the hybrid basic conditions through closed flow and jump sets and continuous system maps.The proofs use the forms of the systems’ sets and the continuity of maps built from the contact-law maps and state projections.Source: A. Background · p. 12
  • For Hf and Hr, the set specified in Lemma V.4 is forward invariant.The supplied passages identify the result but do not state the set’s contents.Source: A. Background · p. 12

B. Passivity

  • If the contact law is passive, then the contact interface Hc is passive with a storage function derived from the law’s storage function.The proof bounds the storage change by input power using nonnegative contact-state and dissipation terms.Source: B. Passivity · p. 12
  • If the contact law is passive, both Hr and Hf are passive under the stated storage functions.The storage for Hf includes relative kinetic energy, center-of-mass kinetic energy, and the contact-law storage.Source: B. Passivity · p. 13

C. Confinement

Confinement is tied to a passive contact law whose storage bounds the internal state when indentation is bounded. The paper gives sufficient storage conditions and transfers confinement to bounded system trajectories over finite time.

C. Confinement

  • A passive contact law is confining when its storage grows without bound as the internal state grows at bounded indentation.Then bounded indentation and bounded storage imply a bounded internal state.Source: C. Confinement · p. 13
  • A storage lower bound that grows with the norm of indentation and internal state is a sufficient condition for confinement.The paper also gives a uniform sufficient condition using a class-K-infinity function.Source: C. Confinement · pp. 13–14
  • For Hr and Hf, boundedness of the stated trajectory sets through a finite time T yields bounded indentation and internal state over that interval.The propositions require a confining law and bounded sets specified in their hypotheses.Source: C. Confinement · p. 14

D. Existence

Under a confining contact law and bounded continuous inputs, both hybrid formulations have complete maximal solutions. The section also establishes dissipativity and notes that completeness does not rule out Zeno behavior.

D. Existence

  • For passive contact laws, Hf and Hr each have an output-bound result tied to the corresponding storage function.Source: D. Existence · p. 14
  • For continuous bounded inputs and initial states in S, maximal solution pairs of Hf and Hr are complete, assuming a confining contact law.Source: D. Existence · pp. 15–16
  • Hf and Hr are dissipative with respect to the input-output supply rate, using their respective storage functions.Source: D. Existence · pp. 15–16
  • Completeness excludes finite-time escape but does not exclude Zeno behavior, so a complete solution may attain only finite ordinary time.Source: D. Existence · p. 16

E. Non-Uniqueness

The formulations need not have unique solutions. Examples show branching at resting contact and separation, while the proposed selection rules are not imposed by the model and do not settle every case.

E. Non-Uniqueness

  • The formulations need not have unique solutions, as illustrated by complete branches at resting contact with different interface behavior.Source: E. Non-Uniqueness · p. 16
  • At separation after the indentation returns to zero, the interface may disengage or remain engaged without force while the bodies recede.Source: E. Non-Uniqueness · p. 17
  • The proposed rules are modeling choices, and branching at other flow-set and jump-set intersections remains unexamined, including their reachability and whether the rules ensure uniqueness.Source: E. Non-Uniqueness · p. 18

A. Background

The numerical section reports three studies spanning the model and contact laws with none, one, or two internal states. Each study reports one solution selected by continuing flow whenever possible.

A. Background

  • Three studies cover a reduced falling-ball system with BWMCL, a full two-ball system with BWSHCCL, and a reduced-system comparison with SHCCL.Source: A. Background · p. 18
  • The studies exercise the model with contact laws carrying none, one, and two internal states.Source: A. Background · p. 18
  • Flow intervals are integrated with SciPy's DOP853 using relative tolerance 10^-11 and absolute tolerance 10^-14, with tests checking hybrid-solution conditions and Euler convergence.Source: A. Background · p. 18
  • Each study reports one solution, with the implementation continuing to flow whenever possible under rule R2.Source: A. Background · p. 18

B. Bouncing Ball

The simulated ball–bitumen impacts evolve from short flights to longer contacts, then terminate in sustained contact and steady sinking. Across collisions, the observables change as the rate-dependent dissipation diminishes and the applied weight becomes increasingly relevant.

B. Bouncing Ball

  • The study simulates a 40 mm hardened steel ball dropped from 20 mm onto fixed bitumen under gravity, using the BWMCL over 0.5 s.The substrate is modeled as an infinite-mass viscoelastic fluid with no equilibrium modulus.Source: B. Bouncing Ball · p. 18
  • The chosen parameters demonstrate the formulation rather than predict material response; identifying them against measurements is outside the article’s scope.The parameters are selected to exhibit residual indentation, transition to permanent contact, and subsequent sinking.Source: B. Bouncing Ball · p. 20
  • The collision sequence terminates when contacts and flights become comparable, after which the ball remains in contact and sinks at rate mg/c.The interface has no configuration in which it supports the ball’s weight without flowing.Source: B. Bouncing Ball · pp. 20–21
  • Contact duration grows and flight duration falls; the coefficient of restitution first rises, then is nearly constant, before falling steeply at the last collision.The final decline reflects the weight impulse becoming no longer negligible beside the contact impulse.Source: B. Bouncing Ball · p. 21
  • The approach velocity, greatest and residual indentation, greatest contact force, and energy surrendered to the interface diminish from collision to collision.The ratio Λ approaches unity in magnitude at the last collisions, as the applied force’s impulse becomes increasingly important.Source: B. Bouncing Ball · p. 21

C. Repeated Binary Collisions

The repeated-impact simulations show contacts lengthening and flights shortening, with the sequence eventually giving way to persistent contact or continued flow, depending on the contact law. The formulation confines switching to an interface coupled to the bodies through contact force and relative velocity.

C. Repeated Binary Collisions

  • In repeated PTFE–polyoxymethylene collisions, indentation, residual indentation, peak force, and interface energy decrease; restitution first rises as approach speed falls, then falls when applied-force impulse becomes non-negligible.The residual indentation’s share of peak indentation becomes negligible by the last collision.Source: C. Repeated Binary Collisions · pp. 21, 23

D. The Approach to Permanent Contact

  • After the last separation, SHCCL approaches the elastic-rest configuration, BWSHCCL settles at an indentation where its hysteretic branch bears part of the weight, and BWMCL continues flowing without coming to rest.Under BWMCL, descent continues at mg/c.Source: D. The Approach to Permanent Contact · p. 23

VII. CONCLUSIONS AND FUTURE WORK

  • The hybrid formulation places the contact law and its state at an interface, leaving body equations free of switching and resets and specifying contact onset, termination, and state disposition at separation.The interface couples to the bodies through contact force and relative velocity; admitted laws need existence and uniqueness and zero force for an undeformed interface.Source: §VII · p. 23
  • Across the simulations, contacts lengthen and flights shorten until bodies spend less time apart than in contact, and the sequence ends when peak contact force is only a small multiple of applied force.The sequence terminates without collision accumulation; retained indentation’s share of peak indentation diminishes over successive impacts.Source: §VII · pp. 23–24
  • The study is limited to convex, axisymmetric bodies moving along one axis and meeting at one point, with small indentation and fresh material loading at each contact.The fresh-loading assumption is exact for a solid that recovers between contacts; simulation parameters are illustrative rather than fitted to measurements.Source: §VII · p. 24
  • The phenomenological interface model is least assured at low incident velocities, where its account diverges from the customary one near sequence termination and measurements are least able to distinguish them.For brief collisions with contact force large relative to applied force, the two accounts agree.Source: §VII · p. 24
  • The proposed next step is to extend the interface to three-dimensional rigid-body systems with multiple contact points and tangential forces, while retaining switching within each interface.The bodies’ equations of motion would retain their present form.Source: §VII · p. 24

APPENDIX A NOTATION, CONVENTIONS, FOUNDATIONS

The appendix sets out standard mathematical notation and conventions used in the article, including set, topology, real-number, function, and calculus notation. The paper also describes its human-in-the-loop use of generative AI in theoretical, computational, and textual work.

APPENDIX A NOTATION, CONVENTIONS, FOUNDATIONS

  • The appendix states that its definitions and results are standard across set theory, topology, analysis, ordinary differential equations, and nonlinear systems and control.It also defines K∞ and notation for gradients with respect to function arguments.Source: Appendix A · pp. 24–25
  • The appendix specifies set-operation notation, topological terms such as closure and connectedness, and conventions for real numbers, intervals, and Euclidean spaces.Source: Appendix A · p. 25
  • It defines conventions for functions and their domains, images and components, as well as integrals, inner products, norms, derivatives, and gradients.Source: Appendix A · p. 25
  • Some notational forms may have different meanings in context, such as (a, b) denoting either a pair or an interval.Source: Appendix A · p. 25

USE OF GENERATIVE ARTIFICIAL INTELLIGENCE (AI)

  • The authors report using generative AI in a human-in-the-loop process for theoretical, computational, textual, and conceptual development, while assuming full responsibility for the publication.Source: USE OF GENERATIVE ARTIFICIAL… · p. 25
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