arxiv: 2604.18494 · v1 · submitted 2026-04-20 · ⚛️ physics.comp-ph

Recognition: unknown

Consistent control of energy dissipation in non-spherical particle contact via a structure-preserving formulation

Y. T. Feng

Authors on Pith no claims yet

Pith reviewed 2026-05-10 02:54 UTC · model grok-4.3

classification ⚛️ physics.comp-ph

keywords non-spherical particle contactenergy dissipationcoefficient of restitutiondamping formulationprojected contact dynamicsstructure-preservingcontact-point restitutiontranslational-rotational coupling

0 comments

The pith

The damping law for non-spherical particle contacts is fixed by the harmonic structure revealed in energy-phase transformed space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how energy dissipation can be controlled consistently in impacts between non-spherical particles. Spherical contact reduces to a simple one-dimensional oscillator, but non-spherical cases have evolving effective mass and stiffness plus coupling across translation, rotation, and tangential directions. Projecting the full particle dynamics onto the instantaneous contact degrees of freedom exposes a configuration-dependent system whose conservative energy admits an exact phase transformation. Aligning dissipation with that transformed harmonic structure produces the only admissible damping law. A reader cares because empirical damping laws used for spheres produce uncontrolled and geometry-dependent energy loss when applied to irregular particles in simulations of granular materials or collisions.

Core claim

By projecting the dynamics onto contact degrees of freedom, the interaction is governed by an instantaneous contact dynamics with a configuration-dependent projected mass and intrinsic translational-rotational coupling. Building on the exact energy-phase transformation for monotone conservative contact, consistent dissipation requires a unique damping structure aligned with the underlying contact energy. The admissible damping law is therefore fixed by the harmonic structure in transformed space, and the appropriate coefficient of restitution is the contact-point value e_cn while the total energy restitution e_E is a geometry-dependent outcome that includes coupling-induced energy transfer.

What carries the argument

The energy-phase transformation for monotone conservative contact, applied after projecting the full dynamics onto the instantaneous contact degrees of freedom.

If this is right

The admissible damping law is determined by the revealed harmonic structure rather than chosen empirically.
Numerical implementations control the contact-point restitution e_cn consistently across all impact configurations.
Observed variation in total energy restitution e_E is explained directly by the coupled translational-rotational dynamics and changing geometry.
Structure-preserving time integrators can now be constructed that maintain the derived dissipation properties for non-spherical particles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The projection approach may extend to simultaneous multi-contact events if the instantaneous contact frames can be composed without loss of the energy-phase property.
Experimental restitution measurements on non-spherical bodies should target the local contact-point value rather than integrated energy loss to match the theory.
Large-scale discrete-element codes could adopt the formulation to reduce spurious energy drift when simulating powders, rocks, or granular flows with irregular shapes.
Incorporating friction would require extending the energy-phase transformation to include tangential coupling while preserving the same structural constraints on damping.

Load-bearing premise

The exact energy-phase transformation derived for simple monotone conservative contact continues to hold after projection onto the coupled, configuration-dependent dynamics of non-spherical particles.

What would settle it

Numerical simulation of a smooth single-contact impact on an ellipsoid showing that the proposed damping law fails to keep the measured contact-point restitution e_cn constant when the impact angle or contact normal is varied.

Figures

Figures reproduced from arXiv: 2604.18494 by Y. T. Feng.

**Figure 2.** Figure 2: Aligned impact (θ = 0◦ , Kn = 103 Pa). (a) Achieved versus target ecn. (b) Phase portrait for ecn = 0.5, showing the damped contact oscillation. Definition Value ecn (contact-point normal velocity) 0.496 eE (total energy) 0.714 Impulsive prediction for eE 0.717 % KE in rotation at exit 51.3% [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: provides detailed diagnostics of this case. Panel (a) shows the energy evolution, with the total energy decreasing by the prescribed amount while the kinetic energy at separation exceeds the target due to rotational energy. Panel (b) displays the breathing mass m∗ n (t) during contact, which starts at its lower bound of 0.648 m and increases as the ellipsoid rotates toward alignment. Panel (c) tracks the… view at source ↗

**Figure 4.** Figure 4: Breathing mass effect on restitution accuracy. (a) Achieved [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 5.** Figure 5: Robustness of ecn control across the parameter space (Kn = 103 Pa, target ecn = 0.5). (a) Aspect ratio sweep at θ = 30◦ : ecn (blue) stays near the target while eE (red) increases with coupling. (b) Impact angle sweep at a/b = 1.67: ecn (blue), eE from compliant simulation (red), and eE from the impulsive formula (black triangles). Kn (Pa) v + z error ω + y error δmax (m) 104 2.40% 0.23% 0.0050 105 0.75% 0… view at source ↗

**Figure 6.** Figure 6: Convergence to the impulsive limit (ecn = 1, θ = 30◦ ). (a) Post-impact v + z and ω + y versus Kn, with impulsive values indicated by dashed lines. (b) Relative errors on a log-log scale, showing O(K −1/2 n ) convergence. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗

**Figure 7.** Figure 7: displays these results, plotting eE against target ecn for four impact angles. The θ = 0◦ curve lies on the diagonal, confirming exact control. The oblique curves lift progressively, with the gap between each curve and the diagonal representing the coupling offset at that angle. Note that the curves for θ = 30◦ and 45◦ virtually coincide (see [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

read the original abstract

The control of energy dissipation in non-spherical particle contact remains an unresolved problem. Unlike spherical contact, where the interaction reduces to a one-dimensional normal oscillator, both the effective inertia and the effective stiffness depend on the evolving contact geometry, and the impact dynamics are intrinsically coupled across translational, rotational, and tangential directions. Classical damping formulations are therefore structurally incompatible with the contact dynamics they are intended to represent. This work addresses the problem from first principles. By projecting the dynamics onto contact degrees of freedom, the interaction is shown to be governed by an instantaneous contact dynamics with a configuration-dependent projected mass and intrinsic translational--rotational coupling. Building on the exact energy--phase transformation for monotone conservative contact, we show that consistent dissipation requires a unique damping structure aligned with the underlying contact energy. The analysis leads to two central consequences. First, the admissible damping law is not empirical but fixed by the harmonic structure revealed in transformed space. Second, the appropriate coefficient of restitution for non-spherical particles is the contact-point restitution $e_{cn}$, whereas the total energy restitution $e_E$ is a geometry-dependent outcome that includes coupling-induced energy transfer. Numerical evidence based on smooth single-contact impacts confirms the theory: the resulting formulation controls $e_{cn}$ consistently across impact configurations, while the apparent variability of $e_E$ follows directly from the coupled dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper projects non-spherical contact dynamics and applies an energy-phase transformation to fix a damping law, but the configuration-dependent inertia during impact leaves the uniqueness claim open to question.

read the letter

The central move is to reduce the rigid-body equations to instantaneous contact coordinates, where both effective mass and stiffness vary with geometry and translation couples to rotation. From there the authors invoke an energy-phase transformation to argue that dissipation must follow a specific structure tied to the contact energy rather than an arbitrary dashpot term. This produces the claim that you should prescribe the contact-point restitution e_cn and treat the total energy restitution e_E as a derived, geometry-dependent quantity that includes coupling losses. That distinction is the clearest practical takeaway and it is new relative to standard DEM damping models that treat the contact as a simple 1D oscillator. The numerical checks on smooth single impacts show that e_cn stays consistent across different approach angles while e_E varies as expected from the coupling, which is useful evidence for the subfield. The derivation starts from first principles without fitted parameters, which is a plus. The soft spot is exactly the one raised in the stress test. The transformation that yields the harmonic structure was originally derived for systems with constant inertia. Here the projected mass matrix changes with the evolving contact geometry, so the reduced equations are no longer autonomous in the required sense. It is not obvious from the abstract how the uniqueness of the damping law survives that time dependence without extra steps or approximations. The tests are limited to single smooth contacts, so they do not yet address whether the same structure holds under multiple contacts or rough surfaces. This work is for people who run discrete-element simulations of non-spherical grains and care about energy balance. A reader who already knows the classical damping problems in DEM will see the point immediately and can judge the math for themselves. It is worth sending to peer review because the problem is real, the approach is grounded, and the central claim is falsifiable with existing codes. A referee can check whether the variable-inertia issue is handled rigorously in the full derivation.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a first-principles derivation for consistent energy dissipation control in non-spherical particle contacts. Rigid-body dynamics are projected onto instantaneous contact coordinates, yielding a reduced system with configuration-dependent projected mass, stiffness, and translational-rotational coupling. Building on an energy-phase transformation previously derived for monotone conservative contact, the authors obtain a unique damping structure aligned with the contact energy. This leads to the claims that the admissible damping law is fixed by the resulting harmonic structure (not empirical) and that the contact-point restitution e_cn is the appropriate controlled quantity, while total energy restitution e_E is a derived, geometry-dependent outcome. Numerical tests on smooth single-contact impacts are said to confirm consistent control of e_cn.

Significance. If the energy-phase transformation extends rigorously to the variable-inertia projected dynamics, the result would be significant for computational granular mechanics: it replaces ad-hoc damping with a structure-preserving law that distinguishes intrinsic contact restitution from observable total-energy loss arising from coupling. This could improve predictive accuracy in DEM simulations of irregular particles without introducing free parameters. The projection approach and the explicit separation of e_cn from e_E are conceptually strong contributions if the central derivation holds.

major comments (2)

[Abstract and central derivation] Abstract, central derivation paragraph: the admissible damping law is obtained by invoking the exact energy-phase transformation for monotone conservative contact to produce a harmonic structure in transformed variables. However, the projected non-spherical dynamics have a configuration-dependent (time-varying during impact) effective mass matrix, as explicitly stated in the abstract ('configuration-dependent projected mass' and 'evolving contact geometry'). The original transformation assumes constant inertia to guarantee autonomy and uniqueness of the aligned damping; with variable inertia the reduced equations are non-autonomous, so the uniqueness and structural alignment of the damping law are not guaranteed by the same argument. This is load-bearing for the claim that dissipation control is 'fixed by the harmonic structure' rather than empirical.
[Numerical evidence] Numerical evidence paragraph: confirmation is restricted to smooth single-contact impacts. While these tests may show control of e_cn for the chosen cases, they do not address whether the damping structure remains consistent when projected inertia variation is strong (e.g., highly eccentric contacts or rapid geometry changes) or when multiple simultaneous contacts occur. The generalization asserted in the abstract therefore rests on an untested extension of the transformation.

minor comments (2)

[Abstract] The abstract introduces e_cn and e_E without a brief parenthetical definition; a short clarification of 'contact-point restitution' versus 'total energy restitution' would improve immediate readability.
[Abstract] The prior energy-phase transformation is referenced but not cited with a specific equation or paper number in the provided text; adding the reference at first mention would allow readers to verify the constant-inertia assumption directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments, which help clarify the scope and foundations of our derivation. We address the two major comments point by point below, indicating where revisions have been made to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract and central derivation] Abstract, central derivation paragraph: the admissible damping law is obtained by invoking the exact energy-phase transformation for monotone conservative contact to produce a harmonic structure in transformed variables. However, the projected non-spherical dynamics have a configuration-dependent (time-varying during impact) effective mass matrix, as explicitly stated in the abstract ('configuration-dependent projected mass' and 'evolving contact geometry'). The original transformation assumes constant inertia to guarantee autonomy and uniqueness of the aligned damping; with variable inertia the reduced equations are non-autonomous, so the uniqueness and structural alignment of the damping law are not guaranteed by the same argument. This is load-bearing for the claim that dissipation control is 'fixed by the harmonic structure' rather than empirical.

Authors: We appreciate the referee's identification of this subtlety. The energy-phase transformation is applied instantaneously to the projected contact dynamics at each configuration, treating the local effective mass matrix as fixed for the purpose of defining the harmonic oscillator structure and the unique damping that aligns with the contact energy. This local alignment fixes the damping law without requiring global autonomy. Nevertheless, we acknowledge that the original transformation was stated for constant inertia and that a more detailed justification for the non-autonomous case strengthens the uniqueness argument. In the revised manuscript we have inserted a clarifying paragraph in the central derivation section that explicitly discusses the instantaneous application, the conditions for local structural preservation, and why the admissible damping remains uniquely determined by the energy form even under slow configuration variation during impact. revision: partial
Referee: [Numerical evidence] Numerical evidence paragraph: confirmation is restricted to smooth single-contact impacts. While these tests may show control of e_cn for the chosen cases, they do not address whether the damping structure remains consistent when projected inertia variation is strong (e.g., highly eccentric contacts or rapid geometry changes) or when multiple simultaneous contacts occur. The generalization asserted in the abstract therefore rests on an untested extension of the transformation.

Authors: The numerical section deliberately isolates smooth single-contact events to provide clean verification of the predicted e_cn control across varying geometries. We agree that additional tests with rapid inertia changes or multiple simultaneous contacts would further support the generalization. The derivation itself is formulated per contact pair and extends directly to multi-contact situations by applying the same damping structure independently to each active contact while the rigid-body equations handle the overall coupling. In the revised manuscript we have added a short discussion paragraph noting this per-contact applicability, the assumptions involved, and the fact that comprehensive multi-contact validation lies beyond the present scope but follows the same structure-preserving principle. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation proceeds from projection and independent transformation

full rationale

The paper begins by projecting rigid-body dynamics onto instantaneous contact coordinates to obtain a reduced system whose effective mass and stiffness are configuration-dependent. It then invokes the energy-phase transformation for monotone conservative contact (previously established) to identify the admissible damping structure aligned with contact energy. No step equates a derived quantity to a fitted parameter by construction, renames an input as a prediction, or reduces the central claim to a self-citation chain whose validity depends on the present result. The distinction between contact-point restitution e_cn and total energy restitution e_E is presented as a direct geometric consequence of the coupled projected equations rather than a redefinition. Numerical evidence is used only for confirmation, not as the source of the damping law.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the energy-phase transformation to projected non-spherical dynamics and the existence of an instantaneous contact dynamics with configuration-dependent mass; limited details available from abstract.

axioms (2)

domain assumption monotone conservative contact admits an exact energy-phase transformation
Invoked to derive the admissible damping law from the harmonic structure in transformed space.
domain assumption dynamics can be projected onto contact degrees of freedom with configuration-dependent projected mass and translational-rotational coupling
Used as the starting point for showing the interaction is governed by instantaneous contact dynamics.

pith-pipeline@v0.9.0 · 5540 in / 1243 out tokens · 36390 ms · 2026-05-10T02:54:03.253371+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 9 canonical work pages · 1 internal anchor

[1]

P. A. Cundall, O. D. L. Strack, A discrete numerical model for granular assemblies, G´ eotechnique 29 (1979) 47–65

1979
[2]

K. H. Hunt, F. R. E. Crossley, Coefficient of restitution interpreted as damping in vibroimpact, Journal of Applied Mechanics 42 (2) (1975) 440–445

1975
[3]

H. M. Lankarani, P. E. Nikravesh, A contact force model with hysteresis damping for impact analysis of multibody systems, Journal of Mechanical Design 112 (3) (1990) 369–376

1990
[4]

Kruggel-Emden, E

H. Kruggel-Emden, E. Simsek, S. Rickelt, S. Wirtz, V. Scherer, Review and extension of normal force models for the discrete element method, Powder Technology 171 (3) (2007) 157–173

2007
[5]

Y. T. Feng, Thirty years of developments in contact modelling of non-spherical particles in DEM - a selective review, Acta Mechanica Sinica 39 (2023) 722343. doi:10.1007/s10409-022-22343-x. 22

work page doi:10.1007/s10409-022-22343-x 2023
[6]

Tsuji, T

Y. Tsuji, T. Tanaka, T. Ishida, Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe, Powder Technology 71 (1992) 239–250

1992
[7]

Labous, A

L. Labous, A. Rosato, R. Dave, Measurements of collisional properties of spheres using high-speed video analysis, Physical Review E 56 (1997) 5717–5725

1997
[8]

Lorenz, C

A. Lorenz, C. Tuozzolo, M. Y. Louge, Measurements of impact properties of small, nearly spherical particles, Experimental Mechanics 37 (1997) 292–298

1997
[9]

S. F. Foerster, M. Y. Louge, H. Chang, K. Allia, Measurements of the collision properties of small spheres, Physics of Fluids 6 (1994) 1108–1115

1994
[10]

Schwager, T

T. Schwager, T. P¨ oschel, Coefficient of restitution for viscoelastic spheres: The effect of delayed recovery, Physical Review E 78 (2008) 051304

2008
[11]

Kuwabara, K

G. Kuwabara, K. Kono, Restitution coefficient in a collision between two spheres, Japanese Journal of Applied Physics 26 (1987) 1230–1233

1987
[12]

N. V. Brilliantov, F. Spahn, J.-M. Hertzsch, T. P¨ oschel, Model for collisions in granular gases, Physical Review E 53 (5) (1996) 5382–5392

1996
[13]

Antypov, J

D. Antypov, J. A. Elliott, On an analytical solution for the damped hertzian spring, EPL 94 (2011) 50004

2011
[14]

Y. T. Feng, Exact phase-space analytical solution for the power-law damped contact oscillator (2026).doi:10.48550/arXiv.2603.27764

work page doi:10.48550/arxiv.2603.27764 2026
[15]

Y. T. Feng, Hidden harmonic structure, universal damping, and stability bounds in nonlinear contact dynamics (2026).doi:10.48550/arXiv.2604.02533

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.02533 2026
[16]

Gilardi, I

G. Gilardi, I. Sharf, Literature survey of contact dynamics modelling, Mechanism and Machine Theory 37 (10) (2002) 1213–1239.doi:10.1016/S0094-114X(02)00045-9

work page doi:10.1016/s0094-114x(02)00045-9 2002
[17]

W. J. Stronge, Impact Mechanics, Cambridge University Press, 2000

2000
[18]

R. M. Brach, Mechanical Impact Dynamics: Rigid Body Collisions, Wiley, 1991

1991
[19]

P. W. Cleary, Dem simulation of industrial particle flows: case studies of dragline excavators, mixing in tumblers and centrifugal mills, Powder Technology 199 (1) (2010) 33–40

2010
[20]

Podlozhnyuk, S

A. Podlozhnyuk, S. Pirker, C. Kloss, Efficient implementation of superquadric par- ticles in discrete element method within an open-source framework, Computational Particle Mechanics 4 (2017) 101–118

2017
[21]

G. Lu, J. R. Third, C. R. M¨ uller, Discrete element models for non-spherical parti- cle systems: From theoretical developments to applications, Chemical Engineering Science 127 (2015) 425–465

2015
[22]

Y. T. Feng, An energy-conserving contact theory for discrete element modelling of arbitrarily shaped particles: basic framework and general contact model, Computer Methods in Applied Mechanics and Engineering 373 (2021) 113454.doi:10.1016/ j.cma.2020.113454. 23

work page arXiv 2021
[23]

Y. T. Feng, An energy-conserving contact theory for discrete element modelling of arbitrarily shaped particles: contact volume based model and computational issues, Computer Methods in Applied Mechanics and Engineering 373 (2021) 113493.doi: 10.1016/j.cma.2020.113493

work page doi:10.1016/j.cma.2020.113493 2021
[24]

Y. T. Feng, Y. Tan, The minkowski overlap and the energy-conserving contact model for discrete element modeling of convex nonspherical particles, International Journal for Numerical Methods in Engineering 122 (2021) 6474–6496.doi:10.1002/nme. 6800

work page doi:10.1002/nme 2021
[25]

Featherstone, Rigid Body Dynamics Algorithms, Springer, 2014

R. Featherstone, Rigid Body Dynamics Algorithms, Springer, 2014

2014
[26]

Pfeiffer, C

F. Pfeiffer, C. Glocker, Multibody Dynamics with Unilateral Constraints, Springer, 1996

1996
[27]

Brogliato, Nonsmooth Mechanics, Springer, 1999

B. Brogliato, Nonsmooth Mechanics, Springer, 1999

1999
[28]

Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids, Edward Arnold, London, 1960

W. Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids, Edward Arnold, London, 1960

1960
[29]

F. G. Wolf, J. E. S. de Lima, M. A. S. S. da Costa, Angle-dependent restitution coef- ficient for collisions between non-spherical particles, Powder Technology 362 (2020) 1–10

2020
[30]

D. A. Gorham, A. H. Kharaz, The measurement of particle rebound characteristics, Powder Technology 112 (3) (2000) 193–202

2000
[31]

D. B. Hastie, Experimental measurement of the coefficient of restitution of irregular shaped particles, Chemical Engineering Science 101 (2013) 828–836

2013
[32]

Wilson, R

J. Wilson, R. Qiao, M. Kappes, J. Loebig, R. Clarkson, The importance of shape in particle rebound behaviors, ASME Journal of Turbomachinery 145 (4) (2023) 041005

2023
[33]

J. Wang, M. Zhang, L. Feng, H. Yang, Y. Wu, G. Yue, The behaviors of particle-wall collision for non-spherical particles: Experimental investigation, Powder Technology 363 (2020) 187–194.doi:10.1016/j.powtec.2019.12.041

work page doi:10.1016/j.powtec.2019.12.041 2020
[34]

H. Dong, M. H. Moys, Experimental study of oblique impacts with initial spin, Powder Technology 161 (1) (2006) 22–31.doi:10.1016/j.powtec.2005.09.001

work page doi:10.1016/j.powtec.2005.09.001 2006
[35]

J. E. Higham, P. Shepley, M. Shahnam, Measuring the coefficient of restitution for all six degrees of freedom, Granular Matter 21 (2) (2019) 15.doi:10.1007/ s10035-019-0871-0. 24

2019