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arxiv: 2606.14288 · v2 · pith:LLXRB3ZPnew · submitted 2026-06-12 · ⚛️ physics.optics · nlin.CD

Characterization of Rocking Block Behaviors with Classical and Alternative Restitution Models

Fernando Gaibor E. , Alexander L\'opez , Esther D. Guti\'errez This is my paper

Pith reviewed 2026-06-27 05:05 UTC · model grok-4.3

classification ⚛️ physics.optics nlin.CD
keywords rocking blocksrestitution modelshybrid dynamical systemsbifurcation analysisLyapunov exponentsbasins of attractionnon-smooth dynamicsslenderness ratio
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The pith

The choice of restitution model strongly influences the predicted dynamics of rocking blocks under harmonic excitation.

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

This paper compares how two different models of energy loss at impacts shape the overall motion of rocking blocks shaken by harmonic forces. It formulates the system as a hybrid non-smooth dynamical model and computes bifurcation diagrams, Lyapunov exponents, and basins of attraction. The alternative restitution formulation produces complex oscillations at lower excitation levels and more frequently than the classical Housner model, while also shifting the types, stability, and reachability of attractors. These differences become smaller as the block slenderness ratio grows, so the two models agree more closely for taller blocks.

Core claim

Formulating the rocking block as a hybrid non-smooth dynamical model and analyzing it via bifurcation diagrams, Lyapunov exponents, and basins of attraction shows that the alternative restitution formulation leads to an earlier onset and greater prevalence of complex oscillations as well as changes in the type, stability, and accessibility of attractors compared with the classical Housner model; the dynamical features produced by both formulations converge as the slenderness ratio increases.

What carries the argument

Comparison of the classical Housner restitution coefficient with the Mao et al. alternative formulation inside a hybrid non-smooth dynamical model of the rocking block.

Load-bearing premise

That the hybrid non-smooth formulation and the chosen numerical diagnostics fully capture and correctly classify all relevant behaviors across the parameter ranges examined.

What would settle it

Running controlled rocking-block experiments with measured impact restitution values and checking whether the observed onset of complex oscillations matches the alternative model's predictions more closely than the classical model's predictions.

read the original abstract

This work investigates how restitution modeling affects the dynamics of rocking blocks subjected to harmonic excitation. While several studies have reported discrepancies between experimentally observed impact behavior and the predictions obtained using the classical Housner restitution coefficient, the implications of adopting alternative restitution formulations on the global dynamics of rocking systems remain largely unexplored. The system is formulated as a hybrid non-smooth dynamical model and analyzed through bifurcation diagrams, Lyapunov exponents, and basins of attraction for different slenderness ratios. By comparing the classical restitution model proposed by Housner with the alternative formulation of Mao et al., we show that the choice of restitution model strongly influences the predicted system response. The alternative formulation leads to an earlier onset and greater prevalence of complex oscillations, as well as changes in the type, stability, and accessibility of attractors compared to the classical model. However, as the slenderness ratio increases, the dynamical features produced by both formulations progressively converge, indicating a reduced sensitivity to the restitution model for taller blocks. These results provide a dynamical perspective on why alternative restitution formulations, which predict impact responses closer to experimental observations, can produce markedly different behaviors from those obtained using the classical Housner model.

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.

Referee Report

2 major / 2 minor

Summary. The paper formulates rocking blocks under harmonic excitation as hybrid non-smooth systems and compares the classical Housner restitution coefficient against the Mao et al. alternative via bifurcation diagrams, Lyapunov exponents, and basins of attraction across slenderness ratios. It claims the alternative model produces earlier onset and greater prevalence of complex oscillations together with changes in attractor type, stability, and accessibility, while both models converge for larger slenderness ratios.

Significance. If the numerical diagnostics are shown to be accurate, the work supplies a concrete dynamical-systems explanation for why restitution models that better match experiments can yield qualitatively different global responses, with direct relevance to the modeling of rocking structures in seismic engineering.

major comments (2)
  1. [Lyapunov exponent calculation] The section describing the computation of Lyapunov exponents does not mention insertion of the saltation matrix at each impact event. In hybrid non-smooth systems the standard variational equations must be corrected by the saltation matrix to obtain the correct linearization across velocity discontinuities; omission of this step renders the reported exponents (and therefore the claimed differences in stability and the onset of complex oscillations) unreliable, especially near grazing or chattering regimes highlighted in the abstract.
  2. [Numerical diagnostics and results] The basins-of-attraction and bifurcation-diagram results are presented without reported checks on integration tolerance, event-detection accuracy, or comparison against an independent event-driven integrator. Because the central claim rests on quantitative differences between the two restitution maps, the absence of such verification leaves open the possibility that the reported attractor changes are sensitive to numerical implementation details.
minor comments (2)
  1. [Abstract] The abstract is information-dense; a short sentence stating the specific ranges of slenderness ratio and excitation amplitude examined would improve readability.
  2. [Model formulation] Notation for the two restitution coefficients should be introduced once with a clear equation reference rather than repeated in prose.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript to improve the description and verification of our numerical methods.

read point-by-point responses

  1. Referee: [Lyapunov exponent calculation] The section describing the computation of Lyapunov exponents does not mention insertion of the saltation matrix at each impact event. In hybrid non-smooth systems the standard variational equations must be corrected by the saltation matrix to obtain the correct linearization across velocity discontinuities; omission of this step renders the reported exponents (and therefore the claimed differences in stability and the onset of complex oscillations) unreliable, especially near grazing or chattering regimes highlighted in the abstract.

    Authors: We acknowledge the omission in the manuscript description. Our implementation follows the standard hybrid-system procedure and inserts the saltation matrix at each impact to linearize across velocity jumps. We will revise the Lyapunov-exponent section to include an explicit description of the saltation matrix, its formula, and its application within the variational equations. This addition will confirm that the reported exponents correctly account for discontinuities, particularly near grazing and chattering. revision: yes

  2. Referee: [Numerical diagnostics and results] The basins-of-attraction and bifurcation-diagram results are presented without reported checks on integration tolerance, event-detection accuracy, or comparison against an independent event-driven integrator. Because the central claim rests on quantitative differences between the two restitution maps, the absence of such verification leaves open the possibility that the reported attractor changes are sensitive to numerical implementation details.

    Authors: We agree that additional numerical verification details will strengthen the paper. We will add a short subsection reporting the integration tolerances, event-detection thresholds for impacts, and direct comparisons of selected bifurcation and basin results against an independent event-driven integrator. These checks will demonstrate that the reported differences between the two restitution models are robust to implementation details. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical comparison of two independent restitution models

full rationale

The paper formulates a hybrid non-smooth rocking-block model and performs direct numerical integration to generate bifurcation diagrams, Lyapunov exponents, and basins of attraction under two distinct restitution rules (Housner classical vs. Mao et al. alternative). No parameters are fitted from the target outputs, no predictions reduce to inputs by construction, and no self-citation chain or uniqueness theorem is invoked to justify the central comparison. The claimed differences in onset of complex oscillations and attractor properties follow from the explicit difference in the impact maps themselves, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the domain assumption that rocking-block motion is adequately described by a hybrid non-smooth system whose only non-smoothness is instantaneous impact governed by a scalar restitution coefficient; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Rocking-block motion is adequately described by a hybrid non-smooth system whose only non-smoothness is instantaneous impact governed by a scalar restitution coefficient.
    Explicitly stated as the modeling framework used for both restitution formulations.

pith-pipeline@v0.9.1-grok · 5742 in / 1230 out tokens · 26507 ms · 2026-06-27T05:05:49.218145+00:00 · methodology

discussion (0)

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Works this paper leans on

27 extracted references · 22 canonical work pages · 2 internal anchors

  1. [1]

    Buildings7(3), 69 (2017) https://doi.org/10.3390/buildings7030069

    Casapulla, C., Giresini, L., Louren¸ co, P.B.: Rocking and Kinematic Approaches for Rigid Block Analysis of Masonry Walls: State of the Art and Recent Devel- opments. Buildings7(3), 69 (2017) https://doi.org/10.3390/buildings7030069 . Accessed 2025-08-27

    work page doi:10.3390/buildings7030069 2017

  2. [2]

    Soil Dynamics and Earthquake Engineering181, 108649 (2024) https://doi.org/10.1016/j.soildyn.2024.108649

    Diamantopoulos, S., Fragiadakis, M.: Seismic response and fragility assessment of freestanding objects with random geometry. Soil Dynamics and Earthquake Engineering181, 108649 (2024) https://doi.org/10.1016/j.soildyn.2024.108649 . Accessed 2025-08-27

    work page doi:10.1016/j.soildyn.2024.108649 2024

  3. [3]

    Engineering Structures31(11), 2723–2734 (2009) https://doi.org/10.1016/j.engstruct.2009.06.021

    Di Egidio, A., Contento, A.: Base isolation of slide-rocking non-symmetric rigid blocks under impulsive and seismic excitations. Engineering Structures31(11), 2723–2734 (2009) https://doi.org/10.1016/j.engstruct.2009.06.021 . Accessed 2025-08-27

    work page doi:10.1016/j.engstruct.2009.06.021 2009

  4. [4]

    Shock and Vibration2020(1), 5498298 (2020) https://doi.org/10.1155/2020/5498298

    Alemzadeh, H., Shakib, H., Khanmohammadi, M.: Development of Rocking Isolation for Response Mitigation of Elevated Water Tanks under Seismic and Wind Hazards. Shock and Vibration2020(1), 5498298 (2020) https://doi.org/10.1155/2020/5498298 . eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1155/2020/5498298. Accessed 2026-03-10

    work page doi:10.1155/2020/5498298 2020

  5. [5]

    Nonlinear Dynamics112(21), 18745–18766 (2024) https://doi.org/10.1007/s11071-024-09998-7

    Ferretti, M., Di Egidio, A.: Effectiveness in protecting rigid-block-like objects through horizontal and vertical seismic isolation. Nonlinear Dynamics112(21), 18745–18766 (2024) https://doi.org/10.1007/s11071-024-09998-7 . Accessed 2026-01-14

    work page doi:10.1007/s11071-024-09998-7 2024

  6. [6]

    Bulletin of the Seismological Society of America53(2), 403–417 (1963) https://doi.org/10.1785/BSSA0530020403

    Housner, G.W.: The behavior of inverted pendulum structures during earth- quakes. Bulletin of the Seismological Society of America53(2), 403–417 (1963) https://doi.org/10.1785/BSSA0530020403 . Accessed 2025-06-12

    work page doi:10.1785/bssa0530020403 1963

  7. [7]

    Nonlinear Dynamics112(20), 17843–17862 (2024) https://doi.org/10.1007/s11071-024-09974-1

    D’Altri, A.M., Vlachakis, G., Miranda, S., Louren¸ co, P.B.: Rocking block simula- tion based on numerical dissipation. Nonlinear Dynamics112(20), 17843–17862 (2024) https://doi.org/10.1007/s11071-024-09974-1 . Accessed 2026-01-12

    work page doi:10.1007/s11071-024-09974-1 2024

  8. [8]

    Journal of Structural Engineer- ing142(12), 06016002 (2016) https://doi.org/10.1061/(ASCE)ST.1943-541X

    Kalliontzis, D., Sritharan, S., Schultz, A.: Improved Coefficient of Restitu- tion Estimation for Free Rocking Members. Journal of Structural Engineer- ing142(12), 06016002 (2016) https://doi.org/10.1061/(ASCE)ST.1943-541X. 0001598 . Accessed 2025-08-27

    work page doi:10.1061/(asce)st.1943-541x 2016

  9. [9]

    Bulletin of Earthquake Engineering15(5), 2305–2319 (2017) https://doi.org/10.1007/ s10518-016-0048-8

    Ther, T., Koll´ ar, L.P.: Refinement of Housner’s model on rocking blocks. Bulletin of Earthquake Engineering15(5), 2305–2319 (2017) https://doi.org/10.1007/ s10518-016-0048-8 . Accessed 2025-08-27 17

    2017

  10. [10]

    Buildings14(7), 2119 (2024) https: //doi.org/10.3390/buildings14072119

    Mao, Q., Deng, T., Shen, B., Wang, Y.: Research on Collision Restitution Coef- ficient Based on the Kinetic Energy Distribution Model of the Rocking Rigid Body within the System of Mass Points. Buildings14(7), 2119 (2024) https: //doi.org/10.3390/buildings14072119 . Accessed 2025-08-27

    work page doi:10.3390/buildings14072119 2024

  11. [11]

    Acta Mechanica Solida Sinica 37(5), 750–761 (2024) https://doi.org/10.1007/s10338-024-00464-w

    Jiang, J., Du, Z.: Strange Nonchaotic Attractors in a Quasiperiodically Excited Slender Rigid Rocking Block with Two Frequencies. Acta Mechanica Solida Sinica 37(5), 750–761 (2024) https://doi.org/10.1007/s10338-024-00464-w . Accessed 2025-04-03

    work page doi:10.1007/s10338-024-00464-w 2024

  12. [12]

    Chaos: An Interdisciplinary Journal of Nonlinear Science 31(7), 073136 (2021) https://doi.org/10.1063/5.0040962

    Liu, Y., P´ aez Ch´ avez, J., Brzeski, P., Perlikowski, P.: Dynamical response of a rocking rigid block. Chaos: An Interdisciplinary Journal of Nonlinear Science 31(7), 073136 (2021) https://doi.org/10.1063/5.0040962 . Accessed 2025-04-03

    work page doi:10.1063/5.0040962 2021

  13. [13]

    International Journal of Bifurcation and Chaos15(06), 1901–1918 (2005) https://doi.org/10.1142/ S0218127405013046

    Lenci, S., Rega, G.: Heteroclinic bifurcations and optimal control in the nonlin- ear rocking dynamics of generic and slender rigid blocks. International Journal of Bifurcation and Chaos15(06), 1901–1918 (2005) https://doi.org/10.1142/ S0218127405013046 . Accessed 2026-04-22

    1901

  14. [14]

    Physica D: Nonlinear Phenomena463, 134163 (2024) https://doi.org/10.1016/j.physd.2024.134163

    Guti´ errez, E.D., P´ aez Ch´ avez, J., L´ opez, A.: Rocking block stability under peri- odic and random perturbations. Physica D: Nonlinear Phenomena463, 134163 (2024) https://doi.org/10.1016/j.physd.2024.134163 . Accessed 2025-06-09

    work page doi:10.1016/j.physd.2024.134163 2024

  15. [15]

    (eds.): Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value problems

    Krauskopf, B., Osinga, H.M., Gal´ an-Vioque, J. (eds.): Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value problems. Understanding Complex Systems. Springer, Dordrecht (2007). https://doi.org/ 10.1007/978-1-4020-6356-5 .http://link.springer.com/10.1007/978-1-4020-6356-5 Accessed 2026-03-06

    work page doi:10.1007/978-1-4020-6356-5 2007

  16. [16]

    A Chapman & Hall book

    Strogatz, S.: Nonlinear Dynamics and Chaos: with Applications to Physics, Biol- ogy, Chemistry, and Engineering, Second edition, first issued in hardback edn. A Chapman & Hall book. CRC Press, Boca Raton London New York (2019)

    2019

  17. [17]

    Investigating the relationship between biodiversity and ecosystem multifunctionality: Challenges and solutions

    T¨ ornqvist, L., Vartia, P., Vartia, Y.O.: How Should Relative Changes be Measured? The American Statistician39(1), 43– 46 (1985) https://doi.org/10.1080/00031305.1985.10479385 . eprint: https://doi.org/10.1080/00031305.1985.10479385. Accessed 2026-03-16

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1080/00031305.1985.10479385 1985

  18. [18]

    In: Vidal, C., Pacault, A

    Feigenbaum, M.J.: Tests of the Period-Doubling Route to Chaos. In: Vidal, C., Pacault, A. (eds.) Nonlinear Phenomena in Chemical Dynamics, pp. 95–102. Springer, Berlin, Heidelberg (1981). https://doi.org/10.1007/978-3-642-81778-6 14

    work page doi:10.1007/978-3-642-81778-6 1981

  19. [19]

    In: Laurea, M.d.B., Champneys, A.R., Budd, C.J., Kowalczyk, P

    Bernardo, M.d., Champneys, A.R., Budd, C.J., Kowalczyk, P.: Bifurcations in general piecewise-smooth maps. In: Laurea, M.d.B., Champneys, A.R., Budd, C.J., Kowalczyk, P. (eds.) Piecewise-smooth Dynamical Systems: The- ory and Applications, pp. 171–217. Springer, London (2008). https://doi. 18 org/10.1007/978-1-84628-708-4 4 .https://doi.org/10.1007/978-1-...

    work page doi:10.1007/978-1-84628-708-4 2008

  20. [20]

    In: Kuznetsov, Y.A

    Kuznetsov, Y.A.: Bifurcations of Equilibria and Periodic Orbits in n- Dimensional Dynamical Systems. In: Kuznetsov, Y.A. (ed.) Elements of Applied Bifurcation Theory, pp. 175–228. Springer, Cham (2023). https://doi. org/10.1007/978-3-031-22007-4 5 .https://doi.org/10.1007/978-3-031-22007-4 5 Accessed 2026-03-09

    work page doi:10.1007/978-3-031-22007-4 2023

  21. [21]

    International journal of control55(3), 531–534 (1992)

    Lyapunov, A.M.: The general problem of the stability of motion. International journal of control55(3), 531–534 (1992). Accessed 2025-07-02

    1992

  22. [22]

    Skokos, C.: The Lyapunov Characteristic Exponents and their computation. vol. 790, pp. 63–135 (2010). https://doi.org/10.1007/978-3-642-04458-8 2 . arXiv:0811.0882 [nlin]. http://arxiv.org/abs/0811.0882 Accessed 2025-12-23

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978-3-642-04458-8 2010

  23. [23]

    Chaos, Solitons & Fractals5(9), 1671–1681 (1995) https://doi.org/ 10.1016/0960-0779(94)00170-U

    M¨ uller, P.C.: Calculation of Lyapunov exponents for dynamic systems with dis- continuities. Chaos, Solitons & Fractals5(9), 1671–1681 (1995) https://doi.org/ 10.1016/0960-0779(94)00170-U . Accessed 2025-12-23

    work page doi:10.1016/0960-0779(94)00170-u 1995

  24. [24]

    International Journal of Bifurcation and Chaos15(06), 2015–2039 (2005) https://doi.org/10.1142/ S0218127405013125

    Ageno, A., Sinopoli, A.: Lyapunov’s exponents for nonsmooth dynamics with impacts: stability analysis of the rocking block. International Journal of Bifurcation and Chaos15(06), 2015–2039 (2005) https://doi.org/10.1142/ S0218127405013125 . Accessed 2025-12-23

    2015

  25. [25]

    International Journal of Non-Linear Mechanics153, 104416 (2023) https://doi.org/10.1016/j.ijnonlinmec.2023.104416

    Frost, P., Cacciola, P.: Rocking of rigid blocks standing on a horizontally-moving compliant base. International Journal of Non-Linear Mechanics153, 104416 (2023) https://doi.org/10.1016/j.ijnonlinmec.2023.104416 . Accessed 2026-01-17

    work page doi:10.1016/j.ijnonlinmec.2023.104416 2023

  26. [26]

    Engineering Structures312, 118245 (2024) https://doi.org/10.1016/j.engstruct.2024.118245

    Frost, P., Cacciola, P.: Rocking of rigid non-symmetric blocks standing on a horizontally-moving compliant base. Engineering Structures312, 118245 (2024) https://doi.org/10.1016/j.engstruct.2024.118245 . Accessed 2025-08-27

    work page doi:10.1016/j.engstruct.2024.118245 2024

  27. [27]

    Journal of Compu- tational Physics229(8), 3019–3045 (2010) https://doi.org/10.1016/j.jcp.2009.12

    Dong, S.: BDF-like methods for nonlinear dynamic analysis. Journal of Compu- tational Physics229(8), 3019–3045 (2010) https://doi.org/10.1016/j.jcp.2009.12. 028 . Accessed 2026-03-02 19

    work page doi:10.1016/j.jcp.2009.12 2010