Hybrid Dynamics of Rocking Blocks Beyond Overturning: Saltation Analysis, Bifurcations, and Stability Characterization

Technical Summary: Hybrid Dynamics of Rocking Blocks Beyond Overturning

Problem Statement
Rocking blocks are fundamental models for understanding the seismic response of rigid structures such as masonry walls, statues, and elevated tanks. While the classical Housner model has long served as the cornerstone for analyzing these systems, recent studies have highlighted discrepancies between its predictions and experimentally observed impact behaviors. Specifically, the classical coefficient of restitution (COR), derived from angular momentum conservation, often fails to accurately capture the energy dissipation mechanics, particularly for blocks with low slenderness ratios. Although alternative formulations, such as the kinetic-energy redistribution model proposed by Mao et al., have been introduced to better match experimental data, the implications of adopting these alternative restitution laws on the global nonlinear dynamics of rocking systems remain largely unexplored. This work investigates how the choice of restitution modeling affects the system's bifurcation structure, stability, and basin of attraction organization, moving beyond the traditional focus on overturning thresholds to characterize complex oscillatory regimes.

Methodology
The study formulates the rocking block as a hybrid non-smooth dynamical system subjected to harmonic base excitation. The system is defined by a rectangular rigid body of mass $M$, width $2b$, and height $2h$, rocking without sliding. The dynamics are governed by piecewise differential equations that switch based on the angular displacement $\theta$, with impacts occurring at $\theta = 0$.

Two distinct restitution models are compared under identical geometric and excitation conditions:

1. Housner Model ($e_H$): The classical formulation where the COR depends solely on the block's geometry (critical angle $\theta_c$).
2. Mao et al. Model ($e_M$): An alternative formulation derived from a vector-based analysis of the velocity field, which accounts for elastic recovery and energy dissipation at the contact interface.

The analysis employs three primary numerical tools to characterize the dynamics across different slenderness ratios ($h/b = 1$ and $h/b = 2$):

• Bifurcation Diagrams: Constructed using Poincaré sections to identify transitions between periodic, multiperiodic, and chaotic regimes as the forcing amplitude ($\alpha$) varies.
• Lyapunov Exponents: The maximum Lyapunov exponent ($\lambda_{max}$) is computed using the Benettin–Wolf algorithm with saltation matrices to handle the discontinuities at impact, distinguishing between stable periodic motion ($\lambda_{max} < 0$) and chaotic dynamics ($\lambda_{max} > 0$).
• Basins of Attraction: The state space is discretized to map initial conditions to their asymptotic outcomes (damped motion, symmetric/asymmetric oscillations, multiperiodic responses, or overturning). This allows for a quantitative assessment of the robustness of bounded motions versus overturning susceptibility.

Key Results
The comparison reveals that the choice of restitution model significantly alters the predicted dynamical landscape, with effects that are highly dependent on the slenderness ratio:

• Divergence in Low-Slenderness Blocks ($h/b = 1$): The Mao et al. formulation predicts a higher coefficient of restitution (lower energy dissipation) compared to the Housner model. This leads to an earlier onset of complex oscillatory behavior and a broader parameter range associated with chaotic dynamics. Crucially, the basin of attraction analysis shows a stark difference: the Mao model drastically reduces the domain of stable symmetric oscillations (from 55.55% to 19.64% of initial conditions) and significantly increases the prevalence of overturning (from 44.45% to 80.36%). The agreement between the two models for this geometry is only 64.09%, meaning over one-third of initial conditions yield qualitatively different outcomes based solely on the impact law.
• Convergence in High-Slenderness Blocks ($h/b = 2$): As the slenderness ratio increases, the discrepancy between the two restitution coefficients diminishes. Consequently, the dynamical features produced by both models progressively converge. While the Mao model still introduces multiperiodic attractors within the bounded region, the overall organization of the phase space and the proportion of overturning events become more similar between the two formulations.
• Nature of Transitions: The Mao model promotes a more abrupt loss of regular periodic motion and shifts transitions toward complex regimes to lower forcing amplitudes. In contrast, the Housner model exhibits a more gradual transition with alternating windows of stability and chaos before overturning.

Significance and Claims
The paper posits that the coefficient of restitution should not be viewed merely as a local impact parameter but as a modeling assumption capable of fundamentally altering the global dynamical structure of rocking systems. The study demonstrates that alternative restitution formulations, while potentially more accurate in predicting local impact responses, can produce markedly different long-term behaviors compared to the classical Housner model.

Specifically, the authors claim that for low-slenderness blocks, the adoption of the Mao et al. formulation suggests a reduced robustness of safe rocking responses, where smaller perturbations in initial conditions are more likely to lead to overturning. Conversely, for taller blocks, the sensitivity to the specific restitution model decreases. These findings provide a dynamical perspective on why discrepancies exist between classical predictions and experimental observations, emphasizing that the treatment of impact dissipation is a critical factor in determining the stability boundaries and the accessibility of attractors in hybrid rocking systems. The work concludes that while the Mao model offers a closer approximation to experimental impact data, the idealized rigid-body assumptions inherent in both models limit their ability to capture complex contact mechanics like local deformation and frictional slip found in real structural systems.