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

Technical Summary: Characterization of Rocking Block Behaviors with Classical and Alternative Restitution Models

Problem Statement
The study addresses the discrepancy between experimentally observed impact behaviors in rocking blocks and the predictions of the classical Housner restitution coefficient. While the Housner model, which assumes conservation of angular momentum about the new pivot, has been a cornerstone for analyzing rigid-body rocking dynamics, it often fails to accurately capture impact responses, particularly for blocks with low slenderness ratios. Although alternative restitution formulations, such as that proposed by Mao et al., have been shown to align better with experimental data, their implications for the global nonlinear dynamics of rocking systems—specifically regarding complex oscillations, attractor stability, and overturning susceptibility—remain largely unexplored.

Methodology
The authors formulate the rocking block as a hybrid non-smooth dynamical system. The physical model consists of a homogeneous rectangular block of mass $M$, width $2b$, and height $2h$, rocking without sliding on a rigid foundation under harmonic base excitation. The equations of motion are derived from the balance of moments about the active pivot, incorporating dimensionless parameters for amplitude ($\alpha$), frequency ($\omega$), and viscous damping ($\xi$).

To analyze the system, the authors compare two restitution models:

1. Classical Housner Model ($e_H$): Derived from angular momentum conservation, dependent solely on geometry ($e_H = 1 - \frac{3}{2}\cos^2(\theta_c)$).
2. Alternative Mao et al. Model ($e_M$): Derived from kinetic-energy redistribution and vector-based velocity field analysis, which generally predicts lower energy dissipation and higher restitution values, especially for low-slenderness blocks.

The dynamical characterization employs three primary tools:

• Bifurcation Diagrams: To identify qualitative changes in asymptotic responses (periodic, multiperiodic, chaotic) as the forcing amplitude $\alpha$ varies.
• Maximum Lyapunov Exponents ($\lambda_{max}$): Computed using the Benettin–Wolf algorithm with saltation matrices to handle impact discontinuities, distinguishing between stable periodic responses ($\lambda_{max} < 0$) and chaotic dynamics ($\lambda_{max} > 0$).
• Basins of Attraction: To map the dependence of long-term dynamics on initial conditions $(\theta_0, \theta'_0)$, classifying outcomes into damped motion, sustained oscillations (symmetric, asymmetric, multiperiodic), and overturning ($|\theta| \ge \pi/2$).

Simulations were conducted for two slenderness ratios ($h/b = 1$ and $h/b = 2$) using Python 3 with implicit BDF integration.

Key Contributions and Results
The study demonstrates that the choice of restitution model significantly influences the predicted system response, particularly for low-slenderness blocks.

1. Divergence in Dynamical Regimes: For $h/b = 1$, the Mao et al. model predicts an earlier onset and greater prevalence of complex oscillatory behaviors compared to the Housner model. The alternative formulation shifts the transition toward irregular behavior to lower forcing amplitudes, exhibiting broader parameter intervals with scattered bifurcation structures and predominantly positive Lyapunov exponents.
2. Basin of Attraction Structure: The restitution model alters the global accessibility of bounded motions. For $h/b = 1$, the Housner model yields a broad, compact basin for symmetric oscillations, indicating robustness against initial condition perturbations. In contrast, the Mao et al. model produces a substantially narrower oscillatory basin with irregular boundaries embedded within a dominant overturning region. This fragmentation indicates increased sensitivity to initial conditions.
3. Quantitative Disagreement: For $h/b = 1$, the agreement between the two models regarding the final dynamical outcome for 10,000 initial conditions is only 64.09%. The Mao et al. formulation significantly increases the proportion of overturning events (80.36% vs. 44.45% for Housner) and reduces the domain of regular symmetric oscillations.
4. Convergence with Slenderness: As the slenderness ratio increases to $h/b = 2$, the dynamical features produced by both formulations progressively converge. The agreement in dynamical outcomes rises to 86.93%, and the differences in bifurcation structures and Lyapunov exponent distributions become less pronounced. This aligns with the convergence of the restitution coefficients $e_H$ and $e_M$ as $\theta_c$ increases.

Significance
The paper claims to provide a dynamical perspective on why alternative restitution formulations, which better match experimental impact data, produce markedly different global behaviors from the classical Housner model. The results suggest that the coefficient of restitution should not be viewed merely as a local impact parameter but as a modeling assumption that fundamentally alters attractor organization, stability boundaries, and overturning susceptibility. The study clarifies how impact dissipation interacts with geometric properties to determine the long-term response of rocking systems, highlighting that for low-slenderness blocks, the choice of restitution model is critical for accurate prediction of stability and complex dynamics.