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Contactgeometry and hysteretic damping in the rebound of a billiard ball from a cushion

This paper presents a three-dimensional contact model incorporating measured cushion geometry and Hunt–Crossley hysteretic damping to accurately predict billiard ball rebound dynamics, successfully explaining complex spin-dependent behaviors and speed-dependent energy dissipation that simpler coefficient-of-restitution models cannot capture.

Original authors: MKM Alharbi

Published 2026-09-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: MKM Alharbi

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

When a billiard ball strikes the padded rail of a pool table, the outcome is a delicate negotiation between geometry and material science. The ball does not simply bounce off a hard wall; it compresses a soft, rubbery cushion, deforms, and then springs back. This interaction determines where the ball will travel next, a factor that professional players rely on for precision. For decades, scientists and engineers have tried to predict this bounce using a single number, known as the coefficient of restitution, which essentially measures how much speed is lost during the collision. However, this simple number fails to capture the full reality of the event. It treats the cushion as a static object and ignores how the rubber's internal friction changes as the ball hits it faster or slower. In the complex world of table sports, where a player might strike the ball with spin or at a sharp angle, these missing details matter immensely. Understanding exactly how the rubber deforms and dissipates energy is the key to unlocking a true prediction of the ball's path.

A researcher at the University of Canberra has taken a fresh look at this problem, moving beyond simple numbers to build a detailed physical model of the collision. Instead of assuming the cushion is a flat, unchanging surface, the study begins with the actual shape of the rubber rail. By tracing the manufacturer's specifications, the researcher mapped the precise, wedge-like profile of the cushion's nose. When a ball hits this shape, the contact area is not a circle, as it would be on a flat wall, but a long, narrow strip. The study found that because of this specific geometry, the cushion is significantly less stiff than a flat surface would be, allowing the ball to sink deeper and interact with the rubber in a more complex way. This geometric detail alone changes the physics of the bounce, proving that the shape of the rail is just as important as the material it is made from.

The investigation then turned to the rubber itself. The researcher tested whether the standard laws of physics, which describe how rubber absorbs energy, could explain why the ball loses more speed as it hits the rail harder. The study calculated a theoretical limit for how much speed dependence any standard rubber model could produce. The results were clear: the measured behavior of the billiard ball exceeded this limit by a factor of five. In other words, the way the ball actually bounces cannot be explained by the usual models of rubber elasticity, no matter how many different types of rubber molecules are mixed together. The existing theories simply cannot account for the sharp increase in energy loss observed when the ball strikes the cushion at higher speeds.

To solve this puzzle, the researcher introduced a different approach called hysteretic damping. This concept treats the rubber not just as a spring that stores energy, but as a material that generates internal friction as it deforms. By adding a single, calibrated constant to the model to represent this friction, the researcher was able to match the real-world data with remarkable accuracy. The model successfully reproduced the results of twenty-four different shots, predicting the rebound speed and angle with an average error of only 3.3 percent. Crucially, this single adjustment allowed the model to work across a wide range of speeds without needing to be re-tuned for each new shot. The calculated properties of the rubber implied by this model align with known values for natural rubber, suggesting that the physics of the bounce is indeed driven by this internal friction.

The power of this new model becomes most apparent when considering spin. Previous models often struggled to predict what happens when a ball hits the rail with sidespin or topspin. The new simulation, however, can predict the outcome for any combination of spin and angle without needing new data. It revealed surprising behaviors, such as a ball striking the rail with left sidespin near a perpendicular angle actually bouncing back toward the side it came from. This effect, where the ball departs on the side of its approach, is a phenomenon that simpler models, which rely on a single friction number, cannot capture. The study shows that the interaction is a three-dimensional event where the ball's rotation and the cushion's shape combine to create outcomes that defy simple intuition.

While the model is highly accurate for the specific conditions tested, the researcher acknowledges that it is not a final, complete solution. The model was calibrated using the same set of twenty-four shots it was tested against, meaning it has not yet been independently verified against a new, separate set of experiments. Furthermore, the specific behavior of the rubber at the exact moment of impact, which happens in a fraction of a second, has not been directly measured in a laboratory setting. The study suggests that the current understanding of rubber properties at these high speeds is incomplete. Nevertheless, the work provides a clear path forward. It demonstrates that to truly understand the game of billiards, one must look at the cushion as a complex, three-dimensional object made of a material that generates friction, rather than a simple, passive wall. This shift in perspective offers a more reliable way to predict the game's most critical moments.

Technical Summary: Contact Geometry and Hysteretic Damping in the Rebound of a Billiard Ball from a Cushion

Problem Statement
Current models for billiard and snooker cushion rebounds typically rely on a fitted coefficient of restitution (COR) that varies with speed or assumes a constant value. These approaches fail to account for the specific geometry of the cushion nose or the viscoelastic properties of the rubber. Previous studies have either treated the rail as rigid, assumed a sphere-plane contact, or modeled the cushion as a triangular body without resolving the actual deformation mechanics. Consequently, discrepancies arise, particularly at higher speeds, where measured energy loss exceeds what linear viscoelastic models can theoretically predict. Furthermore, existing models often lack the capability to predict rebound behavior under arbitrary spin conditions (longitudinal and sidespin) without re-fitting parameters.

Methodology
The study constructs a three-dimensional contact model based on the measured nose profile of a standard cushion and published rubber properties.

Contact Geometry: The cushion nose is digitized from manufacturer specifications, revealing an asymmetric wedge with a filleted tip. The contact region is modeled as a strip rather than a circle, utilizing an active-set boundary-element method for a Boussinesq half-space. A nose load factor ( $K_N$ ) is derived to relate the indentation to the load, accounting for the specific geometry.
Viscoelastic Bounds: The paper derives a theoretical upper bound on the speed dependence of energy dissipation for any linear viscoelastic model (represented as a sum of Maxwell elements). It establishes that for a Hertzian contact where duration scales as $V^{-1/5}$ , the dissipated fraction can increase at most as $V^{1/5}$ .
Hysteretic Damping: To address the measured speed dependence exceeding the linear viscoelastic bound, the model employs Hunt–Crossley hysteretic damping. The damping force is proportional to the elastic force ( $F = F_{el}(1 + \lambda \dot{d})$ ), introducing a single calibrated constant, $\lambda$ .
Spin and Traction: The tangential traction is solved across the contact strip using an annulus-based approach (following Maw et al.), resolving slip direction separately for each annulus to handle arbitrary longitudinal spin and sidespin.
Data Correction: A correction is applied to convert tracked rebound speeds (measured after the ball decelerates on the cloth) to separation speeds, accounting for fixed energy loss due to sliding friction.
Validation: The model is evaluated against 24 measured shots from previous literature (Mathavan et al.), covering a range of incident speeds and angles.

Key Contributions and Results

Geometric Stiffness Reduction: Solving the contact in 3D reveals that the measured cushion nose is 52–62% less stiff than a flat surface. The contact region is a strip (26.3 mm $\times$ 2.2 mm at median indentation) rather than a disc, and the effective load exponent is 1.62, deviating from the Hertzian 1.50.
Theoretical Bound Violation: The measured energy dissipation scales approximately as $V^{1.04}$ (or $V^{1.11}$ when corrected to separation speed). This exceeds the theoretical upper bound of $V^{1/5}$ derived for any linear viscoelastic relaxation spectrum. This finding implies that standard linear viscoelastic models (including Prony series and constant COR models) are fundamentally insufficient to describe the physics of the rebound across the full speed range.
Hysteretic Damping Calibration: A single hysteretic damping constant ( $\lambda = 0.0925 \text{ s m}^{-1}$ ) successfully reproduces the measured speed dependence. This value corresponds to a rebound resilience of 72–79% at standard pendulum striker speeds, which falls within the published range for natural rubber (60–80%).
Predictive Accuracy: The model reproduces the 24 measured shots with a mean error of 3.3% and an RMS error of 0.0494 m/s. While it captures the general trend, it reproduces only about 30% of the measured curvature in the restitution curve, suggesting a mechanism acting primarily below 1 m/s remains unidentified.
Spin Response: Without re-fitting constants, the model predicts rebound over the full plane of longitudinal spin and sidespin. Notably, it predicts that under left sidespin near perpendicular incidence, the ball can depart on the side of approach (rebound angle $> 90^\circ$ ), an effect that cannot be represented by a lumped friction coefficient.

Significance and Limitations
The paper demonstrates that a route exists from a manufacturer's dimensioned drawing and catalogue hardness to a predicted rebound curve, offering a more physical basis for table specification than a single COR value. The study highlights that the coefficient of restitution is inherently speed-dependent and that linear viscoelastic models cannot capture the observed dissipation rates.

However, the author maintains a modest stance regarding validation. The damping constant $\lambda$ and the tracking run-out correction were calibrated jointly against the same 24 data points used for evaluation, meaning the comparison primarily tests the curve shape rather than providing independent validation. Additionally, the predicted spin response has not been experimentally tested, as the source dataset consisted only of pure rolling shots with zero sidespin. The author notes that deriving $\lambda$ independently would require dynamic characterization of cushion rubber at impact rates, for which no data currently exists.

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