[Paper Review] Simple models of bouncing ball dynamics and their comparison

Andrzej Okniński, B. Radziszewski|arXiv (Cornell University)|Jun 7, 2010

Sports Dynamics and Biomechanics7 references3 citations

TL;DR

This paper proposes simplified analytical models of a bouncing ball colliding with a periodically moving table, using quadratic and cubic polynomial approximations of sinusoidal table motion to enable tractable analytical computation of impact times. The cubic model $ Y_{c_2}(T) $ closely replicates the bifurcation structure of the sinusoidal case, including full period-doubling cascades to chaos, while maintaining solvability of the impact time equation via third-order algebraic solutions, thus offering a practical alternative for studying complex nonlinear dynamics in vibro-impacting systems.

ABSTRACT

Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Several simple models of table motion are studied and compared. Dependence of displacement of the table on time, approximating sinusoidal motion and making analytical computations possible, is assumed as quadratic and cubic functions of time, respectively.

Motivation & Objective

• To develop simplified yet accurate models of table motion in bouncing ball dynamics that approximate sinusoidal motion while allowing analytical computation of impact times.
• To compare the dynamical behavior—especially bifurcation structures—of these models with the standard sinusoidal table motion case.
• To identify a polynomial-based model that preserves key nonlinear features such as period-doubling cascades and chaotic dynamics seen in the sinusoidal system.
• To ensure the equation for the next impact time remains analytically solvable, enabling deeper theoretical analysis of the system’s nonlinear dynamics.

Proposed method

• The ball’s motion between impacts is governed by Newton’s law of motion under gravity, with impacts modeled via a coefficient of restitution $ R $.
• The Poincaré map is constructed using Eqs. (5a) and (5b), relating the time and velocity at consecutive impacts via the table’s displacement $ Y(T) $.
• Three polynomial approximations of sinusoidal table motion are introduced: a piecewise quadratic function $ Y_q(T) $, a smooth cubic $ Y_{c_1}(T) $, and a four-piece cubic $ Y_{c_2}(T) $, all periodic with period 1.
• The time of the next impact $ T_{i+1} $ is determined by solving a third-order algebraic equation derived from Eq. (5a), ensuring analytical tractability.
• Bifurcation diagrams are computed for each model to compare the emergence of periodic orbits, period-doubling, and chaotic dynamics against the sinusoidal reference case.
• The models are evaluated based on their ability to reproduce the full period-doubling cascade to chaos observed in the sinusoidal system.

Experimental results

Research questions

• RQ1 Can polynomial approximations of sinusoidal table motion preserve the complex bifurcation structure, including period-doubling cascades, seen in the standard bouncing ball system?
• RQ2 Which polynomial model—quadratic, cubic, or piecewise cubic—most accurately replicates the dynamical behavior of the sinusoidal table motion?
• RQ3 Does the use of a polynomial table displacement allow analytical computation of impact times while maintaining dynamical fidelity to the sinusoidal case?
• RQ4 How do the bifurcation diagrams of the polynomial models compare to that of the sinusoidal system in terms of stability transitions and chaotic bands?

Key findings

• The piecewise quadratic model $ Y_q(T) $ produces a bifurcation diagram with only one period-doubling event and no chaotic bands, differing significantly from the sinusoidal case.
• The smooth cubic model $ Y_{c_1}(T) $ exhibits a full period-doubling cascade to chaos, but some bifurcation paths end abruptly and the stability order differs from the sinusoidal reference.
• The four-piece cubic model $ Y_{c_2}(T) $ produces a bifurcation diagram that is visually and structurally very similar to the sinusoidal case, including the full period-doubling cascade and chaotic bands.
• The equation for the next impact time in the $ Y_{c_2}(T) $ model is a third-order algebraic equation, making it analytically solvable while preserving complex nonlinear dynamics.
• Among the proposed models, $ Y_{c_2}(T) $ is the only one that successfully replicates the key dynamical features of the sinusoidal system, including the onset of chaos via period doubling.