[Paper Review] Simple model of bouncing ball dynamics. Displacement of the limiter assumed as a cubic function of time

Andrzej Okniński, B. Radziszewski|arXiv (Cornell University)|Mar 19, 2012

Quantum chaos and dynamical systems14 references4 citations

TL;DR

This paper presents an analytically tractable model of a bouncing ball colliding with a limiter whose displacement is a periodic cubic function of time, enabling exact computation of impact times and dynamics. The key contribution is the analytical derivation of chattering and grazing conditions, showing chaotic bands emerge via corner-type bifurcations after period-doubling of fixed points.

ABSTRACT

Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the limiter is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifurcation. Equation for the time of the next impact is solved exactly for the case of two subsequent impacts occurring in the same period of limiter's motion making analysis of chattering possible.

Motivation & Objective

• To develop a simple yet analytically solvable model of a bouncing ball with a periodically moving limiter.
• To overcome the intractability of solving impact time equations in standard sinusoidal limiter models by using a cubic displacement function.
• To enable analytical investigation of complex dynamical behaviors such as chattering, grazing, and chaos in nonsmooth dynamical systems.
• To derive exact conditions for multiple impacts within a single limiter period and analyze their stability.
• To characterize the onset of chaos through bifurcation analysis, particularly corner-type bifurcations following period doubling.

Proposed method

• Model the limiter's displacement as a periodic cubic function $ Y_c(T) = 12 ilde{T}( ilde{T} - 1/2)( ilde{T} - 1) $, where $ ilde{T} = T - \lfloor T \rfloor $, to ensure analytical tractability.
• Construct a Poincaré map (2.1) relating post-impact velocity $ V_i $ and impact time $ T_i $, with $ \Delta_{i+1} = T_{i+1} - T_i $, using nondimensional gravity $ \gamma $ and restitution $ R $.
• Derive exact analytical expressions for the time of next impact $ \Delta_{i+1} $ by solving the quadratic equation of motion under constant acceleration, leading to a closed-form nonlinear map (3.12c,d).
• Introduce a relative velocity variable $ v_i = V_i - G(3T_i^2 - 3T_i + 1/2) $ to simplify the dynamics and enable analysis of grazing and chattering.
• Use asymptotic expansion (3.14) and approximate maps (3.16) to model the final stages of chattering when $ v_i \to 0 $, assuming $ \lambda_i < 1 $ for convergence.
• Derive analytical conditions for grazing (infinite impacts in one period) by solving $ T_{(\infty)} < 1 $, yielding bounds $ T_B < T_i < T_C $ (Eqs. 3.21–3.22).

Experimental results

Research questions

• RQ1 How does the use of a cubic limiter displacement function enable analytical computation of impact times in a bouncing ball system?
• RQ2 What are the conditions under which chattering (multiple impacts in one limiter period) occurs, and how can they be analytically characterized?
• RQ3 How do chaotic bands emerge in the system, and what bifurcation mechanism underlies their formation?
• RQ4 What is the role of the grazing manifold in the dynamics, and how does it influence the stability and convergence of chattering?
• RQ5 Can exact or approximate analytical expressions be derived for the time of $ N $-th impact during chattering?

Key findings

• Chaotic bands in the system originate from fixed points via a corner-type bifurcation following the first period-doubling, confirmed by bifurcation diagrams with $ R = 0.85 $ and $ \gamma \in [0, 0.06] $.
• The time of the next impact $ \Delta_{i+1} $ is computed exactly using a quadratic solution, leading to a closed-form nonlinear map (3.12c,d) that enables full analytical treatment.
• Chattering is analytically modeled via approximate equations (3.16), showing that $ \Delta_{i+1} \approx \frac{2v_i}{-3G + 2 + 6GT_i} $, with convergence ensured when $ \lambda_i < 1 $.
• The grazing manifold exists only for $ T_* \geq \max(T_{cr}, 0) $, where $ T_{cr} = \frac{1}{2} - \frac{1}{3G} $, and is stable in one direction with eigenvalue $ \Lambda_1 = 1 $, neutral in the other.
• The condition for infinite impacts (grazing) is derived as $ \max(T_{A_1}, T_{A_2}, T_B) < T_i < T_C $, with $ T_B $ and $ T_C $ given by Eqs. (3.21) and (3.22), ensuring $ T_{(\infty)} < 1 $.
• The time of the $ N $-th impact in chattering is computed exactly as $ T_{(N)} = T_i + \frac{2v_i}{-3G + 2 + 6GT_i} \cdot \frac{1 - R^N}{1 - R} $, showing exponential decay of inter-impact intervals.