Energy level structure of chaotic motion in bouncing ball system

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Highlights

  • Ball’s chaotic trajectories can be classified according to the rescaled flight time.

  • Rescaled kinetic energy of the bouncing ball exhibits an energy level structure.

  • Probability distribution of energy dissipated by impacts has an oscillating profile.

  • Energy dissipation peaks can be attributed to energy level transitions.

Abstract

We report our experimental and numerical results on the chaotic dynamics of a ball bouncing on a vertically vibrating plate. It is demonstrated that if the trajectories between collisions are classified according to the rescaled flight time, the chaotic attractor for the bouncing motion of the ball degrades into a regular structure composed of discrete energy levels in the phase space, and the motion states corresponding to these trajectories evolve into separate “microstates” populated on these levels. The probabilities of level transitions and the selection rules for the transitions are discussed. Then, the probability distribution of the energy dissipated during each collision is analyzed. It is found that there are discrete dissipation peaks in the energy dissipation spectrum. The origination of these dissipation peaks is analyzed from the perspective of level transitions.

Introduction

The behavior of a ball bouncing repeatedly on a vertically vibrating plate has attracted much research attention in the past, due to the rich and interesting dynamics involved. In this system, the plate acts as an energy reservoir, supplying energy for the ball to maintain its continuous bouncing motion. But the collision between the ball and the plate is inelastic, and each collision will dissipate part of the kinetic energy of the ball. The interplay between the gain and dissipation of energy leads to the diversity of motion fashions of the ball, which is manifested in the fact that the trajectories of the ball’s projectile motion between collisions can be regular or chaotic, depending on the frequency and amplitude of the vibration applied [1], [2], [3].

For regular motion, the ball always repeats the same bouncing sequence composed of a finite number of bounces, and the motion of the ball can be characterized by the total flight time consumed on these bounces, which is usually an integer multiple of the vibration period. In chaotic motion, the ball bounces irregularly and shows randomness in its trajectories. But this doesn’t mean the ball is in a completely random motion, since the velocity of the ball doesn’t follow the standard Gaussian distribution [4], [5]. The complexity of the chaotic motion of the ball actually comes from the instability of its trajectories. This instability causes neighboring orbits in the phase space to diverge exponentially fast, usually characterized by a positive Lyapunov exponent [6], [7], [8]. Though strongly sensitive to initial conditions, the trajectories will eventually fall onto a chaotic attractor, which does not fill the phase space in a random manner, but is well-defined structured and exhibits self-similar fractal characteristics [8], [9], [10], [11], [12], [13]. These known features, however, still remain the difficulty of grasping a simple and intuitive picture to represent the chaotic behavior of the bouncing ball.

In this work, we show that the trajectories between collisions can be classified according to the rescaled flight time, and the motion states corresponding to these trajectories evolve into separate “microstates” that populate on discrete energy levels. The “microstates” on each level have the same kinetic energy that has been rescaled by a basic energy unit. Then, the probability distribution of energy dissipation in each collision is determined. It is found that there are discrete dissipation peaks in the dissipation spectrum. According to the probability of the occurrence of energy level transitions, a reasonable explanation is achieved for such discrete peaks.

The organization of the paper is as follows. Section 2 introduces our experimental and theoretical results on the chaotic dynamics of the bouncing ball under different vibration conditions, and gives the chaotic phase diagrams of the flight time and impact velocities. Section 3.1 introduces the definition of the rescaled flight time and impact velocities used in the text, and illustrates the level structure of the rescaled kinetic energy. Then, the degeneracy of the motion states of the bouncing ball is discussed, and based on this, the cause of the uneven structure in the phase diagram is explained. In Section 3.2, we determine the selection rules of level transitions based on experimental and theoretical data. In Section 3.3, we give the energy dissipation spectrum of the bouncing ball system, in which discrete peaks are observed. An explanation is made for the appearance of such discrete peaks based on energy level transitions, and final remarks and conclusions are drawn in Section 4.

Section snippets

Experimental and theoretical results

The experimental setup is the same as in our previous work [11], [12], and the details can be found there. The sketch of the experimental setup is shown in Fig. 1. In the experiment, a 15 mm × 15 mm double-layer composite plate is used, which is composed of a 1.0 mm thick polyethylene piece glued to the upper surface of a 1.5 mm thick stainless steel plate. A bearing-steel (GCr15) ball of 10.0 mm in diameter and 4.078 g in weight is placed on the plate. After the vibration is applied, the ball

Level structure

It is noted that the normalized collision moment ti+1/T and ti+/T can be decomposed into two parts, an integer part and a fractional part ti+1T=p+θi+12π,ti+T=q+θi+2πwhere p,q=0,1,2,3,, and pq. Based on this, a new rescaled flight time tnf can be defined tnf=tifθi+1ω+θi+ω=nTwhere, n=pq=0,1,2,3,.

Combining Eqs. (4), (5), a rescaled velocity vn can be obtained vn=(vi++gωθi+)(vi+1+gωθi+1)=ngTFurthermore, the rescaled kinetic energy En of the ball is defined as En=12mvn2=n2ɛwhere, m is

Conclusions

The classification of the trajectories of the bouncing ball according to the rescaled flight time makes the chaotic attractor degrade into a regular structure composed of discrete levels in the phase space, in which each of these levels is degenerate, for a large number of separate “microstates” with the same rescaled kinetic energy are populated on it. The jumps between these degenerate states caused by level transitions result in a certain degree of randomness in the ball’s bouncing motion.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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