J. Strzałko

Synchronous rotation of the set of double pendula: experimental observations.

We study the occurrence of the synchronous rotation of a set of four uncoupled nonidentical double pendula arranged into a cross structure mounted on a vertically excited platform. Under the excitation, the pendula can rotate in different directions (counter-clockwise or clockwise). It has been shown that after a transient, many different types of synchronous configurations with the constant phase difference between pendula can be observed. The experimental results qualitatively agree with the numerical simulations.

The three-dimensional dynamics of the die throw

A three-dimensional model of a die throw which considers the die bounces with dissipation on the fixed and oscillating table has been formulated. It allows simulations of the trajectories for dice with different shapes. Numerical results have been compared with the experimental observation using high speed camera. It is shown that for the realistic values of the initial energy the probabilities of the die landing on the face which is the lowest one at the beginning is larger than the probabilities of landing on any other face. We argue that non-smoothness of the system plays a key role in the occurrence of dynamical uncertainties and gives the explanation why for practically small uncertainties in the initial conditions a mechanical randomizer approximates the random process.

CAN THE DICE BE FAIR BY DYNAMICS

We consider the dynamics of the three-dimensional model of the die which can bounce with dissipation on the table. It is shown that for the realistic values of the initial energy the probabilities of the die landing on the face which is the lowest one at the beginning is larger than the probabilities of landing on any other face.

Dynamics and Predictability

We present the results of the experimental observations and the numerical simulations of the coin toss, die throw, and roulette run. We give arguments supporting the statement that the outcome of the mechanical randomizer is fully determined by the initial conditions, i.e., no dynamical uncertainties due to the exponential divergence of initial conditions or fractal basin boundaries occur. We point out that although the boundaries between basins of attraction of different final configurations in the initial condition space are smooth, the distance of a typical initial condition from a basin boundary is so small that practically any uncertainty in initial conditions can lead to the uncertainty of the outcome.

Nature of Randomness in Mechanical Systems

We discuss the nature and origin of randomness in mechanical systems. We argue that nonsmoothness of the system plays a key role in the occurrence of dynamical uncertainties. The explanation why for practically small uncertainties in the initial conditions mechanical randomizer approximates the random process is given.

General Motion of a Rigid Body

The tools that allow the description of the motion of the rigid body are recalled. We used both Euler’s angles and Euler’s parameters (normalized quaternions ) to describe the orientations of the body. Precession of the rigid body and air resistance and the dissipation of the energy at successive collisions are discussed.

Equations of the Randomizer’s Dynamics

Basing on the Newton–Euler laws of mechanics we derive the equations which describe the dynamics of the coin toss, the die throw, and roulette run. The equations for full 3D models and for lower dimensional simplifications are given. The influence of the air resistance and energy dissipation at the impacts is described. The obtained equations allow for the numerical simulation of the randomizer’s dynamics and define the mapping of the initial conditions into the final outcome.