טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentKarmi Gleb
SubjectBouncing Ball Inside the Vibrating Circular Fence
Chaotic System Analysis
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Oleg Gendelman
Full Thesis textFull thesis text - English Version


Abstract

Mechanical systems demonstrating chaotic behavior present a great challenge for analysis. In this thesis, we consider a small ball bouncing inside radially pulsating fence and performing inelastic collisions with the fence. The friction between the ball and the fence is considered to be absent; therefore ball angular momentum with respect to the ring center is conserved in the course of each collision. Such a model naturally arises from simplified description of dynamics of two tethered satellites and is relevant for some other applications. In our consideration, the fence pulsations follow the simple harmonic law.

First, the bouncing ball dynamics is simplified through traditional assumptions, typical for this sort of models. As a result, we obtain explicit simplified analytic map (SAM). This SAM is analyzed by common methods, which allow to study periodic regimes, their stability, as well as to reveal peculiar chaotic attractors. Unfortunately, SAM fails to preserve the additional conservation law and under certain conditions reveals completely unphysical behavior (like negative time steps). To correct these irregularities, we consider a complete system model (full model, FM), in which the simplifying assumptions adopted for SAM are relaxed. As a result, we obtain an implicit (unresolvable) map, which conserves the angular momentum, as required. FM reveals important physical feature absent it SAM - chatter behavior. For generic initial conditions, the system is trapped into chatter, which eventually leads to initial conditions “reset”. This FM feature removes the chaotic attractors from a pool of possible response regimes. As for periodic responses, both models provide close results. Current work presents with the following major innovations:

a. new type of Bouncing Ball problem with additional conservation law is stated, analyzed and simulated

b. simple memory reset mechanism is modeled

c. traditional simplified models turn out to have limited applicability - a strange attractor in Simplified Analytic Map model is absent in a Full Model. In the same time, for periodic orbits both models yield close results. So, the SAM is acceptable for the analysis of low-periodic motions.