טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentKedem Shelly
SubjectLocalization in Neutral Dynamical Systems with Time-Delay
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Oleg Gendelman
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


Abstract

Localization of energy is a dynamical phenomenon which causes concentration of energy of oscillations or rotations in certain elements of a multiple-component system.

Localization of energy in linear systems will only occur when a small disorder in the homogeneousness of the structure exists or appears. Localization of energy in non-linear systems, on the other hand, can occur in a perfectly homogeneous system. Large amplitude of movement in a certain area of the system causes a change in the frequency of motion, which acts as an effective disorder, resulting in the energy localization.

Throughout the years, this phenomenon has been investigated in both linear and non-linear systems, and different factors were analyzed to check their influence on the energy localization. For instance, such factors are the damping and the coupling between the elements of the system.

When building dynamical models to describe different systems, one must take into account the time it takes for each component of the systems to response. Nearly all real systems include time-delays. Therefore, many dynamical systems are mathematically modeled with time-delays and there is a great interest exploring the influence of time-delays on energy localization.

Energy localization in time-delayed systems has been studied recently and was found to exist. Yet, only non-conservative time-delayed systems were studied so far. 

Our research focused on investigation of neutral conservative dynamical systems with time-delay. Such models are used for investigation of coupled dynamical systems, when one has to take into account both the time delay and the inertia of the coupling elements.

In this work, the simplest conservative system comprising two weakly nonlinear oscillators coupled by dispersionless elastic rod has been considered.

It was demonstrated that the localization phenomena, well known in counterpart systems without time-delay, persists also in the system under investigation. Still, quite counter-intuitively, time delay opposes nonlinear localization. In particular, our research also showed a clear connection between the “amount” of delay that was put into the system and the level of energy necessary to observe the breakdown of the collective nonlinear normal modes and the localization. The results obtained from the asymptotic approach were verified by independent numeric simulation of the exact system and led us to the following conclusions:

  • The weaker the coupling between the oscillators, the less is the energy level required for the localization.
  • Time-delay opposes energy localization. The higher delay in the system, the more energy is required for localization.