טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentGrinberg Itay
SubjectLocalization and Energy Transport in Vibro-Impact Systems
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Oleg Gendelman
Full Thesis textFull thesis text - English Version


Abstract

This work is dedicated to the study of vibro-impact systems, namely, dynamical systems involving impacts. When the duration of the collision is very short, the impact can be regarded as instantaneous. Under this assumption, the impacts are modeled as simple Newtonian impact. We focus on setups where the impacts are the sole source of nonlinearity which enables finding exact solutions for certain types of responses, and also simplifies the stability analysis.

In the first part of this work, we explore the dynamics of strongly localized periodic solutions (discrete solitons, or discrete breathers) in a finite one-dimensional chain of oscillators. Localization with both single and multiple localization sites (breathers and multi-breathers) is considered. The model involves parabolic on-site potential with rigid constraints (the displacement domain of each particle is finite), and a linear nearest-neighbor coupling. When the particle approaches the constraint, it undergoes the inelastic impact according to Newton's impact model. The rigid non-ideal impact constraints are the only source of nonlinearity and damping in the system. We demonstrate that this vibro-impact model allows derivation of exact analytical solutions for the breathers and multi-breathers with arbitrary set of localization sites, both in conservative and forced-damped settings. We also obtain solutions for asymmetric breathers when the constraints are asymmetric (and for some cases even when they are symmetric) with respect to the parabolic potential. The asymmetric rigid barriers also enable a new type of breather solution when the localization site mass is restricted to collide with only one rigid barrier. Periodic boundary conditions are considered; exact solutions for other types of boundary conditions are also available. Local character of the nonlinearity permits explicit derivation of a monodromy matrix for the breather solutions. Consequently, the stability of all types of derived breather solutions can be efficiently studied in the framework of simple methods of linear algebra, and with rather moderate computational efforts. All three generic scenarios for loss of stability reveal themselves - pitchfork, Neimark-Sacker and period doubling - however, never for the same type of solutions. This model is also extended to conservative multibreather solutions in linearly coupled chain. In this model, we are restricted to an approximate solution, but to the accuracy of our choice. Stability analysis can also be done here in a similar manner without further loss of accuracy.

The second part of this work is dedicated to the study of non-reciprocity in coupled non-dispersive waveguides. The waveguides are coupled by the collisions of the rigid attachments at the ends of each waveguide. Each rigid attachment is grounded by a linear spring. Difference in the stiffness of the grounding springs is sufficient to allow non-reciprocal wave transmission. We derive exact analytical solutions for simple harmonic wave transmission and reveal strong non -reciprocity in terms of efficiency of energy transmission in different directions. Similarly, explicit term for the monodromy matrix can be written, enabling exact stability analysis by means of a simple analytical procedure. The suggested setting demonstrates very strong non-reciprocity, while preserving the frequency of the incoming signal.