Statics and Dynamics of Mechanical Lattices

Abstract

This thesis contributes to the understanding of one dimensional mechanical lattice structures. Structures formed from freely pin jointed rigid links with either vertical or torsional springs at the pivots, or both, are studied under the in uence of an axial load. These studies fall into three parts: static behaviour of a `simple' mechanical system with only vertical springs, dynamic behaviour of this `simple' system, and static behaviour of a compound mechanical lattice with both vertical and torsional springs. The �rst part uses ideas from the �eld of discrete mechanics to derive several discrete boundary value problems that model the static equilibrium states of the `simple' mechanical lattice. This application of discrete mechanics allows us to better understand the relationships between the mechanical system and the discrete boundary value problem used to model it. The resulting discrete boundary value problem is studied in detail and interesting complex behaviour is observed. The study of the dynamic behaviour of the `simple' mechanical lattice concentrates on the existence and stability of time periodic spatially localised solutions called discrete breathers. Discrete breathers are found to exist and to be stable. Also, related solutions called phonobreathers are found to exist and, although the exact phonobreather solutions are unstable, interesting nonlinear dynamic behaviour is observed close to the unstable solutions. Finally, the static behaviour of a new compound mechanical lattice, a discrete version of the strut on a linear foundation, is studied in Chapter 6. We see how the behaviour of two simpler mechanical lattices is manifested in this compound lattice, before presenting analytic and numerical results on the primary, static, bifurcations of this compound lattice. The localised behaviour of the most physically relevant static equilibrium states is also investigated. Extensions to the discrete boundary value problem methods of the earlier chapters are also discussed.

Date of Award	1 Jun 2009
Original language	English
Awarding Institution	University of Bath
Supervisor	Chris Budd (Supervisor) & Giles Hunt (Supervisor)