title: Wave Propagation in Flexural Systems
Creators: name: Madine, Katie
Publisher: name: University of Liverpool Repository
Year: 2025

description: This thesis presents an in-depth study of the dynamic behaviour of flexural systems composed of Euler--Bernoulli beams. Particular attention is paid to the coupling of flexural and torsional rotations between perpendicular beams and the ability to control the dispersive properties of the arrays studied using the properties of the system. Asymmetry is a recurring theme through each study in this thesis, from the asymmetric wave forms on lattices that can be induced by the choice of applied forcing to the inherent chirality and broken symmetries of the perpendicular gyroscope model studied in later chapters. For each system, analytical studies are complemented by numerical computations and finite element simulations, to illustrate the unusual and counter-intuitive effects produced.

The first half of the thesis investigates 1D and 2D flexural lattices of Euler--Bernoulli beams. The canonical object of study is the Green's function, from which information regarding the dynamic response of the lattice under point loading by forces and moments can be obtained. In both the 1D and 2D lattices, the combination of applied point forces and moments is shown to produce strong dynamic anisotropy with a variety of symmetric, anti-symmetric, asymmetric and uni-axial waves. Special consideration is devoted to the interaction between flexural and torsional waves in a square lattice of Euler--Bernoulli beams; by carefully controlling the inertial and elastic properties of the lattice elements, we demonstrate that it is possible to tune the dispersive properties of the lattice. Implementing the resulting control over the preferential directions of wave modes, we investigate the reflection and refraction of waves across interfaces between two dissimilar lattices of beams. The interfaces are used to induce negative refraction, unidirectional reflection, beam splitting and mode trapping inside inclusions.

Later chapters in this thesis study the perpendicular gyroscope model which is formed of two perpendicular beams---with the vertical beam either clamped at the base or mounted on a flexural plate---and a gyroscope placed at the free end of the horizontal beam. Building the analytical model requires careful consideration of the coupling of flexural and torsional rotations between beam junctions and the symmetry-breaking effects induced by the spinning of the gyroscope. We explore the symmetry of, and induced by, individual perpendicular gyroscopes compared to infinite arrays on flexural plates. We investigate the control over the eigenfrequencies and dispersive properties of the array afforded by altering parameters of the system, such as the rate of spin of the gyroscope, the length of the beams and the orientation of the gyroscope axes. 

This work has applications in the development of elastic metamaterials designed to control wave propagation through flexural structures. The study of elastic wave propagation is of particular interest due to the wide variety of applications to physical systems such as buildings, bridges, seismic protection devices, wind farms, cloaking devices and more.