# Seminar: Elastic graphs as a model for propagation of vibrations through complex structures

**Speaker:** Cerian Brewer – PhD student (School of Mathematical Science, University of Nottingham)

**Title:** Elastic graphs as a model for propagation of vibrations through complex structures

**Abstract:** We use networks of beams/plates to model the propagation of noise and vibration in large structures. Using fourth order beam equations on these graph-like structures, we propose an extension of quantum graphs to the elastic case. The fourth order beam equations introduce evanescent waves into the system. Formulating the problem in terms of scattering matrices we note that the transfer operator is non-unitary due to the presence of these evanescent modes. Despite this non-unitarity we are able to derive a functional equation which guarantees real eigenvalues. The transfer operator can also be used to construct a correlation function which we use as a tool for understanding fluctuations about the mean solution. This is done by propagating correlation functions on graphs. Ray tracing methods such as Dynamical Energy Analysis are shown to emerge as a limit of this approach.