## Earthquake Modelling with Differential Equations

By Chloe WIlkins, University of Newcastle

One important application of differential equations is in modelling the motion of multi-storey structures experiencing earthquakes. Over the past few decades, the devastating damage caused by earthquakes has prompted further research into earthquake-building interactions. The focus of my project was to simulate the movement of individual storeys of some simple two-dimensional structures experiencing severe vibrations.

When sudden violent shakings of the Earth’s surface – or earthquakes – occur, elastic body waves referred to as ‘seismic waves’ are generated. Like sound waves, seismic waves travel out in all directions from their source, their velocity depending on the density and elasticity of the materials through which they propagate. During an earthquake, elastic body waves within the Earth are produced. After these body waves arrive, surface waves are generated, and these are the waves responsible for causing the most infrastructural damage.

The interaction of these seismic waves with buildings was modelled using a method called Equivalent Static Lateral Force Analysis. This method involves simulating earthquake vibrations by applying an oscillating force to the ground floor of the building. Motion begins in this first floor, and since the earthquake force is not applied to any other storey of the building, it is the motion of this first floor that determines the motion of subsequent floors.

The structures used in the project were simple rectangular structures based on a spring-mass damper system. This means that each floor of the building was represented by a distinct mass, and the floors were connected elastically so that when they moved, they moved as if a spring was attached between them. When no force is applied, any structure will oscillate freely at its ‘natural frequency’. One important aspect of the project was to compute the natural frequencies of the structures considered. The challenge was to explore the relationship between the frequency of an earthquake and the natural frequency of a building. By considering this, the points at which a structure would give way under a certain earthquake force could be explored.

There are a few extensions that could be made to this project to allow for more realistic situations to be modelled. Most importantly, the project could be extended to consider structures beyond the simple, rectangular, two-dimensional structures I modelled. For example, structures of different shapes, or structures consisting of multiple adjacent buildings could be considered. It would also be interesting to explore structures of different densities or elasticities, as well as structures which use modern earthquake-resistant technology such as Seismic Dampers or Base Isolation Devices.

In undertaking this project, one of the key learning experiences for me was a greater understanding of mathematical modelling and its real-world applications. More specifically, my understanding of the nature and applications of differential equations was furthered. Throughout the research, I gained significant insight into modern studies surrounding earthquake-building interactions, as well as the limitations of certain areas of earthquake modelling which still require much investigation.

Chloe Wilkins was a recipient of a 2018/19 AMSI Vacation Research Scholarship.