This section is the main event in this course. Modal analysis is central to the understanding and practical simulating of real-world structural dynamics. Once you’ve completed this section you’re going to have a far greater understanding of multi-degree of freedom dynamics and how to handle them in the wild.

We’ll start by getting our bearings and discussing at a very high level what modal analysis is. The first lecture in this section is about setting the scene.

In the next lecture, we discuss how to identify the N natural frequencies for an N-DoF system. After discussing mode shapes briefly in the previous lecture - in this lecture we’ll actually calculate them. We’ll also produce some simulation animations to get a better understanding of mode shapes.

Next, we’ll consider the natural frequencies and mode shapes as eigenvectors and eigenvalues. This matrix approach unlocks the door to a much more efficient way of identifying the modal characteristics for larger DoF systems.

After identifying the mode shapes as eigenvectors, we’ll build them into a modal matrix in the next lecture and discuss and demonstrate the concept of mass normalisation. This is a convenient and common form of normalisation that will streamline the modal calculations that follow.

In the next lecture, we’ll cover the topic of mode shape orthogonality and how the modal matrix allows us to decouple the equations of motion. Of all the lectures in this course, this lecture is the key to understanding modal superposition at a fundamental level. Once you digest this lecture, you’ll have a clear understanding of how we can turn a system of N coupled equations of motion into a system of N-uncoupled equations. This operation greatly simplifies the dynamic analysis of MDoF systems by unlocking the door to dynamic analysis by modal superposition.

Now that we understand at a concept level how and why modal superposition works - we’ll demonstrate the process from start to finish in the next lecture by analysing our multi-storey structure. We’ll confirm that a modal superposition approach yields the exact same dynamic behaviour as direct integration of the equations of motion. However, as we’ll see, implementing a modal superposition solution, reveals much more about the underlying dynamic behaviour of the system.