In this section, we’ll address an issue that we’ve carefully avoided up to this point in the course; damping, or more specifically damping orthogonality.

In the previous section, we saw how the modal matrix could be used to uncouple the equations of motion by diagionalising the mass and stiffness matrices. The same is not true for the damping matrix - in other words, we cannot generally uncouple the damping matrix for damped MDoF systems.

Taking a modal analysis approach, this isn’t an issue - we simply apply damping at the modal level and recombine damped modal responses to reveal the damped behaviour of the full system. However, in cases where modal superposition cannot be used, due to non-linearity for example, we must use direct integration to solve for the system behaviour. So we need a way of determining a damping matrix that will yield predictable levels of modal damping.

In this section, we’ll explore how to obtain a damping matrix that yields prescribed levels of modal damping. We’ll start by introducing mass proportional damping, followed by stiffness proportional damping and Rayleigh damping which is a linear combination of both.

We’ll see that Rayleigh damping allows us to control damping in only two modes. In the penultimate lecture in this section, we’ll discuss a method for determining a diagonalisable damping matrix that yields prescribed levels of damping in all modes, allowing us to fully control the level of damping encoded into a damping matrix.

In the final lecture in this section, we’ll demonstrate the equivalence between modal superposition and direct integration for systems that include damping. Once this section is complete, you’ll have two completely different but equivalent methods for analysing MDoF damped systems. This means that you’ll have the tools to handle linear and non-linear dynamic systems.