Using the Exterior Matrix Method to find eigenfrequencies when the governing equations are systems of first-order equations

In this paper, we will apply the Exterior Matrix Method to the case where the governing equations for one piece of the structure is a system of 4 first-order equations. This generalizes a result when the governing equation was a single fourth-order equation. Ironically, the governing equations do not have to be solved in order to find the corresponding exterior matrix, and the exterior matrices can be used to find the eigenfrequencies of the system, even if there are dissipative joints added to the system. We will first look at the well-understood example of Euler–Bernoulli beams to illustrate the concept, and then move on to the more difficult inclined cable problem.