The exterior matrix method for sequentially coupled fourth-order equations

This paper introduces the exterior matrix method as a means to compute asymptotically the eigenfrequencies for the vibrations of serially connected elements which are each governed by fourth-order equations. The Timoshenko beams and a tapered Euler–Bernoulli beams are just two examples that are solved by this method. The technique involves not just lifting the transfer matrices into the higher dimensional exterior algebra space, but also lifting the original fourth-order equation to generally produce a sixth-order exterior equation, but occasionally the exterior equation is only fifth order. The original fourth-order equation does not have to be solvable for this technique to work, since only the approximate solutions for the exterior equation are needed to compute the eigenfrequencies asymptotically.