Dr. Howland conducts his research at Fitzpatrick Hall of Engineering.

Current Research

Accelerated Perturbation Techniques

The practice of transforming a system of differential equations into one which can be readily solved is a well-established one; Hamiltonian mechanics, for example, exemplifies this approach and provides a particularly elegant analytical technique for the study of nonlinear mechanics of conservative multi-degree-of-freedom systems. Even if not amenable to solution in closed form, these systems can often be represented as a soluble one "perturbed" by an additional force of small magnitude, then tractable to one of the broad range of available perturbation methods.

In the investigation of problems in near-linear vibrations, it becomes important to know the frequencies of the perturbed system accurately, since errors in the frequencies introduce a time-linear "drift" of the formal solution away from the "actual" one. While most perturbation methods are "linear", finding at each step of the procedure a single higher order of solution and frequency, an analytical Newtonian transformation approach due to Kolmogorov and Arnold can be implemented to give solutions and frequencies quadratically; problems of transformation and inversion normally present in such an approach are overcome through the use of Lie transforms. In particular, this allows the determination of the new frequencies to twice the order of solution, rendering the latter valid over time intervals literally orders of magnitude greater than those available to linear methods.

If investigation to date has concentrated on the problem of near-linear oscillations, however, the fundamental approach itself is of far broader utility: it has been applied to non- conservative systems (both directly and through formal "Hamiltonization" of the problem), and initial work has even been done in the direction of its extension to the partial differential wave equation (where it is the dispersion relation which can be determined quadratically). Most recently, the general transformation philosophy was used in a non-canonical approach to solution of the Hamiltonian "ideal resonance problem" in celestial mechanics. The last could provide an important tool in the analysis of more general resonance phenomena commonly observed even in our solar system.

Direct comments, questions, and corrections to amedept@nd.edu