Differential Equations Seminar: Kaitlynn Lilly (UMBC)
Undergraduate research talk
Monday, October 5, 2020 · 11 AM - Noon
Title: Spectral Properties of a Non-Self-Adjoint Beam with Applications to Flutter
In this talk we focus on a simplified (1-D) partial differential equation beam model for aeroelastic flutter, a structural instability brought about by the presence of a flow surrounding the beam. We discuss the onset of instability in the problem, related to non-self-adjoint PDE terms. Various methods of simulating the dynamics are presented, and we empirically observe limit cycle oscillations for the solutions. Supported by the abstract theorem that the dynamical system of PDE solutions here has a global attractor, we aim to rigorously construct said periodic solutions, thereby ascertaining some structure of the attractor. By exploiting the 1-D structure of the problem and the explicit root formula for the quartic equation, we use separation of variables to construct eigenvalues and corresponding modes to the non-self-adjoint spatial problem (rather than using in vacuo modes to approximate the perturbed solution). Finally, our explicit eigenpairs yields a cubic-type temporal ODE which can be then be analyzed and solved.