Georgia Institute of Technology College of Sciences

New and proposed interplanetary missions increasingly require the design of trajectories within challenging multi-body environments that stress or exceed the capabilities of the two-body design methodologies typically used over the last several decades. These current methods encounter difficulties because they often require appreciable user interaction, result in trajectories that require significant amounts of propellant, or miss potential mission-enabling options. The use of dynamical systems methods applied to three-body and multi-body models provides a pathway to obtain a fuller theoretical understanding of the problem that can then result in significant improvements to trajectory design in each of these areas. In particular, the computation of periodic Lagrange point and resonant orbits along with their associated invariant manifolds and heteroclinic connections are crucial to finding the dynamical channels that provide new or more optimal solutions. These methods are particularly effective for mission types that include multi-body tours, Earth-Moon transfers, approaches to moons, and trajectories to asteroids. The inclusion of multi-body effects early in the analysis for these applications is key to providing a more complete set of solutions that includes improved trajectories that may otherwise be missed when using two-body methods. This seminar will focus on two representative trajectory design applications that are especially challenging. The first is the design of tours using flybys of planets or moons with a particular emphasis on the Galilean moons and Europa. In this case, the exploration of the design space using the invariant manifolds of resonant and Lyapunov orbits provides information such as the resonance transitions that are required as part of the tour. The second application includes endgame scenarios, which typically involve an approach to a moon with an objective of either capturing into orbit around the moon or landing on the surface. Often, the invariant manifolds of particular orbits may be used in this case to provide a wide set of approach options for both capture and landing analyses. New methods will also be discussed that provide a foundation for rigorously analyzing the transit of trajectories through the libration point regions that is necessary for the approach and capture phase for bodies such as Europa and the Moon. These methods provide a fundamentally new method to search for the invariant manifolds of orbits and hyperbolic invariant sets associated with libration points while giving additional insight into the dynamics of the flow in these regions.

A Jones surface for a knot in the three-sphere is an essential surface whose boundary slopes, Euler characteristic, and number of sheets correspond to quantities defined from the asymptotics of the degrees of colored Jones polynomial. The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran predicts that there are Jones surfaces for every knot. A link diagram D is said to be a Murasugi sum of two links D' and D'' if a state graph of D has a cut vertex, which separates the graph into two state graphs of D' and D'', respectively. We may obtain a state surface in the complement of the link K represented by D by gluing the state surface for D and the state surface for D' along the disk filling the circle represented by the cut vertex in the state graph. The resulting surface is called the Murasugi sum of the two state surfaces. We consider near-adequate links which are certain Murasugi sums of near-alternating link diagrams with an adequate link diagram along their all-A state graphs with an additional graphical constraint. For a near-adequate knot, the Murasugi sum of the corresponding state surface is a Jones surface by the work of Ozawa. We discuss how this proves the Strong Slope Conjecture for this class of knots.

The first part of this talk will review our recent efforts on the electroelastodynamics of smart structures for various applications ranging from nonlinear energy harvesting, bio-inspired actuation, and acoustic power transfer to elastic wave guiding and vibration attenuation via metamaterials. We will discuss how to exploit nonlinear dynamic phenomena for frequency bandwidth enhancement to outperform narrowband linear-resonant devices in applications such as vibration energy harvesting for wireless electronic components. We will also cover inherent nonlinearities (material and internal/external dissipative), and their interactions with intentionally designed nonlinearities, as well as electrical circuit nonlinearities. Electromechanical modeling efforts will be presented, and approximate analysis results using the method of harmonic balance will be compared with experimental measurements. Our recent efforts on phononic crystal-enhanced elastic wave guiding and harvesting, wideband vibration attenuation via locally resonant metamaterials, contactless acoustic power transfer, bifurcation suppression using nonlinear circuits, and exploiting size effects via strain-gradient induced polarization (flexoelectricity) in centrosymmetric elastic dielectrics will be summarized. The second part of the talk, which will be given by Chris Sugino (Research Assistant and PhD Student), will be centered on low-frequency vibration attenuation in finite structures by means of locally resonant elastic and electroelastic metamaterials. Locally resonant metamaterials are characterized by bandgaps at wavelengths that are much larger than the lattice size, enabling low-frequency vibration/sound attenuation. Typically, bandgap analyses and predictions rely on the assumption of waves traveling in an infinite medium, and do not take advantage of modal representations commonly used for the analysis of the dynamic behavior of finite structures. We will present a novel argument for estimating the locally resonant bandgap in metamaterial-based finite structures (i.e. meta-structures with prescribed boundary conditions) using modal analysis, yielding a simple closed-form expression for the bandgap frequency and size. A method for understanding the importance of the resonator locations and mass distribution will be discussed in the context of a Riemann sum approximation of an integral. Numerical and experimental results will be presented regarding the effects of mass ratio, non-uniform spacing of resonators, and parameter variations among the resonators. Electromechanical counterpart of the problem will also be summarized for piezoelectric structures.

We present an algorithm that, with high probability, generates a random spanning treefrom an edge-weighted undirected graph in O (n^{5/3}m^{1/3}) time. The tree is sampled from adistribution where the probability of each tree is proportional to the product of its edge weights.This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, O(n^\omega). For the special case of unweighted graphs, this improves upon thebest previously known running time of ˜O(min{n^\omega, mn^{1/2}, m^{4/3}}) for m >> n^{7/4} (Colbourn et al. ’96, Kelner-Madry ’09, Madry et al. ’15).The effective resistance metric is essential to our algorithm, as in the work of Madry et al., butwe eschew determinant-based and random walk-based techniques used by previous algorithms.Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance ispreserved in the graph resulting from eliminating a subset of vertices (called a Schur complement).As part of our algorithm, we show how to compute -approximate effective resistances for a set Sof vertex pairs via approximate Schur complements in O (m + (n + |S|)/\eps^{ −2}) time, without usingthe Johnson-Lindenstrauss lemma which requires eO(min{(m+|S|) \eps{−2},m+n /eps^{−4} +|S|/eps^{ −2}}) time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn’t sufficiently accurate.Joint work with Rasmus Kyng, John Peebles, Anup Rao, and Sushant Sachdeva