Vibrations of mechanical structures: source localization and nonlinear eigenvalue problems for mode calculation
| dc.contributor.author | Baker, Jonathan Peter | en |
| dc.contributor.committeechair | Embree, Mark P. | en |
| dc.contributor.committeemember | Gugercin, Serkan | en |
| dc.contributor.committeemember | Sarlo, Rodrigo | en |
| dc.contributor.committeemember | Beattie, Christopher A. | en |
| dc.contributor.department | Mathematics | en |
| dc.date.accessioned | 2023-05-20T08:00:44Z | en |
| dc.date.available | 2023-05-20T08:00:44Z | en |
| dc.date.issued | 2023-05-19 | en |
| dc.description.abstract | This work addresses two primary topics related to vibrations in structures. The first topic is the use of a spatially distributed sensor network for localization of vibration events. I use a received signal strength (RSS) framework that presumes exponential energy decay with distance to the source. I derive the Cramér-Rao bound (CRB) for this parameter estimation problem, with the unknown parameters being source location, source intensity, and the energy dissipation rate. In this framework, I show that the CRB matches the variance of maximum likelihood estimators (MLEs) in more computationally expensive Monte Carlo trials. I also compare the CRB to the results of physical experiments to test the power of the CRB to predict spatial areas where MLEs show practical evidence of being ill-conditioned. Supported by this evidence, I recommend the CRB as a simple measure of localization accuracy, which may be used to optimize sensor layouts before installation. I demonstrate how this numerical optimization may be performed for some regions of interest with simple geometries. The second topic investigates modal vibrations of multi-body structures built from simple one-dimensional elements, with networks of elastic strings as the primary example. I introduce a method of using a nonlinear eigenvalue problem (NLEVP) to express boundary conditions of the vibrating elements so that the (infinitely many) eigenvalues of the full structure are the eigenvalues of the finite-dimensional NLEVP. The mode shapes of the structure can then be recovered in analytic form (not as a discretization) from the corresponding eigenvectors of the NLEVP. I show some advantages of this method over dynamic stiffness matrices, which is another NLEVP framework for modal analysis. In numerical experiments, I test several contour integration solvers for NLEVPs on sample problems generated from string networks. | en |
| dc.description.abstractgeneral | This work deals with two primary topics related to vibrations in structures. The first topic is the use of vibration sensors to detect movement or impact and to estimate the location of the detected event. Sensors that are close to the event will record a larger amount of energy than the sensors that are farther away, so comparing the signals of several sensors can approximately establish the event location. In this way, vibration sensors might be used to monitor activity in a building without the use of intrusive cameras. The accuracy of location estimates can be greatly affected by the relative positions of the sensors and the event. Generally, location estimates tend to be most accurate if the sensors closely surround the event, and less accurate if the event is outside of the sensor zone. These principles are useful, but not precise. Given a framework for how event energy and noise are picked up by the sensors, the Cramér-Rao bound (CRB) is a formula for the achievable accuracy of location estimates. I demonstrate that the CRB is usefully similar to the location estimate accuracy from experimental data collected from a volunteer walking through a sensor-rigged hallway. I then show how CRB computations may be used to find an optimal arrangement of sensors. The match between the CRB and the accuracy of the experiments suggests that the sensor layout that optimizes the CRB will also provide accurate location estimates in a real building. The other main topic is how the vibrations of a structure can be understood through the structure's natural vibration frequencies and corresponding vibration shapes, called the "modes" of the structure. I connect vibration modes to the abstract framework of "nonlinear eigenvalue problems" (NLEVPs). An NLEVP is a square matrix-valued function for which one wants to find the inputs that make the matrix singular. But these singular matrices are usually isolated---% distributed among the infinitely many matrices of the NLEVP in places that are difficult to predict. After discussing NLEVPs in general and some methods for solving them, I show how the vibration modes of certain structures can be represented by the solutions of NLEVPs. The structures I analyze are multi-body structures that are made of simple interconnected pieces, such as elastic strings strung together into a spider web. Once a multi-body structure has been cast into the NLEVP form, an NLEVP solver can be used to find the vibration modes. Finally, I demonstrate that this method can be computationally faster than many traditional modal analysis techniques. | en |
| dc.description.degree | Doctor of Philosophy | en |
| dc.format.medium | ETD | en |
| dc.identifier.other | vt_gsexam:37786 | en |
| dc.identifier.uri | http://hdl.handle.net/10919/115129 | en |
| dc.language.iso | en | en |
| dc.publisher | Virginia Tech | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject | mechanical vibrations | en |
| dc.subject | sensor networks | en |
| dc.subject | Cramér-Rao bound | en |
| dc.subject | controllability matrices | en |
| dc.subject | nonlinear eigenvalue problems | en |
| dc.title | Vibrations of mechanical structures: source localization and nonlinear eigenvalue problems for mode calculation | en |
| dc.type | Dissertation | en |
| thesis.degree.discipline | Mathematics | en |
| thesis.degree.grantor | Virginia Polytechnic Institute and State University | en |
| thesis.degree.level | doctoral | en |
| thesis.degree.name | Doctor of Philosophy | en |
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