Browsing by Subject "Stability"

Aeroelastic phenomena, and particularly instabilities and Limit Cycle Oscillations (LCOs), remain an important and complex field of research. Understanding of these instabilities is crucial for the design of aerospace vehicles and turbomachinery, where they may impede performance, reduce efficiency, and induce structural fatigue or failure. The subject of this dissertation – transonic buffet – is one such instability. Transonic buffet is a global flowfield instability characterized by an oscillating shock-wave on the suction-side of an airfoil or wing that produces oscillating forces on the structure. The LCO frequencies scale with theairfoil chord, and are near typical natural frequencies for aerospace structures. Theoretical descriptions of buffet’s mechanism remain incomplete, and its onset is difficult to predict without computationally expensive simulations.

This dissertation seeks to explore two-dimensional buffet using Computational Fluid Dynamics (CFD) and provide insight into its sensitivities to both flow conditions and numerical models. Deeper understanding of the two-dimensional behavior is a critical step in exploring the more complex three dimensional phenomenon. Trends in Reynolds number, angle of attack, and Mach number are explored, as are the impact of wind tunnel walls and airfoil geometry. While most studies included in this document make use of Unsteady Reynolds-Averaged Navier-Stokes (URANS), additional methods of simulation are explored and assessed, including inviscid simulations, Implicit Large Eddy Simulation (ILES), and linearized stability analysis. Buffet shows substantial sensitivity to many of the conditions explored in this work. While the buffet onset point, mean lift coefficient, and frequency tend to be relatively stable, the amplitude of oscillation and the conditions near the offset point are especiallysensitive to many factors. Special care must be taken in discretization and selecting models to apply if these highly sensitive outputs are to be explored. Despite these sensitivities, lower fidelity methods, including inviscid simulations and linearized stability analysis, are successful approaches to studying transonic buffet.

Subtractive machining operations such as milling, turning, and drilling are an essential part of many manufacturing processes. Unfortunately, under certain combinations of machine settings, the motion of the cutting tool can become unstable, due to feedback between consecutive passes of the tool. This phenomenon is known as chatter. Mathematical models, specifically delay differential equations (DDEs), can describe the motion of the cutting tool and predict this instability. While these models are useful, estimates of the models' parameters are necessary in order to apply them to real systems. Unfortunately, estimating the parameters directly can be time-consuming, expensive, and difficult. The objective of this research is to develop automated methods to estimate these parameters indirectly, from time series measurements of the tool's motion which can be collected in a few minutes with sensors attached to the machine. The estimated parameters can then be used to predict when chatter will occur so that the machine operator can select appropriate settings.

One way to estimate the parameters of a dynamics model is to match the characteristic multipliers (CMs) predicted by the model to CMs estimated from time series data. CMs describe the behavior, such as stability, of a dynamical system near a limit cycle. While existing CM estimation methods are available, practical challenges such as measurement noise, limited time series length, and repeated CMs can substantially reduce their accuracy. The first part of this dissertation presents improved methods for estimating CMs from time series. Numerical validation studies demonstrate that the improved methods consistently provide more accurate CM estimates than existing methods in a variety of scenarios.

The second part of this dissertation introduces improvements to CM matching and trajectory matching methods for estimating the parameters of DDEs from noisy time series data. For CM matching, it incorporates the empirical CM estimation improvements from the previous part, and it introduces a way to match multiple CM estimates for each time series. For trajectory matching, it describes how to handle multivariate observations and prior knowledge in a principled way; it uses the spectral element method to provide a convenient representation of the initial interval and reduce the computational cost of computing the objective function; and it fits multiple time series simultaneously. Simulation results demonstrate that these improved methods work well in practice, although CM matching has some limitations which are not a problem for the trajectory matching method.

The final part of this dissertation introduces a new approach to estimate the parameters of a DDE model for milling from noisy time series data, based on the trajectory matching approach described in the previous part. It extends models from the literature to more closely fit the time series data, and it describes a procedure to estimate the unknown parameters in stages, without having to solve a global optimization algorithm for all the parameters simultaneously. Additionally, it adapts the spectral element method to make predictions for this model. Experimental results using time series data collected on an instrumented milling machine demonstrate that the model and fitting procedure successfully estimate parameters for which the predicted stability boundaries approximate the true stability boundaries.

When designing controllers, it is important to consider their stability and safety. However, this can become difficult as the system's complexity increases. This work presents results on the persistent feasibility and stability of externally switching systems. This extends the previous results in the area, which were restricted to narrow classes of systems. The work begins with centralized systems and evolves to explore distributed classes of systems. As with other areas of control system design, distributed methods provide solutions to overcome scaling problems. There are several different tools used in this work to extend the class of systems for which safety and stability assurances can be made. Polytopic sets and their evolution under linear dynamics are used extensively. The parallelization of complex algorithms is another important tool used in these results. Parallelization is a key benefit of distributed systems and prevents the results from falling into the curse of dimensionality. Finally, novel methods of approximating and bounding values are developed, which further reduces the results' numerical complexity. While these approximations do introduce some conservatism, it is often less than the redundancy required in place of formal analysis. The full results of this work advance the study of externally switched systems to applications that, in the past, would not have been possible to study. While fully maturing the field is far too large a goal for any single doctoral dissertation, my contributions include several important advancements that have utility as stand-alone results and provide rich opportunities for future advancements in the field.

Ferroelectric thin films (FETFs) offer a promising new platform for advancing liquid-phase interfacial sensing devices. FETFs are capable of expressing surface charge densities that are an order of magnitude higher than those of traditional charged surfaces in liquid environments (e.g., common oxides, self-assembled monolayers, or electrets). Furthermore, the switchable polarization state of FETFs enables patterning of charge-heterogeneous surfaces whose charge patterns persist over a range of environmental conditions. Integration of FETFs into liquid-phase interfacial sensing devices, however, requires the fabrication of films with nanometer-scale surface roughness, high remnant polarization values, and interfacial stability during prolonged exposure. The objectives of my research were to i) fabricate ferroelectric ultra-smooth lead zirconium titanate (US-PZT) thin films with nanometer-scale surface roughness, ii) establish the interfacial stability of these films after prolonged exposure to aqueous environments, iii) measure the interfacial forces as a function of film polarization and ionic strength, iv) calculate the surface potential of the US-PZT surface using electric double layer (EDL) theory, and v) demonstrate the guided deposition of charged colloidal particles onto locally polarized US-PZT thin films from solution.

I demonstrate the use of ferroelectric US-PZT thin films to manipulate EDL interaction forces in aqueous environments. My work conclusively shows that the polarization state of US-PZT controls EDL formation and can be used to induce the guided deposition of charged colloidal particles in solution.

I present a robust fabrication scheme for making ferroelectric US-PZT thin films from a sol-gel precursor. By optimizing critical thermal processing steps I am able to minimize the in-plane stress of the film and reliably produce US-PZT thin films on the wafer-scale with mean surface roughness values of only 2.4 nm over a 25 μm2 area. I then establish US-PZT film stability in water by measuring changes in film topography, crystallinity, surface chemistry, and electrical properties as a function of exposure duration. My results show that fabrication of crack-free US-PZT thin film is critical for long-term film fidelity in aqueous environments. Furthermore, I found no change in film topography or bulk composition with increasing exposure duration. Prolonged exposure to aqueous environments, however, gradually oxidizes the surface of the US-PZT wich results in a decrease in film resistivity and polarization saturation. Next, I used colloidal probe force microscopy (CPFM) to measure the EDL interaction force as a function of separation distance between polarized US-PZT thin films and a clean borosilicate probe. CPFM measurements were performed on oppositely polarized US-PZT thin films, which expressed either a positive or negative surface charge, and over a range of ionic strengths. The inner-Helmholtz plane (IHP) potential of the US-PZT was determined by fitting the CPFM force-separation data to number of EDL models, including; an analytical EDL model using a constant potential boundary condition with a Stern layer, a charge regulation EDL model, and a numerical EDL model using the non-linear Poisson-Boltzmann equation. Each model provides good agreement with the experimentally measured and predict high IHP surface potential for the polarized US-PZT thin films in solution. Finally, I demonstrate the use of polarized US-PZT to induce the guided deposition of positively or negatively charged colloidal particles from aqueous environments. I explore the effects of ionic strength, particle size, surface roughness, and pH on particle deposition.

Overall, this work demonstrates, for the first time, that FETFs can be used as a platform to manipulate colloidal particles in aqueous environments. The experimental results demonstrate that the surface charge of the FETF is reduced by charge shielding and perform similarly to traditional, charged surfaces in aqueous environments.

In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of points in the extended plane called persistence diagram.

We start with the question of ridding the function of topological noise as suggested by its persistence diagram. We give an algorithm for hierarchically finding such epsilon-simplifications on 2-manifolds as well as answer the question of when it is impossible to simplify a function in higher dimensions.

We continue by examining time-varying functions. The original algorithm computes the persistence pairing from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. We describe how to maintain the pairing in linear time per transposition of consecutive simplices. A side effect of the update algorithm is an elementary proof of the stability of persistence diagrams. We introduce a parametrized family of persistence diagrams called persistence vineyards and illustrate the concept with a vineyard describing a folding of a small peptide. We also base a simple algorithm to compute the rank invariant of a collection of functions on the update procedure.

Guided by the desire to reconstruct stratified spaces from noisy samples, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of a sampled stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample.

Finally, to refine the measurement of local homology the thesis extends the notion of persistent homology to sequences of kernels, images, and cokernels of maps induced by inclusions in a filtration of pairs of spaces. Specifically, we note that persistence in this context is well defined, we prove that the persistence diagrams are stable, and we explain how to compute them. Additionally, we use image persistence to cope with functions on noisy domains.

With the rising level of CO2 in the atmosphere, methods capable of converting CO2 into useful fuels are urgently needed. The electrochemical CO2 reduction has gained significant interest recently due to its ability to use renewable energies. However, the poor stability of catalysts in electrochemical CO2 reduction limit its application in industry. Here we have developed a light-involving method to remove the surface carbonaceous species which are believed to poison the catalysts. By taking advantage of plasmonic properties of the copper catalyst, the stability of the catalysts has apparently improved.

Another problem in electrochemical CO2 reduction is the poor selectivity. One of the main reasons is the existence of the side reaction, hydrogen evolution reaction. Here we have developed a catalyst by dispersing atomic nickel on nitrogen-doped winged carbon nanotubes with the ability to suppress hydrogen evolution during CO2 reduction. The Faradaic Efficiency of CO reached 90% at -1.6 V vs. AgCl/Ag reference electrode while the efficiency of HER had been suppressed to less than 10% in the optimal reaction condition. By comparing with Ni NPs, the suppression of HER can be directly observed in LSV curve. It is suggested that this suppression may result from the lack of adjacent active sites for the Tafel mechanism in HER.

In modern engineering there is a considerable interest in predicting the behavior of post-buckled structures. With current lightweight, aerospace, and high performance applications, structural elements frequently operate beyond their buckled load. This is especially true of plates, which are capable of maintaining stability at loads several times their critical buckling load. Additionally, even structures such as cylindrical shells may be pushed into a post-buckled range in these extreme applications.

Because of the nature of these problems, continuation methods are particularly well suited as a solution method. Continuation methods have been extensively applied to a range of problems in mathematics and physics but have been used to a lesser extent in engineering problems. In the present work, continuation methods are used to solve a variety of buckling and stability problems of discrete dynamical systems, plates and cylinders. The continuation methods, when applied to dynamic mechanical systems, also provide very useful information regarding the modal behavior of the structure, including linearized natural frequencies and mode shapes as a by-product of the solution method.

To verify the results of the continuation calculations, the commercial finite element code ANSYS is used as an independent check. To confirm previously unseen stable equilibrium shapes for square plates, a set of experiments on polycarbonate plates is also presented.

We study how the preimages of a mapping f : X &rarr Y between manifolds vary under perturbations. First, we consider the preimage of a single point and track the history of its connected component as this point varies in Y. This information is compactly represented in a structure that is the generalization of the Reeb graph we call the Reeb space. We study its local and global properties and provide an algorithm for its construction. Using homology, we then consider higher dimensional connectivity of the preimage. We develop a theory quantifying the stability of each homology class under perturbations of the mapping f . This number called robustness is given to each homology class in the preimage. The robustness of a class is the magnitude of the perturbation necessary to remove it from the preimage. The generality of this theory allows for many applications. We apply this theory to quantify the stability of contours, fixed points, periodic orbits, and more.

This work studies the atomistic Thomas-Fermi-Dirac-von Weiszacker model on a Bravais lattice, by establishing relation with the continuum elasticity model, thus provides it a solid microscopic foundation at the atomistic level. More specifically, the stored energy density can be derived from the atomistic TFDW functional by homogenization, by assuming a reasonable stability condition, the discrete deformation function we get from the atomistic model converges to the Cauchy-Born solution from solving the continuum model with a quadratic rate due to underlying inversion symmetry of the lattice. In our analysis, we used two-scale ansatz to construct approximate solutions, discrete Fourier analysis in the consistence estimate, and perturbation technique to analyze the Hessian for the stability analysis.