Subset Of Eigenvalues Research Articles

SBFEM with reduced modal basis for hydrodynamic bearings

AbstractThe numerical effort of transient rotordynamic simulations is often dominated by the computation of nonlinear hydrodynamic bearing forces. These forces are described by the Reynolds equation and need to be computed at every time step. Usually, numerical models, analytical approximations, or look‐up table techniques are employed, depending on the desired tradeoff between accuracy and computational cost. In recent studies, a semi‐analytical approach based on the scaled boundary finite element method (SBFEM) has been developed as an efficient alternative to these methods. The partial differential equation is transformed into a system of ordinary differential equations, leading to an eigenvalue problem. Here, the numerical effort can be further decreased by means of a modal reduction, which is investigated in this study. The shaft eccentricity determines the smoothness of the hydrodynamic pressure field and is identified as an adequate indicator as to what subset of eigenvalues and eigenvectors should be considered in the solution.

Analysis of Multidimensional Time Delay Systems Using Lambert W Function

In this note, the analysis of time delay systems (TDSs) using Lambert W function approach is reassessed. A common canonical (CC) form of time delay systems is defined. We extended the recent results of Cepeda–Gomez and Michiels (2015, “Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using the Matrix Lambert W Function,” Automatica, 53, pp. 339–345) for second-order into nth order system. The eigenvalues of a time delay system are either real or complex conjugate pairs and therefore, the whole eigenspectrum can be associated with only two real branches of the Lambert W function. A new class of time delay systems is characterized to extend the applicability of the above-said method. Moreover, this approach has been exploited to design a controller which places a subset of eigenvalues at desired locations. Stability is guaranteed by using a new algorithm developed in this paper, which is based on the Nyquist plot. The approach is validated through numerical examples.

3D Vascular Decomposition and Classification for Computer-Aided Detection

In this study, we propose a weighted approximate convex decomposition (WACD) and classification methodology for computer-aided detection (CADe) and analysis. We start by addressing the problem of vascular decomposition as a cluster optimization problem and introduce a methodology for compact geometric decomposition. The classification of decomposed vessel sections is performed using the most relevant eigenvalues obtained through feature selection. The method was validated using presegmented sections of vasculature archived for 98 aneurysms in 112 patients. We test first for vascular decomposition and next for classification. The proposed method produced promising results with an estimated 81.5% of the vessel sections correctly decomposed. Recursive feature elimination was performed to find the most compact and informative set of features. We showed that the selected subset of eigenvalues produces minimum error and improved classifier precision. The method was also validated on a longitudinal study of four cases having internal cerebral aneurysms. Volumetric and surface area comparisons were made between expert-segmented sections and WACD classified sections containing aneurysms. Results suggest that the approach is able to classify and detect changes in aneurysm volumes and surface areas close to that segmented by an expert.

Determining If Two Solid Ellipsoids Intersect

An analytical method is presented for determining if two ellipsoids share the same volume. The formulation involves adding an extra dimension to the solution space and examining eigenvalues that are associated with degenerate quadric surfaces. The eigenvalue behavior is characterized and then demonstrated with an example. The same method is also used to determine if two ellipsoids appear to share the same projected area based on an observer' s viewing angle. The following approach yields direct results without approximation, iteration, or any form of numerical search. It is computationally efe cient in the sense that no dimensional distortions, coordinate rotations, transformations, or eigenvector computations are needed. S the U.S. Satellite Catalog transitions from general perturba- tions to special perturbations, the positional accuracy of each space object will be readily available in the form of a covariance matrix. These covariances can be used to determine probability of collision, radio-frequency interference, and/or incidental laser illu- mination. Because the probability calculations can be computation- allyburdensome,itis desirable to prescreen candidate objectsbased on user-dee ned thresholds. Specie cally, each object can be repre- sented by a covariance-based ellipsoid and then processed to deter- mine if its uncertainty volume shares some space in common with another' s. Solid ellipsoids (or their projections) that do not intersect can be eliminated from further processing. This paper presents a simple analytical method to perform such screening. To date, all ellipsoidal prescreening methods involve numerical searches. 1 For computational efe ciency such prescreening is often reduced to spheres or keep-out boxes that have much larger vol- umes but allow for quick distance comparisons. The drawback to suchscreeningisthattheselargervolumescausemanyobjectstobe- come candidates for further (albeit unnecessary) processing. These methods result in increased downstream computational processing and/or increased operator workload to further assess potential satel- lite conjunctions. The following method adds an extra dimension to the solution space. The subset of eigenvalues that are associated with inter- secting degenerate quadric surfaces are then examined. The same method is also used to determine if two ellipsoids appear to share the same projected area based on viewing angle. The approach yields direct results without approximation, iteration, or any form of numerical search. It is computationally efe cient in the sense that no dimensional distortions, coordinate rotations, transforma- tions, or eigenvector computations are needed. This method ex- pands the two-dimensional work of Hill 2 in his formulation of degenerate conics (i.e., the characteristic matrix is singular). It also furthers his work by examining the associated eigenvalue behavior.

On a singular Sturm-Liouville problem involving an advanced functional differential equation

Solutions to a boundary-value problem involving a second-order linear functional differential equation with an advanced argument are investigated in this paper. The boundary conditions imposed on the differential equation are analogous to conditions defining various singular Sturm-Liouville problems, and if an eigenvalue parameter is introduced certain properties of the spectrum can be deduced having analogues with the classical problem. Dirichlet series solutions are constructed for the problem and it is established that the spectrum contains an infinite number of real positive eigenvalues. A Laplace transform analysis of the problem then reveals that the spectrum does not generically consist of isolated points and that there may be an infinite number of eigenfunctions corresponding to a given eigenvalue. In contrast, it is also shown that there is a subset of eigenvalues that correspond to the zeros of an entire function for which the corresponding eigenfunctions are unique.

Perturbing a rectangular membrane with a restorative force: effects on eigenbvalues and eigenfunctions.

Series expansions are obtained for a rich subset of eigenvalues and eigenfunctions of an operator that arises in the study of rectangular membranes: the operator is the 2-D Laplacian with restorative force term and Dirichlet boundary conditions. Expansions are extracted by considering the restorative force term as a linear perturbation of the Laplacian; errors of truncation for these expansions are estimated. Theriteria defining the subset of eigenvalues and eigenfunctions that can be studied depends only on the size and linearity of the perturbation. The results are valid for almost all rectangular domains.

A parallel algorithm for the partial single-input pole assignment problem

For a linear control system, we introduce a parallel algorithm to assign a desired subset of eigenvalues to a single-input linear invariant dynamic system. We obtain a sequential algorithm as a particular case. The proposed algorithms are conceptually simple and are based on the computation of left eigenvectors of the state matrix. In addition, the parallel algorithm parallelizes easily as the numerical examples show.

A finite‐element analysis of sound propagation in a nonuniform moving medium

The problem of sound propagation in shear flow with temperature gradient in a rectangular duct is studied by a finite‐element method. The formulation leads to an eigensystem containing both symmetric and nonsymmetric matrices. A subset of eigenvalues and their corresponding eigenvectors is extracted from this eigensystem by means of a very efficient Block–Stodola iterative technique. Applications of this methodology to various flow problems are presented and comparison of the present results with those of other investigators are made. The present methodology is shown to be computationally economical, accurate, and sufficiently versatile to treat a wide range of duct flow problems.