Using Hankel Singular Values To Analyze Weak Vibrations
Report discusses use of Hankel singular values in identifying mathematical models of dynamics of flexible structure from measurements of vibrations in presence of noise.
Gawronski, Wodek K.↗
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Accurate computation and continuation of homoclinic and heteroclinic orbits for singular perturbation problems
In earlier papers, Doedel and the authors have developed a numerical method and derived error estimates for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in R(exp n). The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using a local (linear or higher order) approximation of the stable and unstable manifolds. A practical limitation for the computation of homoclinic and heteroclinic orbits has been the difficulty in obtaining starting orbits. Typically these were obtained from a closed form solution or via a homotopy from a known solution. Here we consider extensions of our algorithm which allow us to obtain starting orbits on the continuation branch in a more systematic way as well as make the continuation algorithm more flexible. In applications, we use the continuation software package AUTO in combination with some initial value software. The examples considered include computation of homoclinic orbits in a singular perturbation problem and in a turbulent fluid boundary layer in the wall region problem.
Vaughan, William W.↗
The complex structured singular value
A tutorial introduction to the complex structured singular value (mu) is presented, with an emphasis on the mathematical aspects of mu. The mu-based methods discussed here have been useful for analyzing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared, and a simple synthesis problem is studied. Uncertain systems are represented using linear fractional transformations which naturally unify the frequency-domain and state space methods.
Packard, A.↗
Reconstruction of low-dimensional magnetospheric dynamics by singular spectrum analysis
Singular spectrum analysis was used to study global magnetospheric dynamics using time series auroral electrojet (AE) data. The calculated correlation dimension of approximately 2.5 confirms the low dimensionality of earlier results. This technique indicated that global magnetospheric dynamics can be described by three variables whose dynamical features were obtained from the AE data.
Sharma, A. S.↗
A note on singularities of the 3-D Euler equation
In this paper, we consider analytic initial conditions with finite energy, whose complex spatial continuation is a superposition of a smooth background flow and a singular field. Through explicit calculation in the complex plane, we show that under some assumptions, the solution to the 3-D Euler equation ceases to be analytic in the real domain in finite time.
Tanveer, S.↗
Finite-surface method for the Maxwell equations with corner singularities
The finite-surface method for the two-dimensional Maxwell equations in generalized coordinates is extended to treat perfect conductor boundaries with sharp corners. Known singular forms of the grid and the electromagnetic fields in the neighborhood of each corner are used to obtain accurate approximations to the surface and line integrals appearing in the method. Numerical results are presented for a harmonic plane wave incident on a finite flat plate. Comparisons with exact solutions show good agreement.
Vinokur, Marcel↗
Electronic structure and the van Hove singularity scenario in high-T(sub c)H(g)Ba2CuO(4+delta) superconductors
The electronic structure and the hole concentrations in the high Tc superconductor HgBa2CuO(4+delta) (delta = O, 1) has been investigated by employing a first principles full potential self-consistent LMTO method with the local density functional theory. The scalar relativistic effects have been considered. The hole concentrations of the Cu-d and O-p(x,y) orbitals are seen to be larger for the HgBaCuO5 system than those of the HgBaCuO4 solid. However, the van Hove singularity (vHs) induced Cu-d and O-p peak which is seen to lie comparatively away and above the Fermi level in the delta = 1 system shifts towards the Fermi level in the delta = 0 system. Thus, the superconducting behavior appears to originate from the occurrence of the vHs peak at the Fermi level. The Fermi surface nesting area in the delta = 0 compound is seen to be larger than in the delta = 1 compound. The calculation reveals that the increase in pressure on the crystal enhances the hole concentrations but without showing any optimum value, On the other hand, the vHs peak approaches to-wards the Fermi level with pressure and crosses the Fermi surface near V/Vo approximately equals 0.625 (V and Vo are the crystal volumes at high and normal pressures, respectively). Our calculated value of the bulk modulus equal to 0.626 Mbar predicts the occurrence of this crossover at about 24 GPa which is in complete agreement with the experimental value. At this pressure the compound has maximum nesting area and self-doped behavior.
Agrawal, Bal K.↗
On the Stochastic Quantization Method: Characteristics and Applications to Singular Systems
Introducing the generalized Langevin equation, we extend the stochastic quantization method so as to deal with singular dynamical systems beyond the ordinary territory of quantum mechanics. We also show how the uncertainty relation is built up to the quantum mechanical limit with respect to fictitious time, irrespective of its initial value, within the framework of the usual stochastic quantization method.
Kanenaga, Masahiko↗
Vibrating Systems with Singular Mass-Inertia Matrices
Vibrating systems with singular mass-inertia matrices arise in recent continuum models of Smart Structures (beams with PZT strips) in assessing the damping attainable with rate feedback. While they do not quite yield 'distributed' controls, we show that they can provide a fixed nonzero lower bound for the damping coefficient at all mode frequencies. The mathematical machinery for modelling the motion involves the theory of Semigroups of Operators. We consider a Timoshenko model for torsion only, a 'smart string,' where the damping coefficient turns out to be a constant at all frequencies. We also observe that the damping increases initially with the feedback gain but decreases to zero eventually as the gain increases without limit.
Balakrishnan, A. V.↗
Singular Plasma Disturbance in the Low-Latitude F Region
We describe here a new phenomenon characterized by unusual patterns of ion drifts inside ion density depletion regions observed by the AE-E satellite in the low-latitude F region. In about 30 depletions, vertical ion drift relative to the background was upward on the western sides, downward on the eastern sides, and zero near the middle where the density depletion was greatest. These drift characteristics are distinct from those observed in plasma bubble depletions. The structures reported here were observed on circular orbits below 300 km altitude and had density depletions of up to 2 orders of magnitude or more below the ambient ion density. The upward and downward drift excursions were up to 200 m/s relative to the background. Almost all these structures were observed over oceans or near coasts and largely between +/- 10 deg and +/- 30 deg clip latitude. The structures were observed mostly as isolated, single depletion regions with the majority of them about 250 km wide in the east-west direction. They occurred during quiet magnetic conditions with near-equal occurrence frequencies in the premidnight and postmidnight periods. The characteristic density and drift signatures indicate westward propagating disturbances in which the bottomside F layer is first lifted and then returned back to its original position, leaving the ionosphere undisturbed after the disturbance passes by. The estimated speed of these disturbances is of the order of 200 m/s. These unique solitary plasma disturbances, which we designate as singular plasma disturbances, are associated with a propagating source of E x B drift, not driven by neutral perturbations at the altitude of observation.
Singh, Sardul↗
Singular Isothermal Disks and the Formation of Multiple Stars
A crucial missing ingredient in previous theoretical studies of fragmentation is the inclusion of dynamically important levels of magnetic fields. As a minimal model for a candidate presursor to the formation of binary and multiple stars, we therefore consider the equilibrium configuration of isopedically magnetized, scale-free, singular isothermal disks, without the assumption of axial symmetry. We find that lopsided (M = 1) configurations exist at any dimensionless rotation rate, including zero. Multiple-lobed (M = 2, 3, 4, ...) configurations bifurcate from an underlying axisymmetric sequence at progressively higher dimensionless rates of rotation, but such nonaxisymmetric sequences always terminate in shockwaves before they have a chance to fission into separate bodies. We advance the hypothesis that binary and multiple star-formation from smooth (i.e., not highly turbulent) starting states that are supercritical but in unstable mechanical balance requires the rapid (i.e., dynamical) loss of magnetic flux at some stage of the ensuing gravitational collapse.
Galli, Daniele↗
Singular Isothermal Disks: Nonaxiymmetric Bifurcations and Equilibria - Paper 2
We review the difficulties of the classical fission and fragmentation hypotheses for the formation of binary and multiple stars. A crucial missing ingredient in previous theoretical studies is the inclusion of dynamically important levels of magnetic fields. As a minimal model for a candidate presursor to the formation of binary and multiple stars, we therefore formulate and solve the problem of the equilibria of isopedically magnetized, singular isothermal disks, without the assumption of axial symmetry. Considerable analytical progress can be made if we restrict our attention to models that are scale-free, i.e., that have surface densities that vary inversely with distance omega from the rotation axis of the system. In agreement with earlier analysis by Syer and Tremaine, we find that lopsided (M = 1) configurations exist at any dimensionless rotation rate, including zero. Multiple-lobed (M = 2, 3, 4, ...) configurations bifurcate from an underlying axisymmetric sequence at progressively higher dimensionless rates of rotation, but such nonaxisymmetric sequences always terminate in shockwaves before they have a chance to fission into M = 2, 3, 4, ... separate bodies. On the basis of our experience in this paper, we advance the hypothesis that binary and multiple star-formation from smooth (i.e., not highly turbulent) starting states that are supercritical but in unstable mechanical balance requires the rapid (i.e., dynamical) loss of magnetic flux at some stage of the ensuing gravitational collapse.
Galli, Danielle↗
Robust Fault Detection for Aircraft Using Mixed Structured Singular Value Theory and Fuzzy Logic
The purpose of fault detection is to identify when a fault or failure has occurred in a system such as an aircraft or expendable launch vehicle. The faults may occur in sensors, actuators, structural components, etc. One of the primary approaches to model-based fault detection relies on analytical redundancy. That is the output of a computer-based model (actually a state estimator) is compared with the sensor measurements of the actual system to determine when a fault has occurred. Unfortunately, the state estimator is based on an idealized mathematical description of the underlying plant that is never totally accurate. As a result of these modeling errors, false alarms can occur. This research uses mixed structured singular value theory, a relatively recent and powerful robustness analysis tool, to develop robust estimators and demonstrates the use of these estimators in fault detection. To allow qualitative human experience to be effectively incorporated into the detection process fuzzy logic is used to predict the seriousness of the fault that has occurred.
Collins, Emmanuel G.↗
On the Formulation of Weakly Singular Displacement/Traction Integral Equations; and Their Solution by the MLPG Method
In this paper, a very simple method is used to derive the weakly singular traction boundary integral equation based on the integral relationships for displacement gradients. The concept of the MLPG method is employed to solve the integral equations, especially those arising in solid mechanics. A moving Least Squares (MLS) interpolation is selected to approximate the trial functions in this paper. Five boundary integral Solution methods are introduced: direct solution method; displacement boundary-value problem; traction boundary-value problem; mixed boundary-value problem; and boundary variational principle. Based on the local weak form of the BIE, four different nodal-based local test functions are selected, leading to four different MLPG methods for each BIE solution method. These methods combine the advantages of the MLPG method and the boundary element method.
Atluri, Satya N.↗
On the Singularity in the Estimation of the Quaternion-of-Rotation
It has been claimed in the archival literature that the covariance matrix of a Kalman filter, which is designed to estimate the quaternion-of-rotation, is necessarily rank, deficient because the normality constraint of the quaternion produces dependence between the quaternion elements. In reality, though, this phenomenon does not occur. The covariance matrix is not singular, and the filter is well behaved. Several simple examples are presented th at demonstrate the regularity of the covariance matrix. First, a Kalman filter is designed to estimate variables subject to a functional relationship. Then the particular problem of quaternion estimation is analyzed. It is shown that the discrepancy stems from the fact that the functional relationship exists between the elements of the quaternion but not between its estimate elements.
Bar-Itzhack, Itzhack Y.↗
Planning Paths Through Singularities in the Center of Mass Space
The center of mass space is a convenient space for planning motions that minimize reaction forces at the robot's base or optimize the stability of a mechanism. A unique problem associated with path planning in the center of mass space is the potential existence of multiple center of mass images for a single Cartesian obstacle, since a single center of mass location can correspond to multiple robot joint configurations. The existence of multiple images results in a need to either maintain multiple center of mass obstacle maps or to update obstacle locations when the robot passes through a singularity, such as when it moves from an elbow-up to an elbow-down configuration. To illustrate the concepts presented in this paper, a path is planned for an example task requiring motion through multiple center of mass space maps. The object of the path planning algorithm is to locate the bang- bang acceleration profile that minimizes the robot's base reactions in the presence of a single Cartesian obstacle. To simplify the presentation, only non-redundant robots are considered and joint non-linearities are neglected.
Doggett, William R.↗
A Survey of Singular Value Decomposition Methods and Performance Comparison of Some Available Serial Codes
This contractor report describes a performance comparison of available alternative complete Singular Value Decomposition (SVD) methods and implementations which are suitable for incorporation into point spread function deconvolution algorithms. The report also presents a survey of alternative algorithms, including partial SVD's special case SVD's, and others developed for concurrent processing systems.
Plassman, Gerald E.↗
Interannual Length-of-Day Variations and the ENSO Phenomenon: Insights via Singular Spectral Analysis
Singular spectrum analysis (SSA), used in both single channel and complex SSA (CSSA) modes, is applied to time series of Length-of-Day (LOD) variations and the Modified Southern Oscillation Index (MSOI), focusing on interannual periods.
El Nino↗