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All sourcesNASA NTRSDOE OSTI
NASA NTRS1981

Review of analysis methods for rotating systems with periodic coefficients

Two of the more common procedures for analyzing the stability and forced response of equations with periodic coefficients are reviewed: the use of Floquet methods, and the use of multiblade coordinate and harmonic balance methods. The analysis procedures of these periodic coefficient systems are compared with those of the more familiar constant coefficient systems.

Dugundji, J.↗
NASA NTRS1983

Some analysis methods for rotating systems with periodic coefficients

Two of the more common procedures for analyzing the stability and forced response of equations with periodic coefficients are reviewed: the use of Floquet methods, and the use of multiblade coordinate and harmonic balance methods. The analysis procedures of these periodic coefficient systems are compared with those of the more familiar constant coefficient systems. Previously announced in STAR as N82-23702

Dugundji, J.↗
NASA NTRS1974

Periodic Trojan-type orbits in the earth-sun system

Periodic orbits about the triangular equilibrium points are found for the planar restricted three-body problem using the earth-sun system. The maximum semimajor axis for tadpole orbits ranges from the infinitesimal orbit at 1.000 AU to the near-limiting orbit at 1.00285 AU. Horseshoe orbits are found for 1.0029 to 1.0080 AU, larger horseshoes being unstable because of close approaches to the earth. Using stability tests devised by Rabe (1961, 1962), the limit of stability for nonperiodic orbits is found to occur for maximum semimajor axes near 1.0020 AU. In addition, near-periodic tadpole orbits appear to be stable against perturbations by Jupiter and Venus for periods of at least 10,000 yr. The possibility that minor planets actually exist in such orbits is considered.

Weissman, P. R.↗
DOE OSTI2021

Normal stability of slow manifolds in nearly periodic Hamiltonian systems

Kruskal [J. Math. Phys. 3, 806 (1962)] showed that each nearly periodic dynamical system admits a formal U(1) symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly invariant manifolds of each order, near which rapid oscillations are suppressed. We study the nonlinear normal stability of these slow manifolds for nearly periodic Hamiltonian systems on barely symplectic manifolds—manifolds equipped with closed, non-degenerate 2-forms that may be degenerate to leading order. In particular, we establish a sufficient condition for long-term normal stability based on second derivatives of the well-known adiabatic invariant. We use these results to investigate the problem of embedding guiding center dynamics of a magnetized charged particle as a slow manifold in a nearly periodic system. Here, we prove that one previous embedding and two new embeddings enjoy long-term normal stability and thereby strengthen the theoretical justification for these models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

While it is well known that every nearly periodic Hamiltonian system possesses an adiabatic invariant, extant methods for computing terms in the adiabatic invariant series are inefficient. The most popular method involves the heavy intermediate calculation of a non-unique near-identity coordinate transformation, even though the adiabatic invariant itself is a uniquely defined scalar. Here, a less well-known method, developed by S. Omohundro, avoids calculating intermediate sequences of coordinate transformations but is also inefficient as it involves its own sequence of complex intermediate calculations. In order to improve the efficiency of future calculations of adiabatic invariants, we derive generally applicable, readily computable formulas for the first several terms in the adiabatic invariant series. To demonstrate the utility of these formulas, we apply them to charged-particle dynamics in a strong magnetic field and magnetic field-line dynamics when the field lines are nearly closed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Unified analysis of finite-size error for periodic Hartree-Fock and second order Møller-Plesset perturbation theory

Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no comprehensive and rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order Møller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified analysis to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of Lyness [Math. Comp. 30 (1976), pp. 1–23] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations (see X. Xing, X. Li, and L. Lin [J. Chem. Theory Comput. 17 (2021), pp. 4733–4745]). In conclusion, our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Best Practices at the End of Photovoltaic System Performance Period

Responsible and cost-effective dissolution of photovoltaic (PV) system hardware at the end of the performance period has emerged as an important business and environmental consideration. Alternatives include extending the performance period and existing contracts for power purchase, lease, and utility interconnect; refurbishing the plant by correcting any deficiencies; repowering the plant with new PV modules and inverters; or decommissioning the plant and removing all the hardware from the site. Often key decisions are made very early in the project development and might require decommissioning by some certain date after the end of a power purchase agreement. To “abandon in place” is not an alternative acceptable to landowners and regulators, so any financial prospectus should include costs associated with decommissioning, even if those costs are deferred by extending operations, refurbishment, or repowering. Decommissioning costs are driven by regulations regarding the handling and disposal of waste, with reuse and recycling of PV modules and other components preferred as a way to reduce both costs and environmental impact. Each alternative is discussed with order-of-magnitude costs, and recommendations are provided considering site-specific details of that situation, such as estimated costs to refurbish or repower, projected revenue from continued operations, and tax considerations.

14 SOLAR ENERGY↗
DOE OSTI2020

Linear beam stability in periodic focusing systems: Krein signature and band structure

The general question of how a beam becomes unstable has been one of the fundamental research topics among beam and accelerator physicists for several decades. In this study, we revisited the general problem of linear beam stability in periodic focusing systems by applying the concepts of Krein signature and band structure. We numerically calculated the eigenvalues and other associated characteristics of one-period maps, and discussed the stability properties of single-particle motions with skew quadrupoles and envelope perturbations in high-intensity beams on an equal footing. In particular, an application of the Krein theory to envelope instability analysis was newly attempted in this study. Finally, the appearance of instabilities is interpreted as the result of the collision between eigenmodes of opposite Krein signatures and the formation of a band gap.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRS1976

Stability analysis of combined continuous and discrete systems

A discrete, time-invariant system is represented by an equivalent continuous, periodic system. A combined continuous plant and discrete controller may then be formulated as a single continuous, periodic coefficient system. The exact stability solution for the combined system is obtained, as a matrix eigenvalue problem. Periodic system theory also gives some information about the type of instabilities which may be encountered in the combined plant and controller.

Johnson, W.↗
NASA NTRS1972

Stability conditions derived from spectral theory - Discrete systems with periodic feedback.

By use of the spectral theory of linear operators, necessary and sufficient conditions are derived for the stability of a class of discrete time feedback systems with a periodically time-varying feedback gain. These conditions involve a Nyquist plot for an equivalent time-invariant system which may be determined from the frequency response function of the system under consideration. In the special case of a scalar-input scalar-output system, these conditions may be given in a particularly simple form.

Davis, J. H.↗
DOE OSTI2024

Universal transport in periodically driven systems without long-lived quasiparticles

An intriguing regime of universal charge transport at high entropy density has been proposed for periodically driven interacting one-dimensional systems with Bloch bands separated by a large single-particle band gap. For weak interactions, a simple picture based on well-defined Floquet quasiparticles suggests that the system should host a quasisteady state current that depends only on the populations of the system's Floquet-Bloch bands and their associated quasienergy winding numbers. Here we show that such topological transport persists into the strongly interacting regime where the single-particle lifetime becomes shorter than the drive period. Analytically, we show that the value of the current is insensitive to interaction-induced band renormalizations and lifetime broadening when certain conditions are met by the system's nonequilibrium distribution function. We show that these conditions correspond to a quasisteady state. We support these predictions through numerical simulation of a system of strongly interacting fermions in a periodically modulated chain of Sachdev-Ye-Kitaev dots. Our paper establishes universal transport at high entropy density as a robust far from equilibrium topological phenomenon, which can be readily realized with cold atoms in optical lattices. Published by the American Physical Society 2024

Esin, Iliya (ORCID:0000000329590617)↗
NASA NTRS1982

Survey of Spaceflight Safety Systems /Survey period: July 1981 June 1982/

Information from the Survey of Spaceflight Safety Systems considered most applicable to present and future spaceflight activities is summarized. Mission reports for the first three Shuttle Orbiter flights discuss all anomalies which occurred and their resolution, such as the solid rocket booster ignition overpressure spike, and thermal tile damage. Two emergency rescue systems for orbiting space stations are proposed: a dish type rescue capsule and a reusable orbit vehicle. Prebreathing and stage decompression are considered as means of protecting astronauts against decompression effects. The Spacelab Space Sled experiments are described, as is a test program on the assembly of large space structures completed in mid-1981. Finally, collision hazards for space missions are estimated, and operational procedures for minimizing these hazards are discussed.

Brown, N. E.↗