Showing papers on "Floquet theory published in 1983"

Analysis of Periodically Time-Varying Systems

Abstract: Part-I Theory and Techniques.- 1 Historical Perspective.- 1.1 The Nature of Systems with Periodically Time-Varying Parameters.- 1.2 1831-1887 Faraday to Rayleigh-Early Experimentalists and Theorists.- 1.3 1918-1940 The First Applications.- 1.4 Second Generation Applications.- 1.5 Recent Theoretical Developments.- 1.6 Commonplace Illustrations of Parametric Behaviour.- References for Chapter 1.- Problems.- 2 The Equations and Their Properties.- 2.1 Hill Equations.- 2.2 Matrix Formulation of Hill Equations.- 2.3 The State Transition Matrix.- 2.4 Floquet Theory.- 2.5 Second Order Systems.- 2.6 Natural Modes of Solution.- 2.7 Concluding Comments.- References for Chapter 2.- Problems.- 3 Solutions to Periodic Differential Equations.- 3.1 Solutions Over One Period of the Coefficient.- 3.2 The Meissner Equation.- 3.3 Solution at Any Time for a Second Order Periodic Equation.- 3.4 Evaluation of ?(?, 0)m, m Integral.- 3.5 The Hill Equation with a Staircase Coefficient.- 3.6 The Hill Equation with a Sawtooth Waveform Coefficient.- 3.6.1 The Wronskian Matrix with z Negative.- 3.6.2 The Wronskian Matrix with z Zero.- 3.6.3 The Case of ? Negative.- 3.7 The Hill Equation with a Positive Slope, Sawtooth Waveform Coefficient.- 3.8 The Hill Equation with a Triangular Coefficient.- 3.9 The Hill Equation with a Trapezoidal Coefficient.- 3.10 Bessel Function Generation.- 3.11 The Hill Equation with a Repetitive Exponential Coefficient.- 3.12 The Hill Equation with a Coefficient in the Form of a Repetitive Sequence of Impulses.- 3.13 Equations of Higher Order.- 3.14 Response to a Sinusoidal Forcing Function.- 3.15 Phase Space Analysis.- 3.16 Concluding Comments.- References for Chapter 3.- Problems.- 4 Stability.- 4.1 Types of Stability.- 4.2 Stability Theorems for Periodic Systems.- 4.3 Second Order Systems.- 4.3.1 Stability and the Characteristic Exponent.- 4.3.2 The Meissner Equation.- 4.3.3 The Hill Equation with an Impulsive Coefficient.- 4.3.4 The Hill Equation with a Sawtooth Waveform Coefficient.- 4.3.5 The Hill Equation with a Triangular Waveform Coefficient.- 4.3.6 Hill Determinant Analysis.- 4.3.7 Parametric Frequencies for Second Order Systems.- 4.4 General Order Systems.- 4.4.1 Hill Determinant Analysis for General Order Systems.- 4.4.2 Residues of the Hill Determinant for q ? 0.- 4.4.3 Instability and Parametric Frequencies for General Systems.- 4.4.4 Stability Diagrams for General Order Systems.- 4.5 Natural Modes and Mode Diagrams.- 4.5.1 Nature of the Basis Solutions.- 4.5.2 P Type Solutions.- 4.5.3 C Type Solutions.- 4.5.4 N Type Solutions.- 4.5.5 Modes of Solution.- 4.5.6 The Modes of a Second Order Periodic System.- 4.5.7 Boundary Modes.- 4.5.8 Second Order System with Losses.- 4.5.9 Modes for Systems of General Order.- 4.5.10 Coexistence.- 4.6 Short Time Stability.- References for Chapter 4.- Problems.- 5 A Modelling Technique for Hill Equations.- 5.1 Convergence of the Hill Determinant and Significance of the Harmonics of the Periodic Coefficients.- 5.1.1 Second Order Systems.- 5.1.2 General Order Systems.- 5.2 A Modelling Philosophy for Intractable Hill Equations.- 5.3 The Frequency Spectrum of a Periodic Staircase Coefficient.- 5.4 Piecewise Linear Models.- 5.4.1 General Comments.- 5.4.2 Trapezoidal Models.- 5.5 Forced Response Modelling.- 5.6 Stability Diagram and Characteristic Exponent Modelling.- 5.7 Models for Nonlinear Hill Equations.- 5.8 A Note on Discrete Spectral Analysis.- 5.9 Concluding Remarks.- References for Chapter 5.- Problems.- 6 The Mathieu Equation.- 6.1 Classical Methods for Analysis and Their Limitations.- 6.1.1 Periodic Solutions.- 6.1.2 Mathieu Functions of Fractional Order.- 6.1.3 Fractional Order Unstable Solutions.- 6.1.4 Limitations of the Classical Method of Treatment.- 6.2 Numerical Solution of the Mathieu Equation.- 6.3 Modelling Techniques for Analysis.- 6.3.1 Rectangular Waveform Models.- 6.3.2 Trapezoidal Waveform Models.- 6.3.3 Staircase Waveform Models.- 6.3.4 Performance Comparison of the Models.- 6.4 Stability Diagrams for the Mathieu Equation.- 6.4.1 The Lossless Mathieu Equation.- 6.4.2 The Damped (Lossy) Mathieu Equation.- 6.4.3 Sufficient Conditions for the Stability of the Damped Mathieu Equation.- References for Chapter 6.- Problems.- II Applications.- 7 Practical Periodically Variable Systems.- 7.1 The Quadrupole Mass Spectrometer.- 7.1.1 Spatially Linear Electric Fields.- 7.1.2 The Quadrupole Mass Filter.- 7.1.3 The Monopole Mass Spectrometer.- 7.1.4 The Quadrupole Ion Trap.- 7.1.5 Simulation of Quadrupole Devices.- 7.1.6 Non idealities in Quadrupole Devices.- 7.2 Dynamic Buckling of Structures.- 7.3 Elliptical Waveguides.- 7.3.1 The Helmholtz Equation.- 7.3.2 Rectangular Waveguides.- 7.3.3 Circular Waveguides.- 7.3.4 Elliptical Waveguides.- 7.3.5 Computation of the Cut-off Frequencies for an Elliptical Waveguide..- 7.4 Wave Propagation in Periodic Media.- 7.4.1 Pass and Stop Bands.- 7.4.2 The ? - ?r (Brillouin) Diagram.- 7.4.3 Electromagnetic Wave Propagation in Periodic Media.- 7.4.4 Guided Electromagnetic Wave Propagation in Periodic Media.- 7.4.5 Electrons in Crystal Lattices.- 7.4.6 Other Examples of Waves in Periodic Media:.- 7.5 Electric Circuit Applications.- 7.5.1 Degenerate Parametric Amplification.- 7.5.2 Degenerate Parametric Amplification in High Order Periodic Networks.- 7.5.3 Nondegenerate Parametric Amplification.- 7.5.4 Parametric Up Converters.- 7.5.5 N-path Networks.- References for Chapter 7.- Problems.- Appendix Bessel Function Generation by Chebyshev Polynomial Methods.- References for Appendix.

Stark ionization in dc and ac fields: An L2 complex-coordinate approach

Abstract: A finite-dimensional-matrix technique valid for computation of complex eigenvalues and eigenfunctions useful for discussing time evolution in both dc and ac Stark fields is presented. The complex eigenvalue parameters are those of appropriately analytically continued, time-independent Stark Hamiltonians as obtained via the complex scale transformation $r\ensuremath{\rightarrow}r{e}^{i\ensuremath{\theta}}$. Such a transformation distorts the continuous spectrum away from the real axis, exposing the Stark resonances, and also allowing use of finite variational expansions employing ${L}^{2}$ basis functions chosen from a complete discrete basis. The structure of the dc and ac Stark Hamiltonians is discussed and extensive convergence studies performed in both the dc and ac cases to fully document the utility of the method. Sudden and adiabatic dc Stark time evolution is used to illustrate the power of finite-dimensional-matrix methods in describing complex, multiple-time-scale time evolution. The relationship between the ac Stark Hamiltonian used (a time-independent truncated Floquet Hamiltonian) and continued-fraction perturbation theory follows easily via use of matrix partitioning, and provides a particularly straightforward derivation of these results. Finally, some illustrative calculations of off-resonant generalized cross sections are given at low and high intensities, indicating that the method works satisfactorily at intensities the order of internal atomic field strengths. A more detailed discussion of time evolution in two-, three-, and four-photon ionization processes appears in the following paper by Holt, Raymer, and Reinhardt.

Some analysis methods for rotating systems with periodic coefficients

Abstract: 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

Study, extension, and application of Floquet theory for quantum molecular systems in an oscillating field

Abstract: Floquet theory is applied to systems whose Hamiltonians are periodic in time. Specifically, the application deals with the sinusoidal Hamiltonians for the semiclassical approximation of the radiation --- quantum-molecule interaction in an intense field. For these types of Hamiltonians new time symmetries are shown to exist in the Floquet solutions and consequently to yield three useful properties. Three formal approaches to the Floquet solutions are compared. Also the Floquet-state "mean energy" is shown to play the major role in the static part of the molecular-state energy fluctuations. Numerical application to the dynamics of the diatomic HF vibrotor shows multistate participation in both the one- and two-photon excitations, and "resonance interaction" between the one- and two-photon absorption is observed. Also, Magnus approximations to the exact numerical calculations show excellent agreement.

Parametric Instabilities of Internal Gravity Waves in Boussinesq Fluids with Large Reynolds Numbers

Abstract: Previous studies of parametric instabilities of a finite-amplitude internal gravity wave in dissipationless Boussinesq fluids predict the largest growth rates for perturbations with vanishingly small wavelengths, leading to the practical question of which perturbations are most unstable in the presence of molecular dissipation. At high Reynolds numbers, the amplitude of the gravity wave decays on spatial and temporal scales which are large compared to the wavelength and period respectively. Therefore, the interaction equations describing the growth or decay of small perturbations can be solved by a conventional expansion procedure yielding a leading term in the form of a Floquet solution. Numerical calculations show that the competitive processes of parametric growth and viscous decay produce instabilities with maximum growth rates at wavelengths which may be similar or even longer than the wavelength of the basic gravity wave. Two transformations leading to a significant reduction of computation...

Slow-Wave Coplanar Waveguide on Periodically Doped Semiconductor Substrate

Abstract: A metal-insulator-semiconductor (MIS) coplanar waveguide with periodically doped substrate is described. An efficient numerical method is introduced in order to obtain the propagation constants and the characteristic impedances of the constituent sections of each period. Using the results, the characteristic of the periodic MIS coplanar waveguide is analyzed by Floquet's theorem. The theoretical study shows reduction of attenuation and enhancement of the slow-wave factor at certain frequencies, compared to the uniform MIS coplanar waveguide. This structure is experimentally simulated and shows good agreement theory.

Modal control of an unstable periodic orbit

Abstract: Floquet theory is applied to the problem of designing a control system for a satellite in an unstable periodic orbit. Expansion about a periodic orbit produces a time-periodic linear system, which is augmented by a time-periodic control term. It is shown that this can be done such that (1) the application of control produces only inertial accelerations, (2) positive real Poincareexponents are shifted into the left half-plane, and (3) the shift of the exponent is linear with control gain. These developments are applied to an unstable orbit near the earth-moon L(3) point pertubed by the sun. Finally, it is shown that the control theory can be extended to include first order perturbations about the periodic orbit without increase in control cost.

Complex scaling of ac Stark Hamiltonians

Abstract: For a two‐body atom in a temporally periodic, spatially uniform field, it is shown that in an appropriate gauge the essential spectrum of the Floquet Hamiltonian rotates about a certain set of thresholds when subjected to a complex scaling transformation.

Dissipative Quantum Dynamics for Systems Periodic in Time

Abstract: A model of dissipative quantum dynamics (with a nonlinear friction term) is applied to systems periodic in time. The model is compared with the standard approaches based on the Floquet theorem. It is shown that for weak frictions the asymptotic states of the dynamics we propose are the periodic steady states which are usually postulated to be the states relevant for the statistical mechanics of time-periodic systems. A solution to the problem of nonuniqueness of the “quasienergies” is proposed. The implication of a nonlinear evolution for Ludwig's axiomatization is briefly outlined.

Attitude Stability of a Flexible Asymmetric Dual Spin Spacecraft

Abstract: Stability of an attitude motion of a large dual spin spacecraft is studied; the effects of various kinds of asymmetries of the spacecraft and energy dissipations in the spacecraft are examined. Attitude instabilities due to interactions between the asymmetries and the interactions between the asymmetries and the energy dissipations are examined in detail. The analysis is based on the method of multiple time scales. The results are verified by numerical solutions based on the Floquet’s theorem.

Antiplane strain harmonic waves in infinite, elastic, periodically triple‐layered media

Abstract: Propagation of horizontally polarized shear waves in periodically triple‐layered, elastic medium is studied by using Floquet’s theorem. The dispersion relation is characterized. The propagation and attenuation of harmonic waves inside and outside the Brillouin zones are identified. Variation of the spectrum following the modification of the comparative mechanical properties of the three layers is also examined.

Representations of solutions of periodic partial differential equations

Abstract: A variant of the Floquet theory for partial differential equations is constructed. Exponentially increasing solutions of periodic hypoelliptic equations and systems are decomposed into integrals over Floquet solutions. Analogous results are obtained for equations with deviating argument and for boundary-value problems in domains of the periodic wave guide type. The question of nonzero L2(Rn)-solutions of equations in these classes is also examined. Bibliography: 41 titles.

A floquet theory of the linear synchronous machine

Abstract: The differential equation describing the three-phase linear synchronous machine containing an arbitrary stator MMF distribution is reformulated and solved as a perturbation theory problem. The solution algorithm presented also produces a transformation capable of reducing to constant coefficient form the differential equation describing the machine. The theory is illustrated by applying it to a simple two pole machine.

Electromagnetic Scattering by an Infinite Array of Periodic Broken Wires Buried in a Dielectric Sheet

Abstract: A general formulation of the scattering of a plane wave from a periodic broken-wire grid buried in a dielectric sheet is described for a general incidence angle and for arbitrary linear polarisation. The approach is applicable for grids in an infinite dielectric and when located at the boundary interface of two dielectrics. The two-dimensional periodicity of the grid enables the scattered field to be expanded in a set of Floquet modes, whose coefficients are obtained from the appropriate current expansion determined by the method of moments. Predicted results are compared with measured transmission characteristics of a grid buried in a thin dielectric sheet.

Energy stability of thermally modulated Poiseuille flow

Abstract: The nonlinear stability of time-periodic Poiseuille Flow is investigated using the Energy Theory. The time dependency in the basic flow is obtained by periodic modulation of the ground temperature. Energy stability limits, obtained by a combination of Galerkin and Floquet methods, are lowered by the thermal modulation. For both strong and mean energy formulations, the effect of increasing the modulation amplitude is to destabilize the flow which is demonstrated by a decrease in the stability boundary. A shift in the critical wavenumber due to modulation is also observed.