Analysis of Periodically Time-Varying Systems

ISBN-13: 9783642818752 / Angielski / Miękka / 2011 / 176 str.

John A. Richards

Opis

Many of the practical techniques developed for treating systems described by periodic differential equations have arisen in different fields of application; con sequently some procedures have not always been known to workers in areas that might benefit substantially from them. Furthermore, recent analytical methods are computationally based so that it now seems an opportune time for an applications-oriented book to be made available that, in a sense, bridges the fields in which equations with periodic coefficients arise and which draws together analytical methods that are implemented readily. This book seeks to ftll that role, from a user's and not a theoretician's view. The complexities of periodic systems often demand a computational approach. Matrix treatments therefore are emphasized here although algebraic methods have been included where they are useful in their own right or where they establish properties that can be exploited by the matrix approach. The matrix development given calls upon the nomenclature and treatment of H. D'Angelo, Linear Time Varying Systems: Analysis and Synthesis (Boston: Allyn and Bacon 1970) which deals with time-varying systems in general. It is recommended for its modernity and comprehensive approach to systems analysis by matrix methods. Since the present work is applications-oriented no attempt has been made to be complete theoretically by way of presenting all proofs, existence theorems and so on. These can be found in D'Angelo and classic and well-developed treatises such as McLachlan, N. W.: Theory and application of Mathieu functions."

Spis treści

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.