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.
