Simultaneous Systems of Differential Equations and Multi-Dimensional Vibrations

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Diagrams, Notes, and Tables
Preface
Acknowledgments
About the Author
Physical Quantities
7. Simultaneous Differential Equations
	7.1. Reduction of General to Autonomous Systems
		7.1.1. Autonomous System of Differential Equations
		7.1.2. General System of Simultaneous Differential Equations
	7.2. Tangents, Trajectories, and Paths in N-Dimensions
		7.2.1. N-Dimensional Hypercurve Specified by Tangent Vectors
		7.2.2. Families of Curves in the Plane or in Space
		7.2.3. N-Dimensional Curve Lying on the Intersection of M Hypersurfaces
		7.2.4. Space Curves as the Intersection of Two Surfaces
		7.2.5. Hypersurfaces Orthogonal to a Vector Field
	7.3. Order of a Simultaneous System of Differential Equations
		7.3.1. Definition of Order for Simultaneous Differential Equations
		7.3.2. Transformation from a Simultaneous to a Decoupled System
		7.3.3. Constants of Integration and Depression of the Order
	7.4. Linear Simultaneous System with Constant Coefficients
		7.4.1. Linear Simultaneous System with Variable Coefficients
		7.4.2. Linear Forced System with Constant Coefficients
		7.4.3. Characteristic Polynomial of a Simultaneous System
		7.4.4. Non-Degenerate and Degenerate Differential Systems
		7.4.5. Distinct Roots of the Characteristic Polynomial
		7.4.6. Multiple Roots of the Characteristic Polynomial
		7.4.7. General Integral and Linearly Independent Particular Integrals
		7.4.8. General Integral for Distinct Roots
		7.4.9. Arbitrary Constants and Boundary Conditions
		7.4.10. General Integral with Multiple Roots
		7.4.11. Natural Integrals and Diagonal or Banded System
		7.4.12. Block-Banded Diagonal System
		7.4.13. Diagonalization of a Square System
		7.4.14. Transformation from a Non-Diagonal to a Banded System
	7.5. Integrals of Forced and Unforced Systems
		7.5.1. Forcing of a Simultaneous System by an Exponential
		7.5.2. Single and Multiple Resonant Forcing
		7.5.3. Non-Resonant and Resonant Forcing by an Exponential
		7.5.4. Forcing by the Product of an Exponential by a Sine or Cosine
		7.5.5. Forcing by Hyperbolic or Circular Cosines or Sines
		7.5.6. Inverse Matrix of Polynomials of Derivatives
		7.5.7. Power Series Expansion of Inverse Polynomial Operator
		7.5.8. Principle of Superposition and Addition of Particular Integrals
	7.6. Natural Integrals for Simultaneous Homogeneous Systems
		7.6.1. Linear System of Homogeneous Derivatives
		7.6.2. Matrix of Polynomials of Homogeneous Derivatives
		7.6.3. Unforced System and Characteristic Polynomial
		7.6.4. Distinct and Multiple Roots of the Characteristic Polynomial
		7.6.5. Natural Integrals and the General Integral
		7.6.6. Compatibility Conditions for the Dependent Variables
		7.6.7. Arbitrary Constants and Boundary Conditions
		7.6.8. Decoupled or Minimally-Coupled Natural Differential System
		7.6.9. Block Diagonal-Banded System
	7.7. Forced and Unforced Homogeneous Systems
		7.7.1. Analogy of Constant and Homogeneous Coefficients
		7.7.2. Forcing of a Homogeneous System by a Power
		7.7.3. Non-Resonant and Multiply Resonant Particular Integrals
		7.7.4. Power Forcing and Single Resonance
		7.7.5. Double Root and Double Resonance
		7.7.6. Cosine and Sine of Multiples of Logarithms
		7.7.7. Forcing by a Power Multiplied by a Double Product
		7.7.8. Inverse Matrix of Polynomials of Homogeneous Derivatives
		7.7.9. Homogeneous Forcing by a Polynomial of Logarithms
		7.7.10. Complete Integral of the Forced Homogeneous Derivatives
	7.8. Simultaneous Finite Difference Equations
		7.8.1 Non-Linear and Linear Finite Difference Equations
		7.8.2. Operator Forward Finite Difference
		7.8.3. Matrix of Polynomials of Finite Differences
		7.8.4. Simple and Multiple Roots of the Characteristic Polynomial
		7.8.5. General Solution of an Unforced System
		7.8.6. Compatibility Conditions for the Dependent Variables
		7.8.7. Arbitrary Constants and Starting Conditions
		7.8.8. Diagonal or Lower Triangular System
		7.8.9. Block-Diagonal Lower Triangular System
		7.8.10. Diagonalization of a Finite Difference System
	7.9. Unforced and Forced Finite Difference
		7.9.1. Forward, Backward, and Central Differences
		7.9.2. Forcing by a Power with Integer Exponent
		7.9.3. Non-Resonant Forcing by Integral Powers
		7.9.4. Three Cases of Simple Resonance
		7.9.5. Product of Power by Circular and Hyperbolic Functions
		7.9.6. Products of Powers by Cosines of Multiple Angles
		7.9.7. Complete Integral of Forced Finite Differences
		7.9.8. Comparison of Three Matrix Polynomial Systems
	Conclusion 7
8. Oscillations with Several Degrees-of-Freedom
	8.1. Balance of Forces, Energy, and Dissipation
		8.1.1. Restoring, Friction, Inertial and Applied Forces
		8.1.2. Linear Restoring Force and Quadratic Potential
		8.1.3. Friction Force and Dissipation Function
		8.1.4. Coupled and Decoupled Equations of Motion
		8.1.5. Activity/Power and Work of the Applied Forces
		8.1.6. Kinetic, Potential, and Total Energies
	8.2. Modal Frequencies, Damping, Coordinates, and Forces
		8.2.1. Mass, Damping, and Oscillation Matrices
		8.2.2. Friction, Oscillation, and Dispersion Matrices
		8.2.3. Free Undamped Decoupled Oscillations
		8.2.4. Modal Frequencies of Undamped Oscillations
		8.2.5. Modal Dampings of Decaying Oscillations
		8.2.6. Modal Coordinates and Oscillation Frequencies
		8.2.7. Relation between the Physical and Modal Coordinates
		8.2.8. Compatibility Relations and Initial Conditions
		8.2.9. Physical/Modal Coordinates and Forces
		8.2.10. Matrix and Diagonal Dispersion Operators
		8.2.11. Decoupled Damped and Forced Oscillations
		8.2.12. Forcing with Bounded Fluctuation in a Finite Time Interval
		8.2.13. Modal Matrix for Sinusoidal Forcing
		8.2.14. Undamped Multidimensional Sinusoidal Forcing
		8.2.15. Beats and Resonant and Non-Resonant Forcing
		8.2.16. Forcing of a Damped Multidimensional Oscillator
	8.3. Coupled Circuits and Electromechanical Simulations
		8.3.1. Two Masses Linked by Three Spring-Damper Units
		8.3.2. Pair of Electrical Circuits with a Common Branch
		8.3.3. Damped Suspension of a Two-Wheeled Vehicle
		8.3.4. Three Analogue Mechanical and Electrical Circuits
		8.3.5. Mass, Friction, and Resilience Matrices
	8.4. Coupled Natural Frequencies and Dampings
		8.4.1. Translational/Rotational Oscillations of a Rod
		8.4.2. Plane Oscillations of Two Atoms at a Fixed Distance
		8.4.3. Decoupled Rotational and Translational Natural Frequencies
		8.4.4. Coupled Free Undamped Oscillations of a Rod
		8.4.5. Coupled and Decoupled Natural Frequencies of a Homogeneous Rod
		8.4.6. Compatibility Relations between Modal and Physical Coordinates
		8.4.7. Amplitudes, Phases, and Initial Conditions
		8.4.8. Displacement, Rotation, and Linear and Angular Velocities
		8.4.9. Linear Free Oscillations with Dissipation
		8.4.10. Strong Damping of Decoupled Free Oscillations
		8.4.11. Strong Damping of Coupled Oscillators
		8.4.12. Weak Damping of Coupled Oscillations
	8.5. Forced Oscillations, Beats, and Resonance
		8.5.1. Undamped Non-Resonant Forcing
		8.5.2. Undamped Resonant Forcing
		8.5.3. Forcing in Terms of Modal Coordinates and Forces
		8.5.4. Beats at One of the Normal Coordinates
		8.5.5. Forced Damped Oscillations
	8.6. Principle of the Vibration Absorber
		8.6.1. Primary Damped Forced System with Auxiliary Undamped Unforced Circuit
		8.6.2. Suppression of Forced Oscillations in the Primary System
		8.6.3. Transfer of Forced Vibrations to the Secondary System
		8.6.4. Modal Frequencies and Dampings of the Vibration Absorber
		8.6.5. Transient and Forced Oscillatory Components
		8.6.6. Total Oscillations in the Primary and Secondary Circuits
	8.7. A Markov Chain: Radioactive Disintegration
		8.7.1. Sequence of N Elements and N – 1 Distintegration Rates
		8.7.2. Non-Resonant Radioactive Decay at Distinct Rates
		8.7.3. Single Resonance Due to the Coincidence of the Two Disintegration Rates
		8.7.4. Sequential Solution of a Chain of Ordinary Differential Equations
		8.7.5. Totally Non-Resonant Case for Three Distinct Decay Rates
		8.7.6. Coincidence of Two Out of Three Decay Rates
		8.7.7. Double Resonance for the Coincidence of Three Decay Rates
		8.7.8. Higher-Order Resonances along the Disintegration Chain
	8.8. Sequence of Damped and Forced Oscillators
		8.8.1. Sequence of Coupled Mechanical or Electrical Circuits
		8.8.2. Oscillations of Three Masses Coupled by Four Springs
		8.8.3. Limits of Middle Mass Much Larger/Smaller Than the Others
		8.8.4. Comparison of Sequences of Mechanical and Electrical Circuits
		8.8.5. Coupled Modal Frequencies and Dampings
		8.8.6. Amplitudes and Phases of the Coupled Oscillations
		8.8.7. Interactions in Triple/Quadruple Oscillators
	8.9. Passing Bandwidth of a Transmission Line
		8.9.1. Signals and Spectra in Electrical and Mechanical Circuits
		8.9.2. Impedances Due to Selfs/Masses, Resistors/Dampers, and Capacitors/Springs
		8.9.3. Transmission Line with Impedances in Parallel and in Series
		8.9.4. Lossless Transmission or Decay by Reflection
		8.9.5. Six Transmission Lines including Two Lossless Cases
		8.9.6. Frequency Passband and Cut-off Frequency
		8.9.7. Five Regimes of Signal Transmission
	Conclusion 8
Bibliography
Index