Engineers, scientists, and applied mathematicians are habitually curious about behavior of physical systems. More often than not they will model the system and then analyze the model, hoping to expose the system's dynamic secrets. Traditionally, linear methods have been the norm and nonlinear effects were only added peripherally. This bias for linear techniques arises from the consum mate beauty and order in linear subs paces and the elegance of linear indepen dence is too compelling to be denied. And the bias has been, in the past, for tified by the dearth of nonlinear procedures, rendering the study of nonlinear dynamics untidy. But now a new attractiveness is being conferred on that non descript patchwork, and the virtue of the hidden surprises is gaining deserved respect. With a wide variety of individual techniques available, the student and the engineer as well as the scientist and researcher, are faced with an almost overwhelming task of which to use to help achieve an understanding sufficient to reach a satisfying result. If linear analysis predicts system behavior suffi ciently close to reality, that is delightful. In the more likely case where nonlin ear analysis is required, we believe this text fills an important void. We have tried to compile and bring some order to a large amount of information and techniques, that although well known, is scattered. We have also extended this knowledge base with new material not previously published.
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Table of ContentsLinearityA Shangri-La.- I.1 Why Linearity?.- I.2 From Whence Linearity.- I.3 What is Linearity?.- I.4 Things Linear or Not Linear.- I.5 Our Shangri-La.- 1 Four Interesting Equations.- 1.0 The Equations.- 1.1 The SolutionsA Preview.- 1.2 Solving Equation (i): ? + x = 0.- 1.3 Solving Equation (ii): ? + ?x? = 0.- 1.4 Solving Equation (iii): ??? + x = 0.- 1.5 Solving Equation (iv): ??? + ??? = 0.- 1.6 A Variation on Equation (iv): ??? = ???.- 1.7 Existence/Uniqueness of Solutions.- 1.8 Problems.- 1.9 References and Related Literature.- 2 Analytic Solutions to Nonlinear Differential Equations.- 2.0 Introduction.- 2.1 First-Order Base Equations.- 2.2 Second-Order Base Equations.- 2.3 The Ricatti Equation.- 2.4 Nonlinear Base Equations.- 2.5 Derivative and Integral Functional Relations.- 2.6 Matrix Nonlinear Differential Equations.- 2.7 Application to the Calculus of Variations.- 2.8 Problems.- 2.9 References and Related Literature.- 3 Linearization.- 3.0 Introduction.- 3.1 Linearizing Algebraic Functions.- 3.2 Linearizing a Transistor.- 3.3 Linearization of Differential Functions.- 3.4 Linearizing Satellite Motion.- 3.5 Concluding Example.- 3.6 Summary.- 3.7 Problems.- 3.8 References and Related Literature.- 4 The Describing Function.- 4.1 Describing Function.- 4.2 Frequency-Dependent Describing Functions.- 4.3 Digital Simulation Verifies Describing Function Analysis.- 4.4 Asymmetric Describing Functions.- 4.5 Problems.- 4.6 References.- 5 Some Properties of Nonlinear Systems.- 5.0 Introduction.- 5.1 Linear System Characteristics.- 5.2 Nonlinear Equations with Periodic Solutions.- 5.3 Limit Cycles.- 5.4 Nonlinear System Behavior.- 5.5 Some Physically Realizable Nonlinearities.- 5.6 Problems.- 5.7 References and Related Literature.- 6 Liapunov Stability.- 6.0 Introduction.- 6.1 Liapunov Stability: An Overview.- 6.2 Construction of Liapunov Functionals.- 6.3 The Lur’e Problem.- 6.4 The Popov Criterion.- 6.5 Problems.- 6.6 References and Related Literature.- 7 Recursions and their Stability.- 7.1 Recursions.- 7.2 The Mechanic’s Rule.- 7.3 Singularities and Peculiarities.- 7.4 The Logistics Map.- 7.5 Recursing Differential Equations.- 7.6 Problems.- 7.7 References.- 8 Digital Simulation.- 8.0 Background.- 8.1 Recursion Formulae: Fundamental to Digital Simulation.- 8.2 The Sampling Process Creates Discrete Data.- 8.3 An Approach to Digital Simulation: Introduce Samplers.- 8.4 Concept of Pulse Filters Implicit in Digital Simulations.- 8.5 Introducing Z-Transforms.- 8.6 The Pulse Filter Now Becomes a Z-Transform.- 8.7 Introductory Example of Digital Simulation.- 8.8 Digitally Simulating a Feedback System.- 8.9 Simulating Hysteresis Due to Backlash in Gears.- 8.10 Simulating a System with Hysteresis.- 8.11 Problems.- 8.12 References.- 9 Spreadsheet SimulationA Tutorial.- 9.1 Introduction.- 9.2 Use of the “Copy” Command.- 9.3 Using the Graph Command.- 9.4 Summary.- 9.5 Some Heuristic ExercisesBrief Descriptions.- 9.6 Suggested Solutions to the Exercises.- 9.7 References.- 10 An Isobaric Cabin Pressure Control.- 10.0 Introduction.- 10.1 Background.- 10.2 How the Cabin is Pressurized.- 10.3 Basic Numbers and Constraints.- 10.4 Cabin Dynamics.- 10.5 Design Evolution.- 10.6 A Nonlinear Digital Simulation.- 10.7 Limiting Cabin Pressure Rate.- 10.8 Initializing Integrators.- 10.9 Problems.- 10.10 References and Related Literature.