Design Analysis: Mathematical Modeling of Nonlinear Systems

by David E. Thompson

• ISBN13: 9780521621700
• ISBN10: 0521621704
• Format: Hardcover
• Copyright: 1999-01-13
• Publisher: Cambridge University Press

Summary

An integral part of engineering design is the development of models that describe physical behavior or phenomena in mathematical terms. As engineering systems have become more complex, classic linear methods of modeling and analysis have proved inadequate, creating a need for nonlinear models to solve design problems. This text provides an introduction to mathematical modeling of linear and nonlinear systems, with an emphasis on the solution of nonlinear design problems. While encouraging the use of the computer as a tool for modeling and analysis, the aim is to discuss the basic concepts underlying computer techniques and to seek analytical solutions. Among topics covered are exact solution, numerical solution, graphical solution, and approximate solution methods; and the stability of nonlinear systems. Numerous examples show how to apply modeling methods to real engineering systems. The book also includes end-of-chapter problems and case studies of challenging design problems. Intended for senior or beginning graduate students, this text will also serve as a helpful reference for practicing engineers.

Table of Contents

Preface	xi
List of Figures	xiii
List of Tables	xvii
Acknowledgments	xix
The Design Process	1	(9)
Elements of the Design Process	2	(5)
Identification	4	(1)
Defining the Problem	4	(1)
Ideation	5	(1)
Selection	5	(1)
Analysis	6	(1)
Implementation	7	(1)
Evaluation	7	(1)
The Computer as a Design Tool	7	(1)
Summary	8	(2)
Mathematical Modeling	10	(77)
Special Behavior of Nonlinear Systems	13	(1)
Modeling Basics	14	(20)
Purpose	16	(1)
Resources	16	(1)
Expert Knowledge	16	(2)
Model Validation	18	(1)
Constraints	18	(1)
Examples of Modeling	18	(4)
Method of Lagrange's Equations	22	(3)
Vector Notation and Operations	25	(1)
Elementary Operations	25	(1)
∇-Operators	25	(1)
Divergence Theorem	26	(1)
Stokes' Theorem	26	(1)
Fundamental Vector Properties	27	(2)
Introduction to Tensor Notation	29	(1)
Tensor Algebra	30	(1)
Properties of Second-Order Tensors	30	(2)
The Alternating Tensor	32	(1)
Derivatives of Tensors	33	(1)
Isotropic Tensors	33	(1)
Vector-Tensor Equivalence	34	(1)
Integral Equations	34	(7)
Basic Laws for Continuous Matter	34	(1)
General Conservation Formulation	35	(3)
Conservation of Matter	38	(1)
Conservation of Momentum	39	(1)
First Law of Thermodynamics	40	(1)
Differential Equations	41	(2)
Ordinary Differential Equations	42	(1)
Linear Differential Equations	42	(1)
Nonlinear Differential Equations	42	(1)
Partial Differential Equations	43	(1)
Examples of Nonlinear Systems	43	(4)
Heat Transfer	43	(1)
Thermodynamics	43	(1)
Mechanics	44	(1)
Simple Suspension System	44	(1)
Fluid Mechanics	45	(1)
Continuity Equation	45	(1)
Blasius Equation	45	(1)
Vehicle Dynamics	46	(1)
Normalization of Equations	47	(13)
Complex Nonlinear Problems	59	(1)
Infinite and Semi-Infinite Domains	59	(1)
Transformation of Variables	60	(8)
Exponential Transformations	61	(1)
Generalized Transformations	62	(1)
Similarity Transformations	63	(2)
Well-Posed Problems	65	(2)
Problems with Missing Boundary Conditions	67	(1)
Summary	68	(1)
Understanding Differential Equations	68	(4)
Force on a Spring	68	(1)
Force on a Damper	69	(1)
Force on a Mass	69	(1)
Force on a Spring-Damper Pair	69	(1)
Force on a Spring-Mass Pair	70	(1)
Force on a Mass-Dashpot Pair	70	(1)
Force on a Spring-Mass-Dashpot System	71	(1)
Other Forms of Nonlinear Equations	71	(1)
Comparison of Model and Experiment	72	(11)
Concepts of a ``Best'' Model	72	(1)
Error Measures	73	(1)
Minimum (-Maximum)	74	(1)
Least Sum Square Error	75	(2)
Other Natural Error Measures	77	(1)
Nonlinear Problems	77	(1)
Gradient Method	78	(1)
Nongradient Method	78	(2)
Constrained Problems	80	(1)
Variable Mapping	81	(1)
Penalty Functions	81	(2)
Summary	83	(1)
Problems	84	(3)
Exact Solution Methods	87	(31)
Technique of Linearization	87	(9)
Method of Finding Linear Coefficients	88	(2)
Linearization of a Multivariable Function	90	(4)
Taylor-Series Expansion Errors	94	(1)
Classical	94	(1)
First-Term-Neglected Method	95	(1)
Direct Integration	96	(5)
Radiatively Heated Thermal Capacitance	96	(2)
Spring-Dashpot System	98	(1)
Automobile Handling Revisited	99	(1)
Gravitational Attraction	100	(1)
Variation of Parameters	101	(2)
Equations Leading to Elliptic Integrals	103	(5)
Suspended Mass	104	(2)
Simple Pendulum	106	(2)
Power-Series Method	108	(1)
Spring-Damper Problem	108	(1)
Picard's Method	109	(3)
Reversion of Power Series	112	(1)
Summary Comments	113	(1)
Problems	114	(4)
Numerical Solution Methods	118	(15)
Taylor-Series Method	118	(2)
Euler Method	120	(3)
Modified Euler Method	121	(1)
Extension of Euler Method to Higher-Order Systems	122	(1)
Runge-Kutta Method	123	(5)
Solution of Two Simultaneous Equations	126	(1)
Final Remarks on Runge-Kutta Method	127	(1)
Multistep Methods	128	(1)
Step-Size Determination	129	(1)
Summary Comments	130	(1)
Problems	130	(3)
Graphical Solution Methods	133	(15)
First-Order Equations (Method of Isoclines)	133	(4)
Phase-Plane Analysis	135	(1)
Recovery of the Independent Variable	136	(1)
Second-Order Systems	137	(2)
Linear Spring-Mass System	137	(1)
Method of Isoclines	138	(1)
Effect of Normalization	138	(1)
Lienard's Method	139	(3)
Pell's Method	142	(2)
Summary Comments	144	(1)
Problems	144	(4)
Approximate Solution Methods	148	(36)
Method of Perturbation	148	(7)
Intended Area of Applicability	149	(1)
Solution Technique	149	(6)
Iteration Technique	155	(2)
Power-Series Method	157	(6)
Duffing Equation	159	(2)
Generalized Duffing Equation	161	(2)
Method of Harmonic Balance	163	(5)
Equivalent Linear Equation	166	(2)
Galerkin's Method	168	(12)
Secondary Boundary Conditions	171	(9)
Summary Comments	180	(1)
Problems	181	(3)
Stability of Nonlinear Systems	184	(42)
Routh Method	185	(2)
Singular-Point Analysis	187	(15)
General System	188	(1)
Linear Transformations	189	(3)
Classification of Singularities	192	(1)
Real Roots, Same Sign	193	(1)
Real Roots, Different Sign	194	(1)
Complex Roots (Pure Imaginary)	194	(1)
Complex Conjugate Roots	195	(1)
Examples of Singularity Analysis	196	(6)
Poincare Index	202	(4)
Bendixson's First Theorem	206	(4)
Second Method of Lyapunov	210	(7)
Lyapunov Function	211	(1)
First Theorem of Lyapunov	212	(3)
Second Theorem of Lyapunov	215	(2)
Introduction to Chaotic Systems	217	(3)
Phase Space	218	(1)
Poincare Sections	219	(1)
Bifurcation Diagrams	220	(1)
Summary Comments	220	(1)
Problems	221	(5)
Case Studies	226	(41)
Lightbulb Model	226	(3)
Method 1: Linearization	226	(2)
Method 2: Exact Solution of Nonlinear Problem	228	(1)
Method 3: Galerkin Solution	228	(1)
Method 4: Numerical Solution	229	(1)
Summary Comments	229	(1)
Automobile Carburetor Model	229	(2)
Automobile Engine Model	231	(2)
Model Development	232	(1)
Experimental Data	233	(1)
Surge Analysis for a Hydroelectric Power Plant	233	(3)
Development of the Model	234	(1)
Numerical Integration Scheme	235	(1)
Results	235	(1)
Constant-Deceleration Shock Absorber	236	(6)
Equations of Motion for Incompressible Flow	238	(3)
Procedure for Constant-Deceleration Design	241	(1)
Fluid Mechanics of Blood Flow in the Kidney	242	(3)
Exact Solution	244	(1)
Galerkin Solution	244	(1)
Comments	245	(1)
Torque and Motion of the Finger	245	(1)
Summary Comments	246	(1)
Problems	247	(20)
References	267	(4)
Index	271