A First Course in Differential Equations, Modeling, and Simulation

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Emphasizing a practical approach for engineers and scientists, A First Course in Differential Equations, Modeling, and Simulation avoids overly theoretical explanations and shows readers how differential equations arise from applying basic physical principles and experimental observations to engineering systems. It also covers classical methods for obtaining the analytical solution of differential equations and Laplace transforms. In addition, the authors discuss how these equations describe mathematical systems and how to use software to solve sets of equations where analytical solutions cannot be obtained.

Using simple physics, the book introduces dynamic modeling, the definition of differential equations, two simple methods for obtaining their analytical solution, and a method to follow when modeling. It then presents classical methods for solving differential equations, discusses the engineering importance of the roots of a characteristic equation, and describes the response of first- and second-order differential equations. A study of the Laplace transform method follows with explanations of the transfer function and the power of Laplace transform for obtaining the analytical solution of coupled differential equations.

The next several chapters present the modeling of translational and rotational mechanical systems, fluid systems, thermal systems, and electrical systems. The final chapter explores many simulation examples using a typical software package for the solution of the models developed in previous chapters.

Providing the necessary tools to apply differential equations in engineering and science, this text helps readers understand differential equations, their meaning, and their analytical and computer solutions. It illustrates how and where differential equations develop, how they describe engineering systems, how to obtain the analytical solution, and how to use software to simulate the systems.

Table of Contents

Introduction

An Introductory Example

Modeling

Differential Equations

Forcing Functions

Book Objectives

Objects in a Gravitational Field

An Example

Antidifferentiation: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2-1

Another Example

Separation of Variables: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2-5

Equations, Unknowns, and Degrees of Freedom

Classical Solutions of Ordinary Linear Differential Equations

Examples of Differential Equations

Definition of a Linear Differential Equation

Integrating Factor Method

Characteristic Equation

Undetermined Coefficients

Response of First- and Second-Order Systems

Application of the Mathematics to Design

Laplace Transforms

Definition of the Laplace Transform

Properties and Theorems of the Laplace Transform

Solution of Differential Equations Using Laplace Transform

Transfer Functions

Algebraic Manipulations Using Laplace Transforms

Deviation Variables

First- and Second-Order Systems

Mechanical Systems: Translational

Mechanical Law and Experimental Facts

Types of Systems

D’Alembert’s Principle and Free Body Diagrams

Additional Examples

Vertical Systems

Mechanical Systems: Rotational

Mechanical Law, Moment of Inertia, and Torque

Torsion Springs

Rotational Dampening

Gears

Systems with Rotational and Translational Elements

Mass Balances

Conservation of Mass

Flow Rates and Concentrations

Flow Element and Experimental Facts

Examples of Mass Balances

Thermal Systems

Conservation of Energy

Modes of Heat Transfer

Conduction

Convection

Conduction and Convection in Series

Accumulated or Stored Energy

Some Examples

Heat Transfer in a Flow System

Electrical Systems

Some Definitions and Conventions

Electrical Laws and Electrical Components

Examples of Electrical Circuits

Additional Examples

RC Circuits as Filters

Numerical Simulation

Numerical Solution of Differential Equations

Euler’s Method for First Order Ordinary Differential Equations

Euler’s Method for Second Order Ordinary Differential Equations

Step Size

More Sophisticated Methods

Representation of Differential Equations by Block Diagrams

Additional Examples

Index

A Summary and Problems appear at the end of each chapter.

Author/Editor Biography

Carlos A. Smith is a professor emeritus in the Department of Chemical and Biomedical Engineering at the University of South Florida. He has co-authored three editions of Principles and Practice of Automatic Process Control and authored Continuous Automated Process Control.

Scott W. Campbell is a professor in the Department of Chemical and Biomedical Engineering at the University of South Florida, where he has recently incorporated applications and projects into the calculus sequences for engineering and life sciences students. His research interests encompass the areas of thermodynamics and environmental monitoring and modeling.

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