Computer Models of Process Dynamics From Newton to Energy Fields

Comprehensive overview of techniques for describing physical phenomena by means of computer models that are determined by mathematical analysis

Computer Models of Process Dynamics covers everything required to do computer based mathematical modeling of dynamic systems, including an introduction to a scientific language, its use to program essential operations, and methods to approximate the integration of continuous signals.

From a practical standpoint, readers will learn how to build computer models that simulate differential equations. They are also shown how to model physical objects of increasing complexity, where the most complex objects are simulated by finite element models, and how to follow a formal procedure in order to build a valid computer model. To aid in reader comprehension, a series of case studies is presented that covers myriad different topics to provide a view of the challenges that fall within this discipline. The book concludes with a discussion of how computer models are used in an engineering project where the readers would operate in a team environment.

Other topics covered in Computer Models of Process Dynamics include:

• Computer hardware and software, covering algebraic expressions, math functions, computation loops, decision-making, graphics, and user-defined functions
• Creative thinking and scientific theories, covering the Ancients, the Renaissance, Galileo, Newton, electricity and magnetism, and newer sciences
• Uncertainty and softer science, covering random number generators, statistical analysis of data, the method of least squares, and state/velocity estimators
• Flight simulators, covering the motion of an aircraft, the equations of motion, short period pitching motion, and phugoid motion

Established engineers and programmers, along with students and academics in related programs of study, can harness the comprehensive information in Computer Models of Process Dynamics to gain mastery over the subject and be ready to use their knowledge in many practical applications in the field.

Olis Rubin, DSc. Eng., has held many positions in control engineering throughout his career including at Denel, PBMR, ContrOlis, Dentron, and CSIR. He was also an honorary post graduate professor in Control Systems at the University of Pretoria. He previously published Control Engineering in Development Projects (2016) and The Design of Automatic Control Systems (1986) with Artech House.

1.            Introduction

1.1         Engineering Uses of Computer Models

1.1.1      Mission Statement

1.2         The Subject Matter

1.3         Mathematical Material

1.4         Some Remarks

Bibliography

2             From Computer Hardware to Software

2.1         Introduction

2.2         Computing Machines

2.2.1  The Software Interface

2.3         Computer Programming

2.3.1      Algebraic Expressions

2.3.2      Math Functions

2.3.3      Computation Loops

2.3.4      Decisionmaking

2.3.5      Graphics

2.3.6      User Defined Functions

2.4         State Transition Machines

2.4.1      A Binary Signal Generator

2.4.2      Operational Control of an Industrial Plant

2.5         Difference Engines

2.5.1      Difference Equation to Calculate Compound Interest

2.6         Iterative Programming

2.6.1      Inverse Functions

2.7         Digital Simulation of Differential Equations

2.7.1      Rectangular Integration

2.7.2      Trapezoidal Integration

2.7.3      Second Order Integration

2.7.4      An Example

2.8         Discussion

2.9         Exercises

References

3.            Creative Thinking and Scientific Theories

3.1         Introduction

3.2         The Dawn of Astronomy

3.3         The Renaissance

3.3.1      Galileo

3.3.2      Newton

3.4         Electromagnetism

3.4.1      Magnetic Fields

3.4.2      Electromagnetic Induction

3.4.3      Electromagnetic Radiation

3.5         Aerodynamics

3.5.1      Vector Flow Fields

3.6         Discussion

               References

4.            Calculus and the Computer

4.1         Introduction

4.2         Mathematical Solution of Differential Equations

4.3         From Physical Analogs to Analog Computers

4.4         Picard's Method for Solving a Nonlinear Differential Equation

4.4.1      Mechanization of Picard's Method

4.4.2      Feedback Model of the Differential Equation

4.4.3      Approximate Solution by Taylor Series

4.5         Exponential Functions and Linear Differential Equations

4.5.1      Taylor Series to Approximate Exponential Functions

4.6         Sinusoidal Functions and Phasors

4.6.1      Taylor Series to Approximate Sinusoids

4.7         Bessel's Equation

4.8         Discussion

4.9         Exercises

               References

5.            Science and Computer Models

5.1         Introduction

5.2         A Planetary Orbit around a Stationary Sun

5.2.1      An Analytic Solution for Planetary Orbits

5.2.2      A Difference Equation to Model Planetary Orbits

5.3         Simulation of a Swinging Pendulum

5.3.1      A Graphical Construction to Show the Motion of a Pendulum

5.3.2      Truncation and Roundoff  Errors

5.4         Lagrange's Equations of Motion

5.4.1      A Double Pendulum

5.4.2      A few Comments

5.4.3   Modes of Motion of a Double Pendulum

5.4.4   Structural Vibrations in an Aircraft

5.5         Discussion

5.6         Exercises

               References

6.            Flight Simulators

6.1         Introduction

6.2         The Motion of an Aircraft

6.2.1      The Equations of Motion

6.3         Short Period Pitching Motion

6.3.1      Case Study of Short Period Pitching Motion

6.3.2      State Equations of Short Period Pitching

6.3.3      Transfer Functions of Short Period Pitching

6.3.4      Frequency Response of Short Period Pitching

6.4         Phugoid Motion

6.5         User Interfaces

6.6         Discussion

6.7         Exercises

               References

7.            Finite Element Models and the Diffusion of Heat

7.1         Introduction

7.2         A Thermal Model

7.2.1      A Finite Element Model Based on an Electrical Ladder Network

7.2.2      Free Settling from an Initial Temperature Profile

7.2.3      Step Response Test

7.2.4      State Space Model of Diffusion

7.3         A Practical Application

7.4         Two-dimensional Steady-state Model

7.5         Discussion

7.6         Exercises

               References

8.            Wave Equations

8.1         Introduction

8.2         Energy Storage Mechanisms

8.2.1      Partial Differential Equation Describing Propagation in a Transmission Line

8.3         A Finite Element Model of a Transmission Line

8.4         State Space Model of a Standing Wave in a Vibrating System

8.4.1      State Space Model of a Multiple Compound Pendulum

8.5         A Two-dimensional Electromagnetic Field

8.6         A Two-Dimensional Potential Flow Model

8.7         Discussion

8.8         Exercises

               References

9.            Uncertainty and Softer Science

9.1         Introduction

9.2         Empirical and 'Black Box' Models

9.2.1      An Imperfect Model of a Simple Physical Object

9.2.2      Finite Impulse Response Models

9.3         Randomness Within Computer Models

9.3.1      Random Number Generators and Data Analysis

9.3.2      Statistical Estimation and the Method of Least Squares

9.3.3      A State Estimator

9.3.4      A Velocity Estimator

9.3.5      A FIR Filter

9.4         Economic , Geo-, Bio- and other Sciences

9.4.1      A Pricing Strategy

9.4.2      The Productivity of Money

9.4.3      Comments on Business Models

9.5         Digital Images

9.5.1      An Image Processor

9.6         Discussion

9.7         Exercises

               References

10.         Computer Models in a Development Project

10.1       Introduction

10.1.1    The Scope of this Chapter

10.2       A Motor Drive Model

10.2.1    A Concepual Model

10.2.2    The Motor Drive Parameters

10.2.3    Creating the Simulation Model

10.2.4    The Electrical and Mechanical Subsystems

10.2.5    System Integration

10.2.6    Configuration Management

10.3       The Definition Phase

10.3.1  Selection of a Motor

10.3.2    Simulation of Load Disturbances

10.4       The Design Phase

10.4.1    Calculation of Frequency Response

10.4.2    The Current Control Loop

10.4.3    Design Review and Further Actions

10.4.4    Rate Feedback

10.5       A Setback to the Project

10.5.1    Elastic Coupling between Motor and Load

10.6       Discussion

10.7       Exercises

               Reference

11.         Postscript

11.1       Looking Back

11.2       The Operation of a Simulation Facility

11.3       Looking Forward

               References

Appendix A        Frequency Response Methods

A.1         Complex Exponential Functions

A.2         Frequency Response

A.3         Fourier Series

A.3.1      Fourier Spectrum and Laplace transform

A.4         Power Spectra

A.4.1      Effect of Sampling on a Power Spectrum

A.5         Pulse Width Modulation

A.6         Circuit Analysis

A.7         Transfer Functions of Time Delays

A.7.1      Time Delays due to Sampling

A.7.2      Integration of Sampled Signal

A.7.3      Alternative Analysis of Discrete Integration

A.8         Feedback Loops

A.8.1      Digital PI Controllers

A.8.2      Programming a PI Controller

               References

Appendix B        Vector Analysis

B.1         The Use of Vector Analysis

B.2         Vector Analysis of Motion

B.2.1      Rotation of a Rigid Body

B.2.2      Center of Gravity

B.2.3      Moment of Inertia

B.2.4      Equations of Motion for a Rigid Body

Appendix C        Scalar and Vector Fields

C.1         One-dimensional Thermal Systems

C.1.1      The Diffusion Equation

C.2         Harmonic Analysis of a One-dimensional Diffusion Equation

C.2.1      Vibration of a Stretched String

C.3         Transfer Function Analysis of One-dimensional Wave Propagation

C.3.1      A Finite Transmission Line

C.3.2      Transfer Function Analysis of Heat Flow

C.3.3      The Transfer Function of a Transmission Line

C.4         Two-dimensional Thermal Systems

C.4.1      Harmonic Analysis of the Laplace Equation

C.4.2      The Rotation of a Vector Field

C.4.3      Vector Operations in Polar Coordinates

Appendix D        Probability and Statistical Models

D.1   The Laws of Chance

D.2   The Binomial Distributation

D.3   The Poisson Distributation

D.4   The Normal Distributation of Errors (Gaussian Law)

D.5    Inspection by Sampling

References

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