an introduction to numerical methods using mathcad - SDC Publications

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AN INTRODUCTION TO NUMERICAL METHODS USING MATHCAD Mathcad Release 14

Khyruddin Akbar Ansari, Ph.D., P.E. Professor of Mechanical Engineering School of Engineering and Applied Science Gonzaga University

SDC

PUBLICATIONS

Schroff Development Corporation www.schroff.com

www.schroff-europe.com

Table of Contents: i

TABLE OF CONTENTS Preface

v

1.

Basics of Mathcad.

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23

Introduction The Mathcad Screen Exact Answers Variables, Functions and Live math. Feedback Graphics Graphing of Functions and Plotting of Data Animations The Mathcad Tutorials Advantages of Mathcad Computations in Mathcad The Mathcad Window, Toolbars and Palettes Mathcad Regions Entering Math and Text Mathcad Worksheets,Templates and Styles Defining Variables Defining Functions in Mathcad Building and Editing Mathematical Expressions Defining Range Variables Defining Vectors and Matrices Creating Graphs Formatting Math, Text and Results Using Units

1 1 3 3 4 5 5 7 7 7 10 13 15 16 17 18 19 21 23 24 28 35 39

2.

Introduction to Numerical Methods.

43

2.1 2.2 2.3 2.4

The Use of Numerical Methods in Science and Engineering Comparison of Numerical Methods with Analytical Methods Sources of Numerical Errors and their Computation Taylor Series Expansion Problems

43 43 44 44 49

3.

Roots of Equations.

53

3.1 3.2 3.3 3.4. 3.5 3.6

Introduction Methods Available Bisection Method The Regula Falsi or the False Position Method Newton-Raphson Method Use of Mathcad’s root and polyroots Functions

53 53 53 58 65 71

ii AN INTRODUCTION TO NUMERICAL METHODS USING MATHCAD

3.7 3.8. 3.9 3.10. 3.11

Secant Method Method of Successive Substitution Multiple Roots and Difficulties in Computation Solution of Systems of Nonlinear Equations Solving Systems of Equations using Mathcad's Given and Find Functions Applications in Root-Finding 3.12.1 Maximum Design Load for a Column 3.12.2 Natural frequencies of Vibration of a Uniform Beam 3.12.3 Solving the Characteristic Equation in Control Systems Engineering 3.12.4 Horizontal Tension in a Uniform Cable Problems

72 78 80 84

4.

Matrices and Linear Algebra.

103

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Basic Matrix Operations Use of Mathcad in Performing Matrix Operations Solution of Linear Algebraic Equations by Using the Inverse Solution of Linear Algebraic Equations by Cramer’s Rule Solution of Linear Algebraic Equations Using the Function lsolve The Eigenvalue Problem Solving the Eigenvalue Problem with Mathcad Application of the Eigenvalue Problem to Vibration Engineering Application of the Eigenvalue Problem to Stress AnalysisDetermination of Principal Stresses and Principal Directions Repeated Roots in the Determinantal Equation Solution of Nonlinear Simultaneous Equations Problems

103 105 108 110 112 114 116 117

5.

Numerical Interpolation.

141

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Linear Interpolation The Method of Undetermined Coefficients The Gregory-Newton Interpolating Polynomial Interpolation Using Finite Differences Newton’s Method Utilizing Finite Differences The Lagrange Interpolating Polynomial Interpolation Using Linear, Quadratic and Cubic Splines Interpolation with Mathcad Applications in Numerical Interpolation 5.9.1 Stress-Strain data for Titanium 5.9.2 Notch Sensitivity of Aluminum 5.9.3 Speech Interference Level 5.9.4 Load-Deflection Data for Elastomeric Mounts Problems

141 141 144 149 151 155 158 159 168 168 169 172 175 177

3.12

4.10 4.11

86 87 87 89 91 93 96

125 128 131 134

Table of Contents: iii

6.

Curve-Fitting.

183

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

The Need to Fit a Function to Measured Data The Method of Least Squares. Straight Line Regression Curve-Fitting with a Quadratic Function Curve-Fit with a Power Function Curve-Fitting with an Exponential Function Curve-Fitting with a Linear Combination of Known Functions Curve-Fitting with Polynomials. Use of Mathcad's Regression Functions for Curve-Fitting 6.9.1 Linear Regression with Mathcad 6.9.2 Nonlinear Regression with Mathcad 6.9.3 Use of the Function linfit 6.9.4 Use of the Function genfit 6.9.5 Use of the Mathcad Functions logfit, lnfit, pwrfit and expfit 6.9.6 More Examples with Mathcad Applications in Curve-Fitting 6.10.1 Fatigue Failure Curve for Loading in the Finite Life Range 6.10.2 Temperature Response of an Object Placed in a Hot Stream of Air 6.10.3 The Effect of Operating Temperature on the Strength of a Mechanical Element 6.10.4 Drop-Testing of Packaged Articles Problems

183 184 185 188 191 194 199 203 207 207 209 211 213 215 220 236 236

6.10

7. 7.1 7.2 7.3 7.4

Numerical Differentiation Introduction to Numerical Differentiation and the Use of the Mathcad Derivative Operators Method of Finite Differences Interpolating Polynomial Method Applications in Numerical Differentiation 7.4.1 Determination of Velocities and Accelerations from Given Displacement Data 7.4.2 Determination of Shock Absorber Parameters, and Damper and Spring Restoring Forces from Given Vehicle Displacement Data Problems

239 242 245 248 255

255 255 259 262 262 266 271

8.

Numerical Integration

277

8.1

Introduction to Numerical Integration and the Use of the Mathcad Integral Operator The Interpolating Polynomial Method Trapezoidal Rule Simpson’s Rules

277 279 280 283

8.2 8.3 8.4

iv AN INTRODUCTION TO NUMERICAL METHODS USING MATHCAD

8.4.1 Simpson’s One-Third Rule 8.4.2 Simpson’s Three-Eighth Rule 8.4.3 Limitations of Simpson’s Rules Romberg Integration Applications in Numerical Integration 8.6.1 Centroid of a Rod Bent into the Shape of A Parabola 8.6.2 Moment of Inertia of a Beam of Semi-Elliptic Cross Section 8.6.3 Launch of a Projectile 8.6.4 Large Oscillations of a Simple Pendulum Problems

283 286 287 288 301 301 302 303 304 306

9.

Numerical Solution of Ordinary Differential Equations.

311

9.1 9.2 9.3 9.4 9.5

Introduction Taylor Series Method Euler’s Method Modified Euler’s Method Runge- Kutta Methods 9.5.1 Fourth-Order Runge-Kutta Method 9.5.2 Mathcad Solutions to a First-Order Differential Equation Systems of Ordinary Differential Equations Solution of Higher-Order Ordinary Differential Equations Boundary-Value Problems and the Shooting Method Applications in Numerical Solution of Ordinary Differential Equations 9.9.1 Response of an Electric R-L Circuit to a Unit-Step Voltage Input 9.9.2 Deflection Curve of a Cantilevered Beam with a Uniformly Distributed Load 9.9.3 Temperature Response of a Solid Steel Ball Placed in a Hot Stream of Air 9.9.4 Nonlinear Vibration of a Simple Pendulum 9.9.5 Transient Vibration of a Spring-Mass-Damper System Excited by a Pulse Function 9.9.6 Nonlinear Vibration of a Damped System with a Hardening Spring 9.9.7 Temperature Distribution in the Wall of a Pipe Carrying a Hot Fluid 9.9.8 Response of an R-L Circuit with a Nonlinear Resistor 9.9.9 The Effect of Damping on the Step Response of a Second-Order Control System Problems

311 312 317 323 329 329 335 341 349 358 363 363

Bibliography Index

399 401

8.5 8.6

9.6 9.7 9.8 9.9

364 366 367 370 373 378 382 384 386

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an introduction to numerical methods using mathcad - SDC Publications

AN INTRODUCTION TO NUMERICAL METHODS USING MATHCAD Mathcad Release 14 Khyruddin Akbar Ansari, Ph.D., P.E. Professor of Mechanical Engineering School ...

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