Machine Component Analysis with MATLAB

By Dan B. Marghitu and Mihai Dupac

Machine Design Analysis with MATLAB is a highly practical guide to the fundamental principles of machine design which covers the static and dynamic behavior of engineering structures and components. MATLAB has transformed the way calculations are made for engineering problems by computationally generating analytical calculations, as well as providing numerical calculations. Using step-by-step, real world example problems, this book demonstrates how you can use symbolic and numerical MATLAB as a tool to solve problems in machine design. This book provides a thorough, rigorous presentation of machine design, augmented with proven learning techniques which can be used by students and practicing engineers alike.

• Comprehensive coverage of the fundamental principles in machine design
• Uses symbolical and numerical MATLAB calculations to enhance understanding and reinforce learning
• Includes well-designed real-world problems and solutions

• Language: English
• Publisher: Elsevier Science
• Release date: Feb 12, 2019
• ISBN: 9780128042458

Dan B. Marghitu

Dan B. Marghitu is Professor in the Department of Mechanical Engineering at Auburn University. His specialty areas include impact dynamics, biomechanics, nonlinear dynamics, flexible multibody systems and robotics. He is the author of more than 60 journal papers and six books on dynamics, mechanical impact, mechanisms, robots and biomechanics.

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Preface

Dan B. Marghitu; Mihai Dupac     

This book is intended as a supplement for courses in machine component design and as a reference for mechanical engineers. The book uses MATLAB® as a tool to analyze and solve machine component problems.

The solutions of the problems are obtained analytically and numerically using MATLAB. Many figures are generated with MATLAB programs. Specific functions dealing with machine components are created. The book will assist the undergraduate and advanced undergraduate students interested in machine element analysis. The project can be used for classroom instruction and it can be used for a self-study and can also be offered as distance learning.

The chapters of the book are: stress and deflection, fatigue failure, screws, rolling bearings, lubrication and sliding bearings, and spur gears.

Chapter One

Stress and deflection

Abstract

This chapter presents some of the key elements related to the theory of stress and strain, stress analysis and deformation of mechanical parts due to the action of forces (axial, bending, or combined loads) and moments exerted on them. The presented concepts cover the definition of normal and shear stress and strain, normal and shear components, principal stress components, Mohr's circle construction for two and three-dimensional state of stress and its graphical interpretation, Hook's law and Poisson's ratio, deflection and stress energy. Several problems – presenting applications of the theory – are solved analytically and numerically using MATLAB.

Keywords

Stress; Strain; Stress components; Deflection; Mohr's circle; Principal stresses; Stress energy

1.1 Stress components

In the design process, the uniform distribution of stresses is usually considered, that is, the results of forces and moments applied to an element represent pure shear or pure tension. If a force F uniformly distributed at the cut. The normal stress σ can then be expressed as

(1.1)

where A is the bar cross-sectional area. For an element in shear the uniform shear stress distribution is

(1.2)

, is shown in Fig. 1.1A. For the shear stresses, using the static equilibrium, results in

(1.3)

– called either tensile stresses or tension – are oriented as shown in Fig. 1.1A, the stresses are considered positive. The subscripts used in the definition of the normal stresses represent the normal direction to the surface.

Figure 1.1 (A) Three-dimensional stress element and (B) planar element. From Budynas–Nisbett: Shigley's Mechanical Engineering Design, Eighth Edition, McGraw-Hill, 2006. Used with permission from McGraw Hill Inc.

The shear stresses acting in the positive direction of the reference axis are considered positive. The first and second subscripts of the shear stress denote the axes to which it is perpendicular (and subsequently the face on which the stress acts) and respectively parallel.

For the stress element shown in Fig. 1.1B, where only the x and y act in the positive direction. The shear stresses acting in the clockwise (cw) direction are considered positive, otherwise negative and acting counterclockwise (ccw).

Many times it is desirable to calculate stresses on an inclined (or rotated) section acting at an angle ϕ (Fig. 1.2). The stresses τ and σ acting on an inclined plane (section) can be computed considering the equilibrium equations for the force components caused by the stresses by using the formulas

(1.4)

(1.5)

Considering that the derivative with respect to the angle ϕ of Eq. (1.4) equals zero, one can write

(1.6)

or equivalently,

(1.7)

The two solutions of Eq. (1.7) as functions of ϕ represent the principal stresses , respectively named minimum and maximum stresses. The angles ϕ are called principal angles, and the related (or matching) directions that are perpendicular to each other are called principal directions.

Figure 1.2 Stresses σ and τ on an oblique plane. From Budynas–Nisbett: Shigley's Mechanical Engineering Design, Eighth Edition, McGraw-Hill, 2006. Used with permission from McGraw Hill Inc.

Setting the derivative of Eq. (1.5) to zero, one obtains

(1.8)

Solving Eq. (1.8), one can obtain the angles 2ϕ which represent the extreme values of the shear stress τ. Rewriting Eq. (1.7) as

one obtains

(1.9)

Combining Eqs. (1.5) and (1.9), one can write

(1.10)

that is, the shear stress in the principal directions is negligible.

Combining Eqs. (1.4) and (1.8), one gets

(1.11)

that is, the normal stresses related with the maximum shear stresses are equal.

Using Eq. (1.7), one can calculate

(1.12)

From Eqs. as

(1.13)

or equivalently,

(1.14)

are respectively the smaller and larger principal stresses. The planes with no shear stress (when shear stresses are zero) are called principal planes. Similarly, the shear stresses can be written as

(1.15)

The state of stress can be graphically represented using the Mohr's circle diagram method (named after German civil engineer Otto Mohr) shown in Fig. 1.3. The idea behind this graphical representation is that all the possible values σ and τ in Eqs. (1.4) and (1.5) for a given state of stress can be obtained by varying the angle θ shown in Fig. 1.3) and normal compressive stresses are considered positive and (respectively) negative. The shear stresses are considered positive or negative if their orientation is clockwise (cw) or counterclockwise (ccw).

Figure 1.3 Mohr's circle diagram.

For the Mohr's circle diagram presented in are represented by OA and OCby ABby CD. The Mohr's circle center is located at E, points B and D represent the stress conditions on the x- and y-face, respectively. Points B and D , and the angle (on the stress element) between x and y (minimum) are located at points F and G, while the shear stresses' extreme values are located at H and I.

.

To devise Mohr's circle for a three-dimensional state of stresses (. The principal shear stresses are computed