Engineering Applications (eBook)

Analytical and Numerical Calculation with MATLAB

Mihai Dupac, Dan B. Marghitu (Autoren)

eBook Download: EPUB

2021
John Wiley & Sons (Verlag)
978-1-119-09364-0 (ISBN)

Infos

ENGINEERING APPLICATIONS

A comprehensive text on the fundamental principles of mechanical engineering

Engineering Applications presents the fundamental principles and applications of the statics and mechanics of materials in complex mechanical systems design. Using MATLAB to help solve problems with numerical and analytical calculations, authors and noted experts on the topic Mihai Dupac and Dan B. Marghitu offer an understanding of the static behaviour of engineering structures and components while considering the mechanics of materials knowledge as the most important part of their design.

The authors explore the concepts, derivations, and interpretations of general principles and discuss the creation of mathematical models and the formulation of mathematical equations. This practical text also highlights the solutions of problems solved analytically and numerically using MATLAB. The figures generated with MATLAB reinforce visual learning for students and professionals as they study the programs. This important text:

Shows how mechanical principles are applied to engineering design

Covers basic material with both mathematical and physical insight

Provides an understanding of classical mechanical principles

Offers problem solutions using MATLAB

Reinforces learning using visual and computational techniques

Written for students and professional mechanical engineers, Engineering Applications helpshone reasoning skills in order to interpret data and generate mathematical equations, offering different methods of solving them for evaluating and designing engineering systems.

Mihai Dupac, PhD is a Senior Lecturer in Engineering Design at Bournemouth University, UK. He serves as the principal investigator and/or co-investigator for several research projects related to biomechanics and engineering design.

Dan B. Marghitu, PhD is a Professor at Auburn University and the Director of the Impact Dynamics Laboratory, which focuses on the investigation of relevant problems associated with impact dynamics, friction, and locomotion.

Inhalt

1 Forces 1

1.1 Terminology and Notation 1

1.2 Resolution of Forces 3

1.3 Angle Between Two Forces 3

1.4 Force Vector 4

1.5 Scalar (Dot) Product of Two Forces 5

1.6 Cross Product of Two Forces 5

1.7 Examples 6

2 Moments and Couples 15

2.1 Types of Moments 15

2.2 Moment of a Force About a Point 15

2.3 Moment of a Force About a Line 18

2.4 Couples 20

2.5 Examples 21

3 Equilibrium of Structures 55

3.1 Equilibrium Equations 55

3.2 Supports 57

3.3 Free-Body Diagrams 59

3.4 Two-Force and Three-Force Members 60

3.5 Plane Trusses 61

3.6 Analysis of Simple Trusses 62

3.6.1 Method of Joints 62

3.6.2 Method of Sections 65

3.7 Examples 67

4 Centroids and Moments of Inertia 129

4.1 Centre of the Mass and Centroid 129

4.2 Centroid and Centre of the Mass of a Solid Region, Surface or Curve 130

4.3 Method of Decomposition 134

4.4 First Moment of an Area 134

4.5 The Centre of Gravity 135

4.6 Examples 136

5 Stress, Strain and Deflection 185

5.1 Stress 185

5.2 Elastic Strain 185

5.3 Shear and Moment 186

5.4 Deflections of Beams 189

5.5 Examples 193

6 Friction 211

6.1 Coefficient of Static Friction 212

6.2 Coefficient of Kinetic Friction 213

6.3 Friction Models 213

6.3.1 Coulomb Friction Model 214

6.3.2 Coulomb Model with Viscous Friction 216

6.3.3 Coulomb Model with Stiction 217

6.4 Angle of Friction 218

6.5 Examples 219

7 Work, Energy and Power 255

7.1 Work 255

7.2 Kinetic Energy 256

7.3 Work and Power 258

7.4 Conservative Forces 259

7.5 Work Done by the Gravitational Force 259

7.6 Work Done by the Friction Force 260

7.7 Potential Energy and Conservation of Energy 261

7.8 Work Done and Potential Energy of an Elastic Force 261

7.9 Potential Energy Due to the Gravitational Force 262

7.9.1 Potential Energy Due to the Gravitational Force for a Particle 262

7.9.2 Potential Energy Due to the Gravitational Force for a Rigid Body 263

7.10 Examples 264

8 Simple Machines 295

8.1 Load and Effort, Mechanical Advantage, Velocity Ratio and Efficiency of a Simple Machine 295

8.1.1 Load and Effort 295

8.1.2 Mechanical Advantage 296

8.1.3 Velocity Ratio and Efficiency 296

8.2 Effort and Load of an Ideal Machine 297

8.3 The Lever 297

8.4 Inclined Plane (Wedge) 298

8.5 Screws 299

8.6 Simple Screwjack 299

8.6.1 Motion Impending Upwards 301

8.6.2 Motion Impending Downwards 302

8.6.3 Efficiency While Hoisting Load 303

8.7 Differential Screwjack 303

8.8 Pulleys 304

8.8.1 First-order Pulley System 304

8.8.2 Second-order Pulley System 306

8.8.3 Third-order Pulley System 307

8.9 Differential Pulley 308

8.10 Wheel and Axle 309

8.11 Wheel and Differential Axle 310

8.12 Examples 312

References 353

Index 357

Textauszug

1
Forces

1.1 Terminology and Notation

A force exerted on a body tends to change the state of the body, that is, if the body is rigid the force tends to move the body, but when the body is elasto-plastic the force tends to deform the body.

A force can be defined as a vector quantity that is defined by magnitude and direction. The direction of a force is specified by its orientation (also known as the line of action) and sense. The magnitude of a force is a positive scalar. A scalar is a number expressed in specific units of measure.

Vectors (forces) are usually denoted by boldface letters. If the starting point and the end point of a vector (force) are given, the vector (force) could be denoted by or more simply . It is also usual to denote the magnitude of the vector (force) by or by . Some other notations for vectorial quantities could be , , or .

Graphically a force is represented by a straight arrow as shown in Figure 1.1. The point is named the application point or the origin of the force and the line passing through and is named the action line of .

There are some possible operations regarding vectors.

Equality of forces

Two forces and are equal to each other when they have the same magnitude and direction, that is

(1.1)

If the forces and are equal but are acting at different locations on the same body it will not cause identical motion.

Multiplication of a Force by a Scalar

The product between a force and a scalar written as , is a force having the same orientation as , the same sense as if if and opposite sense if , and the magnitude .

Figure 1.1 Vector representation.

Zero Force

A zero force, usually denoted by , has a zero magnitude and an undefined direction.

Unit Vector (Force)

A unit vector has its magnitude equal to unity, that is, . Any force can be written as a product of a unit vector having the same orientation and sense as the force and its magnitude or equivalent

(1.2)

Addition of Forces

The sum of a two forces and is a new force named resultant. The sum of the forces and is the force represented graphically by the diagonal of the parallelogram shown in Figure 1.2 with its tail connecting the tail of the force and head connecting the head of the force .

The sum is named the difference of the two forces as shown in Figure 1.3.

Figure 1.2 Parallelogram law of vector addition.

Figure 1.3 Parallelogram law of vector subtraction.

1.2 Resolution of Forces

If the unit vectors , , have the same application point (origin) and are perpendicular to each other, as shown in Figure 1.4, they form a Cartesian reference frame.

Any force can be expressed with respect to the unit vectors , , by where , , and are the , , components of the force.

The magnitude of can be written as

Addition and subtraction of forces could be easily manipulated using the resolution of forces into components. Considering the forces and , one can calculate

and

1.3 Angle Between Two Forces

The angles between the forces and , and respectively and – in the range between and – are usually denoted by Greek letters such as  and , as shown in Figure 1.5.

The direction of a force in a Cartesian frame is given by the direction cosines (Figure 1.6) of the angles between by the force and the associated unit vectors , , , written as

A unit force (of magnitude 1) having the same direction as can be written as

Figure 1.4 Resolution of a force.

Figure 1.5 The angles and between the forces and , and respectively and .

Figure 1.6 Direction cosines.

1.4 Force Vector

The position force (vector) shown in Figure 1.7 of a point relative to a point can be written as

(1.3)

The position force (vector) shown in Figure 1.7 of the point relative to a point is calculated with

(1.4)

Figure 1.7 Position forces (vectors).

1.5 Scalar (Dot) Product of Two Forces

Definition. The dot product of two forces and is

(1.5)

where is the angle between the forces and .

1.6 Cross Product of Two Forces

The cross product of two forces and is another force defined by (Figure 1.8)

(1.6)

where is a unit force normal to and having its direction given by the right-hand rule.

The magnitude of the cross product is given by

When and the cross product can be calculated using

(1.7)

Figure 1.8 Cross product of two forces and .

1.7 Examples

Example 1.1

Figure 1.9 shows three forces , , and , and the angles of the forces with the horizontal , , and . The forces have the magnitudes  kN,  kN, and  kN. Find the resultant of the planar forces and the angle of the resultant with the horizontal.

Solution

The input data are introduced in MATLAB with:

clear; clc; close all F1 = 1; % kN F2 = 3; % kN F3 = 2; % kN % angle of force F1_ with x-axis theta1 = pi/6; % angle of force F2_ with x-axis theta2 = pi/3; % angle of force F3_ with x-axis theta3 = pi;

Figure 1.9 Graphical representation the forces , and .

The components of the forces on and axes are

(1.8)

or in MATLAB:

% components of forces F1_, F2_, and F3_ F1x = F1*cos(theta1); F1y = F1*sin(theta1); F1_ = [F1x F1y]; F2x = F2*cos(theta2); F2y = F2*sin(theta2); F2_ = [F2x F2y]; F3x = F3*cos(theta3); F3y = F3*sin(theta3); F3_ = [F3x F3y];

The numerical values are:

F1_ = [ 0.866 0.500] (kN) F2_ = [ 1.500 2.598] (kN) F3_ = [-2.000 0.000] (kN)

The resultant is calculated with

(1.9)

and the angle of the horizontal with the horizontal axis is

(1.10)

With MATLAB the resultant and the angle are calculated with:

R_ = F1_+F2_+F3_; phi = atand(R_(2)/R_(1));

and the results are

% R_ = F1_+F2_+F3_ = [ 0.366 3.098] (kN) % phi = atan(Ry, Rx) = 83.262 (deg)

The MATLAB representation of the forces is shown in Figure 1.10 and it is obtained with:

sa = 4; hold on axis([-sa sa -sa sa]) axis square quiver(0,0,F1_(1),F1_(2),0,'Color','k','LineWidth',1.2) quiver(0,0,F2_(1),F2_(2),0,'Color','k','LineWidth',1.2) quiver(0,0,F3_(1),F3_(2),0,'Color','k','LineWidth',1.2) quiver(0,0,R_(1),R_(2),0,'Color','r','LineWidth',2) text(F1_(1),F1_(2),' F_1',... 'fontsize',14,'fontweight','b') text(F2_(1),F2_(2),' F_2',... 'fontsize',14,'fontweight','b') text(F3_(1),F3_(2),' F_3',... 'fontsize',14,'fontweight','b') text(R_(1),R_(2),' R',... 'fontsize',14,'fontweight','b') grid on xlabel('x'), ylabel('y'),

Example 1.2

Figure 1.11 shows a system of spatial forces with the magnitudes  N,  N,  N, and  N. The parallelepiped has the dimensions  m,  m, and  m. Find:

1. (a) the resultant of the system of forces;
2. (b) the angle between the forces and ;
3. (c) the projection of the force on the force ; and
4. (d) calculate .

Figure 1.10 MATLAB representation of the forces , and .

Solution

1. 1)
2. (a) The input data in MATLAB are: clear; clc; close all a = 2; % m b = 3; % m c = 5; % m F1 = 15; % N F2 = 30; % N F3 = 10; % N F4 = 15; % N

   A Cartesian reference frame is selected as shown in Figure 1.11. The position force (vector) of the application point of the force is

   (1.11)

   The position vector of the application point of the force is

   (1.12)

   The position (vector) of the application point of the force is

   (1.13)

   The position vector of the application point of the force is

   (1.14)

   Figure 1.11 System of four spatial forces , , and...

Produkt-Details

• Erscheint lt. Verlag: 24.3.2021
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Chemie
• Schlagworte: chemical engineering • Chemische Verfahrenstechnik • Composites • Computer-aided Engineering • Computergestützte Verfahrenstechnik • Materials Science • Materialwissenschaften • Verbundwerkstoffe • Verfahrenstechnik
• ISBN-10: 1-119-09364-3 / 1119093643
• ISBN-13: 978-1-119-09364-0 / 9781119093640