Theoretical Mechanics for Engineers. Lectures

This book is a written aid to the lectures on Theoretical Mechanics, being given to the international students of Mechanical Engineering at the Faculty of Automotive and Construction Machinery Engineering of the Warsaw University of Technology. The main objective of the book is to provide the undergraduate students with the necessary knowledge and skills to solve mechanical problems that can occur in their engineering job, or earlier – during studies – within semester or diplome projects.

Contents

Preface / 11
Part I. Theoretical Mechanics I / 15

Chapter 1. Introduction / 17
Lecture 1
1.1. What is Mechanics? / 17
1.2. Classification of Mechanics / 17
1.3. Historical outline / 19
1.4. Mechanics as a theory / 20
1.5. Parts of Mechanics / 22
1.6. Mechanical quantities and system of physical units / 22
1.7. Useful knowledge preceding Mechanics / 23
Review questions to Chapter 1 / 24

Chapter 2. Vectors and vector calculus / 25
Lecture 2
2.1. Scalars and vectors in Mechanics / 25
2.2. Geometric and analytic description of vectors / 25
2.3. Vector calculus / 26
2.4. Vector functions of time / 29
Review questions to Chapter 2 / 34

Chapter 3. Geometry of masses / 35
Lecture 3
3.1. What is geometry of masses? / 35
3.2. First moments and center of mass of a multi-particle system and a body / 36
3.3. Calculation of first moments and positions of mass centers of bodies / 38
3.4. Pappus-Guldinus rules / 46
Lecture 4
3.5. Moments and products of inertia (second moments) / 49
3.6. Calculation of second moments / 53
3.7. Transformation of the second moments due to translation of reference frame / 58
Lecture 5
3.8. Transformation of second moments due to rotation of the reference frame / 64
3.9. Principal axes and principal moments of inertia / 68
3.10. Ellipsoid of inertia of a body at a point / 73
Review questions to Chapter 3 / 76

Chapter 4. Statics of mechanical systems / 77
Lecture 6
4.1. Subject, problems and methods of statics / 77
4.2. Classification of forces (and free moments) in statics / 80
4.3. Constraints and supports of mechanical systems / 82
4.4. Equivalent reduction of forces and moments acting on a body / 84
4.5. Geometric conditions of equilibrium of mechanical systems / 92
Lecture 7
4.6. Friction in statics of mechanical systems / 96
4.7. Friction-induced zones of equilibrium and intervals of loads / 98
4.8. Friction-induced static indeterminability / 101
4.9. Duality of loss of equilibrium / 102
4.10. Self-locking and jamming effects / 108
Lecture 8
4.11. Rolling resistance / 114
4.12. Belt friction / 117
4.13. Plane trusses / 119
Review questions to Chapter 4 / 125

Chapter 5. Kinematics of a particle / 127
Lecture 9
5.1. Geometric and analytic description of position of a particle / 127
5.2. Path of a particle in space / 129
5.3. Velocity and acceleration of a particle / 135
5.4. Components of velocity and acceleration in the cylindrical reference frame / 137
Lecture 10
5.5. Components of velocity and acceleration in natural directions / 141
5.6. Rectilinear motion of a particle / 144
5.7. Curvilinear motion of a particle in a uniform field of acceleration / 149
5.8. Motion of a particle in a central field of acceleration / 150
Review questions to Chapter 5 / 152

Chapter 6. Dynamics of a particle / 153
Lecture 11
6.1. Introduction to dynamics / 153
6.2. Dynamics of a free particle / 154
6.3. Motion of a particle under force dependent on position / 155
6.4. Motion of a particle under force dependent on velocity / 162
Lecture 12
6.5. Motion of a particle under the Lorentz force / 167
6.6. Motion of a particle under force dependent on time / 169
6.7. Dynamics of a constrained particle / 171
6.8. Linear momentum law of a particle / 175
6.9. Angular momentum law of a particle / 179
Lecture 13
6.10. Work and power of a force acting on a particle / 183
6.11. Kinetic energy of a particle and kinetic energy law / 186
6.12. Kinetic energy law in a potential field of force / 188
Review questions to Chapter 6 / 194

Chapter 7. Dynamics of multi-particle systems / 195
Lecture 14
7.1. Definition and basic properties of a multi-particle system / 195
7.2. Equations of motion of a multi-particle system / 198
7.3. Linear momentum law of a multi-particle system / 203
7.4. Law of motion of the mass center of a multi-particle system / 205
Lecture 15
7.5. Angular momentum law of a system of particles / 210
7.6. Kinetic energy law of a system of particles / 213
7.7. Kinetic energy law in case of potential forces / 215
Review questions to Chapter 7 / 222
Part II. Theoretical Mechanics II / 223

Chapter 8. Kinematics of a rigid body / 225
Lecture 1
8.1. Description of the position of a body in space / 225
8.2. Classification of motions of a rigid body / 231
8.3. Velocity of points of a body in arbitrary motion / 233
Lecture 2
8.4. Acceleration of points of a body in arbitrary motion / 236
8.5. Velocity and acceleration of points of a body in translatory motion / 240
8.6. Velocity and acceleration of points of a body in rotation about a fixed point / 242
Lecture 3
8.7. Velocity and acceleration of points of a body in plane motion / 248
8.8. Velocity and acceleration of points of a body in screw motion / 256
Review questions to Chapter 8 / 258

Chapter 9. Resultant motion of a particle / 259
Lecture 4
9.1. Description of motion of a particle in different reference frames / 259
9.2. Velocity and acceleration of a particle in resultant motion / 262
Lecture 5
9.3. Dynamics of a particle in relative motion / 269
Review questions to Chapter 9 / 280

Chapter 10. Dynamics of a rigid body / 281
Lecture 6
10.1. Kinetic energy of a rigid body / 281
10.2. Kinetic energy law of a rigid body / 286
Lecture 7
10.3. Linear momentum of a body and linear momentum law / 291
10.4. Angular momentum of a body and angular momentum law / 294
10.5. Equations of motion of a body resulting from linear and angular momentum laws / 300
10.6. Dynamics of a body in translatory motion / 301
Lecture 8
10.7. Dynamics of rotation of a body about a fixed axis / 304
10.8. Reactions in bearings of a body rotating about a fixed axis / 307
Lecture 9
10.9. Dynamics of a body in rotation about a fixed point / 316
10.10. Gyroscopic phenomenon / 322
10.11. Dynamics of a body in plane motion / 324
Lecture 10
10.12. Dynamics of a rolling wheel / 327
10.13. Dynamics of vehicles / 331
Review questions to Chapter 10 / 336

Chapter 11. Elements of analytical mechanics / 337
Lecture 11
11.1. Motivation for studies of analytical mechanics / 337
11.2. Constraints and generalized coordinates of a multi-particle system / 338
11.3. Virtual displacements in a multi-particle system / 341
11.4. Principle of virtual work / 343
11.5. Principle of virtual work in case of a rigid body / 349
Lecture 12
11.6. D’Alembert’s principle / 351
11.7. Lagrange’s equations / 355
Review questions to Chapter 11 / 366

Chapter 12. Elementary impact theory / 367
Lecture 13
12.1. Impact forces / 367
12.2. Dynamics of a particle under impact force / 368
12.3. Collision of a particle with a resting body / 370
12.4. Collision of two particles / 374
12.5. Effect of impact on a rigid body / 377
Lecture 14
12.6. Effect of impact on a body rotating about a fixed axis / 380
12.7. Collision of two bodies in plane motion / 385
Review questions to Chapter 12 / 392

Chapter 13. Dynamics of a particle of continuously varying mass / 393
Lecture 15
13.1. Equation of motion of a particle of variable mass / 393
13.2. Dynamics of a rotating body with a variable moment of inertia / 399
13.3. Equation of motion of a rocket / 401
Review questions to Chapter 13 / 404

References / 405

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