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### Statics and Dynamics, Introduction

Statics and Dynamics, Introduction
Andy Ruina and Rudra Pratap
750 Pages

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Preface

This is a statics and dynamics text for second or third year engineering students with an emphasis on vectors, free body diagrams, the basic momentum balance principles, and the utility of computation. Students often start a course like this thinking of mechanics reasoning as being vague and complicated. Our aim is to replace this loose thinking with concrete and simple mechanics problem-solving skills that live harmoniously with a useful mechanical intuition.

Knowledge of freshman calculus is assumed. Although most students have seen vector dot and cross products, vector topics are introduced from scratch in the context of mechanics. The use of matrices (to tidy-up systems of linear equations) and of differential equations (for describing motion in dynamics) are presented to the extent needed. The set-up of equations for computer solutions is presented in a pseudolanguage easily translated by a student into one or another computation package that the student knows.

TOC

1 What is mechanics? 1
2 Vectors for mechanics 7
2.1 Vector notation and vector addition . . . . . . . . . . . . . . . . . 8
2.2 The dot product of two vectors . . . . . . . . . . . . . . . . . . . . 23
2.3 Cross product, moment, and moment about an axis . . . . . . . . . 32
2.4 Solving vector equations . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Equivalent force systems . . . . . . . . . . . . . . . . . . . . . . . 69
3 Free body diagrams 79
3.1 Free body diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 80

4 Statics 107
4.1 Static equilibrium of one body . . . . . . . . . . . . . . . . . . . . 109
4.2 Elementary truss analysis . . . . . . . . . . . . . . . . . . . . . . 129
4.3 Advanced truss analysis: determinacy, rigidity, and redundancy . . . 138
4.4 Internal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.5 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.6 Structures and machines . . . . . . . . . . . . . . . . . . . . . . . 179
4.7 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.8 Advanced statics . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

5 Dynamics of particles 217
5.1 Force and motion in 1D . . . . . . . . . . . . . . . . . . . . . . . 219
5.2 Energy methods in 1D . . . . . . . . . . . . . . . . . . . . . . . . 233
5.3 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . 240
5.4 More on vibrations: damping . . . . . . . . . . . . . . . . . . . . 257
5.5 Forced oscillations and resonance . . . . . . . . . . . . . . . . . . 264
5.6 Coupled motions in 1D . . . . . . . . . . . . . . . . . . . . . . . . 274
5.7 Time derivative of a vector: position, velocity and acceleration . . . 281
5.8 Spatial dynamics of a particle . . . . . . . . . . . . . . . . . . . . 289
5.9 Central-force motion and celestial mechanics . . . . . . . . . . . . 304
5.10 Coupled motions of particles in space . . . . . . . . . . . . . . . . 314

6 Constrained straight line motion 329
6.1 1-D constrained motion and pulleys . . . . . . . . . . . . . . . . . 330
6.2 2-D and 3-D forces even though the motion is straight . . . . . . . . 343

7 Circular motion 359
7.1 Kinematics of a particle in planar circular motion . . . . . . . . . . 360
7.2 Dynamics of a particle in circular motion . . . . . . . . . . . . . . 371
7.3 Kinematics of a rigid body in planar circular motion . . . . . . . . . 378
7.4 Dynamics of a rigid body in planar circular motion . . . . . . . . . 395
7.5 Polar moment of inertia: I cm zz and I O zz . . . . . . . . . . . . . . . . . 410
7.6 Using I cm zz and I O zz in 2-D circular motion dynamics . . . . . . . . . 420

8 General planar motion of a single rigid body 437
8.1 Kinematics of planar rigid-body motion . . . . . . . . . . . . . . . 438
8.2 General planar mechanics of a rigid-body . . . . . . . . . . . . . . 452
8.3 Kinematics of rolling and sliding . . . . . . . . . . . . . . . . . . . 467
8.4 Mechanics of contacting bodies: rolling and sliding . . . . . . . . . 480
8.5 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

9 Kinematics using time-varying base vectors 517
9.1 Polar coordinates and path coordinates . . . . . . . . . . . . . . . . 518
9.2 Rotating reference frames and their time-varying base vectors . . . . 532
9.3 General expressions for velocity and acceleration . . . . . . . . . . 545
9.4 Kinematics of 2-D mechanisms . . . . . . . . . . . . . . . . . . . 558
9.5 Advance kinematics of planar motion . . . . . . . . . . . . . . . . 572

10 Mechanics of constrained particles and rigid bodies 581
10.1 Mechanics of a constrained particle and of a particle relative to a moving frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
10.2 Mechanics of one-degree-of-freedom 2-D mechanisms . . . . . . . 602
10.3 Dynamics of rigid bodies in multi-degree-of-freedom 2-D mechanisms618

11 Introduction to three dimensional rigid body mechanics 637
11.1 3-D description of circular motion . . . . . . . . . . . . . . . . . . 638
11.2 Dynamics of fixed-axis rotation . . . . . . . . . . . . . . . . . . . 648
11.3 Moment of inertia matrices . . . . . . . . . . . . . . . . . . . . . 661
11.4 Mechanics using the moment of inertia matrix . . . . . . . . . . . . 672
11.5 Dynamic balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

A Units and dimensions 701
A.1 Units and dimensions . . . . . . . . . . . . . . . . . . . . . . . . 701
B Contact: friction and collisions 711
B.1 Contact laws are all rough approximations . . . . . . . . . . . . . . 712
B.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
B.3 A short critique of Coulomb friction . . . . . . . . . . . . . . . . . 716
B.4 Collision mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 721
Homework problems 722