Introduction to STATICS DYNAMICS Chapters 1-10

[Pages:605]Introduction to

STATICS and

DYNAMICS

Chapters 1-10

Rudra Pratap and Andy Ruina Spring 2001

c Rudra Pratap and Andy Ruina, 1994-2001. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without prior written permission of the authors.

This book is a pre-release version of a book in progress for Oxford University Press.

The following are amongst those who have helped with this book as editors, artists, advisors, or critics: Alexa Barnes, Joseph Burns, Jason Cortell, Ivan Dobrianov, Gabor Domokos, Thu Dong, Gail Fish, John Gibson, Saptarsi Haldar, Dave Heimstra, Theresa Howley, Herbert Hui, Michael Marder, Elaina McCartney, Arthur Ogawa, Kalpana Pratap, Richard Rand, Dane Quinn, Phoebus Rosakis, Les Schaeffer, David Shipman, Jill Startzell, Saskya van Nouhuys, Bill Zobrist. Mike Coleman worked extensively on the text, wrote many of the examples and homework problems and created many of the figures. David Ho has brought almost all of the artwork to its present state. Some of the homework problems are modifications from the Cornell's Theoretical and Applied Mechanics archives and thus are due to T&AM faculty or their libraries in ways that we do not know how to give proper attribution. Many unlisted friends, colleagues, relatives, students, and anonymous reviewers have also made helpful suggestions.

Software used to prepare this book includes TeXtures, BLUESKY's implementation of LaTeX, Adobe Illustrator and MATLAB.

Most recent text modifications on January 21, 2001.

Summary of Mechanics

0) The laws of mechanics apply to any collection of material or `body.' This body could be the overall system of study or any part of it. In the equations below, the forces and moments are those that show on a free body diagram. Interacting bodies cause equal and opposite forces and moments on each other.

I) Linear Momentum Balance (LMB)/Force Balance

Equation of Motion

F i = L

Impulse-momentum (integrating in time)

Conservation of momentum (if F i = 0 )

Statics (if L is negligible)

t2

F i ?dt = L

t1

L = 0 L = L2 - L1 = 0

Fi = 0

The total force on a body is equal (I) to its rate of change of linear momentum.

Net impulse is equal to the change in (Ia) momentum.

When there is no net force the linear (Ib) momentum does not change.

If the inertial terms are zero the (Ic) net force on system is zero.

II) Angular Momentum Balance (AMB)/Moment Balance

Equation of motion

MC = H C

Impulse-momentum (angular) (integrating in time)

Conservation of angular momentum (if MC = 0)

t2

MCdt = H C

t1

H C = 0 HC = HC2 - HC1 = 0

Statics (if H C is negligible)

MC = 0

The sum of moments is equal to the (II) rate of change of angular momentum.

The net angular impulse is equal to (IIa) the change in angular momentum.

If there is no net moment about point (IIb) C then the angular momentum about point C does not change. If the inertial terms are zero then the (IIc) total moment on the system is zero.

III) Power Balance (1st law of thermodynamics)

Equation of motion

Q + P = EK + EP + Eint

E

for finite time

Conservation of Energy (if Q = P = 0)

Statics (if EK is negligible)

t2

t2

Q dt + Pdt = E

t1

t1

E = 0

E = E2 - E1 = 0

Q + P = EP + Eint

Pure Mechanics (if heat flow and dissipation are negligible)

P = EK + EP

Heat flow plus mechanical power (III) into a system is equal to its change in energy (kinetic + potential + internal).

The net energy flow going in is equal (IIIa) to the net change in energy.

If no energy flows into a system, (IIIb) then its energy does not change.

If there is no change of kinetic energy (IIIc) then the change of potential and internal energy is due to mechanical work and heat flow.

In a system well modeled as purely (IIId) mechanical the change of kinetic and potential energy is due to mechanical work.

Some Definitions

(Please also look at the tables inside the back cover.)

r or x v dr

dt

a

dv dt

=

d2r dt2

L

mi v i discrete v dm continuous

= mtot vcm

Position Velocity Acceleration Angular velocity Angular acceleration Linear momentum

(e.g., r i r i/O is the position of a point i relative to the origin, O)

(e.g., v i v i/O is the velocity of a point i relative to O, measured in a non-rotating reference frame)

(e.g., ai ai/O is the acceleration of a point i relative to O, measured in a Newtonian frame)

A measure of rotational velocity of a rigid body.

A measure of rotational acceleration of a rigid body.

A measure of a system's net translational rate (weighted by mass).

L

mi ai discrete adm continuous

= HC

mtot a cm ri/C ? mi v i

r /C ? vdm

discrete continuous

Rate of change of linear The aspect of motion that balances the net

momentum

force on a system.

Angular momentum about A measure of the rotational rate of a sys-

point C

tem about a point C (weighted by mass

and distance from C).

H C EK

ri/C ? mi ai discrete

r /C ? adm continuous

1

2

mi vi2 discrete

1

2

v2dm

continuous

Rate of change of angular mo- The aspect of motion that balances the net

mentum about point C

torque on a system about a point C.

Kinetic energy

A scalar measure of net system motion.

Eint = (heat-like terms)

P

F i ?v i + Mi ?i

[I cm]

Ixcxm Ixcym

Ixcym

I

cm yy

Ixczm

I

cm yz

Ixczm

I

cm yz

Izczm

Internal energy Power of forces and torques

The non-kinetic non-potential part of a system's total energy.

The mechanical energy flow into a system. Also, P W , rate of work.

Moment of inertia matrix about A measure of how mass is distributed in

cm

a rigid body.

Contents

1 Mechanics

1

1.1 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 . . . . . . . . . . . . . . . . . . . . 24

2.3 Cross product, moment, and moment about an axis . . . . . . . . . 34

2.4 Equivalent force systems . . . . . . . . . . . . . . . . . . . . . . . 53

2.5 Center of mass and gravity . . . . . . . . . . . . . . . . . . . . . . 62

3 Free body diagrams

77

3.1 Free body diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Statics

105

4.1 Static equilibrium of one body . . . . . . . . . . . . . . . . . . . . 107

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

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 . . . . . . . . . . . . . . . . . . . . 288

5.9 Central-force motion and celestial mechanics . . . . . . . . . . . . 302

5.10 Coupled motions of particles in space . . . . . . . . . . . . . . . . 312

6 Constrained straight line motion

327

6.1 1-D constrained motion and pulleys . . . . . . . . . . . . . . . . . 328

6.2 2-D and 3-D forces even though the motion is straight . . . . . . . . 339

i

ii

CONTENTS

7 Circular motion

353

7.1 Kinematics of a particle in planar circular motion . . . . . . . . . . 354

7.2 Dynamics of a particle in circular motion . . . . . . . . . . . . . . 365

7.3 Kinematics of a rigid body in planar circular motion . . . . . . . . . 372

7.4 Dynamics of a rigid body in planar circular motion . . . . . . . . . 389

7.5 Polar moment of inertia: Izczm and IzOz . . . . . . . . . . . . . . . . . 404 7.6 Using Izczm and IzOz in 2-D circular motion dynamics . . . . . . . . . 414

8 Advanced topics in circular motion

431

8.1 3-D description of circular motion . . . . . . . . . . . . . . . . . . 432

8.2 Dynamics of fixed-axis rotation . . . . . . . . . . . . . . . . . . . 442

8.3 Moment of inertia matrices [I cm] and [I O] . . . . . . . . . . . . . . 455

8.4 Mechanics using [I cm] and [I O] . . . . . . . . . . . . . . . . . . . 467

8.5 Dynamic balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

9 General planar motion of a rigid body

497

9.1 Kinematics of planar rigid-body motion . . . . . . . . . . . . . . . 498

9.2 Unconstrained dynamics of 2-D rigid-body planar motion . . . . . . 508

9.3 Special topics in planar kinematics . . . . . . . . . . . . . . . . . . 513

9.4 Mechanics of contacting bodies: rolling and sliding . . . . . . . . . 526

9.5 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

10 Kinematics using time-varying base vectors

547

10.1 Polar coordinates and path coordinates . . . . . . . . . . . . . . . . 547

10.2 Rotating reference frames . . . . . . . . . . . . . . . . . . . . . . 557

10.3 General expressions for velocity and acceleration . . . . . . . . . . 560

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 tidily set up systems of 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 the student into one or another computation package that the student knows.

Organization

We have aimed here to better unify the subject, in part, by an improved organization. Mechanics can be subdivided in various ways: statics vs dynamics, particles vs rigid bodies, and 1 vs 2 vs 3 spatial dimensions. Thus a 12 chapter mechanics table of contents could look like this

I. Statics

II. Dynamics

A. particles

C. particles

1) 1D 2) 2D 3) 3D

7) 1D 8) 2D 9) 3D

B. rigid bodies

D. rigid bodies

4) 1D 5) 2D 6) 3D

10) 1D 11) 2D 12) 3D

However, these topics are far from equal in their difficulty or in the number of subtopics they contain. Further, there are various concepts and skills that are common to many of the 12 sub-topics. Dividing mechanics into these bits distracts from the unity of the subject. Although some vestiges of the scheme above remain, our book has evolved to a different organization through trial and error, thought and rethought, review and revision, and nine semesters of student testing.

The first four chapters cover the basics of statics. Dynamics of particles and rigid bodies, based on progressively more difficult motions, is presented in chapters five to twelve. Relatively harder topics, that might be skipped in quicker courses, are identifiable by chapter, section or subsection titles containing words like "three dimensional" or "advanced". In more detail:

complexity of objects

rigid body

particle

static dynamic

1D how much

inertia

number of dimensions 2D 3D

iii

iv

PREFACE

Chapter 1 defines mechanics as a subject which makes predictions about forces and motions using models of mechanical behavior, geometry, and the basic balance laws. The laws of mechanics are informally summarized.

Chapter 2 introduces vector skills in the context of mechanics. Notational clarity is emphasized because correct calculation is impossible without distinguishing vectors from scalars. Vector addition is motivated by the need to add forces and relative positions, dot products are motivated as the tool which reduces vector equations to scalar equations, and cross products are motivated as the formula which correctly calculates the heuristically motivated concept of moment and moment about an axis.

Chapter 3 is about free body diagrams. It is a separate chapter because, in our experience, good use of free body diagrams is almost synonymous with correct mechanics problem solution. To emphasize this to students we recommend that, to get any credit for a problem that uses balance laws, a free body diagram must be drawn.

Chapter 4 makes up a short course in statics including an introduction to trusses, mechanisms, beams and hydrostatics. The emphasis is on two-dimensional problems until the last, more advanced section. Solution methods that depend on kinematics (i.e., work methods) are deferred until the dynamics chapters. But for the stretch of linear springs, deformations are not covered.

Chapter 5 is about unconstrained motion of one or more particles. It shows how far you can go using F = m a and Cartesian coordinates in 1, 2 and 3 dimensions in the absence of kinematic constraints. The first five sections are a thorough introduction to motion of one particle in one dimension, so called scalar physics, namely the equation F(x, v, t) = ma. This involves review of freshman calculus as well as an introduction to energy methods. A few special cases are emphasized, namely, constant acceleration, force dependent on position (thus motivating energy methods), and the harmonic oscillator. After one section on coupled motions in 1 dimension, sections seven to ten discuss motion in two and three dimensions. The easy set up for computation of trajectories, with various force laws, and even with multiple particles, is emphasized. The chapter ends with a mostly theoretical section on the center-of-mass simplifications for systems of particles.

Chapter 6 is the first chapter that concerns kinematic constraint in its simplest context, systems that are constrained to move without rotation in a straight line. In one dimension pulley problems provide the main example. Two and three dimensional problems are covered, such as finding structural support forces in accelerating vehicles and the slowing or incipient capsize of a braking car. Angular momentum balance is introduced as a needed tool but without the usual complexities of curvilinear motion.

Chapter 7 treats pure rotation about a fixed axis in two dimensions. Polar coordinates and base vectors are first used here in their simplest possible context. The primary applications are pendulums, gear trains, and rotationally accelerating motors or brakes.

Chapter 8 extends chapter 7 to fixed axis rotation in three dimensions. The key new kinematic tool here is the non-trivial use of the cross product. Fixed axis rotation is the simplest motion with which one can introduce the full moment of inertia matrix, where the diagonal terms are analogous to the scalar 2D moment of inertia and the off-diagonal terms have a "centripetal" interpretation. The main new application is dynamic balance.

Chapter 9 treats general planar motion of a (planar) rigid body including rolling, sliding and free flight. Multi-body systems are also considered so long as they do not involve constraint (i.e., collisions and spring connections but not hinges or prismatic joints).

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