Introduction to statics and dynamics chap1 10

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 Hal- dar, Dave Heimstra, Theresa Howley, Herbert Hui, Michael Marder, Elaina Mc- Cartney, 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 ex- amples 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 home- work 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 implemen- tation 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 The total force on a body is equal to its rate of change of linear momentum. (I) Impulse-momentum (integrating in time)  t 2 t 1   F i ·dt =   L Net impulseis equalto thechange in momentum. (Ia) Conservation of momentum (if   F i =  0 ) ˙  L =  0 ⇒   L =  L 2 −  L 1 =  0 When there is no net force the linear momentum does not change. (Ib) Statics (if ˙  L is negligible)   F i =  0 If the inertial terms are zero the net force on system is zero. (Ic) II) Angular Momentum Balance (AMB)/Moment Balance Equation of motion   M C = ˙ ˙  H C The sum of moments is equal to the rate of change of angular momentum. (II) Impulse-momentum (angular) (integrating in time)  t 2 t 1   M C dt =   H C The net angular impulse is equal to the change in angular mo mentum. (IIa) Conservation of angular momentum (if   M C =  0) ˙  H C =  0 ⇒   H C =  H C 2 −  H C 1 =  0 If there is no net moment about point C then the angular momentum about point C does not change. (IIb) Statics (if ˙  H C is negligible)   M C =  0 If the inertial terms are zero then the total moment on the system is zero. (IIc) III) Power Balance (1st law of thermodynamics) Equation of motion ˙ Q + P = ˙ E K + ˙ E P + ˙ E int    ˙ E Heat flow plus mechanical power into a system is equal to its change in energy (kinetic + potential + internal). (III) for finite time  t 2 t 1 ˙ Qdt +  t 2 t 1 Pdt =E The net energy flow going in is equal to the net change in energy. (IIIa) Conservation of Energy (if ˙ Q = P = 0) ˙ E = 0 ⇒ E = E 2 − E 1 = 0 If no energy flows into a system, then its energy does not change. (IIIb) Statics (if ˙ E K is negligible) ˙ Q + P = ˙ E P + ˙ E int If there is no change of kinetic energy then the change of potential and internal energy is due to mechanical work and heat flow. (IIIc) Pure Mechanics (if heat flow and dissipation are negligible) P = ˙ E K + ˙ E P In a system well modeled as purely mechanical the change of kinetic and potential energy is due to mechanical work. (IIId) Some Definitions  r or  x Position (e.g.,  r i ≡  r i/O is the position of a point i relative to the origin, O)  v ≡ d  r dt Velocity (e.g.,  v i ≡  v i/O is the velocity of apoint i relative to O, measured in a non-rotating reference frame)  a ≡ d  v dt = d 2  r dt 2 Acceleration (e.g.,  a i ≡  a i/O is the acceleration of a point i relative to O, measured in a New- tonian frame)  ω Angular (Please also look at the tables inside the back cover.) velocity A measure of rotational velocity of a rigid body.  α ≡ ˙  ω Angular acceleration A measure of rotational acceleration of a rigid body.  L ≡     m i  v i discrete   vdm continuous Linear momentum A measure of a system’s net translational rate (weighted by mass). = m tot  v cm ˙  L ≡     m i  a i discrete   adm continuous Rate of change of linear momentum The aspect of motion that balancesthe net force on a system. = m tot  a cm  H C ≡      r i/C × m i  v i discrete   r /C ×  vdm continuous Angular momentum about point C A measure of the rotational rate of a sys- tem about a point C (weighted by mass and distance from C). ˙  H C ≡      r i/C × m i  a i discrete   r /C ×  adm continuous Rate of change of angular mo- mentum about point C The aspect of motion that balancesthe net torque on a system about a point C. E K ≡    1 2  m i v 2 i discrete 1 2  v 2 dm continuous Kinetic energy A scalar measure of net system motion. E int = (heat-like terms) Internal energy The non-kinetic non-potential part of a system’s total energy. P ≡   F i ·  v i +   M i ·  ω i Power of forces and torques The mechanical energy flow into a sys- tem. Also, P ≡ ˙ W, rate of work. [I cm ]≡      I cm xx I cm xy I cm xz I cm xy I cm yy I cm yz I cm xz I cm yz I cm zz      Moment of inertia matrix about cm A measure of how mass is distributed in 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: I cm zz and I O zz 404 7.6 Using I cm zz and I O zz 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 pseudo- language 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 A. particles 1) 1D 2) 2D 3) 3D B. rigid bodies 4) 1D 5) 2D 6) 3D II. Dynamics C. particles 7) 1D 8) 2D 9) 3D D. rigid bodies 10) 1D 11) 2D 12) 3D complexity of objects number of dimensions how much inertia 1D 2D 3D static dynamic particle rigid body However,thesetopicsarefarfromequalintheirdifficultyorinthenumberofsubtopics 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: 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. Vectoraddition 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 isabout freebody diagrams. Itis aseparatechapter 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, mecha- nisms, beams and hydrostatics. The emphasis is on two-dimensional problems until the last, more advanced section. Solution methods that depend on kine- matics (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 introductiontomotion of oneparticlein one dimension, socalledscalarphysics, 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 variousforcelaws,andevenwithmultiple particles, is emphasized. Thechapter 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 kinematictoolhereisthenon-trivialuseofthecrossproduct. Fixedaxisrotation 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). PREFACE v Chapter 10 is entirely about kinematics of particle motion. The over-riding theme is the use of base vectors which change with time. First, the discussion of polar coor- dinates started in chapter 7 iscompleted. Then path coordinatesare introduced. The kinematics of relative motion, a topic that many students find difficult, is treated carefully but not elaborately in two stages. First using rotating base vectors connected to a moving rigid body and then using the more abstract notation associated with the famous “five term acceleration formula.” Chapter 11 is about the mechanics of 2D mechanisms using the kinematics from chapter 10. Chapter 12 pushes some of the contents of chapter 9 into three dimensions. In particular, the three dimensional motion of a single rigid body is covered. Rather than emphasize the few problems that are amenable to pencil and paper solution, emphasis is on the basic principles and on the setup for numerical solution. Chapter 13 on contact laws (friction, collisions, and rolling) will probably serve only as a reference for most courses. Because elementary reference material on these topics is so lacking, these topics are covered here with more depth than can be found in any modern text at any level. Chapter 14 on units and dimensions is placed at the end for reference. Because students are immune to preaching about units out of context, such as in an early or late chapter like this one, the main messages are presented by example throughout the book: – All engineering calculations using dimensional quantitiesmust be dimen- sionally ‘balanced’. – Units are ‘carried’ from one line of calculation to the next by the same rules as go numbers and variables. A leisurely one semester statics course, or a more fast-paced half semester prelude to strength of materials shoulduse chapters 1-4. Atypical onesemester dynamics course should cover about two thirds of chapters 5-12 preceded by topics from chapters 1-4, as needed. A one semester statics and dynamics course should cover about two thirds of chapters 1-6 and 8. A full year statics and dynamics course should cover most of the book. Organization and formatting Each subject is covered in various ways. • Every section starts with descriptive text and short examples motivating and describing the theory; • More detailed explanations of the theory are in boxes interspersed in the text. For example, one box explains the common derivation of angular momentum balance form linear momentum balance, one explains the genius of the wheel, and another connects  ω based kinematics to ˆ e r and ˆ e θ based kinematics; • Sample problems (marked with a gray border) at the end of most sections show how to do homework-like calculations. These are meticulous in their use of free body diagrams, systematic application of basic principles, vector notation, units, and checks against intuition and special cases; • Homework problems at the end of each chapter give students a chance to practice mechanics calculations. The first problems for each section build a student’s confidence with the basic ideas. The problems are ranked in approxi- mate order of difficulty, with theoretical questions last. Problems marked with an * have an answer at the back of the book; vi PREFACE • Reference tables on the inside covers and end pages concisely summarize much of the content in the book. These tables can save students the time of hunting for formulas and definitions. They also serve to visibly demonstrate the basically simple structure of the whole subject of mechanics. Notation Clear vector notation helps students do problems. Students sometimes mistakenly transcribe a conventionally printed bold vector F the same way they transcribe a plain-text scalar F. To help minimize this error we use a redundant vector notation in this book (bold and harpooned  F ). As for all authors and teachers concerned with motion in two and three dimen- sions we have struggled with the tradeoffs between a precise notation and a simple notation. Beautifully clear notations are intimidating. Perfectly simple notations are ambiguous. Our attempt to find clarity without clutter is summarized in the box on page 9. Relation to other mechanics books This book is in some ways original in organization and approach. It also contains some important but not sufficiently well known concepts, for example that angular momentum balance applies relative to any point, not just an arcane list of points. But there is little mechanics here that cannot be found in other books, including freshman physics texts, other engineering texts, and hundreds of classics. Mastery of freshman physics (e.g., from Halliday & Resnick, Tipler, or Serway) would encompass some part of this book’s contents. However freshman physics generally leaves students with a vague notion of what mechanics is, and how it can be used. For example many students leave freshman physics with the sense that a free body diagram (or ‘force diagram’) is an vague conceptual picture with arrows for various forces and motions drawn on it this way and that. Even the book pictures sometimes do not make clear what force is acting on what body. Also, because freshman physics tends to avoid use of college math, many students end up with no sense of how to use vectors or calculus to solve mechanics problems. This book aims to lead students who may start with these fuzzy freshman physics notions into a world of intuitive yet precise mechanics. There are many statics and dynamics textbooks which cover about the same material as this one. These textbooks have modern applications, ample samples, lots of pictures, and lots of homework problems. Many are good (or even excellent) in their own ways. Most of today’s engineering professors learned from one of these books. We wrote this book because the other books do not adequately convey the simple network of ideas that makes up the whole of Newtonian mechanics. We intend that through this book book students will come to see not mechanics as a coherent network of basic ideas rather than a collection of ad-hoc recipes and tricks that one need memorize or hope to discover by divine inspiration. There are hundreds of older books with titles like statics, engineering mechan- ics, dynamics, machines, mechanisms, kinematics,orelementary physics that cover aspects of the material here 1  Although many mechanics books written from 1689- 1  One near-classic that we have especially enjoyed is J.P. Den Hartog’s Mechanics originally published in 1948 but still avail- able as an inexpensive reprint. 1960, are amazingly thoughtful and complete, none are good modern textbooks. They lack an appropriate pace, style of speech, and organization. They are too reliant on geometry skills and not enough on vectors and numerical computation skills. They lack sufficient modern applications, sample calculations, illustrations, and homework problems for a modern text book. [...]... machines and machine parts are designed to move something Bicycles, planes, elevators, and hearses are designed to move people; a clockwork, to move clock hands; insect wings, to move insect bodies; and forks, to move potatoes A connecting rod is designed to move a crankshaft; a crankshaft, to move a transmission; and a transmission, to move a wheel And wheels are designed to move bicycles, cars, and skateboards... about vector arithmetic In the rest of this chapter you will learn how to add and subtract vectors, how to stretch them, how to find their components, and how to multiply them with each other two different ways Each of these operations has use in mechanics and, in particular, the concept of vector addition always has a physical interpretation 2.1 Vector notation and vector addition Facility with vectors... cosines can be proved by and C sin a and sin c C sin b sin c In this era of vector algebra and vector components the laws of sines and cosines are seldom used They are here for completeness 16 CHAPTER 2 Vectors for mechanics 2.1 Vector notation and vector addition 17 ˆ SAMPLE 2.1 Drawing vectors: Draw the vector r = 3 ftˆ − 2 ft using ı (a) its components and (b) its magnitude and slope Solution (a)... we could write “Click on WWW.MECH.TOOL today and order your own professional vector calculator and expert free body diagram drawing tool!”, but we can’t After we informally introduce mechanics in the first chapter, the second and third chapters help you build your own set of these two most-important tools Guarantee: if you learn to do clear correct vector algebra and to draw good free body diagrams you... for hand written work The lack of bold face pens and pencils tempts students to transcribe a bold F as F But F with no adornment represents a scalar and not a vector Learning how to work with vectors and scalars is hard enough without the added confusion of not being able to tell at a glance which terms in your equations are vectors and which are scalars 1 Caution: Be careful to distinguish vectors... time when we had to think about the order of our work You also have to think about the order of your work You will find some tips in the text and samples But it is your job to own the material, to learn how to think about it your own way, to become an expert in your own style, and to do the work in the way that makes things most clear to you and your readers What’s in your toolbox? In the toolbox of someone... standard functions (x 3 , cos−1 θ , etc.), to enter and manipulate lists and arrays of numbers, and to write short programs Classical languages, applied packages, and simulators Programming in standard languages such as Fortran, Basic, C, Pascal, or Java probably take too much time to use in solving simple mechanics problems Thus an engineer needs to learn to use one or another widely available computational... are two well worn tools: • A vector calculator that always keeps vectors and scalars distinct, and • A reliable and clear free body diagram drawing tool Because many of the terms in mechanics equations are vectors, the ability to do vector calculations is essential Because the concept of an isolated system is at the core of mechanics, every mechanics practitioner needs the ability to draw a good free... used to define all the others, are the two scalars, mass m and time t, and the two vectors, relative position ri/O , and force F Scalars are typed with an ordinary font (t and m) and vectors are typed in bold with a harpoon on top ( ri/O , F ) All of the other quantities we use in mechanics are defined in terms of these four A list of all the scalars and vectors used in mechanics are given in boxes 2 and. .. theta = X = Y = plot Y vs [1 2 3 100 ] npoints * 2 * pi / 100 cos(theta) sin(theta) X where npoints is the list of numbers from 1 to 100 , theta is a list of 100 numbers evenly spaced between 0 and 2π and X and Y are lists of 100 corresponding x, y coordinate points on a circle • The result of using the laws of dynamics is often a set of differential equations which need to be solved A simple example would . 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 bodies; and forks, to move potatoes. A connecting rod is designed to move a crankshaft; a crankshaft, to move a transmission; and a transmission, to move a wheel. And wheels are designed to move. how to think about it your own way, to become an expert in your own style, and to do the work in the way that makes things most clear to you and your readers. What’s in your toolbox? In the toolbox