OSE800: Analytical and Computational

Dynamics (Spring, 2013)

What we will learn in this course:

1. Kinematics which are essential for the description of motionsof masses.

2. Ability to derive the equations of motion for complex dy-namical systems.

3. Understanding of vibration phenomena for structural ele-ments.

4. Ability to model complex vibration problems by reducednumber of equations.

5. Applications of dynamical principles for engineering designapplications.

Above All, Become Excellent Dynamists!

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Lecture 0: What Isaac Newton (1642-1727) knewfrom his predecessors and Newton’s Principia

Or did he really get an inspiration to develop his theory of grav-itation when he observed an apple falling while he was sittingunder an apple tree?

1.1 An Overview of Dynamics

According to E. Mach and G. Kirchhoff, dynamics is a branchof science that attempts to explain the aggregate phenomena ofmass (force), space (geometry) and time (absolute). While thereis a growing consensus that statics can be subsumed into dy-namics as a special category, historically it was statics that firstreceived attention from many distinguished philosophers. Essen-tial concepts developed over the years in statics include: the prin-ciple of lever and pulley by Archytas (ca 400 B.C.) and Aristotle(384-322 B.C.), the center of gravity by Archimedes (287-212B.C.), the statical moments by Leonardo da Vinci (1452-1519),the principle of plane by Stebinus (1548-1620) and Ubaldo (1545-1607), the principle of virtual velocity by Jean Bernoulli (1667-1748), the rule of parallelogram by Daniel Bernoulli (1700-1782),and the state of stable or unstable equilibrium and the theoryof least action by Pierre L. M. de Maupertuis (1698-1759).

The development of these principles in statics was largely mo-tivated by the questions raised by Aristotle in his book entitledQuestions in Mechanics, perhaps the first known literature onmechanics. His ideas on the principle of balance and a fallingbody were then expanded by Archimedes who left us with amonumental document as compiled by the Cambridge UniversityPress in 1897. It is generally agreed, however, that the modernera of dynamics began with the work of Galileo Galilei (1564-1642) on the motion of projectiles contained in Two New Sci-ences published in 1636 by Elzevir at Leyden, Holland. Historyhas a special coincidence in that the day Michelangelo died on18 February 1564, Galileo was born and the year Galileo died in1642, Issac Newton was born.

Among the contributions of Galileo to dynamics are the conceptof uniformly accelerated motion from his famous experiment on

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falling bodies conducted in 1583, the definition of time (or theprinciple of relativity), the law of inertia which Newton laterincorporated as his first law, the principle of the superpositionof motions, and the establishment of the dynamics of a singlebody. Galileo is also credited to have proposed the design of apendulum clock after he observed the isochronism of a swingingbody. Unfortunately, he did not live to see his idea come true.Following these principles of dynamics due to Galileo, Huygens(1629-1695) introduced the concept of the center of oscillations,determined for the first time the acceleration of gravity by usinghis pendulum device in 1656, established the property of cen-trifugal forces, and the circle of curvature. his idea come true.

Following Galileo and Huygens, Newton (1642-1727) synthesizedthe works of Kepler, Galileo and Huygens to realize that a curvi-linear motion implies deflective acceleration. This observationpaved the way eventually leading to his discovery of universalgravitation. In addition, Newton generalized the idea of force,introduced the concept of mass, accomplished the general formu-lation of the principle of the parallelogram of forces, and estab-lished the law of action and reaction. Thus, he completed theformal enunciation of the mechanical principles now generallyaccepted. His contributions to dynamics, viz., the three laws ofmotions and the law of universal gravitation, are amalgamatedin Philosophiae Naturallis Principia Mathematica published in1687.

The awareness of the rotation of the Earth rekindled an in-terest in the problems of rotational mechanics during the mid-Eighteenth Century, which led both d‘Alembert (1717-1783) andEuler (1707-1783) to publish in 1749 on the problem of the pre-cession of the equinoxes. Subsequently, it was Euler who intro-duced the law of angular momentum which paved the way forthe solution of many rotational motions via the celebrated the-orem on the uniqueness of the axis of rotation. He was also thefirst to introduce the moment of inertia. His ideas and math-ematical treatise on rotational motions were published in 1736and 1765.

In 1788 Lagrange solved the so-called ‘heavy rigid body problem’including a revisitation of the torque-free top problem addressedby Euler. The mathematical treatment he employed therein was

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thought to be simpler than that by Euler, which gave birth toa new school called ‘Lagrangian Dynamics’ by his followers. Inaddition, he attacked the problem of the liberation of the Moon.He was able to combine Newton’s law of the forces of inertiaand Galileo’s principle of the composition of motions to a newprinciple that is now referred to as the principle of virtual workin dynamics: “The sum of the moments[that is, apart from sign,the virtual works] of all the powers[forces] which are in equilib-rium will be taken, and the differential functions which becomezero because of the conditions of the problem will be added toit, after each of these functions has been multiplied by an inde-terminate coefficient[Lagrange multiplier]; then the whole willbe equated to zero. Thus will be obtained a differential equa-tion which will be treated as an ordinary equation of maximis etminimis. From this will be deduced as many equations as thereare variables. These equations, being then rid of the indetermi-nate coefficients by elimination, will provide all the conditionsnecessary for equilibrium.

While Euler/Lagrange approaches continue to be influential inthe description of dynamical systems, Hamilton in his famouslecture of 1834 presented a unifying treatment by which boththe methods of Euler and Lagrange can be brought together todescribe the motions of a system of bodies. Hamilton made thefollowing remark regarding his law of varying action[Hamilton’sprinciple]: “In the method of the present essay, this problem[thedetermination of the motions of the planets] is reduced to thesearch and differentiation of a single function, which satisfiestwo partial differential equations of the first order and of thesecond degree: and every other dynamical problem, respectingthe motions of any system, however numerous, of attracting orrepelling points, is reduced, in like manner, to the study of onecentral function, of which the form marks out and characterizesthe properties of the moving system, and is to be determined bya pair of partial differential equations of the first order, com-bined with some simple consideration.” Since then, not onlyGalilean/Newtonian mechanics owes its dramatic advances toHamilton but also the modern developments of wave and quan-tum mechanics have been greatly facilitated by Hamilton’s prin-ciple.

Thus, the works of Galileo, Huygens, Newton, d‘Alembert, Eu-ler, Lagrange and Hamilton provided an impetus for practical

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applications of dynamics by the beginning of the NineteenthCentury. Perhaps, the most significant application of the prin-ciples of dynamics is none other than the design of the sta-tionary reciprocating steam engine in the balancing of its crankmechanism. Another equally important application was in JamesWatt’s design of fly-ball governor in 1762 to maintain a steadyrotation for the generation of electricity from the steam engines.In modern parlance, these applications can be labeled as thecontrol of vibrations in machinery.

On the design of instruments beyond Huyghens’ pendulum, wecan mention an invention by a certain Englishman named Ser-son in 1742 of a spinning artificial horizon. Then, in 1817 vonBohnenberger in Tubingen invented a two-axis gyroscope, whichwas followed by Foucault’s demonstration of a modern gyro-scope that embodies a spinning body whose spin axis can changein direction. It was Foucault’s gyroscope that enabled for thefirst time to measure the rotation of the Earth. Today’s trans-portation systems such as automobile, train, ships, aircraft andeven sophisticated space vehicles all depend their crucial designfeatures on the sound applications of dynamics, hence our mo-tivation to study dynamics.

Let us recap individual contributions.

1.2 Notable Individual Contributions

A. Aristotle(384 − 322B .C .)

Mhqanik probl mata (Problems of Mechanics),

PerÈ oÌranoυ(Treatise on the Heavens)

Comments by Bertrand Russell (1872-1970): “Aristotle

maintained that women had fewer teeth than men: although he was

twice married, it never occurred to him to verify this statement by

examining his wives’ mouths.”

The four substances on Earth: earth, fire, air and water.

Concept of power ∝ m v;Falling bodies: time of fall ∝ 1

m;

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Acceleration ∝ m;Resistance in the air ∝ density (ρ)

The natural place of heavy bodies is the center of theWorld. The natural place of light bodies is the regioncontiguous with the Sphere of the Moon. Every star isa body as it were divine, moved by its own divinity.

In kinematics, he advanced the composition of motion: Acurved path is generated when v1 and v2 are varied.

In dynamics, he advanced the composition of forces derivedfrom the composition of motions via the law of powers.

He believed the impossibility of a vacuum (no motionoccurs in a vacuum.

The Center of Universe and of the Earth coincide. (Heav-enly bodies are attracted to the center of the Universe,therefore they also are attracted to the Center of theEarth.)

He believed the Earth is a sphere (from the water shape,viz., “it is a property of water to run toward the lowestplaces.” Reasoning by Adrastus (360-317 B. C.): ”Often, dur-ing a voyage, one cannot see the Earth or an approaching shipfrom the deck, while sailors who climb to the top of a mast cansee these things because they are much higher and thus overcomethe convexity of the sea which is an obstacle.”

For all one can tell, Aristotle was ignorant of the conceptof the center of gravity.

B. Archimedes of Syracuse(287 − 212B .C ) , Author of On

the Equilibrium of Planes.

He introduced 8 axioms and 7 propositions to define thecenter of gravity. He employed the method of exhaustionto prove the center of gravity, the principle of lever,and Archimedes’ Principle of buoyancy. He used theword ”Eureka! Eureka! after he discovered the principleof buoyancy.

Other interesting mathematical problems he solved: 31071<

π < 317; 265/153 <

√3 < 1351/780;

∑∞n=0 4−n = 4/3 (for

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computing the area of a parabola)

C. Euclid (circa300B .C .) , Elements (STOIQEIΩN)

D. Pappus of Alexandria (c.290 − 350 ) : He left the so-

called Arabic Manuscript, Synagoge. Of all the cotentswritten about geometry, astronomy, philosophy, and me-chanics, his writings on mechanics are largely preservedin Book VIII of Synagoge (Collection) that he wrote in asystematic arrangement, which both Newton and La-grange quote in their books in reference to mechanicsliterature during the Antiquity.

E. William of Ockham and John I . Buridan

(rector of University of Paris) of XIVth Century:

They advanced the doctrine of Impetus and disputedAristotle’s axiom, viz., the continuous existence of a mo-tive agency in contact with, yet not part of, the projec-tile. That is, the idea of attributing a certain energy toa moving body solely on account of its motion was notaccepted in the Aristotelian dynamics. ”The existenceof impetus seems to be the cause by which the naturalfall of bodies accelerates indefinitely. At the beginningof the fall, indeed, the body is moved by the gravityalone. . . . But before long this gravity imparts a cer-tain impetus to the heavy body – an impetus which iseffective in moving the body at the same time as gravitydoes. ... The more rapid it becomes, the more intensethe impetus becomes. Therefore it can be seen that themotion will be accelerated continuously.”

F. Albert of Saxony and Nicole Oresme of XIV thCentury :He confirmed the sphericity of the Earth and the Oceans.Oresme articulated the concept of uniformly acceler-ated motion (a forerunner of gravity), which in modernmathematical language can be expressed as

e = 12gt2 = (1

2gt) · t = vmean · t

Nicole Oresme as a predecessor of Copernicus: ”No ob-servation could prove the heavens moved with a diurnal motion,

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and that the Earth did not.”

G. Leonard da Vinci (1452 − 1519 ) touched on many top-

ics on mechanics, including: The concept of moment, themotion of a heavy body on an inclined plane, the reso-lution of forces (parallelogram), the energy of a movingbody, the impossibility of perpetual motion, the figureof the Earth, the center of gravity,

H. Nicholas Copernicus (1472 − 1543 ) : The Sorbonne in

the XVI Century remain closed to Copernican ideas andcontinued to teach Ptolemy’s system (the Geocentricsystem of World). He challenged the Geocentric Sys-tem of the World, but in order not to anger the Vatican, hedidn’t use the Earth but mercury and Venus to placethe Sun at the Center of the Planets! This is what hestated, in dedicating his works to Pope Paul III, to pro-tect himself: ”I have believed that I would be readily permit-ted to examine whether, in supposing the motion of the Earth,something more conclusive might not be found in the motion ofcelestial bodies.” Thus he avoided the Vatican Inquisitionuntil 1616, 73 years after his death.

I. Guido Ubaldo (1545 − 1607 ) : He wrote a book enti-

tled, Mechanicorum Liber in which he clarified the defi-nition of the center of gravity by stating, ”All the force,all the gravity of the weight is massed and united at the cen-ter of gravity; it seems to run from all sides toward this point.Because of its gravity, indeed, the weight has a natural desireto pass through the center of the Universe. But it is the centerof gravity that properly tends to the center of the World.” Hisbook remained an authoritative source on mechanicsuntil the beginning of the XVIIIth Century and influ-enced Galileo, Descartes, Lagrange, among others.

J. Tycho Brahe (1546 − 1601 ) : He gathered and mea-

sured himself a vast amount of observation data withprecision, which were the foundation upon which Ke-pler’s laws were based.

K. Johannes Kepler (1571 − 1631 ) : Author of Astronomia

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nova and Harmonices Mundi. A tireless calculator! Basedon careful observations and calculations using TychoBrahe’s data, he formulated the law of areas, the law ofellipticity of planetary trajectories, and the third law,viz., ”One thing is absolutely certain and correct, thatthe ratio between the periods of any two planets is, tothe power 3

2, exactly that of their mean distances, that

is, of their orbits:

T ′

T= (

a′

a)32

L. Galilei Galileo (1564 − 1642 )

(http://en.wikipedia.org/wiki/Galileo Galilei):Early on he studied Aristotle’s De Caelo (On the Heav-ens; PERI OURANOU (PerÈ oÌranìs) . He is consideredthe father of modern science (dynamics). His contri-butions include: the notion of impeto or talento and mo-mento del discentere, the concept of uniform acceleration,the principle of inertia (”A body moving on a level surfacewill continue in the same direction at constant speed unless dis-turbed.”), the motion of projectiles (parabolic motion ofa falling body with initial horizontal velocity), the prin-ciple of virtual work, the isochoronous property of asimple pendulum, inventor of refracting telescope. . .He is credited with the basic principle of relativity, viz.,The laws of physics are the same in any system that is movingat a constant speed in a straight line, regardless of its particularspeed or direction.

He is author of The Starry Messenger (Sidereus Nuncius,1610), Discourse on the tides (Discorso sul flusso e il reflusso delmare, 1616), Discourse on the Comets (Discorso Delle Comete,1619), The Assayer (Il Saggiatore, 1623), Dialogue Concerningthe Two Chief World Systems (Dialogo sopra i due massimi sis-temi del mondo, 1632) which was banned by the Roman In-quisition; and Discourses and Mathematical DemonstrationsRelating to Two New Sciences (Discorsi e dimostrazioni matem-atiche, intorno a due nuove scienze, Leida, 1638) which dealswith small motions, hence was allowed to be published!

M. Rene Descartes (1596 − 1650 ) : the originator of the

Cartesian coordinate system, one of the best knownphilosophical statements (Cogito ergo sum, je pense, donc

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je suis, I think, therefore I am), the concept of infinitesi-mal calculus, an early form of the law of conservationof momentum, ...

N. Christiaan Huyghens(1629 − 1695 ) : He published his

work in Horologium Oscillatorium sive de motu pendulorum adhorologia aptato demonstrasiones geometricae (Paris, 1673). Inthat book he describes: Description of the clock, on thefall and motion of bodies on a cycloid, the evolution anddimensions of curved lines, on the center of oscillationor agitation, on the construction of a new clock witha circular pendulum, the theorems of the centrifugalforces, among others. Using the formula for the pendu-lum period

T = 2π

√`

g

he was able to compute the gravitational acceleration(g).

Also in 1659, he derived the formula for centripetal force

Fc =mv2

r

Equipped with these historical developmentsin mechanics, especially, inheriting the worksof Galileo and Huyghens, now comes IsaacNewton. It may be a historical coincidence:Galilei Galileo died in January 06, 1642. OnDecember 25 of the same year Isaac Newtonwas born.

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1.3 Isaac Newton (1642-1727)

http://en.wikipedia.org/wiki/Isaac Newton

Rene Dugas’ ( author of A History of Mechanics) assess-ment of Newton’s place in the history of mechanics:Thanks to Galileo and Huyghens, mechanics had been emanci-pated from the scholastic discipline (occult and metaphysical).Essential problems like the motion of projectiles in the vacuumand the oscillations of a compound pendulum had been solved.Nevertheless, the task of constructing an organized corpus ofprinciples in dynamics remained. This was the task of Newton,who set his seal on the foundation of classical mechanics at thesame time that he extended its field of application to celestialphenomena.

In the Author’s Preface (p. XVII) of Philosophiae Natu-ralis Principia Mathematica (Mathematical Principles of Natu-ral Philosophy) or simply Newton’s Principia (1687), he states. . . ”Therefore I offer this work as the mathematicalprinciples of philosophy, for the whole burden of phi-losophy seems to consist in this – from phenomena ofmotions to investigate the forces of nature, and thenfrom these forces to demonstrate the other phenomena;and to this end the general propositions in the first andsecond Book are directed.” . .

His description of gravity (Principia, p.547): ”But hith-erto I have not been able to discover the cause of thoseproperties of gravity from phenomena, and I frame nohypothesis; for whatever is not deduced from the phe-nomena is to be called a(an) hypothesis; and hypothe-sis, whether metaphysical or physical, whether of occultqualities or mechanical, have no place in experimentalphilosophy. In this philosophy particular propositionsare inferred from the phenomena, and afterwards ren-dered general by induction.”

Newton’s Eight Definitions (Principia, pp.1-6)

DEFINITION I: The quantity of matter[mass] is themeasure of the same, arising from its density and bulk

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conjointly.

DEFINITION II: The quantity of motion[linear momen-tum] is the measure of the same, arising from the veloc-ity and quantity of matter conjointly.

DEFINITION III: The vis insita, or the innate force ofmatter[inertia force], is a power of resisting, by whichevery body, as much as in it lies, continues in its presentstate, whether it be rest, or of moving uniformly for-wards in a right[straight] line.

DEFINITION IV: An impressed force is an action ex-erted upon a body, in order to change its state, eitherof rest, or of uniform motion in a right line.

DEFINITION V: A centripetal force is that by whichbodies are drawn or impelled, or any way tend, towardsa point as to a centre.

DEFINITION VI: An absolute centripetal force is themeasure of the same, proportional to the efficacy of thecause that propagates it from the centre, through thespaces round about.

DEFINITION VII: The accelerative quantity of a cen-tripetal force is the measure of the same, proportionalto the velocity which it generates in a given time.

DEFINITION VIII: The motive[motion-causing] quan-tity of a centripetal force is the measure of the same,proportional to the motion which it generates in a giventime.

Newton’s Three Laws (Principia, pp.13-14)

Law I: Every body continues in its state of rest, or ofuniform motion in a right line, unless it is compelled to change

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EBOOK - 0

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Philosophiae naturalis principia mathematica (): Sir Isaac Newton,Edmund Halley

•

Google Play

Fig. 1. Title page of Principia

that state by forces impressed upon it.

Law II: The change of motion is proportional to themotive force impressed; and is made in the direction of the rightline in which the force is impressed.

Law III: To every action there is always opposed anequal reaction; or, the mutual actions of two bodies upon eachother are always equal, and directed to contrary parts.

In Book III of Principia (pp.398-400), he lists four rules of

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Fig. 2. English Title page of Principia

reasoning in philosophy:

Rule 1. ”We are to admit no more causes of natural things thansuch as are both true and sufficient to explain their appearance.”(Do not assume other causes than those which are nec-essary to explain the phenomena.)

Rule 2. ”Therefore to the same natural effects we must, as faras possible, assign the same causes.” (Relate as completelyas possible analogous effects to the same cause.)

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Fig. 3. First page of English translation of Principia

Rule 3. The qualities of bodies, which admit neither intensifica-tion nor remission of degrees, and which are found to belong toall bodies within the reach of our experiments, are to be esteemedthe universal qualities of all bodies whatsoever.” (Extend to allbodies the properties which are associated with those onwhich it is possible to make experiments.)

Rule 4. In experimental philosophy we are to look upon propo-

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sitions inferred by general induction from phenomena as accu-rately or very nearly true, till such time as other phenomenaoccur, by which they may either be made more accurate, or li-able to exceptions.” (Consider every proposition obtainedby induction from observed phenomena to be valid untila new phenomenon occurs and contradicts the proposi-tion or limits its validity.)

These four rules provide a guide for how to conductscientific research for subsequent generations. In par-ticular, it is Rule 3 that Newton relied on to formulatethe law of universal gravitation.

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