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K.L.Corum and

Dr. James Cor tint

Give me a long enough lever and a place to stand and I will move the Earth.

Archimedcs, 287-212 BC

The Magnifying Transmitter' is a peculiar transformer specially adapted to excite the Earth.

Nikola Tesla. 1917

The power network3 of the future may have little resemblance to thnse of today.

Richard P. Peynman, 1964

Archimedes was one of civilizalioo's most gified and creative intellects. His striking contributions .panned mathematics, science, invention, and optica! and meclianical engineering. He is commonly rccognized to have been on a level with Isaac Newton, and even Gal ileo c laimed hi m as a mentor. Over a century after his death, the Roman statesman Cicero sought oul Archimedes1 grave slone and was touchcd and saddened to find thai it had been overgrown with thorns and brambles.'

Mechanics is commonly understood to be that branch of physical science which treats the effect of forces on ponderable bodies. As a science, ii appears to have begun with Archimedes, who is ibe first one known to have worked out the theoretical principle of the lever. Other civilizations had used levers, of course, but Archimedes of Syracuse (on the Island of Sicily) was the first to state lhe principle.

Archimedes, the son ofthc astronomer Pheidias, had studied in Alexandria, but returned to his home at Syracuse for a lifetime of creative technical achievement. He solved many difficult problems in geometry, established the concepmai foundations of calculus, calculated the perimeter of the earth (yes, the world was known to be round ai that time), detei mined thai pi is approximately 22/7, authored a number of books on mathematics and theoretical mechanics, and spent his life in vigorous intellectual activity. He created and developed great instruments for agriculture and for military applications. Plutarch, Polybus, and also Livy provide colorful accounts of Archimedes, his directed energy weapons, his thundering mechanical engines of war, and the utter astonishment of both his own king (who was delighted) and the attacking Romans (who were soundly frustrated), during the siege of Syracuse.2-3 Apparently, Archimedes was slain while in the midst of solving an analytical problem of great significance: he didn't respond fast enough to satisfy the whims of a Roman soldier, but begged not to be interrupted for a few more minutes in order to complete a formal solution.

Machines are devices which are used to transmit and modify force and motion and todo work. Machines don't create energy, they receiveenergy from a source (the prime mover) and bring about an advantage for the source in doing work on the load. There are six fundamental "machines" in classical mechanics. Do you remember them from grade school science?4

In their venerable introductory text on Physics, Sears and Zemansky have expressed that "a machine is a force-multiplying device."5 They state the "force multiplication" factor of a machine as the ratio of the weight lifted, W, to the applied force, F, and call it the actual mechanical advantage, R

(I J Force exerted by machine on 3oad _ Output Force

F Force used to operate machine tn*it Force lilis simple magnifying mio reflects the incredible relief which machines have brought to the brows of multitudes of toiling laborers m the progress of civilization. Many ontere of magnitude may tw gotten, even in practical situations, so wonderful are

As recounted above. Archimedes was the first to state the principie of the lever. Although his treatise on "The Lever" has been lost to antiquity, the principle is discussed in his works which are extant.®

I he rigid7 bar and die lever are common mechanical coupling devices for connecting mechanical systems together. The bar pro vides a way oflrans-mining forces and realizing nwiual mass in coupled mechanical systems.

The levee is a pivoted rigid bar used to multiply force or motion. Consider Figure la If force Ft pushes down on one end and it moves with velocity V|, then the (orce and upward velocity at the other end, Fj and Vj, can bi found either ftowi conservation oí"energy (á W = Ft r, A0 * Fif^®' or by summing the moments about the immovable fulcrum (or pi vol)

Rearranging algebraically gives the remarkably simple expression

The rigid bar lever is a simple toree multiplying mechanism.® The force multiplication, or magnification, is giveti by Ihe mechanical advantage

Asan astonishing result, which still delights us today, it is seen thai a very great weight can be moved these remarkable devices. No energy is created, but tremendous advantage is devised for that which is available. In (Jus paper we will show the obvious reason why Tesla called his finest invention a "Mag-ttiiying Transmitter".

Lever with a very small force. Concerning the simplicity of Archimedes^ proofs, Plutarch observed, "No amount of investigation of yours would suoceed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it yourself.*®

In Archimedes' time, many asserted tltat it was as though he was reaching over into a mist enshrouded realm and drawing out Nature's hidden secrets so that anyone of slower intellect could comprehend them. Plutarch indicates that some of his contemporaries believed that incredible effort, toil, and intense contemplation on the part of the invent« bad produced these wonrferful results. Such conflicts to discover ihe hidden possibilities of Nature bring to mind the tragic struggles of Dr. Faust.'0 [ l"he tragedy of Dr. Fausl is thai, in spite of his great natural talent and his years of academic Study, he gave up a sacred and holy quest, and "sold out for guns, girls and a good time" (in the words of C.S Lewis). Archimedes didn'i Ncrdid Tesla.J

The mechanical input power (FjVj) and output power (FjVj) must be equal if there is no (notional loss. Consequently, the velocities are related as

How remarkable the contributions of Archimedes are for mankind. His geometrical proofs, media»* cal devices, and famous hydrostatic principle, so simple and obvious to us today, were, apparendy, obtained al great price.

The Transformer - A Lev« For 20th Century CivilÍMtion

The modem power transformer was invented and developed by William Stanley (\S5B-I916), and used at the first AC power plant inNorth America, atGiwlBarriiigton,Massachusetts, jnlSgfi.11 The physical principles of the electrical transformer go back to Michael Faraday and Joseph Henry. Stanley's plant demonstrated that electrical power could be generated at a low voltage, transformed to a higher voltage for efficient transmission, and retrans formed back to a lower voltage for end-user applications. Tcsla, of course, did not "invent" AC. But he did create and patent the entire system which has made AC commercially viable as a means for powering the 20th, century,'- He contributed the

A( motor, and also the polyphur power generation and distribution system His first AC patents were issued in January of 1886. It w»s his i riven-«1 of an AC motor, along with AC's more efficient i. iirrulion and simpler distribution, which made

■ 'i.mating current of such great commercial value, 1 mught about its triumph ova- Edison's DC sys-

■ m in the I890's, and has powered the wealth and ri ogress of the 20th century.

i. .issessing the importance of electric power gen--iHiion and distribution, and the great i omplishment of the Niagara Falls power plant, 1 >| Chartes F. Scott, Professor Emeritus of Elee-ucal Engineering at Yale University, and past President of the AIEE (now the IEEE) has said, l)if evolution of electric power from the discovery of Faraday in 1831 to the initial great ■wiallation of the Tesla polyphase system in !896 undnuhiedlv the most tremendous event in all wneerinçhîstorv"'! Certainly, not since the days I Archimedes had civilization experienced such a . ant step forward in (to use the ECPD14 défini-Hon for the profession of Engineering) "utilizing ' v resources of the earth fot the benefit of mankind".

I he ideal transformer is the electrical analog to the uttionless mechanical lever and also to the ideal ir ear train,W 6,17,18,1» (See Figure lb.) The opinion of the ideal transformer follows from the Iciinition of magnetic flux, <t>,

(h) <t>=jjBtida vhere the integration is carried out over the transfirmer leg cross-sectional area, and from Faraday's law of induction, where \ is the etnf induced in an N turn coil through which the flux passes. Since the flu* is the same in both legs, we Stave w 3 dt N, ' 1

where one commonly defines the turns ratio as;

The similarity between equations (3) and (9) completes the analogy if force and AC voltage are taken as analogs and if

Depending upon the mechanical advantage, R, the lever magnifies the applied force: F, = R F]. Depending upon the turns ratio, N, a transformer steps the primary AC voltage up to a higher (or down to a lower) secondary voltage: = N^i-

No power is dissipated in an ideal transformer, so that in)

Consequently, the current must be transformed in a manner analogous to equation (5)

The analogy illustrated above is called the f-v (forcevoltage) analog. An al iemat i ve anal ogy20 has been advanced by Firestone.21

As with the lever, the AC voltage has been multiplied by the turns ratio, and the current has been multiplied by the reciprocal ot'the turns ratio. What magnificent leverage has been afforded 20th century civilization by this wonderful appliance.

Figure 1. Classical devices which provide advantage

Figure 1. Classical devices which provide advantage

Conventional power transformers have been splendid electrical devices in the application of electrotechnology. Cascaded transformers, working at power line frequencies have even been used to produce voltages in excess of 1 mega vol t.2223 Such transformers behave as lumped elements and depend entirely upon the turns ratio to achieve voltage rise. They are, however, incapable of attaining the voilage rise» produced through the phenomenon of resonance. The ultimate limitation on lumped-clemenl high voltage transformers is due to the conflicting requirements (hat many tightly wound turns are necessary to produce flux leakage and obtain large step-up, yet great tum-to-tu.rn spacing becomes necessary in order to avoid high voltage breakdown in the appliance.

Mechanical oscillators come in a variety of configurations: mass on a stretched spring, the simple pendulum, torsional oscillators (twisted bars, watch flywheels), floating objects, U-shaped liquid columns, compressed air columns (pistons, shock absorbers, organ pipes, wind instruments). Con-siderthe simple linear mechanical oscillator. The dynamical behavior is to be determined when it is driven by a variable frequency forcing (unction.

(b> The electric analog of the mechanical systemshown in (a). This circuit is based on the f-v (force-vottaga) analog. (See Gardner and Barnes. Reference 8 pg. 62.)

Lei the system be assumed to have a linear restoring force (Hooke's law) and only viscous damping, i.e.- the damping force is proportional to the velocity of the body.24 McLean and Nelson point out two other types of mechanical damping:

" t, Coulomb damping is independent of the velocity and arises because of sliding of the body on dry surfaces (its force is thus proportional to the normal force between the body and the surface on which it 5¡ides.

2. Solid damping occurs as internal friction within the material of the body itself (its fare« is independent of the frequency and proportional to tbe maximum stress induced in the body itscif)."25

In view of the fact thai every basic physics course discusses dry friction, it is remarkable how little space isdevoted to Coulomb damping in advanced texts on classical mechanics. Feynman has observed that the Coulomb friction law "... is another of those semi-empirical laws that are not thoroughly understood."26 We have discussed these issues in greater detail ina previous paper,27 and so we will not consider them further at this time.

Consider the simple mechanical system shown in Figure 2. Suppose that viscous damping is the only loss mechanism present, and the above two types of mechanical damping are negligible. Further, suppose Ihe system is driven by a sinusoidal forcing function of amplitude F„. Equating the sum of the forces to ma leads to the well-known second order linear differential equation for the displacement2®

(14) —r + 2a — + m„x = — coswt dt1 dt m where 2a= g/m and (n03 = k/m

«■I j- the viscous damping contl »111 m ■ the mass of (lie oscillating body k " system spring constant.

I i,i steady slate solution of Equation (14) is

• lit x(t) = Rc{X(<Q)ei<°1} = Re{X(û>) = X(ffl) cos [«^((b)]

fctxre the response function X(oi) is a complex frc-. lie y dependent quantity whose magnitude is (ivra by

•11J whose phase is given by

icre it is customary to introduce the selectivity defined at «„as

2a Energy dissipated per cycle i.i is a measure of the spectra! spread of, Af, of the ■ item response about the resonant frequency. The I hase is the angle by which the displacement logs rhlmi the driving force. Referring to his classic a a an mechanical vibrations, Den Hartog has ailed Equations (16) and (17) "the most important equations in the book".29

i is Of interest to plot these quantities as functions ■I frequency. (See Figure 3a and 3b.) Note that lie effect of viscous damping is to spectrally roaden the response, and to shift fhe frequency of ts maximum from the undamped resonance frequency (!)„, where the phase passes through ji/2, .bwnto (i>m where the displacement is maximum. u>m is analogous to the frequency of maximum ullage across the capacitor in series RLC circuit.)

Quite often older authors in mechanical engineer-ng30 would introduce the term "Magnification i- actor", which was defined as the ratio of the implitnde of the steady state solution X<to> to the italic deflection X(0) = X0:

Figure 3. A plot of the magnitude ana phase response of a linear mechanical oscillator driven by a lime-harmonic forcing function The magnification factor ua normalized frequency is also shewn.

Figure 3. A plot of the magnitude ana phase response of a linear mechanical oscillator driven by a lime-harmonic forcing function The magnification factor ua normalized frequency is also shewn.

This is also the ratio of the magnitude of the force developed across the spring to the amplitude of the applied vibrating force, as shown in the last term. Once again, we have a force multiplication, or magnification, in the form of Equation (4) above. However, as with the transformer, (fie force is "AC".

A plot of the magnification factor verses normalized frequency for various amounts of damping, specified through the parameter Q, is shown in Figure 3(c). The maximum value of X(w) is given by

:*QX0 = QF„/k and force magnification at the maximum issimply

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